Non-Markovian Queueing System, Mx/G/1 with Server Breakdown and Repair Times

This paper deals with the steady state behavior of an MX/G/1 queue with breakdown. It assumed that customers arrive to the system in batches of variable size, but serve one by one. The main new assumption in this paper is that the repair process does not start immediately after a breakdown and there is a delay time waiting for repairs to start. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average waiting time in the queue

Simple bounds for queueing systems with breakdowns

Computationally attractive and intuitively obvious simple bounds are proposed for finite service systems which are subject to random breakdowns. The services are assumed to be exponential. The up and down periods are allowed to be generally distributed. The bounds are based on product-form modifications and depend only on means. A formal proof is presented. This proof is of interest in itself. Numerical support indicates a potential usefulness for quick engineering and performance evaluation purposes

A study of some M[x]/G/1 type queues with random breakdowns and bernouilli schedule server vacations based on a single vacation policy

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Queueing systems arise in modelling of many practical applications related to computer sciences, telecommunication networks, manufacturing and production, human computer interaction, and so on. The classical queueing system, even vacation queues or queues subject to breakdown, might not be sufficiently realistic. The purpose of this research is to extend the work done on vacation queues and on unreliable queues by studying queueing systems which take into consideration both phenomena. We study the behavior of a batch arrival queueing system with a single server, where the system is subject to random breakdowns which require a repair process, and on the other hand, the server is allowed to take a vacation after finishing a service. The breakdowns are assumed to occur while serving a customer, and when the system breaks down, it enters a repair process immediately while the customer whose service is interrupted comes back to the head of the queue waiting for the service to resume. Server vacations are assumed to follow a Bernoulli schedule under single vacation policy. We consider the above assumptions for different queueing models: queues with generalized service time, queues with two-stages of heterogeneous service, queues with a second optional service, and queues with two types of service. For all the models mentioned above, it is assumed that the service times, vacation times, and repair times all have general arbitrary distributions. Applying the supplementary variable technique, we obtain probability generating functions of queue size at a random epoch for different states of the system, and some performance measures such as the mean queue length, mean waiting time in the queue, proportion of server's idle time, and the utilization factor. The results obtained in this research, show the effect of vacation and breakdown parameters upon main performance measures of interest. These effects are also illustrated using some numerical examples and graphs.This work is funded by the Ministry of Education, Kingdom of Bahrain

On single server batch arrival queueing system with balking, three types of heterogeneous service and Bernoulli schedule server vacation

This paper investigates a batch arrival queueing system in which customers arrives at the system in a Poisson stream following a compound Poisson process and the system has a single server providing three types of general heterogeneous services. At the beginning of each service, a customer is allowed to choose any one of the three services and as soon as a service of any type gets completed, the server may take a vacation or may continue staying in the system. The vacation time is assumed to follow a general (arbitrary) distribution and the server vacation is based on Bernoulli schedule under a single vacation policy. During the server vacation period, impatient customers are assumed to balk. This paper described the model as a bivariate Markov chain and employed the supplementary variable technique to find closed-form solutions of the steady state probability generating function of number of customers, the steady state probabilities of various states of the system, the average queue size, the average system size, and the average waiting time in the queue as well as the average waiting time in the system. Further, some interesting special cases of the model are also derived. Keywords Batch Arrivals. Queueing System. Balking. Heterogeneous types of Service. Bernoulli schedule server vacation. Bivariate Markov Processes. MSC2020-Mathematics Subject Classification 34B07, 60G05, 62E1

A class of multi-server queueing systems with unreliable servers: Models and application.

