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A machine interference problem with multiple types offailuresJOHN PALESANO a & JEYA CHANDRA aa Department of Industrial and Management Systems Engineering , The Pennsylvania StateUniversity , University Park, PA, 16802, U.S.APublished online: 02 Apr 2007.

To cite this article: JOHN PALESANO & JEYA CHANDRA (1986) A machine interference problem with multiple types of failures,International Journal of Production Research, 24:3, 567-582, DOI: 10.1080/00207548608919750

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A machine interference problem with multiple types of failures

J O H N PALESANOt and JEYA CHANDRA?

A numerical method for obtaining equilibrium performance measures for a sin- gle group of N identical machines, each subject to k(32) types of failures, is presented. The time intervals between breakdowns of a machine are exponen- tially distributed. The mean time between type i failures is I l l i , i = 1, 2. . . . , k. The service times required to repair any type of failure can be either determin- istic, exponential, hypo-exponential, or hyper-exponential random variables, with different means for different types of failures. The service discipline is a non-preemptive fixed-priority rule, with different priorities assigned to differ- ent types of failures. System performance measures, such as average time spent by the machines waiting for service, average number of idle machines and machine operator utilization are obtained through imbedded Markov chain analysis. The algorithms used to obtain these measures exploit the special structure of the one-step transition probability matrix. The sensitivity of the performance measures to the density function of the service times and the priority assignments given is examined.

Introduction

I n many industrial processes a n operator is required to attend a set of N identical machines, each of which may fail. If a machine fails when another machine is receiving the attention of the operator, the former has to wait, resulting in additional loss of production. The failure may be any one of several types which occur independently of each other. Thus, each machine is either running, being serviced, or waiting for a particular type of service. There is a cost associated with servicing a machine and with each machine in a n idle state. Deci- sion rules, which minimize the total expected cost per unit time, can be formu- lated. I n this paper, we present a numerical method for obtaining equilibrium performance measures for this N machine multiple-type failure repair model. The time between failures is exponentially distributed and the types of failures are mutually independent. A non-preemptive fixed-priority service discipline is used in which the service time of the failure types can follow any of several selected distributions.

Several different approaches have been used to study this model. I n the sup- plementary variable technique, the process is made Markovian by incorporating a supplementary variable, such as the elapsed time of the machine already under service, into the definition of the state probabilities. Using this method, Jaiswal and Thimvengadam (1963) studied the model in which each machine has two types of failures. However, the solution of the equations for joint queue length probabilities is difficult for even two types of failures and generalization to more than two types presents algebraic difficulties. If the service time density functions

Revised received December 1984. Department of Industrial and Management Systems Engineering, The Pennsylvania

State University, University Park, PA 16802, USA.

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568 J . Palesano and J . Chandra

of the machines have rational Laplace transforms, then using the generalized method of stages, the global balance equations can be written and solved to obtain the probabilities of the number of machines of different types at the service facility. Though the equations can be generated easily, the number of equations increases sharply as the number of stages in the representation of the service time density function grows. Benson and Cox (1961) used this technique for machines with two types of failures and exponential service times. Guild and Hartnett (1982) and Reich (1964) used simulation to determine the performance measures for the system with machines having multiple types of failure. However, this method is expensive because long simulation runs are necessary to obtain reasonably precise results.

In this paper we analyse the model using the imbedded Markov chain tech- nique. For this model, service completion epochs constitute a set of regeneration points and the number of idle machines a t these epochs form a Markov chain. The steady-state probabilities of the Markov chain can be obtained from one-step transition probabilities by solving a set of simultaneous linear equations. These steady-state service completion epoch probabilities are then used to obtain the system performance measures using the renewal reward theorem and an extension of Little's formula (Chandra and Sargent 1983). The results obtained using the imbedded Markov Chain technique are used to test the sensitivity of these system performance mewures to the form of service time density function and the pri- orities given to the various failure types. Also,the accuracy of using Palm's model (1958) with one failure type to approximate the multiple-failure type model is examined.

Analysis of the model

The model

The model, represented in Fig. 1, consists of N independent, identical machines each requiring (k 2 2) types of service. A single operator services this group of N machines, which fail due to one of k possible causes. The k failure types occur independently. The time intervals between successive service require-

SINOLE OROUP

n I I \ SERVICE TYPE i t 1 1 1 .h I I I

Figure 1.. Schematic diagram of the model.

