DECOMPOSITIONS OF THE QUEUE LENGTH DISTRIBUTIONS IN THE MAP/G/1 QUEUE UNDER MULTIPLE AND SINGLE VACATIONS WITHN-POLICY

Ho Woo Lee · Boo Yong Ahn · No Ik Park

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Advanced Queuing Theory Analysis Probability and Statistical Research Probability and Risk Models

Abstract

In this paper, we establish an explicit form of matrix decompositions for the queue length distributions of the MAP/G/1 queues under multiple and single vacations with N-policy. We show that the vector generating function Y (z) of the queue length at an arbitrary time and X (z) at departures are decomposed into Y (z) = p idle (z)ζ Y (z) and X (z) = p idle (z)ζ X (z) where p idle (z) is the vector generating function of the queue length at an arbitrary epoch at which the server is not in service, and ζ Y (z) and ζ X (z) are unidentified matrix generating functions.

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References

1984Matrix-Geometric Solutions to Stochastic Models 1991New results on the single server queue with a batch markovian arrival process 1991Structured Stochastic Matrices of M/G/1 Type and Their Applications.1985Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations 1990A single-server queue with server vacations and a class of non-renewal arrival processes 1980The N/G/1 queue and its detailed analysis 1988A stable recursion for the steady state vector in markov chains of m/g/1 type 2005The BMAP/G/1 queue: A tutorial

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