Kinetic Monte-Carlo Simulation to solve master equation

Kinetic monte Carlo can be used to simulate Master equation for Markov processes.For simple 1D random walk the master equation is - $$\frac{\partial P(n,t)}{\partial t}=pP(n-1,t)+qP(n+1,t)-(p+q)P(n,t)$$ where $$p$$ and $$q$$ are forward hopping rate and backward hopping rates respectively.
$$P(n,t)$$ is the probability of random walker to be at position $$n$$ at time $$t$$

This equation can be solved analytically but we can also solve it by Kinetic Monte Carlo method.I want some materials to follow or some guidence for the simulation . It would be really helpful if someone can provide some materials,notes or book chapters to look .

You can calculate the position of a single walker at time $$t$$ using kMC. To calculate the probability of a walker being found in position n at time $$t$$, you will need to run many kMC simulations in order to generate $$P(n,t)$$.
To calculate the position of a single walker using kMC at time $$t$$, you will need to calculate the total rate ($$=p+q$$), from that calculate a time step ($$\tau$$) between hopping events, and then select whether to execute a forward or a backwards hop by selecting one at random, weighted according to their average rates.
Specifically for your case, $$\tau = (1/(p+q)) \ln(1/r_1)$$, where $$r_1$$ is a random number between 0 and 1. To select between executing forward or reverse rate, generate a random number $$r_2$$ between 0 and $$(p+q)$$ (inclusive of 0, excluding $$p+q$$). If $$r_2$$ is less than $$p$$, execute a forward hop. If $$r_2$$ is between $$p$$ and $$p+q$$, execute a backwards hop.
As far as general resources go, you can find information about interarrival times in Poisson distributions in a statistics book, for example, chapter 5 (specifically, section 5.3.3 in the Ninth Edition) of Sheldon Ross's classic text "Introduction to Probability Models," which should let you see how the timestep tau is derived. I would expect choosing $$p$$ or $$q$$ according to their magnitudes is fairly obvious, but you could look, for example, in the 1976 and 1977 Daniel Gillespie papers on chemical kinetics for information on this as well.