Summary The achievable region approach seeks solutions to stochastic optimization problems by characterizing the space of all possible performances (the achievable region) of the system of interest and optimizing the overall system-wide performance objective over this space. This is radically different from conventional formulations based on dynamic programming. The approach is explained with reference to a simple two-class queueing system. Powerful new methodologies due to the authors and co-workers are deployed to analyse a general multi-class queuing system with parallel servers and then to develop an approach to optimal load distribution across a network of interconnected stations. Finally, the approach is used for the first time to analyse a class of intensity control problems.

Herman Chernoff was born in New York City on 1 July 1923. He went to school there and later received the B.S. degree from the City College of New York in 1943, majoring in mathematics with a minor in physics. For a year and a. half, he worked as a junior physicist with the US. Navy, before joining Brown University for graduate work in applied mathematics. His studies were interrupted by a short period in the U.S. Army, and then his interest in statistics led him to complete his Ph.D. thesis at Columbia University under the supervision of Abraham Wald. At Brown University, Herman met Judy Ullman. They have been married since 1947 and have two daughters, Ellen and Miriam.Herman worked for the Cowles Commission at the University of Chicago and then spent three years in the Mathematics Department at the University of Illinois before joining the Department of Statistics at Stanford University in 1952, where he remained for 22 years. He moved to M.I.T. in 1974, where he founded the Statistics Center. Since 1985 he has been in the Department of Statistics at Harvard.Professor Chernoff has been honored for his contributions in many ways. He was President of the Institute of Mathematical Statistics and is an elected member of both the American Academy of Arts and Sciences and the National Academy of Sciences. The hook Recent Advances in Statistics published in honor of his 60th birthday in 1983 contained papers in the fields where his influence as a researcher and teacher has been strong: design and sequential analysis, optimization and control, nonparametrics, large sample theory and statistical graphics.

There have been many papers on biased coin designs and their use in balancing the numbers of patients allocated to different treatments in a clinical trial, without increasing the risk of selection bias.Less attention has been given to the corresponding risk when sequential allocations depend on the previous responses and the aim is to reduce the number of patients on inferior treatments.The ethical requirements may produce a substantial imbalance in the treatment groups.This paper gives a number of examples where selection bias is a serious possibility. Introduction.Selection bias can occur in an experiment designed to compare medical treatments if the experimenter knows, before deciding whether or not to admit a particular patient to the trial, which treatment will be administered next.Blackwell and Hodges (1957) introduced a measure of the bias in a design based on the maximum expected number of correct guesses that an experimenter can achieve when attempting to predict the successive treatment allocations.Their paper and later investigations by Efron (1971), Smith (1984) and many others were concerned with the need to balance the experiment while

Mathematically convenient models for a search can be based on a prior representation of the hidden objects and their values by a mixture of Poisson processes. In simple cases, we can find the corresponding Bayes procedures to maximise expected net gains, allowing for the cost of searching. More generally, optimal stopping rules for the search axe difficult to construct and evaluate. We also investigate a procedure based on the asymptotic behaviour of the system when the number of hidden objects is large. This is shown to provide a reasonably effective stopping rule over a wide range of conditions.

Auditing is based on random sampling from the records of a company. This paper considers a Bayesian model for determining the sample sizes by finding a balance between the cost of sampling and the risk of leaving major faults undiscovered. It leads to a dynamic programming problem which involves substantial computations, but a slightly different approach in which discrete sampling is replaced by a continuous search for faults produces more explicit solutions.

In this paper we assume that an oil company has k areas in which to drill and occurrences of undiscovered oilfields are represented by the model of Beale. The company is seeking a strategy for drilling that maximizes its expected return under the constraint that the total amount spent on drilling in all areas must not exceed R. This leads to an integer programming problem. We establish that the expected return functions for the separate areas are concave and that this property can be used to reduce the computational effort required to find an optimal solution.

