Report Number: CSL-TR-77-129
Institution: Stanford University, Computer Systems Laboratory
Title: On accuracy improvement and applicability conditions of diffusion approximation with applications to modelling of computer systems
Author: Yu, Philip S.
Date: January 1977
Abstract: Starting with single server queueing systems, we find a different way to estimate the diffusion parameters. The boundary condition is handled using the Feller's elementary return process. Extensive comparisons by asymptotic, simulation and numerical techniques have been conducted to establish the superiority of the proposed method compared with conventional methods. The limitation of the diffusion approximation is also investigated. When the coefficient of variation of interarrival time is larger than one, the mean queue length may vary over a wide range even if the mean and variance of interarrival time are kept unchanged. The diffusion approximation is applicable under the condition that the high variation of interarrival time conducted on 2-stage hyperexponential distributions. A similar anomaly is observed in two server closed queueing networks when the service time of any server has a large coefficient of variation. Again, a similar regularity condition on the service time distribution is required in order for the diffusion approximation to be applicable. For general queueing networks, the problems become more complicated. A simple way to estimate the coefficient of variation of interarrival time (when the network is decomposable) is proposed. Besides the anomalies cited before, networks under certain topologies, such as networks with feedback loops, especially self loops, can not be decomposed into separate single servers when the coefficient of variation of service time distributions become large, even if the large variations are due to a large number of short service times. Nevertheless, the decomposability of a network can be improved by replacing each server with a self loop by an equivalent server without a self loop. Finally, we consider the service center with a queue dependent service rate or arrival rate. Generalization to two server closed queueing networks where each server may have a self loop is also considered.