# IT461 Assignment 3 Markov Chains

- Simulate the Land of Oz example given in class for 300 days.
- Simulate the stepping stone model of Example 11
*.*2 of Laurie Snell book with*n*= 10 for two colors. Observe what happen’s at after a long time. Can you explain why this is happening? - Simulate the drunkard’s walk with lattice points {`0
*,*1*,*2*,*3*,*4`} with 0 and 4 being the absorbing states. Start the random walk in state 1. Estimate i)The expected number of times that the drunkard is in state 3 ii)The expected time to absorbtion iii)Probability that the chain will be absorbed in state 4 . Compare with the theoretical formula’s in section 11*.*2 - Simulate the rat’s maze example (Example 11.22) and keep track of the number of times that the rat is in state
*j*for*j*= 1*,*2*,..,*Compare the empirical stationary distribution of this Markov chain to the theoretical one. - A discrete time queueing system of capacity
*n*= 50 consists of the person being served and those waiting to be served. The queue length*x*is observed each second. If 0*< x < n*, then with probability*p*= 0*.*4, the queue size is increased by one by an arrival and, inependently, with probability*r*= 0*.*5, it is decreased by one because the person being served finishes service. If*x*= 0, only an arrival (with probability p) is possible. If*x*= 50, an arrival will depart without waiting for service, and so only the departure (with probability r)of the person being served is possible. This is a Markov chain with states given by the number of customers in the queue. Write a computer program to simulate the queue. Have your program keep track of the proportion of the time that the queue length is*j*for*j*= 0*,*1*,...,n*and the average queue length. Can you estimate the stationary vector ? Test what happens when*p*= 0*.*5 and*r*= 0*.*4?

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