flexible queuing model for number of active users in...

Research ArticleFlexible Queuing Model for Number of Active Users inCognitive Radio Network Environment

Ahsan Tanveer 1 Z U Khan 2 A N Malik2 and I M Qureshi3

1Department of Computer Sciences Iqra University Islamabad Pakistan2Department of Electronic Engineering International Islamic University Islamabad Pakistan3Department of Electrical Engineering Air University Islamabad Pakistan

Correspondence should be addressed to Ahsan Tanveer hafizahsan pkyahoocom

Received 3 September 2018 Revised 18 November 2018 Accepted 24 November 2018 Published 19 December 2018

Guest Editor Sungchang Lee

Copyright copy 2018 Ahsan Tanveer et al This 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 work presents a Soft Queuing Model (SQM) for number of active users present in a cognitive radio network (CRN) at somegiven instant Starting with the existing cellular network where the upper limit for the number of channels and active users is welldefined The idea is then extended to the complicated scenario of CRNs where the upper limit is not deterministic for both thenumber of channels and the active users Accordingly a probabilistic SQM is proposed under the condition that the number ofchannels and active cognitive users are both random variables The proposed model will be useful to offer the level of reliability tothe clients connected with CRN and hence to offer secure communication even on a cooperative CRN The proposed model hasbeen verified theoretically and simulations have been carried out in diversified set ups to evaluate the performance

1 Introduction

Cognitive Radios (CR) seems to be the prominent areaof research in future communication systems including 5Gwhich is based on Massive MIMO and heterogeneous net-works [1 2] A reliable communication model for cognitiveradio is the requirement of the time With the developmentof the idea its implication in almost all the existing commu-nication setups including 4G and mobile communicationand the momentum it has gained it is definite to continue itsresearch importance in future as well [3 4] Recent works onsmart grid for future energy requirements also incorporatethe idea of cognitive radio for communication in differentlayers of networks ie the network of home appliancesthe community and then the mega scale networks up tometropolitan level are supposed to be exploiting the idea ofcognitive radio for communications [5 6]

Surprisingly a reliable channel model is still missing forcognitive radio Specifically if we get the idea of cooperativeusers with a fusion centre operating in community in parallelwith the primary networks and service providers then howmuch resources will be available to cognitive radio network is

a basic question This will also link with the level of reliabilitythat a cognitive radio network may offer to its clients

There have been lot of research in CR however that focuson threemajor areasThe first one is the channel sensing [7 8]while second one is the access of spectrum holes or channels[9] and the third one is optimization of resources to utilizethe available channels for maximum throughout [10]

Now in case of the reliable communication on cognitiveradio network the question will be how many users arepresent at some given instant [11] and how many channelsare available to accommodate these users [12] Answer tothe questions lie in probability theory The available channelsto cognitive radio network will be based on the parameterssuch as the overall availablefree channels in a primarybase station operating in the region and of course on thesensing capability of the cognitive radio network underconsideration

The second question related to the above discussed threemajor areas of CRN will have certain channels available andwill be offering service to its clients then how many clientscan be accommodated at a given instant is a question whoseanswer lies in queuing theory

HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 8349486 6 pageshttpsdoiorg10115520188349486

2 Wireless Communications and Mobile Computing

A queuing model for number of channels occupied by theusers in primary radio network is relatively less complicatedand has been derived similar as in [13] However that is forinfinite resources which has been modified for the limitedavailable resources ie maximum available channels areassumed to be finiteThat seems logical and is being employedin existing primary radio networks

In third phase the idea has been extended to the cognitiveradio networks with flexible upper bound Since the upperlimit for available channels in cognitive radio networks isflexible hence accordingly a Flexible QueuingModel is beingproposed in this paper The model has been implemented insimulations and the results for diversified scenarios have beenpresented

Rest of the paper has been organized as follows Section 2presents mathematical framework and proposed model forexisting networks with fixed upper limit for number of usersIn Section 3 proposed Flexible Queuing Model for cognitiveradio networks is discussed Section 4 presents the wholesystem model Simulation results are given in Section 5 andthe conclusion is included in Section 6

In the document 119901119895 represents probability of 119895 number ofactive users whilen and k are random variables representingrespectively the available number of channels and number ofactive users for cognitive radio network

