The equivalence between processor sharing and service in random order

  • Authors:

  • S. C. Borst;O. J. Boxma;J. A. Morrison;R. NúñEz Queija

  • Affiliations:

  • CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands and B ...;CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;Bell Laboratories, Lucent Technologies, P.O. Box 636, Murray Hill, NJ 07974, USA;CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

  • Venue:
  • Operations Research Letters
  • Year:
  • 2003

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Abstract

In this note we explore a useful equivalence relation for the delay distribution in the G/M/1 queue under two different service disciplines: (i) processor sharing (PS); and (ii) random order of service (ROS). We provide a direct probabilistic argument to show that the sojourn time under PS is equal (in distribution) to the waiting time under ROS of a customer arriving to a non-empty system. We thus conclude that the sojourn time distribution for PS is related to the waiting-time distribution for ROS through a simple multiplicative factor, which corresponds to the probability of a non-empty system at an arrival instant. We verify that previously derived expressions for the sojourn time distribution in the M/M/1 PS queue and the waiting-time distribution in the M/M/1 ROS queue are indeed identical, up to a multiplicative constant. The probabilistic nature of the argument enables us to extend the equivalence result to more general models, such as the M/M/1/K queue and ./M/1 nodes in product-form networks.