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A machine interference problem 569

ments for type i failure are exponentially distributed with a mean l/Li (i = 1 , 2. . . . , k). The sewice times Si for type i failures are iid random variables with an arbitrary density function f ( 4 , and a mean of l/pi. In this paper, the constant, exponential, hyper-exponential, and hypo-exponential density functions for service times are examined. Their respective density functions and coefficients of variation (cv) are as follows (for i = 1, 2, . . . , k):

where 6(. ) is the Dirac impulse function :

= 0, otherwise

cv c 1

Exponential f (s,) = p, e-'"', s, > 0

= 0, otherwise

cv = 1

n i . Hyper-exponential f (8,) = Pil pl,e-'""'

1= 1

= 0, otherwise

cv 2 1

In eqn. (2). K, is the number of successive exponential time intervals (stages) that a failed machine must go through for senrice to be completed. In eqn. (4). Pil is the pro'bability that a type i failure has a mean service time of 1/p, and n; is the number of parallel exponential branches

Each failed machine must complete one of n; possible exponential service time intervals (branches).

Densities 1 4 encompass a wide range of variability about the mean service time. Each type of failure may have different mean service times. A non- preemptive fixed priority service discipline is used by the repairman, with pri- orities increasing with decreasing values of j(j = 1, 2, . . . , k). Within the same type of failure, machines are sewiced in the order of their breakdowns.

Mean value relationships

In a system which hm a single group of machines subjected to a single type of failure, the'following relations, which hold for any work conserving discipline, can be obtained using Little's formula (Kobaywhi 1978) :

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570 J . Palesano and J . Chandra

and

where W, is the mean waiting time in the queue, L is the mean number of idle machines, N is the total number of machines (finite), p is the utilization of the server, 1/p is the mean time spent by a machine in service, 111 is the mean time between failures of a machine, and pp is the rate a t which machines are repaired.

We can extend this to the system studied in this paper, the single group of machines, each subject to k types of failure. Each failure type, i , has its own mean time between failures, l /L i , and mean service time, l / p i . To utilize eqns. (5) and (6) above, we need single equivalent values for the mean time between Tailures (111) and the mean service time ( l / p ) for the system. Since an operating machine can fail due to any of the k independent types of failure and remains in that state until it is serviced, the total failure rate for each machine is

A = 2 1 , . and -=- i n 1 1

i = 1

If pi is the proportion of time the repairman is busy with type i failures

and

From eqns. ( 8 ) and (9) we get k

C P I P ! i = 1

i Pi

p=- 1 i = 1 k and - = -

F k

C P I C pi pi I = 1 i = 1

Relationship of pi to Bi In this finite source multiple-failure type model, the busy cycle, consisting of a

busy period and an idle period of the repairman, is a regenerative cycle. For regenerative processes each p, can be obtained (Ross 1970) using the renewal reward theorem, from the equation

E(Bi) = E(B) + E ( I )

In eqn. ( 1 1 ) E(Bi) is the expected length of one server is busy with type i failures;

k

E(B) = C E(B0 1 = 1

( 1 1 )

busy period, during which the

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A machine interference problem 57 1

is the expected length of one busy period; and E( I ) is <he expected length of one idle period.

The idle period starts a t the instant all of the machines become operational. Since the time that a machine spends in operating condition is exponentially dis- tributed with a mean of

1 - k

from eqn. (10) the mean time until a machine terminates an idle period is also

Any of the N independent machines can terminate the idle period. Therefore, the mean length of the idle period is given by

Relatimhip of B l s to the steady-state probabilities of idle machines at service completion epochs

Let us observe the number of idle machines a t the service completion time epochs. The expected length of a busy period, during which the operator is busy with type i failures, is obtained (Chandra and Sargent 1983) by solving the follow- ing equation.

where pi represents the steady state probability that a machine repaired during a busy cycle has a type i failure, 0 is a vector containing all zero elements, A(0) is the steady state probability that all of'the machines are operating a t the service completion of a machine (i.e. the steady-state probability that a machine which just completed service terminates, a busy period). Therefore, l/A(O) is the expected number of machines served during a busy period.

p, can be subdivided into two parts, pi') and pi2). pj" is the steady state prob- ability that a type i failure of a machine, terminates an idle period.. As the failure types are independent and the times between failures are exponentially distrib- uted,.

is the steady state probability that a type i failure is repaired during a busy period.

