1. The toilets in a bar have 4 cubicles. Assume that the arrivals to the toilets can be modelled as a Poisson process with rate A and that each person does their business with mean time 1/p. Assume that people take any free cubical and line up outside if there are none free. Let the state of the system, X(t), be the number of people using and queuing for the toilets at time t. (a) What is the state space for this model? (b) Sketch the state transition diagram for this model. (c) What are the elements of the transition rate matrix? (d) Let T = inf {t > 0 : X(t) =4; X(0) =0}, i.c. the random time for the toilet to first become full, given that it is empty at t = 0. Code a Monte Carlo simulation to estimate E[T]. Take A = 1 per minute and 1// = 3/4 minutes. You must include a copy of your code with your answer and remember that academic integrity applies to this. Now assume that if all the cubicles are occupied then people don't queue up and just leave to look for another toilet. As before, take A = 1 and 1/p = 3/4. (c) How does this change the state space and transitions of your model? (f) Use matlab, or your favourite language, to calculate the transition function P(t) = eat for this new model as a matrix exponential. What is the probability that if the toilet is unoccupied at time t = 0, then there are 3 people in there at time t = 2? (g) What happens to cach row of P(t) as t -> co. Explain your result. (h) Alter you simulation to estimate the probability that when a person arrives, all cubicles are occupied. Comment on how your result relates to the answer for part (g). [16 marks]