Small switching offices and office information systems share some common properties in the description of load peaks. In both cases, because of the smaller user population, the structure of the underlying random process becomes important, and while the methods of extreme value engineering are applicable, the usual asymptotic distributions are not. The family of gamma distributions is proposed as a flexible process model (which does not imply negative load values), and it is used to model data in the literature on the traffic in small switching offices. New data from trials of office information systems yield extreme values of loads which support this proposal. The gamma model supports an extreme value engineering procedure for the design of peak monitoring methods, the calculation of return period loads, and the calculation of probability curves for extremes.>

The frequency of occurrence of logic functions of binary N-tuples can be observed from sequences of binary patterns. The logic functions considered here are AND, NOR, NAND, OR, and an odd-even parity check; and the frequency parameters are expressed as real matrix transformations on the probabilities of the patterns. Some properties, inverses, and interrelationships among the parameter sets are given, along with fast algorithms to facilitate computational processes. The results permit the outputs of convenient hardware logic operations to be converted into other parameters for smoothing, detection, or inference purposes, or to estimate the pattern probabilities by inversion. A measure of association among binary patterns is given as one characteristic feature which can be derived from the observed parameters.

A statistical distribution is reported for the lengths of Ph.D. programs from which are derived functions analogous to those of elementary reliability theory. These show that the anticipated future duration of these research programs does not decrease uniformly, so that there are diminishing returns from further work almost to the point where the expected date of completion remains a constant distance in the future.

The process of matrix iteration to solve the nth order linear matrix equation Ax + u = 0 for x is considered.The matrix C = I --A, determining the convergence of the process, is assumed to be convergent and to be known without error.A quadratic measure of convergence a(C) = [[ C liE~tin is introduced.For the special case of symmetric C and Tr (C) = 0, a(C) is the standard deviation of the eigenvalues of C. The measure a(C) is the basis for the analysis of the relative error in the iterative vector x(m) for m ~ ~, due to roundoff errors generated at each iteration.If the generated errors are assumed to be independent random variables with mean zero and standard deviation erda, then the errors do not accumulate but stay at a constant level.In fact, if x(m) = (I -~ Am)x8 with x, ffi --A-~u, then the estimated value 8(A) of a(A~) for m --~ ¢¢ is shown to be 8(A) = [~-0 o's(C~)]~'erms > [1 ~(CC)]~.e ....This reduces to 8(A) = erms/(1 --~09 t if cTc = o'0~I and is of the order of e~. for almost all a0 < 1.So the solution obtained by constant iteration is the best possible solution, using s-digit arithmetic and 2s-digit inner product evaluation plus roundoff.This minimum error level is reached after approximately 2 log b(A)/log a-(C) iterations, which is almost proportional to s.The error is proportional to (R 2 --1)½/2 In R if R is the radix of the number system used, which follows from an estimation of e~. in normalized floating point arithmetic.Computer tests on many random systems are included and show good agreement with the proposed error analysis.