Busy period analysis of finite QBD processes

Abstract

We present the number of customers served and the length of a busy period for finite quasi birth and death (QBD) processes where either one or both of the arrival or service processes can be serially correlated or interdependent. Special cases include the G/G/1/K, M/G/1/K, and G/M/1/K queues. The resulting algorithms are linear algebraic in nature and are easily implemented. The solutions allow studies on how the moments and correlations in the arrival and service processes affect the busy period. This includes the probability of serving exactly n customers during a busy period and the moments of the length of the busy period for different system (queue) sizes. We present an example of a QBD process where arrival and service processes are strongly dependent.