A Fixed-size Batch Service Queue with Vacations

The paper deals with batch service queues with vacations in which customers arrive according to a Poisson process. Decomposition method is used to derive the queue length distributions both for single and multiple vacation cases. The authors look at other decomposition techniques and discuss some related open problems .


Introduction
Batch service queues have numerous applications to traffic, transportation, production, and   manufacturing systems.The first study on batch service queues was due to Bailey [1].He obtain- ed the transform solution to the fixed-size batch service queue with Poisson arrivals.Miller [23] studied the batch arrival batch service queues and Jaiswal [14] considered batch service queues in which service size is random.Neuts [24] proposed the "general bulk service rule" in which service initiates only when a certain number of customers in the queue is available.His general bulk ser- vice rule was extended by Borthakur and Medhi [2].Studies on waiting time in a batch service queue were also rendered by Downton [8], Cohen [5], Medhi [22]  and Powell [25].Fakinos [11]  de-  rived the relation between limiting queue size distributions at arrival and departure epochs.Briere and Chaudhry [3], Grassmann and Chaudhry [13], and Kambo and Chaudhry [15] used numerical approaches to obtain the performance measures.Numerical methods were found to be effective especially for batch service queues, because the transform solution of the queue length (number of customers in the system including those in service) in batch service systems contains 1This is a part of the research supported by Korea Science and Engineering Foundation (KOSEF) Grant #921-0900-003-2.206 ttO WOO LEE, SOON SEOK LEE, AND K.C. CHAE some unknown values.For more extensive study on batch arrival/service queues, refer to Chaud- hry and Templeton [4].
Vacation queues have been extensively studied by many researchers.Comprehensive surveys can be found in Doshi [7] and Takagi [26].Most of the studies on vacation queues have been con- cerned with single-unit service systems such as M/G/1 or MX/G/1 queues, h well-known result concerning vacation queues is the "decomposition property" (Fuhrmann and Cooper [12]) which states that the probability generating function (PGF) of the queue length of a vacation system can be factorized into the queue length of ordinary queue without vacation and "something else", the "something else" depends on the system characteristics.Lee et al. [18] and Lee et al. [19]   analyzed the operating characteristics of batch arrival queues with N-policy and vacations, and obtained the queue length and waiting time distributions.
For batch service queues with vacations, there have been a few related works.Dhas [6] consi- dered Markovian batch service systems and obtained the queue length distributions by matrix-geo- metric methods.Lee et al. [16] obtained various performance measures for M/GB/1 queue with single vacation.Dshalalow and Yellen [10] considered a non-exhaustive batch service system with multiple vacations in which the server starts a multiple vacation whenever the queue drops below a level r and resumes service at the end of a vacation segment when the queue accumulates to at least r.They called such a system (r,R)-quorum system, R >_ r) being the service capacity of the server.They applied the theory of the first excess level (Dshalalow [9]).Lee et al. [17] showed that for some batch service queues, mean queue length may even de- crease in systems with server vacations.This has an implication that for some batch service queues, customers do not have to complain about unavailability of the server.Instead, they would rather force the server to take a vacation.
In this paper, we are going to concentrate on a very specific batch service queues called the fixed-size batch service queues with vacations.We first analyze the fixed-size batch service queue without vacations.
2. The M/GK/1/FS Queue In this section we consider the fixed-size batch service queue without vacations.Consider a batch service queueing system in which the server can take in a maximum of K customers into his service.If less than K customers are in the queue just after a service completion, the server waits in the system until the queue size reaches K (Figure 1).We will denote this queueing sys- tem by M/GK/1/FS queue in which 'FS' stands for 'fixed-size'.We are going to derive the de- composition of the queue length distribution at an arbitrary time point.By queue length we mean the number of customers in the system including those in service.The PGF of the steady-state queue length of M/GK/1/FS queueing system can be found in Chaudhry and Templeton [4] and is given by K-1 (z K-1)S*(A-Az) Pn,o zn p(z) z in which S*(. is the Laplace-Stieltjes transform (LST) of the service time distribution, and P is the joint probability that the server is idle and there are n customers in the system, n-1,2, ...,K-1.Pn, o can be seen as the probability of state n before a busy period begins.Note that P(z) contains K unknown values, Pn,0, n-0,...,K-1.These unknowns occur in any type of batch service queues (see Chaudhry and Templeton [4]) and can be found by applying the well- known Rouche's theorem.
