The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

This paper is concerned with the solution of an initial and boundary value problem for a parabolic differential equation driven by a stochastic point process.

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Traffic noise depends significantly on statistical properties of highway flow. The noise heard by an off-highway observer is the sum of the contributions of all the vehicles on the highway. We assume that vehicle spacings on a single-lane infinite straight road form a Markov renewal process (MRP) with N states or vehicle types. The noise impact is then a two-sided filtered MRP. We give explicit formulae for the mean and variance in the case N = 2 and exponential headways. Jewell's (1965) study (cars vs. trucks in the southbound curb lane of U.S. 40) gives parameters of the MPR for a numerical example. Even with only a modest degree of truck clustering and variability in vehicle spacings and noise emission, the variation in noise level is much greater than usually predicted. The coefficient of variation of arithmetic noise intensity is greater than unity at distances from the roadway less than 700 feet. The logarithmic noise level parameters L50 and L10 usually computed in engineering practice may be in error by 2 to 3 decibels if these sources of variability are ignored.