Linear Transformations between Embedded Processes Associated with M/M/1 Queueing Systems

Toshikane ODA  Aurel A. LAZAR  

Publication
IEICE TRANSACTIONS on Communications   Vol.E75-B   No.12   pp.1308-1314
Publication Date: 1992/12/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8516
Type of Manuscript: Special Section PAPER (Special Issue on Teletraffic)
Category: 
Keyword: 
Markovian queueing systems,  linear transformations,  embedded Markov chains,  arrival and departure theorems,  

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Summary: 
The embedded Markov processes associated with Markovian queueing systems are closely related, and their relationships are important for establishing an analytical basis for performance evaluation techniques. As a first step, we analyze the embedded processes associated with a general M/M/1 queueing system. Linear transformations between the infinitesimal generators and the transition probability matrices of embedded processes at arrival and departure times are explicitly derived. Based upon these linear transformations, the equilibrium distributions of the system states at arrival and departure times are obtained and expressed in terms of the equilibrium distribution at arbitrary times. The approach presented here uncovers an underlying algebraic structure of M/M/1 queueing systems, and establishes an algebraic methodology for analyzing the equilibrium probabilities of the system states at arrival and departure times for more general Markovian queueing systems.