Theorem 2.1: The PGF of the queue length distribution given by equation (2.1) decomposes into P(z) Pu(z).Pi(z), where and pu(z ( 1 IIere r j is the probability that the idle process ever enters state j, j 0,1,..., K-1. Proof: From the definition of Pn, o we see that (2.4) where p is the probability that the server is busy.Then Pn, o/.P j, o is the probability that 3--0 the server is idle with n customers in the system under the condition that the server is idle.
Define Ij as Ij-{ 1 0 if the idle process ever enters state j, j 0, 1, 2,..., K-1, O/W.Then defining r. as the probability that the idle process ever enters state j, we see that .= 3 (K-1 K-1 3 Pr(Ij 1), and E E I j} E j is the mean number of states the idle process enters until t [,,SJ= j j=0 he server begins tooe busy.Since the arrival process is Poisson with rate A, the mean time for K-1 the idle process to stay in a state is l/A, and thus j/A becomes the mean length of the idle K-1 j=0 (rnl)) --Pn,ol Pj,o" The statement follows from equations (2.1)   period.Thus we have K-1 jl -o j=0 and (2.5).Remark 2.1: Equation (2.2) shows that at an arbitrary point of time, the queue length of the M/GK/1/FS queue is the sum of two random variables.The behavior of the system represented by the PGF, Pv(z) is not clear at this point of our analysis and it will be left as an open problem for future study.Pu(z) plays a very important role in the subsequent analysis and will be called "the basic stochastic system (BSS)." Remark 2.2: (Decomposition of the queue length of M/GK/1/FS queue) Since Pj,o is K-1 3=0 the probability that the server is idle, Pn, o/., P j,o is the probability that there are n customers 3----0 in the system under the condition that the server is idle.Thus Pi(z), given by equation (2.4), can be interpreted as the PGF of the queue length given that the server is idle.Then we see that the PGF of the queue length distribution of the M/GK/1/FS queue decomposes into: 1. the BSS represented by Pu(z), and 2. the queue length during the idle period represented by Pi(z).
It is well known that the queue length of an M/G/1 vacation queue decomposes into two random variables one of which is the queue length of the ordinary M/G/1 queue (Fuhrmann and  Cooper [12]).We will see in the upcoming sections that the role of the ordinary M/G/1 queue is played by the BSS in the decomposition of the fixed-size batch service queue.
Remark 2.3: In the M/GK/1/FS queue, let P+(z) be the PGF of the queue length embedded at departure epochs.Then from Chaudhry and Templeton [4], K-1 S*(-Az) E zK-zn)Pn + nO P + (z) z K S*(A Az) In order for the system state to enter n during the idle period, it suffices to have n or less customers nt a departure point.Thus we have n-p/i-.Then, after some manipu]ations, Therefore, we have the following relationship between the queue length at an arbitrary time point and a departure point: (2.6) This agrees with the result of Fakinos [11].Observe that (1-zK)/[K(1z)] is the PGF of the backward recurrence time of a renewal interval H in a discrete renewal process with Pr(H K) 1. Equation (2.6) states that a departing customer is more likely to find the sys- tem empty than an arriving customer.
