Earlier stochastic analyses of chemical reactions have provided formal solutions which are unsuitable for most purposes in that they are expressed in terms of complex algebraic functions. Normal approximations are derived here for solutions to a variety of reactions. Using these, it is possible to investigate the level at which the classical deterministic solutions become inadequate. This is important in fields such as radioimmunoassay.

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

Suppose that a device is subjected to shocks and that denotes the probability of surviving k shocks. Then is the probability that the device will survive beyond t, where is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.

The distribution and its moments are given of the length of the intercept made by a random probe through a box having perpendicular edge lengths a, b and c. Three types of random probe are considered.

If a rumour is spread in a population through random contacts between its members, to how many people, on average, will the rumour be told? What can be said about the variations in the number of people who have heard the rumour? Does it matter very much whether a single person or a group of persons starts spreading the rumour? Questions such as these are discussed below, using the framework of a simple stochastic model for snowball sampling.

Asymptotic normality and the weak invariance principle are studied for an iterated renewal process of order A martingale decomposition is used in this context to provide a simple proof.

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.

We consider the estimation of the parameter of the best interpolator from a restricted class of linear, time-invariant interpolators for the weakly stationary process {xt}. The estimation is supposed to be based on a part-realisation x1, · ··, xn and on certain prior information regarding the spectral density of the process. Two methods of estimation are proposed — one which uses x1, · ··, xn and the prior information and another which uses only x1, · ··, xn. For the methods proposed it is reasonable to conjecture that the method which utilises the prior information as well as x1, · ··, xn, will be more efficient. Reasonable as this conjecture may seem, its proof is by no means obvious and is the main subject of the paper.