The processes of the title have frequently been used to represent situations involving numbers of individuals in different categories or colonies. In such processes the state at any time is represented by the vector n = (n1, n2, …, nk), where nt is the number of individuals in the ith colony, and the random evolution of n is supposed to be that of a continuous-time Markov chain. The jumps of the chain may be of three types, corresponding to the arrival of a new individual, the departure of an existing one, or the transfer of an individual from one colony to another.

A problem of interest to many population geneticists is the process of change in a gene frequency. A popular model used to describe the change in a gene frequency involves the assumption that the gene frequency is Markovian. The probabilities in a Markov process can be approximated by the solution of a partial differential equation known as the Fokker-Planck equation or the forward Kolmogorov equation. Mathematically this equation is where subscripts indicate partial differentiation. In this equation, f(p, x; t) is the probability density that the frequency of a gene is x at time t, given that the frequency was p at time t = o. The expressions MΔX and VΔx are, respectively, the first and second moments of the change in the gene frequency during one generation. A rigorous derivation of this equation was given by Kolmogorov (1931).

One area of science in which probabilistic models are beginning to play a role is in neuron firing. The firing process may be thought of in terms of electrical potentials: there is a potential difference across the membrane surrounding a neuron; this potential difference is determined by the concentrations of sodium, potassium, and chlorine ions on the two sides of the membrane. When electrical impulses arrive along the many input fibers, the potential difference changes; when it crosses a certain threshold, the neuron “fires”, that is, an impulse is discharged along the axon. Arriving impulses are of two types, those which raise and those which lower the potential inside the neuron; these are called stimuli and inhibitors, respectively. The mechanism by which stimuli and inhibitors interact in changing the neuron potential is not well understood; however, it appears that some of the stimuli cause the neuron to respond, or fire, and the inhibitors play a role in preventing the other stimuli from causing a response. The ability of stimuli to cause responses is further weakened by the decay, in the absence of arriving impulses, of the transmembrane potential toward a “resting level”, the level to which the potential is reset after a firing. While several models for the firing process have been proposed ([3], [4], [5], [6], [7], [11]), the present paper is devoted to introducing some modifications into a model proposed by Ten Hoopen and Reuver ([9], [10]). In particular, we place a time limit on the ability of inhibitors to prevent a stimulus from producing a response. Hopefully, these modifications bring their model closer to physiological reality.

In many chemical and biochemical situations it is of interest to know the distribution that results from the breaking up of long molecules into shorter segments under certain hypotheses. For example, Montroll and Simha (1940) assumed that polymers consist of discrete units—monomers—connected by bonds all of which have an equal chance of breaking in the depolymerization process. Charlesby (1954) used a different approach to essentially the same model and in effect obtained differential equations for the moments of the ensuing distributions. More recently, Daniels (1967) has given a more thorough mathematical exposition, especially with regard to the effect of the initial length distribution, while Blatt (1967) has considered a model in which not all links are susceptible to breakage.

In the present paper we consider the r-molecular reversible reaction rA⇌B from several viewpoints. The deterministic theory for integral reaction orders is considered first and is subsequently extended to cover the case of fractional order reactions. Stochastic models are then proposed, the analyses being carried through by spectral methods and, in the case of first order reactions, the first passage time problem is also examined. Finally, we use a diffusion theory approach to the problem to obtain results which are valid for a large number of molecules.

Consider an infinite capacity dam in which the input and release occur continuously in time. Write X(t) for the total input up to time t starting from X(0) = 0 at t = 0. Let Z(t) be the content of the dam at time t and R(u) (0 ≦ u < ∞) a release function such that in any interval of time (t, t + dt), the amount of water released is R(Z(t))dt + o(t) for any bounded realisation of the process {Z(t)}. Thus R(u) can be regarded as a “rate of release”.

