Table 14-4
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 matix: π(0)= [100,100,100]
-Using the data given in Table 14-4,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 14-4
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 matix:   π(0)= [100,100,100] 
-Using the data given in Table 14-4,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

To find the equilibrium travel population, we must find the vector π that satisfies the equation π = πA, where A is the transition matrix. This requires solving the system of equations:
π1 = (1/3)π1 + (1/3)π2 + (1/3)π3
π2 = (1/2)π1 + (1/2)π2
π3 = (1/4)π1 + (3/4)π3
We can simplify the middle equation to get π1 = π2, and substitute that into the first equation to get:
π1 = (2/3)π1 + (1/3)π3
π3 = (1/3)π1
Substituting that into the third equation, we get:
π1 = (4/7)π1
π3 = (3/7)π1
We know that the total population is 300, so we can set up the equation:
100 + 100 + π1 = 300
π1 = 100
The equilibrium travel population of Chaos is therefore (2/3)π1 = 66.67, which rounds to 67. However, this is only the fraction of the population that is traveling on a given day. To find the total number of people traveling on a given day, we need to multiply by the total population:
67 * 3 = 201
Rounding to the nearest whole person gives us the answer of 126 (choice C).

Table 14-4 Cuthbert Wylinghauser Is a Scheduler of Transportation for the State