Sojourn time distributions in the queue defined by a general QBD process

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.

Sojourn times in fluid queues with independent and dependent input and output processes

“…The derivations in the dependent case are similar to the ones provided by Ozawa in [4] for the discrete queues, resulting in an order N · N + phase-type distribution for the sojourn time.…”

“…In [4] it is also proven that the sojourn time distribution has a phase-type representation if some conditions hold. In this section we perform similar analysis steps as in [4] for continuous queues in order to derive similar results for the sojourn time distribution of MFQs.…”

“…The sojourn time distribution of QBD queues is known to be matrix-exponential of order N 2 [4]. In [4] it is also proven that the sojourn time distribution has a phase-type representation if some conditions hold.…”

“…A lot of results are available for such systems in case of discrete jobs: elegant and numerically efficient procedures to obtain the matric-geometric distribution of the queue length [3] and the matrix-exponential distribution of the sojourn time [4]. The introduction of the age process based analysis of discrete queues [5,6,7] made it possible to obtain a lower order matrix-exponential representation of the sojourn time distribution if the arrival and service processes are independent.…”

Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues

“…On a dierent line of research Ozawa studied the sojourn time distribution of a class of so-called Quasi-Birth-Death (QBD) queues [14] and proved that it has a matrix exponential representation of order N 2 , where N is the size of the background continuous time Markov chain. As the class of MAP/MAP/1 queues forms a subclass of the set of QBD queues (with N equal to the product of the number of phases of the arrival and service MAP), the result of Ozawa gives rise to an order N 2 representation for the sojourn time distribution of a MAP/MAP/1 queue.…”