Part I [21] treated the case of a finite number of independent random uniform s-flats in an ‘admissible’ subset of Ed (s = 0, · · ·, d − 1). In this second part, the natural and fruitful ‘Poisson extension’ to a ‘countable number of independent random uniform s-flats in Ed itself” is considered. It is worth mentioning at the outset that to have read Part I is not a prerequisite for reading the present paper. Although results of that part are often applied here, they serve only in an auxiliary capacity, thereby allowing the main thread of the theory to be developed without interruption.

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

Many service systems exhibit variations of a random nature in the intensity of the arrival process or of the speed of service or of both. Changes in work shifts, rush hours, interruptions in the arrival process, server breakdowns, etc. all fall into this category.

The present study deals with a generalization of the classical M/G/1 queue by considering an extraneous process of phases which can be in one of the states {1, …, m}. During any interval spent in phase i, the arrivals are according to a homogeneous Poisson process of rate λi and any service initiated during such an interval has a duration distributed according to Hi(·). The process of phases is assumed to be an irreducible Markov chain in continuous time and is fully characterized by its initial conditions, by an irreducible stochastic matrix P and by the mean sojourn times σ1-1, …, σm-1 in each phase.

Independently of the queueing aspects, this arrival process is a generalization of the classical Poisson process which can be of interest in modelling simple point processes with randomly fluctuating “arrival” rate.

Two approaches to the time dependent study of this queue are presented; one generalizes the imbedded semi-Markov process obtained by considering the queue immediately following departure points; the other approach exploits the relationship between this queue and branching processes. The latter is more elegant from a purely theoretical viewpoint and involves iterates of a general type of matrix function introduced by the author. By making extensive use of the Perron-Frobenius theory of positive matrices the equilibrium condition of the queue is obtained. While retaining a similar intuitive interpretation the equilibrium condition is substantially more complicated than for the M/G/1 model.

The recurrence relations which yield the joint distribution of the phase state at time t, the queue length, the total number served and the virtual waiting time at t are exhibited in detail. Via transform techniques a number of limiting and marginal distributions are discussed. The discussion relies heavily on the theory of Markov renewal processes.

Throughout the paper and in a final section the author advocates the use of the structural properties of the queue and the resulting recurrence relations to organize the numerical analysis of complex queueing models such as the present one.

More explicit results for the case of two phases are given and are compared to results obtained by Yechiali and Naor for a closely related two-phase generalization of the M/M/1 queue.

The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.

In this paper, some results on the asymptotic behavior of Markov renewal processes with auxiliary paths (MRPAP's) proved in other papers ([28], [29]) are applied to queueing theory. This approach to queueing problems may be regarded as an improvement of the method of Fabens [7] based on the theory of semi-Markov processes. The method of Fabens was also illustrated by Lambotte in [18], [32]. In the present paper the ordinary M/G/1 queue is generalized to allow service times to depend on the queue length immediately after the previous departure. Such models preserve the MRPAP-structure of the ordinary M/G/1 system. Recently, the asymptotic behaviour of the embedded Markov chain (MC) of this queueing model was studied by several authors. One aim of this paper is to answer the question of the relationship between the limiting distribution of the embedded MC and the limiting distribution of the original process with continuous time parameter. It turns out that these two limiting distributions coincide. Moreover some properties of the embedded MC and the embedded semi-Markov process are established. The discussion of the M/G/1 queue closes with a study of the rate-of-convergence at which the queueing process attains equilibrium.

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.