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.

Impulse Control of Brownian Motion

Abstract: Consider a storage system, such as an inventory or cash fund, whose content fluctuates as a (μ, σ2) Brownian motion in the absence of control. Holding costs are continuously incurred at a rate proportional to the storage level and we may cause the storage level to jump by any desired amount at any time except that the content must be kept nonnegative. Both positive and negative jumps entail fixed plus proportional costs, and our objective is to minimize expected discounted costs over an infinite planning horizon. A control band policy is one that enforces an upward jump to q whenever level zero is hit, and enforces a downward jump to Q whenever level S is hit (0 < q < Q < S). We prove the existence of an optimal control band policy and calculate explicitly the optimal values of the critical numbers (q, Q, S).

Dynamic Control of a Queue with Adjustable Service Rate

Abstract: We consider a single-server queue with Poisson arrivals, where holding costs are continuously incurred as a nondecreasing function of the queue length. The queue length evolves as a birth-and-death process with constant arrival rate ? = 1 and with state-dependent service rates µ nthat can be chosen from a fixed subset A of [0, 8). Finally, there is a nondecreasing cost-of-effort functionc(·) on A, and service costs are incurred at ratec(µ n ) when the queue length isn. The objective is to minimize average cost per time unit over an infinite planning horizon. The standard optimality equation of average-cost dynamic programming allows one to write out the optimal service rates in terms of the minimum achievable average costz*. Here we present a method for computingz* that is so fast and so transparent it may be reasonably described as an explicit solution for the problem of service rate control. The optimal service rates are nondecreasing as a function of queue length and are bounded if the holding cost function is bounded. From a managerial standpoint it is natural to comparez*, the minimum average cost achievable with state-dependent service rates, against the minimum average cost achievable with a single fixed service rate. The difference between those two minima represents the economic value of a responsive service mechanism, and numerical examples are presented that show it can be substantial.

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. This paper was accepted by Gerard P. Cachon, stochastic models and simulation.

Brownian models of open processing networks: canonical representation of workload

Abstract: 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 insures that $M$ admits a factorization related to the notion of resource pooling.

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