In this thesis, we consider two applications of queueing theory. In the first part, we study a two-stage security-check system which is motivated by the border-crossing stations between United States and Canada. The main tradeoff in such a system is good customer service (short waiting time) and high security level. We establish a two-stage queueing model and derive the diffusion approximations under nondegenerate slowdown regime for the system. Some numerical results are demonstrated to show the good performance of our approximation formulas. We also consider the staffing problem under guaranteed security and service levels based on the diffusion approximations, with the decision variables being the proportion of travelers from stage 1 to stage 2 and the number of staff membe...[ Read more ]

In this thesis, we consider two applications of queueing theory. In the first part, we study a two-stage security-check system which is motivated by the border-crossing stations between United States and Canada. The main tradeoff in such a system is good customer service (short waiting time) and high security level. We establish a two-stage queueing model and derive the diffusion approximations under nondegenerate slowdown regime for the system. Some numerical results are demonstrated to show the good performance of our approximation formulas. We also consider the staffing problem under guaranteed security and service levels based on the diffusion approximations, with the decision variables being the proportion of travelers from stage 1 to stage 2 and the number of staff members in each stage. An efficient algorithm is proposed to solve the staffing problem.

In the second part, we consider the problem of server scheduling in an overloaded multi-class queueing system with multiple homogeneous servers and general customer abandonment. In the case of exponential reneging, an indexed priority policy, called the cμ/θ rule, is known to be asymptotically optimal in the many-server heavy traffic regime, where the arrival rates and number of servers increase proportionally. For general patience distributions, we aim to find an asymptotically optimal control policy among the class of fixed priority rules. We first describe a fluid model, which is known to be the limit of the stochastic system under priority rules, and show its convergence to an equilibrium state in the case of exponential service. We then formulate an optimization problem in terms of these equilibrium states and derive a closed from expression for the optimal solution when the hazard rate function of patience distribution is increasing and the cost function is concave.