New Releases by Michael Harrison

Here is a systematic discussion of Brownian motion and Ito stochastic calculus. Develops the mathematical methods needed to analyze stochastic processes related to Brownian motion and shows how these methods are used to model and analyze various stochastic flow systems such as queueing and inventory systems. Emphasizes stochastic calculus and models used in engineering, economics, and operations research. Topics include stochastic models of buffered flow, the backward and forward equations, hitting time problems, regulated Brownian motion, optimal control of Brownian motion, and optimizing flow system performance.

Suggests the real people and events which inspired Sir Arthur Conan Doyle to create Sherlock Homes and his adventures.

More than 100 poems, old and new, to celebrate the Christmas season.

We consider the diffusion limits of several closely related production planning problems. Each involves a make-to-stock producer who faces IID demands over a sequence of future periods. In the simplest formulation, a production capacity or workforce level must be fixed at time zero, and thereafter the actual production rate is adjusted dynamically in response to inventory fluctuations. The capacity decision fixes certain operating costs and constrains subsequent decisions regarding production rates. Fixing the demand process, we consider a sequence of cost structures in which the total inventory carrying cost approaches zero. (This requires that both the physical cost of carrying inventory and the interest rate earned on external investment vanish). Under this condition, the production planning problem approaches a two-stage optimal control problem for Brownian motion. The first stage of the limiting problem involves drift rate selection for a Brownian motion, and its second stage is the instantaneous control problem formulated and solved by Harrison-Taylor (1977). With small holding costs, we find that an optimal capacity decision calls for near equality of the average production and demand rates, which in turn justifies a diffusion approximation for the subsequent problem of inventory control through production rate adjustment.

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector if distributed approximately as a random vector Z under heavy traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian Motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case. (Author).