Conditional convex orders and measurable martingale couplings

Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.Comment: 21 page

Stabilization of an overloaded queueing network using measurement-based admission control

Admission control can be employed to avoid congestion in queueing networks subject to overload. In distributed networks the admission decisions are often based on imperfect measurements on the network state. This paper studies how the lack of complete state information affects the system performance by considering a simple network model for distributed admission control. The stability region of the network is characterized and it is shown how feedback signaling makes the system very sensitive to its parameters.Comment: Published at http://dx.doi.org/10.1239/jap/1143936256 in the Journal of Applied Probability (http://projecteuclid.org/jap) by the Applied Probability Trust (http://www.appliedprobability.org/

Comparison and scaling methods for performance analysis of stochastic networks

Stochastic networks are mathematical models for traffic flows in networks with uncertainty. The goal of this thesis is to develop new methods for analyzing performance and stability of stochastic networks, helping to better understand and control uncertainty in complex distributed systems. The thesis considers three instances of stochastic networks, each representing a specific challenge for analytical modeling. The first case studies the impact of incomplete information to a queueing network with distributed admission control. Stability conditions for various admission policies are derived, together with a numerical algorithm for performance evaluation. In the second case, stochastic comparison is used to derive performance bounds for multiclass loss networks with overflow routing. The third model is a spatial random field generated by a large number of noninteracting sources, for which scaling and renormalization are used to show how the level of randomness of the individual sources may critically affect the macroscopic statistical properties of the field. The results of the thesis illustrate the feasibility of stochastic comparison and stochastic analysis in deriving approximations and performance bounds for complex physical networks with uncertainty. Approximations and performance bounds based on exact mathematical methods have the advantage that they explicitly state the type of circumstances required for the accuracy of the estimates. The resulting analytical formulas can sometimes reveal interesting properties that are not easily detected using numerical simulation.reviewe