Читать книгу Queueing Theory 1 - Nikolaos Limnios - Страница 13

Consider the simple Geo/Geo/1 system where the probability of arrival is a and of service completion is b. Let Xn(a, b), n = 0,1,2, ∙∙∙ be the number in the system at time n for a specified pair (a, b), with for the case of a stable system. Based on the results in Alfa (2016), the associated discrete time Markov chain (DTMC) has the transition matrix P written as

For a stable system with , we have

and

where is the expected value of .

Next we consider the cases of a = f(b). We require that the following condition applies all the time in this section:

[1.1]

We consider two main cases: one when f(∙) is a decreasing function and the second one when it is an increasing function.