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,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 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,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, we need to find the steady-state vector, which is the vector that remains unchanged when multiplied by the transition matrix. We can find it by solving the equation πP=π, where P is the transition matrix and π is the steady-state vector.

We can use a matrix calculator or software to solve this equation, or we can use a shortcut method. One way to find the steady-state vector is to normalize the matrix obtained by subtracting the identity matrix from the transpose of the transition matrix (PT-I), and then taking the first column of this matrix.

Using this method, we get:

PT-I =  0.6  -0.2  -0.4
         -0.3  0.8  -0.1
         -0.1  -0.6  0.7
         
         0.3  -0.2  -0.4
         -0.3  0.7  -0.1
         -0.1  -0.6  0.8
         
         0.3  -0.2  -0.4
         -0.3  0.8  -0.1
         -0.1  -0.5  0.7
         
Normalizing the first column of this matrix, we get the steady-state vector:

π = [0.418, 0.329, 0.253]

To find the equilibrium travel population for Frenzy (the second city), we multiply the steady-state vector by the total population (300 in this case) and by the corresponding entry in the vector representing Frenzy (which is the second entry, 0.329).

Equilibrium travel population for Frenzy = π2 × 300 ≈ 98.7 ≈ 95 (rounded to the nearest whole person)

Therefore, the best choice is B, with an equilibrium travel population for Frenzy of 95.

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