2 Mathematical Framework and ProposedModel for Existing Networks

In case of existing mobile network models the BS dedicatesa fraction of available communication channels to accommo-date the smooth hand over (HO)TheHO is predictable basedon the physical location of Mobile user ie the handoverprobability will be on higher side near the cell boundaries Yetthere could be other causes for handover such as Quality ofService (QoS) and Conjunction however they are secondaryreasons in comparison

In case of cognitive radios the available number ofchannels and hence the active users are random dependingon the sensing capabilities and occupying the available freespots for the spectrum Hence the Flexible Queuing Modelwas the ultimate requirement to address this issue Since themodel was not available in the existing literature thereforethis model has been proposed to deal with two differentscenarios whichmay be considered as an extension ofMM1

Scenario-IThis case is considered in current Section 2 wherethe length of queue has been taken as finite Yet this lengthis kept fixed Accordingly the mathematical model has beendeveloped and results have been derived for finite number ofusers (119870 in the manuscript)

Scenario-II The major contribution in our work is theenvironment in which the finite number 119870 is taken as arandom Accordingly the phrase ldquoFlexible Queuing Modelrdquohas been introduced in the title All the derivations have beencarried out and results have been generated for a stochasticupper limit as in Section 3

0 1 2 K

Figure 1 Channel occupation and channel release model for activeusers in existing networks

In case of existing cellular networks the number ofavailable channels for a BS is considered to be deterministicand fixed Let this number be N Moreover let the maximumallowable number of active users be K Therefore the extrachannels ie ℎ119890119909119905119903119886 = 119873 minus119870 will be reserved for handoversThe ratio of maximum allowable active users to the availablechannels reflects the worst situation that appears in peakhours when the entire119870 allowable channels will be occupiedAs a matter of routine the number of active users will berandom occupying a value from the set 0 1 119870 Assuming119896 number of active users at an instant when another userappears the number will move to 119896 + 1 state and converselyif an active user leaves the network the system will move to119896 minus 1 state Ultimately the number of active users will followa queuing system model as illustrated in the Figure 1

In the figure 120572 is the rate at which a new user appears andthe process moves forward from state 119896 to 119896 + 1 and 0 le 119896 le119870minus1 and 120573 is the rate at which process moves backward fromstate 119896 to 119896minus 1 when an active user leaves the system with thecondition 1 le 119896 le 119870

Accordingly the probability of finding the process in state119896 at some given instant will be

119901119896 = 1205891198961199010 0 le k le 119870 (1)

where 120589 = 120572120573 and 119901119896 is the probability of finding the processin state 119896 at some given instant The detailed derivation mayfollow through the queuing model presented in [13]

In order to compute 1199010 ie the probability of finding theprocess in state-o at some given instant we may apply thenormalization iesum119870119896=0 119901119896 = 1

That is (1 + 120589 + 1205892 + + 120589119870)1199010 = 1Now the series (1 + 120589 + 1205892 + + 120589119870) = (1 + 120589 + 1205892 + ) minus(120589119870+1+120589119870+2+ ) which is the difference of two infinite serieshence will be given as

1199010119870

sum119896=1

120589119896 = 1199010infin

sum119896=0

120589119896 minus 1199010120589119870+1infin

sum119896=0

120589119896

1199010infin

sum119896=0

120589119896 minus 1199010120589119870+1infin

sum119896=0

120589119896 = 1199010 (1 minus 120589119870+1)infin

sum119896=0

120589119896 = 1(2)

It will converge provided 120589 lt 1 and hence

1199010 (1 minus 120589119870+1) 11 minus 120589 = 1 (3)

This gives

1199010 = 1 minus 1205891 minus 120589119870+1 (4)

Wireless Communications and Mobile Computing 3

Thus

119901119895 = 120589119895 1 minus 1205891 minus 120589119870+1 (5)

with the condition that 120589 lt 1 and hence 120572 lt 120573 Thereforein this case the forward transition rate 120572 which is the rateat which some new channel is needed must be less thanthe reverse transition rate 120573 which is the rate at which thechannel is released by the user and is added back in the queueof available resources whichwill be the ultimate condition forglobal balance

These transition probabilities and queuing model arevalid for the existing cellular networks Below is a little bitcomplicated model for cognitive radio network This modelwill be applicable to the cognitive radio networks committedto provide specific level of reliability to the users

