# 2Single-Server Queues Embedded at Arrival Epochs

In this chapter, we look at queueing systems embedded at arrival epochs and study them under *GI/M/*1-paradigm. Recall the basics of this paradigm from Chapter 7 of Volume 1. This chapter is organized as follows. In section 2.1, we look at the *GI/PH/*1 queue and in sections 2.2, 2.3 and 2.4, respectively, we look at the special cases, namely, *GI/M/*1, *M/PH/*1 and *M/M/*1 queues.

## 2.1. *GI/PH/*1 queue

In this section, we briefly discuss a queueing model in which the inter-arrival times are *i.i.d.* random variables with a common distribution function, say, *F* (*x*), and with probability density function, *f* (*x*), and its Laplace transform (LT) is denoted by *f*^{∗}(*s*). It is assumed that the mean and variance are finite. Let *λ* denote the average rate of arrivals per unit of time. The service times are assumed to follow a continuous phase type (CPH) distribution with representation given by (* β, S*) of order

*ν*, and with the mean given by

*(*

**β***S*)

^{−1}

*. For use in the following, let*

**e***µ*= [

*(*

**β***S*)

^{−1}

*]*

**e**^{−1}denote the average service rate. We need to look at the system at points of arrivals.

It should be pointed out that most of the fundamental ideas needed in the analysis of *GI/M/*1-queues were presented in sections 7.3 and 7.4 of Volume 1 and only the items pertaining to the queues are discussed here. Thus, the results needed for getting the expressions in the queueing context will be stated without repeating the details from those sections.

*Embedded process at arrivals ...*

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