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

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  • Jan A. Van Mieghem

    (Kellogg School of Management, Northwestern University, Evanston, Illinois 60208-2009)

Abstract

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 (tau) k must not exceed a given delay or lead-time D k . In a stochastic system, the constraint (tau) k (le) D 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 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(mu) (G c(mu) ) scheduling rule is known to be asymptotically optimal. The G c(mu) 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, (lambda) 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 (theta) : GLQ( (theta) ) serves FIFO within class and prioritizes the class with highest index (theta) k N k , whereas GLD( (theta) ) uses index (theta) k (lambda) 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( (theta) * ) and GLD( (theta) * ), where (theta) * ,k = 1/ (lambda) k D k , asymptotically minimize the probability of maximal due-date violation in heavy traffic. Specifically, they minimize \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$\mbox{lim}\, \mbox{inf}_{n\to\infty}\mbox{Pr}\{\max_k \mbox{sup}_{s\in [0,t]}\tfrac{\tau_k(ns)}{n^{1/2}D_k}\geq x\}$\end{document} for all positive t and x , where (tau) k ( s ) is the delay of the most recent class k job that arrived before time s . GLQ with appropriate parameter (theta) (alpha) also reduces “total variability” because it asymptotically minimizes a weighted sum of (alpha)th delay moments. Properties of GLQ and GLD, including an expression for their asymptotic delay distributions, are presented.

Suggested Citation

  • Jan A. Van Mieghem, 2003. "Due-Date Scheduling: Asymptotic Optimality of Generalized Longest Queue and Generalized Largest Delay Rules," Operations Research, INFORMS, vol. 51(1), pages 113-122, February.
  • Handle: RePEc:inm:oropre:v:51:y:2003:i:1:p:113-122
    DOI: 10.1287/opre.51.1.113.12793
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    References listed on IDEAS

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    3. Yichuan Ding & Eric Park & Mahesh Nagarajan & Eric Grafstein, 2019. "Patient Prioritization in Emergency Department Triage Systems: An Empirical Study of the Canadian Triage and Acuity Scale (CTAS)," Manufacturing & Service Operations Management, INFORMS, vol. 21(4), pages 723-741, October.
    4. Dong Li & Kevin D. Glazebrook, 2010. "An approximate dynamic programing approach to the development of heuristics for the scheduling of impatient jobs in a clearing system," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(3), pages 225-236, April.
    5. Junfei Huang & Boaz Carmeli & Avishai Mandelbaum, 2015. "Control of Patient Flow in Emergency Departments, or Multiclass Queues with Deadlines and Feedback," Operations Research, INFORMS, vol. 63(4), pages 892-908, August.
    6. Achal Bassamboo & Ramandeep S. Randhawa & Jan A. Van Mieghem, 2012. "A Little Flexibility Is All You Need: On the Asymptotic Value of Flexible Capacity in Parallel Queuing Systems," Operations Research, INFORMS, vol. 60(6), pages 1423-1435, December.
    7. Rami Atar & Chanit Giat & Nahum Shimkin, 2010. "The c(mu)/(theta) Rule for Many-Server Queues with Abandonment," Operations Research, INFORMS, vol. 58(5), pages 1427-1439, October.
    8. Maglaras, Constantinos & Van Mieghem, Jan A., 2005. "Queueing systems with leadtime constraints: A fluid-model approach for admission and sequencing control," European Journal of Operational Research, Elsevier, vol. 167(1), pages 179-207, November.
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    11. Romero-Silva, Rodrigo & Shaaban, Sabry & Marsillac, Erika & Hurtado, Margarita, 2018. "Exploiting the characteristics of serial queues to reduce the mean and variance of flow time using combined priority rules," International Journal of Production Economics, Elsevier, vol. 196(C), pages 211-225.
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