Matrix-Geometric Solution for Two Node Tandem Queueing Systems with Phase-Type Servers Subject to Blocking and Failures.

Abstract

A two node tandem queueing system with phase-type servers and Bernoulli arrivals is considered in discrete-time when servers are subject to bloclring and failuree. The invariant probability vector of the underlying finite state Quasi-Birth-and-Death process is shown to admit a matrix-geometric representation for all values of the arrival rate A. The corresponding rate matrix is given explicitly in terms of the model parameters and the resulting closed-form expression provides the basis for an efficient calculation of the invariant probability vector. The cases LAMBDA = 1 and LAMBDA < 1 are studied separately and the irreducibility of the underlying Markov chain is discussed for each case. The continuous-time formulation is briefly discussed and only major differences with the discrete-time results are pointed out. Some numerical examples are also provided.