Question

In a large metropolitan area, 25% of the commuters currently use the public transportation system, whereas the remaining 75% commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now 40% of those who are now commuting to work via automobile will switch to public transportation, and 60% will continue to commute via automobile. At the same time, it is expected that 20% of those now using public transportation will commute via automobile and 80% will continue to use public transportation.

(a) Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters.

			P              A
T	=	P A

(b) Find the initial distribution vector for this Markov chain.

X0

=

(c) What percentage of the commuters are expected to use public transportation 6 months from now? (Round your answer to the nearest percent.)

Answer 1

Let the states of the Markov chain be P and A denoting public transportation and automobile respectively.

The transition probability from P to P and A are 0.8 and 0.2 respectively.

The transition probability from A to P and A are 0.4 and 0.6 respectively.

The transition matrix for the Markov chain is,

transportation used by these commuters.

			P              A
T	=	P A	0.8 0.2 0.4 0.6

(b)

Given, 25% of the commuters currently use the public transportation system, whereas the remaining 75% commute via automobile.

So, the initial distribution vector for this Markov chain is

X0 = [ 0.25 , 0.75]

(c)

State after 6 months

X1 = X0 P

= [ 0.25 0.75]  

0.8 0.2 0.4 0.6

= [0.25 * 0.8 + 0.75 * 0.4   0.25 * 0.2 + 0.75 * 0.6]

= [0.5 0.5]

Percentage of the commuters are expected to use public transportation 6 months from now = 0.5 = 50%