Stationary Characteristics Of Homogenous Geo/Geo/2 Queue With Resequencing In Discrete Time

Resequencing problem is a crucial issue in communication systems, databases, production and information networks because correct processing of information by them may often be performed only if original order of packets, queries, jobs is preserved. In this paper consideration is given to one of the queueing systems that may model processes of discrete nature where resequencing phenomenon may arise. SpecificallyGeo/Geo/2/∞ queueing system with reordering buffer of infinite capacity is being analyzed. Expressions for stationary sojourn time distribution and joint stationary distribution of the number of customers in system and reordering buffer are given in explicit form and in terms of generating functions. Illustrative numerical example is presented. INTRODUCTION It is well-known that resequencing problem is a crucial issue in packet switching networks, parallel and distributed systems (see e.g. Baccelli et al. (1981)). The resequencing delay deteriorates the performance of delaysensitive applications. For example, voice over IP services that gain popularity nowadays suffer from this problem because data packets at receiver’s side need to be played out in the same sequence they left the sender’s side. Thus the more disordering communication network introduces into the stream of packets the more time it takes for data packets to be resequenced (in de-jitter buffer) and the more difficult it is for network operators to satisfy stringent constraints on end-to-end delay and jitter. Resequencing influences design, performance and optimization of distributed computing systems where subtasks need to be put in sequential order before they can be assembled into one task. Reordering takes place also in modern microprocessors to achieve high application performance at an acceptable level of power dissipation. Here an entity (reordering buffer) is needed for out-of-order instructions to be committed in-order (see e.g. Min Choi et al. (2012)). In order to analyze impacts of resequencing various analytical methods and models have been proposed. 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). The studies that deal with packet disordering in communication networks typically consider communication system, consisting of disordering network and reordering buffer. Disordering network is a prototype of the real network. Data packets, sent from sender’s side through real network can be disordered (or even lost) for various well-known reasons and since many applications can only accept packets in the same order they were sent from the sender’s side, reordering buffer is needed at receiver’s side to reestablish the initial packets’ sequence. As reported in Leung et al. (2010), existing papers in the considered field of study can be grouped into two major categories. The first category consists of papers that characterize the disordering network as a queueing system with several servers sharing a single queue (see, e.g. Matyushenko (2010)). In the second category of papers, the disordering network is described as 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 Leung et al. (2010) and Ye Xia et al. (2008). Typically studies are mostly concerned with finding the distribution of number of packets in reordering buffer; distribution and/or mean of the resequencing delay, end-to-end (i.e. sender–receiver) delay; large deviations of the queue size in reordering buffer, asymptotics of the resequencing delay etc. Among the latest papers on the subject one can cite Leung et al. (2010), Jun Li et al. (2010), Zheng et al. (2010) and Wen-Fen (2011), where authors consider some new problems and improve previous known results. In Leung et al. (2010) authors propose a framework that allows estimation of resequencing delay and reordering buffer occupancy distribution under an orderly dispersion of traffic on multiple disjoint paths in disordering network. The asymptotic properties of the steady-state probability disProceedings 27th European Conference on Modelling and Simulation ©ECMS Webjørn Rekdalsbakken, Robin T. Bye, Houxiang Zhang (Editors) ISBN: 978-0-9564944-6-7 / ISBN: 978-0-9564944-7-4 (CD) tribution of the queue length in reordering buffer are studied in Jun Li et al. (2010). In Zheng et al. (2010) consideration is given to mean resequencing delay for an average packet in a multipath transfer scenario where path delay is assumed to be constant but distinct from path to path. Paper Wen-Fen (2011) is devoted to analysis of the influence of traffic intensity, types and number of paths, that packets traverse, on queue size in reordering buffer. For reordering network that is represented by m parallel M |M |1 queues in Gao et al. (2012) a large deviation result is proved for the reordering buffer. All the systems and papers mentioned above deal with queueing systems that function in continuous time. But for more that a decade it is known that discrete time queueing systems mostly correspond to the discreteness of the real processes in packet networks. Significant number of papers are devoted to the study of the discrete time queueing systems but to our knowledge a little number of them analyze resequencing schemes. Among the latest papers related to resequencing one can mention Li et al. (2010) and De Clercq et al. (2010). In Li et al. (2010) authors propose a novel discrete-time priority queueing network