Naoki Makimoto

GEOMETRIC DECAY OF THE STEADY-STATE PROBABILITIES IN A QUASI-BIRTH-AND-DEATH PROCESS WITH A COUNTABLE NUMBER OF PHASES

A sufficient condition is proved for geometric decay of the steady-state probabilities in a quasi-birth-and-death process having a countable number of phases in each level. If there is a positive number η and positive vectors x = (x i) and y = (y j ) satisfying some equations and inequalities, the steady-state probability π mi decays geometrically with rate η in the sense π mi ∼ cη m x i as m → ∞. As an example, the result is applied to a two-queue system with shorter queue discipline.

A unified approach to gi/m(n)/l/k and m(n)/g/1/k queues via finite quasi-birth-death processes

In this paper, we provide a unified approach to study GI/M/LK Queses with state-dependent services and M/G/1/K Queueswith state dependent arrivals First,we describe such queses in term of finite quasi-birth-death (QBD) arrivalS. First, describe wen this paper, we treat GI/M/l/K queues with state-dependent services and M/G/l/K queues with state-dependent arrivals. First,we describe such queues in term of finite quasi-birth-death (QBD) processes by approximating the general distribution on term of phase-type distributions. Then by solving the corresponding Matrix equation for the QBD processes and by showing that the results obtained are free from the representation we drive various existing results together with some new result for important characteistics in queuesing systems.

Computation of the quasi-stationary distributions in M(n)/GI/ 1/ K and GI/M(n)/ 1/ K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.

Stochastic minimization of the makespan in flow shops with identical machines and buffers of arbitrary size

Stochastic permutation flow shops of m identical machines and n jobs are considered. There exist buffers of arbitrary size between two consecutive machines. For particular types of job processing times, we present optimal schedules which minimize the makespan in the stochastic sense. Moreover, it will be shown that the optimal schedules are closely related to those of the no buffer case. Finally, our results are recaptured in the context of tandem queues.

Quasi-stationary distributions in a PH/PH/c queue

Explicit representations of the quasi-stationary (QS) distributions in a multi-server phase-type queue are obtained by extending the results for a single server queue in Kijima (1993). Two types of QS distributions are treated, one for fully busy periods and one for partially busy periods. It is shown that these distributions are given as positive solutions of vector equations where Q m is the lossy generator governing the queueing process with at least m customers and γ m is its decay parameter. We first develop the method to determine γ m , and then we obtain explicit expressions for the QS distributions. It turns out that the QS distribution for partially busy periods has a matrix-geometric structure in some cases. By investigating the asymptotic behaviors of these distributions, it is also shown that the QS distributions have longer tails than the stationary distribution

Quasi-Stationary Distributions of Markov Chains Arising from Queueing Processes: A Survey

There are a variety of random processes that evanesce either by dying out or by exploding, yet the time it takes for this to occur is very long, and over any reasonable time, these processes reach some apparent equilibrium. For example, in some chemical processes, there are a number of reactions in which one or more species can become exhausted, yet these reactions appear to settle down quickly to a stable equilibrium (see, e.g., Parsons and Pollett [49] and Pollett [52]). The quasi-stationary distribution (QSD) is a stationary distribution conditioned to stay in some states of interest and has proved to be a potent tool in modeling and analyzing such phenomena. The idea can be traced back to Yaglom [69] (see Pollett and Stewart [53] for the history and references applied to a variety of contents).

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

AbstractThis paper studies the geometric decay property of the joint queue-length distribution {p(n1,n2)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c1,c2,d1 and d2, an upper bound

On interchangeability for exponential single-server queues in tandem

\overline{\eta}(c_{1},c_{2})

An efficient algorithm for finding the minimum norm point in the convex hull of a finite point set in the plane

of the decay rate is derived in the sense