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

The purpose of this paper is to illustrate the usefulness of statedependent birth and death processes in reducing the dimensions of 557 stochastic systems. The principal application is the reduction of a twodimensional retrial model, proposed by R. I. Wilkinson and R. C. Radnik, 1 to an approximate one-dimensional model, which is then readily solved. While algorithms 2 exist for the numerical solution of the two-dimensional Wilkinson-Radnik model, the large number of states that are often needed can result in convergence difficulties. Section II presents the history of and motivation for the techniques used throughout the paper. The equations for the reduced one-dimensional retrial model are described in Section III. Their solution is discussed in Section IV. Section V contains numerical results and comparisons, and Section VI discusses the theoretical accuracy of our one-dimensional approximation. The main points of this paper are the dimension reduction of stochastic models using a state-dependent birth-and-death process and a method of solution of the resulting approximate model. However, the retrial problem considered here as an example is of interest in itself and has been extensively studied in the past. L. Kosten' and J. Riordan 4 considered retrials coming back in a secondary, uncorrelated Poisson stream. J. W. Cohenr> allowed for negative exponential distributions in the interarrival times of calls, the holding times of calls, the duration of the time interval between two successive attempts by a subscriber whose call was blocked at the first attempt, and the time during which a subscriber continues to make repeated attempts.