Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment

We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle.

[1]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[2]  R. Azencott,et al.  Mélanges d'équations différentielles et grands écarts à la loi des grands nombres , 1977 .

[3]  P. Dupuis.  Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets , 1987 .

[4]  P. Lions,et al.  Hamilton-Jacobi equations with state constraints , 1990 .

[5]  Nadine Guillotin-Plantard,et al.  Distributed algorithms with dynamical random transitions , 2002, Random Struct. Algorithms.

[6]  Robert Azencott,et al.  Ecole d'eté de probabilités de Saint-Flour VIII-1978 , 1980 .

[7]  P. Lions.  Optimal control of reflected diffusion processes : An example of state constraints , 1986 .

[8]  Irina Ignatiouk-Robert.  Large deviations for processes with discontinuous statistics , 2004, math/0409392.

[9]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[10]  Andrew C. Yao.  An Analysis of a Memory Allocation Scheme for Implementing Stacks , 1981, SIAM J. Comput..

[11]  P. Lyons.  Neumann Type Boundary Conditions for Hamilton-Jacobi Equations. , 1985 .

[12]  Irina Ignatiouk-Robert,et al.  Sample Path Large Deviations and Convergence Parameters , 2001 .

[13]  G. Louchard.  Some distributed algorithms revisited , 1995 .

[14]  Paolo Baldi,et al.  Large deviations and stochastic homogenization , 1988 .

[15]  P. Dupuis,et al.  The large deviation principle for a general class of queueing systems. I , 1995 .

[16]  René Schott,et al.  Distributed algorithms in an ergodic Markovian environment , 2005, Random Struct. Algorithms.

[17]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[18]  R. Azencott,et al.  Grandes deviations et applications , 1980 .

[19]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[20]  P. Dupuis.  Large Deviations Analysis of Some Recursive Algorithms with State Dependent Noise , 1988 .

[21]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[22]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[23]  Philippe Flajolet,et al.  The Evolution of Two Stacks in Bounded Space and Random Walks in a Triangle , 1996, MFCS.

[24]  Guy Louchard,et al.  Random walks, heat equation and distributed algorithms , 1994 .

[25]  P. Dupuis,et al.  A time-reversed representation for the tail probabilities of stationary reflected Brownian motion , 2002 .

[26]  T. Kurtz,et al.  Large Deviations for Stochastic Processes , 2006 .

[27]  Paul Dupuis,et al.  Large deviations and queueing networks: Methods for rate function identification , 1998 .

[28]  Robert S. Maier.  Colliding Stacks: A Large Deviations Analysis , 1991, Random Struct. Algorithms.

[29]  A. Veretennikov,et al.  Large Deviations for Discrete-Time Processes with Averaging , 1993 .

[30]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[31]  Guy Louchard,et al.  Probabilistic Analysis of Some Distributed Algorithms , 1990, Random Struct. Algorithms.

[32]  E. Olivieri,et al.  Large deviations and metastability , 2005 .