[Show abstract][Hide abstract] ABSTRACT: We consider a single-server retrial queue with infinite capacity of the primary buffer and finite capacity of the orbit to which customers arrive according to a Poisson process, and the service time follows phase-type distribution. The customer-induced interruption while in service occurs according to a Poisson, process. The self-interrupted customers enter into orbit. Any interrupted customer, finding the orbit full, is considered lost. The interrupted customers retries for service after the interruption is completed. We investigate the behavior of this queueing system. Several performance measures are evaluated. Numerical illustrations of the system behavior are also provided.
American Journal of Mathematical and Management Sciences 10/2015; 34(4). DOI:10.1080/01966324.2015.1042562
[Show abstract][Hide abstract] ABSTRACT: In this paper we discuss a queueing system with service interruption. The service gets interrupted due to different environmental factors. Here it is assumed that interruption due to only one factor is allowed at a time. Further we assume that while in interruption no other interruption befalls the system. Even though any number of interruptions can occur during the service of a customer, the maximum number of interruptions is restricted to a finite number K and if the number of interruptions exceeds the maximum, the customer leaves the system without completing service. The difference between the model under discussion and those considered earlier in literature is that the customer/server is unaware of the interruption until a random amount of time elapses from the moment interruption strikes. At the moment the interruption occurs, a random clock and a superclock start ticking. The interruption is identified only when the random clock is realized. The superclock measures the total interruption time during the service of a customer. On realization of superclock the customer goes out of the system without completing service. The kind of service to be started after the interruption depends on the environmental factor that caused the interruption. Here we first analyze the service process to find the response time and to compute the stability condition. The optimal values of K for a suitable cost function is investigated. Numerical investigations indicates the cost function as convex/increasing/decreasing in K.
Annals of Operations Research 08/2015; 233(1). DOI:10.1007/s10479-015-1931-4 · 1.22 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we model a queueing-inventory system that has applications in railway and airline reservation systems. Maximum items in the inventory is \(S\) which have a random common life time; this includes those that are sold in particular cycle. A customer, on arrival to an idle server with at least one item in inventory, is immediately taken for service; or else he joins the buffer of maximum size \(S\) depending on number of items in the inventory (the buffer capacity varies and is, at any time, equal to the number of items in the inventory). The arrival of customers constitutes a Poisson process, demanding exactly one item each from the inventory. If there is no item in the inventory, the arriving customer first queue up in a finite waiting space of capacity \(K\) . When it overflows an arrival goes to an orbit of infinite capacity with probability \(p\) or is lost forever with probability \(1-p\) . From the orbit he retries for service according to an exponentially distributed inter-occurrence time. The service time follows an exponential distribution. Cancellation of sold items before its expiry is permitted. Inventory gets added through cancellation of purchased items, until the expiry time. Cancellation time is assumed to be negligible. We analyze this system. Several performance characteristics are computed; expected sojourn time of the system in a cycle with “no inventory” and also “maximum inventory” are computed. Some illustrative numerical examples are presented. An optimization problem is numerically analyzed.
Annals of Operations Research 04/2015; DOI:10.1007/s10479-015-1849-x · 1.22 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We attempt to derive the steady-state distribution of the queueing-inventory system with positive service time. First we analyze the case of servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair and the corresponding expected minimum cost are computed. As in the case of retrial queue with , we conjecture that for , queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutive to transitions of the inventory level (i.e., the first return time to ) is computed. We also obtain several system performance measures.
Advances in Operations Research 01/2015; 2015:1-16. DOI:10.1155/2015/747328
[Show abstract][Hide abstract] ABSTRACT: A queueing-inventory system, with the item given with probability γ to a customer at his service completion epoch, is considered in this paper. Two control policies, (s,Q) and (s,S) are discussed. In both cases we obtain the joint distribution of the number of customers and the number of items in the inventory as the product of their marginals under the assumption that customers do not join when inventory level is zero. Optimization problems associated with both models are investigated and the optimal pairs (s,S) and (s,Q) and the corresponding expected minimum costs are obtained. Further we investigate numerically an expression for per unit time cost as a function of γ. This function exhibit convexity property. A comparison with Schwarz et al. (Queueing Syst. 54:55–78, 2006) is provided. The case of arbitrarily distributed service time is briefly indicated.
