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Maximum queue lengths during a fixed time interval in the M/M/c retrial queue
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DOI：10.1016/j.amc.2014.02.074.png)
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Abstract
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We are concerned with the problem of characterizing the distribution of the maximum number Z(t(0)) of customers during a fixed time interval [0, t(0)] in the M/M/c retrial queue, which is shown to have a matrix exponential form. We present a simple condition on the service and retrial rates for the matrix exponential solution to be explicit or algorithmically tractable. Our methodology is based on splitting methods and the use of eigen-values and eigenvectors. A particularly appealing feature of our solution is that it allows us to obtain global error control. Specifically, we derive an approximating solution p(x; t(0)) = p(x; t(0); epsilon) verifying [P(Z(t(0)) <= x vertical bar X(0) = (i,j)) - p(x; t(0))] < epsilon uniformly in x >= i + j, for any epsilon > 0 and initial numbers i of busy servers and j of customers in orbit. (C) 2014 Elsevier Inc. All rights reserved.
Keywords:
Absorbing Markov chain
Eigenvalues/eigenvectors
Maximum queue length
Retrial queue
Splitting method
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