A two-echelon multi-server markovian queueing system with a limited intermediate buffer

We consider a two-echelon multi-server tandem queueing system fed by a Poissonian stream of arrivals. There are S_k servers at stage k where service times are exponentially distributed with parameter μ_k (k = 1, 2). The first stage has an unlimited waiting capacity, while the second stage has a finite buffer of size M—S₂. Upon service completion at the first stage, a customer leaves the system with probability q, or moves on to request additional service at the second stage. If the intermediate buffer is full, the customer leaves the system with probability 1. Such a model describes, for example, a telephone information system where a customer, after being served by a regular operator, may request a higher-echelon-service from a supervisor. We formulate the problem as a two-dimensional continuous-time Markov chain and derive a set of linear equations, A(z)·II(z) = b(z), where II(z) is an M + 1-dimensional vector of unknown partial generating functions which determine the steady-state probabilities of the system. The tridiagonal matrix A(z), which contains the parameters, is transformed into an Hessenberg matrix whose determinants yield polynomials with interesting interlacing properties. These properties are exploited to obtain a set of equations in “boundary” probabilities that are required for the complete specification of the generating functions. An example and numerical results are presented.