Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 834731 11 pageshttpdxdoiorg1011552013834731

Research ArticleA Discrete-Time 1198661198901199001198661 Retrial Queue with 119869 Vacations andTwo Types of Breakdowns

Feng Zhang and Zhifeng Zhu

College of Science North University of China Taiyuan 030051 China

Correspondence should be addressed to Feng Zhang gigigi69163com

Received 12 July 2013 Revised 21 September 2013 Accepted 21 September 2013

Academic Editor Ram N Mohapatra

Copyright copy 2013 F Zhang and Z ZhuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with a discrete-time 1198661198901199001198661 retrial queueing model with 119869 vacations and two types of breakdowns If theorbit is empty the server takes at most 119869 vacations repeatedly until at least one customer appears in the orbit upon returning from avacation It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair We analyzethe Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit sizeand the system size distributions in terms of their generating functions Then we obtain some performance measures through thegenerating functions Moreover the stochastic decomposition property and the corresponding continuous-time queueing systemare investigated Finally some numerical examples are provided to illustrate the effect of vacations and breakdowns on severalperformance measures of the system

1 Introduction

During the past few decades retrial queueing systems havebeen widely studied due to their important applications inmany practical systems such as inventory systems computersystems and telecommunication networks Retrial queueingsystems are characterized by the fact that arriving customerswho find the server busymay join a retrial group to retry theirrequests after some random time For excellent bibliographieson retrial queues the readers are referred to [1ndash3]

Althoughmany continuous-time retrial queueingmodelshave been studied extensively in the past years only fewresearchers studied the discrete-time retrial queues In factthe discrete-time retrial queueing systems are suitable forthe description of the various phenomena in slotted timecomputer and communication systems such as broad-bandintegrated services digital network (B-ISDN) and time-division multiple access (TDMA) systems Yang and Li [4]firstly studied a discrete-time 1198661198901199001198661 retrial queue andobtained some performance measures of the systems Thiswork was generalized to discrete-time 1198661198901199001198661 retrial queuewith general retrial times by Atencia and Moreno [5] Inrecent years there has been a growing interest in the investi-gation of discrete-time retrial queue with server breakdowns

Such phenomena occur in day-to-day life for example thecomponents of the computer and communication systems aresubject to random breakdowns Discrete-time retrial queuesthat take into account server breakdowns were studied byAtencia and Moreno [6 7] Moreno [8] Wang and Zhao [9]Wang [10] Wang and Zhang [11] and Atencia et al [12]

On the other hand queueing systems with server vaca-tions have been widely used to analyze the performance ofsome systems such as manufacturing systems and computersystems In recent years retrial queues with server vacationshave drawn more and more attention Li and Yang [13]studied a 1198721198661 retrial queue with a finite number ofinput sources and Bernoulli vacations by the method ofsupplementary variables Since the work of Li and Yang theretrial queues with Bernoulli vacations have been studied bymany authors see Kumar and Arivudainambi [14] Kumar etal [15] and Choudhury [16] In addition the retrial queueswith exhaustive service vacations were also studied by severalauthors For example Artalejo [17] analyzed an1198721198661 retrialqueue with multiple vacations and obtained two stochasticdecomposition results for the system size As a generalizationof single andmultiple vacation policy Chang and Ke [18] andKe and Chang [19] introduced the concept of 119869 vacations into1198721198661 retrial queueing systems Ke and Chang [20] andWu

2 Journal of Applied Mathematics

[21] studied the models concerning both server breakdownsand vacations simultaneously

The majority of articles studied the continuous-timeretrial queues with server vacations Most recently Wang[22] extended the continuous-time1198721198661 retrial queue withBernoulli vacations to discrete-time counterpart Li et al [23]Liu and Song [24] considered the discrete-time 1198661198901199001198661198901199001

retrial queue with working vacation Zhang et al [25] studieda discrete-time 1198661198901199001198661 retrial queue with single vacationand starting failure To the best of our knowledge no workappeared till now concerning the discrete-time retrial queuewith exhaustive 119869 vacations policy and serverrsquos different typesof breakdowns In fact the discrete-time retrial queues with119869 vacations and server breakdowns are more appropriateto analyze some real situations such as e-mail systems IPaccess networks and computer systems This motivates us toanalyze the discrete-time retrial queue with 119869 vacations andand general retrial times In this work we consider a discrete-time1198661198901199001198661 retrial queue with 119869 vacations where the servermay be subjected to two different types of breakdowns

The rest of the paper is organized as follows In Section 2the description of our model is given In Section 3 We derivethe generating functions of the number of customers inthe orbit and in the system The closed-form expressions ofsome performance measures of the system are also obtainedIn Section 4 we give two stochastic decomposition resultsfor the system size In Section 5 the relationship betweenour model and the continuous-time counterpart is given InSection 6 we present some numerical results to show theinfluence of the parameters on some performance measuresof the system

2 Description of the Mathematical Model

We consider a discrete-time 1198661198901199001198661 retrial queue with 119869

vacations and two types of server breakdowns where the timeaxis is segmented into slots of equal length and all queueingactivities occur at the slot boundaries Specifically let the timeaxis be marked by 0 1 119898 Consider the epoch 119898 andassume that the departures the end of the vacations and theend of repairs occur in the interval (119898minus 119898) while arrivalsretrials the beginning of the vacations and the beginning ofrepairs occur in the interval (119898119898

+

) in sequence That is weconsider an early arrival system (EAS) policy in our model

Customers arrive at the system according to a geometricalarrival process with parameter 119901 where 119901 (0 lt 119901 lt 1) isthe probability that an arrival occurs in a slot If an arrivingcustomer finds that the server is idle he commences hisservice immediately Otherwise if the server is busy underrepair or on vacation the arriving customer joins the retrialorbit It is assumed that only the customer at the head of theorbit is permitted to access to the server The time betweenretrials is assumed to follow a general probability distribution119886119894

infin

119894=0

with the generating function 119860(119909) = suminfin

119894=0

119886119894

119909119894

The service times are independent and identically dis-tributed with arbitrary distribution 119904

119894

infin

119894=1

generating func-tion 119878(119909) = sum

infin

119894=1

119904119894

119909119894 and 119899th factorial moment 119878

119899

It isassumed that the server is subject to two different types of

breakdowns The first type of breakdown is starting failureIn this model an arriving customer who finds the server idlemust turn on the server If the server is activated successfully(with a probability 120579

1

= 1 minus 1205791

) the customer beginshis service immediately otherwise if the server is startedunsuccessfully (with a complementary probability 120579

1

) theserver is sent to repair immediately and the customer beingserved must join the orbit The second type of breakdown isnormal breakdown that is the servermay break downduringserving customers The customer just being served beforebreakdown waits until the server is repaired to completehis remaining service time The lifetime of the server isassumed to be geometrically distributed with parameter 120579

2

The repair times for the server of two types of breakdownsare independent and identically distributed with arbitrarydistribution 119903

119897119894

infin

119894=1

generating functions 119877119897

(119909) = suminfin

119894=0

119903119897119894

119909119894

and 119899th factorial moment are 119877119897119899

119897 = 1 2 respectivelyAs soon as the system is empty the server takes a

vacation immediatelyThe vacation time is assumed to followa general probability distribution V

119894

infin

119894=1

with generatingfunction 119881(119909) = sum

infin

119894=1

V119894

119909119894 and 119899th factorial moment 119881

119899

Ifthere is at least one customer in the orbit at the end of avacation the server begins to provide service for customersimmediately Otherwise if no customers appear in the orbitwhen the server returns from the vacation it leaves foranother vacation This pattern continues until the server hastaken a maximum number 119869 of vacations If the system isempty at the end of the 119869th vacation the server becomes idleand waits for new arriving customers

Finally we assume that the interarrival times the servicetimes the retrial times the repair times and the vacationtimes are mutually independent

3 The Markov Chain and SomePerformance Measures

At time 119898+ the system can be described by the process

119884119898

119898 ge 1 with 119884119898

= (119862119898

120585119898

120577119898

119871119898

119873119898

) where 119862119898

denotes the state of the server and 119873119898

be the number of thecustomers in the orbit It is assumed that 119862

119898

= 0 1 2 3 or4 according to whether the server is free busy under repairfor first type of breakdown under repair for second typeof breakdown or on vacations respectively If 119862

119898

= 0 120585119898

represents the remaining retrial time If119862119898

= 1 120585119898

representsthe remaining service time of the customer currently beingserved If 119862

119898

= 2 120585119898

represents the remaining repair timefor the first type of breakdown If 119862

119898

= 3 120585119898

represents theremaining service time of a customer currently being servedand 120577

119898

represents the remaining repair time of the secondtype of breakdowns If 119862

119898

= 4 let 119871119898

= 119899 represent thatthe server takes the 119899th vacation and let 120585

119898

represent theremaining vacation time 119899 = 1 2 119869 It can be shown that119884119898

119898 ge 1 is a Markov chain with the following state space

Ω = (0 0) (0 119894 119896) 119894 ge 1 119896 ge 1

(1 119894 119896) 119894 ge 1 119896 ge 0 (2 119894 119896) 119894 ge 1 119896 ge 1

Journal of Applied Mathematics 3

(3 119894 119895 119896) 119894 ge 1 119896 ge 0 (4 119894 119896) 119894 ge 1

119896 ge 0 119899 = 1 2 119869

(1)

Our object is to find the stationary distribution of theMarkov chain which is defined as follows

12058700

= lim119898rarrinfin

119875 119862119898

= 0119873119898

= 0

1205870119894119896

= lim119898rarrinfin

119875 119862119898

= 0 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 1

1205871119894119896

= lim119898rarrinfin

119875 119862119898

= 1 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 0

1205872119894119896

= lim119898rarrinfin

119875 119862119898

= 2 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 1

1205873119894119895119896

= lim119898rarrinfin

119875 119862119898

= 3 120585119898

= 119894 120577119898

= 119895119873119898

= 119896

119894 ge 1 119895 ge 1 119896 ge 0

120596119899119894119896

= lim119898rarrinfin

119875 119862119898

= 4 119871119898

= 119899 120585119898

= 119894119873119898

= 119896

119894 ge 1 119896 ge 0 119899 = 1 2 119869

(2)

The Kolmogorov equations for the stationary distribution ofthe system are

12058700

= 11990112058700

+ 11990112059611986910

(3)

1205870119894119896

= 1199011205870119894+1119896

+ 119901119886119894

12058711119896

+ 119901119886119894

12058721119896

+ 119901119886119894

119869

sum

119899=1

1205961198991119896

119894 ge 1 119896 ge 1

(4)

1205871119894119896

= 1205750119896

1199011205791

1205792

119904119894

12058700

+ 1205750119896

1199011205791

1205792

119904119894

infin

sum

119895=1

1205870119895119896

+ 1199011205791

1205792

119904119894

12058701119896+1

+ 1199011205791

1205792

119904119894

1205961198691119896

+ 1199011198860

1205791

1205792

119904119894

12058711119896+1

+ 1205750119896

1199011205791

1205792

119904119894

12058711119896

+ 1199011205792

1205871119894+1119896

+ 1205750119896

1199011205792

1205871119894+1119896minus1

+ 1205750119896

1199011205791

1205792

119904119894

12058721119896

+ 1199011198860

1205791

1205792

119904119894

12058721119896+1

+ 1199011205792

12058731198941119896

+ 1205750119896

1199011205792

12058731198941119896minus1

+ 1199011198860

1205791

1205792

119904119894

119869

sum

119899=1

1205961198991119896+1

+ 1205750119896

1199011205791

1205792

119904119894

119869minus1

sum

119899=1

1205961198991119896

119894 ge 1 119896 ge 0

(5)

1205872119894119896

= 1205751119896

1199011205791

1199031119894

12058700

+ 1205751119896

1199011205791

1199031119894

infin

sum

119895=1

1205870119895119896minus1

+ 1199011205791

1199031119894

12058701119896

+ 1199011198860

1205791

1199031119894

12058711119896

+ 1205751119896

1199011205791

1199031119894

12058711119896minus1

+ 1205751119896

1199011205872119894+1119896minus1

+ 1199011205872119894+1119896

+ 1199011198860

1205791

1199031119894

12058721119896

+ 1205751119896

1199011205791

1199031119894

12058721119896minus1

+ 1199011198860

1205791

1199031119894

119869

sum

119899=1

1205961198991119896

+ 1205751119896

1199011205791

1199031119894

119869minus1

sum

119899=1

1205961198991119896minus1

+ 1199011205791

1199031119894

1205961198691119896minus1

119894 ge 1 119896 ge 1

(6)

1205873119894119895119896

= 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058700

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

infin

sum

119897=1

1205870119897119896

+ 1199011205791

1205792

119904119894

1199032119895

12058701119896+1

+ 1199011205791

1205792

119904119894

1199032119895

1205961198691119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

12058711119896+1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058711119896

+ 1199011205792

1199032119895

1205871119894+1119896

+ 1205750119896

1199011205792

1199032119895

1205871119894+1119896minus1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058721119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

12058721119896+1

+ 1199011205873119894119895+1119896

+ 1199011205792

1199032119895

12058731198941119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

1205961198991119896+1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

1205961198991119896

+ 1205750119896

1199011205792

1199032119895

12058731198941119896minus1

+ 1205750119896

1199011205873119894119895+1119896minus1

119894 ge 1 119895 ge 1 119896 ge 0

(7)

1205961119894119896

= 1199011205961119894+1119896

+ 1205750119896

1199011205961119894+1119896minus1

+ 1205750119896

119901V119894

120587110

+ 1205751119896

119901V119894

120587110

119894 ge 1 119896 ge 0

(8)

120596119899119894119896

= 119901120596119899119894+1119896

+ 1205750119896

119901120596119899119894+1119896minus1

+ 1205750119896

119901V119894

120596119899minus110

+ 1205751119896

119901V119894

120596119899minus110

119894 ge 1 119896 ge 0 119899 = 2 3 119869

(9)

where 120575119894119895

is the Kroneckerrsquos symbol 120575119894119895

= 1 minus 120575119894119895

and thenormalizing condition is

12058700

+

infin

sum

119894=1

infin

sum

119896=1

(1205870119894119896

+ 1205872119894119896

) +

infin

sum

119894=1

infin

sum

119896=0

1205871119894119896

+

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

+

119869

sum

119899=1

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

= 1

(10)

