On the accuracy of approximating loss probabilities in finite queues by probabilities to exceed queue levels in infinite queues

Abstract

Finite queues are often hard or impossible to analyze in an exact analytical way. In such cases, the often preferred solution is to approximate the loss probability in a finite queue by the probability that the corresponding infinite queue (with the same service process) exceeds the finite queue length. This approximation is attractive since infinite queues are often amenable to exact analytical solutions. Little work has been done, however to investigate the accuracy of this approximation, especially in the light of long range dependence (LRD) (or self-similarity) which is present in today's data network traffic. We investigate the accuracy of the above-mentioned approximation for a number of different arrival processes, finite queue lengths, and load situations. We compare exact results for the finite queues with approximate results obtained by the corresponding infinite queues. We consider M/G/1, discrete-time GI/G/1, discrete-time AR(1)/D/1, and FBM/D/1 queues. Exact analytical solutions for the finite and infinite M/G/1 and discrete-time GI/G/1 queues are readily available. For the discrete-time AR(1)/D/1 queue, we developed a new solution method that determines the exact loss probabilities in both the finite and infinite queue cases. There does not yet exist an exact solution for the finite fractional Brownian motion (FBM) queue. Thus, we relied on simulation results for the finite and infinite FBM/D/1 queues. The numerical results show that the approximation is slightly on the conservative side for the non-LRD arrival processes (M/G/1, discrete-time GI/G/1, and discrete-time AR(1)/D/1) as long as the load and the loss probabilities are low. For higher loads (and loss probabilities) the approximation overestimates the exact loss probability by several orders of magnitude.