3. A Fixed-Size Batch Service Queue with Vacations In this section, we analyze the fixed-size batch service queues with server vacations.We consider two types of server vacations: single and multiple.The systems are described as follows: 1. Single vacation queue (Figure 2): everytime a service is finished, if less than K customers are in the queue, the server leaves for a vacation of random length V.When he returns from the vacation, and finds K or more customers waiting, he begins to process K of them.Otherwise, he A Fixed-Size Batch Service Queue with Vacations 209 remains dormant in the system until the queue length reaches K.
This system will be denoted as 2. Multiple vacation queue (Figure 3): every time a service is finished, and there are less than K customers in the queue, the server leaves for a vacation of random length V 1.If there are less than K customers in the queue upon his return from the vacation, he immediately leaves for another vacation of random length V2, and so on until he finally finds K or more customers in the queue.We assume that {Vj, j > 1} constitutes iid sequence with generic representation V.
This system will be denoted by M/GR/1/FS/MV where 'MV' stands for 'multiple vacation'.We will use the following notations and probabilities: IIO WOO LEE, SOON SEOK LEE, AND K.C. pdf, DF, LST of S pdf, DF, LST of V remaining service time for the customer in service at time t remaining vacation time for the server on vacation at time if the server is in dormancy if the server is busy in the system if the server is on vacation system size at time t probability that n customers arrive during a vacation 3.1 The M/GK/1/FS/SV queue In this section, we analyze the M/GK/1/FS/SV queue.First we model the system by using the residual service and vacation times as supplementary variables.Using the above notations, we easily derive the following steady-state system of equations: O---AR n+AR n_l+Qn(0), (n-l,2,...,K-1), dPK(x) APK(X) + P2K(O)s(x) + As(X)RK 1 + QK(O)s(x), dd-xPK+ n(x) APK + n(x) + APK + n-1(x) dd-:Qm(x) AQm(x -4-AQ m l(X) --PK + m(O)v(x)' (m 1,2,...,K 1), dQn(x AQn(x + AQn_ l(x), (n >_ K).
We need to take a closer look at Y(z).First, Qn(O) can be expressed in terms of the number of customers at a vacation initiation point and the number of customers that arrive during the vacation as Qn(0)-Evi'PK+n-i(O), (n-O, 1,2,...,K-1).i=0 Then, Y(z) becomes Y(z) E ZnPK + n (0 [V*(A-/kz)-1] + (z 1) zmAm n=0 m=0 (3.1.12)The PGF of the queue length distribution given by equation (3.1.11)can be P where n K- Proof: From Y(z) given by (3.1.13),we get E(Y) E P K+n( 0 AE(V) + A m n=0 m=0 The theorem follows from equations (3.1.11)and (3.1.13).
l{emark 3.1.1"Since there are n customers in the queue right after leaving for a vacation with K + n customers in the system just before the service completion, it is easily seen that n equals the probability that there are n customers in the queue just after leaving for a vacation.
To interpret the terms in the bracket in equation (3.1.14),we need the following theorems.Theorem 3.1.2:Let F n be the event that there are n (n-O, 1,2,...,K-1) customers in the system just after the server leaves for a vacation, i.e., Pr(Fn)-n from remark 3.1.1.Then under F Am(m-O, 1,2,...,K-n-1) is the probability that the system ever enters state n + m n during the dormant period.
Proof: Define if the server is dormant with n + m customers in the queue,

O/Wo
In order for the system to enter state n + m during the dormant period, it is necessary that m m or less customers arrive during the vacation.Thus we have Pr(I m 1) v Am.

K-n-1
Proof: From theorem 3.1.2,it is easily seen that I m is the number of states that are m--0 {K-I )K-n-I entered under F n before the server gets busy.Thus E Im -E Pr(I m -1) A m is the mean number of states that are entered under F n. Since the arrival process is m:0 K-n-1 Poisson, Am/A is the mean length of the dormant period under F n. Thus r--O K-n-1 N(V) + .Am/A is the mean length of the idle period under 1-' n.Relaxing the condition on 1-" n rn--O completes the proof.