To solve (3)ndash(10) we introduce the following generatingfunctions

1205930

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205870119894119896

119909119894

119911119896

1205931

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

1205871119894119896

119909119894

119911119896

4 Journal of Applied Mathematics

1205932

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205872119894119896

119909119894

119911119896

1205933

(119909 119910 119911) =

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

119909119894

119910119894

119911119896

120595119899

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

119909119894

119911119896

119899 = 1 2 119869

1205934

(119909 119911) =

119869

sum

119899=1

120595119899

(119909 119911)

(11)

and the following auxiliary generating functions

1205930119894

(119911) =

infin

sum

119896=1

1205870119894119896

119911119896

1205931119894

(119911) =

infin

sum

119896=0

1205871119894119896

119911119896

1205932119894

(119911) =

infin

sum

119896=1

1205872119894119896

119911119896

120595119899119894

(119911) =

infin

sum

119896=1

120596119899119894119896

119911119896

1205933119894119895

(119911) =

infin

sum

119896=0

1205873119894119895119896

119911119896

1205933119894

(119910 119911) =

infin

sum

119896=0

1205873119894119895119896

119910119895

119911119896

1206011

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

12058731198941119896

119909119894

119911119896

(12)

Now we can solve (3)ndash(10) by using the generating functiontechniqueWe first give some lemmaswhichwill be used lateron and their proof which can be readily obtained Thus theyare omitted The following inequalities hold

Lemma 1 If 1205791

1205881

+1205791

+1205791

1205882

lt 119901+119901119860(119901) then the inequality

[119911 + (1 minus 119911) 119901119860 (119901)]Ω (119911) minus 119911120574 (119911) gt 0 (13)

holds for 0 le 119911 lt 1 where

120574 (119911) = 119901 + 119901119911 1205881

= 1199011198781

(1 +1205792

1205792

11987721

)

1205882

= 11990111987711

Ω (119911) = 1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911))

120591 (119911) =1205792

119911

1 minus 1205792

1198772

(119911)

(14)

Lemma 2 The following limits exist if 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 +

119901119860(119901)

lim119911rarr1

120574 (119911) Γ (119911) minus 1205783

Ω (119911)

Λ (119911)

=

1199011198811

1205782

+ 1199011205783

[1205791

1198781

(1 + (1205792

1205792

) 11987721

) minus 1205791

+ 1205791

11987711

]

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

lim119911rarr1

Θ (119911)

Λ (119911)=

1199011198811

1205782

+ 119901119860 (119901) 1205783

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

(15)

where

120585 = 119881 (119901) 1205781

= 1 minus 120585119869+1

1205782

= 1 minus 120585119869

1205783

= (1 minus 120585) 120585119869

Θ (119911) = 1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

Γ (119911) = 1205781

minus 1205782

119881 (120574 (119911))

Λ (119911) = [119911 + 119901119860 (119901) (1 minus 119911)]Ω (119911) minus 119911120574 (119911)

(16)

By using Lemmas 1 and 2 we can obtain the generatingfunctions of the stationary distribution of the system whichare given by the following theorem

Theorem 3 If 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 + 119901119860(119901) the stationarydistribution of the Markov chain 119884

119898

119898 = 0 1 2 has thefollowing generating functions

1205930

(119909 119911) =119860 (119909) minus 119860 (119901)

119909 minus 119901

119901119909119911 [120574 (119911) Γ (119911) minus 1205783

Ω (119911)]

1205783

Λ (119911)12058700

1205931

(119909 119911) =119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

Θ (119911) 1199011199091205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

1205932

(119909 119911) =1198771

(119909) minus 1198771

(120574 (119911))

119909 minus 120574 (119911)

Θ (119911) 1199011199091205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

1205933

(119909 119910 119911)

=119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

times1198772

(119910) minus 1198772

(120574 (119911))

119910 minus 120574 (119911)

Θ (119911) 1199011199091199101205791

120591 (120574 (119911))

1199011205783

Λ (119911)

1205792

1205792

12058700

1205934

(119909 119911) =1199011205782

119909120574 (119911) [119881 (119909) minus 119881 (120574 (119911))]

1199011205783

[119909 minus 120574 (119911)]12058700

(17)

where

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(18)

Journal of Applied Mathematics 5

Proof Multiplying (3)ndash(9) by 119911119896 and summing over 119896 we getthe following equations

1205930119894

(119911) = 1199011205930119894+1

(119911)

+ 119901119886119894

[12059311

(119911) + 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus 119901119886119894

120587110

minus 119901119886119894

119869

sum

119899=1

12059611989910

119894 ge 1

(19)

1205931119894

(119911) = 1199011205791

1205792

119904119894

12058700

+ 1199011205791

1205792

119904119894

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

12059301

(119911) + 1205792

120574 (119911) 1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

[12059311

(119911)

+ 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

120587110

minus1199011205791

1205792

119904119894

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911) 119894 ge 1

(20)

1205932119894

(119911) = 1199011205791

1199111199031119894

12058700

+ 1199011205791

1199111199031119894

1205930

(1 119911)

+ 1199011205791

1199031119894

12059301

(119911) + 120574 (119911) 1205932119894+1

(119911)

+ (119901119911 + 1199011198860

) 1205791

1199031119894

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

minus (119901119911 + 1199011198860

) 1205791

1199031119894

120587110

minus 1199011198860

1205791

1199031119894

119869

sum

119899=1

12059611989910

minus 1199011205791

1199031119894

119911

119869minus1

sum

119899=1

12059611989910

119894 ge 1

(21)

1205933119894119895

(119911) = 1199011205791

1205792

119904119894

1199032119895

12058700

+ 1199011205791

1205792

119904119894

1199032119895

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

1199032119895

12059301

(119911) + 1205792

120574 (119911) 1199032119895

1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

+ 120574 (119911) 1205933119894119895+1

(119911) minus 1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911)

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

120587110

119894 ge 1

(22)1205951119894

(119911) = 1199011205951119894+1

(119911) + 1199011199111205951119894+1

(119911)

+ 120574 (119911) V119894

120587110

119894 ge 1 119896 ge 0

(23)

120595119899119894

(119911) = 119901120595119899119894+1

(119911) + 119901119911120595119899119894+1

(119911)

+ 120574 (119911) V119894

120596119899minus110

119894 ge 1 119896 ge 0 2 le 119899 le 119869

(24)

Multiplying (23) and (24) by 119909119894 and summing over 119894 we get

119909 minus 120574 (119911)

1199091205951

(119909 119911) = minus120574 (119911) 12059511

(119911) + 120574 (119911) 119881 (119909) 120587110

(25)

119909 minus 120574 (119911)

119909120595119899

(119909 119911) = minus120574 (119911) 1205951198991

(119911) + 120574 (119911) 119881 (119909) 120596119899minus110

2 le 119899 le 119869

(26)

Setting 119909 = 120574(119911) in (25) and (26) we get

12059511

(119911) = 119881 (120574 (119911)) 120587110

(27)

1205951198991

(119911) = 119881 (120574 (119911)) 120596119899minus110

2 le 119899 le 119869 (28)

Substituting (27) and (28) into (25) and (26) respectively weobtain

1205951

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120587110

120595119899

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120596119899minus110

2 le 119899 le 119869

(29)

Taking the derivative of (29) with respective to 119909 and letting119909 = 119911 = 0 we get

120596110

= 120585120587110

(30)

12059611989910

= 120585120596119899minus110

= 120585119899

120587110

2 le 119899 le 119869 (31)

Substituting (31) into (3) we get

120587110

=11990112058700

119901120585119869 (32)

6 Journal of Applied Mathematics

Multiplying (19) by 119909119895 summing over 119895 and taking into

account (30)ndash(32) we obtain

119909 minus 119901

1199091205930

(119909 119911)

= 119901 (119860 (119909) minus 1198860

) (12059311

(119911) + 12059321

(119911))

minus 11990112059301

(119911) minus 119901 (119860 (119909) minus 1198860

)1205781

minus 1205782

119881 (120574 (119911))

1205783

12058700

(33)

Note that we can find 1205930

(1 119911) by setting 119909 = 1 in (33) thenmultiplying (20) and (21) by 119909

119895 summing over 119895 and takinginto account (30)ndash(32) we obtain

119909 minus 120574 (119911) 1205792

1199091205931

(119909 119911)

=119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

12059311

(119911) +119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 12059301

(119911)

+ 120574 (119911) 1205792

1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 12058700

(34)

119909 minus 120574 (119911)

1199091205932

(119909 119911) = [119911 + 1199011198860

(1 minus 119911)] 1205791

1198771

(119909)

times (12059311

(119911) + 12059321

(119911)) minus 120574 (119911) 12059321

(119911)

+ 1199011205791

1198771

(119909) (1 minus 119911) 12059301

(119911)

minus119870 (119911)

1199011205783

1199011205791

1198771

(119909) 12058700

(35)

where119870(119911) = 1199111205782

[1 minus 119881(120574(119911))] + 1199011198860

(1 minus 119911)Γ(119911)Multiplying (22) by 119910

119895 and 119909119894 summing over 119895 and 119894

respectively and taking into account (30)ndash(32) we obtain

119910 minus 120574 (119911)

1199101205933

(119909 119910 119911) =119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772

(119910)

times [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

1198772

(119910) 12059311

(119911)

+119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772(119910)

12059301

(119911)

+ 120574 (119911) (1 minus 1205792

1198772

(119910)) 1206011

(119909 119911)

minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 1198772

(119910) 12058700

+120574 (119911) 120579

2

1198772

(119910)

1199091205931

(119909 119911)

(36)

Note that we can find 1206011

(119909 119911) by setting 119910 = 120574(119911) in (36) andsubstituting the expression of 120601

1

(119909 119911) into (34) we get

119909 minus 1205911

(120574 (119911))

119909[1 minus 120579

2

1198772

(120574 (119911))] 1205931

(119909 119911)

= 1205792

119911 + 119901119886

0

(1 minus 119911)

1199111205791

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 12059311

(119911) +119901 (1 minus 119911)

1199111205791

119878 (119909) 12059301

(119911)

+120574 (119911) 1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

119878 (119909) 12058700

(37)

Setting119909 = 119901 in (33)119909 = 120574(119911) in (35) and119909 = 120591(120574(119911)) in (37)respectively we can get the equations for 120593

01

(119911) 12059311

(119911) and12059321

(119911) By solving these equations we get the the generatingfunctions as follows

12059301

(119911) = 119901119911 (A (119901) minus 1198860

)120574 (119911) Γ (119911) minus 120578

3

Ω (119911)

1199011205783

Λ (119911)12058700

12059311

(119911) =Θ (119911) 119901120579

1

119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

12059321

(119911) =Θ (119911) 119901120579

1

119911119877 (120574 (119911))

1199011205783

Λ (119911)12058700

(38)

Using Lemmas 1 and 2 it is easy to show that 12059301

(119911)12059311

(119911) and 12059321

(119911) are defined for 119911 isin [0 1) and can beextended by continuity in 119911 = 1 if 120579

1

1205881

+ 1205791

+ 1205791

1205882

lt

119901 + 119901119860(119901) Now substituting (30)ndash(32) into (29) we can get1205934

(119909 119911) Similarly substituting (38) into (33) (35)ndash(37) wederive 120593

0

(119909 119911) 1205931

(119909 119911) 1205932

(119909 119911) and 1205933

(119909 119910 119911) Using thenormalizing condition we can find the unknown constant12058700

This completes the proof

Based on Theorem 3 we can easily obtain the marginalgenerating functions of the number of customers when theserver is in various states and some performance measuresThey are summarized in the following CorollaryTheir proofsare very easy and thus are omitted For convenience wedefine variable119873 be the orbit size and 119871 be the system size

Corollary 4

(1) The marginal generating function of the number ofcustomers in the orbit when the server is idle or vacationis given by

12058700

+ 1205930

(1 119911)

=119911 (1 minus 119860 (119901)) [120574 (119911) Γ (119911) minus 120578

3

Ω (119911)] + 1205783

Λ (119911)

1205783

Λ (119911)12058700

(39)

(2) The marginal generating function of the number ofcustomers in the orbit when the server is busy is givenby

1205931

(1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

Θ (119911) 1205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(40)

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

[21] studied the models concerning both server breakdownsand vacations simultaneously

The majority of articles studied the continuous-timeretrial queues with server vacations Most recently Wang[22] extended the continuous-time1198721198661 retrial queue withBernoulli vacations to discrete-time counterpart Li et al [23]Liu and Song [24] considered the discrete-time 1198661198901199001198661198901199001

retrial queue with working vacation Zhang et al [25] studieda discrete-time 1198661198901199001198661 retrial queue with single vacationand starting failure To the best of our knowledge no workappeared till now concerning the discrete-time retrial queuewith exhaustive 119869 vacations policy and serverrsquos different typesof breakdowns In fact the discrete-time retrial queues with119869 vacations and server breakdowns are more appropriateto analyze some real situations such as e-mail systems IPaccess networks and computer systems This motivates us toanalyze the discrete-time retrial queue with 119869 vacations andand general retrial times In this work we consider a discrete-time1198661198901199001198661 retrial queue with 119869 vacations where the servermay be subjected to two different types of breakdowns

The rest of the paper is organized as follows In Section 2the description of our model is given In Section 3 We derivethe generating functions of the number of customers inthe orbit and in the system The closed-form expressions ofsome performance measures of the system are also obtainedIn Section 4 we give two stochastic decomposition resultsfor the system size In Section 5 the relationship betweenour model and the continuous-time counterpart is given InSection 6 we present some numerical results to show theinfluence of the parameters on some performance measuresof the system

2 Description of the Mathematical Model

We consider a discrete-time 1198661198901199001198661 retrial queue with 119869

vacations and two types of server breakdowns where the timeaxis is segmented into slots of equal length and all queueingactivities occur at the slot boundaries Specifically let the timeaxis be marked by 0 1 119898 Consider the epoch 119898 andassume that the departures the end of the vacations and theend of repairs occur in the interval (119898minus 119898) while arrivalsretrials the beginning of the vacations and the beginning ofrepairs occur in the interval (119898119898