3 Proposed Flexible Queuing Model forCognitive Radio Networks

In above section we have proposed a queuing model forcellular networks In this case a state is an indicator of theactive number of users at some given instant The advantagein case of above derivation was that the maximum allowableactive users and number of channels ie 119870 and 119873 wereknown to us and HO occurs with relatively less probabilitythat was only 119901119895 However in case of CRN the situationwill be little bit more complicated ie the available numberof channels in this case will be a random variable n whichmay take a value from the set 0 1 2 119873 with respectiveprobabilities changing with outcome Evaluation of theseprobabilities may not be simple specifically when there isthe existence of multiple CRNs operating in parallel oncooperative basis Yet having n available channels the CRNwill be in position to accommodate k users which will bedecided based on the level of reliability ie

ℎ119903119886119905119894119900 = ℎ119890119909119905119903119886ℎ119886119888119905119894V119890 =n minus kk997904rArr k = n

ℎ119903119886119905119894119900 + 1 (6)

Ratio in (6) is decided by the CRN fusion centre Accordinglyfor k active usersm = ℎ119903119886119905119894119900k wherem = k le nwill be upperbounded where n is the total number of channels available toCRN after sensing and it will be random as well

k +m = n 997904rArrk + kℎ119903119886119905119894119900 = n 997904rArr

k = n1 + ℎ119903119886119905119894119900

(7)

This k represents the maximum allowable users Obviouslyk will also be a random variable in this case Depending onthis set up the queuing system as developed in case of abovecellular model will need to be flexible The updated Figure 2is given below

The transition rates 120572119888 and 120573119888 in this scenario will begreater than their corresponding counterparts in previoussection ie 120572119888 gt 120572 and 120573119888 gt 120573 which indicates more vibrantenvironment in this case

0 1 2 K

c cc

cc c

Figure 2 State model for channel occupation and release by activeusers in cognitive radio network

Theorem 1 In given scenario the state probabilities 119901119895 andcondition for convergence are derived as below

Proof Given fixed k ie k = 119870 the state probabilities 119901119895 willbecome

119901119895 = 120582119895 1 minus 1205821 minus 120582119870+1 0 le 119895 le 119870 (8)

The scenario of expression (8) is similar to that discussed inequation (5) However present scenario is flexible and 120582 =120572119888120573119888 Similarly the condition for convergence will be 120573119888 gt120572119888 Since k is random the above probability in this case willbecome

119901119895 =119870

sum119896=1

120582119895 (1 minus 120582)1 minus 120582119896+1 119875 (k = 119896) (9)

where 119875(k = 119896) will be derived from 119875(n) availablechannels for CRN which is based on the sensing strategiesthe environment temperature and number of active CRNs ingiven area The expression in (9) is proposed with the help ofprobability theory and specifically total probability theorem

Rearranging expression (9) we get

119901119895 = 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 = 120582

119895 (1 minus 120582)

sdot (119875 (k = 0)1 minus 120582 +119875 (k = 1)1 minus 1205822 + +

119875 (k = 119870)1 minus 120582119870+1 )

(10)

4 Whole System Model

Following flowchart in Figure 3 describes the whole systemmodel including the architecture of primary and cognitiveradio user environment

5 Simulations

In order to verify the performance of proposed modelssimulations are carried out in MATLAB For this purposedifferent cases under different situations are considered

Case 1 In Case 1 the probability of number of active usersfor existing cellular networks is determined For this purposethe number of available channels and maximum number of


BS

Model for existing network systems

Fixed Upper Limit

Flexible Model for Cognitive systems

Stochastic Upper Limit

Available channels=NMaximum active channels=K

Fig 1Expression (5)

Available channels in cellular network= TOTALNNo of active PUs=NMax Channels for CUs=No of CU networks= CRNM

Average channels for CU=

Fig 2Expression (10)

N = NTOTAL minus N

N MCRN

Figure 3 Model for channel occupation and release by active users in existing and CR networks

G=075G=085G=095

0

005

01

015

02

025

03

pj (p

roba

bilit

y of

act

ive

user

s)

10 150 5j (number of active users)

Figure 4 Probabilities of different number of active users for 120589 =075 085 and 095

active users for base station are taken as 20 and 15 respectively(119873 = 20 119870 = 15) Random variable k representingnumber of active users will occupy values from the set0 1 15 Required probabilities are determined usingexpression (5) for different values of 120589 ie 075 085 and 095The corresponding probabilities are shown in Figure 4

Case 2 This case deals with probability of active users forcognitive radio network For this purpose let 119873 = numberof active users for primary BS=100

