Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration

Attention has recently focussed on stochastic population processes that can undergo total annihilation followed by immigration into state j at rate αj. The investigation of such models, called Markov branching processes with instantaneous immigration (MBPII), involves the study of existence and recurrence properties. However, results developed to date are generally opaque, and so the primary motivation of this paper is to construct conditions that are far easier to apply in practice. These turn out to be identical to the conditions for positive recurrence, which are very easy to check. We obtain, as a consequence, the surprising result that any MBPII that exists is ergodic, and so must possess an equilibrium distribution. These results are then extended to more general MBPII, and we show how to construct the associated equilibrium distributions.

Parallel implementation of the recurrence method for computing the power-spectral density of thin avalanche photodiodes

A simulation program has been developed to calculate the power-spectral density of thin avalanche photodiodes, which are used in optical networks. The program extends the time-domain analysis of the dead-space multiplication model to compute the autocorrelation function of the APD impulse response. However, the computation requires a large amount of memory space and is very time consuming. We describe our experiences in parallelizing the code using both MPI and OpenMP. Several array partitioning schemes and scheduling policies are implemented and tested Our results show that the OpenMP code is scalable up to 64 processors on an SGI Origin 2000 machine and has small average errors.

The M/M/1 queue with mass exodus and mass arrivals when empty

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate {αr; r≥ 1} when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

Markovian bulk-arriving queues with state-dependent control at idle time

This paper considers a Markovian bulk-arriving queue modified to allow both mass arrivals when the queue is idle and mass departures which allow for the possibility of removing the entire workload. Properties of queues which terminate when the server becomes idle are developed first, since these play a key role in later developments. Results for the case of mass arrivals, but no mass annihilation, are then constructed with specific attention being paid to recurrence properties, equilibrium queue-size structure, and waiting-time distribution. A closed-form expression for the expected queue size and its Laplace transform are also established. All of these results are then generalised to allow for the removal of the entire workload, with closed-form expressions being developed for the equilibrium size and waiting-time distributions.