In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order convergence results.

A Markov-modulated Poisson process (MMPP) is a Poisson process whose rate is a finite Markov chain. The Poisson process is a simple MMPP. An MMPP/M/1 queue is a queue with MMPP arrivals, an infinite capacity, and a single exponential server. We prove that the output of an MMPP/M/1 queue is not an MMPP process unless the input is Poisson. We derive this result by analyzing the structure of the non-linear filter of the state given the departure process of the queue. The practical relevance of the result is that it rules out the existence of simple finite descriptions of queueing networks with MMPP inputs.

Consider a network of queues in equilibrium. When are the flows observed on two given links of that network equivalent, i.e., when do they have the same law?

The verification of the equivalence of two such point processes is first reduced to an algebraic problem by a technique based on the filtering theory.

This method is then used to show that the arrival and departure processes at an M/M/1 node in a Jackson network are not always equivalent, thereby contradicting a conjecture made in [8]. An example where that equivalence holds is also given; it provides new results on the time reversibility of some familiar processes.

Controls of the perturbation type are evolved for the purpose of monitoring the random movement of a response curve using noise-corrupted observations of the response. Dynamic programming methods are used to obtain approximations to the optimum control system when the response is quadratic. Further results are briefly described.