1198731015840 = number of vacant channels for cognitive users= 1198732 = 50119872119862119877119873 = number of cognitive networks = 5

lambda=075lambda=085lambda=095

0

002

004

006

008

01

012

014

016

018

02pj

(pro

babi

lity o

f act

ive u

sers

)

86 100 2 12 144j (number of active users)

Figure 5 Probabilities of active cognitive users for Case 2 whenvariance of 120590 = 3

120572 = average number of channels per cognitivenetwork = 1198731015840119872119862119877119873 = 10Suppose 119899 = variance of 120590 = 3

The probability of active users for CRN is evaluated usingPoisson random variable distribution in expression (10) andthe resulting expression becomes

119901119895 = 120582119895 (1 minus 120582) 119890120572120572+119899

sum119896=120572minus119899

11 minus 120582119896+1

120572119896119896 (11)

The resultant probabilities for 120582 = 075 085 095 are shownin Figure 5

Case 3 For Case 3 simulations are performed under the samecondition as Case 2 except the variance of 120590which is taken tobe as 5 and the simulation results are shown in Figure 6



0

005

01

015

02

025

pj (p

roba

bilit

y of a

ctive

use

rs)




0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

pj (p

roba

bilit

y of

chan

nels)

86 100 2 12 144j (number of channels)


Case 4 Case 4 is same as Case 2 except that the probabilityof active users for CRN is evaluated using uniform randomvariable distribution in expression (10) and the resultingexpression becomes

119901119895 = 120582119895 (1 minus 120582) 1119873 + 1120572+119899

sum119896=120572minus119899

11 minus 120582119896+1 (12)


0 2 4 6 8 10 12 14

Uniform VS Poisson distribution for lambda=055

UniformPoisson

0 2 4 6 8 10 12 14


UniformPoisson

j (number of channels)0 2 4 6 8 10 12 14


UniformPoisson

0

005

01

015

02

0

005

01

015

pj (p

roba

bilit

y of

chan

nels)

0

005

01

Figure 8 Comparison of Uniform and Poisson distributions for 120582 =055 065 075

Case 5 Case 5 presents simulation for the comparisonbetween numerical results for Uniform and Poisson distribu-tions using different values of 120582 These results are shown inFigure 8

Above simulation results show that an exponential-likedecay in probabilities is observed in almost all the plots byincreasing the number of users This is quite in line with theexpectation ie considering the expression (9) and 120582 is keptconstant which is less than 1 Hence 120582119870+1 will be negligible incomparison with 1 and

119901119895 asymp 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 asymp 120582

119895 (1 minus 120582) = 120582119895 (13)

wheresum119870119896=0 119875(k = 119896) asymp 1Hence by increasing 119895 119901119895 decreases exponentially as is

evident from plotsIn case we take the value of 120582 closer to unity the PDF

becomes almost linear ie a straight line It may be observedthat the area under the curve which can be easily computed


in case of straight line is approximately unity that is thenormalization of PDF

6 Conclusion

Methods of finding probabilities of number of active users forexisting cellular networks and for cognitive radio networksare presented For existing cellular networks the number ofchannels for base station and maximum allowable numberof active users are considered to be fixed The state of thesystem at any time is defined by the number of activeusers at that time The probability of the system to be ina particular state is evaluated on the basis of the rate 120572 atwhich a new user appears in the system and the rate 120573 atwhich an active user leaves the systemThen flexible queuingstate model addresses the complicated scenario for cognitiveradio networks where the available channels are actually thespare channels of primary BS Hence both the number ofchannels for cognitive network and the maximum number ofactive cognitive users are not deterministic The probabilityof the cognitive network system to be in a particular state isevaluated on the basis of the rate 120572119888 at which a new cognitiveuser may appear in the system the rate 120573119888 at which an activecognitive user may leave the system and pmf of number ofavailable channels

The work is important to deal with handover problem fornumber of active users in existing cellular networks and incognitive radio networks on the basis of their probabilities

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

All the authors declare that there are no conflicts of interest

References

[1] Y-J Liu S-M Cheng and P-Y Huang ldquoCognitive verticalhandover in heterogeneous networksrdquo in Proceedings of the 11thEAI International Conference on Heterogeneous Networking forQuality Reliability Security and Robustness QSHINE 2015 pp392ndash397 Taiwan August 2015

[2] E M Malathy and V Muthuswamy ldquoVertical handover facili-tation through queuing model in heterogeneous wireless net-worksrdquo International Journal of Advanced Engineering Technol-ogy vol 7 pp 346ndash348 2016

