# The achievable region method in the optimal control of queueing systems; formulations, bounds and policies

@article{Bertsimas1995TheAR,
title={The achievable region method in the optimal control of queueing systems; formulations, bounds and policies},
author={Dimitris Bertsimas},
journal={Queueing Systems},
year={1995},
volume={21},
pages={337-389}
}
• D. Bertsimas
• Published 1 September 1995
• Computer Science
• Queueing Systems
We survey a new approach that the author and his co-workers have developed to formulate stochastic control problems (predominantly queueing systems) asmathematical programming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable…
47 Citations
• Computer Science
• 1999
This approach seeks to obtain performance bounds and scheduling policies from the solution of a mathematical program over a relaxation of the system's performance region to address the performance optimization problem in a single-station multiclass queueing network with changeover times.
• Computer Science
• 1998
We address the problem of scheduling a multi-station multiclass queueing network (MQNET) with server changeover times to minimize steady-state mean job holding costs. We present new lower bounds on
• Computer Science
• 2000
A formal definition of strong asymptotic optimality in the context of design and control of queueing systems that can be modeled as single or multiple stage queueing networks is provided.
• Computer Science
• 1999
The achievable region approach seeks solutions to stochastic optimization problems by characterizing the space of all possible performances (the achievable region) of the system of interest and
• 1999
Our object of study is a multiclass queueing network (MQNET) which consists of a collection of (connected) single-server stations. Exogenous arrivals into the system form independent Poisson streams,
This paper surveys computable performance bounds for sequencing and scheduling control in open networks to minimize the long run average number of customers, or a weighted average over different customer classes.
• Computer Science, Mathematics
ArXiv
• 2011
The proposed unified framework provides a new set of tools for stochastic optimization and control over multi-class queueing systems with time average constraints and implement weighted priority policies in every busy period.
• Computer Science
Queueing Syst. Theory Appl.
• 2014
Combining the achievable region approach in queueing systems and the Lyapunov drift theory suitable to optimize renewal systems with time-average constraints, this paper shows that convex optimization problems can be solved by variants of adaptive $$c\mu$$cμ rules.
• Mathematics, Computer Science
Math. Methods Oper. Res.
• 2002
It is shown that any optimal, nonpreemptive policy is of threshold type, i.e., it assigns a customer to server Si, if this server is the fastest server available and the number of customers in the queue is mi or more.
Ann. Oper. Res.
• 1999
A primal‐dual approachexploits the fact that the system satisfies (approximate) conservation laws to yield perform-ance guarantees for a natural index‐based scheduling heuristic, and is able to analyse the performance of an arbitrary priority policy.

## References

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The proposed nonlinear programming relaxations provide strong lower bounds to the optimal cost for all static policies and also obtain lower bounds for dynamic policies as well, which are primarily useful under light traffic conditions and/or small switch-over times.
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Using a powerful technique involving quadratic or higher order potential functions, methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies are proposed and found that the first order approximation of the method is at least as good as simulation-based existing methods.
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The performance space, the set of performance vectors which are achievable under some preemptive work conserving rule, is a polyhedron described by 2J-1 inequalities, which allows for efficient procedures to minimize any separable convex function of the performance vector.
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The goal of this paper is to assess the improvement in performance that might' be achieved by optimally scheduling a multiclass open queueing network. A stochastic process is defined whose
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