Sojourn Times in Markov Queueing Networks: Little's Formula Revisited.
MICHIGAN UNIV ANN ARBOR COMPUTER INFORMATION AND CONTROL ENGINEERING PROGRAM
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It is commonly supposed that L lambda W applies to almost any queueing system, with lambda the mean customer entrance rate, L the asymptotic expectation of the number of customers in the system, and W the asymptotic sojourn time expectation. We study the formula for irreducible positive recurrent Markov queueing systems whose state vector Z consists of entries representing queue lengths at the respective service stations such a model permits blocking, finite capacities, jockeying, state-dependent or random routing, bulk andor Erlang service, and variable arrival and service rates. To define waiting times under various queueing disciplines, Z is augmented by a customer location process to yield the new Markov process Y Z,U. It is shown that the standard regenerative process proof of Littles equality fails in the absence of further hypotheses however, additional assumptions assure the validity of L lambda W for a broad variety of queueing disciplines.
- Statistics and Probability