Statistical queuing theory with some applications (original) (raw)
2016
Queueing Theory is one of the most commonly used mathematical tool for the performance evaluation of systems. The aim of the book is to present the basic methods, approaches in a Markovian level for the analysis of not too complicated systems. The main purpose is to understand how models could be constructed and how to analyze them. It is intended not only for students of computer science, engineering, operation research, mathematics but also those who study at business, management and planning departments, too. It covers more than one semester and has been tested by graduate students at Debrecen University over the years. It gives a very detailed analysis of the involved queueing systems by giving density function, distribution function, generating function, Laplace-transform, respectively. Furthermore, Java-applets are provided to calculate the main performance measures immediately by using the pdf version of the book in a WWW environment. I have attempted to provide examples for ...
Computing the Performance Measures in Queueing Models via the Method of Order Statistics
Journal of Applied Mathematics, 2011
This paper focuses on new measures of performance in single-server Markovian queueing system. These measures depend on the moments of order statistics. The expected value and the variance of the maximum minimum number of customers in the system as well as the expected value and the variance of the minimum maximum waiting time are presented. Application to an M/M/1 model is given to illustrate the idea and the applicability of the proposed measures.
Measures of Performance in the M/M/1/N Queue via the Methods of Order Statistics
Life Science Journal
This paper computes new measures of performance in Markovian queueing model with single-server and finite system capacity. The expected value and the variance of the minimum (maximum) number of customers in the system (queue) as well as the r th moments of the minimum (maximum) waiting time in the queue are derived. The computations of the proposed measures are depending on the methods of order statistics. The regular performance measures of the M/M/1/N model are considered as special cases of our results. [Yousry H. Abdelkader and A. I. Shawky Measures of Performance in the M/M/1/N Queue via the Methods of Order Statistics] Life Science Journal. 2012; 9(1):945-953] (ISSN: 1097-8135). http://www.lifesciencesite.com. 138
Comments on: Queueing models for the analysis of communication systems
TOP, 2014
In the article, authors restrict their attention to the class of discrete-time single-server queueing models useful for performance analysis of communication systems. This is quite natural because the leader of the author's group professor Herwig Bruneel has the great experience in analysis of discrete-time queues, see his book published twenty years ago and an impressive amount of journal and conference papers published by the authors after this book. As the authors mention, queueing models in discrete-time are very appropriate to describe traffic and congestion phenomena in digital communication systems, since these models reflect in a natural way the synchronous nature of modern transmission systems, whereby time is segmented into intervals ("slots") of fixed length and information packets are transmitted at slot boundaries only, i.e., at a discrete sequence of time points. Because it is well recognized that correlation is a typical feature of the traffic in modern telecommunication networks; authors concentrate their efforts in this paper on analysis of queues with correlation in the arrival process. Taking into account popularity of so-called BMAP (Batch Markov Arrival Process), see , for modeling correlated traffic in continuous-time queueing systems, see, e.g., ; , the authors might use the discrete counterpart of the BMAP, so-called DBMAP (Discrete Batch Markov Arrival Process) for description of the arrival process in their models. However, as authors mention in the text, in the case of the DBMAP usually it is possible to develop
Response Time Distributions in Networks of Queues
International Series in Operations Research & Management Science, 2010
The problem of computing the response (sojourn) time distribution in queuing networks has been researched extensively during the past few decades. (For a somewhat dated survey see .) In case of open queuing networks, a considerable amount of work has been done in computing the response time distribution in the domain of Jackson networks. Closed form solutions have been derived for the (Laplace-Stieltjes transform of) response time distributions through a particular path in product-form queuing networks .
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
Rejoinder on: Queueing models for the analysis of communication systems
TOP, 2014
Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks, for instance to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discretetime models come natural. We start this paper with a review of suitable discretetime queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter,. . .). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.
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.
Queueing models for the analysis of communication systems
Top, 2014
Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discretetime models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival This invited paper is discussed in the comments available
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
There are three topics in the thesis. In the first topic, we addressed a control problem for a queueing system, known as the "N-system", under the Halfin-Whitt heavy traffic regime and a static priority policy was proposed and is shown to be asymptotically optimal, using weak convergence techniques. In the second topic, we focused on the hospitals, where faster servers(nurses), though work more efficiently, have the heavier workload, and the Randomized Most-Idle (RMI) routing policy was proposed to tackle this unfairness issue, trying to reward faster servers who serve more with less workload. we extended the existing result to show that this desirable property of the RMI policy holds under a system with multiple customer classes using theoretical exact analysis as well as numerical simulations. In the third topic, the problem was to decide an appropriate number of representatives over time according to the prescribed service quality level in the call center. We examined the stability of two methods which were designed to generate appropriate staffing functions on a simulated data and real call center data from an actual bank.
Asymptotic properties of queuing networks
1997
A new approach to the analysis of asymptotic properties of closed queuing networks with both constant service rates and, in certain cases, load-dependent service rates is developed. The method is based on a decomposition of the generating function of the normalising constant into simpler node functions which are easily inverted term by term. An exact closed form is obtained for the normalising constant in some cases and an approximation, based on an integral formula, in others. These results are applied to model a large computer system with terminals, which is also used to illustrate the main properties of the normalising constant and the system throughput function as the population increases. The authors' method is compared with others in terms of both accuracy and efficiency. Finally, it is indicated how multiclass networks can be handled, essentially by reduction to a collection of single class networks.
Statistical inference for G/M/1 queueing system
Operations Research Letters, 1988
In this paper, maximum likelihood estimates of the parameters are derived for the G/M/1 queueing model with variable arrival rate. A simulated numerical example is used to illustrate its application for estimating the parameter when the interarrival time distribution is exponential. Problems of hypothesis testing are also investigated.
Queuing Theory and Telecommunications: Networks and Applications. Springer NY, 2nd edition, 2014
2014
This book provides a basic description of current networking technologies and protocols as well as important tools for network performance analysis based on queuing theory. The second edition adds selected contents in the first part of the book for what concerns: (i) the token bucket regulator and traffic shaping issues; (ii) the TCP protocol congestion control that has a significant part in current networking; (iii) basic satellite networking issues; (iv) adding details on QoS support in IP networks. The book is organized so that networking technologies and protocols (Part I) are first and are then followed by theory and exercises with applications to the different technologies and protocols (Part II). This book is intended as a textbook for master level courses in networking and telecommunications sectors.
PERFORMANCE ANALYSIS OF QUEUING AND COMPUTER NETWORKS
Humanitarian & Natural Sciences Journal, 2024
Queuing is one of the most usable tools that help in analyzing the performance of complex telecommunication and system networks. Thus, this term paper presents the performance measurements of computer networks with queuing technique. The paper covers the detail introduction of queuing theory and its various applications widely used for complex network/system environment.
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.
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.