Suppose in a particular town that there is a gas station at every intersection and it can be one of two possible brands: A or B.
For simplicity model all of the possible intersections in the town via the 2 dimensional square lattice $\mathbb{Z}^2 = \{f(x; y), x, y = 0, \pm1, \pm2, \cdots\}$.
Each station has an independent exponential clock (rate $\lambda_A$ for brand A and rate $\lambda_B$ for brand B), and when the clock goes off the station replaces one of its 4 nearest neighbors (chosen at random) with a station of its own type.
Note if an A station replaces an A station, the system does not change, however if an A station replaces a B station the number of B stations goes down by 1 and the number of A stations goes up by 1.
Assume that the city starts out with one brand B station and every other intersection is a brand A station.
Note, the four nearest neighbors of a station at $(x; y)$ are $(x + 1; y)$; $(x - 1; y);$ $(x; y + 1)$; and $(x; y - 1)$.
Find the probability that the brand B stations die out if
- $\lambda_A \geq \lambda_B$
- $\lambda_B > \lambda_A$
I have no idea how to deal with the probability since I think I have to consider location of new $B$s when they become more and more and thus I cannot find a iteration formula which can capture the pattern of location since they are random.
Can anyone tell me what's the type of such problem and is there any material? It would be better if you could give me some ideas about how to solve the problem.