Point Process Observations Research Articles

Flows in Queueing Networks: A Martingale Approach

This paper applies the filtering theory for point process observations to the analysis of flows in networks. An abstract model for the study of Markovian queueing systems is developed and used to derive the filtering equations satisfied by the conditional probability distribution of the state given the past of an observed set of flows. These filtering equations are used to obtain a necessary and sufficient condition for that observation to be independent of the value of the state of the system at any time. This approach also yields a probabilistic interpretation of a necessary condition for a given flow in a multi-class Jackson network to be Poisson. These results are combined to give a proof of the “loop criterion” for single-class Jackson network and to show that this criterion does not apply to multi-class networks. Examples of loop-free Jackson networks are given for which some of the flows are not Poisson.

Optimal information processing using counting point process observations (Ph.D. Thesis abstr.)

Optimal information processing using counting point process observations (Ph.D. Thesis abstr.)

Recursive nonlinear estimation of a diffusion acting as the rate of an observed Poisson process

A conditional Poisson process is observed, whose rate is a diffusion process of known structure. The problem is to estimate the rate from the observed point process. Recursive equations are given for the conditional moment generating function and for an unnormalized conditional probability density of the rate. By studying these equations separately in between jumps and at the jumps, series expansions are obtained for these generating functions and densities in a number of examples that arise in applications to optical modulation and communications networks.

Optimal Thinning of a Point Process

It is shown that a problem of control in which decisions occur at the jump times of an observed point process can be reformulated as an intensity control problem, and also that the methods of dynamic programming can accommodate the case of random controls. The particular example considered was addressed by Rishel [9; Optimal control of a Poisson source, Proceedings JACC Conference at Purdue, 1976] and is relative to the regulation of a point process to a given rate. The question of equivalence of various information patterns (whether input only, regulated input only, or both input and regulated input are observed) is answered by the affirmative.

Estimation and control performance for space-time point-process observations

Estimation and control problems are examined for a class of models involving a linear system, a quadratic cost, and observations that include a space-time point as well as the familiar signal in additive Wiener process measurements. Motivation for this class of models is given in terms of position sensing and tracking for quantum-limited optical communication problems. These models include as special eases several simpler ones considered previously. As in the simpler cases, the optimum estimator is finite-dimensional and nonlinear, and the optimum controller separates into the optimum estimator followed by the certainty-equivalent control law. Although the optimum estimator and the optimum controller are finite-dimensional, the corresponding expected error covariance and optimum cost require infinite-dimensional calculations. This motivates the derivation of easily-computed upper and lower bounds on estimator and controller performance. The upper bounds are derived by evaluating exactly the performance of a parametrized family of suboptimum designs; one of these is identified as having smaller performance than any other, thus providing a minimal upper bound within this family. The lower bounds are obtained directly by calculations involving inequalities.

A separation theorem for stochastic control problems with point-process observations

The exact solution is derived for a stochastic optimal control problem involving a linear stochastic plant, quadratic costs, and nonlinear, nongaussian observations. The observations are in the form of a point process in which each point has both a temporal and a spatial coordinate. The state of the stochastic plant influences the intensity of the observed time-space point process. The solution to this dual control problem can be realized with a separated estimator-controller in which the estimator is nonlinear, mean-square optimal, and finite dimensional, and the controller is the certainty equivalent linear controller. Motivation for the stochastic optimal control problem studied here is given in terms of position sensing and tracking for quantum-limited optical communication problems.