It is shown how a triangle can be recognised from the odd moments of the length of random chords and how a quadrilateral can be recognised from the derivative of its chord length density.

Let X1 ··· Xn be i.i.d.r.v.'s uniformly distributed on [0, l]. Let X(1) ··· X(n) be the ordered r.v.'s 0≦ X(1) ≦ X(2)···≦ X(n) ≦ l. We obtain the n + 1 intervals (X(i), X(i+1))ni=0, with X(0) = 0, X(n+1)= l of length Li = X(i+1) – X(i). Let N(c) be the number of intervals of length Li < c. We have where The study of Nc is of some interest in molecular biology and was studied in [3] where the author found a complicated expression for the probability for a randomly chosen segment to have a length less than or equal to c; he then obtains an approximation. From the experimental point of view, it seems more reasonable to work with the mean since one can obtain an unbiased estimator that way.

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj <∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.

The asymptotic distribution of Hotelling's one-sample T2 in multivariate Edgeworth populations is expanded to terms of the first order. Comparison with published simulation results indicates that the result is quite useful, even in cases where the underlying population is not well represented by an Edgeworth expansion.

We consider a random field {Xij, i, j = 1, ···, n} where the random variables Xij takes on values 1 or 0. The collection {Xij} can be viewed as a random graph with nodes {1, ···, n} by interpreting Xij = 1 as the existence of an arc emanating from the node i to the node j. Such a representation will enable us to study ordered and unordered graphs, being also the general representation of a random graph. In this note the probability that the graph is connected is computed under the condition that ΣiXki=l for k = 1, · ··, n. This result extends Ross's recent theorems on connectivity of random graphs.

In the study of null-persistent Markov chains the sequence plays an important role. (See for example, Orey (1971), Athreya and Ney (1980).) In this paper we consider the rate of growth of UN to ∞ as N → ∞. This rate of growth is related to the rate of growth of as N → ∞ and also to the rate at which as n → ∞.

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.

This paper considers functional dependence from a general standpoint. Some basic ideas are explored and the extent of functional dependence studied. These are illustrated by means of simple linear regression.

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.

The extended block replacement policy with used items as advocated by Tango (1978) uses used items of age T only although the policy creates used items of lower age. In this note we modify Tango's policy so as to make use of all used items created by the policy.

The standard linear regression model is analysed using a method called functional least squares which yields a family of estimators for the slope parameter indexed by a real variable t, |t| ≦ T. The choice t = 0 corresponds to ordinary least squares, non-zero values being appropriate if the error distribution is long-tailed, and it is argued that the approach is a natural extension of least squares methodology. It emerges that the asymptotic normal distribution of these estimators has a covariance matrix characterised by a scalar function of t, called the variance function, which is determined by the error distribution. Properties of this variance function suggest graphical criteria for detecting departures from normality.

Consider a network of queues in equilibrium. When are the flows observed on two given links of that network equivalent, i.e., when do they have the same law?

The verification of the equivalence of two such point processes is first reduced to an algebraic problem by a technique based on the filtering theory.

This method is then used to show that the arrival and departure processes at an M/M/1 node in a Jackson network are not always equivalent, thereby contradicting a conjecture made in [8]. An example where that equivalence holds is also given; it provides new results on the time reversibility of some familiar processes.

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

The changes in diversity of competitive strategies in a Maynard Smith population model with mixed strategies are related to the changes in population mean strategy. The effects of slight mutations in strategy frequencies, and of slight perturbations of the contest payoff rules are then investigated, and found to increase and decrease diversity respectively (to a third-order approximation). A relation among mutational effects, payoff perturbation effects and stable population diversity is suggested.

A fundamental task of philologists is to construct the family tree (stemma) of preserved copies of ancient manuscripts. A simple probabilistic model based on random rooted trees is proposed to assist in the identification of the number of terminal copies. The model provides the distribution of the number of terminal vertices in a random tree. An application to stemma construction is given.

A new approach to the minimization of polyhedral convex functions is applied to give a finite algorithm for the rank regression problem. Numerical results for the Daniel and Wood example are presented.

In this note we consider the marginal effect of adding an additional spare to a system containing operating units and their spares. We shall see that under the circumstance that the number of spares is greater than the number of repair facilities, the marginal effect will assume a simple interesting structure.