An M/M/1/N Queueing Model with Retention of Reneged Customers and Balking

Abstract

The concept of customer balking and reneging has been exploited to a great extent in recent past by the queu- ing modelers. Economically, if we see, the customer impatience (due to balking and reneging) leads to the loss of potential customers and thereby results into the loss in the total revenue. Taking into consideration this customers' loss due to impa- tience, a new queuing model has been developed that deals with retention of reneged customers. According to this model, a reneged customer can be convinced in many cases by employing certain convincing mechanism to stay in the queue for completion of his service. Thus, a reneged customer can be retained in the queuing system with some probability (say, q) and it may leave the queue without receiving service with probability p (=1-q). This process is referred to as customer re- tention. We consider a single server, finite capacity queuing system with customer retention and balking in which the inter- arrival and service times follow negative-exponential distribution. The reneging times are assumed to be exponentially dis- tributed. An arriving customer may not join the queue if there is at least one customer in the system, i.e. the customer may balk. The steady state solution of the model has been obtained. Some performance measures have been computed. The sen- sitivity analysis of the model has been carried out. The effect of probability of retention on the average system size has been studied. The numerical results show that the average system size increases proportionately and steadily as the prob- ability of retention increases. Some particular cases of the model have been derived and discussed.