Question

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train next year.

What percent of commuters will be using the transit system in the long run?

Answer 1

Let T denote transit system and M denote metro train.

Here again, there are four cases.

(T T) ⇒ those who use the transit system will continue to use the transit system.

(T M) ⇒ those who use the transit system will switch over to the metro train.

(M T) ⇒ those who use metro train will change to the transit system.

(M M) ⇒ those who use metro train will continue to use the metro train.

From the question,

(T T) =70% = 0.7.

(T M) = 30% = 0.3

(M T) = 30% = 0.3

(M M) = 70% = 0.7

The transition probability matrix is given by

\[{\begin{matrix} & \begin{matrix}T&&M\end{matrix} \\ \begin{matrix}T\\M\end{matrix} & \begin{pmatrix}0.7&0.3\\0.3&0.7\end{pmatrix}\\ \end{matrix}}\]

The current position is given by T = 60% and M = 40%

⇒ `(("T", "M"),(0.6, 0.4))`

We have to predict the values of T and M after one year.

At equilibrium which will be reached in the long run

T + M = 1

We have, `("T"  "M") ((0.7, 0.3),(0.3, 0.7)) = (("T", "M"))`

By matrix multiplications,

(0.7T + 0.3M 0.3 T + 0.7 M) = (T M)

Equating the corresponding elements,

0.7T + 0.3M = T

0.7T + 0.3(1 – T) = T  ......(Using T + M = 1)

0.7T + 0.3 – 0.3T = T

0.3 = 0.6T

T = 0.5

T = 50%

Thus in the long run, 50% of the commuters will be using transit system and 50% will be using a metro train.