Joint Stationary Distribution Of Queues In Homogenous M|M|3 Queue With Resequencing

Resequencing issue is a crucial issue in simultaneous processing systems where the order of customers (jobs, units) upon arrival has to be preserved upon departure. In this paper stationary characteristics of M/M/3/∞ queueing system with reordering buffer of infinite capacity are being analyzed. Noticing that customer in reordering buffer may form two separate queues, focus is given to the study of their size distribution. Expressions for joint stationary distribution are obtained both in explicit form and in terms of generating functions. Numerical example is presented. INTRODUCTION Resequencing issue is a crucial issue in simultaneous processing systems where the order of customers (jobs, units, et.c.) upon arrival has to be preserved upon departure. Various analytical methods and models have been proposed to study the impacts of resequencing. A general survey of queueing theoretic methods and early models for the modeling and analysis of parallel and distributed systems with resequencing can be found in Boxma et al. (1994). Survey on the resequencing problem that covers period up to 1997 can be found in Dimitrov (1997). Queueing-theoretic approach to resequencing problem implies that the system under consideration is represented as interconnected queueing systems/networks. Following Leung et al. (2010) existing papers can be grouped into categories: papers that characterize the disordering process through a queueing system with several servers sharing a single queue (see, e.g. Agrawal and Ramaswami (1987)) and papers where disordering is modeled by a queueing system with several parallel servers and queues, and each server has its own dedicated queue (see, e.g. Ye Xia et al. (2008)). For a short survey of these two categories see e.g. Leung et al. (2010). Following this approach various problems setting have been considered and solved including distribution of number of packets in reordering buffer and in system under different assumptions about arrival and service process (see, e.g. Jain and Sharma (2011), Lelarge (2008), Chakravarthy (1998), Takine et al. (1994), De Nicola C. et al. (2013)); distribution and/or mean of the resequencing delay (see, e.g. Huisman and Boucherie (2002), Ding et al. (1991)), end-to-end (i.e. sender– receiver) delay (see, e.g. Chowdhury (1991)); large deviations of the queue size in reordering buffer (see, e.g. Gao et al. (2012)), asymptotics of the resequencing delay (see, e.g. Jun Li et al. (2010)), optimal allocation of customers to servers (Gogate and Panwar (1999), optimization issues (Dimitrov et al. (2002)). Among practical related papers one can also cite Leung et al. (2010), Zheng et al. (2010) andWen-Fen (2011), Li et al. (2010), Huisman and Boucherie (2001), Min Choi et al. (2012), Rubin et al. (1991). In this paper we propose new problem statement for systems with resequencing that are modeled by multiserver queues followed with infinite resequencing buffer. New problem is motivated by noticing that customers awaiting in resequencing buffer may form separate queues. The most convenient way to explain how queues are separated in resequencing buffer is with the example. Consider a queueing system with three servers, infinite capacity main buffer and reordering buffer. Let the state of the system at some instant be as depicted in Fig. 1. In squares one can see customers’ sequential numbers. White (black) squares in Fig. 1 mean that customers with these sequential numbers have received (have not yet received) service. Here one can distinguish two queues: one which is formed by customers awaiting customer no. 18 (queue #1), another is formed by customers awaiting customer no. 15 (queue #2). Three cases need to be considered. Case 1. Now on if customer no. 21 is next to complete its service then it joins queue #1 and stays there until service of customer no. 18 is complete. Customer no. 22 joins idle server. Case 2. If customer no. 15 is next to complete its service then it goes through queue #1 without waiting and joins queue #2. Meanwhile customer no. 22 joins idle server. As there is no customer in the system with sequential number smaller than any sequential number in Proceedings 28th European Conference on Modelling and Simulation ©ECMS Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani (Editors) ISBN: 978-0-9564944-8-1 / ISBN: 978-0-9564944-9-8 (CD) queue #2, then all customers from queue #2 leave the system. Resequencing buffer “sees“, that queue #2 is empty and moves its contents to queue #2. Now there are three options. First: if customer no. 18 is next to complete service, then it goes through queue #1 without waiting and joins queue #2. Customer no. 23 joins idle server. Again there is no customer in the system with sequential number smaller than any sequential number in queue #2. Thus all customers from queue #2 leave the system. Resequencing buffer becomes empty. Now if customer no. 21 is next to complete service, it leaves the system. If customer no. 22 is next to complete service, it goes through queue #1 without waiting and joins queue #2 where it waits for customer no. 21. Finally, if customer no. 