Mathematical Challenges in Stochastic Networks

Abstract

The workshop was devoted to the discussion of recent progress in modern stochastic network theory and to the exploration of open mathematical challenging problems in the field. The workshop covered a wide range of mathematical topics; while being centered around applied probability, it also included a substantial amount of graph theory and (combinatorial) optimization. Mathematics Subject Classification (2000): 60xx. Introduction by the Organisers Stochastic Networks is a flourishing area in the very heart of modern Applied Probability. Its primary aim is to obtain a fundamental understanding of the mathematical properties of complex interacting random systems. Among the main goals are the design, analysis and evaluation of important benchmark systems, as well as the development of efficient tools for the optimization and simulation of networks. Among the 52 participants there were many leading experts in the field as well as many very promising young scientists. For about a half of the participants the workshop was their first opportunity to visit Oberwolfach. The programme included three 2-hour lectures, nineteen 40-minute talks, and two 90-minute open problems sessions. The lectures formed the organizational backbone of the workshop. They were delivered by Bruce Hajek on ”Peer to peer communication in networks issues, models, and analysis”, by Alexandre Proutiere on ”Short and long-term behaviors of Markov processes through mean 2746 Oberwolfach Report 48/2010 field asymptotics” and by Volker Schmidt on ”Distribution of cost functionals in spatial network models: Scaling limits and MonteCarlo methods”. All the lectures have received a very warm welcome from the audience. Bruce Hajek gave an impressive overview on the modelling and analysis of peer to peer networks and, in particular, on problems related to the so-called ”singlechunk syndrome” problem. These models are very new, they are influenced by the practice, and their analysis is very challenging and practically important. Alexandre Proutiere was speaking about modern problems in understanding and analysis of the dynamics of high dimensional Markov processes using the mean-field approximation. He also discussed two practical examples, of the epidemic diffusion of viruses and of large wireless networks with random access. Volker Schmidt lectured on a stochastic geometry approach to the modeling and performance analysis of spatial stochastic networks. The focus was on telecommunication networks involving road systems. The contributed talks were typically linked to one or more issues covered by these lectures. In more detail, the topics discussed during the workshop were related to the following areas. One main theme was the probabilistic analysis of stochastic processes (uniand multivariate Levy processes, random walks, Markov additive processes, etc.) relevant to stochastic networks (with talks by V. Anantharam, S. Asmussen, E. Baurdoux, O. Kella, L. Leskela, V. Wachtel, R. Szekli). Another direction was around various scaling techniques and related asymptotic analysis of stochastic networks. In addition to the lecture by A. Proutiere, there were several other talks (T. Dieker, D. Gamarnik, R. Johari, D. McDonald, A. Stolyar) on scaling in general Markov processes, parallel server systems, stochastic games, large deviations, and back-pressure algorithms. Another topic was the optimization, complexity, and scheduling problems in communication and spatial networks, and related asymptotic analysis (B. Hajek, V. Schmidt, I. Norros, D. Shah, P. Thiran). There were also several presentations (M. Lelarge, G. Hooghiemstra, J. Salez) on routing, matching and related problems in random graphs. Finally, C. Wichelhaus gave a talk on non-parametric inference for general stochastic networks. An integral part of the programme was provided by two problem sessions organized by S. Foss and M. Mandjes. These sessions provided space for participants to bring up new ideas and discuss open problems in an informal manner. There were 8 presentations which covered a broad range of problems and, in particular, on classical stability analysis of Markov processes (P. Glynn), on greedy server and vacuum cleaner models (T. Konstantopoulos), and on sequential algorithms for solving linear equations (D. Wischik). Finally, Balaji Prabhakar gave a very informal presentation on how to use incentives to make ”Societal Networks” more efficient. For example, how to decongest the roads? How to increase the use of public transit? How to reduce energy waste? He described the results of actual deployments and illustrated the role played by computer communication networks and incentive algorithms in these deployments.