Lerche (1986) investigated a sequential testing problem concerned with deciding the sign of a normal mean. He demonstrated that a parabolic boundary which corresponds to Keywords: asymptotic expansionBrownian motionfree boundary problemheat equationoptimal stoppingsequential analysistruncated test

It is well known how to combine the significance levels observed in a number of independent experiments. When this number is a random variable determined by a stopping rule, the observed significance level can still be calculated if there is an acceptable ordering of the points in the extended sample space. But what can be said if the stopping time is ill defined This paper obtains explicit lower bounds on the level of significance by considering orderings based on a family of alternative hypotheses. These bounds give some measure of the effect of failing to specify the stopping rule in advance.

There are good reasons for using sequential methods in some statistical decision problems, but a stopping rule that is helpful for deciding whether or may not be so good for estimating This paper considers the construction of confidence bounds on a real parameter and investigates the relation between the ordering of boundary points that are accessible under the stopping rule and the natural ordering of the parameter space.

SUMMARY The risk involved in a trial to compare two medical treatments is shared by patients who receive the inferior treatment during the experimental phase and those remaining after the experiment who might all receive the inferior treatment if the results are misleading. We consider the maximum of this risk with respect to the unknown probabilities of success and seek allocation rules that minimize this quantity, for a given total of patients. It needs extensive computations to find such minimax procedures, but there are simple and almost equally effective allocation rules based on a truncated sequential probability ratio test.

Given a finite number of different experiments with unknown probabilities p 1 , p 2 , ···, p k of success, the multi-armed bandit problem is concerned with maximising the expected number of successes in a sequence of trials. There are many policies which ensure that the proportion of successes converges to p = max ( p 1 , p 2 , ···, p k ), in the long run. This property is established for a class of decision procedures which rely on randomisation, at each stage, in selecting the experiment for the next trial. Further, it is suggested that some of these procedures might perform well over any finite sequence of trials.

This paper is concerned with the problem of selecting the transition intensities for a Markov chain in continuous time so as to minimise the long-term average cost. Sufficient conditions are established for an optimal stationary policy using unbounded solutions of the optimality equation. This is a development of recent work on Markovian decision processes in discrete time. The theory is illustrated by considering a simple birth and death process with controlled immigration.

This paper is concerned with the general problem of choosing an optimal stopping time for a Brownian motion process, where the cost associated with any trajectory depends only on its final time and position.

This paper is a sequel to [1] and considers a more realistic formulation of the same question: that of finding an optimal policy for controlling the path of a space-ship as it moves towards its target. The difference here is that we no longer suppose there is an infinite quantity of fuel, always available at a fixed price, for modifying the current direction of motion. This complicates the problem of reducing the final miss distance, by introducing an extra variable. As before, we shall be particularly concerned to find a control procedure which always minimizes the mean square terminal miss. From the theoretical point of view we are also interested to see whether the techniques used to approximate the optimal policy can be extended, and how far we shall be forced to adopt a new approach. Results are derived which provide bounds on the form of the optimal policy. The derivation depends on a comparison technique whose validity is intuitively obvious, but which is still only a conjecture. However, further confirmation is obtained in the quadratic case from asymptotic expansions giving the form of the solution both when the space-ship is far away from its target and during its final approach.

: Imagine a spaceship travelling towards a certain planet with predetermined speed, in a direction which will bring it close to the target after a known period of time. Observations on the position of the target, relative to the present course, are made continuously and lead to a gradually improving prediction of the eventual miss distance. On the other hand, the fuel available in the spaceship for making minor changes in the direction of motion, is gradually losing its effectiveness. This is because the final change of position caused by a small velocity imposed perpendicular to the present motion, is roughly proportional to the remaining time. Thus we have a control problem which is essentially one of compromise between the extremes of using the fuel early and perhaps in the wrongway, because of poor information; or waiting too long for more precise information, so that the fuel becomes ineffective.