Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cµ-Rule

  • Authors:

  • Avishai Mandelbaum;Alexander L. Stolyar

  • Affiliations:

  • -;-

  • Venue:
  • Operations Research
  • Year:
  • 2004

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Abstract

We consider a queueing system with multitype customers and flexible (multiskilled) servers that work in parallel. IfQ iis the queue length of typei customers, this queue incurs cost at the rate ofC i ( Q i ), whereC i (.) is increasing and convex. We analyze the system in heavy traffic (Harrison and Lopez 1999) and show that a very simple generalizedc脗µ-rule (Van Mieghem 1995) minimizes both instantaneous and cumulative queueing costs, asymptotically, over essentially all scheduling disciplines, preemptive or non-preemptive. This rule aims at myopically maximizing the rate of decrease of the instantaneous cost at all times, which translates into the following: when becoming free, serverj chooses for service a typei customer such thati ? arg max i C' i ( Q i )脗µ ij , where 脗µ ijis the average service rate of typei customers by serverj.An analogous version of the generalizedc脗µ-rule asymptotically minimizes delay costs. To this end, let the cost incurred by a typei customer be an increasing convex functionC i ( D) of its sojourn timeD. Then, serverj always chooses for service a customer for which the value ofC' i ( D) 脗µ ijis maximal, whereD andi are the customer's sojourn time and type, respectively.