On an unreliable-server retrial queue with customer feedback and impatience
TL;DR: This system is analyzed as a process of quasi-birth-and-death (QBD) where the quasi-progression algorithm is applied to compute the rate matrix of QBD model, and a recursive solver algorithm for computing the stationary probabilities is developed.
About: This article is published in Applied Mathematical Modelling.The article was published on 2018-03-01. It has received 49 citations till now. The article focuses on the topics: Retrial queue.
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TL;DR: The reliability function and mean-time-to-failure (MTTF) based on the Laplace transform technique is developed and numerical examples are given to assess the effects of system parameters on the system reliability, MTTF, and steady-state availability.
56 citations
TL;DR: A new efficient iterative algorithm for approximate computation of the stationary distribution of an asymptotically quasi-Toeplitz Markov chain that computes many times faster and requires much fewer computer resources than the standard algorithm is developed.
35 citations
TL;DR: Investigation of the customers’ balking behavior in a double orbit retrial queueing system having two types of customers namely, ordinary and premium class customers, using parametric nonlinear programing (P-NLP) approach to determine fuzzified indices.
19 citations
TL;DR: The current investigation deals with customers’ impatient behavior in case of unreliable server Markovian queueing system operating under N-policy, and quasi-Newton technique is applied to decrease the complete cost associated with multiple activities in the queueed system.
Abstract: The current investigation deals with customers’ impatient behavior in case of unreliable server Markovian queueing system operating under N-policy. The broken down server is repaired in accordance with the threshold policy i.e. repair begins when the system reaches its capacity. To obtain the steady-state distribution of queue length, recursive technique has been used. Different performance indices are established and numerical experiments are conducted to characterize the behavior of the system indices by varying different system parameters. The numerical results obtained are compared with the soft computing technique based on adaptive neuro-fuzzy inference system. A cost function is formed and quasi-Newton technique is applied to decrease the complete cost associated with multiple activities in the queueing system.
14 citations
TL;DR: Markovian queue with working vacation, retrial and impatient customers has been investigated by including the feature of imperfect service during working vacation and a cost function is constructed and minimized using quasi-Newton method and genetic algorithm.
Abstract: Markovian queue with working vacation, retrial and impatient customers has been investigated by including the feature of imperfect service during working vacation. If the server is occupied, an incoming customer may either decide to balk from the system or join the retrial orbit following the classical retrial policy. When there is no customer in the system, the server switches over to a working vacation (WV) mode. The customers who enter into the system during working vacation are also served at a slower speed. The probability generating functions (PGFs) are derived for evaluating the distributions of the number of customers in the system and orbit, which are further utilized to derive several performance metrics in closed form. The sensitivity analysis has been done to explore the sensitiveness of various descriptors on the system performance measures. Finally, a cost function is constructed and minimized using quasi-Newton method (QNM) and genetic algorithm (GA).
11 citations
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TL;DR: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point.
Abstract: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.
27,271 citations
"On an unreliable-server retrial que..." refers methods in this paper
...The NM method, proposed by Nelder & Mead (1965), is an enormously popular direct search method for multidimensional unconstrained minimization....
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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.
2,846 citations
TL;DR: This chapter discusses quasi-Birth-and-Death Processes, a large number of which are based on the Markovian Point Processes and the Matrix-Geometric Distribution, as well as algorithms for the Rate Matrix.
Abstract: Preface Part I. Quasi-Birth-and-Death Processes. 1. Examples Part II. The Method of Phases. 2. PH Distributions 3. Markovian Point Processes Part III. The Matrix-Geometric Distribution. 4. Birth-and-Death Processes 5. Processes Under a Taboo 6. Homogeneous QBDs 7. Stability Condition Part IV. Algorithms. 8. Algorithms for the Rate Matrix 9. Spectral Analysis 10. Finite QBDs 11. First Passage Times Part V. Beyond Simple QBDs. 12. Nonhomogeneous QBDs 13. Processes, Skip-Free in One Direction 14. Tree Processes 15. Product Form Networks 16. Nondenumerable States Bibliography Index.
1,940 citations
TL;DR: This book deals with regression analysis and suggests to shift the focus to problem solving, using existing (or developing new) mathematical-statistical and subject-matter theory rather than developing new theory to solve problems that could arise in the future.
Abstract: (2001). Introduction to Matrix Analytic Methods in Stochastic Modeling. Technometrics: Vol. 43, No. 3, pp. 379-380.
701 citations
"On an unreliable-server retrial que..." refers background in this paper
...(7) Here, R is gotten by lim n n R (Neuts, 1981; Latouche and Ramaswami, 1999), where the sequence nR is given by 0 R 0 and 1 2 1 1n N n N N R BA R C A for 0n ....
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...(8) 9 / 21 Noted that the sequence nR converges to matrix R monotonically (Neuts, 1981; Latouche and Ramaswami, 1999)....
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TL;DR: This work proves global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.
Abstract: We present a convergence theory for pattern search methods for solving bound constrained nonlinear programs. The analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints. This analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems. We prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.
512 citations
"On an unreliable-server retrial que..." refers methods in this paper
...The PS method are defined through a finite set of directions used at each iteration to create a conceptual mesh on which trial points are selected, evaluated and compared against the current best solution (see Lewis and Torczon, 1999, 2000)....
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