On queues with dependent interarrival and ... - Wiley Online Library

ON QUEUES WITH DEPENDENT INTERARRIVAL A N D SERVICE TIMES Shun-Chen Niu School of Management The University of Texas at Dallas Richardson. Texas ABSTRACT .An M / C / l queueing svsleni IS siudied iii which l l i e w r v i c c tinic required cusionicr is dependent o n the intcriirriviil l i m e b e i w c c n this arrival and i1i;ii ol' his predecessor Assuming ihc iwo variables are ";issocialed," w e prove 1h;ii [ l i e e x p e c ~ c ddelav i n [his svsicni is less than o r equal io i h i i i 01' Li c o n v c n iion;il M / G / I queue This coiicltision h:is heen verified vi;i s i n i u l ~ i t i o t i hv M i i c h e l l .ind I'LiuIsnn 191 l o r ii spcci;iI class 01' dcpendcni M / . I I / I queue 'Their m o d e l is ii spccial ciisc 01' ihc one we consider here. We iilso siudv i i n o i h c r iiiodilietl G//C/ I qucue. where i h c arriviil proccs\ ;ind/or t h e set-vice process ;ire individu;illv ";issnci,iied bv

;I

"

1. INTRODUCTION

The conventional G I / G / l queueing model considered in most papers in the literature assumes that the sequences of interarrival times and service times are i.i.d. (independent and identically distributed) random variables. However, the independence assumption may not be realistic for many real world problems. The purpose of this note is to investigate the effect on average delay (queueing time only) of customers in systems whose interarrival and service times are associafed, a concept of positive dependence developed by Esary, Proschan and Walkup [61. In a recent paper, Mitchell and Paulson 191 studied via simulation an M / M / l queue with t h e modification that a customer's service time and the interarrival time between his arrival and that of his predecessor are positively correlated random variables having a bivariate exponential distribution. Their simulation results indicate that this type of dependency reduces the mean waiting time of customers as compared to the usual M / M / 1 queue. Some related results also appeared in Conolly [2l, Conolly and Hadidi [3, 41. Motivated by Mitchell and Paulson's simulation results, we shall show here in Section 3 that their conclusion can be proven analytically under weaker conditions. In Section 2, we briefly summarize the useful concept of association of random variables. Another variant of G I / G / l queues with dependent interarrival times or service times will also be discussed in Section 4.

2. ASSOCIATION OF RANDOM VARIABLES The concept of association of random variables was developed by Esary. Proschan and Walkup [61. It is a very useful tool in the study of reliability systems (see Barlow and Proschan VOL. 28. NO. 3, SEPTEMBER 1981

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[ I ] , Esary and Proschan [51), queueing systems (see Niu [lo]), and simulation (see Heidelberger and Iglehart [71). DEFINITION 1: Random variables X = (-XI, . . . , X,) are called associated if cov 2 0 for all pairs of nondecreasing functions f a n d g.

[ f ( x ) , g (X)l

The following is a partial list of properties enjoyed by associated random variables: (P1):The set consisting of a single random variable is associated. (P2): Nondecreasing functions of associated random variables are associated. (P3): Any subset of associated random variables are associated. (P4): If two sets of associated random variables are independent of each other, then their union is a set of associated random variables.

(PS): A set of independent random variables are associated. The last property, P5, is a direct consequence of P1 and P4. As an illustration, we will next show that the bivariate exponential distribution discussed in Mitchell and Paulson [91 is generated from a pair of associated random variables. Consider the random vector (X, Y ) defined by

where X,,i = 1, 2, . . . , are i.i.d. random variables, Y,, i ables, and N is an integer-valued positive random variable.

=

1, 2, . . . , are i.i.d. random vari-

PROPOSITION 1 : (X, Y ) is associated.

N

=

PROOF: Let f a n d g be an arbitrary pair of nondecreasing functions. Conditioning on n, we have cov

Now, given N

UYx,

=

Y ) , R(X, Y)1

n, the vector

(XI,. . . , X,, Y t , . . . , Y,) is a

random variables and hence is associated (by P5). By P2, this ated. Therefore, the first term follows by observing that both nondecreasing functions of n. N A V A L RESEARCH LOGISTICS QUARTERLY

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REMARK: When both XI,i = 1, 2 , . . . , and Y,,, i = 1, 2, . .. , have exponential distributions and N is geometric, then (X, Y) has the bivariate exponential distribution described in

[91. 3. A N M/G/l Q U E U E WITH D E P E N D E N T ARRIVAL A N D SERVICE Consider the following model of a single server queue with infinite waiting room. Denote by T, the interarrival time between customers C, and C,+, with E ( T , ) = l / A . Let S , be the service time of customer C, with E ( S , ) = 1 / ~and Alp < 1. We assume (T,, S,+l), n = 0 , 1, . . . , is a sequence of i.i.d. random vectors and (T,,, S , + , ) is associated for each n. It should be noted that this model includes both the conventional G I I G I l queue (where T,, and Sn+lare independent) and the M / M / 1 queue considered by Mitchell and Paulson [91 by letting to have their bivariate exponential distribution (see remark after Proposition 1). (T,,, Denoting the delay of customer C,, by D,, it is well-known that (1)

D,,]

=

max[O, D,

+ S, - T,],

or equivalently,

Dn+l- A,, = D,, + S, - T,, where A , = - min 10, D, + S, - T,]. We will assume that t h e first customer arrives at time 0 and the system is initially empty, i.e., D ,= 0. The following key lemma will be needed: (2)

LEMMA 1: cov [D,, S,]

< 0 for all n

2 1.

