Topic
Matrix geometric method
About: Matrix geometric method is a research topic. Over the lifetime, 109 publications have been published within this topic receiving 4693 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond is presented, which is a book of applicable mathematics, which avoids the approach of listing only the techniques, followed by a few examples.
Abstract: This is a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond. It is a book of applicable mathematics. It avoids the approach of listing only the techniques, followed by a few examples, without explaining why the techniques work. Thus, it provides not only the know-how but also the know-why. Equally, the text has not been written as a book of pure mathematics with a list of theorems followed by their proofs. The authors' aim is to help students develop an understanding of mathematics and its applications. They have refrained from using clichés like “it is obvious” and “it can be shown”, which may be true only to a mature mathematician. On the whole, the authors have been generous in writing down all the steps in solving the example problems.The book comprises ten chapters. Each chapter contains several solved problems clarifying the introduced concepts. Some of the examples are taken from the recent literature and serve to illustrate the applications in various fields of engineering and science. At the end of each chapter, there are assignment problems with two levels of difficulty. A list of references is provided at the end of the book.This book is the product of a close collaboration between two mathematicians and an engineer. The engineer has been helpful in pinpointing the problems which engineering students encounter in books written by mathematicians.
3,550 citations
TL;DR: A new, simple algorithm for the matrix-geometric rate matrix has quadratic convergence and is shown theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.
Abstract: Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.
411 citations
TL;DR: The results illustrate that under the FCFS discipline, the status update frequency needs to be carefully chosen according to the service rate and energy arrival rate in order to minimize the average penalty.
Abstract: Timely status updates are crucial to enabling applications in the massive Internet of Things (IoT). This paper measures the data-freshness performance of a status update system with an energy-harvesting transmitter, considering the randomness in information generation, transmission, and energy harvesting. The performance is evaluated by a non-linear function of age of information (AoI) that is defined as the time elapsed since the generation of the most up-to-date status information at the receiver. The system is formulated as two queues with status packet generation and energy arrivals both assumed to be Poisson processes. With negligible service time, both first-come-first-served (FCFS) and last-come-first-served (LCFS) disciplines for arbitrary buffer and battery capacities are considered, and a method for calculating the average penalty with non-linear penalty functions is proposed. The average AoI, the average penalty under exponential penalty function, and the AoI’s threshold violation probability are obtained in a closed form. When the service time is assumed to follow exponential distribution, a matrix geometric method is used to obtain the average peak AoI. The results illustrate that under the FCFS discipline, the status update frequency needs to be carefully chosen according to the service rate and energy arrival rate in order to minimize the average penalty.
115 citations
TL;DR: This paper presents a matrix geometric method for deriving the steady-state probabilities, using which various system performance measures that can be obtained are obtained based on assumed numerical values given to the system parameters.
Abstract: We consider the machine repair problem in which failed machines balk (do not enter) with a constant probability (1 – b) and renege (leave the queue after entering) according to a negative exponential distribution. A group of identical automatic machines are maintained by R servers which themselves are subject to breakdowns. Failure and service times of the machines, and breakdown and repair times of the servers, are assumed to follow a negative exponential distribution. Each server is subject to breakdown even if no failed machines are in the system. This paper presents a matrix geometric method for deriving the steady-state probabilities, using which various system performance measures that can be obtained. A cost model is developed to determine the optimum number of servers. The minimum expected cost, the optimal number of servers, and various system performance measures are provided based on assumed numerical values given to the system parameters. Also the sensitivity analysis is investigated.
81 citations
TL;DR: The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates, and the particle swarm optimization algorithm is employed to solve the optimization problem.
76 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 7 |
| 2020 | 7 |
| 2019 | 8 |
| 2018 | 5 |
| 2017 | 3 |
| 2016 | 5 |