A rumour model due to Maki and Thompson (1973) is slightly modified to incorporate a continuous-time random contact process and varying individual behaviours in front of the rumour. Two important measures of the final extent of the rumour are provided by the ultimate number of people who have heard the rumour, and the total personal time units during which the rumour is spread. Our purpose in this note is to derive the exact joint distribution of these two statistics. That will be done by constructing a family of martingales for the rumour process and then using a particular family of Gontcharoff polynomials.

We prove that the expected number of particles of the critical contact process on a homogeneous tree is bounded above. This is the first graph for which the behavior of the expected number of particles of the critical contact process is known. As an easy corollary of our result we get that the critical contact process dies out on any homogeneous tree. This completes the work of Pemantle (1992).

Let {ξn, n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξn, n ≧1 and let ξ1 have an arbitrary distribution. Define Xn+1 = k max(Xn, ξ n+1), Yn+ 1 = max(Yn, ξ n+1) – c, Un+1 = l min(Un, ξ n+1), Vn+1 = min(Vn, ξ n+1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X1 = Υ1= U1 = V1 = ξ1. We establish conditions under which the limit law of max(X1, · ··, Xn) coincides with that of max(ξ2, · ··, ξ n+1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y1, · ··, Yn), min(U1,· ··, Un) and min(V1, · ··, Vn).

An observer watches one of a set of Poisson streams. He may switch from one stream to another instantaneously. If an arrival occurs in a stream while the observer is watching another stream, he does not see the arrival. The experiment terminates when the observer sees an arrival. We derive a formula which states essentially that the expected total time that the observer watches a stream is equal to the probability that he sees the arrival in this stream divided by the intensity of the stream. This formula is valid independently of the observation policy. We also discuss applications of this formula.

Given a family of Markov chains with a single recurrent class, we present a potential application of Schweitzer's exact formula relating the steady-state probability and fundamental matrices of any two chains in the family. We propose a new policy iteration scheme for Markov decision processes where in contrast to policy iteration, the new criterion for selecting an action ensures the maximal one-step average cost improvement. Its computational complexity and storage requirement are analysed.

Consider a workstation with one server, performing jobs with a service time, Y, having distribution function, G(t). Assume that the station is unreliable, in that it occasionally breaks down. The station is instantaneously repaired, and the server restarts the uncompleted job from the beginning. Let T denote the time it takes to complete each job. If G(t) is exponential with parameter A, then because of the lack-of-memory property of the exponential, P (T > t) = Ḡ(t) =exp(−γt), irrespective of when and how the failures occur. This property also characterizes the exponential distribution.

We consider the Mx/G/∞ queue in which customers in a batch belong to k different types, and a customer of type i requires a non-negative service time with general distribution function Bi(s) (1 ≦ i ≦ k). The number of customers in a batch is stochastic. The joint probability generating function of the number of customers of type i being served at a fixed time t > 0 is derived by the method of collective marks.