+

) in sequence That is weconsider an early arrival system (EAS) policy in our model

Customers arrive at the system according to a geometricalarrival process with parameter 119901 where 119901 (0 lt 119901 lt 1) isthe probability that an arrival occurs in a slot If an arrivingcustomer finds that the server is idle he commences hisservice immediately Otherwise if the server is busy underrepair or on vacation the arriving customer joins the retrialorbit It is assumed that only the customer at the head of theorbit is permitted to access to the server The time betweenretrials is assumed to follow a general probability distribution119886119894

infin

119894=0

with the generating function 119860(119909) = suminfin

119894=0

119886119894

119909119894

The service times are independent and identically dis-tributed with arbitrary distribution 119904

119894

infin

119894=1

generating func-tion 119878(119909) = sum

infin

119894=1

119904119894

119909119894 and 119899th factorial moment 119878

119899

It isassumed that the server is subject to two different types of

breakdowns The first type of breakdown is starting failureIn this model an arriving customer who finds the server idlemust turn on the server If the server is activated successfully(with a probability 120579

1

= 1 minus 1205791

) the customer beginshis service immediately otherwise if the server is startedunsuccessfully (with a complementary probability 120579

1

) theserver is sent to repair immediately and the customer beingserved must join the orbit The second type of breakdown isnormal breakdown that is the servermay break downduringserving customers The customer just being served beforebreakdown waits until the server is repaired to completehis remaining service time The lifetime of the server isassumed to be geometrically distributed with parameter 120579

2

The repair times for the server of two types of breakdownsare independent and identically distributed with arbitrarydistribution 119903

119897119894

infin

119894=1

generating functions 119877119897

(119909) = suminfin

119894=0

119903119897119894

119909119894

and 119899th factorial moment are 119877119897119899

119897 = 1 2 respectivelyAs soon as the system is empty the server takes a

vacation immediatelyThe vacation time is assumed to followa general probability distribution V

119894

infin

119894=1

with generatingfunction 119881(119909) = sum

infin

119894=1

V119894

119909119894 and 119899th factorial moment 119881

119899

Ifthere is at least one customer in the orbit at the end of avacation the server begins to provide service for customersimmediately Otherwise if no customers appear in the orbitwhen the server returns from the vacation it leaves foranother vacation This pattern continues until the server hastaken a maximum number 119869 of vacations If the system isempty at the end of the 119869th vacation the server becomes idleand waits for new arriving customers

Finally we assume that the interarrival times the servicetimes the retrial times the repair times and the vacationtimes are mutually independent

3 The Markov Chain and SomePerformance Measures

At time 119898+ the system can be described by the process

119884119898

119898 ge 1 with 119884119898

= (119862119898

120585119898

120577119898

119871119898

119873119898

) where 119862119898

denotes the state of the server and 119873119898

be the number of thecustomers in the orbit It is assumed that 119862

119898

= 0 1 2 3 or4 according to whether the server is free busy under repairfor first type of breakdown under repair for second typeof breakdown or on vacations respectively If 119862

119898

= 0 120585119898

represents the remaining retrial time If119862119898

= 1 120585119898

representsthe remaining service time of the customer currently beingserved If 119862

119898

= 2 120585119898

represents the remaining repair timefor the first type of breakdown If 119862

119898

= 3 120585119898

represents theremaining service time of a customer currently being servedand 120577

119898

represents the remaining repair time of the secondtype of breakdowns If 119862

119898

= 4 let 119871119898

= 119899 represent thatthe server takes the 119899th vacation and let 120585

119898

represent theremaining vacation time 119899 = 1 2 119869 It can be shown that119884119898

119898 ge 1 is a Markov chain with the following state space

Ω = (0 0) (0 119894 119896) 119894 ge 1 119896 ge 1

(1 119894 119896) 119894 ge 1 119896 ge 0 (2 119894 119896) 119894 ge 1 119896 ge 1

Journal of Applied Mathematics 3

(3 119894 119895 119896) 119894 ge 1 119896 ge 0 (4 119894 119896) 119894 ge 1

119896 ge 0 119899 = 1 2 119869

(1)

Our object is to find the stationary distribution of theMarkov chain which is defined as follows

12058700

= lim119898rarrinfin

119875 119862119898

= 0119873119898

= 0

1205870119894119896

= lim119898rarrinfin

119875 119862119898

= 0 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 1

1205871119894119896

= lim119898rarrinfin

119875 119862119898

= 1 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 0

1205872119894119896

= lim119898rarrinfin

119875 119862119898

= 2 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 1

1205873119894119895119896

= lim119898rarrinfin

119875 119862119898

= 3 120585119898

= 119894 120577119898

= 119895119873119898

= 119896

119894 ge 1 119895 ge 1 119896 ge 0

120596119899119894119896

= lim119898rarrinfin

119875 119862119898

= 4 119871119898

= 119899 120585119898

= 119894119873119898

= 119896

119894 ge 1 119896 ge 0 119899 = 1 2 119869

(2)

The Kolmogorov equations for the stationary distribution ofthe system are

12058700

= 11990112058700

+ 11990112059611986910

(3)

1205870119894119896

= 1199011205870119894+1119896

+ 119901119886119894

12058711119896

+ 119901119886119894

12058721119896

+ 119901119886119894

119869

sum

119899=1

1205961198991119896

119894 ge 1 119896 ge 1

(4)

1205871119894119896

= 1205750119896

1199011205791

1205792

119904119894

12058700

+ 1205750119896

1199011205791

1205792

119904119894

infin

sum

119895=1

1205870119895119896

+ 1199011205791

1205792

119904119894

12058701119896+1

+ 1199011205791

1205792

119904119894

1205961198691119896

+ 1199011198860

1205791

1205792

119904119894

12058711119896+1

+ 1205750119896

1199011205791

1205792

119904119894

12058711119896

+ 1199011205792

1205871119894+1119896

+ 1205750119896

1199011205792

1205871119894+1119896minus1

+ 1205750119896

1199011205791

1205792

119904119894

12058721119896

+ 1199011198860

1205791

1205792

119904119894

12058721119896+1

+ 1199011205792

12058731198941119896

+ 1205750119896

1199011205792

12058731198941119896minus1

+ 1199011198860

1205791

1205792

119904119894

119869

sum

119899=1

1205961198991119896+1

+ 1205750119896

1199011205791

1205792

119904119894

119869minus1

sum

119899=1

1205961198991119896

119894 ge 1 119896 ge 0

(5)

1205872119894119896

= 1205751119896

1199011205791

1199031119894

12058700

+ 1205751119896

1199011205791

1199031119894

infin

sum

119895=1

1205870119895119896minus1

+ 1199011205791

1199031119894

12058701119896

+ 1199011198860

1205791

1199031119894

12058711119896

+ 1205751119896

1199011205791

1199031119894

12058711119896minus1

+ 1205751119896

1199011205872119894+1119896minus1

+ 1199011205872119894+1119896

+ 1199011198860

1205791

1199031119894

12058721119896

+ 1205751119896

1199011205791

1199031119894

12058721119896minus1

+ 1199011198860

1205791

1199031119894

119869

sum

119899=1

1205961198991119896

+ 1205751119896

1199011205791

1199031119894

119869minus1

sum

119899=1

1205961198991119896minus1

+ 1199011205791

1199031119894

1205961198691119896minus1

119894 ge 1 119896 ge 1

(6)

1205873119894119895119896

= 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058700

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

infin

sum

119897=1

1205870119897119896

+ 1199011205791

1205792

119904119894

1199032119895

12058701119896+1

+ 1199011205791

1205792

119904119894

1199032119895

1205961198691119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

12058711119896+1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058711119896

+ 1199011205792

1199032119895

1205871119894+1119896

+ 1205750119896

1199011205792

1199032119895

1205871119894+1119896minus1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058721119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

12058721119896+1

+ 1199011205873119894119895+1119896

+ 1199011205792

1199032119895

12058731198941119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

1205961198991119896+1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

1205961198991119896

+ 1205750119896

1199011205792

1199032119895

12058731198941119896minus1

+ 1205750119896

1199011205873119894119895+1119896minus1

119894 ge 1 119895 ge 1 119896 ge 0

(7)

1205961119894119896

= 1199011205961119894+1119896

+ 1205750119896

1199011205961119894+1119896minus1

+ 1205750119896

119901V119894

120587110

+ 1205751119896

119901V119894

120587110

119894 ge 1 119896 ge 0

(8)

120596119899119894119896

= 119901120596119899119894+1119896

+ 1205750119896

119901120596119899119894+1119896minus1

+ 1205750119896

119901V119894

120596119899minus110

+ 1205751119896

119901V119894

120596119899minus110

119894 ge 1 119896 ge 0 119899 = 2 3 119869

(9)

where 120575119894119895

is the Kroneckerrsquos symbol 120575119894119895

= 1 minus 120575119894119895

and thenormalizing condition is

12058700

+

infin

sum

119894=1

infin

sum

119896=1

(1205870119894119896

+ 1205872119894119896

) +

infin

sum

119894=1

infin

sum

119896=0

1205871119894119896

+

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

+

119869

sum

119899=1

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

= 1

(10)

To solve (3)ndash(10) we introduce the following generatingfunctions

1205930

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205870119894119896

119909119894

119911119896

1205931

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

1205871119894119896

119909119894

119911119896

4 Journal of Applied Mathematics

1205932

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205872119894119896

119909119894

119911119896

1205933

(119909 119910 119911) =

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

119909119894

119910119894

119911119896

120595119899

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

119909119894

119911119896

119899 = 1 2 119869

1205934

(119909 119911) =

119869

sum

119899=1

120595119899

(119909 119911)

(11)

and the following auxiliary generating functions

1205930119894

(119911) =

infin

sum

119896=1

1205870119894119896

119911119896

1205931119894

(119911) =

infin

sum

119896=0

1205871119894119896

119911119896

1205932119894

(119911) =

infin

sum

119896=1

1205872119894119896

119911119896

120595119899119894

(119911) =

infin

sum

119896=1

120596119899119894119896

119911119896

1205933119894119895

(119911) =

infin

sum

119896=0

1205873119894119895119896

119911119896

1205933119894

(119910 119911) =

infin

sum

119896=0

1205873119894119895119896

119910119895

119911119896

1206011

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

12058731198941119896

119909119894

119911119896

(12)

Now we can solve (3)ndash(10) by using the generating functiontechniqueWe first give some lemmaswhichwill be used lateron and their proof which can be readily obtained Thus theyare omitted The following inequalities hold

Lemma 1 If 1205791

1205881

+1205791

+1205791

1205882

lt 119901+119901119860(119901) then the inequality

[119911 + (1 minus 119911) 119901119860 (119901)]Ω (119911) minus 119911120574 (119911) gt 0 (13)

holds for 0 le 119911 lt 1 where

120574 (119911) = 119901 + 119901119911 1205881

= 1199011198781

(1 +1205792

1205792

11987721

)

1205882

= 11990111987711

Ω (119911) = 1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911))

120591 (119911) =1205792

119911

1 minus 1205792

1198772

(119911)

(14)

Lemma 2 The following limits exist if 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 +

119901119860(119901)

lim119911rarr1

120574 (119911) Γ (119911) minus 1205783

Ω (119911)

Λ (119911)

=

1199011198811

1205782

+ 1199011205783

[1205791

1198781

(1 + (1205792

1205792

) 11987721

) minus 1205791

+ 1205791

11987711

]

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

lim119911rarr1

Θ (119911)

Λ (119911)=

1199011198811

1205782

+ 119901119860 (119901) 1205783

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

(15)

where

120585 = 119881 (119901) 1205781

= 1 minus 120585119869+1

1205782

= 1 minus 120585119869

1205783

= (1 minus 120585) 120585119869

Θ (119911) = 1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

Γ (119911) = 1205781

minus 1205782

119881 (120574 (119911))

Λ (119911) = [119911 + 119901119860 (119901) (1 minus 119911)]Ω (119911) minus 119911120574 (119911)

(16)

By using Lemmas 1 and 2 we can obtain the generatingfunctions of the stationary distribution of the system whichare given by the following theorem

Theorem 3 If 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 + 119901119860(119901) the stationarydistribution of the Markov chain 119884

119898

119898 = 0 1 2 has thefollowing generating functions

1205930

(119909 119911) =119860 (119909) minus 119860 (119901)

119909 minus 119901

119901119909119911 [120574 (119911) Γ (119911) minus 1205783

Ω (119911)]

1205783

Λ (119911)12058700

1205931

(119909 119911) =119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

Θ (119911) 1199011199091205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

1205932

(119909 119911) =1198771

(119909) minus 1198771

(120574 (119911))

119909 minus 120574 (119911)

Θ (119911) 1199011199091205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

1205933

(119909 119910 119911)

=119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

times1198772

(119910) minus 1198772

(120574 (119911))

119910 minus 120574 (119911)

Θ (119911) 1199011199091199101205791

120591 (120574 (119911))

1199011205783

Λ (119911)

1205792

1205792

12058700

1205934

(119909 119911) =1199011205782

119909120574 (119911) [119881 (119909) minus 119881 (120574 (119911))]

1199011205783

[119909 minus 120574 (119911)]12058700

(17)

where

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(18)

Journal of Applied Mathematics 5

Proof Multiplying (3)ndash(9) by 119911119896 and summing over 119896 we getthe following equations

1205930119894

(119911) = 1199011205930119894+1

(119911)

+ 119901119886119894

[12059311

(119911) + 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus 119901119886119894

120587110

minus 119901119886119894

119869

sum

119899=1

12059611989910

119894 ge 1

(19)

1205931119894

(119911) = 1199011205791

1205792

119904119894

12058700

+ 1199011205791

1205792

119904119894

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

12059301

(119911) + 1205792

120574 (119911) 1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

[12059311

(119911)

+ 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

120587110

minus1199011205791

1205792

119904119894

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911) 119894 ge 1

(20)

1205932119894

(119911) = 1199011205791

1199111199031119894

12058700

+ 1199011205791

1199111199031119894

1205930

(1 119911)

