STATS 320 Exam 2021
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STATS 320
1. (8 marks)
Suppose {Ⅹn．n 2 0} is a Markov chain with state space ³ = {1．2．3．4．5} and transition
(a) Draw the transition diagram for this chain.
(b) Find P(Ⅹ3 = 1IⅩ〇 = 3).
(c) Find h41 = P (Ⅹn = 1 for some n 2 0IⅩ〇 = 4), the probability that the chain ever reaches state 1, given that it starts in state 4 at time 0.
2. (12 marks)
During lockdown Delta took a walk every day. She had a choice of three walks that she could take, and her choices on successive days formed a Markov chain with state space ³ = {1．2．3} and transition matrix
Let Ⅹn denote her choice on day n, n = 1．2．3．4．．．．．．, where day 1 is the first day of lockdown.
(a) Write down equations that the equilibrium distribution satisfies, and find the equilibrium distribution for this chain.
(b) Is the equilibrium distribution also the limiting distribution? Justify your answer briefly.
(c) Delta starts with walk 1 on day 1 of lockdown. What is the expected number of days until she does walk 3?
3. (8 marks)
A service facility has two streams of customers arriving at it. Stream A customers arrive as a Poisson process at rate 5 per hour. Stream B customers arrive as a Poisson process at rate 3 per hour. The two arrival streams are independent of each other. Suppose there were 5 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) Suppose 5 customers arrived at the facility between 11 a.m. and 12 noon. What is the probability that 2 of those 5 arrivals occurred between 11 a.m. and 11.30 a.m.? Show your working clearly.
4. (22 marks)
A very small takeaway cafe with 2 servers has customers arriving at it as a Poisson process of rate 40 per hour. It takes each customer 3 minutes, on average, to be served, and the service times are exponentially distributed. Interarrival times and service times are all independent of each other. There is room for at most 5 customers in the cafe, including those in service. Whenever the cafe is full (i.e. has 5 customers in it) arriving customers don’t go in and are turned away. Let N(t) be the number of customers in the cafe at time t, including any in service.
(a) Draw the transition diagram and give the birth and death rates, 入n and μn , for the process N(t).
(b) If there is one customer already in the facility, what is the probability that the current customer finishes her service before another customer joins the queue?
(c) Find the equilibrium distribution {πn．0 < n < 5} for N(t)．
(d) What is L, the expected number of customers in the system in equilibrium?
(e) If a customer joins the queue, what is the expected time that she spends in the system before completing service?
(f) What is the expected length of an idle period for the cafe (that is, when both servers are free and not working)? What is the expected length of a busy period for the cafe?
(g) 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.
2023-06-08