Equilibria for diffusion models of pairs of communicating computers--Symmetric case

The steady-state queueing behavior of interacting computers is basic to computer network theory. Equilibrium properties are determined for the diffusion model for the special case of a symmetrically disposed pair of communicaling computers. In the mathematical model, arrivals to the system can represent the presentation of programming jobs, and each service represents the completion of a stage of processing. The exogenous arrivals and the services during the busy period are permitted to be arbitrary renewal processes. Computer interaction arises as follows. When each stage of processing is completed, a certain probability, exists that the entire job is complete. However, with the complementary probability., the job is input to the other computer for an additional stage of processing. In the space of all systems we find an infinite set of curves whereon the joint equilibrium density reduces to a very simple form. For the general case, a direct formal infinite series representation of the joint equilibrium density, is found. Using conformal mapping the moment generating function is obtained. Rational expressions for the mean queue size (and delay) are given. The mean queue size for Poisson arrivals is interpreted. A simple expression for the tail exponent for the queue size marginal is found. The nature of the joint density, in the neighborhood of the origin is delineated. The connection to delay equilibria is mentioned. Extensions to a larger class of communicating computer pairs is indicated.