Transient solution to the time-dependent multiserver Poisson queue

Abstract

We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express pĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in pĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.

Transient and periodic solution to the time-inhomogeneous quasi-birth death process

“…We also derive formulae for the moments of the process as a function of time within the period. This is an extension and generalization of results obtained for the transient solution for the time-dependent Poisson queue with exponential service given in [17] and the asymptotic periodic number in an M t /M t /c t queue in [18].…”

“…Other references include [17,25] (transient case), and [18] (periodic asymptotic). We derive formulae for the idle probability, the mean and the variance for the general time-varying single server Poisson queue and then apply them to an example used in the recent paper by Zeifman et al [24].…”

“…, c, and more generally, the ith component of the vector p k (t) will give the probability that there are ck + i − 1 in the system (waiting or in service). The ith component of the vector q (1) 0 (t) gives the probability that the number in the system is equal to ck + i for some nonnegative integer k. Solving for performance measures for the time-varying multi-server queue can be done more efficiently using methods outlined in [17] and [18] which involve scalar integral functions, but we provide a numerical example for the M t /M t /2 queue using the method of this paper for the purpose of illustration.…”

Performance analysis of time-dependent queueing systems: Survey and classification

“…A multi-server MðtÞ=MðtÞ=c system is analyzed by Margolius [154]. Margolius [155] derives integral equations for the probability distribution of jobs in an MðtÞ=MðtÞ=cðtÞ system. By considering quasi-birth-and-death processes, Margolius [156] generalizes her results to phase-type distributions and establishes a connection with matrix analytic methods [157].…”