N N N N N 1:

pj2' = 1 1 . . . 1 . . . 1 A(u), 1 vi < N - 1 v1=0 v2=0 " , = I * + 1 = 0 v t = o i = 1

(15)

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572 J . Palesam and J . Chundm

where A(v) refers to the steady state probability that a t the service completion of a machine the state of the system is v = { v , , v , , . . . , v,). vi represents the number of machines with failure type i (and priority i ) a t the service facility.

Calculation of the steady-state service completion epoch probabilities

The values of the steady-state service completion epoch probabilities, A(v) in eqn. (15). can be calculated using imbedded Markov chain analysis. Let the total number of these steady state probabilities be m' and the row vector consisting of all these m' probabilities be A. hen A can be obtained by solving the following set of simultaneous equations;

subject to the normalizing condition A, = 1 , which can be written as I = 1

Aa' = 1 (17)

where a' is the column vector consisting of m' ones. In eqn. (16). P is the (m' x mi) transition probability matrix, I is an (m' x m') identity matrix, and 0 is the row vector containing m' zeros. Replacing the last column of the matrix ( I - P ) with the normalizing equation yields

A R = a (18)

where a is the ( 1 x m') row vector with all zeros, except one as the last element, and R is the ( I - P ) matrix with the last column replaced by the vector a', given in eqn. (17). From eqn. (18) we get

The number of elements m' is obtained by the following combinatorial,

m' becomes large with increases in N and k. A technique developed by Shanthiku- mar and Sargent (1980) is used to obtain these m' steady-state probabilities. Let R be decomposed into lower and upper triangular matrices as follows:

R = L + U (20)

If we set R-' = FL-' , the eqn. (19) becomes

A = aFL-' (21)

As the elements of a are all zeros except the last element, which is equal to one,

A = f ;L - ' (22)

where f; is th8 last row in the matrix F. Multiplying both sides of eqn. (20) by the inverse of R, we obtain I = R - l L + R-IU and substituting KL-' for R-I yields

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A machine interference problem 573

Considering only the last rows of the matrices on both sides of eqn. (23) yields,

In eqn. (24) the only unknown elements are those in f b which can be obtained by solving this system of equations. Then the values in A can be obtained by using eqn. (22). I n eqn. (24) L-' is lower triangular because it is the inverse of the lower triangular matrix L.

By suitable rearrangement of the column and row states of U, it is possible to group all its non-zero elements in the last I' columns, where

leaving all zero elements in the first (m' - 1') columns. Therefore, from eqn. (24), only the last 1' rows of L-' need to be calculated in order to obtain L-'U. Also, the product L-'U will contain non-zero elements in the last 1' columns and zero elements in the rest of the columns. So the sum (I + L-'U) will be identical to the first (m' - 2') columns of I . Since a t least the first (m' - 1') elements of a on the left hand side of eqn. (24) are zeros, the first (m' - 1') elements off :. will also be zeros.

Using the technique explained above, a system of equations containing only 1' unknown variables are to be solved for vector f in eqn. (24). Table 1 illustrates the resultant savings in computation.

Table 1. Computational savings

Generation of the elemenh of P

The (m' x m') transition matrix P must be obtained to solve the system of equations given in the previous section. Let X, and X,+, be the states of the system representing the number of idle machines after the nth and (n + 1)th service completion epochs, respectively. Then the elements of the matrix P are the transition probabilities, P[X,+, = u/X, = j]. Row vectors j = (j, , j, , . . . , j,) and u = (u, , u 2 , . . . , uk) represent the number of idle machines due to the various failure types a t the nth and (n + 1)th service completions, respectively. The calculation of these transition probabilities depends upon whether the vector j contains a t least one non-zero element or all zero elements.

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574 J . Palesano and J . Chandra

If j contains a t least one non-zero element and 1 = min {ilj, > 0, 1 C i C k} (1 is the highest priority failure type waiting for service), then a machine with type 1 failure is being repaired during the (n + 1)th service time and

number of breakdowns during the (n + 1)th P [ X , + , = u/X, = j] = Prob service time is (u, - j, + 1) for type 1

and (ui - ji) for i = 1, 2, . . . , k # 1/X, = j I In this model each machine can breakdown due to any one of the k independent failure types, and the time between failures of type i is exponentially distributed with a mean l / l i . Hence, the probability that a given machine does not break down during a time interval t is

and the probability that a given machine breaks down during this time interval is

Because of independent, exponential inter-arrival times, the probability that a type i failure causes a given machine to break down is

