Optimal control of queueing networks: an approach via fluid models

Abstract

We consider a general control problem for networks which includes the special cases of scheduling in multiclass queueing networks and routing problems. The fluid approximation of the network is used to derive new results about the optimal control for the stochastic network. The main emphasis lies on the average cost criterion, however the β-discounted as well as the finite cost problem are also investigated. One of our main results states that the fluid problem provides a lower bound to the stochastic network problem. For scheduling problems in multiclass queueing networks we show the existence of an average cost optimal decision rule, if the usual traffic conditions are satisfied. Moreover, we give under the same condition a simple stabilizing scheduling policy. Another important issue that we address is the construction of simple asymptotically optimal decision rules. Asymptotic optimality is here seen w.r.t. fluid scaling. We show that every minimizer of the optimality equation is asymptotically optimal. And what is more important for practical purposes, we outline a general way to identify fluid optimal feedback rules as asymptotically optimal ones. Last but not least for routing problems an asymptotically optimal decision rule is given explicitly, namely a so-called least-loaded-routing rule.

Liquidation in Limit Order Books With Controlled Intensity

“…Remark Results somewhat similar to Theorem appeared in Bäuerle (, , ); Day (); Piunovskiy (); Piunovskiy and Zhang () which are on the optimal control of queueing networks. (Among these papers only Day considered optimal time‐to‐empty queueing control problems.)…”

“…On a formal level, our control problem is equivalent to a controlled death process and is closely related to fluid approximations of some queueing problems. We refer to Bäuerle (, , ), Day (), Piunovskiy (), Piunovskiy and Zhang () and references therein for the most relevant strand of this rich literature. In contrast with the previous literature, which uses probabilistic arguments, we utilize viscosity techniques to show convergence (both of the value functions and the corresponding optimal controls) from the discrete‐ to the continuous‐state problems.…”

A Restless Bandit Model for Resource Allocation, Competition and Reservation

“…Asymptotically optimal policies for cost-minimization problems in network systems using a fluid approximation technique have been considered in [Bäuerle et al, 2000], [Bäuerle, 2002], [Stolyar, 2004], [Nazarathy and Weiss, 2009] and [Bertsimas et al, 2015]. The fluid approximation to the stochastic optimization problem can be much simpler than the original.…”

Continuous-Time Mean Field Markov Decision Models

“…Fluid models have also been used to approximate the optimal control in these networks, see e.g. [10][11][12][13][14]. In [15] different scales of time are treated for the approximation and some components may be replaced by differential equations.…”