[3] Y Shim and D Shin ldquoAnalyzing the development of 4thgeneration mobile network in China actor network theoryperspectiverdquo Emerald Insight vol 17 no 1 pp 22ndash38 2015

[4] L A Magagula H A Chan and O E Falowo ldquoHandoverapproaches for seamless mobility management in next gener-ation wireless networksrdquo Wireless Communications and MobileComputing vol 12 no 16 pp 1414ndash1428 2012

[5] S Alam M F Sohail S A Ghauri I M Qureshi andN AqdasldquoCognitive radio based Smart Grid Communication Networkrdquo

Renewable amp Sustainable Energy Reviews vol 72 pp 535ndash5482017

[6] S Alam A N Malik I M Qureshi S A Ghauri andM Sarfraz ldquoClustering-based channel allocation scheme forneighborhood area network in a cognitive radio based smartgrid communicationrdquo IEEE Access vol 6 pp 25773ndash257842018

[7] H T Cheng and W Zhuang ldquoSimple channel sensing orderin cognitive radio networksrdquo IEEE Journal on Selected Areas inCommunications vol 29 no 4 pp 676ndash688 2011

[8] A Ghasemi and E S Sousa ldquoOptimization of spectrum sensingfor opportunistic spectrum access in cognitive radio networksrdquoin Proceedings of the 4th Annual IEEE Consumer Communica-tions and Networking Conference pp 1022ndash1026 January 2007

[9] K W Choi W S Jeon and D G Jeong ldquoAdaptive anddistributed access to spectrum holes in cognitive radio systemrdquoWireless Personal Communications vol 70 no 1 pp 207ndash2262013

[10] G I Tsiropoulos O A Dobre M H Ahmed and K EBaddour ldquoRadio resource allocation techniques for efficientspectrum access in cognitive radio networksrdquo IEEE Communi-cations Surveys amp Tutorials vol 18 no 1 pp 824ndash847 2014

[11] W-C Wu and K-C Chen ldquoIdentification of active users insynchronous CDMA multiuser detectionrdquo IEEE Journal onSelected Areas in Communications vol 16 no 9 pp 1723ndash17351998

[12] A S Cacciapuoti M Caleffi L Paura and M A RahmanldquoChannel availability for mobile cognitive radio networksrdquoJournal of Network and Computer Applications vol 47 pp 131ndash136 2015

[13] A L-Garcia Probability Statistics and Random Processes forElectrical Engineering Chapter 12 Prentice Hall 3rd edition2008

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A queuing model for number of channels occupied by theusers in primary radio network is relatively less complicatedand has been derived similar as in [13] However that is forinfinite resources which has been modified for the limitedavailable resources ie maximum available channels areassumed to be finiteThat seems logical and is being employedin existing primary radio networks

In third phase the idea has been extended to the cognitiveradio networks with flexible upper bound Since the upperlimit for available channels in cognitive radio networks isflexible hence accordingly a Flexible QueuingModel is beingproposed in this paper The model has been implemented insimulations and the results for diversified scenarios have beenpresented

Rest of the paper has been organized as follows Section 2presents mathematical framework and proposed model forexisting networks with fixed upper limit for number of usersIn Section 3 proposed Flexible Queuing Model for cognitiveradio networks is discussed Section 4 presents the wholesystem model Simulation results are given in Section 5 andthe conclusion is included in Section 6

In the document 119901119895 represents probability of 119895 number ofactive users whilen and k are random variables representingrespectively the available number of channels and number ofactive users for cognitive radio network

2 Mathematical Framework and ProposedModel for Existing Networks

In case of existing mobile network models the BS dedicatesa fraction of available communication channels to accommo-date the smooth hand over (HO)TheHO is predictable basedon the physical location of Mobile user ie the handoverprobability will be on higher side near the cell boundaries Yetthere could be other causes for handover such as Quality ofService (QoS) and Conjunction however they are secondaryreasons in comparison

In case of cognitive radios the available number ofchannels and hence the active users are random dependingon the sensing capabilities and occupying the available freespots for the spectrum Hence the Flexible Queuing Modelwas the ultimate requirement to address this issue Since themodel was not available in the existing literature thereforethis model has been proposed to deal with two differentscenarios whichmay be considered as an extension ofMM1

