Priority queues with semiexhaustive service and class-dependent setup times

This paper considers a new type of priority queues with semiexhaustive service and setup times, which operates as follows. A single server continues serving class-n messages in queue n until the number of messages decreases to one less than that found upon the serverʹs last arrival at queue n, where 1 ≤ n ≤ N. In succession, messages of the highest class present in the system, if any, will be served according to this semiexhaustive service. Two types of class-dependent setup times are considered: Setup Time I (the first service after a setup time is given to the first arrival during an idle period) and Setup Time II (the first service after a setup time is given to the highest class message in the system at the end of the setup time). We derive the Laplace-Stieltjes transforms of the message waiting times for each kind of setup times and explicit formulae for the mean waiting times. We also give some numerical results to assess the influence of setup times.