On the Conditional Probability of a Successful Retrial in Retrial Queues

Customer retrials are very common phenomena in industrial engineering and business management. The number of retrials required before receiving a success is an important measure for evaluating system performance. Focusing on the M/M/1 and M/M/2 retrial queues, we study the conditional probability of a successful retrial given that all previous retrials have been denied. The idea is to obtain the recursive expressions based on the properties of continued fractions and the transient solutions of systems. The result leads to computational algorithms for the probability of the number of retrials made by a blocked primary customer. Through implementation of the proposed algorithms, numerical examples and analysis are presented to show computational efficiency and properties of the conditional probability.