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GIMIl-FCFS and GIGI&FCFS queueing models N Chapters 5 and 6 we have addressed queues with generally distributed service time distributions, but still with Poisson arrivals. In this chapter we focus on queues with more general interarrival time distributions. In Section 7.1 we address the GlMll queue, the important “counterpart” of the MIGil q ueue. Then, in Section 7.2, we present an exact result for the GIG11 q ueue. Since this result is more of theoretical than of any practical interest, we conclude in Section 7.3 with a well-known approximate result for the GIG11 queue. . | Performance of Computer Communication Systems A Model-Based Approach. Boudewijn R. Haverkort Copyright 1998 John Wiley Sons Ltd ISBNs 0-471-97228-2 Hardback 0-470-84192-3 Electronic Chapter 7 G M 1-FCFS and G G 1-FCFS queueing models IN Chapters 5 and 6 we have addressed queues with generally distributed service time distributions but still with Poisson arrivals. In this chapter we focus on queues with more general interarrival time distributions. In Section 7.1 we address the G M 1 queue the important counterpart of the M G 1 queue. Then in Section 7.2 we present an exact result for the G G 1 queue. Since this result is more of theoretical than of any practical interest we conclude in Section 7.3 with a well-known approximate result for the G G 1 queue. It should be noted that most of the exact results presented in this chapter are less easy to apply in practical performance evaluation. For a particular subclass of G G 1 queueing models namely those where the interarrival and service times are of phase-type easy-applicable computational techniques have been developed known as matrix-geometric techniques. These techniques will be studied in Chapter 8. 7.1 The G M 1 queue For the analysis of the G M 1 queue one encounters similar problems as for the analysis of the M G 1 queue. As before the state of the G M 1 queue consist of two parts a continuous and a discrete part since the state is given by the number of customers in the system and the time since the last arrival. Unfortunately the intuitively appealing method of moments followed for the M G 1 queue based on average values the PASTA property and knowledge about the residual service time cannot be used in this case because the PASTA property does not hold. Instead we will simply state a number of important results and discuss 134 7 G M 1-FCFS and G G 1-FCFS queueing models their meaning in detail. We first have to define some notation. The interarrival time distribution is denoted Fyi i and has as first moment
