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Numbers of Served and Lost Customers in Busy-Periods of M/M/1/n Systems with Balking

Ferreira, Fátima; Pacheco, António; Ribeiro, Helena

Lecture Notes in Computer Science, 12251 (2020), 147-156

In this work we analyse single server Markovian queueing systems with finite capacity and balking, that is M/M/1/n systems with balking. In these systems, the admission of customers is modulated by the state of the system at the instants of customer arrivals. Depending on the size of the queue upon arrival, customers that find place to join the system decide to enter the system with a certain probability. The number of customers in the system amounts to a Markov chain whose transition probabilities incorporate the balking probabilities. Using the Markovian regenerative property of the chain embedded at the instants of arrival or departure of customers, we characterise the joint probability distribution of the number of customers served and the number of customers lost in busy-periods, that is, during continuous occupation periods of the server. This is accomplished implementing a priori a recursive algorithmic procedure for computing the respective probability-generating function. Finally, a numerical illustration of the derived results is presented for different balking policies.