Analysis of the M/Gi/1 →/M/1 queueing model

Abstract

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The cyclic queue and the tandem queue

“…Ω(y) is shown to satisfy a Fredholm integral equation of the second kind ((4.20) in [2]), and Ω(y) is determined in Section 5 of [2]. First of all, we conclude that (cf.…”

“…The arrival process at Q G and the service times at Q G and Q M are all independent. Blanc, Iasnogorodski and Nain [2] have determined the (transform of the) steady-state joint queue length distribution for this tandem queue, and in particular also the (transform of the) probabilities P(X G = i, R ∈ (t, t + dt), X M = j), where X G (X M ) denotes the steady-state queue length in Q G (Q M ) and R denotes the residual service time of the customer in service at Q G . By PASTA, this joint distribution is the same at an arrival epoch of a customer in Q G .…”

“…This is well known, and follows from the reversibility of the queue length process in the first queue, which now is an M/M/1 queue (and its output process is a Poisson process, turning the second queue into an M/G/1 queue). The paper of Blanc et al [2] filled an important gap in the classical queueing literature by determining the joint queue length distribution in the M/G/1 − ·/M/1 queue; we aim to fill another gap in that literature by determining the joint sojourn time distribution. Our starting-point is the following expression for the joint sojourn time LST, which is obtained by conditioning on X a G and X a M :…”

“…(ii) Ω(y) is not explicitly given in [2]; the authors of [2] only need the real part of Ω(y) on a circle, but there is analytic continuation. (iii) In (5.5) we need the real part of ω G − ω M to be non-negative.…”

“…We evaluate T 31 in a similar way: Finally, using the definition in (5.8), we can write: This LST is expressed in contour integrals of terms which are known: A 1 (x, ω G , ω M ) and A(x, y, ω G , ω M ) were derived in the previous section, while Ω(y) and Φ(r 1 , ω, r 2 ) are in principle known from [2]. Evaluation of the contour integrals that appear in the joint LST expression can be done explicitly once the service time LST is specified.…”