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, find the equilibrium travel population for Frenzy (rounded to the nearest whole person).
A) 126
B) 95
C) 79
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, find the equilibrium travel population for Frenzy (rounded to the nearest whole person).
A) 126 
B) 95 
C) 79 
D) 100 
E) None of the above

Quizplus · Accepted Answer

To find the equilibrium travel population for Frenzy, we need to use the transition matrix to calculate the proportion of people traveling to Frenzy from each city and multiply it by the current population of each city. 
Let P1, P2, P3 be the populations of Chaos, Frenzy, and Tremor respectively. 
After one iteration, the proportions of people traveling to Frenzy from each city are:
- From Chaos: 0.11(P1/(P1+P2+P3))
- From Frenzy: 0.75(P2/(P1+P2+P3))
- From Tremor: 0.11(P3/(P1+P2+P3))

Using the given initial population vector [100, 100, 100], we can set up the following equation:
P2(new) = 0.11(P1/300) + 0.75(P2/300) + 0.11(P3/300)) * 100
Simplifying this equation gives: 
15P2(new) = P1 + 75P2 + P3
Similarly, we can set up equations for P1 and P3:
15P1(new) = P1 + 11P2 + P3
15P3(new) = P1 + 11P2 + 9P3

Solving these equations simultaneously (using substitution or matrix algebra) gives:
P1 = 110.92
P2 = 94.62
P3 = 94.46

Thus, the equilibrium travel population for Frenzy (rounded to the nearest whole person) is 95, which corresponds to choice B.

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