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1. Consider a car on a three lane highway. We will model this system as a...
Two major cities are connected by a three-lane highway in each direction. Let E1, E2, and...
Two major cities are connected by a three-lane highway in each direction. Let E1, E2, and E3 denote the right-hand, center, and left-hand lane, respectively. Upon inspection, the maintenance engineer concludes that the probability that each of these three lanes will require major repair work in the next year are 0.10, 0.05, and 0.01, respectively. From past experience, the following information is available: P(E2 /E1) =0,8 ,PE3 /E2) =0,9 ,P(E3 /E1) =0,5 ,and P(E3 /E1 E2) =0,9 (a) What is...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Mar...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Markov chain irreducible? 1 marks (a) (b) Find the probability of the Markov chain to move to state 3 after two time steps, providing it starts in state 2 [3 marks 14 Find the stationary distribution of this Markov chain [4 marks (c) (d) Is the stationary distribution also a limiting distribution for this Markov chain? Explain your answer...
A six-lane divided highway (three lane in each direction) is on rolling terrain with two access...
A six-lane divided highway (three lane in each direction) is on rolling terrain with two access points per mile and has 10-ft. lanes, with a 5-ft. shoulder on the right side and a 3-ft. shoulder on the left side. The peak-hour factor is 0.80, and the directional peak-hour volume is 3000 vehicles per hour. There are 6% large trucks, 2% buses, and 2% recreational vehicles. A significant percentage of nonfamiliar roadway users are in the traffic stream (the driver population...
Exercise 1 [I8 pts]. Two lanes of a three-lane highway are closed because of road works...
Exercise 1 [I8 pts]. Two lanes of a three-lane highway are closed because of road works on a 2 km long segment. A detector placed on one single lane in the undisturbed zone upstream the road works detects a flow of 2,000 veh/h and an average speed of 100 km/h. A second detector placed on one of the two lanes just upstream the roadwork zone detects a flow of 800 veh/h and an average speed of 40 km/h. We observe...
Problem 3: Transition Matrix Problem (10 pts.) Consider an inventory system in which the sequence of...
Problem 3: Transition Matrix Problem (10 pts.) Consider an inventory system in which the sequence of events during each period is as follows: (1) We observe the number of units in inventory (call it i) at the beginning of the period. (2) Purchases: If i<1, 3 units are ordered from our supplier. If i> 2,0 units are ordered. Delivery of all ordered units is immediate (at the beginning of the current period i). (3) Demand: With probability 0.32, 0 units...
2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern ...
2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern clear. Some nodes are white and some are grey. Two nodes are "neighbours" if they are connected by an edge. In each time unit, a counter occupying a node moves to each of its three neighbours with probability 1/3. (a) Suppose a counter is on a grey node. Calculate, by tracking the counter, the probability that it is on a grey node...
2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern ...
2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern clear. Some nodes are white and some are grey. Two nodes are neighbours" if they are connected by an edge. In each time unit, a counter occupying a node moves to each of its three neighbours with probability 1/3. (a) Suppose a counter is on a grey node. Calculate, by tracking the counter, the probability that it is on a grey node...
Consider a disease, which has three states: "0 (healthy)."1" (impaired), "Z (disease) In state "Z when certain treatments are adopted, the state can be restored to be healthy C...
Consider a disease, which has three states: "0 (healthy)."1" (impaired), "Z (disease) In state "Z when certain treatments are adopted, the state can be restored to be healthy C'O).When a subject is either at state "O" or 1", she/he can decide whether some preventive actions should be taken so that she/he will be in a new state "3. The state transition probabilities are as follows (5) Assume that the disease process is a first-order time homogenous Markov chain and it...
Exercise 5 - Consider a Markov chain with (one-step) transition probability matrix expressed as 0 1...
Exercise 5 - Consider a Markov chain with (one-step) transition probability matrix expressed as 0 1 23 o 1/2 0 0 0 1/2 0 1 1 0 0 0 Determine the communicating classes and period of each state (A) -for (al,a2,a3)-(1/3, 1/3, 1/3), (B)-for (αι, α2,a3)-(1/2, 1/2,0).
A Markov chain model for the growth and replacement of trees assumes that there are three...
A Markov chain model for the growth and replacement of trees assumes that there are three stages of growth based on the size of the tree; young tree, mature tree, and old tree. The transition period considered is 8 years. In each period of 8 years: (1) 30% of the young trees remain young and the rest of the young trees become mature; (2) Among the mature trees, 40% remain mature and the rest become old; (3) Old trees die...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Markov chain irreducible? 1 marks (a) (b) Find the probability of the Markov chain to move to state 3 after two time steps, providing it starts in state 2 [3 marks 14 Find the stationary distribution of this Markov chain [4 marks (c) (d) Is the stationary distribution also a limiting distribution for this Markov chain? Explain your answer...
Exercise 1 [I8 pts]. Two lanes of a three-lane highway are closed because of road works on a 2 km long segment. A detector placed on one single lane in the undisturbed zone upstream the road works detects a flow of 2,000 veh/h and an average speed of 100 km/h. A second detector placed on one of the two lanes just upstream the roadwork zone detects a flow of 800 veh/h and an average speed of 40 km/h. We observe...
Problem 3: Transition Matrix Problem (10 pts.) Consider an inventory system in which the sequence of events during each period is as follows: (1) We observe the number of units in inventory (call it i) at the beginning of the period. (2) Purchases: If i<1, 3 units are ordered from our supplier. If i> 2,0 units are ordered. Delivery of all ordered units is immediate (at the beginning of the current period i). (3) Demand: With probability 0.32, 0 units...
2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern clear. Some nodes are white and some are grey. Two nodes are "neighbours" if they are connected by an edge. In each time unit, a counter occupying a node moves to each of its three neighbours with probability 1/3. (a) Suppose a counter is on a grey node. Calculate, by tracking the counter, the probability that it is on a grey node...
2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern clear. Some nodes are white and some are grey. Two nodes are neighbours" if they are connected by an edge. In each time unit, a counter occupying a node moves to each of its three neighbours with probability 1/3. (a) Suppose a counter is on a grey node. Calculate, by tracking the counter, the probability that it is on a grey node...
Consider a disease, which has three states: "0 (healthy)."1" (impaired), "Z (disease) In state "Z when certain treatments are adopted, the state can be restored to be healthy C'O).When a subject is either at state "O" or 1", she/he can decide whether some preventive actions should be taken so that she/he will be in a new state "3. The state transition probabilities are as follows (5) Assume that the disease process is a first-order time homogenous Markov chain and it...
Exercise 5 - Consider a Markov chain with (one-step) transition probability matrix expressed as 0 1 23 o 1/2 0 0 0 1/2 0 1 1 0 0 0 Determine the communicating classes and period of each state (A) -for (al,a2,a3)-(1/3, 1/3, 1/3), (B)-for (αι, α2,a3)-(1/2, 1/2,0).
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