Critical random walks on two-dimensional complexes with applications to polling systems

Abstract

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.