Response Time Distributions in Networks of Queues

2012

An overview of theory of queues, single node and in network, is presented in this paper. In addition some applications are outlined. Some very well-known and others more uncommon.

Performance Evaluation, 2002

The study presented in this paper is motivated by the performance analysis of response times in distributed information systems, where transactions are handled by iterative server and database actions. We model system response times as sojourn times in a two-node open queueing network with a processor sharing (PS) node and a first-come-first-served (FCFS) node. External customers arrive at the PS node according to a Poisson process. After departing from the PS node a customer proceeds to the FCFS node with probability p, and with probability 1 − p the customer departs from the system. After a visit to the FCFS node, customers are fed back to the PS node. The service requirements at both nodes are exponentially distributed. The model is a Jackson network, admitting a product-from solution for the joint number of customers at the nodes, immediately leading to a closed-form expression for the mean sojourn times in steady-state. The variance of the sojourn times, however, does not admit an exact expression-the complexity is caused by the possibility of overtaking. In this paper we propose a methodology for deriving simple, explicit and fast-to-evaluate approximations for the variance of the sojourn times. Numerical results demonstrate that the approximations are very accurate in most model instances. and his master of technological design degree in Mathematics from the University of Twente. In 1996 he joined KPN Research, Leidschendam, The Netherlands, where he is currently working at the Quality of Service Center of Excellence. His research interests include performance Modeling and evaluation of fixed and mobile packet switched communication networks. Recently, his focus extended to performance modelling and evaluation of distributed information systems at the borders of communication networks.

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.

In this paper, we consider a discrete-time multiserver queueing system with inÞnite buer size, geometric service times and a FCFS (Þrst-come-Þrst-served) queueing discipline. A relationship between the probability distributions of the system contents and the packet delay is established. The relationship is general in the sense that it doesn't require knowledge of the exact nature of the arrival process. By means of the relationship, results for the characteristics of the system contents for various discrete-time queueing models can be transformed into corresponding results for the delay characteris- tics, thus making a separate delay analysis superßuous.

Performance Evaluation, 2006

This paper is motivated by the response-time analysis of distributed information systems, where transactions are handled by a sequence of front-end server and back-end server actions. We study sojourn times in an open queueing network with a single Processor Sharing (PS) node and an arbitrary number of M multi-server First-Come-First-Served (FCFS) nodes. Customers arrive at the PS according to a Poisson process. After departing from the PS node a customer jumps to FCFS node k with probability p k , and departs from the system with probability 1 − p, where p = M k−1 p k (0 < p < 1). After receiving service at a FCFS node, a customer jumps back to the PS node. For this model, we focus on the mean and the variability of the sojourn time of an arbitrary customer in the system. The model is a product-form network, which immediately leads to a closed-form expression for the mean sojourn times. The variance of the sojourn times, however, does not admit an exact expression; the complexity is caused by the possibility of overtaking. To this end, we propose a new methodology for deriving closed-form approximations for the variance of sojourn times in queueing networks with feedback. Numerical results from extensive experimentation with simulations demonstrates that the approximations are highly accurate for a wide range of parameter values. , focusing on performance modeling and analysis of a variety of AT&T networks and services. In 1999 he moved back to his home country to rejoin KPN Research as a senior researcher. He is particularly working on performance modelling, evaluation and dimensioning of fixed and mobile communication networks. In these fields, he has co-operated within many European research programmes. Since July 2003 he has a part-time full professorship within the

1989

In this paper we derive an approximation for the mean response time of a multiple queue system in which shortest queue routing is used. We assume there are K identical queues with infinite capacity and service times that are exponentially distributed. Arrivals of jobs to this system are Poisson and are routed to a queue of minimal length. We develop an approximation which is based on both theoretical and experimental considerations and, for K 5 8, has an relative error of less than one half of one percent when compared to simulation. For K = 16, the relative error is still acceptable, being less than 2 percent. An application to a model of parallel processing and a comparison of static and dynamic load balancing schemes are presented.

OR Spectrum, 2005

In this paper, we consider acyclic networks of queues as a model to support the design of a dynamic production system. Each service station in the network represents a manufacturing or assembly operation. Only one type of product is produced by the system, but there exist several distinct production processes for manufacturing this product, each one corresponding with a directed path in the network of queues. In each network node, the number of servers in the corresponding service station is either one or infinity. The service time in each station is either exponentially distributed or belongs to a special class of Coxian distribution. Only in the source node, the service system may be modeled by an M/G/∞ queue. The transport times between every pair of service stations are independent random variables with exponential distributions. In method proposed in this paper, the network of queues is transformed into an equivalent stochastic network. Next, we develop a method for approximating the distribution function of the length of the shortest path of the transformed stochastic network, from the source to the sink node. Hence, the method leads to determining the distribution function of the time required to complete a product in this system (called the manufacturing lead time). This is done through solving a system of linear differential equations with non-constant coefficients, which is obtained from a related continuous-time Markov process. The results are verified by simulation.

Mathematics

In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distri...

SSRN Electronic Journal, 2000

In this article we give a new derivation for the waiting time distributions in an M/M/c queue with multiple priorities and a common service rate by using elementary lattice paths counting. An advantage of the approach is that it does not require inversion of the Laplace-Stieltjes transform.

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

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.

Performance Evaluation, 1993

Nelson, R.D. and T.K. Philips, An approximation for the mean response time for shortest queue routing with general interarrival and service times, Performance Evaluation 17 (1993) 123-139. In this paper we derive an approximation for the mean respone time of a multiple queue system in which shortest queue routing is used. We assume there are K identical queues with infinite capacity. Interarrival and service times are generally distributed, and an arriving job is routed to a queue of minimal length. Our approximation is a simple closed form equation that requires only the mean and coefficient of variation of job's interarrival and service times. The approximation is extensively compared to simulated values for values of K ~< 8, and has small relative errors, typically less than 5%, for systems where the coefficient of variation of interarrival and serivce times are both ~< 1. For the system consisting of Poisson arrivals and exponential service times, we extend the approximation so that the error is less than one half of one percent for K~<8.

Journal of Industrial and Management Optimization, 2021

We study the \begin{document}$ MAP/M/s+G $\end{document} queueing model that arises in various multi-server engineering problems including telephone call centers, under the assumption of MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest via the steady-state solution of a certain Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid, for which case discretization of the patience time distribution is required giving rise to a computational complexity depending linearly on the number of discretization levels. Additionally, a novel method is proposed to accurately obtain the first passage time d...

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

2011

This paper deals with approximate calculation methods for delay link systems with several input queues. The calculations are based on the idea of &quot;equivalent systems&quot; where the link system is equated with 1. a full availability group, and 2. Erlang&#39;s ideal grading. An equivalent system also includes an equivalent queueing structure which might be completely different from the original one • The equivalent system is constructed to have the same probability of waiting, W, as the link system. In the latter case, W m&#39;ight be calculated from the modified methods of Lotze or Jacobaeus, but the possibility of improving the approximation for W is also suggested. This improvement leads to a better approximation of the mean waiting time T than found by Hieber. Finally, an equivalent queueing structure for the ideal grading is constructed and approximate formulas for the waiting time distribution are founq. 1. INI&#39;RODUCTION A modification of the theories of Lotze [lJ and ...