A Diffusion Model for Queues in Randomly Varying Environment

Haruhisa TAKAHASHI  Haruo AKIMARU  

Publication
IEICE TRANSACTIONS (1976-1990)   Vol.E69   No.1   pp.13-20
Publication Date: 1986/01/25
Online ISSN: 
DOI: 
Print ISSN: 0000-0000
Type of Manuscript: PAPER
Category: System Theory
Keyword: 


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Summary: 
Consider the diffusion approximation for queues in randomly varying environment. The environmental variation may be the changes of arrival rates and/or changes of service rates caused by changes in message transmission under occasional crisis. The problem is to seek the queue size distribution conditional on the process being under different environment. The problem arises, for example, when we consider the delay distribution for each type of bursty customers in MG/G/1 queues, where MG represents nonhomogeneous renewal process GIi with i being two-state Markov process. To solve the problem, we define and analyze a diffusion process on R++R+ with the elementary returning boundaries. The forward equation and its stationary solutions are derived. Then it is applied to GI+IPP/G/1 queues and queues with service interruption, where IPP represents the interrupted Poisson process. Numerical examples are given in order to evaluate the accuracy of our approach. Finally a corrected diffusion parameter diminishing errors for general service time distributions is proposed.