0. For n 2 2, (T,-i, S,) is, PROOF: The assertion is clearly true for n = 1 since D , --(D,-] S , , - ] ) . Therefore, by assumption, associated and independent of (-D,-] - S , , - , , Tn-l, S,) is associated by P4. Observe that -D, and S , are nondecreasing S,) is associated and consefunctions of (-D,-l - S n P l rT,,-I, S , ) . Hence, by P2, (-D,,, S,I 0. quently cov [Do, Q.E.D.

+


0) is the idle period ended by the arrival of customer C , + , . Therefore, by t h e memoryless property of exponential distributions, ( A , l l A , l > 0) is also exponential with rate A. Hence,

EM,;) - E(A,,ZIA,, > 0) . PL4, > 01 2 E ( A , I A , , > 0) . P { A , , > 01

2E(A,)

- E(A,,ZIA, > 0)

2 E ( A , , l A , , > 0)

=

1/A Q.E.D.

and from which ( 5 ) follows.

For arrival processes other than Poisson, t h e analysis of (4) becomes more difficult because we do not know the distribution of (A,,IA,, > 0). However, various bounds for it may be obtained by considering special classes of arrival processes (see Marshall [Sl). We shall not pursue this further here except for mentioning that all upper bounds for expected delay in special classes of G I I G I l queues obtained by Marshall [81 can also be applied to our modified GI1 G I 1 queues. 4. SOME RELATED RESULTS

In this section, we will consider another modification of G I I G I l queues which is different from the one discussed in Section 3. We shall assume that the sequences of service times and interarrival times, { S , , n 1) and { T , , n 3 11, are identically distributed and associated, i.e., ( S l , S 2 , . . . , S,> and ( T I , T I , . . . ,T,) are associated vectors for all n 3 1 . Of course, this is again a generalization of the conventional G I / G / 1 queue because independent random variables are associated. Now, it is easy to see that

>

-T,> are associated vectors for all n 3 1 . It follows that Hence, (D,,S , ) and (Dn, E(D,,S,,) 3 E ( D , ) E ( S , , ) and E(D,T,) E(D,)E ( T , ) . An argument similar to Section 3 will then lead to




THEOREM 2: If {T,,, n 11 are i.i.d. exponential random variables, i.e., the arrival process is Poisson, then, under stationary conditions, we have N A V A L R E S E A R C H LOGISTICS Q U A R T E R L Y

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In other words, associated service times in an M / G / I queue tend to increase t h e expected delay of a customer. This is a somewhat expected result. When (T,,, n >, 1 ) is a renewal process (not necessarily Poisson), lower bounds for E(D,,) may be found through (6) for several special classes of interarrival times as in Marshall [SI. As a final remark, we note that the departure process of a conventional G t / D / l queue (the service times are deterministic) is an associated process. So, (6) may be used to find a lower bound for the expected delay of a subsequent station.

REFERENCES 111 Barlow, R.E. and F. Proschan, Statistical Theory of Reliability and Life Testing: Probabi1it.v Models, (Holt, Rinehart, and Winston, New York, 1975). [21 Connolly, B.W., “The Waiting Time Process for a Certain Correlated Queue,” Operations Research, 16, 1006-1015 (1968). [3] Connolly, B.W. and N. Hadidi, “ A Correlated Queue,” Journal of Applied Probability, 6, 122-136 (1969). [41 Connolly, B.W. and N. Hadidi, “ A Comparison of the Operational Features of Conventional Queues with a Self-Regulating System,” Applied Statistics, 18, 41-53 (1974). [51 Esary, J.D. and F. Proschan, “ A Reliability Bound for Systems of Maintained Interdependent Components,” Journal of American Statistical Association, 65, 329-338 (1 970). [61 Esary, J.D., F. Proschan and D.W. Walkup, “Association of Random Variables, with Applications,” Annals of Mathematical Statistics, 38, 1466-1474 (1967). [71 Heidelberger, P. and D.L. Iglehart, “Comparing Stochastic Systems Using Regenerative Simulation with Common Random Numbers,” Advances in Applied Probability, 11, 804-819 (1979). [SI Marshall, K.T., “Some Inequalities in Queueing,” Operations Research, 16, 6 5 1-665 (1968). 191 Mitchell, C.R. and A S . Paulson, “ M / M / 1 Queues with Interdependent Arrival and Service Processes,” Naval Research Logistics Quarterly, 26, 47-56 ( I 979). 1101 Niu, S.C., ”Bounds for the Expected Delays in Some Tandem Queues,” Journal of Applied Probability, 17, 831-838 (1980).

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