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SHBI-GB.7301 Stochastic Modeling & Simulation Summer 2022 Homework 3
发布时间:2022-08-02
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SHBI-GB.7301
Stochastic Modeling & Simulation
Summer 2022
Homework 3
1. Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i = 0, 1, 2, 3, if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth step. Explain why {Xn, n = 0, 1, 2, ...} is a Markov chain and calculate its probability transition matrix.
2. A store that stocks a certain commodity uses the following (s, S) ordering policy. If its supply at the beginning of a time period is x, then it orders
o(x) = .0
if
if
x > s,
x < s.
The order quantity is immediately filled. The daily demands are independent and equal to j with probability αj for j = 0, 1, 2, .... All demands that cannot be immediately met are lost. Let Xn denote the inventory level at the end of the nth time period. Argue that {Xn, n > 1} is a Markov chain and compute its transition probabilities.
3. Let X = {Xn, n > 0} be a time-homogeneous Markov chain with state space s = {1, 2, 3, 4} and transition matrix
╱ .2 .3 .5 0 、
『 0 .2 .3 .5 │
P =
『 .5 0 .2 .3 │
( .3 .5 0 .2 .
Given that at time zero we start out in state 1, what is the expected value of X2? In other words, calculate E[X2IXó = 1].
4. Specify the communicating classes of the following Markov chains.
╱ 0 Pf = 『 6(f)
( |
3 4 0 f 3 |
0(6) . , |
P2 = 『(『) 0 『 『 0
|
f 2 f 2 0 0 0 |
0 0 1 f 3 0 |
0 0 0 2 3 0 |
0 │ │ 0 │(│) , │ 0 │ 0 . |
╱『『
0(0) 0(0) 、││
、│
││. 『(0(0)
0(1).│
5. An organization has N employees where N is a large number. Each employee has one of three possible job classifications and changes classifications (independently) according to a Markov chain with transition probabilities
╱0.7 0.2 0.1、
『 0.2 0.6 0.2 │
(0.1 0.4 0.5. .
What is the long run percentage of employees in each classification?