UvA/CWI Seminar: Phil Whiting (Macquarie University)

Probability and Breaking the Enigma Codes (Part 1) / Measure Valued Mean Field Limits for Random Access Networks (Part 2)

Phil Whiting (Macquarie University)

Part 1: Probability and Breaking the Enigma Codes

Part 2: Measure Valued Mean Field Limits for Random Access Networks

Abstract

Part 1: This talk gives a brief description of the 3 rotor wheel version of the Enigma machine and a probability model for the encoder. It is then explained how breaks could be obtained due to a particular quirk that occurred because of a certain German signalling protocol.  These breaks were made well before the start of the Turing-Welchman bombe which began in late summer 1940.

Part 2: Random-access algorithms such as the CSMA protocol provide a popular mechanism for distributed medium access control in wireless networks. We investigate the asymptotic performance of such networks in unsaturated buffer scenarios.  These   arise  in an Internet of Things (IoT)  context with highly intermittent traffic sources.

Our results are  obtained as a spatial mean field limit, which serves as an approximation to many node random access networks. This limit emerges as a (vector) time dependent measure, and the Partial Differential Equation (PDE) for the  underlying  densities is presented. As we show, the mean field limit emerges from a sequence of "cluster''  approximations to the interference graph of the underlying prelimit network. Subsequently we develop numerical results for the special case of nodes uniformly distributed on a circle. We conclude with some remarks as to
generalisations of the above construction which might be applied in modelling such networks with additional controls and under more general
assumptions than those presented here.