Annals of Operations Research 08/2013; 233(1). DOI:10.1007/s10479-013-1437-x · 1.22 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider a multiserver Markovian queuing model where each server provides service only to one customer. Arrival of customers is according to a Poisson process and whenever a customer leaves the system after getting service, that server is also removed from the system. Here the servers are considered as an inventory that will be replenished according to the standard (s,S) policy. Behavior of this system is studied using a two dimensional QBD process. The condition for checking ergodicity, the steady state solutions and average inventory cycle time are obtained using matrix analytic methods. Also we have studied an optimization problem that minimizes the total cost induced by the waiting cost of arrivals, holding cost of the inventory of servers and ordering cost. Some numerical illustrations are provided.
Annals of Operations Research 06/2013; 233(1). DOI:10.1007/s10479-013-1405-5 · 1.22 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we study a MAP/PH/1 queueing model in which the server is subject to taking vacations and offering services at a lower rate during those times. The service is returned to normal rate whenever the vacation gets over or when the queue length hits a specific threshold value. This model is analyzed in steady state using matrix analytic methods. An illustrative numerical example is discussed.
[Show abstract][Hide abstract] ABSTRACT: A queueing model consisting of two multi-server service systems is considered. Primary customers arrive at a multi-server queueing system-1 having an infinite buffer. The input flow is described by a MAP (Markovian Arrival Process). The service time of a primary customer has a PH (Phase-type) distribution. Besides the primary customers, a MAP of interruptions arrives to the system. An interruption removes one of the primary customers from the service if the state (phase) of its PH service process does not belong to some given set of so called protected phases when an interruption is successful. The interrupted customer leaves the system permanently with some probability. With complementary probability, the interrupted primary customer moves for service to system-2. This system consists of K independent identical servers and has no buffer. If all K servers are busy at the moment of a primary customer interruption, this customer will be lost. Otherwise, this primary customer starts the service in an arbitrary idle server of system-2. It is assumed that the service time of a primary customer by a server of system-2 has a PH distribution. Upon completion of the service at system-2, the customer becomes priority customer. If, at the service completion moment, there are free servers at system-1, the priority customer immediately starts getting the service at system-1. It is assumed that the service time of a priority customer by a server of system-1 has a PH distribution and this service can not be interrupted. If, at the moment when the primary customer finished the service at system-2, there are no idle servers at system-1, this customer is placed into the finite buffer for priority customers of capacity K. The customers will be picked up for the service according to the FIFO discipline. When a server of system-1 becomes free, it takes for service a priority customer from this buffer, if any. Type-1 customers are picked-up from the infinite buffer only if the buffer for priority customers is empty at the service completion moment at system-1.
Behavior of this system is described by a multi-dimensional Markov chain. Algorithms for checking ergodicity condition and computing the stationary distribution are presented. Formulas for computing important performance measures of the system are derived.
Annals of Operations Research 01/2013; 233(1). DOI:10.1007/s10479-013-1318-3 · 1.22 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Server induced interruptions such as server break downs, server attending a high priority customer, and server taking a vacation in queues have been extensively studied in the literature. However, customer-induced interruptions such as customers leaving in the middle of a service due to not having enough information for completing a service and customer breakdowns have not been studied so far. The purpose of this work is to introduce customer interruptions in queueing systems. We consider an infinite capacity queueing system with a single server to which customers arrive according to a Poisson process and the service time follows an exponential distribution. The customer interruption while in service occurs according to a Poisson process and the interruption duration follows an exponential distribution. The self-interrupted customers will enter into a finite buffer of size K. Any interrupted customer, finding the buffer full, is considered lost. Those interrupted customers who complete their interruptions will be placed into another buffer of same size. The interrupted customers waiting for service are given non-preemptive priority over new customers. We investigate the behavior of this queuing system. Several performance measures are evaluated. Numerical illustrations of the system behavior are also provided. An optimization problem of interest will be discussed through an illustrative example.
