Stationary regime for a queue of the type M|M|c|c + m with constant retrial rate

Abstract

A bivariate Markov process {X(t) = (X₁(t), X₂(t))^T , t ⩾ 0} whose phase space is a lattice semistrip S(X) = {0,1,....,c +m}xZ is considered. The first component X1(t) ∈ {0,1,....,c+m} indicates the summarized number of busy servers and calls in the queue at the instant t ⩾ 0 , whereas the second one X₂ (t) ∈ {0, 1, ...} is the number of retrial sources. Parameter c ∈ Z₊ is a number of servers and m ∈ Z is a maximal size of the queue. Local rates of X(t) are defined in such a way that X(t) describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of repeated calls. First, using tools of the theory for the QBD-processes (quasi-birth-and-death processes), we study the ergodicity conditions. Then, under these conditions, we consider a problem of finding the steady state probabilities for X(t) . A vector-matrix representation of the probabilities via the model parameters is obtained. The applied technique uses an approximation of the initial model by a truncated one and the direct passage to the limit. The obtained formulae are the adequate method to calculate the steady state probabilities. An application of the main result is demonstrated via numerical examples in which we can see relation graphs of the blocking probability and the average number of retrials versus system parameters.