Tandem queues with impatient customers

This paper studies a Markovian two-station tandem queueing network with impatient customers. Queueing networks with abandonment are common in many industries, e.g., call centers and healthcare. Therefore, their management has received much attention. The resulting model is a level-dependent quasi-birth-and-death (LDQBD) process. Such models are considered analytically intractable and require numerical methods for their solution. We study a specific type of LDQBD process, where the total abandonment rate increases with the number of waiting customers leading to the level-dependent feature. We analyze an equivalent last-come-first-serve system to develop a recursive relation in our LDQBD process, reducing the problem to solving quadratic matrix equations, for which efficient and exact numerical methods exist. We further simplify the analysis by combining the recursive renewal reward theorem with Queueing and Markov chain decomposition (QMCD), so that we only need to solve one quadratic matrix equation instead of infinite ones caused by the system’s level-dependent feature. We develop an exact numerical method to evaluate various performance measures of a tandem queueing network with abandonment. Our method is applicable to the analysis of queueing networks with abandonment under settings with diverse features and in various service disciplines.