Due-Date Scheduling: Asymptotic Optimality of Generalized Longest Queue and Generalized Largest Delay Rules

Consider the following due-date scheduling problem in a multiclass, acyclic, single-station service system: any class k job arriving at time t must be served by its due date t D_{k}. Equivalently, its delay ¦O_{k} must not exceed a given delay or lead-time D_{k}. In a stochastic system the constraint ¦O_{k}iUD_{k} must be interpreted in a probabilistic sense. Regardless of the precise probabilistic formulation, however, the associated optimal control problem is intractable with exact analysis. This article proposes a new formulation that incorporates the constraint through a sequence of convex-increasing delay cost functions. This formulation reduces the intractable optimal scheduling problem into one for which the Generalized c¦I (Gc¦I) scheduling rule is known to be asymptotically optimal. The Gc¦I rule simplifies here to a generalized longest queue (GLQ) or generalized largest delay (GLD) rule, which are defined as follows. Let N_{k} be the number of class k jobs in system, ¦E_{k} their arrival rate and a_{k} the age of their oldest job in the system. GLQ and GLD are dynamic priority rules, parameterized by ¦E: GLQ(¦E) serves FIFO within class and prioritizes the class with highest index ¦E_{k}N_{k}, whereas GLD(¦E) uses index ¦E_{k}¦E_{k}a_{k}. The argument is presented first intuitively, but is followed by a limit analysis that expresses the cost objective in terms of the maximal due-date violation probability. This proves that GLQ(¦E_{∗}) and GLD(¦E_{∗}), where ¦E_{∗,k}=1/¦E_{k}D_{k}, asymptotically minimize the probability of maximal due-date violation in heavy traffic. Specifically, they minimize liminf_{niuiÞ}Pr{max_{k}sup_{siE[0,t]}((¦O_{k}(ns))/(n^{1/2}D_{k}))iÝx} for all positive t and x, where ¦O_{k}(s) is the delay of the most recent class k job that arrived before time s. GLQ with appropriate parameter ¦E_{¦A} also reduces total variability because it asymptotically minimizes a weighted sum of ¦A^{th} delay moments. Properties of GLQ and GLD, including an expression for their asymptotic delay distributions, are presented.

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