Where queueing systems with unreliable servers are concerned, most research that has been done focuses on one-server systems or systems with a Poisson arrival process and exponential service time. However, in some situations we need to consider non-exponential service time or service rate changes with the number of available servers. These are the queueing systems that are discussed in this thesis, none of which has ever been discussed in the literature. Since the phase type distribution is more general than the exponential distribution and captures most features of a general distribution, the phase type distributed service time is considered in unreliable queueing systems such as M/PH/n and M/PH/n/c. For the M/PH/n queueing system with unreliable servers, the mathematical model, stability condition analysis, stationary distribution calculation, computer programs and examples are all presented. For the M/PH/n/c queueing system with server failures, a finite birth-and-death mathematical model is built and the stationary distribution and performance evaluation measurements are calculated. Computer programs are developed and an example is given to demonstrate the application of this queueing system. (Abstract shortened by UMI.)Dept. of Industrial and Manufacturing Systems Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2003 .Y375. Source: Masters Abstracts International, Volume: 43-01, page: 0295. Adviser: Attahiru S. Alfa. Thesis (M.A.Sc.)--University of Windsor (Canada), 2004

Performance and reliability modelling of computing systems using spectral expansion

PhD ThesisThis thesis is concerned with the analytical modelling of computing and other discrete event systems, for steady state performance and dependability. That is carried out using a novel solution technique, known as the spectral expansion method. The type of problems considered, and the systems analysed, are represented by certain two-dimensional Markov-processes on finite or semi-infinite lattice strips. A sub set of these Markov processes are the Quasi-Birth-and-Death processes. These models are important because they have wide ranging applications in the design and analysis of modern communications, advanced computing systems, flexible manufacturing systems and in dependability modelling. Though the matrixgeometric method is the presently most popular method, in this area, it suffers from certain drawbacks, as illustrated in one of the chapters. Spectral expansion clearly rises above those limitations. This also, is shown with the aid of examples. The contributions of this thesis can be divided into two categories. They are, • The theoretical foundation of the spectral expansion method is laid. Stability analysis of these Markov processes is carried out. Efficient numerical solution algorithms are developed. A comparative study is performed to show that the spectral expansion algorithm has an edge over the matrix-geometric method, in computational efficiency, accuracy and ease of use. • The method is applied to several non-trivial and complicated modelling problems, occuring in computer and communication systems. Performance measures are evaluated and optimisation issues are addressed

Non-Cooperative Spectrum Access -- The Dedicated vs. Free Spectrum Choice

We consider a dynamic spectrum access system in which Secondary Users (SUs) choose to either acquire dedicated spectrum or to use spectrum-holes (white spaces) which belong to Primary Users (PUs). The trade-off incorporated in this decision is between immediate yet costly transmission and free but delayed transmission (a consequence of both the possible appearance of PUs and sharing the spectrum holes with multiple SUs). We first consider a system with a single PU band, in which the SU decisions are fixed. Employing queueing-theoretic methods, we obtain explicit expressions for the expected delays associated with using the PU band. Based on that, we then consider self-interested SUs and study the interaction between them as a non-cooperative game. We prove the existence and uniqueness of a symmetric Nash equilibrium, and characterize the equilibrium behavior explicitly. Using our equilibrium results, we show how to maximize revenue from renting dedicated bands to SUs and briefly discuss the extension of our model to multiple PUs. Finally, since spectrum sensing can be resource-consuming, we characterize the gains provided by this capability.National Science Foundation (U.S.) (Grant CNS-0915988)National Science Foundation (U.S.) (Grant CNS-0916263)National Science Foundation (U.S.) (Grant CNS-1054856)National Science Foundation (U.S.). Engineering Research Centers Program (Center for Integrated Access Networks Grant EEC-0812072)United States. Office of Naval Research (Grant N00014-12-1-0064)United States. Army Research Office. Multidisciplinary University Research Initiative (Grant W911NF-08-1-0238

An M/G/1 Queue with Server Breakdown and Multiple Working Vavation

This paper deals with the steady state behavior of an M=G=1 multiple working vacation queue with server breakdown. The server works with different service times rather than completely stopping service during a vacation. Both service times in a vacation period and in a regular service period are assumed to be generally distributed random variables. The system may breakdown at random and repair time is arbitrary. Further, just after completion of a customer’s service the server may take a multiple working vacation. Supplementary variable technique is employed to find the probability generating function for the number of customers in the system. The mean number of customers in the system is calculated. Some particular cases of interest are discussed. Numerical results are also presented