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1.
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.  相似文献   

2.
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.  相似文献   

3.
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  相似文献   

4.
This paper studies the operating characteristics of the variant of an M[x]/G/1 vacation queue with startup and closedown times. After all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time, and then takes at most J vacations of constant time length T repeatedly until at least one customer is found waiting in the queue upon returning from a vacation. If at least one customer is present in the system when the server returns from a vacation, then the server reactivates and requires a startup time before providing the service. On the other hand, if no customers arrive by the end of the J th vacation, the server remains dormant in the system until at least one customer arrives. We will call the vacation policy modified T vacation policy. We derive the steady‐state probability distribution of the system size and the queue waiting time. Other system characteristics are also investigated. The long‐run average cost function per unit time is developed to determine the suitable thresholds of T and J that yield a minimum cost. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

5.
This paper deals with an N policy M/G/1 queueing system with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches N(N ? 1), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. We analyze various system performance measures and investigate some designated known expected cost function per unit time to determine the optimal threshold N at a minimum cost. Sensitivity analysis is also studied.  相似文献   

6.
《Applied Mathematical Modelling》2014,38(21-22):5113-5125
This paper deals with the (p, N)-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1  p). When the server returns from vacation and the system size exceeds N, the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N. It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes.  相似文献   

7.
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.  相似文献   

8.
This paper studies the operating characteristics of an M[x]/G/1 queueing system with N-policy and at most J vacations. The server takes at most J vacations repeatedly until at least N customers returning from a vacation are waiting in the queue. If no customer arrives by the end of the Jth vacation, the server becomes idle in the system until the number of arrivals in the queue reaches N. We derive the system size distribution at a random epoch and departure epoch, as well as various system characteristics.  相似文献   

9.
In this paper a MX/G (a, b)/1 queueing system with multiple vacations, setup time with N-policy and closedown times is considered. On completion of a service, if the queue length is ξ, where ξ < a, then the server performs closedown work. Following closedown the server leaves for multiple vacations of random length irrespective of queue length. When the server returns from a vacation and if the queue length is still less than ‘N’, he leaves for another vacation and so on, until he finds ‘N’ (N > b) customers in the queue. That is, if the server finds at least ‘N’ customers waiting for service, then he requires a setup time ‘R’ to start the service. After the setup he serves a batch of ‘b’ customers, where b  a. Various characteristics of the queueing system and a cost model with the numerical solution for a particular case of the model are presented.  相似文献   

10.
We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.  相似文献   

11.
12.
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.  相似文献   

13.
This paper considers two types of setup/closedown policies: interruptible and insusceptible setup/closedown policies. When all customers are served exhaustively in a system under the interruptible setup/closedown policy, the server shuts down (deactivates) by a closedown time. When the server reactivates since shutdown, he needs a setup time before providing service again. If a customer arrives during a closedown time, the service is immediately started without a setup time. However, in a system under the insusceptible setup/closedown policy, customers arriving in a closedown time can not be served until the following setup time finishes. For the systems with interruptible setup/closedown times, we assume both the fully and almost observable cases, then derive equilibrium threshold strategies for the customers and analyze the stationary behavior of the systems. On the other hand, for the systems with insusceptible setup/closedown times, we only consider the fully observable case. We also illustrate the equilibrium thresholds and the social benefits for systems via numerical experiments. As far as we know, there is no work concerning equilibrium behavior of customers in queueing systems with setup/closedown times.  相似文献   

14.
In this paper, we study an M/M/c queue with a three threshold vacation policy denoted by (e, d, N). With such a policy, the servers keep serving the customers until the number of idle servers reaches d and then e of d servers start taking a vacation together. These e servers keep taking vacations until the number of customers in the system is at least N at a vacation completion instant, then the e servers return to serve the queue again. Using the matrix analytic method, we obtain the stationary performance measures and prove the conditional stochastic decomposition properties for the waiting time and queue length. This model is a generalization of previous multi-server vacation models and offers a useful performance evaluation and system design tool in multi-task server queueing systems.  相似文献   

15.
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 (pJ) at a minimum cost, and some numerical examples are presented for illustrative purpose.  相似文献   

16.
An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.  相似文献   

17.
This paper analyzes the F-policy M/M/1/K queueing system with working vacation and an exponential startup time. The F-policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K, the optimal threshold F and the optimal service rates (μB,μV) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.  相似文献   

18.
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.  相似文献   

19.
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.  相似文献   

20.
A discrete time Geo/Geo/1 queue with (mN)-policy is considered in this paper. There are three operation periods being considered: high speed, low speed service periods and idle periods. With double thresholds policy, the server begins to take a working vacation when the number of customers is below m after a service and there is one customer in the system at least. What’s more, if the system becomes empty after a service, the server will take an ordinary vacation. Otherwise, high speed service continues if the number of customers still exceeds m after a service. At the vacation completion instant, servers resume their service if the quantity of customers exceeds N. Vacations can also be interrupted when the system accumulate customers more than the prefixed threshold. Using the quasi birth-death process and matrix-geometric solution methods, we derive the stationary queue length distribution and some system characteristics of interest. Based on these, we apply the queue to a virtual channel switching system and present various numerical experiments for the system. Finally, numerical results are offered to illustrate the optimal (mN)-policy to minimize cost function and obtain practical consequence on the operation of double thresholds policy.  相似文献   

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