Scheduling for a Processor Sharing System with Linear Slowdown

We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a "local search", where we use the problem structure in a novel manner. The second is a one-coordinate "global search", used in coordinate pivot iteration. We then merge these two heuristics into a unified "local-global" heuristic, and numerically illustrate the effectiveness of this heuristic

Near optimal control of queueing networks over a finite time horizon

We propose a method for the control of multi-class queueing networks over a finite time horizon. We approximate the multi-class queueing network by a fluid network and formulate a fluid optimization problem which we solve as a separated continuous linear program. The optimal fluid solution partitions the time horizon to intervals in which constant fluid flow rates are maintained. We then use a policy by which the queueing network tracks the fluid solution. To that end we model the deviations between the queuing and the fluid network in each of the intervals by a multi-class queueing network with some infinite virtual queues. We then keep these deviations stable by an adaptation of a maximum pressure policy. We show that this method is asymptotically optimal when the number of items that is processed and the processing speed increase. We illustrate these results through a simple example of a three stage re-entrant line

Polynomial Approximations for Infinite-Dimensional Optimization Problems

Many real-life decision problems in management science and engineering involve decisions that are functions of time and/or uncertainty. The resulting optimization models are therefore naturally formulated on infinite-dimensional function spaces. However, such infinite-dimensional optimization problems are notoriously difficult, and to solve them one usually has to resort to approximation methods. The objective of this thesis is to devise polynomial approximations for solving continuous linear programs and multi-stage stochastic programs, both of which constitute important classes of infinite-dimensional optimization problems with manifold practical applications. Approximating the functional decision variables by polynomials allows us to apply sum-of-squares techniques from algebraic geometry to reformulate the resulting problems as tractable semidefinite programs, which can be solved efficiently with interior point algorithms. Continuous linear programs represent deterministic optimization problems whose decision variables are functions of time subject to pointwise and dynamic linear constraints. They have attracted considerable interest due to their potential for modelling manufacturing, scheduling and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (non-separated) problem instances. In this thesis we propose a more generic approximation scheme for non-separated continuous linear programs, which are believed to be NP-hard. We approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. To estimate the approximation error, we also compute a lower bound by solving a dual continuous linear program in (piecewise) polynomial decision rules. Multi-stage stochastic programming provides a versatile framework for optimal decision making under uncertainty, but it gives rise to hard functional optimization problems since the adaptive recourse decisions must be modelled as functions of some or all uncertain parameters. We propose to approximate these recourse decisions by polynomial decision rules and show that the best polynomial decision rule of a fixed degree can be computed efficiently. Again, we show that the suboptimality of the best polynomial decision rule can be estimated efficiently by solving a dual version of the stochastic program in polynomial decision rules. Recent progress in the theory of dynamic risk measures has found a strong echo in stochastic programming, where the time-consistency of dynamic decision making under uncertainty is currently under scrutiny. We extend the concepts of coherence and time consistency to stochastic programming models subject to distributional ambiguity, which motivates us to introduce robust dynamic risk measures. We discuss conditions under which these robust risk measures inherit coherence and time-consistency from their nominal counterparts. We also propose an approximation scheme based on polynomial decision rules for solving linear multi-stage stochastic programs involving robust dynamic risk measures