The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p, independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.

A particular random walk on the integers leads to a new, tractable Markov chain. Of the stationary probabilities, we discuss the existence, some analytic properties and a factorization which leads to an algorithmic procedure for their numerical computation. We also consider the positive recurrence of some variants which each call for different mathematical arguments. Analogous results are derived for a continuous version on the positive reals.

A Markov chain of M/G/1 type, which arose in a problem of communications engineering, is analyzed by a combination of matrix analysis and appropriate series expansions. The highly explicit results obtainable for this model owe much to analytic methods introduced by Professor Takács. Special features of the numerical implementation of the proposed solution are also discussed. The simplifications proper to this model can also be used in some related, more complex models to be discussed elsewhere.

Consider a slotted communication channel which carries packetized voice and which can transmit exactly one packet every timeslot. Assume that every conversation routed over this channel generates exactly one packet every N timeslots. We study, for the case of an infinite buffer and an N-periodic strongly interchangeable input process, buffer behavior and packet delays as functions of the number of calls routed over the channel, as long as that number is less than or equal to N. Among our results are a simple algorithm which computes the marginal distribution of the buffer content and an algorithm with complexity of order N4 which computes the distribution of the maximal buffer content (over a period of N timeslots).