We shall consider a many server (multiple channels in parallel) queueing process in which customers arrive at the queue according to a Poisson process. The service times are assumed to be independent and exponentially distributed. As usual, the service times are independent of the arrival process. We assume that no server is idle if there is a customer waiting, but that otherwise the service discipline is arbitrary. If there are n servers and we let Xn(t) denote the number of customers waiting or being served at time t, then it is well known that Xn(t) is a birth and death process with stationary transition probabilities. A very comprehensive analysis of this many server queue from the point of view of birth and death processes has been carried out by Karlin and McGregor [5].

Certain patches of ground are favorable to the formation of fair weather cumulus clouds. These are patches which reflect rather than absorb solar heat. As the air above these patches is warmed, it will rise and, if the humidity is sufficient, the cooling effect of higher elevation will cause formation of a cloud of the type known as the fair weather cumulus. This fact is used in the present paper to develop a stochastic model for the temporal evolution of the contribution to cloud cover by fair weather cumuli.

We consider a queueing system in which arrivals occur at times , and after every kth arrival a servicing of k arrivals is begun. We assume that the number of servers is infinite. Initially, at t0= 0, the system is empty and the arrival process {tn} is about to start. The batch service times are independently and identically distributed with distribution function No assumption is made about the process {tn}.

For the queueing system GI/G/1 with both waiting-line and service-line customer impatience an integral equation for the limiting waiting-time distribution function W(x) is derived and the existence of W(x) proved when the system satisfies certain conditions. Two counter examples are constructed showing that W(x) does not necessarily exist. A sufficient condition for its existence in GI/G/1 with deterministic waiting-line impatience is given. Formal solutions for M/G/l are derived for deterministic impatience and negative exponential impatience.

Suppose X1, X2, X3,… are independent and identically distributed positive nonlattice random variables with common distribution function F(x).

Suppose that we observe an infinite realisation of a point process Π, which is a superposition of a number of mutually independent recurrent point processes Πi, such that

A well-known theorem (Moran, 1959) relating to the equilibrium condition of a semi-infinite “Moran reservoir” with stationary independent inputs and unit output, gives (a) the probability of emptiness, and (b) the generating function of the distribution of levels, in terms of the input distribution. A further theorem (Prabhu, 1958) points out that, in a finite reservoir, the ratio of the probabilities of any two comparable levels is independent of the size of the reservoir, and and is in fact the same as the corresponding ratio for the semi-infinite reservoir. In the present note the theorems are extended to deal with the case of Markovian inputs.

Let us consider a stream of homogeneous random events which we call signals and let us denote by the probability that exactly k signals will appear in the time interval . We assume that the stream is not empty, i.e. the probability P0(a, t) is not identically equal to 1.

In a previous paper [2] the author has studied the single-server queue with non-homogeneous Poisson input and general service time, with particular emphasis on the case when the parameter of the Poisson input is of the form

Transition distribution functions (d.f.) of the stochastic process u + t − X(t), where X(t) has a compound Poisson distribution, are used to derive explicit results for the transition d.f.s of the waiting time processes in the queueing systems M/G/1 and GI/M/1.

Gani (1955) finds the form of the multinomial probabilities which admit a sufficient estimator for the single parameter θ to be where λ θ is a positive function of θ, the Ki, are constants and the αi are nonnegative constants.