A single-server discrete-time retrial <Emphasis Type="Italic">G</Emphasis>-queue with server breakdowns and repairs

This paper concerns a discrete-time Geo/Geo/1 retrial queue with both positive and negative customers where the server is subject to breakdowns and repairs due to negative arrivals. The arrival of a negative customer causes one positive customer to be killed if any is present, and simultaneously breaks the server down. The server is sent to repair immediately and after repair it is as good as new. The negative customer also causes the server breakdown if the server is found idle, but has no effect on the system if the server is under repair. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating function of the number of customers in the orbit and in the system are also obtained, along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, we present some numerical examples to illustrate the influence of the parameters on several performance characteristics of the system.     

Optimal control of a removable and non-reliable server in an M/M/1 queueing system with exponential startup time

We study a single removable and non-reliable server in the N policy M/M/1 queueing system. The server begins service only when the number of customers in the system reaches N (N1). After each idle period, the startup times of the server follow the negative exponential distribution. While the server is working, it is subject to breakdowns according to a Poisson process. When the server breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. The steady-state results are derived and it is shown that the probability that the server is busy is equal to the traffic intensity. Cost model is developed to determine the optimal operating N policy at minimum cost.     

Cost analysis of the unloader queueing system with a single unloader subject to breakdowns

In this paper, we deal with an unloader queueing model in which N identical trailers are unloaded by a single unloader. We consider two different types of a single unloader, which are subject to breakdowns. In type 1, the unloader can break down only when there is at least one trailer in the system, while in type 2, the unloader can break down even if no trailers are in the system. Analytic closed-form solutions of the unloader queueing system are derived. A cost model is developed in order to determine the optimal value of the number of trailers to be assigned to the unloader for both types. Under the optimal operating condition, numerical results are presented in which several system performance measures are evaluated based on assumed numerical values given to the system parameters. Sensitivity analysis is also investigated.     

Geo/Geo/1 retrial queue with working vacations and vacation interruption

Consider a Geo/Geo/1 retrial queue with working vacations and vacation interruption, and assume requests in the orbit try to get service from the server with a constant retrial rate. During the working vacation period, customers can be served at a lower rate. If there are customers in the system after a service completion instant, the vacation will be interrupted and the server comes back to the normal working level. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.     

带有N策略,启动时间和服务台故障的M/M/1排队的均衡策略

本文考虑带有N策略,启动时间和服务台故障的M/M/1排队的顾客的策略行为.当系统为空时服务台关闭,并且只有当系统中的顾客数达到一个给定的阈值时才会被激活,启动时间服从指数分布.服务台在工作时可能会故障,一旦发生故障,它立即被维修,维修的时间服从指数分布.我们得到了不同状态的均衡到达率并且给出了均衡社会收益函数.最后对均衡到达率和均衡社会收益进行了数值研究.     

Batch arrival queues under vacation policies with server breakdowns and startup/closedown times

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under vacation policies with startup/closedown times, where the vacation time, the startup time, and the closedown time are generally distributed. When all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time. After shutdown, the server operates one of (1) multiple vacation policy and (2) single vacation policy. When the server reactivates since shutdown, he needs a startup time before providing the service. If a customer arrives during a closedown time, the service is immediately started without a startup time. The server may break down according to a Poisson process while working and his repair time has a general distribution. We analyze the system characteristics for the vacation models.     

Optimal strategy policy in batch arrival queue with server breakdowns and multiple vacations

This paper considers a like-queue production system in which server vacations and breakdowns are possible. The decision-maker can turn a single server on at any arrival epoch or off at any service completion. We model the system by an M^[x]/M/1 queueing system with N policy. The server can be turned off and takes a vacation with exponential random length whenever the system is empty. If the number of units waiting in the system at any vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N units in the system, he immediately starts to serve the waiting units. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. We derive the distribution of the system size through the probability generating function. We further study the steady-state behavior of the system size distribution at random (stationary) point of time as well as the queue size distribution at departure point of time. Other system characteristics are obtained by means of the grand process and the renewal process. Finally, the expected cost per unit time is considered to determine the optimal operating policy at a minimum cost. The sensitivity analysis is also presented through numerical experiments.     

带启动时间的Geom/Geom/1多重工作休假排队

本文中研究了一个带有启动时间的Geom/Geom/1多重工作休假排队模型。服务台在休假期间,不停止服务,而是以较低的服务率为顾客提供服务。运用拟生灭过程和矩阵几何解的方法,给出了该模型的稳态队长分布,并求出了平均队长以及顾客的平均逗留时间。     

ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

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Geo/G/1 queues with disasters and general repair times

This paper discusses discrete-time single server Geo/G/1 queues that are subject to failure due to a disaster arrival. Upon a disaster arrival, all present customers leave the system. At a failure epoch, the server is turned off and the repair period immediately begins. The repair times are commonly distributed random variables. We derive the probability generating functions of the queue length distribution and the FCFS sojourn time distribution. Finally, some numerical examples are given.     