23 is next to complete service, it joins queue #1 and does not proceed to queue #1 because it needs customer no. 22 to complete its service before both of them may join queue #2. Second: if customer no. 21 is next to complete service, then it goes through queue #1 again without waiting, joins queue #2 and waits there with other customer for service completion of customer no. 18. Third: if customer no. 22 is next to complete service, then customer no. 23 joins idle server, customer no. 22 joins queue #1 and stops there because “sees” gap between its sequential number and largest sequential number in queue #2. It waits there for customer no. 21. Case 3. If customer no. 18 is the first to complete its service then it joins queue #1 and customer no. 22 joins idle server. Resequencing buffer “sees“, that there is no gap in the middle of sequence and moves the content of queue #1 to queue #2 (queue #1 becomes empty). Now there are again three options. First: if customer no. 15 is next to complete service, then it goes through queue #1 without waiting, joins queue #2 and immediately (because the sequence is complete) leaves the system with all contents of queue #2. Second: if customer no. 21 is next to complete service, then it goes through queue #1 again without waiting, joins other customers in queue #2 that wait for service completion of customer no. 15. Third: if customer no. 22 is next to complete service, then it joins queue #1 and stops there, because “sees” gap between its sequential number and the largest sequence number in queue #2. The operation of the system proceeds along the line. Figure 1: Scheme of the model Clearly, when the number of server is n there are (n−1) queues in resequencing buffer. Sum their contents is the total number of customers in resequencing buffer. The main contribution of this paper are algorithm and probability generating function of joint stationary probabilities of the number of customers in buffer, queue #1 and queue #2. The paper is organized as follows. In the next section we give detailed description of the system. Then we find joint stationary distribution both algorithm-wise and in terms of probability generating functions. The last section is devoted to numerical results. DESCRIPTION OF THE SYSTEM Consider a queueing system with three servers, infinite capacity buffer, incoming poisson flow of customers (of intensity λ) and exponential distribution of service time at each server (with parameter μ) and resequencing buffer (RB) of infinite capacity. Customers upon entering the system obtain sequential number and join buffer. Without loss of generality we suppose that the sequence starts from 1 and coincides with the row of natural numbers, i.e. the first customer upon entering the (empty) system receives number 1, the second — number 2 and so on and so forth. Customers leave the system strictly in order of their arrival (i.e. in the sequence order). Thus after customer’s arrival it remains in the buffer for some time and then receives service when one of the servers becomes idle. If at the moment of its service completion there are no customers in the system or all other customers present at that moment in the queue and the rest two servers have greater sequential numbers it leaves the system. Otherwise it occupies one place in the RB. Customer from RB leaves it if and only if its sequential number is less than sequential numbers of all other customers present in system. Thus customers may leave RB in groups. Let us call “1 level” customer the one which is in service and was the last to enter server; “2 level” customer is the one which is in service and was the penultimate to enter server; finally, “3 level” the customer is the one which is in service and was the first to enter server. If the number of busy servers is 3, then customers that entered RB between “1 level” and “2 level” customer form queue #1; customers which entered RB between “2 level” and “3 level” customer form queue #2. If the number of busy servers is 2, then customers which entered RB after “1 level” customer form queue #1; customers which entered RB between “1 level” and “2 level” customer form queue #2. When there is only one busy server all customers in RB form queue #2. The operation of the considered queueing system can be completely described by Markov process ζ(t) = {(ξ(t), η(t), ν(t)), t ≥ 0} with three components: ξ(t) — number of customers in buffer and server at time t, η(t) — number of customers in queue #1 of RB at time t, ν(t) — number of customers in queue #2 of RB at time t. In case ξ(t) = 0, the second and third component of ζ(t) are omitted; in case ξ(t) = 1, the second is omitted. The state space of ζ(t) is X = {0}∪{(1, i) , i ≥ 0}∪{(n, i, j) , n ≥ 2, i ≥ 0, j ≥ 0}. Henceforth it is assumed that service and arrival processes are mutually independent and necessary and sufficient condition of stationarity ρ/3 < 1, where ρ = λ/μ holds for the system. STATIONARY JOINT PROBABILITY DISTRIBUTION Note that the total number of customers in servers and buffer of the considered QS with resequencing coincides with the total number of customers in M/M/3/∞ queue. Therefore, its stationary distribution {pi, i ≥ 0}, has the form