Two competing queues with time-critical customers: optimal server assignment

The issue of scheduling heterogeneous time-critical customers at a single node is addressed. A simple priority assignment problem between two queues with time-constrained customers is considered. A slotted system that operates in discrete time is assumed. For queue k ( k=1, 2), arrivals are independent from slot to slot and geometrically distributed with parameter λ_k. Also, the arrivals to the two queues are independent. It is shown that the minimization given previously by the authors (IEEE Trans. Control vol.34, no.7, p.721-8, July 1989) can be restricted, without loss of generality, to the class of policies that are nonidling and are allowed to choose only between the customers with smallest residual time in each queue. Indeed, idling does not pay because, by assumption, arrivals at a slot are not available until the beginning of the next slot. The main result here is a certain type of monotonicity of the optimal control law