Performance Analysis of a Markovian Working Vacations Queue with Impatient Customers

Performance Analysis of a Markovian Working Vacations Queue with Impatient Customers

P. Vijaya Laxmi (Andhra University, India), Veena Goswami (KIIT University, India) and K. Jyothsna (Andhra University, India)
DOI: 10.4018/978-1-4666-5958-2.ch013


This chapter analyzes a steady-state finite buffer M/M/1 working vacation queue wherein the customers can balk or renege. Unlike the classical vacation queues, the server can still render service to customers during the working vacations, at a different rate rather than completely terminating the service. The inter-arrival times of customers follow exponential distribution. The arriving customers either decide not to join the queue (that is, balk) with a probability or leave the queue after joining without getting served due to impatience (that is, renege) according to negative exponential distribution. The service times during a regular busy period, service times during a working vacation period, and vacation times are all independent and exponentially distributed random variables. Using Markov process, the steady-state equations are set and the steady-state system length distributions at arbitrary epoch are derived using blocked matrix method. A cost model is formulated to determine the optimum service rate. Sensitivity analysis is carried out to investigate the impact of the system parameters on various performance indices.
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Performance modeling of Markovian queues with balking and reneging has attracted many researchers due to their applications in real life congestion problems such as impatient telephone switchboard customers, the hospital emergency rooms handling critical patients, perishable goods storage in inventory systems, etc. Balking and reneging are common phenomena in queues; as a consequence, customers either decide not to join the queue or depart after joining the queue without getting service due to impatience, respectively. Modeling balking and reneging is worthwhile because one obtains new managerial insights. The lost revenues due to balking and reneging in various industries can be enormous. While making decision for the number of servers needed in the service system to meet time-varying demand, the balking and reneging probabilities can be used to estimate the amount of lost business in more practical consideration for the managers.

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 it shows the line 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 customers’ decision making can be predicted, a powerful set of tools will be available to customer service managers.

Queueing models with customers’ impatience 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 effect of balking and reneging in an M/M/1/N queue has been reported in Ancker and Gafarian (1963a, 1963b). Abou-El-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. Drekic and Woolford (2005) discussed a preemptive priority Markovian queue with state-dependent service and lower priority balking customers. An M/M/1 queueing system with random balking has been studied by Manoharan and Jose (2011). For further works on balking and reneging queues, one may refer Abou-El-Ata (1991), Altman and Yechiali (2008) and Al-seedy et al. (2009).

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