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 stochastic matrix P that is the transition matrix from vector p_k to p_k+1.

Answer 1

The required transition matrix is

0.92 0.04 0.02 0.05 0.895 0.015 0.03 0.065 0.965

This was found as

Ak+10.92.A 0.05 Bk0.03Ck Bk+1 0.05.Ak 0.895Bk 0.015Ck CK+10.03.Ak 0.065Bk0.965Ck

Based on the transition dynamics of the consumers

$\blacksquare$

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