Since each machine is independent of the other machines, the number of break- downs of all failure types within a given time interval follows a multinomial dis- tribution. Let p(m, n', 1) denote the probability that the number of failures of each type within a time period t is given in the vector rn = {ml , m,, . . . , m,},

k - when there are n' idle machines a t the beginning of the time period. If m = 1 mi,

i = 1

the total number of failures of all types during t, then

1 p(m, n', t) = (N - n')! * , * 1

( N - n ' - m ) ! n (mi)! , = I

* [ i - fi 1 {bi/ i - 1 5 li)[1- exp (- i = l 5 l f t ) ] r ] * [exp (- i = i 1 A~~)-J-"'-'" (25)

where

Now for j, # 0 and ji = 0 for i < 1 (there are no idle machines with higher priority failure types) the transition probability can be written as

k

where f (5,) is the distribution of the service time of type 1 failures and n' = j i . i = l

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A machine interference problem 575

In m, mi = pi - ji for all i # 1 and m, = p, - j, + 1. The evaluation of this integral, eqn. (26), for the different sewice time density functions used in this paper is given in appendix A.

If all the elements of j are zero, then the (n + 1)th service time depends upon which type of failure terminates the idle period initiated by the nth service com- pletion. Therefore, the conditional probability

li idle period is terminated P[X,+, = u p , = 01 = Prob

i = 1 by a type i failure I

Arrmgement of rows and columns

The row and column states of the transition probability matrix are arranged in order to obtain the special structure of U, given earlier. Also, all of the diago- nal elements must be non zero to apply the algorithm for inverting the lower triangular part (L) of matrix R. The algorithm for arranging the m' row states of the matrix is given in Appendix B.

Results and sensitivity analysis

The entire numerical solution procedure is depicted in Fig. 2. The results of using this procedure to solve systems having various service time density func- tions are given in Tables 2, '3, and 4. The cases examined have machines with three failure types. The hypo-exponential service density function used has four stages and the hyper-exponential function used has two parallel exponential branches (PIJPI, + P1Jp12 = l/p,).

From these numerical results it can be obsewed that the performance mea- sures L and W, increase as the coefficient of variation of the sewice time distribu- tion increases. The constant service time has the lowest values of L and W,, followed by the hypo-exponential, exponential and hyper-exponential, in that order. When the seyer utilization was low (approximately 50% for N equal to 5 machines) the maximum difference in L for all selected sewice distributions is 14%. However, as the server utilization approaches unity, the performance mea- sures become insensitive to the form of service time density function. For instance, when the server utilization equalled about 95%, the maximum differ- ence in L is only 8.35%.

The effect of changing priorities is illustrated in Table 4. From the results for the example given, i t is seen that the mean number of machines waiting L, and the mean waiting time in queue W, increase, when the failure types with the higher mean service times are given a high priority. This increase is less than 5.0%. which confirms the observation made by Jaiswal and Thiruvengadam (1963) that the order of priorities does not affect much the system performance measures. However, a detailed analysis of a large number of cases is necessary in

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J. Paleaano and J. Chandra

I I N P U T Syatem Porometers N; X i ,yi , ( i 4 . 2 , . . . . k 1 I

ArrOngem~nt of R o w and Columns of P (Section 2.7 ond Appendix B )

I Generation of the Element* of P (Section 2.6 and Appendlx A ) 1

Calculating Stationary Probobilitiss (Section 2 . 5 1

i Colculotion of E(Bi); i = l , 2 , ..., k

(Section 2 . 4 )

(Section 2.3 )

Calculation of System Performonce Meaaurea L and Wq (Section 2 . 2 )

Figure 2. Sequence of the algorithms.

Service distribution P w a L P W q ,5

Constant 0.506 0.331 0732 0-969 2.18 3.76 Hypo- (k = 4) 0.502 0389 0.765 0964 2.23 3.86

exponential Ex~onential 0493 0.536 0.848 0.936 2.51 4.11

Table 2. Results for constant, hypo-exponential and exponential service distributions. .

A, = 0.1 pll = 1.5 p12 = 7.5 A, = 0.05 p,, = 1.75 pll = 0.7 a, = 0.01 P,, = 0.30 pal = 0.42

PI, = 0.50 P,, = 0.50

N = 5 N = 12 Service

distribution P w , L P W, . L

Hyper- 0489 0589 '0877 0.925 2.62 4.19 exponential

Table 3. Results for hyper-exponential service distribution.

