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Due-date scheduling: asymptotic optimality of generalized longest queue and generalized largest delay rules
- Operations Research , 2003
"... 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 � D ..."
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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 �k�ns � of maximal due-date violation in heavy traffic. Specifically, they minimize lim inf n→ � Pr�maxk sups∈�0�t � n1/2 � x � for all positive t and Dk 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.