We consider branching processes for which the first three moments of the distribution of offspring exist. Let f(t) andz be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f(t) = t. Then if M = f'(z) and fn(t) is equal to the generating function of the distribution of nth generation descendants of a single individual, it is known that, quite generally, (z – fn(0)) / Mn tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even ifzis less than 1. Numerical values are given for the Poisson and negative binomial cases.

In a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).

Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.

For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

Suppose that a new hospital is to be planned or an existing one reorganized according to the modern Progressive Patient Care System (Hartley et al. (1968)). In this system, there are three types of wards, viz.

A number of items are to be distributed among n individuals, arranged in a line and labelled by the integers 1, 2, ···, n, in such a way that the distance between any two members presented should be at least d, a given number, but otherwise any subset of the population should have the same probability of each of its members being presented. The number of items thus distributed is a random variable X(d, n) and it turns out that X(d, n) is asymptotically a fixed percentage of the population size n. This fact makes it possible to apply the model in a number of situations. For example, if in a foreign aid program a number of tools are offered and to be distributed in a fair way but, for lack of sufficient quantity, one tool is expected to be used by a number of neighbours, the model is applicable if the amount of aid is not fixed in advance but may vary within certain limits. A similar situation arises in cases of disaster (floods, earthquakes, etc.) when the victims are sent gifts, again in an insufficient quantity, hence for the sake of justice, a family of a smaller size is supposed to receive only one gift. In many other similar cases, the model can be used.

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t0 at which the visit to state T begins.

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

As stated elsewhere [9], the two experimenter-controlled events learning model of Bush and Mosteller [1] may be described by a Markov process X0,X1, ···satisfying the following conditions.

1. (i) Xo has an arbitrary distribution on (0, 1).

2. (ii) If Xn is given, then with probability π1 and with probability π2

3. (ii)

In Jennrich (1969) the model is considered, where x(n) is a sequence of i.i.d. (0, σ2) random variables and z(n; θ) is a continuous but possibly non-linear function of θ∈ Θ, Θ being a compact set in Rp. We shall use a second subscript when referring to a particular coordinate of θ0 so that θ0j is the jth coordinate. Jennrich establishes, under suitable conditions on z(n; θ) and x(n), the strong consistency and asymptotic normality of the least squares estimates of θ. Our main purpose here is to extend these results to the case where x(n) is generated by a stationary time series.

When is the product of the d.f.'s H1(·), ···, Hm(·) attracted to an extreme value law φ(x)? We associate with each Hi(·) its A-function Hi(x) is attracted to φ(x) if each Hi(x) is in the domain of attraction of φ(x) and Ai(z) ~ Aj(z), 1 ≦ i, j ≦ m. Equivalence of A-functions determines an equivalence relation which partitions the domain of attraction of φ(x)into one or more convex sets. These sets fail to be closed under passages to the limit (complete convergence).

It is shown that the extreme stable laws have one-sided moment generating functions with interesting mathematical forms. The fact that one of these forms, zcz, is a moment generating function is used to establish two interesting statistical results; first, that rr is a moment sequence for a density with decreasing failure rate, and secondly, that the likelihood ratio test for testing a simple null hypothesis in a multinomial distribution is admissible and Bayes.

Leslie (1969) obtained the Laplace transform for the recurrence time of clusters of Poisson processes, which can be thought of as yielding the interspike interval distribution for a neuron that receives Poisson excitatory inputs subject to decay. Here, several extensions of this model are derived, each including Poisson inhibitory inputs. Expressions for the mean and variance are derived for each model, and the results for the different models are compared.

In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of a GI/G/s queue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the cases M/M/s (s = 1, 2, ···, ∞) and M/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for the M/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady state M/G/∞ queue are independent at any fixed time; this is a result I have been unable to find in the literature.

In problems of traffic theory, we are frequently concerned with the queues of vehicles which form at an intersection. A standard model is the vehicle queue whose length is increased during unit time intervals [t, t + 1) by non-negative integer inputs which form a sequence of i.i.d. or Markovian random variables; see Gani (1970) and Lehoczky ((1969), (1971)). The output from the queue at the end of each unit of time is one vehicle if the queue is non-empty, and no vehicle if there are none waiting.

The Laplace-Stieltjes transform of the distribution of the busy period for the M/G/1 system with infinite waiting room can be obtained by using an argument from branching theory. In the present paper it is shown that by applying this argument it is rather easy to derive the expression for the joint distribution of the busy period and the maximum number of customers present simultaneously during this busy period for the M/G/1 system with infinite waiting room as well as the expression for the distribution of the busy period for the M/G/1 system with finite waiting room.

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold:

1. (a) no server sits idle while there are customers waiting to be served;

2. (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one);

In [4], the limiting behaviour of a stochastic system with two types of input was investigated by reducing the problem to the solution of an integral equation. In this note we use the same approach to study the equilibrium waiting time problem for the general single server queue with preemptive service interruptions. (For a comprehensive account of the existing literature on queues with service interruptions we refer to [2] and [3].)