Other Descriptors
This chapter ends up the discussion on performance descriptors in retrial queues through various measures of intrinsic interest, but less studied in the literature. In Section 6.1, we give an alternative Markovian description of the M/G/1 retrial queue th
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In this last chapter, we turn the attention to the GI/M/1 and M/G/1 structures. In order to illustrate the former, we present an exhaustive analysis of the Geo/Geo/c retrial queue, which includes the study of the stationary distribution of the system state, the busy period and the waiting time analysis. In Section 9.2, we discuss the BM AP/SM/1 retrial queue. We first analyze the system state at the departure epochs. Then, we extend the study at any arbitrary time.
9.1 The Geo/Geo/c Retrial Queue In Section 3.3, we studied the limiting probabilities of the Geo/G/1 retrial queue. The aim of the present section is to investigate the Geo/Geo/c queue with geometric retrials; that is, the discrete-time analogue of the M/M/c retrial queue. We propose several algorithmic procedures for the efficient computation of the main performance measures. More specifically, we focus on the stationary distribution of the system state, the busy period and the waiting time. Although the efficient computational analysis of non-homogeneous GI/M/ 1-type processes remains still an open problem, we shall show that it is possible to exploit the matrix structure and sparsity of the underlying blocks to proceed algorithmically. There are several methods for an efficient solution of the stationary probabilities and the rate matrix R. For the sake of brevity, we present in detail only those methods that we use for obtaining the numerical results. 9.1.1 Stationary Distribution of the System State We start with the description of the mathematical model. We assume that the time axis has been divided into equal intervals of unit length called slots. In the Geo/Geo/c retrial queue, primary customers arrive to the system according
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9 Selected Retrial Queues with GI/M/1 and M/G/1 Structures
to a geometric arrival process with probability p. We denote the complementary probability 1 − p by p¯. Service is rendered by c identical servers with service times geometrically distributed with probability mass function q q¯k−1 , for k ≥ 1, where q¯ = 1 − q. This means that any busy server finishes the undergoing service in the next slot with probability q. Customers in orbit behave independently of each other and retry with probability s at every time slot; that is, their retrial times are independent geometric random variables with probability mass function s¯ sk−1 , for k ≥ 1, where s¯ = 1 − s. Any arriving customer who finds a free server begins immediately to be served. Otherwise, he joins the orbit and reapplies for service later.
Fig. 9.1. Various epochs in G-EAS
We consider a G-EAS policy in which all events occur around the slot boundaries. We recall that, at a given slot boundary t, departures occur in (t−, t), while primary arrivals and retrials occur in (t, t+), see Figure 9.1. For analyzing the waiting time distribution, we must be even more specific about which customers are selected for service at a given slot. If we have a primary arrival and/or many retrials that exceed the number of free servers, then we assume that primary arrivals have priority
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