Abstract
We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.
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Ozawa, T. Sojourn time distributions in the queue defined by a general QBD process. Queueing Syst 53, 203–211 (2006). https://doi.org/10.1007/s11134-006-7651-3
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DOI: https://doi.org/10.1007/s11134-006-7651-3