A Discrete-Time Single-Server Queueing System Under Multiple Vacations and Setup-Closedown Times

Abstract

This article concerns a Geo/G/1/∞ queueing system under multiple vacations and setup-closedown times. Specifically, the operation of the system is as follows. After each departure leaving an empty system, the server is deactivated during a closedown time. At the end of each closedown time, if at least a customer is present in the system, the server begins the service of the customers (is reactivated) without setup; however, if the system is completely empty, the server takes a vacation. At the end of each vacation, if there is at least a customer in the system, the server requires a startup time (is reactivated) before beginning the service of the customers; nevertheless, if there are not customers waiting in the system, the server takes another vacation. By applying the supplementary variable technique, the joint generating function of the server state and the system length together with the main performance measures are derived. We also study the length of the different busy periods of the server. The stationary distributions of the time spent waiting in the queue and in the system under the FCFS discipline are analysed too. Finally, a cost model with some numerical results is presented.     

On M/G/1 system under NT policies with breakdowns,startup and closedown

This paper studies the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times, in which the length of the vacation period is controlled either by the number of arrivals during the vacation period, or by a timer. After all the customers are served in the queue exhaustively, the server is shutdown (deactivates) by a closedown time. At the end of the shutdown time, the server immediately takes a vacation and operates two different policies: (i) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the closedown time. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. If some customers arrive during this closedown time, the service is immediately started without leaving for a vacation and without a startup time. We analyze the system characteristics for each scheme.     

Discrete time analysis of MAP/PH/1 vacation queue with gated time‐limited service

We analyse a single‐server queue in which the server goes through alternating periods of vacation and work. In each work period, the server attends to the queue for no more than a fixed length of time, T. The system is a gated one in which the server, during any visit, does not attend to customers which were not in the system before its visit. As soon as all the customers within the gate have been served or the time limit has been reached (whichever occurs first) the server goes on a vacation. The server does not wait in the queue if the system is empty at its arrival for a visit. For this system the resulting Markov chain, of the queue length and some auxiliary variables, is level‐dependent. We use special techniques to carry out the steady state analysis of the system and show that when the information regarding the number of customers in the gate is not critical we are able to reduce this problem to a level‐independent Markov chain problem with large number of boundary states. For this modified system we use a hybrid method which combines matrix‐geometric method for the level‐independent part of the system with special solution method for the large complex boundary which is level‐dependent. This revised version was published online in June 2006 with corrections to the Cover Date.     

带负顾客启动期N策略的Geom/Geom/1工作休假排队

将带RCH抵消策略的负顾客、启动期和N策略引入离散时间排队.休假策略为空竭服务多重工作休假.负顾客一对一抵消队首正在接受服务的正顾客,若系统中无正顾客时,到达的负顾客自动消失,负顾客不接受服务.利用拟生灭过程和矩阵几何解方法,给出了稳态队长分布及其随机分解.通过数值例子表现了启动率和负顾客到达率对稳态队长的影响.     

Scheduling service in tandem queues attended by a single server

Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the i^th queue is an exponentially distributed random variable with rate μ_i. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit     

An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

Discrete NT-policy single server queue with Markovian arrival process and phase type service

We consider a discrete time single server queueing system in which arrivals are governed by the Markovian arrival process. During a service period, all customers are served exhaustively. The server goes on vacation as soon as he/she completes service and the system is empty. Termination of the vacation period is controlled by two threshold parameters N and T, i.e. the server terminates his/her vacation as soon as the number waiting reaches N or the waiting time of the leading customer reaches T units. The steady state probability vector is shown to be of matrix-geometric type. The average queue length and the probability that the server is on vacation (or idle) are obtained. We also derive the steady state distribution of the waiting time at arrivals and show that the vacation period distribution is of phase type.     

The performance measures and randomized optimization for an unreliable server M/G/1 vacation system

This paper examines an M^[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 − p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. Whenever one or more customers arrive at server idle state, the server immediately starts providing service for the arrivals. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the distributions of important system characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, the distributions of idle period, busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗, J^∗) at a minimum cost, and some numerical examples are presented for illustrative purpose.     

Exponential Queues with Server Vacations

In this paper, we analyse a queueing system where the server may take a vacation. The customers arrive at the service facility according to a Poisson process, and are served if the server is available (not on vacation). We consider two models: when the server vacation cycle is independent of and dependent on the number of customers in the system. The infinitesimal generators of the underlying Markov processes have a block tri-diagonal structure, and we provide a matrix geometric solution. When the vacation cycle is independent of the customer queue length, we present a simple load-dependent approximation that is fairly accurate.     

