DISCRETE-TIME QUEUES WITH DELAYED INFORMATION

We study the behavior of a single-server discrete-time queue with batc h arrivals, where the information on the queue length and possibly on service completions is delayed. Such a model describes situations aris ing in high speed telecommunication systems, where information arrives in messages, each comprising a variable number of fixed-length packet s, and it takes one unit of time (a slot) to transmit a packet. Since it is not desirable to attempt service when the system may be empty, w e study a model where we assume that service is attempted only if, giv en the information available to the server, it is certain that there a re messages in the queue. We characterize the probability distribution of the number of messages in the queue under some general stationarit y assumptions on the arrival process, when information on the queue si ze is delayed K slots, and derive explicit expressions of the PGF of t he queue length for the case of i.i.d. batch arrivals and general inde pendent service times. We further derive the PGF of the queue size whe n information on both the queue length and service completion is delay ed K = 1 units of time. Finally, we extend the results to priority que ues and show that when all messages are of unit length, the c mu rule remains optimal even in the case of delayed information.