Optimal Design and Control of Finite-Population Queueing Systems

CHAO DENG: Optimal Design and Control of Finite-Population Queueing Systems (Under the direction of Professor Nilay Tanık Argon and Professor Vidyadhar G. Kulkarni) We consider a service system with a finite population of customers (or jobs) and a service resource with finite capacity. We model this finite-population queueing system by a closed queueing network with two stages. The first stage, which represents the arrivals of customers for service, consists of an automated station with ample capacity. The second stage, which represents the service for customers, consists of multiple service stations which share the finite service resource. We consider both discrete and continuous service resources. We are interested in static or dynamic allocation of the service resource to the service stations in the second stage in order to optimize a given system measure. Specifically, a static allocation refers to a design problem, while a dynamic allocation refers to a control problem. In this thesis, we study both. For control problems, we specify a parallel-series structure for service stations. We first consider dynamically allocating a single flexible server under both preemptive and nonpreemptive policies. We characterize the optimal policies of dynamically scheduling this single server in order to maximize the long-run average throughput of the system. In the special case of a series system, we show that the optimal policy is a sequential policy where each customer is served by the single server sequentially from the first station until the last one. For a parallel system, we show that there exists an optimal policy which gives the highest priority to the station that has the largest service rate. We also propose an index policy heuristic for the general parallel-series system and compare its performance as opposed to the optimal policy by a numerical study. Finally, we study dynamically allocating a finite amount of continuous service resource for the parallel system. For design problems, we consider allocating a finite amount of service resource which is continuously divisible and can be used at any of the service stations. Suppose that service times at a service station are exponentially distributed and their mean is a strictly increasing and concave function of the allocated service resource. We characterize the optimal allocation of the continuous resource in order to maximize the long-run average throughput of the system. We first show that the system throughput is non-decreasing in the number of customers. Then, we study the optimization problem in three cases depending on the population size of customers in the system. First, when there is a single customer, we show that the optimal allocation is given by a set of optimality equations. Secondly, when the number of customers approaches infinity, we show that the optimal allocation approaches to a limit. Finally, for any finite number of customers, we show that the system throughput is bounded up by a limit. Moreover, under a certain condition, we show that the system throughput function is Schur-concave.