A certain salesman, called Mark O.V. delivers his goods to three cities: C1, C2, & C3, staying only for one day at a time in each of them.
His stay in C1 is always followed by going next to C2. If he delivers in either C2 or C3 he is thrice as likely to continue to C1, than to the other city.

Provide your estimate of the number of working days in each of the cities
within a period of 12,000 working days.

I'm going to call the cities A,B,C and think of this as a recursive definition:
A(n+1)=(3/4)B(n)+(3/4)C(n)
B(n+1)=(1/4)C(n)
C(n+1)=A(n)+(1/4)B(n)

Since we want this to settle down to a proportion A:B:C lets just assume it has after enough (infinite) steps.  That means  A(n+1)=A(n) etc. Further, to set a scale for now let B(n)=B=1

Solving is a piece of cake.  B=1 makes C=4 and A=15/4.
Scale by 4/35 to make then sum to 1.
A:B:C = 3/7 : 4/35: 16/35
And so our long term estimate should be asymptotic to these proportions out of 12,000

A=5143, B=1371, C=5486
Is this the Markov Chain way?

Edited on May 28, 2016, 10:24 pm

Posted by Jer on 2016-05-28 22:19:25