We consider a time-homogeneous random walk $\Xi = \{\xi(t)\}$ on a
two-dimensional complex. All of our results here are formulated in a
constructive way. By this we mean that for any given random walk we
can, with an expression using only the first and second moments of
the jumps and the return probabilities for some transient
one-dimensional random walks, conclude whether the process is
ergodic, null-recurrent or transient. Further we can determine when
$p$th moments of passage times $\tau_K$ to sets $S_K = \{x \dvtx \|x\|
\leq K\}$ are finite ($p >0$, real). Our main interest is in a new
critical case where we will show the long-term behavior of the
random walk is very similar to that found for walks with zero mean
drift inside the quadrants. Recently a partial case of a polling
system model in the critical regime was investigated by Menshikov
and Zuyev who give explicit results in terms of the parameters of
the queueing model. This model and some others can be interpreted as
random walks on two-dimensional complexes.
Publié le : 2003-11-14
Classification:
Random walk,
two-dimensional complex,
transience,
recurrence,
passage time moments,
polling systems,
60G42,
60J10,
90B22
@article{1069786503,
author = {MacPhee, I. M. and Menshikov, M. V.},
title = {Critical random walks on two-dimensional complexes with applications to polling systems},
journal = {Ann. Appl. Probab.},
volume = {13},
number = {1},
year = {2003},
pages = { 1399-1422},
language = {en},
url = {http://dml.mathdoc.fr/item/1069786503}
}
MacPhee, I. M.; Menshikov, M. V. Critical random walks on two-dimensional complexes with applications to polling systems. Ann. Appl. Probab., Tome 13 (2003) no. 1, pp. 1399-1422. http://gdmltest.u-ga.fr/item/1069786503/