A diffusion model of scheduling control in queueing systems with many servers

This paper studies a diffusion model that arises as the limit of a queueing system scheduling problem in the asymptotic heavy traffic regime of Halfin and Whitt. The queueing system consists of several customer classes and many servers working in parallel, grouped in several stations. Servers in different stations offer service to customers of each class at possibly different rates. The control corresponds to selecting what customer class each server serves at each time. The diffusion control problem does not seem to have explicit solutions and therefore a characterization of optimal solutions via the Hamilton-Jacobi-Bellman equation is addressed. Our main result is the existence and uniqueness of solutions of the equation. Since the model is set on an unbounded domain and the cost per unit time is unbounded, the analysis requires estimates on the state process that are subexponential in the time variable. In establishing these estimates, a key role is played by an integral formula that relates queue length and idle time processes, which may be of independent interest.Comment: Published at http://dx.doi.org/10.1214/105051604000000963 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamic fluid-based scheduling in a multi-class abandonment queue

International audienceWe investigate how to share a common resource among multiple classes of customers in the presence of abandonments. We consider two different models: (1) customers can abandon both while waiting in the queue and while being served, (2) only customers that are in the queue can abandon. Given the complexity of the stochastic optimization problem we propose a fluid model as a deterministic approximation. For the overload case we directly obtain that the c˜µ/θ rule is optimal. For the underload case we use Pontryagin’s Maximum Principle to obtain the optimal solution for two classes of customers; there exists a switching curve that splits the two-dimensional state-space into two regions such that when the number of customers in both classes is sufficiently small the optimal policy follows the c˜µ-rule and when the number of customers is sufficiently large the optimal policy follows the c˜µ/θ-rule. The same structure is observed in the optimal policy of the stochastic model for an arbitrary number of classes. Based on this we develop a heuristic and by numerical experiments we evaluate its performance and compare it to several index policies. We observe that the suboptimality gap of our solution is small

Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Temporal Concatenation for Markov Decision Processes

We propose and analyze a Temporal Concatenation (TC) heuristic for solving large-scale finite-horizon Markov decision processes (MDP). The Temporal Concatenation divides a finite-horizon MDP into smaller sub-problems along the time horizon, and generates an overall solution by simply concatenating the optimal solutions from these sub-problems. As a "black box" architecture, Temporal Concatenation works with a wide range of existing MDP algorithms with the potential of substantial speed-up at the expense of minor performance degradation. Our main results characterize the regret of Temporal Concatenation, defined as the gap between the expected rewards from Temporal Concatenation's solution and that from the optimal solution. We provide upper bounds that show, when the underlying MDP satisfies a bounded-diameter criterion, the regret of Temporal Concatenation is bounded by a constant independent of the length of the horizon. Conversely, we provide matching lower bounds that demonstrate that, for any finite diameter, there exist MDP instances for which the regret upper bound is tight. We further contextualize the theoretical results in an illustrative example of dynamic energy management with storage, and provide simulation results to assess Temporal Concatenation's average-case regret within a family of MDPs related to graph traversal.Comment: Added references and updated the theoretical result in Section