Abstract
Two decompositions are established for the probability transition function of the queue length process in the M/M/1 queue by a simple probabilistic argument. The transition function is expressed in terms of a zero-avoiding probability and a transition probability to zero in two different ways. As a consequence, the M/M/1 transition function can be represented as a positive linear combination of convolutions of the busy-period density. These relations provide insight into the transient behavior and facilitate establishing related results, such as inequalities and asymptotic behavior.
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Abate, J., Kijima, M. & Whitt, W. Decompositions of theM/M/1 transition function. Queueing Syst 9, 323–336 (1991). https://doi.org/10.1007/BF01158470
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DOI: https://doi.org/10.1007/BF01158470