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

Modern highways, particularly the freeways of large cities, carry a considerable volume of traffic during certain times of day. Thus if any interruption or retardation of flow occurs, a large reaction in the shape of a monumental and time-consuming traffic jam soon appears. For example, when an accident or mechanical breakdown gives rise to a severe flow restriction or stoppage, many other vehicles may be quickly halted, and remain stopped until the impediment is cleared away. In addition, the flow of traffic may be slowed considerably even after the original stoppage is removed owing to the existence of a queue. Consequently, vehicles that arrive long after the original restriction is removed experience prolonged, and sometimes seemingly inexplicable, delay. Our purpose is to develop a probability model for the situation described, with the aim of estimating the consequence of a temporary flow restriction. Various measures of (in)effectiveness are worth considering. We consider primarily the total vehicle-hours waited while the jam dissipates; the latter may roughly measure total social cost. The total number of vehicles involved in the jam is also of interest, as are other figures of merit.

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:

1. 1. The duration of the busy period.

2. 2. The number of customers served during the busy period.

3. 3. The maximum number of customers in the queue during the busy period.

A weighted Markov chain technique is used to determine the general distribution of the number of tie positions in a two candidate ballot. This may be interpreted as the distribution of the number of returns to the origin in an asymmetric random walk, when the particle at each step moves μ units in the positive direction and γ units in the negative direction, given the total number of positive steps and the number of negative steps. Explicit results are obtained when γ = 1, whereas the case of μ and γ ≠ 1 is solved in the form of a generating function.

Un théorème de Lewis [3] montre que les seuls facteurs indécomposables de la mesure de Lebesgue m sur S = [–π, π] sont certaines lois discrètes πn,r (r est premier et divise n), et que les décompositions en facteurs premiers étudiées dans [5] constituent toutes les décompositions possibles (en facteurs premiers).1 De ces décompositions se déduisent toutes les propriétés de divisibilité de m.2 Nous développons ce point de vue et posons des problèmes analogues pour le cas de la circonférence γ.

There are now many examples in various fields where the behaviour of ‘particles' as exhibited by their transition from one state to another is described by a multidimensional stochastic process. The linear migration model in which particles move independently of one another through a number of states has been dealt with by Bartlett (1949). This process has been used by Siegert (1949) in considering the approach to equilibrium of non-interacting gas molecules and by Krieger and Gans (1960) and Gans (1960) to examine the distribution of a multicomponent system disturbed from its equilibrium distribution and relaxing by first-order processes to another equilibrium. The correspondence between the deterministic model based on the principle of mass action and the stochastic model has been discussed by Darvey and Staff (1966) in the context of unimolecular multicomponent chemical reactions.

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σk=0∞pksk and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

We consider a branching process for which the offspring distribution has the generating function f(t) and mean f'(1) = m < 1. The probability that a line descended from an individual still survives in generation n is asymptotically of the form cmn. A method is derived whereby good bounds for c may be obtained. This method makes use of the first three moments of the distribution of offspring.

In this note we consider an age-dependent branching process in which the life-spans of sister cells are correlated as well as the numbers of offspring of sister cells, but otherwise cells live and reproduce independently. One might surmise that in many populations there would be some positive correlation among siblings due to similar characteristics inherited from parents. Data of Powell (1955) seem to corroborate this conjecture. Powell's data indicate that in certain bacterial populations the life-spans of sister cells have significant positive correlations while the correlation between the life-spans of mother and daughter cells is negligible. More recently positive correlations have been observed in cell populations among the life-spans of first cousins and also among second cousins (Kubitschek (1967)). However, the model presented in this note permits correlations only among siblings.

In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation (where f(y − z) = f(0), y < z) provided:

1. 1. g(x) ≧ 0, x ≧ 0;

2. 2. 0 < a < 1;

3. 3. h(x) is monotonically nondecreasing, h(0) = 0;

4. 4. F is a distribution function on [0, ∞);

   (In [1], 1–4 were denoted collectively as (A).)

   and either

5. 5a. g(x) is continuous for all x ≧ 0;

6. 5b. limx→∞g(x) = ∞;

7. 5c h(x) is continuous for all x > 0 (Theorem 2 of [1]);

   or

8. 6. g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.