Scenario-IThis case is considered in current Section 2 wherethe length of queue has been taken as finite Yet this lengthis kept fixed Accordingly the mathematical model has beendeveloped and results have been derived for finite number ofusers (119870 in the manuscript)

Scenario-II The major contribution in our work is theenvironment in which the finite number 119870 is taken as arandom Accordingly the phrase ldquoFlexible Queuing Modelrdquohas been introduced in the title All the derivations have beencarried out and results have been generated for a stochasticupper limit as in Section 3

0 1 2 K

Figure 1 Channel occupation and channel release model for activeusers in existing networks

In case of existing cellular networks the number ofavailable channels for a BS is considered to be deterministicand fixed Let this number be N Moreover let the maximumallowable number of active users be K Therefore the extrachannels ie ℎ119890119909119905119903119886 = 119873 minus119870 will be reserved for handoversThe ratio of maximum allowable active users to the availablechannels reflects the worst situation that appears in peakhours when the entire119870 allowable channels will be occupiedAs a matter of routine the number of active users will berandom occupying a value from the set 0 1 119870 Assuming119896 number of active users at an instant when another userappears the number will move to 119896 + 1 state and converselyif an active user leaves the network the system will move to119896 minus 1 state Ultimately the number of active users will followa queuing system model as illustrated in the Figure 1

In the figure 120572 is the rate at which a new user appears andthe process moves forward from state 119896 to 119896 + 1 and 0 le 119896 le119870minus1 and 120573 is the rate at which process moves backward fromstate 119896 to 119896minus 1 when an active user leaves the system with thecondition 1 le 119896 le 119870

Accordingly the probability of finding the process in state119896 at some given instant will be

119901119896 = 1205891198961199010 0 le k le 119870 (1)

where 120589 = 120572120573 and 119901119896 is the probability of finding the processin state 119896 at some given instant The detailed derivation mayfollow through the queuing model presented in [13]

In order to compute 1199010 ie the probability of finding theprocess in state-o at some given instant we may apply thenormalization iesum119870119896=0 119901119896 = 1

That is (1 + 120589 + 1205892 + + 120589119870)1199010 = 1Now the series (1 + 120589 + 1205892 + + 120589119870) = (1 + 120589 + 1205892 + ) minus(120589119870+1+120589119870+2+ ) which is the difference of two infinite serieshence will be given as

1199010119870

sum119896=1

120589119896 = 1199010infin

sum119896=0

120589119896 minus 1199010120589119870+1infin

sum119896=0

120589119896

1199010infin

sum119896=0

120589119896 minus 1199010120589119870+1infin

sum119896=0

120589119896 = 1199010 (1 minus 120589119870+1)infin

sum119896=0

120589119896 = 1(2)

It will converge provided 120589 lt 1 and hence

1199010 (1 minus 120589119870+1) 11 minus 120589 = 1 (3)

This gives

1199010 = 1 minus 1205891 minus 120589119870+1 (4)


Thus

119901119895 = 120589119895 1 minus 1205891 minus 120589119870+1 (5)






ℎ119903119886119905119894119900 + 1 (6)



k = n1 + ℎ119903119886119905119894119900

(7)



0 1 2 K

c cc

cc c




119901119895 = 120582119895 1 minus 1205821 minus 120582119870+1 0 le 119895 le 119870 (8)


119901119895 =119870

sum119896=1

120582119895 (1 minus 120582)1 minus 120582119896+1 119875 (k = 119896) (9)



119901119895 = 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 = 120582

119895 (1 minus 120582)


119875 (k = 119870)1 minus 120582119870+1 )

(10)



5 Simulations




BS


Fixed Upper Limit




Fig 1Expression (5)




N = NTOTAL minus N

N MCRN


G=075G=085G=095

0

005

01

015

02

025

03

pj (p

roba

bilit

y of

act

ive

user

s)







0

002

004

006

008

01

012

014

016

018

02pj

(pro

babi

lity o

f act

ive u

sers

)





119901119895 = 120582119895 (1 minus 120582) 119890120572120572+119899

sum119896=120572minus119899

11 minus 120582119896+1

120572119896119896 (11)





0

005

01

015

02

025

pj (p

roba

bilit

y of a

ctive

use

rs)




0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

pj (p

roba

bilit

y of

chan

nels)




119901119895 = 120582119895 (1 minus 120582) 1119873 + 1120572+119899

sum119896=120572minus119899

11 minus 120582119896+1 (12)


