## Related questions with answers

In an office complex of 1000 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 85% chance that she will be at work tomorrow, and if the employee is absent today, there is a 60% chance that she will be absent tomorrow. Suppose that today there are 760 employees at work. (a) Find the transition matrix for this scenario. (b) Predict the number that will be at work five days from now. (c) Find the steady-state vector.

Solution

Verified(a) Let the columns of the matrix contain the percentage of workers today and the rows the percentage of workers tomorrow (at work/absent).

$A=\begin{bmatrix} 0.85 & 0.6 \\ 0.15 & 0.4 \end{bmatrix}$

(b) The number of people ar work/absent in 5 days are given by $A^5\bold{x}_0$:

$A^5\begin{bmatrix} 760\\ 240\end{bmatrix}=\begin{bmatrix} 800\\ 200\end{bmatrix}$

Thus there are 800 people at work.

(c) The steady-state vector is a solution of $(A-I_n)\bold{x}=\bold{0}$.

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