Exact and Asymptotic Solutions to Applied Stochastic Models Arising in Queueing Theory and Other Areas

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The purpose of our research is to develop and apply new mathematicalmethods for constructing solutions to difference, differential-difference,integral, and integro- differential equations. These asymptotic andsingular perturbation techniques are based on the WKB method, boundarylayer analysis and matched asymptotic expansions. New nontrivialextensions of these methods have previously been developed to solveproblems in a wide variety of applications, which led to remarkablyaccurate results. We plan to continue the development of these methods,and to apply them in particular to models arising in the performanceevaluation of queueing systems. Several aspects of the project aredirectly related to problems arising in the analysis of high-speedcommunication systems. We also propose to study problems in mathematicalphysics, related to diffusion-convection, bifurcation, and tunnelingthrough potential barriers.An important aspect of our research program is the study of queues orlines. In its simplest form, a queueing system consists of a server and aqueue or waiting room. Customers arrive, possibly at random times, andrequest service of random length from the server. If the system is empty,an arriving customer is sent directly to the server. Otherwise, thecustomer is relegated to wait in the queue. One is interested indeveloping mathematical models of the queueing system and computingperformance measures, such as the average number of customers waiting, theaverage waiting time, the utilization, etc., to help understand thebehavior of the system. For complicated queueing systems, the performancemeasures are difficult to compute and a major goal of our research is toconstruct useful approximations to them. The main application of our workin queueing is to high-speed communication networks such as the Internet. Here the customers are packets of information (bits), such as data files,voice messages, or video. The server is hardware such as a router orswitch, which transmits the packets along the Internet. At each switch,the packets are stored in a queue, if needed, before transmission. Thesebuffers can only a fixed number of packets. Arrivals finding a fullbuffer are lost and hence the system performance is degraded. Theperformance measures, average buffer size, average packet delays, and theprobability of losing information, is important in the design and tuningof high-speed communication networks. Our research provides networkdesigners with mathematical tools for approximating these performancemeasures.