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A model of a queueing network is proposed which leads to a stochastic equation generalizing a standard Lindley equation for a single FCFS server. We study the problem of the existence and uniqueness of a stationary solution to this equation and its connection with random processes on a Cayley tree.

A Brownian particle walks on a line, with negative drift coefficient. After an exponential lifetime it dies giving birth to a certain number of ‘descendants’ each of which then proceeds independently, following the same rule. We first re-derive a necessary and sufficient condition for P(X’ < ∞) = 1 and P(X < ∞) = 1 where X and X’ are the suprema of the deviation of particles in the positive direction taken, respectively, over the infinite time interval ℝ + = [0, ∞) and over all epochs of death (in a slightly weaker form, a similar result may be deduced from an earlier paper by (McKean 1975)). We then check that functions P(X < x) and P(X’ < x) give (unique) maximal solutions to certain stochastic equations and we prove the existence of other solutions to these equations.

One gives a limit theorem for the joint distribution of the stationary waiting times of customers in the queues of a multiphase queueing system, functioning in a heavy traffic regime. One proves that the joint distribution function of the waiting times is a solution of a problem with a directional derivative for an elliptic differential equation in a polyhedral angle.