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

• Authors: Jan A. Van Mieghem
• Venue: Operations Research
• Year: 2003

Consider the following due-date scheduling problem in a multiclass, acyclic, single-station service system: Any classk job arriving at timet must be served by its due datet+ D k . Equivalently, its delayt kmust not exceed a given delay or lead-timeD k . In a stochastic system, the constraintt k=D kmust 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 Generalizedc脗µ (G c脗µ) scheduling rule is known to be asymptotically optimal. The G c脗µ rule simplifies here to a generalized longest queue (GLQ) or generalized largest delay (GLD) rule, which are defined as follows. LetN kbe the number of classk jobs in system,? ktheir arrival rate, anda kthe 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? kN k , whereas GLD( ?)uses index? k? ka 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( ?*)and GLD( ?*), where?*, k= 1/ ? kD k , asymptotically minimize the probability of maximal due-date violation in heavy traffic. Specifically, they minimize lim inf n ?8 Pr{max k sup s?[0,t ]t k ( ns)/ n 1/2D k=x} for all positivet andx, wheret k ( s) is the delay of the most recent classk job that arrived before times. GLQ with appropriate parameter ? a also reduces "total variability" because it asymptotically minimizes a weighted sum of ath delay moments. Properties of GLQ and GLD, including an expression for their asymptotic delay distributions, are presented.