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Fundamentals of Stochastic Networks by Oliver C. Ibe

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6

APPROXIMATIONS OF QUEUEING SYSTEMS AND NETWORKS

6.1 INTRODUCTION

The exact solutions of many queueing models are available only under restrictive assumptions that are usually not consistent with real life, thereby making approximate solutions a practical necessity. Most of the proposed approximate solutions are based on heavy traffic assumption. That is, the system is operating at such a point that the server is almost continuously busy. Alternatively, the arrival rate of customers is almost equal to the service rate.

Two of the widely used heavy traffic approximate solution techniques are the fluid approximation and the diffusion approximation, both of which are used to provide conversion from discrete state space to continuous state space. The fluid approximation of a queueing system attempts to smooth out the randomness in the arrival and departure processes by replacing these processes with continuous flow functions of time. Thus, fluid approximation can be considered as a first-order approximation that replaces the arrival and departure processes by their mean values, thereby creating a deterministic continuous process.

The diffusion approximation attempts to improve on the fluid approximation by permitting the arrival and departure processes to fluctuate about their means. Thus, it is a second-order approximation that replaces a jump process N(t) with a continuous diffusion process X(t) whose incremental changes dX(t) = X(t + dt) – X(t) are normally distributed with finite ...

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