Analysis of Finite Buffer Markovian Queue with Balking, Reneging and Working Vacations

Introduction

In recent years, there is an emerging trend to study queueing systems from an economic view point. Traditional models of queueing systems do not consider the economic costs and therefore, these models have been criticized as being inappropriate for service providers. Many queueing situations arise in real life wherein the customers are discouraged by a longer queue resulting in a long wait anticipated by him upon arrival. As a result, the customers either decide not to join the queue or depart after joining the queue without receiving service due to impatience. Balking means the phenomenon of customers arriving for service into a non-empty queue and leaving without joining the queue. On the other hand, a customer is said to have reneged if after joining the queue he gets impatient and leave the system without receiving service. The losses in revenues in various industries due to balking and reneging are enormous and thus need to be studied in suitable context.

When customers have to wait for long more dissatisfaction is likely to arise. Perhaps, service providers may be interested in allowing certain amount of balking if it leads to a higher level of customer satisfaction. It is more important to reduce reneging as renegers make the line seem longer for those contemplating joining and may prompt some to balk. At the same time customers become dissatisfied because of the total length of time they have to spend before being served. If customer’s decision making can be predicted, a powerful set of tools will be available to customer service managers.

Models with customers’ impatience in queues have been studied extensively due to their versatility and applicability. An M/M/1 queue with customers balking and reneging has been discussed in Haight (1957) and (1959), respectively. The combined effects of balking and reneging in an M/M/1/N queue has been reported in Ancker and Gafarian (1963a, 1963b). Analytical solutions of the single-server Markovian overflow queue with balking, reneging and an additional server for longer queue have been discussed in Abou-El-Ata and Shawky (1992). Al-Seedy and Kotb (1991) studied the transient solution of the state-dependent M/M/1 queue with balking. Drekic and Woolford (2005) discussed a preemptive priority Markovian queue with state-dependent service and lower priority balking customers. Abou-EI-Ata and Hariri (1992) discussed the finite buffer multiple server queueing system with balking and reneging. Choudhury (2004) considered a single server finite buffer queueing system assuming reneging customers. Some of its variations have been studied by several authors including Abou-EI-Ata (1991), Altman and Yechiali (2008), Baccelli et al. (1984), Kok and Tijms (1985) and Shawky (1997).