Repairable Queueing System Research Articles

Analysis of single server repairable queueing system with many kinds of service, single working vacation and standby server

We consider a single server queueing model in which the customers arrive according to a Markovian Arrival Process (MAP). We modeled this queueing system with many kinds of service where the customers can choose any kind from the server. Each kind of service follows Phase-type (PH) distribution. We analyzed the model with breakdown, repair, single working vacation and standby server. A standby server works only when the main server is breakdown while serving the customer during a busy or vacation period. Using the matrix-geometric approach, we examined the model. A few real-world examples have been given, and several graphical and numerical examples have also been provided.

Reliability analysis of an M/G/1 repairable queueing system with vacations and patient server

Reliability analysis of an M/G/1 repairable queueing system with vacations and patient server

Single Server Repairable Queueing System With Variable Service Rate and Failure Rate

This paper studies the single server repairable queueing system with variable service rate and failure rate. The rule of variation of the service rate is that the service rate changes when the number of customers in the system reaches a certain value. The server may fail at any time, and the failure rates of idle periods and busy periods are different. The system has one reliable repairman to repair the failure server. The steady-state joint distribution of the customers number and server states is obtained by the matrix geometric solution method, the steady-state availability of the server and significant queue performances measures are evaluated. Two special cases are analyzed, and some numerical experiments are given for the illustrations of the parameters effect.

Analysis of G-Queue with Pseudo-Fault and Multiple Working Vacations

This paper presents a new model of discrete time Geo/Geo/1 repairable queueing system with pseudo-fault, negative customers and multiple working vacations. The authors assume that system service may be interrupted by breakdown or pseudo-fault, this system may become disabled only when it is in a regular busy period, and negative customers adopt two types of typical killing strategies. In this paper, the authors know that the evolution of the system can be described by a two-dimensional Markov chain, and the two-dimensional Markov chain satisfies the condition of quasi birth and death chains. Based on the method of matrix-geometric solution, the authors obtain distributions for the stationary queue length in RCH and RCE strategy, respectively. Moreover, the reliability of the system is analyzed and the number of customers and waiting time of a customer in the system in steady state are obtained. The authors analyze the impact of two killing strategies on the system comparatively. This paper studies the individually and socially optimal behaviors of positive customers, and presents a pricing policy for positive customers, therefore, the authors obtain the socially optimal arrival rate. Various numerical results are provided to show the change of performance measures.

Performance and reliability analysis of a repairable discrete‐time Geo/G/1 queue with Bernoulli feedback and randomized policy

This paper is concerned with a discrete‐time Geo/G/1 repairable queueing system with Bernoulli feedback and randomized ‐policy. The service station may be subject to failures randomly during serving customers and therefore is sent for repair immediately. The ‐policy means that when the number of customers in the system reaches a given threshold value N, the deactivated server is turned on with probability p or is still left off with probability 1−p. Applying the law of total probability decomposition, the renewal theory and the probability generating function technique, we investigate the queueing performance measures and reliability indices simultaneously in our work. Both the transient queue length distribution and the recursive expressions of the steady‐state queue length distribution at various epochs are explicitly derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and the steady‐state unavailability of the service station, the expected number of the service station breakdowns during the time interval and the equilibrium failure frequency of the service station are also discussed. Finally, an operating cost function is formulated, and the direct search method is employed to numerically find the optimum value of N for minimizing the system cost. Copyright © 2017 John Wiley & Sons, Ltd.

Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with <i>N</i>-policy, setup time and multiple working vacations

In this paper, we consider a discrete time Geo/Geo/1 repairable queueing system with a pseudo-fault, setup time, $N$-policy and multiple working vacations. We assume that the service interruption is caused by pseudo-fault or breakdown, and occurs only when the server is busy. If the pseudo-fault occurs, the server will enter into a vacation period instead of a busy period. At a breakdown instant, the repair period starts immediately and after repaired the server is assumed to be as good as new. Using a quasi birth-and-death chain, we establish a two-dimensional Markov chain. We obtain the distribution of the steady-state queue length by using a matrix-geometric solution method. Moreover, we analyze the considered queueing system and provide several performance indices of the system in steady-state. According to the queueing system, we first investigate the individual and social optimal behaviors of the customer. Then we propose a pricing policy to optimize the system socially, and study the Nash equilibrium and social optimization of the proposed strategy to determine the optimal expected parameters of the system. Finally, we present some numerical results to illustrate the effect of several parameters on the systems.

Analysis of an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility

This paper considers an M/G/1 repairable queueing system with N-policy and single vacation, in which the service station is subject to random breakdowns. Once the service station breaks down, it is repaired by a repair facility. Moreover, the repair facility may fail during the repair period which results in repair interruptions. Failed repair facility resumes repair after a random period of time. Applying the renewal process theory and the probability decomposition technique, the probability that the service station is broken, the rate of occurrence of breakdowns of the service station, the probability that the repair facility is being replaced, the rate of occurrence of failures of the repair facility and the stochastic decomposition property of the reliability measures are obtained. Following the construction of the long-run expected cost function per unit time, we numerically determined the optimal threshold N ∗ for minimizing the cost function.

Probabilistic Decomposition Method on the ServerIndices of anMξ/G/1 Vacation Queue

This paper develops a probabilistic decomposition method for anMξ/G/1 repairable queueing system with multiple vacations, in which the customers who arrive during server vacations enter the system with probabilityp. Such a novel method is used to analyze the main performance indices of the server, such as the unavailability and the mean failure number during(0,t]. It is derived that the structures of server indices are two convolution equations. Further, comparisons with existing methods indicate that our method is effective and applicable for studying server performances in single-serverMξ/G/1 vacation queues and their complex variants. Finally, a stochastic order and production system with a multipurpose production facility is numerically presented for illustrative purpose.

Research of a Repairable Queueing System with Three Kinds of States

The resolvent set of the operator corresponding to a repairable queueing system with three kinds of states has been studied.Through studying the resolvent set of the adjoint operator of this operator the resolvent set of this operator has been obtained.

The SM/PH/1 queue with repairable server of phase type life time

We consider the reliability problem of the SM/PH/1 queue with repairable server of PH life time and PH repair time. The generalized service times are proved to be dependent on each other and not to have the same distribution if the system is at transient state. For this reason the transient behavior of the system is so complex that we can not give any available results. On the contrary, the repairable queueing system is shown to be equivalent to the classical SM/SM(PH)/1 queue when the system is at steady state, where SM(PH) denotes Markovian semi-Markov process which was discussed in Latouche [10] and Sengupta [23]. By means of the equivalent model SM/SM(PH)/1, the explicit expressions of the reliability quantities are derived, for instance, the steady state availability and the steady state failure frequency.

Probability analysis of the repairable queueing systemM/G(E k /H)/1

The repairable queueing system (RQS) in which the server has an exponential lifetime distribution has been studied in several articles [1–4]. Here, we deal with the new RQSM/G(Ek/H)/1 in which the lifetime distribution of the server is Erlangian. By forming a vector Markov process, i.e. by using the method of supplementary variables, we obtained some system characters, the reliability indices of the server, and the time distribution of a customer spent on the server. For this RQS, the generalized service time distribution of each customer will depend on the remainder life of the server. Based on this, a new kind of queues, for which the service time distributions are chosen by the customers in some stochastic manner, appears in queueing theory.