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A machine interference problem 577

Exponential distribution

Failure N = 5 N = 1 2

Table 4. Results for piority changes.

order to arrive a t a definite conclusion in this regard. In cases where the order of priorities significantly affects the system performance measures, a problem of interest is to find the optimum assignment of priorities which will minimize the mean waiting time W , .This objective can be written as,

Minimize W, = N ( i=1 t Pi) 1

(28) (tpl14)-(iPill) i= 1 1= 1 (L) f= 1

from eqns. (5) . (7). (9) and (10). But as p i , i = 1, 2, . . . , k, depends upon the priority allocation and as no closed-form expression is available for pi in terms of the system input parameters and the priority assignments, it is very difficult to obtain the optimum assignment of priorities, analytically. Exhaustive enumer- ation which calculates W , for all possible priority assignments is one method which could be used to solve this problem.

In Table 5, the accuracy of using Palm's model (1958) with one failure type to approximate the multiple failure type model is tested. The input parameter for

Exact results 0848 2.56 4.1 1 using algorithm

Palm's 0782 1.72 3.57 approximation

Table 5. Palm's approximation.

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578 J . Palesano and J . C?mndra

this equivalent model with one type of service is the ratio

This approximation is inaccurate for the three failure type system, especially as N increases. For given values of It's and pi's, Palm's approximation differed from the exact numerical solution by as much as 30% when N equals 10 and 25% when N equals 12.

The results given in Table 2, 3, 4, and 5 are valid for the specific system parameters chosen. A change in these parameters can significantly alter the resultant performance measures.

Conclusions

In this paper, imbedded Markov chain analysis is used to develop a numerical method for obtaining the system performance measures for the finite source, multiple-failure type queueing model. By utilizing its special structure, the tran- sition probability matrix is inverted to yield these mean performance measures. From our results, we conclude that the measures, L and W,, become insensitive to the type of service distribution and order of priorities as the server utilization approaches unity. However, for systems with low server utilization, the per- formance measures ( L and W,) increase, as the coefficient of variation of the service distribution increases and if the high priority failure types have the higher mean service times, though the increase in the latter case is found to be less than 5%.

The development of approximate methods of obtaining the performance mea- sures might reduce the number of computations. In this paper only mean values are obtained for the performance measures. Techniques which obtain or estimate the variability of the number in the system and waiting time in the repair queue might yield further interesting results. A system having a service time density function with a high coefficient of variation, such as the hyper-exponential, will probably yield highly variable measures.

Appendix A: Evaluation of the integral in the elements of P The integral for each one step transition probability is given by

where

1 p(m, n', sl) = (N - n')! * ,-- +

1 ( N - n ' - m ) ! n (mi)!

i = 1

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A machine interference problem

and

Exponential service time distribution

f (sl) = pl e-'"l, s, > 0

Using the substitutions

the final form of the integral is given by

p(m. nf. s f ) f (8,) dq = D * ( - l) '[l + a I * ( [ N - m - nf] + h)]-I

where

1 D = ( N -nf)! *,--*

( N - m - n')! i= l n (mi) ! 1-1

Hypo-ezponatial service time distribution ( K , pf)Kls~"l - 1 )e -KICISI

f (81) = ( K 1 - l ) ! 9 S l > O

Using the substitutions

= e - ' ~ ~ ~ a1 = (i S / k l P , . 1=1

i = 1, 2, . . . , k, and the integral relation

if (m + 1 ) and n > 0 , the final form of the integral is given by m

[ p ( m , n'. 8,) f (81) dsi = D * 1 ( y ) ( - 1)&[1 + a , * ( [ N - m - n'] + h)]-'I, h = O

where D is as listed above.

Hyper-exponential service time distribution .I'

f (a1) = 1 P f l pil e-""", S, > 0 I = 1

where

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580 J . Palesano and J . Chandra

Writing the integral as a sum of n; integrals and using 'the substitutions,

the integral becomes

p(m, n', 8,) f (a,) ds, = D * [ I + a, , * ([N - m - nt] + D ) ] - I

where D is as listed earlier.

Appendix B: Algorithm for arranging the row states of P The algorithm is divided into two blocks. In the first block, the first (m' - l')

elements are obtained and in the second block, the remaining Z' elements are - obtained.

Arrangement of thejirst (m' - l') states

See Fig. 3.

Figure 3. Flow diagram for arranging the first (m' - l') states.

