The Annals of Applied Probability

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

I. M. MacPhee and M. V. Menshikov

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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.

Article information

Ann. Appl. Probab., Volume 13, Number 4 (2003), 1399-1422.

First available in Project Euclid: 25 November 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 90B22: Queues and service [See also 60K25, 68M20]

Random walk two-dimensional complex transience recurrence passage time moments polling systems


MacPhee, I. M.; Menshikov, M. V. Critical random walks on two-dimensional complexes with applications to polling systems. Ann. Appl. Probab. 13 (2003), no. 4, 1399--1422. doi:10.1214/aoap/1069786503.

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