K Iyer and K Sikdar Department of Mathematics Indian Institute of Science Bangalore India Abstract We derive the waiting time distribution in a GIM 1 queueing system with de pendence between the service time of each customer and the subsequent inte ID: 28047 Download Pdf

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K Iyer and K Sikdar Department of Mathematics Indian Institute of Science Bangalore India Abstract We derive the waiting time distribution in a GIM 1 queueing system with de pendence between the service time of each customer and the subsequent inte

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On the waiting time in the GI/M/1 vacations queues with dependency between interarrival and service times S. K. Iyer and K. Sikdar Department of Mathematics, Indian Institute of Science, Bangalore, India. Abstract – We derive the waiting time distribution in a GI/M/ 1 queueing system with de- pendence between the service time of each customer and the subsequent interarrival times. In addition, the server takes exponentially distributed vacations when there are no customers left to serve in the queue. Keywords: dependence; multiple vacations; queue; single

server. 1 Introduction Most queueing models considered in the literature assume independence between service times and the interarrival times. In practice, however, they are not independent. In many practical situations, a signiﬁcant dependence has been observed. This happens, for example, when the arrival of a customer with a long service time discourages the next arrival. Fendick et al. (1989) give a review of the various types of dependencies that exist in packet queues in packet communication networks and their impact on system performance. Cidon et al. (1993, 1996), study a family

of queues with random proportional dependency with additive delay between the interarrival and service times. The model they consider is = where is the service time of the th customer and is the subsequent interarrival time. In this model, the distributions of the random variables that appear on the right of the above equation are speciﬁed. is assumed to have a discrete distribution. They describe how their models can describe the working of a queue in the presence of a spacer controller. More recently, Iyer and Manjunath (2005) study queues in which the interarrival and service times

are dependent and are assumed to follow exponential, Erlang, hyperexponential or Coxian distributions. Dependence between the service time of the th customer, and the subsequent interarrival time is modeled Correspondence: Tel. +91-80-22932265. Fax: +91-80-23600146. Email addresses : skiyer@math.iisc.ernet.in (S. K. Iyer), karabi@math.iisc.ernet.in (K. Sikdar)

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using the equation aX where is a positive constant and is assumed to be independent of Thus, given the distributions of and , the distribution of can be characterized. In contrast to the models of Cidon et al. (1993,

1996), this model gives rise to a sort of non-intrusive control, in the sense that the marginal distribution of the interarrival and service times are left unchanged. We refer the reader to Iyer and Manjunath (2004), where the r.v.s have been characterized when , X belong to the class of distributions mentioned above. In Iyer and Manjunath (2006), dependence between and is induced by assuming that are discrete r.v.s taking ﬁnitely many values. Given , the distribution of is assumed to be either discrete or exponential, with parameters of the conditional distribution depending on In both

the above papers, the authors derive the steady state waiting time distribution of customers in the queue. We refer the reader Cidon et al. (1993, 1996), Chao (1995), Muller (2000), Iyer and Manjunath (2005, 2006) for detailed discussions and further references on dependence queues. The GI/M/ 1 queue with multiple vacations has been independently studied by Tian et al. (1989), Chatterjee and Mukharjee (1990). Tian et al. (1989) analyzed this queue for exponential vacations using matrix geometric solutions whereas Chatterjee and Mukharjee (1990) incorrectly claim that their solution is valid

for generally distributed vacation time. This has been pointed out by Karaesmen and Gupta (1996, appendix 1). Recently, Chae et al. (2006a, 2006b) analyzed the GI/M/ 1 queue with multiple and single vacation. In the former they obtained the trans- form of the joint distribution of the length, the number of customers served, and the residual interarrival time at the end, of a busy period. Later in 2006b, they showed that both the queue length and the waiting time can be stochastically decomposed into meaningful quantities. The ﬁnite capacity GI/M/ 1 queue with multiple vacations has been

analyzed by Karaesmen and Gupta (1996) whereby they obtained the system length distributions and waiting time. In the above papers on queues with server vacations, the interarrival and service time sequences are assumed to be independent. In this paper we consider the GI/M/ 1 queueing system, considered in Iyer and Manjunath (2005). In addition to the dependence between the service and subsequent interarrival times, the server begins a “vacation” of random length each time the system becomes empty. If the server returns from a vacation to ﬁnd one or more customers waiting, she works

until the system empties, then begins another vacation. If the server returns from a vacation to ﬁnd no customers waiting, she begins another vacation immediately. This vacation policy is referred to as ‘exhaustive service discipline’. Throughout the paper we will assume the service times to be exponentially distributed while the interarrival times could be exponential, Erlang, hyperexponential or Coxian. The vacation times are assumed to be independent exponentially distributed random variables. The organization of this paper is as follows. In the next section we discuss the model and

the notations used. In section 3 we discuss the queue length distributions at pre-arrival epochs. Finally, we will derive the waiting time distribution in section 4.

