Several queueing processes may be modeled as random walks on a multidimensional grid. In this paper the equilibrium distribution for the case of a two-dimensional grid is considered. In previous research it has been shown that for some two-dimensional random walks the equilibrium distribution has the form of an infinite series of products of powers which can be constructed with a compensation procedure. The object of the present paper is to investigate under which conditions such an elegant solution exists and may be found with a compensation approach. The conditions can be easily formulated in terms of the random behaviour in the inner area and the drift on the boundaries.

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A compensation approach for two-dimensional Markov processes

Many real-world situations involve queueing systems in which customers wait for service for a limited time only and leave the system if service has not begun within that time. This paper considers a multiserver queueing system with impatient customers, where the customers arrive according to a Poisson process and the service requirements have a general distribution. A simple and insightful solution is presented for the loss probability. The solution is exact for exponential services and is an excellent heuristic for general service times.

A Multiserver Queueing System with Impatient Customers

Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

Most models for location of immobile congested facilities assume exponentially distributed service time at the facilities. Although the resulting formulations are tractable, they do not adequately represent service time distributions with small variances, as often occur in practice. In a recent paper, the authors utilized an order r Erlang distribution for the service time, applied to the simple case of single-server facilities. We generalize this approach to multiple-server facilities, which need a different mathematical treatment. The constraint on service availability is cast as a linear constraint on the proportion of time the servers are busy, and its right-hand side parameter is provided for different situations. Extensive analysis is offered on the influence of the parameters of the service time and the capacity of the facilities on the performance of the system. Numerical results are given for a data set relating to the municipality of Rio de Janeiro.

https://link.springer.com/content/pdf/10.1057%2Fpalgrave.jors.2602606.pdf

Optimal location of multi-server congestible facilities operating as M/E r /m/N queues

We discuss various aspects of phase service queueing models. A large number of models have been developed in the area of queueing theory incorporating the concept of phase service. These phase service queueing models have been investigated for resolving the congestion problems of many day-to-day as well as industrial scenarios. In this survey paper, an attempt has been made to review the work done by the prominent researchers on the phase service queues and their applications in several realistic queueing situations. The methodology used by several researchers for solving various phase service queueing models has also been described. We have classified the related literature based on modeling and methodological concepts. The main objective of present paper is to provide relevant information to the system analysts, managers, and industry people who are interested in using queueing theory to model congestion problems wherein the phase type services are prevalent.

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Mathematical Analysis of Queue with Phase Service: An Overview

Analyzing Ek|Er|c queues

In this paper we study a system consisting of c parallel identical servers and a common queue. The service times are Erlang-r distributed and the interarrival times are Erlang-k distributed. The service discipline is first-come first-served. The waiting process may be characterised by (n_{-1}, n_0, n_1, ..., n_c) where n_{-1} represents the number of remaining arrival stages, n_0 the number of waiting jobs and n_i, i = 1, ... , c, the number of remaining service stages for server i. Bertsimas has proved that the equilibrium probability for a saturated state (all n_i > 0, i = 1, ... , c) can be written as a linear combination of geometric terms with no as exponent. In the present paper it is shown that the coefficients also have a geometric form with respect to n_{-1}, n_1, ... , n_c .It is also shown how the factors may be found efficiently. The present paper uses a direct approach for solving the equilibrium equations rather than a generating function approach as Bertsimas does. The direct approach has been inspired by previous work of two of the authors on the shortest queue problem in particular and the two-dimensional random walk more generally. Although the paper extends results of Bertsimas it is self-contained.

Analysing $E_k|E_r|c$ queues

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WA van de Waarsenburg

Papers

Analyzing Ek|Er|c queues

Analysing $E_k|E_r|c$ queues

Analysing $E_k|E_r|c$ queues