The Information-Theoretic Capacity of Discrete-Time Queues

The information-theoretic capacity of continuous-time single-server queues has been analyzed recently by Anantharam and Verdu 1]. They observed that information is transmitted through a queue not only by the contents of the packets but also by the arrival times of these packets. The sequence of packet interarrival times can be used as a code to convey information to the receiver, who observes the sequence of packet interdeparture times and attempts to infer the transmitted codeword. The random service times experienced by these packets act as noise in distorting the interarrival times. Anantharam and Verdu 1] analyzed the maximum rate at which information can be transmitted with arbitrarily small probability of error through a continuous-time single server queue, using only the timing of the packets. We refer to this maximum rate as the timing capacity of the queue, and to the maximum information rate achievable using codes with an average packet departure rate of packets/sec as the-timing capacity. For a queue with general service time distribution, upper and lower bounds to the-timing capacity were established in 1]. It was also shown that among all queues with an average service rate of packets/sec, the queue with exponentially distributed service time has the least-timing capacity, which is given by C() = ln nats/sec: This capacity can be achieved by a random code generated by independent, exponentially distributed interarrival times. The timing capacity was shown in 1] to be the supremum of-timing capacities over 0 <. Consequently, the timing capacity of the queue with exponential service time distribution is C = e nats/sec: In this paper, we consider the timing capacity of two models of discrete-time single-server queues. The rst model is the discrete-time analogue of the continuous-time model analyzed in 1]. In this model, packets arrive and depart in discrete time slots. We allow at most one packet arrival and at most one packet departure in each slot. Service times of packets are independent, identically distributed (i.i.d.), integer-valued random variables with mean 1== slots. We obtain upper and lower bounds to the-timing capacity using the analytical techniques of 1]. In particular, we show that among all queues within this model, the queue with geometric service time distribution has the smallest-timing capacity, which is given by C() = h() ? h() nats/slot where h() is the binary entropy function. For the geometric server, it is shown, using the results of 2], that the-timing capacity …