Diffusion Models for Double-ended Queues with Renewal Arrival Processes

We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits

Three stochastic models for order book dynamics

In this dissertation, we study three stochastic models for order book dynamics. We first consider a double-ended queue with renewal arrivals where buyers and sellers arrive to conduct trades. Because the analytical results of the limiting behavior of the double ended queue is intractable except in very special cases, we apply the diffusion approximation method. We find that the queue length process of the double-ended queue converges to an asymmetric Ornstein-Uhlenbeck process with drift. We use simulation to evaluate the goodness of our approximations. Next we consider the double-ended queue where the arrival processes of the buyers and sellers are state-dependent. We assume that traders arrive at the queue according to a phasetype renewal process (PH-renewal process), and an arriving trader is a buyer or a seller according to state-dependent probabilities. We derive an explicit algorithm to compute the limiting distribution of this double-ended queue. We study two special cases withDoctor of Philosoph

Operations management for double-ended queues

In this paper we have presented the double-ended queueing model with its generic form, abstract modeling extensions, different types of controls to be assumed by the management of the system, as well as optimization via various methodologies. Since its inception in 1950s, double-ended queuing model has had widespread use and recently it is observed that literature is focusing on social welfare perspectives on queues under strategically acting customers and enhanced information exchange due to advanced technology. It is also reviewed that for the complex and analytically tractable versions of double-ended queues it is possible to obtain approximate or near-optimal results via methodologies such as simulation and fluid and diffusion approximations.No sponso

STOCHASTIC MODELS FOR SERVICE AND TAXI SYSTEMS

This dissertation consists of two topics: inventory systems and service systems. The first topic involves a single-item inventory system with two demand classes with backorders. We introduce a four-parameter (A; B; C1; C2) policy to manage such a system. Under this policy, we place an order of size A if the on-hand inventory level is less than or equal to B, and reject demands of class k if the inventory level is below or at Ck (k = 1; 2). We develop methods of computing the long-run average cost that can be used to numerically obtain the four parameters that minimize this average cost. When the demands arrive according to Poisson processes and the production lead times are exponential and the order size A is fixed, we formulate the problem as a Markov Decision Process. We prove structural properties when A = 1, and numerically show that the four-parameter policies are optimal when A > 1. We also study the cases where demand interarrival times or production lead times are generally distributed. The numerical optimality is done using the Genetic Algorithm. The second topic considers a system of customers and taxis with Poisson arrivals and exponential patience times and delayed matching. We formulate the system as a Continuous Time Markov Chain and study the fluid and diffusion approximations. We consider Kurtz's Approximation and Gaussian Approximation and compare their performance numerically with simulation. We next formulate an optimal control problem to maximize the total net revenue over a fixed time horizon by controlling the arrival rate of taxis. We solve the optimal control problem numerically and compare its performance to simulation. We propose a heuristic control policy. We show that the expected regret of the heuristic policy is a bounded function of the horizon.Doctor of Philosoph

Simulation of sample paths for Gauss-Markov processes in the presence of a reflecting boundary

Algorithms for the simulation of sample paths of Gauss–Markov processes, restricted from below by particular time-dependent reflecting boundaries, are proposed. These algorithms are used to build the histograms of first passage time density through specified boundaries and for the estimation of related moments. Particular attention is dedicated to restricted Wiener and Ornstein–Uhlenbeck processes due to their central role in the class of Gauss–Markov processes