to model selective repeat automatic repeat request protocol and study the performance of the reordering buffer in terms of the mean packet resequencing delay. A discrete-time queueing system with a single server and single queue, in which N types of packets of different priorities enter is being analyzed in De Clercq et al. (2010). Customers that enter the system during the same frame are reordered such that the high-priority customers are served first. The efficiency of this mechanism is under consideration. In discrete time model time is assumed to be slotted, i.e. consists of concatenation of fixed length intervals (slots). Events are constrained to take place during these slots. A discrete time queue may accept more than one packet during a slot and service more than one packet during a slot. That is multiple event may occur during each slot (which cannot happen in continuous time case). This what makes analysis of discrete time systems more complicated and sometimes intractable, requiring development of special computationally efficient methods and utilization of high performance simulation. In this paper we build and analyze system with reordering buffer analogous to the one studied in Takine et al. (1994) but functioning in discrete time. The main contributions of this paper are algorithms for obtaining main stationary characteristics of the system. The paper is organized as follows. In the next section we give detailed description of the system under consideration. Then we analyze stationary sojourn time distribution and after that joint stationary distribution. The last section is devoted to illustrative numerical results. DESCRIPTION OF THE SYSTEM Without loss of generality we normalize the length of a slot to unit time. Slots are sequentially numbered by nonnegative integers so that slot n is located in time interval [n−1, n), n = 1, 2, . . . . During one slot events occur in the following sequence (see, e.g. Bruneel et al. (1993)): the customer whose service is completed in slot n leaves the system at instant (n− 0); a customer is chosen from the queue at instant n and immediately taken for service; a customer whose generation was completed in the n-th cycle arrives at the system at instant (n + 0) (figure 1). Such convention corresponds to Late arrival system rule (LAS). Figure 1: Discrete-time conventions Consider a discrete-time queueing system with two servers and infinite waiting room (buffer) for customers that wait for service. Customers arrive at the system and obtain sequential number. 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. 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 second server have greater sequential number it leaves the system. Otherwise it occupies one place in the reordering buffer which has infinite capacity. Customer from reordering buffer leaves it if and only if its sequential number is less than sequential numbers of all other customers present in the buffer, two servers and reordering buffer. That is customers may leave reordering buffer in groups (figure 2). The departure of customers from reordering buffer (if any) happens immediately after the moment of service completion. Service of customers can start only at slot boundaries. This means that, when the system is empty at the beginning of a slot, newly arriving customer will enter server but its service will start not in this but in the next time slot. Without loss of generality we assume that if at some time slot j there are no customers in the buffer, one of the servers completes service in slot j and there is an arrival in slot j, then newly arriving customer is put into service. We also assume that only one customer may arrive at the system during each time slot. Denote the probability of customer’s arrival in a slot by a (i.e. the probability of no arrival in a slot is a = 1 − a) and the probability of service completion in a slot by b (i.e. the probability of no service completion in a slot is b = 1− b). Finally, it is assumed that service and arrival processes are mutually Figure 2: Scheme of the model independent. In Kendall’s shorthand notation the considered system system is Geo/Geo/2/∞ with reordering buffer of infinite capacity. Let the necessary and sufficient condition of stationarity ρ/2 < 1, where ρ = a/b hold for the system. STATIONARY SOJOURN TIME DISTRIBUTION Consider two customers that are being served. Henceforth we will call “early customer” one of those two customers which entered the system earlier than the other one. The latter we respectively call “late customer”. Denote by {pi, i ≥ 0} the stationary distribution of number of customers in the buffer and two servers at slot boundaries (i.e. immediately at the beginning of a random slot) and by {pi , i ≥ 0} the stationary distribution of number of customers in the buffer and two servers as seen by new arrivals (i.e. just before the arrival of a random customer). These distributions coincide with corresponding stationary distributions of number of customers in Geo/Geo/2/∞ queue i.e. are given by (see e.g. Chan et al. (1978), Rubin et al. (1991), Artalejo et al. (2003), Alfa (2010)): p0 = ab a2b (1 + ab+ abz2)p2, (1) p1 = b ab (b+ 2ab+ abz2)p2, p‘i = zi−2 2 p2, i ≥ 3, p0 = p0 + p1b+ p2b , p1 = p1b+ p22bb+ p3b , p2 = p2b 2 + p32bb+ p4b, pi = z i−2 2 p ∗ 2, i ≥ 3, where z2 is the maximum root of the equation abz + b(b+ 2ab)z − ab = 0, i.e. has the form z2 = −b(b+ 2ab) + √ b2(b+ 2ab)2 + 4aab2b 2