具有负顾客到达的M/G/1可修排队系统

本文考虑一个具有负顾客到达的M／G／1可修捧队系统．所有顾客(包括正顾客和负顾客)的到达都是泊松过程,服务器是可修的．Harrison和Pitel研究过具有负顾客到达的M／G／1捧队系统．这里我们推广到有可修服务器情形,系统的稳态解最后可以通过Fredholm积分方程解出．     

服务台不可靠的重试排队系统均衡分析

本文研究服务台不可靠的M/M/1常数率重试排队系统中顾客的均衡进队策略, 其中服务台在正常工作和空闲状态下以不同的速率发生故障。在该系统中, 服务台前没有等待空间, 如果到达的顾客发现服务台处于空闲状态, 该顾客可占用服务台开始服务。否则, 如果服务台处于忙碌状态, 顾客可以选择留下信息, 使得服务台在空闲时可以按顺序在重试空间中寻找之前留下信息的顾客进行服务。当服务台发生故障时, 正在被服务的顾客会发生丢失, 且系统拒绝新的顾客进入系统。根据系统提供给顾客的不同程度的信息, 研究队长可见和不可见两种信息情形下系统的稳态指标, 以及顾客基于收入-支出函数的均衡进队策略, 并建立单位时间内服务商的收益和社会福利函数。比较发现, 披露队长信息不一定能提高服务商收益和社会福利。     

Analysis of the M1,M2/G/1 G-queueing system with retrial customers

We consider a single server retrial queue with waiting places in service area and three classes of customers subject to the server breakdowns and repairs. When the server is unavailable, the arriving class-1 customer is queued in the priority queue with infinite capacity whereas class-2 customer enters the retrial group. The class-3 customers which are also called negative customers do not receive service. If the server is found serving a customer, the arriving class-3 customer breaks the server down and simultaneously deletes the customer under service. The failed server is sent to repair immediately and after repair it is assumed as good as new. We study the ergodicity of the embedded Markov chains and their stationary distributions. We obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law, the busy period of the system and the virtual waiting times. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analyzed numerically.     

Geo/Geo/1 retrial queue with non-persistent customers and working vacations

In this paper, we consider a Geo/Geo/1 retrial queue with non-persistent customers and working vacations. The server works at a lower service rate in a working vacation period. Assume that the customers waiting in the orbit request for service with a constant retrial rate, if the arriving retrial customer finds the server busy, the customer will go back to the orbit with probability q (0≤q≤1), or depart from the system immediately with probability $\bar{q}=1-q$ . Based on the necessary and sufficient condition for the system to be stable, we develop the recursive formulae for the stationary distribution by using matrix-geometric solution method. Furthermore, some performance measures of the system are calculated and an average cost function is also given. We finally illustrate the effect of the parameters on the performance measures by some numerical examples.     

带负顾客和非空竭服务随机休假的M~([x])/G/1可修排队系统

研究了带负顾客和非空竭服务随机休假的M~([X])/G/1可修排队系统.负顾客不仅仅移除一个正在接受服务的正顾客,而且还使得服务器损坏而立即进行修理.通过构造一个具有吸收态的马尔可夫链求得了系统稳态存在的充分必要条件.利用补充变量法求得了系统的排队指标和可靠性指标.最后我们还给出了一个数值实例.     

A queueing model for a nonreliable multiterminal system with polling scheduling

This paper deals with a nonhomogeneous finite-source queueing model to describe the performance of a multiterminal system subject to random breakdowns under the polling service discipline. The model studied here is a closed queueing network which has three service stations. a CPU (single server), terminals (infinite server), a repairman (single server), and a finite number of customers (jobs) that have distinct service rates at the service stations. The CPU's repair has preemptive priority over the terminal repairs, and failure of the CPU stops the service of the other stations, thus the nodes are not independent. It can be viewed as a continuation of papers by the authors (see references), which discussed a FIFO (first-in, first-out) and a PPS (priority processor sharing) serviced queueing model subject to random breakdowns. All random variables are assumed to be independent and exponentially distributed. The system behavior can be described by a Markov chain, but the number of states is very large. The purpose of this paper is to give a recursive computational approach to solve steady-state equations and to illustrate the problem in question using some numerical results. Supported by the Hungarian National Foundation for Scientific Research (grant Nos. OTKA T014974/95 and T016933/95) Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló, Hungary, 1997, Part, II.     

On the single server retrial queue subject to breakdowns

This paper deals with a single server retrial queueing system subject to active and independent breakdowns. The objective is to extend the results given independently by Aissani [1] and Kulkarni and Choi [15]. To this end, we introduce the concept of fundamental server period and an auxiliary queueing system with breakdowns and option for leaving the system. Then, we concentrate our attention on the limiting distribution of the system state. We obtain simplified expressions for the partial generating functions of the server state and the number of customers in the retrial group, a recursive scheme for computing the limiting probabilities and closed-form formulae for the second order partial moments. Some stochastic decomposition results are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs

Retrial queues have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, etc. It is of basic importance to study reliability of retrial queues with server breakdowns and repairs because of limited ability of repairs and heavy influence of the breakdowns on the performance measure of the system. However, so far the repairable retrial queues are analyzed only by queueing theory. In this paper we give a detailed analysis for reliability of retrial queues. By using the supplementary variables method, we obtain the explicit expressions of some main reliability indexes such as the availability, failure frequency and reliability function of the server. In addition, some special queues, for instance, the repairable M/G/1 queue and repairable retrial queue can be derived from our results. These results may be generalized to the repairable multi-server retrial models.     

On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper examines a discrete-time Geo/G/1 queue, where the server may take at most J − 1 vacations after the essential vacation. In this system, messages arrive according to Bernoulli process and receive corresponding service immediately if the server is available upon arrival. When the server is busy or on vacation, arriving messages have to wait in the queue. After the messages in the queue are served exhaustively, the server leaves for the essential vacation. At the end of essential vacation, the server activates immediately to serve if there are messages waiting in the queue. Alternatively, the server may take another vacation with probability p or go into idle state with probability (1 − p) until the next message arrives. Such pattern continues until the number of vacations taken reaches J. This queueing system has potential applications in the packet-switched networks. By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers. A cost model, developed to determine the optimum values of p and J at a minimum cost, is also studied.