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 (e.g., loss‐delay systems). © 1979 The Bell System Technical Journal