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A machine interference problem

Arrangemen4 of the remaining 1' slates

See Fig. 4. I n Figs. 3 and 4 priorities increase with decreasing values of i ( i = 1 , 2 , . . . , k). The arrangement of the column states is similar to the row state arrangement,

with minor modifications.

v = v - l fi

Figure 4. Flow diagram for arranging the last I' states.

. Dans cet article, nous pzkentons une m6thode numBrique pourle calcul des mesures de rendement en Bquilibre pour un seul groupe de N machines iden- tiques, chacune 6tsnt assujettie Q k (>2) types de pannes. ,les intervallea de temps entre les pannes d'une machine sont B distribution exponentielle. Le temps moyen entre des pannes de type i est 111,. i= l , 2 , . . . , k. Les temps de dkpannage requis pour &parer tout type de panne peuvent 6tre des variables soit dBterministiques exponentielles, hypo-exponentielles ou hyper-exponentielles avec des moyennes - dBBrentes pour diffkrenta types d e pannes. La discipline dr depannage est une &gle de prioritk fixe non pkmptive, diffe- rentes prioritks ktant attribubs a diffkrenta types de pannes. Des mesures de rendement du sysbime, telles que la du& moyenne pae& par les machines qui attendent un dkpannage, le nombre moyen de machines non prcductives et l'utilisation des opkateun des machines sont calculbs par une analyse en chaine Markovienne encastke. Les algorithmes utili&s pour le calcul de ces mesures exploitent la structure sp6ciale de la matrice de probabiliti: tran- sition a une ktape. Nous examinons la sensibilitk des mesurea de rendement a la fonction de densiti: des dukes de dkpannage et aux attributions de priorit& qui sont donn6es.

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582 A machine interference problem

Mit dieser Abhandlung w i d eine numerische Methode fur die Enielung von GleichgewichtsleistungsmaDnahmen fur eine einzelne Gmppe von N- identischen Maschinen, die alle zu Versagungserten von k ( 2 2 ) neigen, angefuhrt. Die Zeitintemalle zwischen Maschinenausfiillen werden exponential verteilt. Die mittlere Zeit zwischen St6mngen . des Typs i ist 114, i = 1, 2, . . . , k. Die Wartungszeiten, die fur die Reparatur aller Versagensar- ten benotigt werden, konnen entweder deterministische, exponentiale, hypo- exponentiale oder hyperexponentiale Zufallsvariablen mit unterschiedlichen MittelgroDen fur verschiedene Versagensarten sein. Der Wartungsdienst ist eine nicht preemptive festgesetzte Regel, wobei unterschiedliche Prioritaten den verschiedenen Versagensarten zugeschrieben werden. SystemleistungsmaDnahmen wie die I)urchschnittszeit, die benotigt wid , bis die Maschinen gewartet werden, die Durchachnittszahl der leerlaufenden Mas- chinen und die Nutzbarmachung des Maschinenbedieners werden durch die eingefugte Kettenanalyse von Markov erhalten. Die Algorithmuse, die zur Erhaltung dieser MaDnahmen benutzt werden, nutzen die spezielle Stmktur der Einschrittubergangswahrscheinlichkeitamx. Die Empfindlichkeit der Leistungsmahahmen gegeniiber der Dichtefunktion der Wartungszeiten und der gegebenen Prioritetszuteilungen wird untersucht.

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random intervals, Journal of the R q a i Statistied Society. 13.65. CHAUDRA. M. J., and SAROENT, R. G., 1983, A numerical method to obtain the equilibrium

results for the multiple finite source priority queueing model, Management Science, 29, 1298.

GUILD, R. D., and H A R T N E ~ , R. J., 1982, A simulation study of a machine interference problem with M machinea under the care of N operators having two types of stop- page , I d e r n a t i d Journal of Produdion Research, 20.47.

JAISWAL, N. K., and THIEWENOADAM. K., 1963, Simple machine interference with two types of failure, Operations Research, 11,624.

KOBAYASHI. H . , 1978. Modelling and Analysis: An Introduction Lo Syslem and Perfonname EvalwJion Methodology, (Reading, Mass: Addison-Wesley).

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SHANTHIKUMAR, J. G., and SAROENT, R. G., 1980, An algorithmic approach for a special case of Markov chains arising in queueing theory, Working Paper 80-015, Depart- ment of Industrial Engineering and Operations Research, Syracuse University, Syra- cuse. D

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