In the range of single server queues that can be studied with the stochastic tools introduced in this book, after the M/M/1 queue which will be discussed in Chapter 8, the following ones in term of generality are the GI/M/1 and M/GI/1 queues. In the latter, the inter-arrival times (respectively, service times) are independent and identically distributed, but not necessarily of exponential distribution.
Unfortunately, in both cases the process counting the number of customers in the system is no longer Markov. In fact, at a given time we cannot repeat the argument of example 7.1, as the exponential distribution is the only one that satisfies Theorems 6.6 and 7.3.
In order to circumvent this difficulty, the system is observed in discrete time, at instants suitably chosen. In the case of the M/GI/1 queue, these are the departure times of the customers. To keep this chapter as simple as possible, we only address the embedded Markov chain, which gives the most important results.
Let us recall the main notation concerning this queue. The arrivals form a Poisson process of intensity λ and the service times are independent and of the same distribution Pσ, of mean 1/μ. We denote ρ = λ/μ the traffic load. For all n ≥ 1, let σn be the service time of customer n and Xn be the number of customers in the system just after the departure of customer n. We have seen in the introduction that Xn satisfies the recurrence equation ...