Asymptotic Analysis of a State-Dependent $M/G/1$ Queueing System

Abstract

We present new asymptotic methods for the analysis of queueing systems. These methods are applied to a state-dependent $M/G/1$ queue. We formulate problems for and compute approximations to (i) the stationary density of the unfinished work; (ii) the mean length of time until the end of a busy period; (iii) the mean length of a busy period; and (iv) the mean time until the unfinished work reaches or exceeds a specified capacity. The methods are applied to the full Kolmogorov equations, scaled so that the arrival rate is rapid and the mean service is small. Thus, we do not truncate equations as in diffusion approximations. For state-independent $M/G/1$ queues, our results are shown to agree with the known exact solutions. We include comparisons, both analytic and numerical, between our results and those obtained from diffusion approximations.