The Diffusion Approximation for Tandem Queues in Heavy Traffic.
Abstract
Consider a pair of single server queues arranged in series. A limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting time distribution. Specifically the waiting time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian Motion, and the general problem of determining F was left unsolved. It is shown that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian Motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated. It is shown that the limit distribution F is the solution of a first passage problem for a certain dual diffusion process Z*. The generator of Z* is calculated, and the analytical theory of Markov process is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components).
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 10, 1977
- Accession Number
- ADA044911
Entities
People
- J. Michael Harrison
Organizations
- Stanford University