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

Estimating customer impatience in a service system with unobserved balking

This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information they decide to wait for service or to leave the system. The main objective is to estimate the customers' patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication, and distinguishing feature of our setup, lies in the fact that customers who decide not to join are not observed, but, remarkably, we manage to devise a procedure to estimate the load they would generate. We express our system in terms of a multi-server queue with a Poisson stream of customers, which allows us to evaluate the corresponding likelihood function. Estimating the unknown parameters relying on a maximum likelihood procedure, we prove strong consistency and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. The performance of our approach is further assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized-hyperexponential distributions our method provides a robust estimation framework for any continuous patience-level distribution

Stochastic approximation of symmetric Nash equilibria in queueing games

We suggest a novel stochastic-approximation algorithm to compute a symmetric Nash-equilibrium strategy in a general queueing game with a finite action space. The algorithm involves a single simulation of the queueing process with dynamic updating of the strategy at regeneration times. Under mild assumptions on the utility function and on the regenerative structure of the queueing process, the algorithm converges to a symmetric equilibrium strategy almost surely. This yields a powerful tool that can be used to approximate equilibrium strategies in a broad range of strategic queueing models in which direct analysis is impracticable