ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 4, pp. 331–343.
Pleiades Publishing, Inc., 2007.
Original Russian Text
A.A. Zamyatin, V.A. Malyshev, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 4, pp. 68–82.
Accumulation on the Boundary for a One-Dimensional
Stochastic Particle System
A. A. Zamyatin and V. A. Malyshev
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Received June 21, 2007; in ﬁnal form, August 29, 2007
Abstract—We consider an inﬁnite particle system on the positive half-line, with particles mov-
ing independently of each other. When a particle hits the boundary, it immediately disappears
and the boundary moves to the right by some ﬁxed quantity (the particle size). We study the
speed of the boundary movement (growth). Possible applications are dynamics of traﬃc jam
growth, growth of a thrombus in a vessel, and epitaxy. Nontrivial mathematics concerns the
correlation between particle dynamics and boundary growth.
Boundary hitting for a random walk is a classical problem of probability theory. More com-
plicated is the problem of hitting a moving boundary; see . Here we consider the case where
the movement of the boundary is correlated with the movement of particles. Namely, we consider
inﬁnitely many particles performing random walks on the positive half-line and sticking to the
boundary when hitting it. Moreover, when a particle hits the boundary, the boundary moves to
the right by some quantity, the “size” of the particle. Thus, the boundary moves due to accumu-
lation of particles on it. As far as we know, such problems were not previously considered.
One often encounters similar problems in applications such as growth of a thrombus, epitaxy,
and other methods of coating a surface with metal particles. In many cases, the eﬀect of correlation
between movements of particles and of the boundary can be neglected. Here, on the contrary, we
consider the inﬂuence of this correlation on the growth speed of the boundary. Such eﬀects can be
observed in quick formation of traﬃc jams.
In the paper we present two results. The ﬁrst concerns nonzero drift of the particles, and the
second is for the zero drift. In the ﬁrst case one can ﬁnd the exact asymptotic rate of growth; in the
second case, only the order of growth.
2. FORMULATION OF THE PROBLEM AND RESULTS
We consider an inﬁnite system of particles on a half-line; when hitting the boundary, the particles
adhere to it (accumulate on it). As a result, the boundary moves to the right. Now we give an
Assume that at time 0 a random conﬁguration of particles at points
0 ≤ x
(0) < ... <x
(0) < ...
on the half-line R
is distributed as a point Poisson process with density λ.