1. (12 marks)

Alex has three green balls (numbered 1 to 3) and three blue balls (numbered 4 to 6), plus a bin with space for exactly 3 balls. At each step:

• He rolls a fair 6-sided die (equally likely to roll each number) and finds the ball with the corresponding number.

• If that ball is currently in the bin, remove it and replace it with a ball of the opposite colour. Otherwise, do nothing.

Let X_n be the number of green balls in the bin after n steps. Then (X_n, n = 0,1,2,...) is a Markov chain with state space S = (0,1, 2,3}. Start with all the green balls in the bin and all the blue balls outside, so that X_o = 3.

(a) Part of the transition matrix is given below. Find the remaining entries, explaining how you obtained them.

* * * \

* * *

1/3 1/2 1/6
0 1/2 1/2/

(b) Write down equations that the equilibrium distribution satisfies, and find the equilibrium distribution for this chain.

(c) Is the limiting distribution the same as the equilibrium distribution? Justify your answer.

2. (12 marks)

A Markov chain (X_n, n = 0,1, 2,...) with state space S = {1, 2,3,4, 5} has transition matrix

(a) Draw the transition diagram for this Markov chain.

(b) Find P(X₃ = 1|X° = 4, Xi = 3,X₂ = 2).

(c) Find the two-step transition probability = P(%2 = 2|X()= 3).

(d) Find = P(X_n = 1 for some n > 0|X()= 3), the probability that the chain ever reaches state 1, given that it starts in state 3 at time 0..

3. (10 marks)

A service facility has two streams of customers arriving at it. Stream A customers arrive as a Poisson process at rate 1 per hour. Stream B customers arrive as a Poisson process at rate 2 per hour. The two arrival streams are independent of each other. Suppose there were 6 arrivals at the facility between 9 a.m. and 11 a.m.

(a) What is the expected number of customers to arrive at the facility between 11.00 a.m. and 11.30 a.m.?

(b) What is the probability that the time between two successive arrivals from stream A is greater than 30 minutes?

(c) What is the probability that exactly 3 customers from stream A arrive at the service desk between 11.00 a.m. and 11.30 a.m.?

(d) What is the probability that of the first 4 customers to arrive at the facility after 11.00 a.m., 3 are from stream A, and 1 is from stream B?

(e) Suppose 3 customers arrived at the facility between 11 a.m. and 12 noon. What is the probability that exactly 1 of those 3 arrivals occurred between 11 a.m. and 11.30 a.m.? Show your working clearly.

4. (21 marks)

Suppose two repair staff are always available 24 hours a day to fix broken photocopiers in a very large office building. Assume that breakdowns occur as a Poisson process at rate 12 per day, and it takes an exponentially distributed time with mean 4 hours for a single repairperson to repair a single machine. Let N(t) denote the number of photocopiers that are broken down at time i, including any that are currently being repaired. If the number of broken down machines ever exceeds 10, then they are replaced, rather than repaired, so they do not enter the queue for repair. Suppose 7V(i) is modelled as a birth and death process with state space S = {0,1,2,3,4,5,6,7,8,9,10}.

(a) This birth and death process model makes a number of assumptions that might not be justified here. Write down two assumptions that you think might make this model unrealistic, and briefly explain why. One or two sentences for each is ample here.

(b) Draw the transition diagram for this system and write down the transition rates.

(c) Obtain the equilibrium distribution for the number of broken down machines.

(d) What is L, the expected number of broken down machines (including the one currently being repaired)?

(e) For those machines that are repaired rather than replaced, what is W_qy the expected time from a machine breaking down until repairs on it commence?

(f) A busy period has just started. A busy period is a period during which at least one repairperson is busy. What is the expected time until both repair staff are next free to have a cup of coffee together (that is, what is the expected length of a busy period)?

5. (5 marks) A single server queue has arrivals as a Poisson process at rate 2 per hour, with service times that are uniformly distributed. The minimum service time is 5 minutes and maximum is 25 minutes, so the service times are distributed as Uniform[5,25] random variables. All service times and interarrival times are independent of each other.

(a) Find the expected number of customers in the system (including customers in service).

(b) What is the probability that an arriving customer finds the server busy?