Optimal Control of Arrivals to Queues with Delayed Queue Length Information

We consider discrete time versions of two classical problems in the optimal control of admission to a queueing system: (i) optimal routing of arrivals t o two parallel queues and (ii) optimal acceptancefrejection of arrivals to a single queue. We extend the formulation of these problems to permit a k step delay in the observation of the queue lengths by the controller. For geometric inter-arrival times and geometric service times the problems are formulated as Controlled Markov Chains with expected total discounted cost as the minimization objective. For problem (i) we show that when k = 1, the optimal policy is t o allocate an arrival to the queue with the smaller ezpected queue length (JSEQ: Join the Shortest Expected Queue) For k 2 2 , however, JSEQ is not optimal. For problem (ii) we show that when k = 1, the optimal policy is a threshold policy. There are, however, two thresholds mo 2 ml > 0, such that mo is used when the previous action was to reject, and mi is used when the previous action was to accept