+ 1199011205791

1199031119894

12059301

(119911) + 120574 (119911) 1205932119894+1

(119911)

+ (119901119911 + 1199011198860

) 1205791

1199031119894

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

minus (119901119911 + 1199011198860

) 1205791

1199031119894

120587110

minus 1199011198860

1205791

1199031119894

119869

sum

119899=1

12059611989910

minus 1199011205791

1199031119894

119911

119869minus1

sum

119899=1

12059611989910

119894 ge 1

(21)

1205933119894119895

(119911) = 1199011205791

1205792

119904119894

1199032119895

12058700

+ 1199011205791

1205792

119904119894

1199032119895

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

1199032119895

12059301

(119911) + 1205792

120574 (119911) 1199032119895

1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

+ 120574 (119911) 1205933119894119895+1

(119911) minus 1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911)

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

120587110

119894 ge 1

(22)1205951119894

(119911) = 1199011205951119894+1

(119911) + 1199011199111205951119894+1

(119911)

+ 120574 (119911) V119894

120587110

119894 ge 1 119896 ge 0

(23)

120595119899119894

(119911) = 119901120595119899119894+1

(119911) + 119901119911120595119899119894+1

(119911)

+ 120574 (119911) V119894

120596119899minus110

119894 ge 1 119896 ge 0 2 le 119899 le 119869

(24)

Multiplying (23) and (24) by 119909119894 and summing over 119894 we get

119909 minus 120574 (119911)

1199091205951

(119909 119911) = minus120574 (119911) 12059511

(119911) + 120574 (119911) 119881 (119909) 120587110

(25)

119909 minus 120574 (119911)

119909120595119899

(119909 119911) = minus120574 (119911) 1205951198991

(119911) + 120574 (119911) 119881 (119909) 120596119899minus110

2 le 119899 le 119869

(26)

Setting 119909 = 120574(119911) in (25) and (26) we get

12059511

(119911) = 119881 (120574 (119911)) 120587110

(27)

1205951198991

(119911) = 119881 (120574 (119911)) 120596119899minus110

2 le 119899 le 119869 (28)

Substituting (27) and (28) into (25) and (26) respectively weobtain

1205951

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120587110

120595119899

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120596119899minus110

2 le 119899 le 119869

(29)

Taking the derivative of (29) with respective to 119909 and letting119909 = 119911 = 0 we get

120596110

= 120585120587110

(30)

12059611989910

= 120585120596119899minus110

= 120585119899

120587110

2 le 119899 le 119869 (31)

Substituting (31) into (3) we get

120587110

=11990112058700

119901120585119869 (32)

6 Journal of Applied Mathematics

Multiplying (19) by 119909119895 summing over 119895 and taking into

account (30)ndash(32) we obtain

119909 minus 119901

1199091205930

(119909 119911)

= 119901 (119860 (119909) minus 1198860

) (12059311

(119911) + 12059321

(119911))

minus 11990112059301

(119911) minus 119901 (119860 (119909) minus 1198860

)1205781

minus 1205782

119881 (120574 (119911))

1205783

12058700

(33)

Note that we can find 1205930

(1 119911) by setting 119909 = 1 in (33) thenmultiplying (20) and (21) by 119909

119895 summing over 119895 and takinginto account (30)ndash(32) we obtain

119909 minus 120574 (119911) 1205792

1199091205931

(119909 119911)

=119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

12059311

(119911) +119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 12059301

(119911)

+ 120574 (119911) 1205792

1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 12058700

(34)

119909 minus 120574 (119911)

1199091205932

(119909 119911) = [119911 + 1199011198860

(1 minus 119911)] 1205791

1198771

(119909)

times (12059311

(119911) + 12059321

(119911)) minus 120574 (119911) 12059321

(119911)

+ 1199011205791

1198771

(119909) (1 minus 119911) 12059301

(119911)

minus119870 (119911)

1199011205783

1199011205791

1198771

(119909) 12058700

(35)

where119870(119911) = 1199111205782

[1 minus 119881(120574(119911))] + 1199011198860

(1 minus 119911)Γ(119911)Multiplying (22) by 119910

119895 and 119909119894 summing over 119895 and 119894

respectively and taking into account (30)ndash(32) we obtain

119910 minus 120574 (119911)

1199101205933

(119909 119910 119911) =119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772

(119910)

times [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

1198772

(119910) 12059311

(119911)

+119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772(119910)

12059301

(119911)

+ 120574 (119911) (1 minus 1205792

1198772

(119910)) 1206011

(119909 119911)

minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 1198772

(119910) 12058700

+120574 (119911) 120579

2

1198772

(119910)

1199091205931

(119909 119911)

(36)

Note that we can find 1206011

(119909 119911) by setting 119910 = 120574(119911) in (36) andsubstituting the expression of 120601

1

(119909 119911) into (34) we get

119909 minus 1205911

(120574 (119911))

119909[1 minus 120579

2

1198772

(120574 (119911))] 1205931

(119909 119911)

= 1205792

119911 + 119901119886

0

(1 minus 119911)

1199111205791

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 12059311

(119911) +119901 (1 minus 119911)

1199111205791

119878 (119909) 12059301

(119911)

+120574 (119911) 1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

119878 (119909) 12058700

(37)

Setting119909 = 119901 in (33)119909 = 120574(119911) in (35) and119909 = 120591(120574(119911)) in (37)respectively we can get the equations for 120593

01

(119911) 12059311

(119911) and12059321

(119911) By solving these equations we get the the generatingfunctions as follows

12059301

(119911) = 119901119911 (A (119901) minus 1198860

)120574 (119911) Γ (119911) minus 120578

3

Ω (119911)

1199011205783

Λ (119911)12058700

12059311

(119911) =Θ (119911) 119901120579

1

119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

12059321

(119911) =Θ (119911) 119901120579

1

119911119877 (120574 (119911))

1199011205783

Λ (119911)12058700

(38)

Using Lemmas 1 and 2 it is easy to show that 12059301

(119911)12059311

(119911) and 12059321

(119911) are defined for 119911 isin [0 1) and can beextended by continuity in 119911 = 1 if 120579

1

1205881

+ 1205791

+ 1205791

1205882

lt

119901 + 119901119860(119901) Now substituting (30)ndash(32) into (29) we can get1205934

(119909 119911) Similarly substituting (38) into (33) (35)ndash(37) wederive 120593

0

(119909 119911) 1205931

(119909 119911) 1205932

(119909 119911) and 1205933

(119909 119910 119911) Using thenormalizing condition we can find the unknown constant12058700

This completes the proof

Based on Theorem 3 we can easily obtain the marginalgenerating functions of the number of customers when theserver is in various states and some performance measuresThey are summarized in the following CorollaryTheir proofsare very easy and thus are omitted For convenience wedefine variable119873 be the orbit size and 119871 be the system size

Corollary 4

(1) The marginal generating function of the number ofcustomers in the orbit when the server is idle or vacationis given by

12058700

+ 1205930

(1 119911)

=119911 (1 minus 119860 (119901)) [120574 (119911) Γ (119911) minus 120578

3

Ω (119911)] + 1205783

Λ (119911)

1205783

Λ (119911)12058700

(39)

(2) The marginal generating function of the number ofcustomers in the orbit when the server is busy is givenby

1205931

(1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

Θ (119911) 1205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(40)

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

(3 119894 119895 119896) 119894 ge 1 119896 ge 0 (4 119894 119896) 119894 ge 1

119896 ge 0 119899 = 1 2 119869

(1)

Our object is to find the stationary distribution of theMarkov chain which is defined as follows

12058700

= lim119898rarrinfin

119875 119862119898

= 0119873119898

= 0

1205870119894119896

= lim119898rarrinfin

119875 119862119898

= 0 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 1

1205871119894119896

= lim119898rarrinfin

119875 119862119898

= 1 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 0

1205872119894119896

= lim119898rarrinfin

119875 119862119898

= 2 120585119898

= 119894119873119898

= 119896 119894 ge 1 119896 ge 1

1205873119894119895119896

= lim119898rarrinfin

119875 119862119898

= 3 120585119898

= 119894 120577119898

= 119895119873119898

= 119896

119894 ge 1 119895 ge 1 119896 ge 0

120596119899119894119896

= lim119898rarrinfin

119875 119862119898

= 4 119871119898

= 119899 120585119898

= 119894119873119898

= 119896

119894 ge 1 119896 ge 0 119899 = 1 2 119869

(2)

The Kolmogorov equations for the stationary distribution ofthe system are

12058700

= 11990112058700

+ 11990112059611986910

(3)

1205870119894119896

= 1199011205870119894+1119896

+ 119901119886119894

12058711119896

+ 119901119886119894

12058721119896

+ 119901119886119894

119869

sum

119899=1

1205961198991119896

119894 ge 1 119896 ge 1

(4)

1205871119894119896

= 1205750119896

1199011205791

1205792

119904119894

12058700

+ 1205750119896

1199011205791

1205792

119904119894

infin

sum

119895=1

1205870119895119896

+ 1199011205791

1205792

119904119894

12058701119896+1

+ 1199011205791

1205792

119904119894

1205961198691119896

+ 1199011198860

1205791

1205792

119904119894

12058711119896+1

+ 1205750119896

1199011205791

1205792

119904119894

12058711119896

+ 1199011205792

1205871119894+1119896

+ 1205750119896

1199011205792

1205871119894+1119896minus1

+ 1205750119896

1199011205791

1205792

119904119894

12058721119896

+ 1199011198860

1205791

1205792

119904119894

12058721119896+1

+ 1199011205792

12058731198941119896

+ 1205750119896

1199011205792

12058731198941119896minus1

+ 1199011198860

1205791

1205792

119904119894

119869

sum

119899=1

1205961198991119896+1

+ 1205750119896

1199011205791

1205792

119904119894

119869minus1

sum

119899=1

1205961198991119896

119894 ge 1 119896 ge 0

(5)

1205872119894119896

= 1205751119896

1199011205791

1199031119894

12058700

+ 1205751119896

1199011205791

1199031119894

infin

sum

119895=1

1205870119895119896minus1

+ 1199011205791

1199031119894

12058701119896

+ 1199011198860

1205791

1199031119894

12058711119896

+ 1205751119896

1199011205791

1199031119894

12058711119896minus1

+ 1205751119896

1199011205872119894+1119896minus1

+ 1199011205872119894+1119896

+ 1199011198860

1205791

1199031119894

12058721119896

+ 1205751119896

1199011205791

1199031119894

12058721119896minus1

+ 1199011198860

1205791

1199031119894

119869

sum

119899=1

1205961198991119896

+ 1205751119896

1199011205791

1199031119894

119869minus1

sum

119899=1

1205961198991119896minus1

+ 1199011205791

1199031119894

1205961198691119896minus1

119894 ge 1 119896 ge 1

(6)

1205873119894119895119896

= 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058700

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

infin

sum

119897=1

1205870119897119896

+ 1199011205791

1205792

119904119894

1199032119895

12058701119896+1

+ 1199011205791

1205792

119904119894

1199032119895

1205961198691119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

12058711119896+1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058711119896

+ 1199011205792

1199032119895

1205871119894+1119896

+ 1205750119896

1199011205792

1199032119895

1205871119894+1119896minus1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

12058721119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

12058721119896+1

+ 1199011205873119894119895+1119896

+ 1199011205792

1199032119895

12058731198941119896

+ 1199011198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

1205961198991119896+1

+ 1205750119896

1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

1205961198991119896

+ 1205750119896

1199011205792

1199032119895

12058731198941119896minus1

+ 1205750119896

1199011205873119894119895+1119896minus1

119894 ge 1 119895 ge 1 119896 ge 0

(7)

1205961119894119896

= 1199011205961119894+1119896

+ 1205750119896

1199011205961119894+1119896minus1

+ 1205750119896

119901V119894

120587110

+ 1205751119896

119901V119894

120587110

119894 ge 1 119896 ge 0

(8)

120596119899119894119896

= 119901120596119899119894+1119896

+ 1205750119896

119901120596119899119894+1119896minus1

+ 1205750119896

119901V119894

120596119899minus110

+ 1205751119896

119901V119894

120596119899minus110

119894 ge 1 119896 ge 0 119899 = 2 3 119869

(9)

where 120575119894119895

is the Kroneckerrsquos symbol 120575119894119895

= 1 minus 120575119894119895

and thenormalizing condition is

12058700

+

infin

sum

119894=1

infin

sum

119896=1

(1205870119894119896

+ 1205872119894119896

) +

infin

sum

119894=1

infin

sum

119896=0

1205871119894119896

+

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

+

119869

sum

119899=1

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

= 1

(10)

To solve (3)ndash(10) we introduce the following generatingfunctions

1205930

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205870119894119896

119909119894

119911119896

1205931

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

1205871119894119896

119909119894

119911119896

4 Journal of Applied Mathematics

1205932

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205872119894119896

119909119894

119911119896

1205933

(119909 119910 119911) =

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

119909119894

119910119894

119911119896

120595119899

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

119909119894

119911119896

119899 = 1 2 119869

1205934

(119909 119911) =

119869

sum

119899=1

120595119899

(119909 119911)

(11)

and the following auxiliary generating functions

1205930119894

(119911) =

infin

sum

119896=1

1205870119894119896

119911119896

1205931119894

(119911) =

infin

sum

119896=0

1205871119894119896

119911119896

1205932119894

(119911) =

infin

sum

119896=1

1205872119894119896

119911119896

120595119899119894

(119911) =

infin

sum

119896=1

120596119899119894119896

119911119896

1205933119894119895

(119911) =

infin

sum

119896=0

1205873119894119895119896

119911119896

1205933119894

(119910 119911) =

infin

sum

119896=0

1205873119894119895119896

119910119895

119911119896

1206011

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

12058731198941119896

119909119894

119911119896

(12)

Now we can solve (3)ndash(10) by using the generating functiontechniqueWe first give some lemmaswhichwill be used lateron and their proof which can be readily obtained Thus theyare omitted The following inequalities hold

