Performance Analysis of a Markovian Queuing System with Reneging and Retention of Reneged Customers

Performance Analysis of a Markovian Queuing System with Reneging and Retention of Reneged Customers

Rakesh Kumar (Shri Mata Vaishno Devi University, India)
DOI: 10.4018/978-1-5225-1793-1.ch030
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In this chapter a finite capacity single server Markovian queuing system with reneging and retention of reneged customers is considered. It is envisaged that a reneging customer may be convinced to stay for his service if some customer retention mechanism is employed. Thus, there is a probability that a reneging customer may be retained. Steady-state balance equations of the model are derived using Markov chain theory. The steady-state probabilities of system size are obtained explicitly by using iterative method. The performance measures like expected system size, expected rate of reneging, and expected rate of retention are obtained. The effect of probability of retaining a reneging customer on the performance measures is studied. The economic analysis of the model is performed by developing a cost model. The optimum service rate and optimum system capacity are obtained using classical optimization and pattern search techniques. The optimization carried out helps to identify the optimum customer retention strategy from among many.
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The notion of customer impatience appeared in queuing theory in the work of Haight (1957). He considered a model of balking for M/M/1 queue in which there was a greatest queue length at which an arrival would not balk. This length was a random variable whose distribution was same for all customers. Haight (1959) studied a queue with reneging. Ancker and Gafarian (1963a) studied M/M/1/N queuing system with balking and reneging, and derived its steady state solution. Ancker and Gafarian (1963b) obtained results for a pure balking system (no reneging) by setting the reneging parameter equal to zero. Gavish and Schweitzer (1977) considered a deterministic reneging model with the additional assumption that arrivals can be labeled by their service requirements before joining the queue, and they are admitted only if their waiting time in the system does not exceed some fixed amount. Robert (1979) discussed in detail the reneging phenomenon of single channel queues. Baccelli et al. (1984) considered customer impatience in which a customer gives up whenever his patience or waiting time is larger than a random threshold.

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