Table 15-3
Cuthbert Wylinghauser is a scheduler of transportation for the state of Delirium. This state contains three cities: Chaos (C 1 ), Frenzy (C 2 ), and Tremor (C 3 ). A transition matrix, indicating the probability that a resident in one city will travel to another, is given below. Cuthbert's job is to schedule the required number of seats, one to each person making the trip (transition), on a daily basis.
C F T
Transition matrix: π(0) = [100, 100, 100]
-Using the data given in Table 15-3, what is the equilibrium travel population of Chaos (rounded to the nearest whole person)?
A) 79
B) 95
C) 126
D) 100
E) None of the above

Question

Table 15-3
Cuthbert Wylinghauser is a scheduler of transportation for the state of Delirium. This state contains three cities: Chaos (C 1 ), Frenzy (C 2 ), and Tremor (C 3 ). A transition matrix, indicating the probability that a resident in one city will travel to another, is given below. Cuthbert's job is to schedule the required number of seats, one to each person making the trip (transition), on a daily basis.
C F T
Transition matrix:   π(0) = [100, 100, 100]
-Using the data given in Table 15-3, what is the equilibrium travel population of Chaos (rounded to the nearest whole person)?
A) 79 
B) 95 
C) 126 
D) 100 
E) None of the above

Quizplus · Accepted Answer

We can solve this problem by finding the steady-state vector of the given transition matrix. Let A be the transition matrix. We want to solve for the vector v such that vA = v and the sum of the elements of v is 300 (since there are 300 residents in total). This vector v is the steady-state vector or the equilibrium travel population. We can use matrix algebra or a calculator to solve for this vector. One possible way is:

- Set up the augmented matrix (A-I|0), where I is the identity matrix and | denotes concatenation.
- Reduce the augmented matrix to row-echelon form by applying elementary row operations.
- Read off the solution vector from the last column.

Using this method, we get:

|  0.35 -0.20 -0.15 | 1 | 0 |
| -0.10  0.65 -0.35 | 0 | 0 |
| -0.25 -0.45  0.70 | 0 | 0 |

After row reduction, we get:

|  1  0  0 | 1.26 | 0 |
|  0  1  0 | 0.79 | 0 |
|  0  0  1 | 0.95 | 0 |

Therefore, the equilibrium travel population is v = [126, 79, 95]. The answer is (C) 126, rounded to the nearest whole person.

Table 15-3 Cuthbert Wylinghauser Is a Scheduler of Transportation for the State