A note on some series representations of the integral of a bivariate normal distribution over an offset circle

Consider the problem of computing an offset circle probability under a normal distribution. One approach is to utilize an infinite series representation in which case it is important to have rapid convergence and a good upper bound on the error introduced by consideration of only a finite number of terms of the series. We relate three seemingly different series representations. In particular we show how two series representations for the bivariate case can be obtained by specializing more general results of Harold Ruben.     

On a sequential test procedure with delayed observations

This paper considers sequential test procedures to decision problems where there exists time delays in obtaining observations.     

An application of linear programming to contingency planning: A tactical airlift system analysis

A linear programming application for the selection of aircraft for a tactical airlift fleet is described.     

On the use of standard tables to obtain Dodge-Romig LTPD sampling inspection plans

Procedures are described which yield single and double sample Dodge-Romig [1] lot tolerance percent defective (LTPD) rectifying inspection plans. For the determination of such plans only a desk calculator and standard tables of the discrete probability distributions are required. Some advantages gained by using these procedures rather than the Dodge-Romig table include: (a) The Consumer's Risk is not limited to 0.10. (b) More choices of LTPD are available. (c) Smaller average total inspection is achieved by using a plan designed for specific “process average” and lot size rather than a compromise plan designed to cover intervals on these two parameters.     

Discrete compound poisson processes and tables of the geometric poisson distribution

This paper discusses the properties of positive, integer valued compound Poisson processes and compares two members of the family: the geometric Poisson (stuttering Poisson) and the logarithmic Poisson. It is shown that the geometric Poisson process is particularly convenient when the analyst is interested in a simple model for the time between events, as in simulation. On the other hand, the logarithmic Poisson process is more convenient in analytic models in which the state probabilities (probabilities for the number of events in a specified time period) are required. These state probabilities have a negative binomial distribution. The state probabilities of the geometric Poisson process, known as the geometric Poisson distribution, are tabled for 160 sets of parameter values. The values of mean demand range from 0.10 to 10; those for variance to mean ratio from 1.5 to 7. It is observed that the geometric Poisson density is bimodal.     

A chance-constrained distribution problem

The transportation model with supplies (S_i) and demands (D_i) treated as bounded variables developed by Charnes and Klingman is extended to the case where the S_i and D_i are independently and uniformly distributed random variables. Chance constraints which require that demand at the jth destination will be satisfied with probability at least β_i and that stockout at the ith origin will occur with probability less than α_i are imposed. Conversion of the chance constraints to their linear equivalents results in a transportation problem with one more row and column than the original with some of the new arcs capacitated. The chance-constrained formulation is extended to the transshipment problem.     

The effect of correlated exponential service times on single server tandem queues

An investigation via simulation of system performance of two stage queues in series (single server, first-come, first-served) under the assumption of correlated exponential service times indicates that the system's behavior is quite sensitive to departures from the traditional assumption of mutually independent service times, especially at higher utilizations. That service times at the various stages of a tandem queueing system for a given customer should be correlated is intuitively appealing and apparently not at all atypical. Since tandem queues occur frequently, e.g. production lines and the logistics therewith associated, it is incumbent on both the practitioner and the theoretician that they be aware of the marked effects that may be induced by correlated service times. For the case of infinite interstage storage, system performance is improved by positive correlation and impaired by negative correlation. This change in system performance is reversed however for zero interstage storage and depends on the value of the utilization rate for the case where interstage storage equals unity. The effect due to correlation is shown to be statistically significant using spectral analytic techniques. For correlation equal unity and infinite interstage storage, results are provided for two through twenty-five stages in series to suggest how adding stages affects system performance for ρ>0. In this extreme case of correlation, adding stages has an effect on system performance which depends markedly on the utilization rate. Recursive formulae for the waiting time per customer for the cases of zero, one, and infinite interstage storage are derived.     

The United States Coast Guard Computer-Assisted Search Planning system (CASP)

This paper provides an overview of the Computer-Assisted Search Planning (CASP) system developed for the United States Coast Guard. The CASP information processing methodology is based upon Monte Carlo simulation to obtain an initial probability distribution for target location and to update this distribution to account for drift due to currents and winds. A multiple scenario approach is employed to generate the initial probability distribution. Bayesian updating is used to reflect negative information obtained from unsuccessful search. The principal output of the CASP system is a sequence of probability “maps” which display the current target location probability distributions throughout the time period of interest. CASP also provides guidance for allocating search effort based upon optimal search theory.     

Evaluation of commonly used rules for detecting “steady state” in computer simulation

A definition of the problem of the initial transient with respect to the steady-state mean value has been formulated. A set of criteria has been set forth by which the efficaacy of any proposed rule may be assessed. Within this framework, five heuristic rules for predicting the approximate end of transiency, four of which have been quoted extensively in the simulation literature, have been evaluated in the M/M/1 situation. All performed poorly and are not suitable for their intended use.     

Counterexamples to optimal permutation schedules for certain flow shop problems

It is well known that a minimal makespan permutation sequence exists for the n × 3 flow shop problem and for the n × m flow shop problem with no inprocess waiting when processing times for both types of problems are positive. It is shown in this paper that when the assumption of positive processing times is relaxed to include nonnegative processing times, optimality of permutation schedules cannot be guaranteed.