We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t0 = 0, finds a free server, and experiences a service time v0. The nth customer arrives at time tn and experiences a service time vn. Let the interarrival times tn - tn-1 = un, n ≧ 1, and define the random vectors Xn = (vn-1, un), n ≧ 1. We assume the sequence of random vectors {Xn : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{un} = λ-1 and E{vn} = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v02} < ∞ and that the deterministic system in which both vn and un are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.

The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and conditional on non-extinction when there is no immigration, the number of individuals remaining in the population after a large number of generations has approximately a gamma distribution. The error estimates are uniform within a specified class of offspring distributions, and are independent of whether the critical situation is approached from above or below. These results parallel those given for continuous time branching processes by Sevast'yanov (1959), and extend recent work by Nagaev and Mohammedhanova (1966), Quineand Seneta (1969), and Seneta (1970).

Consider the well known Galton-Watson branching process (Harris (1963)) in which individuals reproduce independently of each other and have probability aj (j = 0, 1, · · ·) of giving rise to j individuals in the next generation. In recent years some attention has been given to the branching process in which there is an independent immigration component at each generation, bj (j = 0, 1, · · ·) being the probability of j immigrants entering each generation. For a review of this work see Seneta (1969), and Pakes (1971 a, b and c) for further results.

In the homogeneous process X(t) defined on a finite irreducible Markov chain R(t) was studied. The process was characterized by an overall transition rate v per unit time, a matrix β of transition probabilities for the chain and a linear growth for X(t) dependent on the chain; viz. dX(t)/dt|R(t) = i = vi. The central limit behavior of the process was exhibited in. For the case when the chain had only two states, the ruin and ergodic problems were considered in for the bounded process. The object of this paper is to investigate the ruin and ergodic problems for the process of in the presence of boundaries.

For the linear growth birth-death process with parameters λn = nλ, μn = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0tX(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional Wx = ∫0Txg{X(τ)}dτ, where Tx is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form Wx arise naturally in traffic and storage theory; for example Wx may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

Some obscure, yet fundamental, formulae of integral geometry are re-considered. They are applied to determine all the moments of the random volume of various isotropic random r-dimensional simplices in En (r = 1, …, n).

Let X = {1, …, a} and Y = {1, …, a} be the input and output alphabets, respectively. In the (unsynchronized) channels studied in this paper, when an element of X is sent over the channel, the receiver receives either nothing or a sequence of k letters, each a member of Y, where k, determined by chance, can be 1, or 2, or … or L, a given integer. The channel is called unsynchronized because the sequence received (for each letter sent) is not separated from the previous sequence or the following sequence, so that the receiver does not know which letters received correspond to which letter transmitted.

In Sections 1 and 2 we give the necessary definitions and auxiliary results. In Section 3 we extend the results of Dobrushin [2] by proving a strong converse to the coding theorem1 and making it possible to compute the capacity to within any desired accuracy.

In Section 4 we study the same channel with feedback, prove a coding theorem and strong converse, and give an algorithm for computing the capacity.

In Section 5 we study the unsynchronized channel where the transmission of each word is governed by an arbitrary element of a set of channel probability functions. Again we obtain the capacity of the channel, prove a coding theorem and strong converse, and give an algorithm for computing the capacity.

In Section 6 we apply results of Shannon [4] and supplement Dobrushin's results on continuous transmission with a fidelity criterion.

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients for F. Let us denote Xn+ = XnI[Xn > 0] and Xn− = − XnI[Xn<0], so that Xn = Xn+ - Xn−. Let F+ and F− denote the distribution functions of X1+ and X1− respectively, and let Sn(+) = X1+ + · · · + Xn+, Sn(-) = X1− + · · · + Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+) + an, Bn-1Sn(-) + bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫ x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x2dF(x) = ∞, then at least one of F+, F− is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.