An Interpolation Approximation for Queueing Systems with Poisson Input

For many queueing systems of practical interest, it is difficult to obtain exact analytical results. It is, however, often possible to obtain asymptotic results for light and heavy traffic. Heavy traffic limit theorems yield expressions for normalized quantities of interest. In light traffic, we can obtain, in addition to limits, more sensitive information by calculating what are effectively derivatives (of the quantity of interest) with respect to the arrival rate. We can then combine the light and heavy traffic results to yield a polynomial (in the arrival rate) as an approximation to the normalized quantity of interest. For instance, by utilizing the heavy traffic limit, the light traffic limit, and the first derivative, we can obtain a quadratic approximation. Then, by reversing the normalization process, we can obtain an approximation for the original quantity of interest. In this paper we present the details of the above approximation, focusing, via several examples, on applications of the method. We then compare the results of the approximation, for the examples considered, with exact and simulation results. In addition, we compare our results to some related approximations. For the examples considered, the approximation works extremely well.