Lemma 1 If 1205791

1205881

+1205791

+1205791

1205882

lt 119901+119901119860(119901) then the inequality

[119911 + (1 minus 119911) 119901119860 (119901)]Ω (119911) minus 119911120574 (119911) gt 0 (13)

holds for 0 le 119911 lt 1 where

120574 (119911) = 119901 + 119901119911 1205881

= 1199011198781

(1 +1205792

1205792

11987721

)

1205882

= 11990111987711

Ω (119911) = 1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911))

120591 (119911) =1205792

119911

1 minus 1205792

1198772

(119911)

(14)

Lemma 2 The following limits exist if 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 +

119901119860(119901)

lim119911rarr1

120574 (119911) Γ (119911) minus 1205783

Ω (119911)

Λ (119911)

=

1199011198811

1205782

+ 1199011205783

[1205791

1198781

(1 + (1205792

1205792

) 11987721

) minus 1205791

+ 1205791

11987711

]

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

lim119911rarr1

Θ (119911)

Λ (119911)=

1199011198811

1205782

+ 119901119860 (119901) 1205783

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

(15)

where

120585 = 119881 (119901) 1205781

= 1 minus 120585119869+1

1205782

= 1 minus 120585119869

1205783

= (1 minus 120585) 120585119869

Θ (119911) = 1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

Γ (119911) = 1205781

minus 1205782

119881 (120574 (119911))

Λ (119911) = [119911 + 119901119860 (119901) (1 minus 119911)]Ω (119911) minus 119911120574 (119911)

(16)

By using Lemmas 1 and 2 we can obtain the generatingfunctions of the stationary distribution of the system whichare given by the following theorem

Theorem 3 If 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 + 119901119860(119901) the stationarydistribution of the Markov chain 119884

119898

119898 = 0 1 2 has thefollowing generating functions

1205930

(119909 119911) =119860 (119909) minus 119860 (119901)

119909 minus 119901

119901119909119911 [120574 (119911) Γ (119911) minus 1205783

Ω (119911)]

1205783

Λ (119911)12058700

1205931

(119909 119911) =119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

Θ (119911) 1199011199091205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

1205932

(119909 119911) =1198771

(119909) minus 1198771

(120574 (119911))

119909 minus 120574 (119911)

Θ (119911) 1199011199091205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

1205933

(119909 119910 119911)

=119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

times1198772

(119910) minus 1198772

(120574 (119911))

119910 minus 120574 (119911)

Θ (119911) 1199011199091199101205791

120591 (120574 (119911))

1199011205783

Λ (119911)

1205792

1205792

12058700

1205934

(119909 119911) =1199011205782

119909120574 (119911) [119881 (119909) minus 119881 (120574 (119911))]

1199011205783

[119909 minus 120574 (119911)]12058700

(17)

where

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(18)

Journal of Applied Mathematics 5

Proof Multiplying (3)ndash(9) by 119911119896 and summing over 119896 we getthe following equations

1205930119894

(119911) = 1199011205930119894+1

(119911)

+ 119901119886119894

[12059311

(119911) + 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus 119901119886119894

120587110

minus 119901119886119894

119869

sum

119899=1

12059611989910

119894 ge 1

(19)

1205931119894

(119911) = 1199011205791

1205792

119904119894

12058700

+ 1199011205791

1205792

119904119894

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

12059301

(119911) + 1205792

120574 (119911) 1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

[12059311

(119911)

+ 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

120587110

minus1199011205791

1205792

119904119894

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911) 119894 ge 1

(20)

1205932119894

(119911) = 1199011205791

1199111199031119894

12058700

+ 1199011205791

1199111199031119894

1205930

(1 119911)

+ 1199011205791

1199031119894

12059301

(119911) + 120574 (119911) 1205932119894+1

(119911)

+ (119901119911 + 1199011198860

) 1205791

1199031119894

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

minus (119901119911 + 1199011198860

) 1205791

1199031119894

120587110

minus 1199011198860

1205791

1199031119894

119869

sum

119899=1

12059611989910

minus 1199011205791

1199031119894

119911

119869minus1

sum

119899=1

12059611989910

119894 ge 1

(21)

1205933119894119895

(119911) = 1199011205791

1205792

119904119894

1199032119895

12058700

+ 1199011205791

1205792

119904119894

1199032119895

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

1199032119895

12059301

(119911) + 1205792

120574 (119911) 1199032119895

1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

+ 120574 (119911) 1205933119894119895+1

(119911) minus 1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911)

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

120587110

119894 ge 1

(22)1205951119894

(119911) = 1199011205951119894+1

(119911) + 1199011199111205951119894+1

(119911)

+ 120574 (119911) V119894

120587110

119894 ge 1 119896 ge 0

(23)

120595119899119894

(119911) = 119901120595119899119894+1

(119911) + 119901119911120595119899119894+1

(119911)

+ 120574 (119911) V119894

120596119899minus110

119894 ge 1 119896 ge 0 2 le 119899 le 119869

(24)

Multiplying (23) and (24) by 119909119894 and summing over 119894 we get

119909 minus 120574 (119911)

1199091205951

(119909 119911) = minus120574 (119911) 12059511

(119911) + 120574 (119911) 119881 (119909) 120587110

(25)

119909 minus 120574 (119911)

119909120595119899

(119909 119911) = minus120574 (119911) 1205951198991

(119911) + 120574 (119911) 119881 (119909) 120596119899minus110

2 le 119899 le 119869

(26)

Setting 119909 = 120574(119911) in (25) and (26) we get

12059511

(119911) = 119881 (120574 (119911)) 120587110

(27)

1205951198991

(119911) = 119881 (120574 (119911)) 120596119899minus110

2 le 119899 le 119869 (28)

Substituting (27) and (28) into (25) and (26) respectively weobtain

1205951

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120587110

120595119899

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120596119899minus110

2 le 119899 le 119869

(29)

Taking the derivative of (29) with respective to 119909 and letting119909 = 119911 = 0 we get

120596110

= 120585120587110

(30)

12059611989910

= 120585120596119899minus110

= 120585119899

120587110

2 le 119899 le 119869 (31)

Substituting (31) into (3) we get

120587110

=11990112058700

119901120585119869 (32)

6 Journal of Applied Mathematics

Multiplying (19) by 119909119895 summing over 119895 and taking into

account (30)ndash(32) we obtain

119909 minus 119901

1199091205930

(119909 119911)

= 119901 (119860 (119909) minus 1198860

) (12059311

(119911) + 12059321

(119911))

minus 11990112059301

(119911) minus 119901 (119860 (119909) minus 1198860

)1205781

minus 1205782

119881 (120574 (119911))

1205783

12058700

(33)

Note that we can find 1205930

(1 119911) by setting 119909 = 1 in (33) thenmultiplying (20) and (21) by 119909

119895 summing over 119895 and takinginto account (30)ndash(32) we obtain

119909 minus 120574 (119911) 1205792

1199091205931

(119909 119911)

=119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

12059311

(119911) +119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 12059301

(119911)

+ 120574 (119911) 1205792

1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 12058700

(34)

119909 minus 120574 (119911)

1199091205932

(119909 119911) = [119911 + 1199011198860

(1 minus 119911)] 1205791

1198771

(119909)

times (12059311

(119911) + 12059321

(119911)) minus 120574 (119911) 12059321

(119911)

+ 1199011205791

1198771

(119909) (1 minus 119911) 12059301

(119911)

minus119870 (119911)

1199011205783

1199011205791

1198771

(119909) 12058700

(35)

where119870(119911) = 1199111205782

[1 minus 119881(120574(119911))] + 1199011198860

(1 minus 119911)Γ(119911)Multiplying (22) by 119910

119895 and 119909119894 summing over 119895 and 119894

respectively and taking into account (30)ndash(32) we obtain

119910 minus 120574 (119911)

1199101205933

(119909 119910 119911) =119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772

(119910)

times [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

1198772

(119910) 12059311

(119911)

+119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772(119910)

12059301

(119911)

+ 120574 (119911) (1 minus 1205792

1198772

(119910)) 1206011

(119909 119911)

minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 1198772

(119910) 12058700

+120574 (119911) 120579

2

1198772

(119910)

1199091205931

(119909 119911)

(36)

Note that we can find 1206011

(119909 119911) by setting 119910 = 120574(119911) in (36) andsubstituting the expression of 120601

1

(119909 119911) into (34) we get

119909 minus 1205911

(120574 (119911))

119909[1 minus 120579

2

1198772

(120574 (119911))] 1205931

(119909 119911)

= 1205792

119911 + 119901119886

0

(1 minus 119911)

1199111205791

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 12059311

(119911) +119901 (1 minus 119911)

1199111205791

119878 (119909) 12059301

(119911)

+120574 (119911) 1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

119878 (119909) 12058700

(37)

Setting119909 = 119901 in (33)119909 = 120574(119911) in (35) and119909 = 120591(120574(119911)) in (37)respectively we can get the equations for 120593

01

(119911) 12059311

(119911) and12059321

(119911) By solving these equations we get the the generatingfunctions as follows

12059301

(119911) = 119901119911 (A (119901) minus 1198860

)120574 (119911) Γ (119911) minus 120578

3

Ω (119911)

1199011205783

Λ (119911)12058700

12059311

(119911) =Θ (119911) 119901120579

1

119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

12059321

(119911) =Θ (119911) 119901120579

1

119911119877 (120574 (119911))

1199011205783

Λ (119911)12058700

(38)

Using Lemmas 1 and 2 it is easy to show that 12059301

(119911)12059311

(119911) and 12059321

(119911) are defined for 119911 isin [0 1) and can beextended by continuity in 119911 = 1 if 120579

1

1205881

+ 1205791

+ 1205791

1205882

lt

119901 + 119901119860(119901) Now substituting (30)ndash(32) into (29) we can get1205934

(119909 119911) Similarly substituting (38) into (33) (35)ndash(37) wederive 120593

0

(119909 119911) 1205931

(119909 119911) 1205932

(119909 119911) and 1205933

(119909 119910 119911) Using thenormalizing condition we can find the unknown constant12058700

This completes the proof

Based on Theorem 3 we can easily obtain the marginalgenerating functions of the number of customers when theserver is in various states and some performance measuresThey are summarized in the following CorollaryTheir proofsare very easy and thus are omitted For convenience wedefine variable119873 be the orbit size and 119871 be the system size

Corollary 4

(1) The marginal generating function of the number ofcustomers in the orbit when the server is idle or vacationis given by

12058700

+ 1205930

(1 119911)

=119911 (1 minus 119860 (119901)) [120574 (119911) Γ (119911) minus 120578

3

Ω (119911)] + 1205783

Λ (119911)

1205783

Λ (119911)12058700

(39)

(2) The marginal generating function of the number ofcustomers in the orbit when the server is busy is givenby

1205931

(1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

Θ (119911) 1205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(40)

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

1205932

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=1

1205872119894119896

119909119894

119911119896

1205933

(119909 119910 119911) =

infin

sum

119894=1

infin

sum

119895=1

infin

sum

119896=0

1205873119894119895119896

119909119894

119910119894

119911119896

120595119899

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

120596119899119894119896

119909119894

119911119896

119899 = 1 2 119869

1205934

(119909 119911) =

119869

sum

119899=1

120595119899

(119909 119911)

(11)

and the following auxiliary generating functions

1205930119894

(119911) =

infin

sum

119896=1

1205870119894119896

119911119896

1205931119894

(119911) =

infin

sum

119896=0

1205871119894119896

119911119896

1205932119894

(119911) =

infin

sum

119896=1

1205872119894119896

119911119896

120595119899119894

(119911) =

infin

sum

119896=1

120596119899119894119896

119911119896

1205933119894119895

(119911) =

infin

sum

119896=0

1205873119894119895119896

119911119896

1205933119894

(119910 119911) =

infin

sum

119896=0

1205873119894119895119896

119910119895

119911119896

1206011

(119909 119911) =

infin

sum

119894=1

infin

sum

119896=0

12058731198941119896

119909119894

119911119896

(12)

Now we can solve (3)ndash(10) by using the generating functiontechniqueWe first give some lemmaswhichwill be used lateron and their proof which can be readily obtained Thus theyare omitted The following inequalities hold

Lemma 1 If 1205791

1205881

+1205791

+1205791

1205882

lt 119901+119901119860(119901) then the inequality

[119911 + (1 minus 119911) 119901119860 (119901)]Ω (119911) minus 119911120574 (119911) gt 0 (13)

holds for 0 le 119911 lt 1 where

120574 (119911) = 119901 + 119901119911 1205881

= 1199011198781

(1 +1205792

1205792

11987721

)

1205882

= 11990111987711

Ω (119911) = 1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911))

120591 (119911) =1205792

119911

1 minus 1205792

1198772

(119911)

(14)

Lemma 2 The following limits exist if 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 +

119901119860(119901)

lim119911rarr1

120574 (119911) Γ (119911) minus 1205783

Ω (119911)

Λ (119911)

=

1199011198811

1205782

+ 1199011205783

[1205791

1198781

(1 + (1205792

1205792

) 11987721

) minus 1205791

+ 1205791

11987711

]

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

lim119911rarr1

Θ (119911)

Λ (119911)=

1199011198811

1205782

+ 119901119860 (119901) 1205783

119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

(15)

where

120585 = 119881 (119901) 1205781

= 1 minus 120585119869+1

1205782

= 1 minus 120585119869

1205783

= (1 minus 120585) 120585119869

Θ (119911) = 1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

Γ (119911) = 1205781

minus 1205782

119881 (120574 (119911))

Λ (119911) = [119911 + 119901119860 (119901) (1 minus 119911)]Ω (119911) minus 119911120574 (119911)

(16)

By using Lemmas 1 and 2 we can obtain the generatingfunctions of the stationary distribution of the system whichare given by the following theorem

Theorem 3 If 1205791

1205881

+ 1205791

+ 1205791

1205882

lt 119901 + 119901119860(119901) the stationarydistribution of the Markov chain 119884

119898

119898 = 0 1 2 has thefollowing generating functions

1205930

(119909 119911) =119860 (119909) minus 119860 (119901)

119909 minus 119901

119901119909119911 [120574 (119911) Γ (119911) minus 1205783

Ω (119911)]

