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 probable market share of each company after the first month.

2) Find the probable market share of each company after six months.

Answer 1

Taihally A 11 8D0 6200 2ODD Lach nont Fanm A to C 37 47 Famm 2 7 to A : Af ter 1 nonth : Change in A -0.08 A 0-04 B I656 So A LL800- 6 56 Change in B -3) O.0150 +0.05 A - 6169 13= 6200- 3 Sc Crange in C O.065 B 687 O.035C 0.03 A 687 2000 t68 So Sivnai Lealy Caladahons pesfosauig 6 Months same 5482 for 8712 : 13 5806 wuitka yaur Mpeacaie AC prognan is alo

C program code:

#include <stdio.h>

int main()
{
float t, a, b, c, a_, b_, c_, i;
a_ = 11800;
b_ = 6200;
c_ = 2000;
scanf("%f", &t);
for(i = 0; i < t; i++)
{
a = -0.08*a_ + 0.04*b_ + 0.02*c_;
b = -0.105*b_ + 0.05*a_ + 0.015*c_;
c = -0.035*c_ + 0.03*a_ + 0.065*b_;
a_ = a_ + a;
b_ = b_ + b;
c_ = c_ + c;
}
printf("%f %f %f", a_, b_, c_);
return 0;
}