An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server

This paper considers the bi-level control of an M/G/1 queueing system, in which an un-reliable server operates N policy with a single vacation and an early startup. The server takes a vacation of random length when he finishes serving all customers in the system (i.e., the system is empty). Upon completion of the vacation, the server inspects the number of customers waiting in the queue. If the number of customers is greater than or equal to a predetermined threshold m, the server immediately performs a startup time; otherwise, he remains dormant in the system and waits until m or more customers accumulate in the queue. After the startup, if there are N or more customers waiting for service, the server immediately begins serving the waiting customers. Otherwise the server is stand-by in the system and waits until the accumulated number of customers reaches or exceeds N. Further, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. We obtain the probability generating function in the system through the decomposition property and then derive the system characteristics     

带有负顾客的N策略工作休假M/M/1排队

考虑带有正、负顾客的N策略工作休假M/M/1排队。负顾客一对一抵消队尾的正顾客(若有),若系统中无正顾客,到达的负顾客自动消失,负顾客不接受服务。在休假期间,服务员并未完全停止工作而是以较低的服务率为顾客服务。用拟生灭过程和矩阵几何解方法,我们给出了稳态队长和稳态等待时间的分布。此外,我们也证明了稳态条件下的队长和等待时间的条件随机分解并得到了附加队长和附加延迟的分布。     

Pseudo-conservation laws in cyclic-service systems with a class of limited service policies

This paper considers a cyclic-service system with a class of limited service policies that consists of exhaustive limited, gated limited and general decrementing policies. Under these policies, the number of customers served consecutively during a server visit is limited by a vector of integers. The major results in this paper are derivations of expected amount of work left in the queues at the server departures for these three policies. Exact expressions of weighted sum of mean waiting times, known as pseudo-conservation laws, are subsequently obtained. The conservation laws for this class of policies contain unknown boundary probabilities. We estimate these probabilities using corresponding server vacation models. Numerical results presented for the exhaustive limited policy are noted to be very accurate compared with simulation results. Moreover, we have obtained analytical bounds for the weighted sums. Finally, we present a conservation law with mixed service policies, and mean waiting times for symmetric systems.This work was completed while the author was in the Ph.D. program at Rensselaer Polytechnic Institute.This work was partially supported by the Center for Advanced Technologies of the New York State.     

带RCE抵消策略的负顾客M/M/1工作休假排队系统

考虑服务员在休假期间不是完全停止工作,而是以相对于正常工作时低些的速率服务顾客的M/M/1工作休假排队模型.在此模型基础上,笔者针对现实的M/M/1排队模型中可能出现的外来干扰因素,提出了带RCE(Removal of Customers at the End)抵消策略的负顾客M/M/1工作休假排队这一新的模型.服务规则为先到先服务.工作休假策略为空竭服务多重工作休假.抵消原则为负顾客一对一抵消队尾的正顾客,若系统中无正顾客时,到达的负顾客自动消失,负顾客不接受服务.使用拟生灭过程和矩阵几何解方法给出了系统队长的稳态分布,证明了系统队长和等待时间的随机分解结果并给出稳态下系统中正顾客的平均队长和顾客在系统中的平均等待时间.     

Discrete Time Geo/G/1 Queue with Multiple Adaptive Vacations

This paper treats the discrete time Geometric/G/1 system with vacations. In this system, after serving all customers in the system, the server will take a random maximum number of vacations before returning to the service mode. The stochastic decomposition property of steady-state queue length and waiting time has been proven. The busy period, vacation mode period, and service mode period distributions are also derived. Several common vacation policies are special cases of the vacation policy presented in this study.     

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.     

Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N(∞) under gated-limited discipline

This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process (BMAP) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations. The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms.     

A simple approach for analyzing feedback vacation queues with levy input process

We introduce a simple approach for modeling and analyzing a SII/G/I queue where the server may take repeated vacations. Whenever a busy period ends the server takes a vacation of random duration. At the end of each vacation the server may either take a new vacation or resume service; if the queue is found empty the server always takes a new vacation. Furthermore, the queuing system allows Bernoulli feedback of customers. Three classes of service disciplines, random gated, 1-limited and exhaustive, are considered. The random gated service discipline generalizes several known service disciplines. The customers arrival process is assumed to be a Levy process (i.e., satisfies the stationary and independent increments (SII property). We obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, the mean queue length at the beginning of a cycle of service and the expected delay observed by a customer. Furthermore, our analysis provides a uniform method to get several results previously obtained by Baba, Chiarawongse and Sriniwasan, and Takine, Takagi and Hasegawa.     

Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.