STATS 720 Test 1
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Department of Statistics
STATS 720
Test 1
Due: 3 p.m. Wednesday 13 April 2022
Answer ALL questions. Total number of marks = 40.
Marks for each question are shown in brackets.
Show all working.
This is an open book exam.
The test will run from 3.00 p.m. Tuesday for 24 hours until 3.00 p.m. Wednesday.
Your work in this test should be yours alone. Do not communicate with others in your class about the contents of the test while it is in progress.
1. Let x = {Xn ; n = 0, 1, 2, . . .} be a Markov chain with state space S = {1, 2, 3, 4, 5} and transition matrix
|
╱ 0 ． 1 ． P = ．(．) 1/4
0 |
1 0 0 0 0 |
0 0 1/4 1 0 |
0 0 1/4 0 0 |
0 、 0 ． 1/4 ．(．) 0 ．(．) 1 |
(a) Identify the communicating classes for this chain, and classify the states of the chain. For each state, say whether it is transient, null-recurrent or positive-recurrent and identify its period.
(b) Find h35 = P (Xn = 5 for some n 2 0IX0 = 3), the probability the process ever reaches state 5, given that it is in state 3 at time 0.
(c) What are h31 and h32 ? (You should be able to write these down without extensive futher calculation.) (11 marks)
2. The weather on a remote island is either good, bad, or indifferent. A pe- riod of indifferent weather is equally likely to be succeeded by good or bad weather, and bad weather is equally likely to be succeeded by good or indif- ferent weather, but good weather is always followed by bad. The durations of periods of good, bad or indifferent weather are independent, exponentially distributed random variables with mean duration 3,1 and 2 days respectively. Let the state space be S = {1, 2, 3}, where 1 denotes good, 2 denotes bad, and
3 denotes indifferent.
(a) Write down the Q-matrix for this process.
(b) Write down equations which the equilibrium distribution must satisfy, and hence find the equilibrium distribution, π, for this process.
(c) Find the equilibrium distribution for the jump chain, π J .
(d) If the weather is bad at time 0, find the expected number of days until it first becomes good after time 0. (13 marks)
3. Consider a queue in continuous time with 4 servers. Customer arrivals occur as a Poisson process with rate λ > 0 and service times have mean 1/µ . The queue has infinite capacity. Let X(t) be the number in the queue at time t, and consider the process X = {X(t), t 2 0}
(a) What further assumptions are needed in order for X to be a Markov process? In the following you may assume that these assumptions hold and that X is a Markov process.
(b) Find the equilibrium distribution, π = {πi , i = 0, 1, 2, . . .}, for the process X, for those values of λ and µ for which it exists.
(c) For what values of λ and µ does π exist? Justify your answer briefly.
(d) Is the process, X, reversible? Justify your answer briefly.
(e) Now suppose the queue has finite capacity N 2 4, that is, there can’t be more than N individuals in the system, including any customer currently in service. Any arrivals to the system when the queue is full do not enter the queue and are turned away without being served. Let Y (t) be the number in the queue at time t.
i. Write down an expression for π N = {πi(N), i = 0, 1, 2, . . . , N - 1, N}, the equilibrium distribution for Y = {Y (t), t 2 0}, for those values of λ and µ for which it exists.
ii. For what values of λ and µ does π N exist? Justify your answer briefly.
2023-04-15