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2 Description of the model and notations The dependent GI/M/ 1 queueing model with server vacations following the exhaustive service rule can be described as follows. Let represents the arrival point of the th customer with = 0, therefore +1 . The sequence of random vectors ( , X ) are assumed to be an i.i.d. sequence distributed as ( A, X Dependence between and is induced via the linear equation aX , where is a

positive constant and is independent of is assumed to be exponential with mean We will denote the distribution function and Laplace-Stieltjes (L.S.T) transform of a generic r.v. by and respectively. The system has a single server. As soon as the system becomes empty, the server has a vacation distributed as , which is exponentially distributed with mean . When a vacation is over, if the system is still empty then the server has another independent vacation of length distributed as ; otherwise she returns from the vacation. Let denote the remaining vacation time at the time of a random arrival

that occurs at a time when the server is on vacation. By the memoryless property, is exponential with mean Let denote the probability that exceeds the subsequent interarrival time, therefore V > A ) = γv dF ) = ). Further, let us assume that := ( be the r.v. denoting the amount by which exceeds . Finally, the vacation times are mutually independent and are independent of the service and interarrival times and the service discipline is ﬁrst-come-ﬁrst-served. Let us now consider some examples of possible distributions for the interarrival times and the resulting distribution

of Case (i) : Let be exponential with rate and let λ/ with a 1. In this case, the L.S.T. of is given by ) = as = (1 a aρ. (1) From the form of ), we observe that it is a product of two independent r.v.s - a Bernoulli r.v. with mean (1 a ) and an exponential r.v. with parameter . Thus the distribution function of will be a + (1 a )(1 λz ). Case (ii) : If is Erlang with parameter and , the L.S.T. of will be ) = as as This is the sum of two r.v.s - a Erlang( , ) distributed r.v. and another is the product of a Bernoulli and exponential like the in eq. (1). Case (iii) : If is

hyperexponential, then the L.S.T. of is given by ) = =1 [(1 a a where 0 1, =1 = 1 0 and / . Evidently, is a mixture of r.v.s with L.S.T. of the form given by (1).

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Case (iv) : If is a stage Coxian with being the probability that stage is entered and the rate of the exponential of stage , the L.S.T. of will be ) = as =1 (1 +1 =1 with 0 1 for = 1 , k . For simplicity we take = 2 with = 1 and +1 = 0 and rewrite the L.S.T. of as ) = + (1 Let / , then we can write ) = as + (1 a + (1 a )( Thus with probability is a product of a Bernoulli and an exponential r.v.s (see eq. (1)), and

with probability (1 ) it is a two stage Coxian. For more details, see Iyer and Manjunath (2004, 2005). 3 Queue length distributions at pre-arrival epoch Let ) denotes the number of customers in the system at time (queue length). We chose as the embedded points and denote by 0) the queue length just prior to the th arrival epoch. Let if an arrival occurs in a vacation if an arrival occurs in a busy period. Clearly is a r.v. Because both service and vacation times have memoryless property, the process ( , q ) is an embedded Markov chain. Deﬁne the limiting probabilities, ij = lim i, q , i

= 0 1; = 0 By observing the system at two consecutive pre-arrival epochs under steady state, we get 11 =1 ,r + (1 =0 ,r (2) =0 ,j + (1 =0 ,j , j (3) 00 =1 ,r + (1 =0 ,r (4) ωp ,j , j (5)

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where ) is the probability that customers are served during time , given that there are exactly + 1 (more than + 1) customers at the time the vacation ends. Similarly, represents the probability that customers are served during an interarrival time given there are exactly + 1 (more than + 1) customers in the queue when the server is busy. Further, let us assume that ) is the probability

that all the customers served during time (an interarrival time). Let , H , G ) and ) be the probability generating functions of and , respectively and are given by ) = (1 1 + a )( (1 (1 (1 )( + (1 (1 + (1 ))( + (1 )( (1 (1 (6) ) = (1 a 1 + (1 (1 )) (7) ) = + (1 )(1 + (1 )) (1 )) (1 + (1 (8) ) = 1 + (1 (1 )) (9) For derivation of equations (6)-(9), see appendix. Equation (5) yields 00 , j (10) Let be the forward displacement operator, that is, Ep ,j . Using this and (10), (3) can be rewritten as )) = (1 00 , j (11) Let ) = ). We will show that ) = 0 has exactly one root of between 0 and 1.

