Stochastics and Statistics Calculation of delay characteristics for multiserver queues with constant service times Peixia Gao, Sabine Wittevrongel * , Joris Walraevens, Marc Moeneclaey, Herwig Bruneel Department of Telecommunications and Information Processing (TELIN), Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium article info Article history: Received 28 June 2007 Accepted 21 October 2008 Available online 29 October 2008 Keywords: Queueing Discrete time Multiple servers Constant service times Delay analysis abstract We consider a discrete-time infinite-capacity queueing system with a general uncorrelated arrival pro- cess, constant-length service times of multiple slots, multiple servers and a first-come-first-served queueing discipline. Under the assumption that the queueing system can reach a steady state, we first establish a relationship between the steady-state probability distributions of the system content and the customer delay. Next, by means of this relationship, an explicit expression for the probability gener- ating function of the customer delay is obtained from the known generating function of the system con- tent, derived in previous work. In addition, several characteristics of the customer delay, namely the mean value, the variance and the tail distribution of the delay, are derived through some mathematical manipulations. The analysis is illustrated by means of some numerical examples. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Discrete-time queueing models have received considerable attention during the past years, see e.g. the books [1,5,12,14, 16,18] and the references therein. A main reason is the applicabil- ity of these models in the performance evaluation of packet-based high-speed telecommunication networks, where buffers are used for the temporary storage of information packets which cannot be transmitted to their destination immediately. The information packets then constitute the customers of the queueing system and the transmission of packets corresponds to the service of cus- tomers. In discrete-time queueing models, the time axis is divided into fixed-length intervals, referred to as slots, and the service of customers can start or end at slot boundaries only. The latter im- plies that the service times of the customers consist of an integer number of slots. Usually, the performance of a queueing system is expressed in terms of such quantities as the system content (i.e., the total num- ber of customers present in the queueing system) and the delay of a customer (i.e., the time (in slots) spent by a customer in the sys- tem). Especially when multimedia applications in packet-based networks are concerned, it is important to be able to accurately predict the characteristics of the packet delay, such as the mean delay and the delay jitter, in order to guarantee acceptable delay boundaries for the admitted network traffic. The analysis of delay characteristics in the current internet thus is an important research topic. There are a number of performance analysis techniques for discrete-time systems, ranging from computer simulation to the numerical solution of the associated set of balance equations and various types of analytical methods. Computer simulation often suffers from long run times and requires a new run for each param- eter setting. Hence, for performance engineering purposes, analyt- ical methods are preferred [11], since these lead to closed-form expressions for the performance measures of interest and therefore allow a fast performance prediction. If we focus our attention on analytical performance studies, many results have been obtained for both the system content and the customer delay in a single-server environment. In case sys- tems with multiple servers are considered, fewer analytical results are however available, although such systems occur in many prac- tical applications, for instance, in output-buffering switches in the nodes of packet-based networks (see Section 6 for more details). Most studies of multiserver systems assume constant service times equal to one slot, see e.g. [2 and 17]. Multiserver systems with geo- metrically distributed service times have been considered in [7–9 and 15]. In [3 and 4], discrete-time queueing models with multiple servers and constant service times of multiple slots have been studied, but only results in connection with the system content have been derived. This deterministic service-time distribution has several applications, for instance, in the performance analysis of packet switches with a different internal and external transfer mode, as explained in [3]. In this paper, we will extend the analysis of [3] in order to investigate the characteristics of the delay, which is one of the most important performance metrics from a user perspective [11]. First, a relationship between the steady-state probability distributions of the customer delay and the system content is 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.10.017 * Corresponding author. Tel.: +32 9 264 89 01; fax: +32 9 264 42 95. E-mail address: sw@telin.UGent.be (S. Wittevrongel). European Journal of Operational Research 199 (2009) 170–175 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor