Diffusion approximation for equilibrium and transient queue behavior

Abstract

The difficulty in solving diffusion equations generally increases with the number of dimensions. Although the evolution of the joint density f(x, y; t) for both the arrivals and the departures would give a fairly complete description of what is happening and why, it is much easier to analyze the behavior of only Q(t). We have seen that if the mean and variance rates for A(t) and D(t) depend only upon the queue length and time, then f _Q (l;t) itself satisfies a diffusion equation (7.19). If there is a negligible probability that Q(t) is on a boundary, then f _Q (l;t) also satisfies the boundary conditions (7.21) and F _Q (l;t) satisfies (7.22).