Question

Markov Chains

Three companies provide internet service in a city of 20,000 people. At the beginning of the year, the

market shares of each company are as follows: 11,800 people use company A, 6200 people use company

B, and only 2000 people use company C. Each month, 5% of company A’s customers switch to company

B, and 3% of company A’s customers switch to company C. During the same time, 4% of company B’s

customers switch to A, and 6.5% switch to C. Also during the same time, 2% of company C’s customers

switch to company A, and 1.5% switch to company B. Let A_k represent the probability that a randomly

chosen consumer on day k will be using company A’s service, B_k be the probability they will be using

company B’s service, and C_k be the probability that they go with company C.

1) Find the steady state vector q of the Markov chain using the fact that P has an eigenvalue of 1 with

corresponding eigenvector q. What are the probable long-term market shares of each company?

Answer 1

1) The matrix of the Markov Chain is

0.92 0.04 0.02 M 0.05 0.895 0.015 0.03 0.065 0.965

For steady state vector, we must have

-0.08 0.04 0.02 0 a (M I 0 0.05 0.105 0.015 b 10 0.03 0.065 0.035

Along with (as these are steady state probabilities)

abc1

Which has the solution

27c 22c abc 1 b 64 a = 64

And so

22 15 27 b 64 64 a 64

So the steady state vector is

27 1 22 64 15 q =

$\blacksquare$

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