J. Michael Harrison

Bio: J. Michael Harrison is an academic researcher from Stanford University. The author has contributed to research in topics: Queueing theory & Heavy traffic approximation. The author has an hindex of 45, co-authored 86 publications receiving 15644 citations. Previous affiliations of J. Michael Harrison include University of Florida & University of Bristol.

Correction: Brownian models of open processing networks: canonical representation of workload

Abstract: Due to a printing error the above mentioned article had numerous equations appearing incorrectly in the print version of this paper. The entire article follows as it should have appeared. IMS apologizes to the author and the readers for this error. A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an “equivalent workload formulation” of a Brownian network model. Denoting by Z(t) the state vector of the original Brownian network, one has a lower dimensional state descriptor W(t)=MZ(t) in the equivalent workload formulation, where M can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of “heavy traffic” for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix M. To be specific, rows of the canonical M are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix M is shown to be nonnegative, and another natural condition is identified which ensures that M admits a factorization related to the notion of resource pooling.

On the distribution of multidimensional reflected

Abstract: Let Z ={Z(t), t_>0} be a reflected Brownian motion on the K-dimensional nonnegative orthant, with the direction of reflection constant over each boundary surface. Such processes arise in heavy traffic theory for K-station networks of queues. This paper continues our study of the diffusion process Z, with attention restricted to the case K 2 for simplicity. (Most of the results extend directly to higher dimensions; our notation and style of argument are designed to suggest appropriate generalizations for arbitraryK wherever possible.) The backward equation for the transition density function (with boundary and initial conditions), the corresponding forward equation and the equation for the steady-state distribution are all derived informally. Also presented are various calculations relating to steady-state distributions, including a moment formula and the derivation of a condition (involving the drifts and directions of reflection), that we conjecture to be necessary and sufficient for existence of a steady-state. Finally, the relevance of reflected Brownian motion for approximate analysis of queuing networks is reviewed, and some further references on the associated differential equations are given. 1. Introduction. This paper is devoted to further study of the multidimensional diffusion processZ introduced in (5). This process, which we shall simply call reflected Brownian motion, arises in the heavy traffic theory for networks of queues (4), (15). Its state space is the K-dimensional nonnegative orthant, whereK is a positive integer. On the interior of this state space,Z behaves like an ordinaryK-dimensional Brownian motion, with given covariance matrix and drift vector. At each of the (K- 1)-dimensional hyperplanes that form the boundary of its state space, Z reflects instantaneously in a direction that is constant over that boundary hyperplane. Aswe have said, reflected Brownian motion is the natural diffusion approximation for certain multidimensional stochastic processes occurring in the theory of queues. Such approximations have no practical significance, however, unless one can compute interesting quantities associated with the process Z. This paper is aimed at computation of the time-dependent and steady-state distributions ofZ in the caseK 2, but it will fall far short of that objective. We present an assortment of preliminary results, formal calculations and conjectures, hoping to set the stage for later work and to stimulate the interest of others in this rich, complex and potentially important analytical problem. By restricting attention to the case K 2, we simplify certain aspects of the analysis, but most ofwhat will be said extends easily to higher dimensions. Our notation and style of argument are designed to suggest, wherever possible, the appropriate generalizations for arbitrary K. Because Z is a diffusion process, its generator is a partial differential operator, and hence, its time-dependent and steady-state distributions satisfy certain partial differential equations (PDE's). Our primary goal here is to derive the backward equation (with boundary and initial conditions) for the transition density of the process, the corresponding forward (or Fokker-Planck) equation and the PDE associated with the steady-state distribution. In doing this, we shall not attempt to prove rigorous mathematical results. Many questions of regularity are suppressed in order to minimize technical minutae, and, to be honest, there are some fine points that we simply do not

Bayesian Dynamic Pricing Policies: Learning and Earning Under a Binary Prior Distribution

Abstract: Motivated by applications in financial services, we consider a seller who offers prices sequentially to a stream of potential customers, observing either success or failure in each sales attempt. The parameters of the underlying demand model are initially unknown, so each price decision involves a trade-off between learning and earning. Attention is restricted to the simplest kind of model uncertainty, where one of two demand models is known to apply, and we focus initially on performance of the myopic Bayesian policy (MBP), variants of which are commonly used in practice. Because learning is passive under the MBP (that is, learning only takes place as a by-product of actions that have a different purpose), it can lead to incomplete learning and poor profit performance. However, under one additional assumption, a constrained variant of the myopic policy is shown to have the following strong theoretical virtue: the expected performance gap relative to a clairvoyant who knows the underlying demand model is bounded by a constant as the number of sales attempts becomes large.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Controlled Markov processes and viscosity solutions

Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an