Vacation Of Random Duration Research Articles

Analysis of a production inventory system with MAP arrivals, phase-type services and vacation to production facility

In this paper, we discuss a production inventory system with service times. Customers arrive in the system according to a Markovian arrival process. The service times follow a phase-type distribution. We assume that there is an infinite waiting space for customers. Arriving customers demand only one unit of item from the inventory. The production facility produces items according to an (s, S)-policy. Once the inventory level becomes the maximum level S, the production facility goes on a vacation of random duration. When the production facility returns from the vacation, if the inventory level depletes to the fixed level s, it is immediately switched on and starts production until the inventory level becomes S. Otherwise, if the inventory level is greater than s on return from the vacation, it takes another vacation. The vacation times are exponentially distributed. The production inventory system in the steady-state is analyzed by using the matrix-geometric method. A numerical study is performed on the system performance measures. Besides, an optimization study is discussed for the inventory policy.

Matrix-geometric solution of multi-server queueing systems with Bernoulli scheduled modified vacation and retention of reneged customers: A meta-heuristic approach

ABSTRACT In the present article, we deal with the multi-server finite capacity queueing system with Bernoulli’s scheduled modified vacation policy and the realistic retaining policy of reneged customers. After completion of the service of any customer, the server decides whether to go for the vacation of random duration or to continue facilitating the service to the waiting customer, if any, present in the queue. All servers, homogeneous in nature, provide the state-dependent service following the threshold policy to reduce the overload of the system. The impatience behavior of customers like balking and reneging is also considered in the stochastic modeling of the studied problem. The matrix analytic approach is employed to obtain the steady-state probabilities with which various system performance measures are also developed with practical justification. Finally, the expected cost minimization problem is formulated and dealt with the meta-heuristic approach: particle swarm optimization (PSO). All results of numerical simulation and optimal analysis are summarized in tables and graphs to provide quick insight. The concluding remarks and future scopes have also been discussed.

A Production Inventory System with Service Time and Production Vacations

In this paper, we consider a production inventory system with service time and production vacations. Customers arrive in the system according to a Poisson process requiring service from a single server. The single production facility produces items according to an (s, S) policy, and it takes a vacation of random duration once the inventory level becomes (S. It is assumed that all arriving customers are lost during the stock out period. We first derive the stationary joint distribution of the queue length and the on-hand inventory level in product form. Then, we compute explicitly some performance measures, and develop a cost function based on these performance measures. Finally, some numerical results are presented.

Analysis of State Dependent Vacation Queues with Threshold Gated Service Policy

In this article, the authors introduce a simple approach for modeling and analyzing a queue where the server may take repeated vacations. When a busy period ends, the server takes a vacation of random duration. At the end of each vacation, the server may either start a new vacation or resume service. If a queue is found of less than customers, the server will always take a new vacation. If there are at least customers in queue, the server provides services to those customers after a brief set-up time. The authors obtain several performance measures of the system, including the mean and second moment of the cycle time, the number of customers in a cycle of service, and the expected delay experienced by a customer.

On an (s, S) Inventory Policy with Service Time, Vacation to Server and Correlated Lead Time

We consider an inventory with random positive service time. The control policy is of (s, S) type. Customer demands constitute a Markovian arrival process (MAP). Service times have phase type distribution. Lead time for replenishment of inventory follows a correlated process similar to the customer arrival process. At a service completion epoch if no customer is waiting or the inventory level is zero then the server goes on a vacation of random duration. The vacation time is also phase type distributed. On return from vacation if the server finds no customer waiting or inventory level is zero, the server goes on another vacation of random duration having the same phase type distribution as the earlier one; we assume that the successive vacation times are independent of each other. This goes on until on return he finds at least one customer and at least one item in the inventory at which he starts serving. The customer arrival process is subject to balking. While waiting for service, customers may become impatient and leave the system (renege). Under these assumptions the underlying level dependent quasi birth-death process (LDQBD) is analysed. Several particular cases are considered in which the system is exhaustively studied. Numerical illustrations are provided. A cost function is introduced to numerically evaluate the optimal values of s and S.

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 (S II 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.

Optimal control of the M/G/1 queue with repeated vacations of the server

An M/G/1 queue where the server may take repeated vacations is considered. Whenever a busy period terminates, 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. The cost structure includes a holding cost per unit of time and per customer in the system and a cost each time the server is turned on. One discounted cost criterion and two average cost criteria are investigated. It is shown that the vacation policy that minimizes the discounted cost criterion over all policies (randomized, history dependent, etc.) converges to a threshold policy as the discount factor goes to zero. This result relies on a nonstandard use of the value iteration algorithm of dynamic programming and is used to prove that both average cost problems are minimized by a threshold policy.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stochastic petri net analysis of finite-population vacation queueing systems

We consider queueing systems in which the server occasionally takes a vacation of random duration. The vacation can be used to do additional work; it can also be a rest period. Several models of this problem have been analyzed in the past assuming that the population of the system is infinite. Similarly, it is generally assumed that the capacity of the system is infinite. In this paper we show how the finite-population system can be modeled by the stochastic Petri net. We also extend the model to the finite-capacity system.

Bulk arrival queues with server vacations

We consider M/G/1 bulk arrival queues with server vacations. Server vacation queues are those in which the server leaves for a vacation as soon as the system becomes empty, and returns after a vacation of random duration. We consider two types of vacation policies: (1) if the server finds no customer upon return from a vacation, he immediately leaves for another one, and so on, until he finally finds at least one customer; (2) if the server finds no customer upon return, he waits for the first customer group to arrive. We derive the server idle probabilities, waiting time distributions, and the mean system sizes.