1205783

Λ (119911)12058700

1205931

(119909 119911) =119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

Θ (119911) 1199011199091205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

1205932

(119909 119911) =1198771

(119909) minus 1198771

(120574 (119911))

119909 minus 120574 (119911)

Θ (119911) 1199011199091205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

1205933

(119909 119910 119911)

=119878 (119909) minus 119878 (120591 (120574 (119911)))

119909 minus 120591 (120574 (119911))

times1198772

(119910) minus 1198772

(120574 (119911))

119910 minus 120574 (119911)

Θ (119911) 1199011199091199101205791

120591 (120574 (119911))

1199011205783

Λ (119911)

1205792

1205792

12058700

1205934

(119909 119911) =1199011205782

119909120574 (119911) [119881 (119909) minus 119881 (120574 (119911))]

1199011205783

[119909 minus 120574 (119911)]12058700

(17)

where

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(18)

Journal of Applied Mathematics 5

Proof Multiplying (3)ndash(9) by 119911119896 and summing over 119896 we getthe following equations

1205930119894

(119911) = 1199011205930119894+1

(119911)

+ 119901119886119894

[12059311

(119911) + 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus 119901119886119894

120587110

minus 119901119886119894

119869

sum

119899=1

12059611989910

119894 ge 1

(19)

1205931119894

(119911) = 1199011205791

1205792

119904119894

12058700

+ 1199011205791

1205792

119904119894

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

12059301

(119911) + 1205792

120574 (119911) 1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

[12059311

(119911)

+ 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

120587110

minus1199011205791

1205792

119904119894

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911) 119894 ge 1

(20)

1205932119894

(119911) = 1199011205791

1199111199031119894

12058700

+ 1199011205791

1199111199031119894

1205930

(1 119911)

+ 1199011205791

1199031119894

12059301

(119911) + 120574 (119911) 1205932119894+1

(119911)

+ (119901119911 + 1199011198860

) 1205791

1199031119894

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

minus (119901119911 + 1199011198860

) 1205791

1199031119894

120587110

minus 1199011198860

1205791

1199031119894

119869

sum

119899=1

12059611989910

minus 1199011205791

1199031119894

119911

119869minus1

sum

119899=1

12059611989910

119894 ge 1

(21)

1205933119894119895

(119911) = 1199011205791

1205792

119904119894

1199032119895

12058700

+ 1199011205791

1205792

119904119894

1199032119895

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

1199032119895

12059301

(119911) + 1205792

120574 (119911) 1199032119895

1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

+ 120574 (119911) 1205933119894119895+1

(119911) minus 1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911)

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

120587110

119894 ge 1

(22)1205951119894

(119911) = 1199011205951119894+1

(119911) + 1199011199111205951119894+1

(119911)

+ 120574 (119911) V119894

120587110

119894 ge 1 119896 ge 0

(23)

120595119899119894

(119911) = 119901120595119899119894+1

(119911) + 119901119911120595119899119894+1

(119911)

+ 120574 (119911) V119894

120596119899minus110

119894 ge 1 119896 ge 0 2 le 119899 le 119869

(24)

Multiplying (23) and (24) by 119909119894 and summing over 119894 we get

119909 minus 120574 (119911)

1199091205951

(119909 119911) = minus120574 (119911) 12059511

(119911) + 120574 (119911) 119881 (119909) 120587110

(25)

119909 minus 120574 (119911)

119909120595119899

(119909 119911) = minus120574 (119911) 1205951198991

(119911) + 120574 (119911) 119881 (119909) 120596119899minus110

2 le 119899 le 119869

(26)

Setting 119909 = 120574(119911) in (25) and (26) we get

12059511

(119911) = 119881 (120574 (119911)) 120587110

(27)

1205951198991

(119911) = 119881 (120574 (119911)) 120596119899minus110

2 le 119899 le 119869 (28)

Substituting (27) and (28) into (25) and (26) respectively weobtain

1205951

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120587110

120595119899

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120596119899minus110

2 le 119899 le 119869

(29)

Taking the derivative of (29) with respective to 119909 and letting119909 = 119911 = 0 we get

120596110

= 120585120587110

(30)

12059611989910

= 120585120596119899minus110

= 120585119899

120587110

2 le 119899 le 119869 (31)

Substituting (31) into (3) we get

120587110

=11990112058700

119901120585119869 (32)

6 Journal of Applied Mathematics

Multiplying (19) by 119909119895 summing over 119895 and taking into

account (30)ndash(32) we obtain

119909 minus 119901

1199091205930

(119909 119911)

= 119901 (119860 (119909) minus 1198860

) (12059311

(119911) + 12059321

(119911))

minus 11990112059301

(119911) minus 119901 (119860 (119909) minus 1198860

)1205781

minus 1205782

119881 (120574 (119911))

1205783

12058700

(33)

Note that we can find 1205930

(1 119911) by setting 119909 = 1 in (33) thenmultiplying (20) and (21) by 119909

119895 summing over 119895 and takinginto account (30)ndash(32) we obtain

119909 minus 120574 (119911) 1205792

1199091205931

(119909 119911)

=119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

12059311

(119911) +119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 12059301

(119911)

+ 120574 (119911) 1205792

1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 12058700

(34)

119909 minus 120574 (119911)

1199091205932

(119909 119911) = [119911 + 1199011198860

(1 minus 119911)] 1205791

1198771

(119909)

times (12059311

(119911) + 12059321

(119911)) minus 120574 (119911) 12059321

(119911)

+ 1199011205791

1198771

(119909) (1 minus 119911) 12059301

(119911)

minus119870 (119911)

1199011205783

1199011205791

1198771

(119909) 12058700

(35)

where119870(119911) = 1199111205782

[1 minus 119881(120574(119911))] + 1199011198860

(1 minus 119911)Γ(119911)Multiplying (22) by 119910

119895 and 119909119894 summing over 119895 and 119894

respectively and taking into account (30)ndash(32) we obtain

119910 minus 120574 (119911)

1199101205933

(119909 119910 119911) =119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772

(119910)

times [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

1198772

(119910) 12059311

(119911)

+119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772(119910)

12059301

(119911)

+ 120574 (119911) (1 minus 1205792

1198772

(119910)) 1206011

(119909 119911)

minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 1198772

(119910) 12058700

+120574 (119911) 120579

2

1198772

(119910)

1199091205931

(119909 119911)

(36)

Note that we can find 1206011

(119909 119911) by setting 119910 = 120574(119911) in (36) andsubstituting the expression of 120601

1

(119909 119911) into (34) we get

119909 minus 1205911

(120574 (119911))

119909[1 minus 120579

2

1198772

(120574 (119911))] 1205931

(119909 119911)

= 1205792

119911 + 119901119886

0

(1 minus 119911)

1199111205791

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 12059311

(119911) +119901 (1 minus 119911)

1199111205791

119878 (119909) 12059301

(119911)

+120574 (119911) 1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

119878 (119909) 12058700

(37)

Setting119909 = 119901 in (33)119909 = 120574(119911) in (35) and119909 = 120591(120574(119911)) in (37)respectively we can get the equations for 120593

01

(119911) 12059311

(119911) and12059321

(119911) By solving these equations we get the the generatingfunctions as follows

12059301

(119911) = 119901119911 (A (119901) minus 1198860

)120574 (119911) Γ (119911) minus 120578

3

Ω (119911)

1199011205783

Λ (119911)12058700

12059311

(119911) =Θ (119911) 119901120579

1

119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

12059321

(119911) =Θ (119911) 119901120579

1

119911119877 (120574 (119911))

1199011205783

Λ (119911)12058700

(38)

Using Lemmas 1 and 2 it is easy to show that 12059301

(119911)12059311

(119911) and 12059321

(119911) are defined for 119911 isin [0 1) and can beextended by continuity in 119911 = 1 if 120579

1

1205881

+ 1205791

+ 1205791

1205882

lt

119901 + 119901119860(119901) Now substituting (30)ndash(32) into (29) we can get1205934

(119909 119911) Similarly substituting (38) into (33) (35)ndash(37) wederive 120593

0

(119909 119911) 1205931

(119909 119911) 1205932

(119909 119911) and 1205933

(119909 119910 119911) Using thenormalizing condition we can find the unknown constant12058700

This completes the proof

Based on Theorem 3 we can easily obtain the marginalgenerating functions of the number of customers when theserver is in various states and some performance measuresThey are summarized in the following CorollaryTheir proofsare very easy and thus are omitted For convenience wedefine variable119873 be the orbit size and 119871 be the system size

Corollary 4

(1) The marginal generating function of the number ofcustomers in the orbit when the server is idle or vacationis given by

12058700

+ 1205930

(1 119911)

=119911 (1 minus 119860 (119901)) [120574 (119911) Γ (119911) minus 120578

3

Ω (119911)] + 1205783

Λ (119911)

1205783

Λ (119911)12058700

(39)

(2) The marginal generating function of the number ofcustomers in the orbit when the server is busy is givenby

1205931

(1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

Θ (119911) 1205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(40)

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Proof Multiplying (3)ndash(9) by 119911119896 and summing over 119896 we getthe following equations

1205930119894

(119911) = 1199011205930119894+1

(119911)

+ 119901119886119894

[12059311

(119911) + 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus 119901119886119894

120587110

minus 119901119886119894

119869

sum

119899=1

12059611989910

119894 ge 1

(19)

1205931119894

(119911) = 1199011205791

1205792

119904119894

12058700

+ 1199011205791

1205792

119904119894

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

12059301

(119911) + 1205792

120574 (119911) 1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

[12059311

(119911)

+ 12059321

(119911) +

119869

sum

119899=1

1205951198991

(119911)]

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

120587110

minus1199011205791

1205792

119904119894

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911) 119894 ge 1

(20)

1205932119894

(119911) = 1199011205791

1199111199031119894

12058700

+ 1199011205791

1199111199031119894

1205930

(1 119911)

+ 1199011205791

1199031119894

12059301

(119911) + 120574 (119911) 1205932119894+1

(119911)

+ (119901119911 + 1199011198860

) 1205791

1199031119894

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

minus (119901119911 + 1199011198860

) 1205791

1199031119894

120587110

minus 1199011198860

1205791

1199031119894

119869

sum

119899=1

12059611989910

minus 1199011205791

1199031119894

119911

119869minus1

sum

119899=1

12059611989910

119894 ge 1

(21)

1205933119894119895

(119911) = 1199011205791

1205792

119904119894

1199032119895

12058700

+ 1199011205791

1205792

119904119894

1199032119895

1205930

(1 119911)

+119901

1199111205791

1205792

119904119894

1199032119895

12059301

(119911) + 1205792

120574 (119911) 1199032119895

1205931119894+1

(119911)

+ (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

[12059311

(119911) + 12059321

(119911)

+

119869

sum

119899=1

1205951198991

(119911)]

+ 120574 (119911) 1205933119894119895+1

(119911) minus 1199011205791

1205792

119904119894

1199032119895

119869minus1

sum

119899=1

12059611989910

minus119901

1199111198860

1205791

1205792

119904119894

1199032119895

119869

sum

119899=1

12059611989910

+ 1205792

120574 (119911) 12059331198941

(119911)

minus (119901 +119901

1199111198860

)1205791

1205792

119904119894

1199032119895

120587110

119894 ge 1

(22)1205951119894

(119911) = 1199011205951119894+1

(119911) + 1199011199111205951119894+1

(119911)

+ 120574 (119911) V119894

120587110

119894 ge 1 119896 ge 0

(23)

120595119899119894

(119911) = 119901120595119899119894+1

(119911) + 119901119911120595119899119894+1

(119911)

+ 120574 (119911) V119894

120596119899minus110

119894 ge 1 119896 ge 0 2 le 119899 le 119869

(24)

Multiplying (23) and (24) by 119909119894 and summing over 119894 we get

119909 minus 120574 (119911)

1199091205951

(119909 119911) = minus120574 (119911) 12059511

(119911) + 120574 (119911) 119881 (119909) 120587110

(25)

119909 minus 120574 (119911)

119909120595119899

(119909 119911) = minus120574 (119911) 1205951198991

(119911) + 120574 (119911) 119881 (119909) 120596119899minus110

2 le 119899 le 119869

(26)

Setting 119909 = 120574(119911) in (25) and (26) we get

12059511

(119911) = 119881 (120574 (119911)) 120587110

(27)

1205951198991

(119911) = 119881 (120574 (119911)) 120596119899minus110

2 le 119899 le 119869 (28)

Substituting (27) and (28) into (25) and (26) respectively weobtain

1205951

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120587110

120595119899

(119909 119911) =119901119909 (119881 (119909) minus 119881 [120574 (119911)])

119909 minus 120574 (119911)120596119899minus110

2 le 119899 le 119869

(29)

Taking the derivative of (29) with respective to 119909 and letting119909 = 119911 = 0 we get

120596110

= 120585120587110

(30)

12059611989910

= 120585120596119899minus110

= 120585119899

120587110

2 le 119899 le 119869 (31)

Substituting (31) into (3) we get

120587110

=11990112058700

119901120585119869 (32)

6 Journal of Applied Mathematics

Multiplying (19) by 119909119895 summing over 119895 and taking into

account (30)ndash(32) we obtain

119909 minus 119901

1199091205930

(119909 119911)

= 119901 (119860 (119909) minus 1198860

) (12059311

(119911) + 12059321

(119911))

minus 11990112059301

(119911) minus 119901 (119860 (119909) minus 1198860

)1205781

minus 1205782

119881 (120574 (119911))

1205783

12058700

(33)

Note that we can find 1205930

(1 119911) by setting 119909 = 1 in (33) thenmultiplying (20) and (21) by 119909

119895 summing over 119895 and takinginto account (30)ndash(32) we obtain

119909 minus 120574 (119911) 1205792

1199091205931

(119909 119911)

=119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

12059311

(119911) +119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 12059301

(119911)

+ 120574 (119911) 1205792

1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 12058700

(34)

119909 minus 120574 (119911)

1199091205932

(119909 119911) = [119911 + 1199011198860

(1 minus 119911)] 1205791

1198771

(119909)

times (12059311

(119911) + 12059321

(119911)) minus 120574 (119911) 12059321

(119911)

