We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

Consider a machine, which may or may not have a defect, and the probability q that this machine is defective is unknown. In order to determine whether the machine is defective, it is tested. On each test, the defect is found with probability p, if it has not been found yet. Performing n tests costs cn dollars and there is a fine of 1 dollar if there is a defect and it is not found on the tests. When should we stop testing, in order to minimize the cost?

This problem is treated in a minimax setting: we try to find a strategy that works well, even for ‘bad’ q's. It turns out that the minimax optimal stop rule can be unexpectedly complicated. For example, if p = 1/2 and cn = cn = 0.25n, then the optimal rule is to start by performing one test. If a defect is found we stop, otherwise we perform a second test. If a defect is found, then again we stop, else toss a coin and stop if this shows heads. If we still have not stopped, a third and last test is performed.

A spectrum of self-organizing rules including the move-to-front rule and the transposition rule are applied to the communication problem. The stationary distributions under these rules are obtained. Cost comparison between them is considered. In the special case of three paths, it is shown that the transposition rule always outperforms the move-to-front rule.

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.

We show that the GEM process has strong ordering properties: the probability that one of the k largest elements in the GEM sequence is beyond the first ck elements (c > 1) decays superexponentially in k.