A Tandem Storage System and Its Diffusion Limit.
Abstract
We consider a two-dimensional diffusion process Z(t) = (Z sub 1(t), Z sub 2 (t)) that lives in the half strip (0 < or = Z sub 1 < or = 1, 0 < or = Z sub 2 < infinity. On the interior of this state space, Z behaves like a standard Brownian motion (independent components with zero drift and unit variance), and there is instantaneous reflection at the boundary. The reflection is in a direction normal to the boundary at Z sub 1 = 1 and Z sub 2 = 0, but at Z sub 2 = 0, but as Z sub 1 = 0 the reflection is at an angle theta below the normal (0 < theta < pi/2). This process Z is shown to arise as the diffusion limit of a certain tandem storage or queueing system. It is shown that Z(t) has a non-defective limit distribution F as t yields infinity, and the marginal distributions of F are computed explicitly. The marginal limit distribution for Z sub 1 is uniform (this result is essentially trivial), but that for Z sub 2 is much more complicated.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 01, 1982
- Accession Number
- ADA120644
Entities
People
- J. Michael Harrison
- L. A. Shepp
Organizations
- Stanford University