Analysis of Stochastic Models through Multi-Layer Markov Modulated Fluid Flow Processes

Abstract

This thesis is concerned with the multi-layer Markov modulated fluid flow (MMFF) processes and their applications to queueing systems with customer abandonment.

For the multi-layer MMFF processes, we review and refine the theory on the joint distribution of the multi-layer MMFF processes and develop an easy to implement algorithm to calculate the joint distribution. Then, we apply the theory to three quite general queueing systems with customer abandonment to show the applicability of this approach and obtain a variety of queueing quantities, such as the customer abandonment probabilities, waiting times distributions and mean queue lengths.

The first application is the MAP/PH/K+GI queue. The MMFF approach and the count-server-for-phase (CSFP) method are combined to analyze this multi-server queueing system with a moderately large number of servers. An efficient and easy-to-implement algorithm is developed for the performance evaluation of the MAP/PH/K +GI queueing model. Some of the queueing quantities such as waiting time distributions of the customers abandoning the queue at the head of the waiting queue are difficult to derive through other methods.

Then the double-sided queues with marked Markovian arrival processes (MMAP) and abandonment are studied. Multiple types of inputs and finite discrete abandonment times make this queueing model fairly general. Three age processes related to the inputs are defined and then converted into a multi-layer MMFF process. A number of aggregate queueing quantities and quantities for individual types of inputs are obtained by the MMFF approach, which can be useful for practitioners to design stochastic systems such as ride-hailing platforms and organ transplantation systems.

The last queueing model is the double-sided queues with batch Markovian arrival processes (BMAP) and abandonment, which arise in various stochastic systems such as perishable inventory systems and financial markets. Customers arrive at the system with a batch of orders to be matched by counterparts. The abandonment time of a customer depends on the batch size and the position in the queue of the customer. Similar to the previous double-sided queueing model, a multi-layer MMFF process related to some age processes is constructed. A number of queueing quantities including matching rates, fill rates, sojourn times and queue length for both sides of the system are derived. This queueing model is used to analyze a vaccine inventory system as a case study in the thesis.

Overall, this thesis studies the joint stationary distribution of the multi-layer MMFF processes and shows the power of this approach in dealing with complex queueing systems. Four algorithms are presented to help practitioners to design stochastic systems and researchers do numerical experiments.