Simple relationships among moments of queue lengths in product form queueing networks (original) (raw)

An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions

Operations Research, 2004

For a broad class of discrete-and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some of multi-server queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the stationary distributions of the numbers in queue and in system as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with BMAP (Batch Markovian Arrival Process) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

The mathematics of product form queuing networks

ACM Computing Surveys, 1993

Markov processes that have a product form solution have become an important computer performance modeling tool. The fact that such a simple solutlon exists for seemingly complex Markov processes M surprlsmg at first encounter and can be 340 CON .

Special volume on ‘Current Trends in Queueing Theory’ of the second ECQT conference

Queueing Systems, 2017

The nine papers in this special issue have been chosen, revised, and edited for publication through a careful refereeing process. There will be a second, companion volume, with a few more papers, shortly. QUESTA also generously published a special issue of papers selected from the first ECQT meeting (vol. 82, issues 1 and 2, 2016). These special issues would not have been possible without the help of the Technical Program Committee of ECQT and other anonymous referees, and the support of Editorin-Chief Sergey Foss. The next ECQT meeting will be in Jerusalem, July 24, 2018. The papers in this volume can be grouped into three general areas. Three of the papers concentrate on extensions to single-server queueing models, three study multiserver and multi-queue models, and three focus on systems with strategic customers. Abhishek, Boon, Boxma and Núñez Queija study service systems with correlated service times. Special attention is paid to the classical single-server queue with batch arrivals and semi-Markov service times, where the sequence of service times is governed by a modulating process. The authors use generating function techniques to study the transient and stationary queue-length distributions. Numerical evidence shows B Rhonda Righter

Joint Distribution of Instantaneous and Averaged Queue Length in an M/M/1/K System

We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue. A system of ordinary dierential equations is derived for the joint stationary distribution of the instantaneous and the exponen- tially averaged queue length. The solution of the system of equations is obtained in a few special cases. Three dierent numerical approaches are presented to find the stationary distribution in the general case. Some results obtained with the numerical methods are presented and the eciency of the numerical approaches is discussed. In addition, we describe how the model can be extended to a more complex situa- tion which contains a rejection mechanism that randomly drops incoming customers with a dropping probability that depends on the current state of the averaged queue length.

Analytical Results on the Stochastic Behaviour of an Averaged Queue Length

2000

The joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue is studied. A system of ordinary dierential equa- tions is derived for the joint stationary distribution of the instantaneous and the exponentially averaged queue length. The equations are similar to those gov- erning an MMRP driven uid queue. An analytical solution to the equations is

Bounds for the mean system size in M/G/1/K-queues

Journal of Computational and Applied Mathematics, 1995

Contrary to their infinite capacity counterparts, the moments of the distribution of the number in a M/G/l/K-system cannot be determined by means of the Pollaczek-Khinchine equation. If the finite capacity K is small the distribution under study can be obtained as the steady-state probability distribution related to the transition probability matrix. For larger capacities, we derive upper and lower bounds on the mean system size in an M/G/l/K-queue for which the first two moments of the number in the system of the infinite capacity queue are known. Numerical examples for the M/D/l/l-and M/D/1/3-queues are given. * Corresponding author. 0377~0427/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0377-0427(95)00012-7

An Approximate Analytical Method for General Queueing Networks

IEEE Transactions on Software Engineering, 2000

In this paper, we present an approximate solution for the asymptotic behavior of relatively general queueing networks. In the particular case of networks with general service time distributions (i.e., fixed routing matrix, one or many servers per station, FIFO discipline), the application of the method gives relatively accurate results in a very short time. The approximate stationary state probabilities are identified with the solution of a nonlinear system. The proposed method is applicable to a larger class of queueing networks (dependent routing matrix, stations with fimite capacity, etc.). In this case, the structure of the network studied must satisfy certain decomposability conditions.

European Conference on Queueing Theory 2016

HAL (Le Centre pour la Communication Scientifique Directe), 2016

Kleinrock (1964) proposed a queueing discipline for a single-server queue in which customers from different classes accumulate priority as linear functions of their waiting time. When the server becomes free, it selects the waiting customer with the highest amount of accumulated priority at that instant, provided that the queue is nonempty. For such a queue, Kleinrock developed a recursion for calculating the expected waiting time of customers from each class. More recently, Stanford, Taylor and Ziedins (2014) took another look at this queue, which they termed the Accumulating Priority Queue (APQ), and derived the waiting time distributions for each class. Kleinrock and Finkelstein (1967) also studied an accumulating priority system in which customers' priorities increase as a power-law function of their time in the queue. They established that it is possible to associate a particular linear accumulating priority queue with such a power-law accumulating priority queue, in such a way that the expected waiting times of customers from the different classes are preserved. In this paper, we extend their analysis to characterise the class of nonlinear accumulating priority queues for which an equivalent linear APQ can be found, in the sense that the waiting time distributions for each of the classes are identical in both the linear and nonlinear systems.