Brownian models of closed queueing networks with homogeneous customer populations

Abstract

We study a diffusion process Z whose state space is the K-dimensional unit simplex, K2. This process arises as the heavy traffic approximation for a K-station closed queueing network with a homogeneous customer population. The last phrase means that customers occupying any given node or station of the queueing network are essentially indistinguishable from one another. The classical closed network model studied by J. R. Jackson and by Gordon and Newell fits this description, as do other more general types of systems, but multiclass network models do not. After firstreviewing exactly how one fits the parameters of Z so as to model a given closed queueing network,we state and prove some new results regarding the stationary distribution of Z. Although no general formula has yet been found for the stationary distribution, a number of important foundational issues are resolved here, and a necessary and sufficient condition is found for the stationary distribution to have an exponential density function. When that condition is met, all important performance measures can be easily computed

Keywords:

• Brownian system models
• heavy traffic
• performance analysis
• closed queueing networks
• stationary distributions
• product form solutions
• partition functions

^*Research supported in part by NSF Grants ECS-8603857, DMS-8319562, and DMS-8657483, and Semiconductor Research Corporation Grant 84-01-046.

^*Research supported in part by NSF Grants ECS-8603857, DMS-8319562, and DMS-8657483, and Semiconductor Research Corporation Grant 84-01-046.

Notes

^*Research supported in part by NSF Grants ECS-8603857, DMS-8319562, and DMS-8657483, and Semiconductor Research Corporation Grant 84-01-046.