Inequalities for Queues with a Learning Server

We study a multi-class queue with a learning server who becomes stochastically faster with each subsequent customer served of the same type in a row, and returns to some baseline speed each time he switches to a different type of customer. We show under some conditions that customer waiting time is larger (in the increasing convex ordering sense) with server learning than in a queue with iid service times having the same marginal service distribution as the learning server. An easy to evaluate inequality for the mean stationary waiting time is derived from this in the case of Poisson arrivals, and results in more general settings are given. The primary tool used in the proofs is the supermodularity of the delay in queue as a function of previous service times.