+ 1199011205791

1198771

(119909) (1 minus 119911) 12059301

(119911)

minus119870 (119911)

1199011205783

1199011205791

1198771

(119909) 12058700

(35)

where119870(119911) = 1199111205782

[1 minus 119881(120574(119911))] + 1199011198860

(1 minus 119911)Γ(119911)Multiplying (22) by 119910

119895 and 119909119894 summing over 119895 and 119894

respectively and taking into account (30)ndash(32) we obtain

119910 minus 120574 (119911)

1199101205933

(119909 119910 119911) =119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772

(119910)

times [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

1198772

(119910) 12059311

(119911)

+119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772(119910)

12059301

(119911)

+ 120574 (119911) (1 minus 1205792

1198772

(119910)) 1206011

(119909 119911)

minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 1198772

(119910) 12058700

+120574 (119911) 120579

2

1198772

(119910)

1199091205931

(119909 119911)

(36)

Note that we can find 1206011

(119909 119911) by setting 119910 = 120574(119911) in (36) andsubstituting the expression of 120601

1

(119909 119911) into (34) we get

119909 minus 1205911

(120574 (119911))

119909[1 minus 120579

2

1198772

(120574 (119911))] 1205931

(119909 119911)

= 1205792

119911 + 119901119886

0

(1 minus 119911)

1199111205791

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 12059311

(119911) +119901 (1 minus 119911)

1199111205791

119878 (119909) 12059301

(119911)

+120574 (119911) 1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

119878 (119909) 12058700

(37)

Setting119909 = 119901 in (33)119909 = 120574(119911) in (35) and119909 = 120591(120574(119911)) in (37)respectively we can get the equations for 120593

01

(119911) 12059311

(119911) and12059321

(119911) By solving these equations we get the the generatingfunctions as follows

12059301

(119911) = 119901119911 (A (119901) minus 1198860

)120574 (119911) Γ (119911) minus 120578

3

Ω (119911)

1199011205783

Λ (119911)12058700

12059311

(119911) =Θ (119911) 119901120579

1

119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

12059321

(119911) =Θ (119911) 119901120579

1

119911119877 (120574 (119911))

1199011205783

Λ (119911)12058700

(38)

Using Lemmas 1 and 2 it is easy to show that 12059301

(119911)12059311

(119911) and 12059321

(119911) are defined for 119911 isin [0 1) and can beextended by continuity in 119911 = 1 if 120579

1

1205881

+ 1205791

+ 1205791

1205882

lt

119901 + 119901119860(119901) Now substituting (30)ndash(32) into (29) we can get1205934

(119909 119911) Similarly substituting (38) into (33) (35)ndash(37) wederive 120593

0

(119909 119911) 1205931

(119909 119911) 1205932

(119909 119911) and 1205933

(119909 119910 119911) Using thenormalizing condition we can find the unknown constant12058700

This completes the proof

Based on Theorem 3 we can easily obtain the marginalgenerating functions of the number of customers when theserver is in various states and some performance measuresThey are summarized in the following CorollaryTheir proofsare very easy and thus are omitted For convenience wedefine variable119873 be the orbit size and 119871 be the system size

Corollary 4

(1) The marginal generating function of the number ofcustomers in the orbit when the server is idle or vacationis given by

12058700

+ 1205930

(1 119911)

=119911 (1 minus 119860 (119901)) [120574 (119911) Γ (119911) minus 120578

3

Ω (119911)] + 1205783

Λ (119911)

1205783

Λ (119911)12058700

(39)

(2) The marginal generating function of the number ofcustomers in the orbit when the server is busy is givenby

1205931

(1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

Θ (119911) 1205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(40)

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Multiplying (19) by 119909119895 summing over 119895 and taking into

account (30)ndash(32) we obtain

119909 minus 119901

1199091205930

(119909 119911)

= 119901 (119860 (119909) minus 1198860

) (12059311

(119911) + 12059321

(119911))

minus 11990112059301

(119911) minus 119901 (119860 (119909) minus 1198860

)1205781

minus 1205782

119881 (120574 (119911))

1205783

12058700

(33)

Note that we can find 1205930

(1 119911) by setting 119909 = 1 in (33) thenmultiplying (20) and (21) by 119909

119895 summing over 119895 and takinginto account (30)ndash(32) we obtain

119909 minus 120574 (119911) 1205792

1199091205931

(119909 119911)

=119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

12059311

(119911) +119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 12059301

(119911)

+ 120574 (119911) 1205792

1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 12058700

(34)

119909 minus 120574 (119911)

1199091205932

(119909 119911) = [119911 + 1199011198860

(1 minus 119911)] 1205791

1198771

(119909)

times (12059311

(119911) + 12059321

(119911)) minus 120574 (119911) 12059321

(119911)

+ 1199011205791

1198771

(119909) (1 minus 119911) 12059301

(119911)

minus119870 (119911)

1199011205783

1199011205791

1198771

(119909) 12058700

(35)

where119870(119911) = 1199111205782

[1 minus 119881(120574(119911))] + 1199011198860

(1 minus 119911)Γ(119911)Multiplying (22) by 119910

119895 and 119909119894 summing over 119895 and 119894

respectively and taking into account (30)ndash(32) we obtain

119910 minus 120574 (119911)

1199101205933

(119909 119910 119911) =119911 + 119901119886

0

(1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772

(119910)

times [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 1205792

1198772

(119910) 12059311

(119911)

+119901 (1 minus 119911)

1199111205791

1205792

119878 (119909) 1198772(119910)

12059301

(119911)

+ 120574 (119911) (1 minus 1205792

1198772

(119910)) 1206011

(119909 119911)

minus119870 (119911)

1199111199011205783

1199011205791

1205792

119878 (119909) 1198772

(119910) 12058700

+120574 (119911) 120579

2

1198772

(119910)

1199091205931

(119909 119911)

(36)

Note that we can find 1206011

(119909 119911) by setting 119910 = 120574(119911) in (36) andsubstituting the expression of 120601

1

(119909 119911) into (34) we get

119909 minus 1205911

(120574 (119911))

119909[1 minus 120579

2

1198772

(120574 (119911))] 1205931

(119909 119911)

= 1205792

119911 + 119901119886

0

(1 minus 119911)

1199111205791

119878 (119909) [12059311

(119911) + 12059321

(119911)]

minus 120574 (119911) 12059311

(119911) +119901 (1 minus 119911)

1199111205791

119878 (119909) 12059301

(119911)

+120574 (119911) 1206011

(119909 119911) minus119870 (119911)

1199111199011205783

1199011205791

119878 (119909) 12058700

(37)

Setting119909 = 119901 in (33)119909 = 120574(119911) in (35) and119909 = 120591(120574(119911)) in (37)respectively we can get the equations for 120593

01

(119911) 12059311

(119911) and12059321

(119911) By solving these equations we get the the generatingfunctions as follows

12059301

(119911) = 119901119911 (A (119901) minus 1198860

)120574 (119911) Γ (119911) minus 120578

3

Ω (119911)

1199011205783

Λ (119911)12058700

12059311

(119911) =Θ (119911) 119901120579

1

119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

12059321

(119911) =Θ (119911) 119901120579

1

119911119877 (120574 (119911))

1199011205783

Λ (119911)12058700

(38)

Using Lemmas 1 and 2 it is easy to show that 12059301

(119911)12059311

(119911) and 12059321

(119911) are defined for 119911 isin [0 1) and can beextended by continuity in 119911 = 1 if 120579

1

1205881

+ 1205791

+ 1205791

1205882

lt

119901 + 119901119860(119901) Now substituting (30)ndash(32) into (29) we can get1205934

(119909 119911) Similarly substituting (38) into (33) (35)ndash(37) wederive 120593

0

(119909 119911) 1205931

(119909 119911) 1205932

(119909 119911) and 1205933

(119909 119910 119911) Using thenormalizing condition we can find the unknown constant12058700

This completes the proof

Based on Theorem 3 we can easily obtain the marginalgenerating functions of the number of customers when theserver is in various states and some performance measuresThey are summarized in the following CorollaryTheir proofsare very easy and thus are omitted For convenience wedefine variable119873 be the orbit size and 119871 be the system size

Corollary 4

(1) The marginal generating function of the number ofcustomers in the orbit when the server is idle or vacationis given by

12058700

+ 1205930

(1 119911)

=119911 (1 minus 119860 (119901)) [120574 (119911) Γ (119911) minus 120578

3

Ω (119911)] + 1205783

Λ (119911)

1205783

Λ (119911)12058700

(39)

(2) The marginal generating function of the number ofcustomers in the orbit when the server is busy is givenby

1205931

(1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

Θ (119911) 1205791

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(40)

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

(3) The marginal generating function of the number ofcustomers in the orbit when the server is under the firsttype of repair

1205932

(1 119911) =1 minus 1198771

(120574 (119911))

1 minus 119911

Θ (119911) 1205791

119911120574 (119911)

1199011205783

Λ (119911)12058700

(41)

(4) The marginal generating function of the number ofcustomers in the orbit when the server is under thesecond type of repair

1205933

(1 1 119911) =1 minus 119878 (120591 (120574 (119911)))

1 minus 120591 (120574 (119911))

1 minus 1198772

(120574 (119911))

1 minus 119911

timesΘ (119911) 120579

1

120591 (120574 (119911))

1199011205783

Λ (119911)12058700

(42)

(5) The marginal generating function of the number ofcustomers in the orbit when the server is on vacationis given by

1205934

(1 119911) =1205782

120574 (119911) [1 minus 119881 (120574 (119911))]

1199011205783

(1 minus 119911)12058700

(43)

(6) The generating function of the number of customers inthe orbit is given by

Ψ (119911) = 12058700

+ 1205930

(1 119911) + 1205931

(1 119911)

+ 1205932

(1 119911) + 1205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911)

1199011205783

Λ (119911)12058700

(44)

(7) The probability generating function of the number ofcustomers in the system is given by

Φ (119911) = 12058700

+ 1205930

(1 119911) + 1199111205931

(1 119911)

+ 1205932

(1 119911) + 1199111205933

(1 1 119911) + 1205934

(1 119911)

=Θ (119911) 120579

1

120574 (119911) 119878 (120591 (120574 (119911)))

1199011205783

Λ (119911)12058700

(45)

Corollary 5

(1) The system is idle with probability

12058700

=

1199011205783

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(46)

(2) The probability that the server is free is

12058700

+ 1205930

(1 1) + 1205934

(1 1) = 1 minus 1205881

minus1205791

1205791

1205882

(47)

(3) The probability that the server is busy is

1205931

(1 1) = 1199011198781

(48)

(4) The probability that the server is under the first type ofrepair is

1205932

(1 1) = 11990111987711

1205791

1205791

(49)

(5) The probability that the server is under the second typeof repair is

1205933

(1 1 1) = 1199011198781

11987721

1205792

1205792

(50)

(6) The probability that the server is on vacation is

1205934

(1 1) =

1199011205782

1198811

[119901 + 119901119860 (119901) minus 1205791

1205881

minus 1205791

minus 1205791

1205882

]

1205791

[1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(51)

(7) The mean number of customers in the orbit is

119864 (119873) = Ψ1015840

(119911)10038161003816100381610038161003816119911=1

=

Ω10158401015840

(1) + 2119901 (1 minus 119860 (119901)) [Ω1015840

(1) minus 119901]

2 [119901 + 119901119860 (119901) minus Ω1015840 (1)]

+

1205782

[21199011198811

(1 minus 119901119860 (119901)) + 1199012

1198812

]

2 [1199011198811

1205782

+ 119901119860 (119901) 1205783

]

(52)

where

Ω1015840

(1) = 1205791

1205881

+ 1205791

+ 1205791

1205882

(53)

Ω10158401015840

(1) = 1205791

1199012

1198782

(1 +1205792

1205792

)

2

+ 1199012

1198781

times [1205792

1205792

11987722

+ 21205792

1205792

11987721

(1 +1205792

1205792

)]

+ 21205791

11990111987711

+ 1205791

1199012

11987712

(54)

(8) The mean number of customers in the system is

119864 (119871) = Φ1015840

(119911)10038161003816100381610038161003816119911=1

= 119864 (119873) + 1205881

(55)

Remark 6 Consider some special cases as follows

(i) When 119869 = 0 1205791

= 1205792

= 0 our model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes In this case Φ(119911) reduces to

Φ (119911) =(1 minus 119911) 120574 (119911) [119901 + 119901119860 (119901) minus 119901119878

1

] 119878 [120574 (119911)]

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911) (56)

which coincides with results of the model studied byAtencia and Moreno [5]

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

(ii) When 119869 = 0 1205792

= 0 the present model becomesa discrete-time 1198661198901199001198661 queue with general retrialtimes and starting failure In this case Φ(119911) reducesto

Φ (119911) = ( (1 minus 119911) 120574 (119911) 119878 (120574 (119911))

times [119901 + 119901119860 (119901) minus 1205791

1199011198781

minus 1205791

minus 1205791

11990111987721

] )

times ( [119911 + 119901119860 (119901) (1 minus 119911)]

times [1205791

119878 (120574 (119911)) + 1199111205791

1198771

(120574 (119911))] minus 119911120574 (119911))minus1

(57)

(iii) When 1205791

= 1205792

= 0 our model becomes a discrete-time 1198661198901199001198661 queue with 119869 vacations In this caseΦ(119911) reduces to

Φ (119911)

=1199111205782

(1 minus 119881 (120574 (119911))) + 119901119860 (119901) (1 minus 119911) Γ (119911)

[119911 + 119901119860 (119901) (1 minus 119911)] 119878 (120574 (119911)) minus 119911120574 (119911)119878 [120574 (119911)] 120587

00

(58)

where

12058700

=1199011205783

[119901 + 119901119860 (119901) minus 1199011198781

]

1199011198811

1205782

+ 119901119860 (119901) 1205783

(59)

4 Stochastic Decomposition

The property of stochastic decomposition was proposedfirstly in queueing system with vacations This property hasalso been studied for retrial queues with vacations byArtalejo[17] The property of stochastic decomposition shows thatthe steady-state system size at an arbitrary point can berepresented as the sum of two independent random variableone of which is the system size of the corresponding queueingsystem without server vacations and the other is the orbitsize given that the server is on vacations In this section wepresent two different stochastic decompositions of the systemsize distribution in our model