Observe that 0 < K (0) = 1 and (1) = =0 = 1. Since, ) = =1 n k and 00 ) = =2 1) ) is non-decreasing and convex. Since service times are exponential, all the , n > are 0; hence ) is strictly convex. Thus ) = ) has only one zero inside = 1. Let be the root of ) = 0 satisfying 1. Thus the solution of (11) is C (1 00 , j (12)

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where is a constant which is determined by using (2) and (12) as 00 (13) Using (13) in (12), we get = (1 00 , j (14) Now, the only remaining unknown quantity 00 can be obtained using the normalizing condition =0 =1 = 1. 4 Waiting time distribution Let denotes

the waiting time of a customer in the queue (excluding service). On arrival a customer may ﬁnd either the server busy or on vacation with 0) customers in the queue. If a customer ﬁnds that the server is busy, then the waiting time will consist of service completion of customers. Similarly, if the server on vacation, then her waiting time will be sum of remainder vacation time and service completion. Thus the distribution function of is given by ) = ) = 00 ) + =1 ) + )) (15) where star denotes the convolution. Thus the L.S.T. of by ) be obtain from (15) ) = 00 =1 The expected

waiting time in the queue is ) = =0 =1 5 Conclusion In this paper we have integrated two generalizations of single server queuing systems, namely queues in which the server goes on vacations and queues in which there is dependence between the service and interarrival times. Server vacations are useful in enabling the server to multi-task and utilize the idle periods productively. Queues with dependent service and interarrival times are useful in congestion control. Such queuing models also to capture the dependence observed in many traﬃc traces which are known to have a

signiﬁcant impact on the waiting times. We have assumed throughout this paper that the service and vacation times are exponentially distributed. Further, the dependence between the service and interarriva time extends only upto one single

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lag, that is the service time of any customer has an eﬀect only on the subsequent interarrival. Modeling dependence over higher lags in ways which will keep analysis tractable is a possible way to extend this work. Queues with bulk arrivals is another possible direction for future research. Acknowledgements- The second author

wishes to thank National Board of Higher Mathe- matics (NBHM), Mumbai, India, for their ﬁnancial support. Appendix The expression of 0) is given by X > A A > Z < (1 =0 =0 =0 e x dx ) ( γe γv dv dF (1 (1 (1 (16) Let =1 where is the service time of the th customer. So, is the sum of consecutive service time, i.e. Erlang k, ). Therefore ) = 1 x =0 x . Thus, X < Y < A A > , r (1 V < Y < Z aX (0 ax )) (0 (1 )) e x dx ) ( γe γv dv dF (17) where ax =0 ax ) ( γe γv dv )( e x dx dF setting ax v, ax =0

γe γp dp ) ( e a dx γz dF ax =0 p =0 p γe γp dp e a dx γz dF ax e a dx γz dF

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= 1 a (18) and =0 (1 =0 (1 ) ( γe γv dv )( e x dx dF setting (1 v, =0 (1 =0 γe γq dq ) ( e (1 dx γz dF =0 (1 =0 q =0 q γe γq dq e (1 dx γz dF =0 (1 e (1 dx γz dF = 1 + (1 (1 (1 (19) Further, ax =0 γe =0 p dp e a dx γz dF and (1 =0 γe =0 q dq e (1 dx γz dF Using (18) and (19) in

(17), we get a )( (1 (1 (1 , r (20) Again, is given by X < A A > < Z (1 > Z (1 Proceeding in similar way, the expression of 1) and are given by X < Y < A (1 X < Y < Z aX ax (0 (1 )) e x dx dF (21) where ax )( e x dx dF setting ax p,

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dp dF p =0 p dp dF = 1 =0 (1+ dp dF (22) and =0 (1 )( e x dx dF setting (1 q, =0 dq dF =0 q =0 q dq dF = 1 =0 =0 (1 dq dF (23) Thus, using (22) and (23) in (21) we get =0 =0 (1 dq dF =0 (1+ dp dF , r (24) and X < A < Z (1 = 1 > Z (1 = 1 Multiplying (20) by , summing

over all possible values of and adding (16), we get the probability generating function of as ) = =0 (1 ) + a )( (1 (1 (1 (25) where ax =0 γe =1 =0 p dp e a dx γz dF

