Modified vacation policy for M/G/1 retrial queue with balking and feedback

Abstract

This paper studies a general retrial queue with balking and Bernoulli feedback, where the server operates a modified vacation policy. If the server is busy or on vacation, an arriving customer either enters an orbit with probability b, or balks (does not enter) with probability 1 − b. Otherwise the service of the arriving customer commences immediately. At any service completion epoch, the test customer may either enter the orbit for another service with probability p or leave the system with probability 1 − p. If the orbit is empty, the server takes at most J vacations until at least one customer is recorded in the orbit when the server returns from a vacation. This retrial system has potential applications in e-mail system and WWW server. By applying the supplementary variable technique, some important performance measures are derived. The effect of various retrial/vacation time distributions and different values of J on the system performance measures is also investigated. The analyses and results presented in this paper may be useful for network system designers and software system engineers.

Introduction

Retrial queueing system is characterized by the feature that the arriving customers who encountering the server busy join a trial queue called orbit. An arbitrary customer in the orbit generates a stream of repeated requests that is independent of the rest of customers in the orbit. Such queueing systems play important roles in the analysis of many telephone switching systems, telecommunication networks and computer systems. Review of retrial queue literature could be found in Yang and Templeton, 1987, Falin and Templeton, 1997, Artalejo, 1999. A number of applications of retrial queues in science and engineering can be found in Kulkarni and Liang (1997). Diamond and Alfa (1999) constructed a method for approximating the stationary distribution and waiting time moments of an M/PH/1 retrial queue with phase type inter-retrial times.

In a retrial queue, an arriving customer who finds the server busy has to leave and may retry later. Such queues model many real world situations, including web access, call centers, telecommunication networks and computer systems. Artalejo and Lopez-Herrero (2000) studied the M/G/1 retrial queue with balking probabilities depending on the number of customers in the system upon arrival, where the limiting distribution of the number of customers in the system is determined with the help of a recursive approach based on the theory of regenerative processes. Lopez-Herrero (2002) presented the explicit formulae for the probabilities of the number of customers (denoted by I) being served in a busy period, and an explicit expression for the second moment of I for the M/G/1 retrial queueing system was also been given. Lopez-Herrero further employed the principle of maximum entropy to estimate the distributions of I. An M/G/1 retrial queueing system with additional phase of service and possible preemptive resume service discipline was investigated by Kumar, Vijayakumar, and Arivudainambi (2002a).

Many queueing situations have the feature that the customers may be served repeatedly for a certain reason. When the service of a customer is unsatisfied, it may be retried again and again until a successful service completed. These queueing models arise in the stochastic modeling of many real-life situations. For example, in data transmission, a packet transmitted from the source to the destination may be returned and it may go on like that until the packet is finally transmitted. Kumar, Madheswari, and Vijayakumar (2002b) examined the M/G/1 retrial queue with feedback including some applications, where the server is subjected to starting failure. Recently, The BMAP/G/1 retrial system with search for customers immediately on termination of service was studied by Dudin, Krishnamoorthy, Joshua, and Tsarenkov (2004), in which the inter-retrial time is followed an exponential distribution and the duration of search is characterized by a generally distributed random variable.

The modeling analysis for the queueing systems with vacations has been done by a considerable amount of work in the past and successfully used in various applied problems such as production/inventory systems, communication systems, computer systems, etc. (see survey paper by Doshi (1986)). A comprehensive and excellent study on the vacation models can be found in Takagi (1991). For related literature of retrial queues with vacations, Li and Yang (1995) developed an M/G/1 retrial system with server vacations and M independent, identical input sources. Li and Yang’s vacation policy is described as the following: in case when the server is idle at the time of the arrival of a fresh or returning customer, he starts to serve a customer with probability α_k or takes a vacation with probability 1 − α_k (k denotes the number of returning customers present in the system at the time of the arrival). They further used a CSMA/CD based local area network as a numerical example to explain how their model can be applied to study network properties which have interesting implications to system control and design. Later, Artalejo (1997) analyzed an M/G/1 retrial queue with exhaustive server vacations, i.e. the server takes a vacation only when there are no customers in the system. Kumar et al., 2002a, Kumar et al., 2002b dealt with an M/G/1 retrial queue where the server operates according to a Bernoulli vacation policy as described by Keilson and Servi (1986). Recently, Atencia, Fortes, Moreno, and Sánchez (2006) analyzed an M/G/1 retrial queue with active breakdowns where the customer being served during the failure decides, with probability q, to join the orbit (impatient customer) and, with complementary probability p, to remain in the server for the repair in order to conclude his remaining service (patient customer). They derived various system performance measures and evaluated some characteristic quantities of such queueing system using the supplementary variable technique.

In this paper, we consider an M/G/1 retrial system with balking and feedback under preemptive resume discipline, where the server applies a modified vacation policy when no customers are recorded in the orbit. The vacation policy of our model varies from other approaches and it is when no customers are recorded in the orbit, the server goes on at most J vacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. The server remains dormant in the system to wait for customers whenever the orbit is not empty upon returning from a vacation or at least one customer appears after he returns from the Jth vacation. More special, so far very few authors have studied the comparable work on the variant vacation policy for the retrial queueing models which the server may take a sequence of finite vacations in his idle time. This motivates us to develop the modified vacation policy for M/G/1 retrial system with balking and feedback, in which the server may take at most J vacations when the orbit is empty.

This paper is organized as follows. We first describe the model with two examples. Secondly, we construct the mathematical model for the M/G/1 retrial queue with balking and feedback, where the server takes at most J vacations utilizing his idle time. After that, the steady-state distribution of the server state and the number of customers in the system/orbit are obtained. We also derive some important performance measures of this system and compute some characteristic quantities. Finally, we show that a general decomposition law also holds for this variant vacation system.