The Camp-On Problem for Multiple-Address Traffic

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Abstract
A communication system for multiple-address messages is described, in which a message waits in parallel queues until it can be transmitted simultaneously to all the addressed receivers. An idealized mathematical model of this system leads to a nonlinear integral equation for the stationary distribution of delays in receiver queues. A phase-plane analysis shows this equation to have a one-parameter family of solutions, one member of which is found to be the unique limiting distribution of receiver delays. Even though service times (message lengths) are not bounded, the receiver queues in this model can operate in the steady state at critical load. Under these conditions, the probability that a server is idle is positive; and all moments of the delay distribution are finite. Computation of the delay distribution is discussed, numerical examples are given, and the behavior of the transmitter queues is analyzed. Predictions of this model are compared with performance parameters of simulated systems. The model is shown to be very accurate up to its critical load. For higher loads, performance depends strongly upon the number of receivers in the system. The model's discontinuity in receiver occupancy is not physically realizable, but is approached asymptotically as the number of receivers tends to infinity.

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