Chapter 4Stationary Queues
In Chapters 8 and 9, it is assumed that the generic distributions of the sequences (ξn, n ∈ N) and (σn, n ∈ N) of service and inter-arrival times are exponential. This allows us to represent many of the models by Markov processes in continuous time. Many quantitative results can then be obtained regarding the performances of these systems.
Unfortunately, this hypothesis is unrealistic in many cases, and we are led to consider sequences of random variables which are independent and identically distributed with general distributions (GI/GI/… queues). The architectures of the systems under consideration often lead to further weaken these hypotheses. In fact, a queue often models the traffic in a node that is integrated within a network, and it is desirable that the probabilistic characteristics of a queue are the same as that of the following one, in other words that the input traffic of a queue be the same type as the output traffic. However, aside from the particular case where the input is Poissonian (then, the output is also Poissonian - see Theorem 8.8), it is easy to see that the inter-arrivals time in the second queue (which are the intervals between the departure times from the first queue) are not independent in general, even if the inter-arrivals in the first queue are independent, since their order, for example, depends on the order of service in the first queue.
It is therefore of crucial interest to consider queuing models where stationarity, ...