Abstract—The transient analysis of a queuing system with fixed-size batch Poisson arrivals and a single server with exponential service times is presented. The focus of the paper is on the use of the functions that arise in the analysis of the transient behaviour of the queuing system. These functions are shown to be a generalization of the modified Bessel functions of the first kind, with the batch size B as the generalizing parameter. Results for the case of single-packet arrivals are obtained first. The similarities between the two families of functions are then used to obtain results for the general case of batch arrival queue with a batch size larger than one. Keywords—batch arrivals, generalized Bessel functions, queue transient analysis, time-varying probabilities. I. INTRODUCTION HE transient analysis of a queuing system with fixed-size batch Poisson arrivals and a single server with exponential service times is presented. One can envision messages arriving at a buffer in fixed size batches of packets and released one packet at time [1]. Transient performance measures for queues have long been recognized as being complementary to the steady-state analysis [2-5], and justifications for transient analysis of traffic in telecommunication systems abound [6]. There often exists a need to understand the initial behaviour of a system. In general queuing systems, arrivals at a service point (e.g. a switch) may occur in batches of different sizes. Depending on the network conditions, the arrivals may be queued for later forwarding. Additionally, there may be cases where the network traffic is diverted suddenly to cope with faults, as in automatic protection switching in which the transport system re-directs traffic when faults and failures occur in subcomponents of the network. In such cases a service point may experience a sudden increase in its load, and this may continue until the original fault has been cleared. After the fault is corrected, traffic reverts to the previous distribution, and this presents another perturbation in the network. There exists now a reduction in the load in parts of the network. The restored element experiences an increased load transient starting from an empty state. This scenario presents the service points with transient conditions that require the kind of analysis attempted here and elsewhere [7]. Numerical inversion of Laplace transforms or generating functions have been used to obtain the transient behaviour of queuing systems [8,9]. Other methods that are equally V. K. Oduol is with the Department of Electrical and Information Engineering, University of Nairobi, Nairobi, Kenya (+254-02-318262 ext.28327, vkoduol@uonbi.ac.ke) Cemal Ardil is with the National Academy of Aviation, Baku, Azerbaijan time-consuming are based on recursive computations. What is presented here is a method that uses a family of functions arising in the analysis of the queuing system. It is shown that these functions are related to the modified Bessel functions of first kind. The relevant properties of the Bessel functions are available in [10] from which a select few are taken for the purposes of this paper. The relationship between the function families is exploited to obtain the results presented. The main key in this exercise is the fact that the empty system probability can be obtained explicitly in closed form for single-packet arrivals. The analysis proceeds here by first considering single-packet Poisson arrivals, i.e. a batch size of one (B=1). Results for this case can also be found in [11]. In Appendix A the present paper includes additional steps to those of [11] and then proceeds in Appendix B to the general case (B > 1) by using the inherent correspondence between the modified Bessel functions of the first kind and the batch arrival functions. Incidentally this analysis provides a method of implicitly solving the integral equation that is encountered, which elsewhere [7] is solved by a combination of modelling and signal processing. There are other methods [12-14] of solution to such integral equations. The paper is organized as follows. Section II presents the system model, and sets the main point of reference for the rest of the presentation. Section III presents a discussion of the time-varying probabilities that in the limit of large t eventually yield the steady state results. The section relies on the derivations given Appendix A and Appendix B concerning the relationships between the function families. Section IV presents the expressions for the queuing system statistics needed to assess performance. Section VI gives the results and conclusion. Appendix A gives a detailed derivation of the relevant results for single-packet arrivals (B=1), and Appendix B relies on the results of Appendix B and proceeds to the general case B>1). The appendices carry material that is necessary to support the development of the paper but which, if placed in the main body, would compromise readability. II. SYSTEM MODEL Packets arrive at a service point in fixed size batches of B packets according to a Poisson process of mean rate arrivals per second. The single server completes the service at the rate of packets per second. The probability flow rates are as shown in Fig.1 for a batch size of B packets. Denote by P k (t) the probability that there are k packets in the system at time t. The probability flow balance equations are given by Transient Analysis of a Single-Server Queue with Fixed-Size Batch Arrivals Vitalice K. Oduol and Cemal Ardil T World Academy of Science, Engineering and Technology International Journal of Electrical and Computer Engineering Vol:6, No:2, 2012 253 International Scholarly and Scientific Research & Innovation 6(2) 2012 scholar.waset.org/1307-6892/6091 International Science Index, Electrical and Computer Engineering Vol:6, No:2, 2012 waset.org/Publication/6091