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+Dk. Equivalently, its delay k must not exceed a given delay or lead-time Dk. In a stochastic system, the constraint k Dk 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 which 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 (Gc ) scheduling rule is known to be asymptotically optimal. The Gc rule simplifies here to a generalized longest queue (GLQ) or generalized largest delay (GLD) rule, which are defined as follows. Let Nk be the number of class k jobs in system, k their arrival rate, and ak the age of their oldest job in the system. GLQ and GLD are dynamic priority rules, parameterized by : GLQ( ) serves FIFO within class and prioritizes the class with highest index kNk, whereas GLD( ) uses index k kak. 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( ∗) and GLD( ∗), where ∗ k = 1/ kDk, asymptotically minimize the probability of maximal due-date violation in heavy traffic. Specifically, they minimize lim infn→ Pr maxk sups∈ 0 t k ns nDk x for all positive t and x, where k s is the delay of the most recent class k job that arrived before time s. GLQ with appropriate parameter also reduces “total variability” because it asymptotically minimizes a weighted sum of th delay moments. Properties of GLQ and GLD, including an expression for their asymptotic delay distributions, are presented.