Forest Fire Model on $\mathbb{Z}_{+}$ with Delays

TL;DR

This paper extends the forest fire model on the positive integers by incorporating delays in fire spread and burning times, revealing new phenomena such as infinite fires not seen in the original model.

Contribution

It introduces delays into the forest fire model on $ Z_+$ and analyzes their effects, including the emergence of infinite fires.

Findings

Delays can cause infinite fires in the model.

The model exhibits new behaviors not present in the original.

General properties of the delayed model are established.

Abstract

We consider a generalization of the forest fire model on $\mathbb{Z}_+$ with ignition at zero only, studied in [arXiv:0907.1821]. Unlike that model, we allow delays in the spread of the fires as well as the non-zero burning time of individual ``trees''. We obtain some general properties for this model, which cover, among others, the phenomena of an ``infinite fire'', not present in the original model.