Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t1)w(t2)}= min (t1, t2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

The concepts of distribution operator, stochastic independence, convergence in distribution and normal distribution are formulated for pairs of canonically conjugate quantum-mechanical momentum and position operators. It is shown that if the sequence (pn, qn), n = 1, 2, ··· is stochastically independent and identically distributed with finite covariance and zero mean then the sequence of pairs of canonical observables converges in distribution to a normal limit distribution.

The stochastic model consists of a two-dimensional random walk with unit steps, each step being a Bernoulli trial. Two sets of possible conditions arising from astronomical theory give rise to two different barriers in the plane of the walk which cause a return to the origin. Another barrier indicates the formation of a planet and also causes a return to the origin. Consideration of the composition of the planets formed at this barrier allows their orbital distances and direction of rotation to be calculated. The predictions of one of the two models were found to be in close agreement with the observed planetary distances and masses in the solar system.

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

Little [8] gave conditions sufficient to obtain the formula L = λW, where L is the time average queue size, W is the expected wait in queue, and λ is the arrival rate of customers. This formula is applicable to a very general class of queuing models and is one of the most useful results in queuing theory.

This paper describes two one-for-one ordering (S − 1, S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function of S in order to obtain an optimal value of S.

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationary GI/G/m queue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for the GI/G/1 queue. An upper bound is also determined for the kth factorial moment of the number found in the queue by an arrival in terms of the kth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for the GI/G/1 queue.

This paper develops expressions for the derivatives with respect to a parameter μ of the stable probabilities of a class of Markov chains whose transition matrices are of the form Q + μW. These expressions lead to iterative schemes for calculation which in turn suggest gradient algorithms for finding locally optimal chains.

Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

Results similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

In dealing with Markov chains, we are sometimes forced by circumstances to make observations on lumped states. We may wish, however, to deal with a random variable which is a function of the numbers in the various (unlumped) states at various times and whose moments, therefore, will involve the probabilities of transition between the unlumped states.

This paper demonstrates that, although these probabilities are immeasurable directly, we can, provided that we restrict our attention to a certain well-defined class of functions as our random variables, reduce all moments and product-moments of such random variables to functions of lumped state probabilities.

Search for an object whose motion is a symmetrical two-dimensional diffusion process is considered in the case where a given amount of search of constant density is distributed, during a given interval of time, over a fixed circular area. An expression for the probability of detection of the object during this interval of time is derived in the special case where the location of the object at time zero is given. The application of this expression would involve numerical work.

A realization of a doubly stochastic Poisson field is assumed to be observed in a Borel set S ⊂ Rk. The best linear estimate of the realization of the intensity field (at an arbitrary but fixed point x ∈ S) which generated the observation is obtained.

In a recent paper Rényi (1964) has obtained the following characterization of the Poisson process.

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

The game of pure strategy, sometimes called Goofspiel or Gops (see [2] and [3]), is played by two players, using a normal deck of cards, as follows. The 13 clubs are first taken out of the deck and of the remaining 39 cards the 13 hearts are given to Player I, the 13 diamonds to Player II, and the 13 spades are placed face down in the center. The spades are shuffled and one is turned face up. At this point, the two players choose one of their cards and then simultaneously discard it. The one who discards the higher card (ace being low, king high) wins from the other an amount equal to the value of the upturned spade (ace = 1, king = 13). If both players discard the same card, then neither wins. The three cards are then thrown away, a new spade upturned and the game continues. After 13 plays, there are no remaining cards and the game ends.

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x1, ···, xn) in Rn by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.