Optimal quantization of periodic task requests on multiple identical processors

We simplify the periodic tasks scheduling problem by making a trade off between processor load and computational complexity. A set N of periodic tasks, each characterized by its density ρ_i, contains n possibly unique values of ρ_i. We transform N through a process called quantization, in which each ρi ∈ N is mapped onto a service level s_j ∈ L, where |L| = l ≪ n and ρ_i ≤ s_j, (this second condition differentiates this problem from the p-median problem on the real line). We define the periodic task quantization problem with deterministic input (PTQ-D) and present an optimal polynomial time dynamic programming solution. We also introduce the problem PTQ-S (with stochastic input) and present an optimal solution. We examine, in a simulation study, the trade off penalty of excess processor load needed to service the set of quantized tasks over the original set, and find that, through quantization onto as few as 15 or 20 service levels, no more than 5 percent processor load is required above the amount requested. Finally, we demonstrate that the scheduling of a set of periodic tasks is greatly simplified through quantization and we present a fast online algorithm that schedules quantized periodic tasks.