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August, 1995 Dynamic Scheduling with Convex Delay Costs: The Generalized $c|mu$ Rule
Jan A. van Mieghem
Ann. Appl. Probab. 5(3): 809-833 (August, 1995). DOI: 10.1214/aoap/1177004706


We consider a general single-server multiclass queueing system that incurs a delay cost $C_k(\tau_k)$ for each class $k$ job that resides $\tau_k$ units of time in the system. This paper derives a scheduling policy that minimizes the total cumulative delay cost when the system operates during a finite time horizon. Denote the marginal delay cost function and the (possibly nonstationary) average processing time of class $k$ by $c_k = C'_k$ and $1/\mu_k$, respectively, and let $a_k(t)$ be the "age" or time that the oldest class $k$ job has been waiting at time $t$. We call the scheduling policy that at time $t$ serves the oldest waiting job of that class $k$ with the highest index $\mu_k(t)c_k(a_k(t))$, the generalized $c\mu$ rule. As a dynamic priority rule that depends on very little data, the generalized $c\mu$ rule is attractive to implement. We show that, with nondecreasing convex delay costs, the generalized $c\mu$ rule is asymptotically optimal if the system operates in heavy traffic and give explicit expressions for the associated performance characteristics: the delay (throughput time) process and the minimum cumulative delay cost. The optimality result is robust in that it holds for a countable number of classes and several homogeneous servers in a nonstationary, deterministic or stochastic environment where arrival and service processes can be general and interdependent.


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Jan A. van Mieghem. "Dynamic Scheduling with Convex Delay Costs: The Generalized $c|mu$ Rule." Ann. Appl. Probab. 5 (3) 809 - 833, August, 1995.


Published: August, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0843.90047
MathSciNet: MR1359830
Digital Object Identifier: 10.1214/aoap/1177004706

Primary: 90B35
Secondary: 60J70 , 60K25 , 90B22 , 93E20

Keywords: $c\mu$ rule , asymptotic optimality , dynamic priorties , heavy traffic limit , production control , queueing systems , scheduling

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.5 • No. 3 • August, 1995
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