Scheduling multiclass queueing networks on parallel servers : approximate and heavy-traffic optimality of Klimov’s rule

We address the problem of scheduling a multiclass queueing network on M parallel servers to minimize the time-average holding cost. We analyze a heuristic index rule, based on Klimov’s solution to the single-server model: when a server becomes free it selects a customer with largest Klimov’s 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 c?-rule. Our analysis is based on comparing the cost of the heuristic to the value of (the dual of) a strong linear programming relaxation, which equals the optimal cost for the “corresponding” single-server network. This relaxation follows from a set of approximate conservation laws 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.