Multicriteria problems of optimal control for queueing systems

Queueing systems arise in different areas of human activity, especially in managing and controlling of production systems. Queues appear in modern communication systems, by maintenance of complicated equipment, and so on. The main problem here is to choose the optimal rate of servicing: on the one hand, the small rate results in the bad work of the system as a whole, on the other hand, the big rate of servicing can occur too expensive. So, the problems of optimal control very often are multicriteria. The most proper mathematical model for investigation of any queueing system is jump stochastic process. Of course, if the investigated system is controlled, the jump process must be controlled too. So, the description of the multicriteria optimization problem is presented for jump stochastic processes. The suggested method for solving the problem is the method of constraints. The main results are based on general theorems of convex analysis. The theoretical results are used for the solving of specific optimization problem for the single-channel queueing system with refusals. The controlling parameter (action) is the service rate, and one should minimize the penalty for refusals and the losses conditioned by the service.