Scheduling multiclass queueing networks on parallel servers: Approximate and heavy-traffic optimality of Klimov's priority rule

Abstract

We address the problem of scheduling a multiclass queueing network on M parallel servers to minimize time-average linear holding costs. We analyze a heuristic priority-index rule, based on Klimov's solution to the single-server model: Compute the indices given by Klimov's adaptive greedy algorithm and, when a server becomes free, select a customer with largest index. We present closed-form performance guarantees for this heuristic, with respect to (1) the optimal cost in the original parallel-servers network, and (2) the optimal cost in a “corresponding” single-server network, attended by a server working M times faster. Simpler expressions are derived for the special case that there is no customer feedback, where the heuristic becomes the cµ-rule. Our analysis is based on a primal-dual approach: We compare the cost of the heuristic to the value of (the dual of) a strong linear programming (LP) relaxation, which represents the optimal cost for the “corresponding” single-server network. This relaxation follows from a set of approximate conservation laws (ACLs) satisfied by the network. Our proof of these laws relies on the first set of work decomposition laws known for this model, which we obtain from a classical flow conservation law.