ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 3, pp. 234–250.
Pleiades Publishing, Inc., 2006.
Original Russian Text
A.G. Malyshkin, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 3, pp. 78–96.
Limit Dynamics for Stochastic Models of Data
Exchange in Parallel Computation Networks
A. G. Malyshkin
Lomonosov Moscow State University
Received December 20, 2005; in ﬁnal form, April 18, 2006
Abstract—We study limit dynamics of a system of interacting particles, which is one of
possible models for the parallel and distributed computation process. For a rather wide class of
multi-particle interactions, we prove that the stochastic process describing the conﬁguration of
a particle system weakly converges in the ﬂuid-dynamic limit to a deterministic process, which
is a solution of a certain partial diﬀerential equation.
Interacting particles provide a convenient tool for modeling various processes and events where
objects interact with each other in a certain way. Models similar to the one we consider in the
present paper occur in the theory of parallel and distributed computations. In this case, particles
correspond to processors. Each processor performs its own part of a task, but it has to exchange
information with other processors during the computation. In the language of particles, a lot of
diﬀerent models corresponding to this problem can be proposed (see, e.g., [1–4]). The coordinate of
the ith particle is usually interpreted as the internal time of the ith processor or as the number of
“microtasks” performed by this processor. Free motion of the particles (random jumps [1,2] or con-
stant velocity movement [3,4]) correspond to computations made by each processor independently
of the others, whereas particle interaction corresponds to processor synchronization. In practice,
one of the synchronization methods is the so-called rollback procedure (see [5, ch. 8]). In [1, 3, 4],
the rollback procedure in terms of particles is realized by the pairwise interaction where a particle
with a larger coordinate rolls back to that with a smaller coordinate. In the present paper, we make
an attempt to extend the class of possible interactions. For simplicity, we mainly consider three-
particle interaction, but at the end of the paper we also present results for an arbitrary n-particle
It should be noted that the particle interaction considered in the present paper is a mean-ﬁeld
interaction. Mean-ﬁeld models comprise not only interacting particle systems (see, e.g., [1,3, 6–8])
but also information networks (see, e.g., ) and queueing systems .
2. MODEL AND THE MAIN RESULT
We consider a system of N interacting particles on the integer line Z. Changes in the system
occur at discrete time instants t ∈ Z
. Dynamics of the whole system is formed by free motions of
each particle and by interaction in an arbitrarily chosen triple of particles. At each time instant,
with probability α
< 1, the following interaction takes place: a triple of particles is
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00662.