Martingale Methods in Queueing Theory

Mats Pihlsgård

Handledare: Sören Asmussen

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2001:E5


Abstract
Let t_n denote the first passage time to level n in a
M/M/c queue The main aim of this thesis is to investigate the expectation and Laplace transform of t_n. The cases c=1 and c>1 are treated separately. Important tools in the analysis, in both cases, are martingales; the Wald martingale and the Kella-Whitt martingale. These are treated in a separate chapter. Also a chapter on
regenerative processes is included, and the  theory is used to obtain approximate values of Et_n in two special cases. Although the practical aspects of the problems are not emphasised, a few equations, rather well suited for computer visualization, are produced and presented graphically. Finally, the queue length process and the work load process are simulated in a few cases.