Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies

Abstract

Obtains a variety of linear programs to conduct the performance analysis and stability/instability determination of queueing networks and scheduling policies. The authors exhibit a strong duality relationship between the performance of a system, and its stability analysis via mean drift. A Performance LP bounds the performance of all stationary non-idling scheduling policies. If it is bounded, then its dual, called the Drift LP, has a feasible solution, which is a copositive matrix. The quadratic form associated with this copositive matrix has a negative drift, allowing the authors to conclude that all stationary non-idling scheduling policies are stable in the very strong sense of having a geometrically converging exponential moment. Some systems satisfy an auxiliary set of linear constraints. Their performance is also bounded by a Performance LP, provided that they are stable, i.e., have a finite first moment for the number of parts. If the Performance LP is infeasible, then the system is unstable. Any feasible solution to the dual of the Performance LP provides a quadratic function with a negative drift. If this quadratic form is copositive, then the system is strongly stable as above. If not, the system is either unstable, or else is highly non-robust in that arbitrarily small perturbations can lead to an unstable system. These results carry over to fluid models, allowing the study of networks with non-exponential distributions. Another LP test of stability avoids a copositivity check. If a Monotone LP is bounded, then the system is stable for all smaller arrival rates. Finally, a Finite Time LP provides transient bounds on the performance of the system.<>