Approximations to Stochastic Service Systems, with an Application to a Retrial Model

This paper illustrates the usefulness of state-dependent, birth-death processes in reducing the dimensions of stochastic service systems. The approximation techniques introduced have wide applicability to general (finite) multidimensional, state-dependent, birth-death processes. These techniques are introduced by considering the “classical” telephony problems dealing with trunk group overflow traffic from the point of view of state-dependent, birth-death processes. The main part of the paper then applies these techniques to a two-dimensional trunk group retrial model of Wilkinson and Radnik. The method, which reduces the W-R model to an approximate, easily-solved, one-dimensional model, makes use of the transition probabilities for state-dependent, birth-death processes. These are obtained via a simple extension of known results. We use the one-dimensional results to compute blocking for a range of parameter values (trunk group sizes and retrial rates) exceeding the computational limits of the W-R model. Maximum relative errors do not exceed 10 to 15 percent, while for most cases of practical interest the relative errors are less than 5 percent. The approximation also provides insight into the region of applicability of even simpler retrial models. This one-dimensional retrial model actually applies to more general (finite) state-dependent, birth-death processes (eg., loss-delay systems).