K competing queues with customer abandonment: optimality of a generalised $$c \mu $$cμ-rule by the Smoothed Rate Truncation method

We consider a K-competing queues system with the additional feature of customer abandonment. Without abandonment, it is optimal to allocate the server to a queue according to the $$c \mu $$cμ-rule. To derive a similar rule for the system with abandonment, we model the system as a continuous-time Markov decision process. Due to impatience, the Markov decision process has unbounded jump rates as a function of the state. Hence it is not uniformisable, and so far there has been no systematic direct way to analyse this. The Smoothed Rate Truncation principle is a technique designed to make an unbounded rate process uniformisable, while preserving the properties of interest. Together with theory securing continuity in the limit, this provides a framework to analyse unbounded rate Markov decision processes. With this approach, we have been able to find close-fitting conditions guaranteeing optimality of a strict priority rule.