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ax =0 γe =0 +1 p dp e a dx γz dF ax =0 γe =0 =1 p dp e a dx γz dF ax =0 γe (1 dp e a dx γz dF (1 e a e (1+ (1 )) (1 γz dF (1 a 1 + (1 (1 )) (26) and (1 =0 γe =1 =0 q dq e (1 dx γz dF (1 (1 + (1 (1 )) (27) Now using (26) and (27) in (25), we get (6). Similarly, multiplying

(24) by , summing over all possible values of , we get the probability generating function of as ) = =1 =0 ((1 dq dF ((1 )+ dp dF + (1 (1 )) 1 + (1 (1 )) + (1 )(1 + (1 )) (1 )) (1 + (1 Next we will derive the expressions for and . Therefore > A A > where is the service time of ( + 2) th customer > Z aX =0 =0 ax =0 ax e x dx )( γe γv dv )( e x dx dF =0 =0 ax =0 ax γe γv dv )( e x dx dF (28) 10

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Again, < Y < A A > aX < Y < Z aX (0 aX (0 aX =0 =0 ax =0 ax ) ( γe γv dv )( e x dx dF (29)

where =0 =0 ax =0 ax =0 ax )( e x dx )( γe γv dv )( e x dx dF =0 =0 ax =0 +1 ax ) ( γe γv dv )( e x dx dF (30) Using (30) in (29), we get =0 =0 ax =0 ax ax )) γe γv dv )( e x dx dF (31) Thus, combining (28) and (31), for 0 we get =0 =0 ax =0 ax ax )) γe γv dv )( e x dx dF (32) Further > A > Z aX =0 =0 ax e x dx )( e x dx dF =0 =0 ax e x dx dF (33) and < Y < A aX < Y < Z aX aX (0 aX )) aX =0 =0 ax =0 ax )( e x dx

)( e x dx dF aX +1 aX =0 =0 ax ax )) e x dx dF (34) 11

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Thus, combining (33) and (34), for 0, we get =0 =0 ax ax )) e x dx dF (35) Now multiplying (32) and (35) by and summing over all possible values of , we get the result of (8) and (9), respectively. References [1] Fendick KW, Saksena VR and Whitt W (1989). Dependence in packet queues. IEEE Trans. Commun. 37: 1173-1183. [2] Tian N, Zhang D and Cao C (1989). The GI/M/ 1 queue with exponential vacations. Queueing Syst. 5: 331-344. [3] Chatterjee U and Mukherjee SP (1990). GI/M/

1 queue with server vacations. J. Oper. Res. Soc. 41: 83-87. [4] Cidon I, Guerin R, Khamisy A and Sidi M (1993). Analysis of a correlated queue in communication systems. IEEE Trans. Inform. Theory IT- 39: 456-465. [5] Chao X (1995). Monotone eﬀect of dependency between interarrival and service times in a simple queueing system. Oper. Res. Lett. 17: 47-51. [6] Cidon I, Guerin R, Khamisy A and Sidi M (1996). On queues with interarrival times proportional to service times. Probab. Engrg. Inform. Sci. 10: 87-107. [7] Karaesmen F and Gupta SM (1996). The ﬁnite capacity GI/M/ 1 queue

with server vacations. J. Oper. Res. Soc. 47: 817-828. [8] Muller A (2000). On the waiting time in queues with dependency between interarrival and service times. Oper. Res. Lett. 26: 43-47. [9] Iyer SK and Manjunath D (2004). Correlated bivarite sequences for queueing and reli- ability applications. Communications in Statistics 33: 331-350. [10] Iyer SK and Manjunath D (2005). Correlated queues using bivariate mixtures. Stochas- tic Models and Applications 8: [11] Iyer SK and Manjunath D (2006). Queues with dependency between interarrival and service times using mixture of bivariates.

Stochastic Models 22: 3-20. [12] Chae KC and Kim SJ (2006a). Busy period analysis for the GI/M/ 1 queue with expo- nential vacations. Oper. Res. Lett. [13] Chae KC, Lee SM and Lee HW (2006b). On stochastic decomposition in the GI/M/ queue with single exponential vacation. Oper. Res. Lett. 12

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