0 2 4 6 8 10 12 14


UniformPoisson

0 2 4 6 8 10 12 14


UniformPoisson



UniformPoisson

0

005

01

015

02

0

005

01

015

pj (p

roba

bilit

y of

chan

nels)

0

005

01




119901119895 asymp 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 asymp 120582

119895 (1 minus 120582) = 120582119895 (13)






6 Conclusion



Data Availability




References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Thus

119901119895 = 120589119895 1 minus 1205891 minus 120589119870+1 (5)






ℎ119903119886119905119894119900 + 1 (6)



k = n1 + ℎ119903119886119905119894119900

(7)



0 1 2 K

c cc

cc c




119901119895 = 120582119895 1 minus 1205821 minus 120582119870+1 0 le 119895 le 119870 (8)


119901119895 =119870

sum119896=1

120582119895 (1 minus 120582)1 minus 120582119896+1 119875 (k = 119896) (9)



119901119895 = 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 = 120582

119895 (1 minus 120582)


119875 (k = 119870)1 minus 120582119870+1 )

(10)



5 Simulations




BS


Fixed Upper Limit




Fig 1Expression (5)




N = NTOTAL minus N

N MCRN


G=075G=085G=095

0

005

01

015

02

025

03

pj (p

roba

bilit

y of

act

ive

user

s)







0

002

004

006

008

01

012

014

016

018

02pj

(pro

babi

lity o

f act

ive u

sers

)





119901119895 = 120582119895 (1 minus 120582) 119890120572120572+119899

sum119896=120572minus119899

11 minus 120582119896+1

120572119896119896 (11)





0

005

01

015

02

025

pj (p

roba

bilit

y of a

ctive

use

rs)




0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

pj (p

roba

bilit

y of

chan

nels)




119901119895 = 120582119895 (1 minus 120582) 1119873 + 1120572+119899

sum119896=120572minus119899

11 minus 120582119896+1 (12)


0 2 4 6 8 10 12 14


UniformPoisson

0 2 4 6 8 10 12 14


UniformPoisson



UniformPoisson

0

005

01

015

02

0

005

01

015

pj (p

roba

bilit

y of

chan

nels)

0

005

01




119901119895 asymp 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 asymp 120582

119895 (1 minus 120582) = 120582119895 (13)






6 Conclusion



Data Availability




References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



BS


Fixed Upper Limit




Fig 1Expression (5)




N = NTOTAL minus N

N MCRN


G=075G=085G=095

0

005

01

015

02

025

03

pj (p

roba

bilit

y of

act

ive

user

s)







0

002

004

006

008

01

012

014

016

018

02pj

(pro

babi

lity o

f act

ive u

sers

)





119901119895 = 120582119895 (1 minus 120582) 119890120572120572+119899

sum119896=120572minus119899

11 minus 120582119896+1

120572119896119896 (11)





0

005

01

015

02

025

pj (p

roba

bilit

y of a

ctive

use

rs)




0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

pj (p

roba

bilit

y of

chan

nels)




119901119895 = 120582119895 (1 minus 120582) 1119873 + 1120572+119899

sum119896=120572minus119899

11 minus 120582119896+1 (12)


0 2 4 6 8 10 12 14


UniformPoisson

0 2 4 6 8 10 12 14


UniformPoisson



UniformPoisson

0

005

01

015

02

0

005

01

015

pj (p

roba

bilit

y of

chan

nels)

0

005

01




119901119895 asymp 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 asymp 120582

119895 (1 minus 120582) = 120582119895 (13)






6 Conclusion



Data Availability




References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




0

005

01

015

02

025

pj (p

roba

bilit

y of a

ctive

use

rs)




0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

pj (p

roba

bilit

y of

chan

nels)




119901119895 = 120582119895 (1 minus 120582) 1119873 + 1120572+119899

sum119896=120572minus119899

11 minus 120582119896+1 (12)


0 2 4 6 8 10 12 14


UniformPoisson

0 2 4 6 8 10 12 14


UniformPoisson



UniformPoisson

0

005

01

015

02

0

005

01

015

pj (p

roba

bilit

y of

chan

nels)

0

005

01




119901119895 asymp 120582119895 (1 minus 120582)119870

sum119896=0

119875 (k = 119896)1 minus 120582119896+1 asymp 120582

119895 (1 minus 120582) = 120582119895 (13)






6 Conclusion



Data Availability




References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




6 Conclusion



Data Availability




References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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