[Show abstract][Hide abstract] ABSTRACT: We study a multi-server queueing system with customer induced interruption of service to which customers arrive according to Poisson process and service time follows an exponential distribution. The customer induced interruption occurs according to Poisson process and the interruption duration follows an exponential distribution. The self-interrupted customer will enter into a buffer of finite capacity. Any self-interrupted customer, finding the buffer full, is considered lost for ever. Those self-interrupted customers who complete their interruption will be placed into another buffer of same size. The self-interrupted customers waiting for service are given non-preemptive priority over new customers. We investigate the behavior of this queueing system. Several performance measure are evaluated. Numerical illustrations of the system behavior are also provided. Optimization problem to maximize the revenue with respect to number of servers to be employed and optimal buffer size for the self-interrupted customers are discussed through two illustrative examples.
Neural, Parallel and Scientific Computations 06/2012; 20(2):153-172.
[Show abstract][Hide abstract] ABSTRACT: This paper analyzes an M / M /2 queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.
International Journal of Stochastic Analysis 01/2012; 2012(2). DOI:10.1155/2012/145867
[Show abstract][Hide abstract] ABSTRACT: A detailed review of inventory models involving positive service time is given. These include classical and retrial cases. A detailed review of inventory models involving positive service time is given. These include classical and retrial cases.
Also contributions to production inventory with service time is indicated towards the end. In addition directions for future Also contributions to production inventory with service time is indicated towards the end. In addition directions for future
work are indicated. work are indicated.
KeywordsInventory with positive service time–Classical inventory–Retrial inventory with/without production KeywordsInventory with positive service time–Classical inventory–Retrial inventory with/without production
[Show abstract][Hide abstract] ABSTRACT: This paper considers a production inventory with positive service time. The time for producing each item is assumed to follow a “Markovian production scheme”. Because of this, correlation automatically gets into the production process. The customer arrival process follows a Markovian arrival process. When inventory falls to s, the production process is switched on, and is switched off when the on-hand inventory reaches S. The service time to each customer has a phase-type distribution. We investigate the system stability. Under the condition of stability, we investigate the system state distribution. Next, several performance measures are computed, such as the fraction of time server is on vacation; fraction of time the inventory level is zero; fraction of time the production process is On; expected number of customers in the system while server is on vacation and also while it is busy. Numerical results indicate, among other things, the effect of the control variables s and S on the fraction of time the system goes out of inventory and on expected loss rate of customers.
IMA Journal of Management Mathematics 12/2010; 21(3). DOI:10.1093/imaman/dpp025 · 0.50 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A Markovian single server feedback retrial queue with linear retrial rate and collisions of customers is studied. Using generating function technique, the joint distribution of the server state and the orbit length under steady-state is investigated. Some interesting and important performance measures of the system are obtained. Finally, numerical illustrations are provided.
[Show abstract][Hide abstract] ABSTRACT: We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability ÃŽÂ³ and is lost forever with probability (1Ã¢ÂˆÂ’ÃŽÂ³). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability ÃŽÂ´ and is lost forever with probability (1Ã¢ÂˆÂ’ÃŽÂ´). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability ÃŽÂ³ and is lost forever with probability (1Ã¢ÂˆÂ’ÃŽÂ³). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability ÃŽÂ´ and is lost forever with probability (1Ã¢ÂˆÂ’ÃŽÂ´). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.
Journal of Applied Mathematics and Stochastic Analysis 01/2008; 2007. DOI:10.1155/2007/37848
[Show abstract][Hide abstract] ABSTRACT: This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N(<M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.
[Show abstract][Hide abstract] ABSTRACT: A transient solution is obtained analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and non-zero repair time. The steady state probability of the system
size is present. Some key performance measures, namely, throughput, loss probability and response time for the system under
consideration are investigated. Further, reliability and availability of the system are analysed. Finally, numerical illustrations
are used to discuss the system performance measures.
Queueing Systems 08/2007; 56(3):133-141. DOI:10.1007/s11134-007-9014-0 · 0.84 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of
a positive customer who initiates a busy period is G
(x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution
(x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences.
This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers
are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various
parameters on the mean system size and the probability that the system is empty are also analysed numerically.
Annals of Operations Research 03/2006; 143(1):277-296. DOI:10.1007/s10479-006-7388-8 · 1.22 Impact Factor