Optimal control of arrivals to queues with delayed queue length information

The authors consider discrete-time versions of two classical problems in the optimal control of admission to a queuing system: (i) optimal routing of arrivals to two parallel queues and (ii) optimal acceptance/rejection of arrivals to a single queue. They extend the formulation of these problems to permit a k step delay in the observation of the queue lengths by the controller. For geometric interarrival 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) it is shown that, when k=1, the optimal policy is to allocate an arrival to the queue with the smaller expected queue length. For problem (ii) it is shown that, when k=1, the optimal policy is a threshold policy