Optimization of Multiclass Queueing Networks: Polyhedral and Nonlinear Characterizations of Achievable Performance

Abstract

We consider open and closed multiclass queueing networks, with Poisson arrivals (for open networks), exponentially distributed class dependent service times and class dependent deterministic or probabilistic routing. The performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies. By optimizing over these sets, we obtain lower bounds on achievable performance. In the special case of single station networks (multiclass queues and Klimov's model) and homogeneous multiclass networks, the polyhedron derived is exactly equal to the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. We apply the same approach to closed networks to obtain upper bounds on the optimal throughput. We check the tightness of our bounds by simulating heuristic policies and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with a number of variables and constraints that is polynomial (quadratic) in the number of classes in the network. The $i$th order approximation leads to a convex programming problem in dimension $O(R^{i+1})$, where $R$ is the number of classes in the network, and can be solved efficiently using techniques from semidefinite programming.