Theorem 7 The total number of customers (119871) in the systemcan be represented as the sum of two independent randomvariables 119871 = 119871

1

+ 1198721

1198711

is the number of customers inthe 1198661198901199001198661 queue with server breakdowns and 119872

1

is thenumber of repeated customers given that the server is idleunder the first type of repair or on vacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (60)

where

Φ1

(119911) =(1 minus 120588

1

) 119878 (120591 (120574 (119911))) (1 minus 119911)

119878 (120591 (120574 (119911))) minus 119911(61)

is the generating function of the 1198661198901199001198661 queue server withbreakdowns and

Φ2

(119911) =119878 (120591 (120574 (119911))) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205881

) Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

p1198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205932

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205932

(1 1) + 1205934

(1 1)

(62)

is the generating function given that the server is idle underfirst type of repair or on vacations This completes the proof

Theorem 8 The total number of customers (119871) in the systemcan be expressed as the sum of two independent randomvariables 119871 = 119871

2

+ 1198722

1198712

is the number of customers in the1198661198901199001198661 queue with two types of breakdowns and 119872

2

is thenumber of repeated customers given that the server is idle or onvacation

Proof After some algebra operation Φ(119911) can be expressedby

Φ (119911) = Φ1

(119911)Φ2

(119911) (63)

where

Φ1

(119911) =

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

) 119878 (120591 (120574 (119911))) (1 minus 119911)

1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

(64)

is the generating function of the unreliable 1198661198901199001198661 queueand

Φ2

(119911) =1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

(120574 (119911)) minus 119911

1 minus 119911

times1199111205782

[1 minus 119881 (120574 (119911))] + 119901119860 (119901) (1 minus 119911) Γ (119911)

(1 minus 1205791

1205881

minus 1205791

minus 1205791

1205882

)Λ (119911)

times119901 + 119901119860 (119901) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1199011198811

1205782

+ 119901119860 (119901) 1205783

=12058700

+ 1205930

(1 119911) + 1205934

(1 119911)

12058700

+ 1205930

(1 1) + 1205934

(1 1)

(65)

is the generating function given that the server is idle or onvacations This completes the proof

5 Relationship to the CorrespondingContinuous-Time Model

In this section we investigate the relationship between ourdiscrete-time model and the corresponding continuous-time

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

counterpart We consider a continuous-time 1198721198661 generalretrial queue with 119869 vacations and two types of breakdownsThe customers arrive according to a Poisson process with rate120582 If an arriving customer finds that the server is busy underrepair or on vacations the customer joins the orbit and onlythe first customer in the orbit is permitted to retry for serviceThe server is subject to two types of different breakdownsAs soon as the server fails it is repaired immediately Thefirst type of breakdown is starting failure that is an arrivingcustomer must start the server which takes negligible timeand the server is started successfully with a probability 120579

1

Inaddition to starting failure the servermay break down duringserving an customer and the lifetime of the server followsexponential distribution with rate 120579

2

Once the system becomes empty the server takes vaca-

tions immediately according to 119869 vacation policy The retailtimes the service times the repair times for the two types ofbreakdowns and the vacation times are all assumed to followgeneral continuous-time distributions denoted by119860(119909) 119878(119909)1198771

(119909) 1198772

(119909) and 119881(119909) respectively and their Laplace-Stieltjes transforms denoted by 119860(119904) 119878(119904) 119877

1

(119904) 1198772

(119904) and(119904) respectively and their finite means are denoted by 119860

1

1198781

and 11987711

11987721

and 1198811

respectivelyWe suppose that the time is divided into sufficiently

small intervals of equal length Δ Then the continuous-time1198721198661 general retrial queue with 119869 vacations and two typesof breakdowns can be approximated by the following

119901 = 120582Δ 119886119894

= int

(119894+1)Δ

119894Δ

119889119860 (119909) 119904119894

= int

119894Δ

(119894minus1)Δ

119889119878 (119909)

119877119897119894

= int

119894Δ

(119894minus1)Δ

119889119877119897

(119909) 119897 = 1 2 V119894

= int

119894Δ

(119894minus1)Δ

119889119881 (119909)

(66)

By using the definition of Lebesgue integration we can getthe following equalities

limΔrarr0

1205882

= 12058211987721

= 1205882

limΔrarr0

119901119881119894

= 1205821198811

limΔrarr0

1205881

= 1205821198781

(1 +1205792

1205792

11987721

) = 1205881

limΔrarr0

119878 (120591 (120574 (119911))) = 119878 [120582 (1 minus 119911) + 1205792

1198772

(120582 (1 minus 119911))]

= 119878 (120591 (120574 (119911)))

limΔrarr0

1198812

(119901) = 1198812

(120582) limΔrarr0

1198771

(120574 (119911)) = 1

[120582 (1 minus 119911)]

limΔrarr0

119860 (119901) = 119860 (120582) limΔrarr0

119881 (120574 (119911)) = [120582 (1 minus 119911)]

limΔrarr0

120578119894

= 120578119894

119894 = 1 2 3

(67)

1 2 3 4 5 6 7 8 9 10

045

04

035

03

025

02

015

01

005

1205791 = 05

1205791 = 03

1205791 = 01

PV

J

Figure 1 The curve of 119875119881

with 119869

From the above relations we obtain the generating functionof the system size for the corresponding continuous-timemodel by

limΔrarr0

Φ (119911) = (1199111205782

(1 minus [120582 (1 minus 119911)])

+ 119860 (120582) (1 minus 119911) (1205781

minus 1205782

[120582 (1 minus 119911)]))

times ([1205791

119878 (120591 (120574 (119911))) + 1199111205791

1198771

[120582 (1 minus 119911)] minus 119911]

times [119911 + 119860 (120582) (1 minus 119911)] minus 119911)minus1

times119860 (120582) minus 120579

1

1205881

minus 1205791

minus 1205791

1205882

1205821198811

1205782

+ 119860 (120582) 1205783

119878 (120591 (120574 (119911)))

(68)

6 Numerical Examples

In this section we present some numerical examples toillustrate the effect of the parameters on the main perfor-mance measures of the system We consider two importantperformance measures the probability that server is onvacation 119875

119881

and the mean orbit size 119864(119873) It is assumedthat the retrial time is geometric distribution with generatingfunction 119860(119909) = (1 minus 119903)(1 minus 119903119909) The service time therepair times of the server for the two types of breakdownsand the vacation time are also assumed to follow geometricdistribution with generating functions 119878(119909) = (1 minus 120583)119909(1 minus

120583119909) 119877119897

(119909) = (1 minus 119903119897

)119909(1 minus 119903119897

119909) 119897 = 1 2 and 119881(119909) =

(1 minus V)119909(1 minus V119909) respectively In general we have chosenthe arrival rate 119901 = 01 the retrial rate 119903 = 025 the servicerate 120583 = 05 and the vacation rate V = 05 and we havepresented three curves which correspond to 120579

1

= 01 03 and05 respectively in Figures 1ndash4 where 120579

1

is the probabilitythat the server experience the first type of breakdown

Figure 1 describes 119875119881

with varying values of the maxi-mum number of the serverrsquos vacations 119869 As illustrated in

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

0 01 02 03 04 0502

025

03

035

04

045

05

055

1205792

1205791 = 05

1205791 = 03

1205791 = 01

PV

Figure 2 The curve of 119875119881

with 1205792

1 2 3 4 5 6 7 8 9 1005

06

07

08

09

1

11

12

13

14

E(N

)

1205791 = 01

1205791 = 03

1205791 = 05

J

Figure 3 The curve of 119864(119873) with 119869

Figure 1 119875119881

is increasing as a function of 119869 For differentvalues of 120579

1

we observe that 119875119881

decreases with increasingvalues of 120579

1

That is because the mean service time of acustomer is getting longer with increasing values of 120579

1

and119875119881

decreases accordingly Figure 2 describes the effect of 1205792

on 119875119881

The curve in Figure 2 shows that 119875119881

is decreasing as afunction of 120579

2

which agrees with intuitive expectationsFigures 3 and 4 describe the influence of the parameter

119869 and 1205792

on the mean orbit size 119864(119873) respectively As to beexpected 119864(119873) increases with increasing value of 119869 and 120579

2

that is there are more customers in the orbit as 119869 and 120579

2

increaseAs we can observe in Figures 3 and 4 the mean orbit

size 119864(119873) increases with increasing values of 1205791

As the thevalue of 120579

1

increases the mean sojourn time time for ancustomer increases and therefore the number of customersin the system increases Moreover the parameter 120579

1

affects119864(119873)more apparently when the value of the parameter 120579

1

isgetting greater

16

14

12

1

08

06

04

020 01 02 03 04 05

E(N

)

1205792

1205791 = 01

1205791 = 03

1205791 = 05

Figure 4 The curve of 119864(119873) with 1205792

Acknowledgments

This research is partially supported by the Youth ScienceFoundation of Shanxi Province (No 2012021001-1)

References

[1] T Yang and J G C Templeton ldquoA survey of retrial queuesrdquoQueueing SystemsTheory andApplications vol 2 no 3 pp 201ndash233 1987

[2] A Gomez-Corral ldquoA bibliographical guide to the analysis ofretrial queues through matrix analytic techniquesrdquo Annals ofOperations Research vol 141 pp 163ndash191 2006

[3] G Falin ldquoA survey of retrial queuesrdquo Queueing Systems Theoryand Applications vol 7 no 2 pp 127ndash167 1990

[4] T Yang and H Li ldquoOn the steady-state queue size distributionof the discrete-time 1198661198901199001198661 queue with repeated customersrdquoQueueing Systems Theory and Applications vol 21 no 1-2 pp199ndash215 1995

[5] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with general retrial timesrdquoQueueing SystemsTheory andApplications vol 48 no 1-2 pp 5ndash21 2004

[6] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with server breakdownsrdquo Asia-Pacific Journal of Opera-tional Research vol 23 no 2 pp 247ndash271 2006

[7] I Atencia and P Moreno ldquoA discrete-time 1198661198901199001198661 retrialqueue with the server subject to starting failuresrdquo Annals ofOperations Research vol 141 pp 85ndash107 2006

[8] P Moreno ldquoA discrete-time retrial queue with unreliable serverand general server lifetimerdquo Journal of Mathematical Sciencesvol 132 no 5 pp 643ndash655 2006

[9] JWang andQ Zhao ldquoDiscrete-time1198661198901199001198661 retrial queuewithgeneral retrial times and starting failuresrdquo Mathematical andComputer Modelling vol 45 no 7-8 pp 853ndash863 2007

[10] J Wang ldquoOn the single server retrial queue with prioritysubscribers and server breakdownsrdquo Journal of Systems Scienceamp Complexity vol 21 no 2 pp 304ndash315 2008

[11] J Wang and P Zhang ldquoA discrete-time retrial queue with nega-tive customers and unreliable serverrdquo Computers and IndustrialEngineering vol 56 no 4 pp 1216ndash1222 2009

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

[12] I Atencia I Fortes and S Sanchez ldquoA discrete-time retrialqueueing system with starting failures Bernoulli feedback andgeneral retrial timesrdquo Computers and Industrial Engineeringvol 57 no 4 pp 1291ndash1299 2009

[13] H Li and T Yang ldquoA single-server retrial queue with servervacations and a finite number of input sourcesrdquo EuropeanJournal of Operational Research vol 85 no 1 pp 149ndash160 1995

[14] B K Kumar and D Arivudainambi ldquoThe1198721198661 retrial queuewith Bernoulli schedules and general retrial timesrdquo Computersamp Mathematics with Applications vol 43 no 1-2 pp 15ndash302002

[15] B K Kumar R Rukmani and VThangaraj ldquoAnMMC retrialqueueing system with Bernoulli vacationsrdquo Journal of SystemsScience and Systems Engineering vol 18 no 2 pp 222ndash2422009

[16] G Choudhury ldquoA two phase batch arrival retrial queueingsystem with Bernoulli vacation schedulerdquo Applied Mathematicsand Computation vol 188 no 2 pp 1455ndash1466 2007

[17] J R Artalejo ldquoAnalysis of an 1198721198661 queue with constantrepeated attempts and server vacationsrdquo Computers amp Opera-tions Research vol 24 no 6 pp 493ndash504 1997

[18] F-M Chang and J-C Ke ldquoOn a batch retrial model with 119869

vacationsrdquo Journal of Computational and Applied Mathematicsvol 232 no 2 pp 402ndash414 2009

[19] J-C Ke and F-M Chang ldquoModified vacation policy forMG1 retrial queue with balking and feedbackrdquoComputers andIndustrial Engineering vol 57 no 1 pp 433ndash443 2009

[20] J-C Ke and F-M Chang ldquo119872[119909]1198661

1198662

1 retrial queue underBernoulli vacation schedules with general repeated attemptsand starting failuresrdquo Applied Mathematical Modelling vol 33no 7 pp 3186ndash3196 2009

[21] J Wu and X Yin ldquoAn MG1 retrial G-queue with non-exhaustive random vacations and an unreliable serverrdquo Com-puters ampMathematics with Applications vol 62 no 5 pp 2314ndash2329 2011

[22] J Wang ldquoDiscrete-time 1198661198901199001198661 retrial queues with generalretrial time and Bernoulli vacationrdquo Journal of Systems Scienceamp Complexity vol 25 no 3 pp 504ndash513 2012

[23] T Li Z Wang and Z Liu ldquoGeoGeo1 retrial queue with work-ing vacations and vacation interruptionrdquo Journal of AppliedMathematics and Computing vol 39 no 1-2 pp 131ndash143 2012

[24] Z Liu and Y Song ldquoGeoGeo1 retrial queue with non-persistent customers and working vacationsrdquo Journal of AppliedMathematics and Computing vol 42 no 1-2 pp 103ndash115 2013

[25] F Zhang D Q Yue and Z F Zhu ldquoA discrete time GeoG1retrial queue with second multi-optional service and servervacationrdquo ICIC Express Letters vol 6 pp 1871ndash1876 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of