Question

(2 points) Is an office complex of 1010 empos, on any given day some are at work and the rest ane acsent it is known that i that she wal be absent tomorow Suppose that today there are 808 employees at work Remainingime: min sec Problem 2. 2 points) lmanence complex of 1010 employees, on any given day some ar" at-Ork and terest are absent "as known tat at she willbe absent temerrow Suppose th loday here ane 808 emplayees at won an empley" is at work today thereゅan TO% chance that ue we be at werk tormon ow, and ifte empley" i, absent today, there chance a (a) Find te transition matrix for this scenario (asume that state is at work" and stane 24abver b) Predict the number hat will be at work ve days fom new

Answer 1

a) Here, the transition matrix is : A =

70% 40% 30% 60%

i.e., A =

0.7 0.4 0.3 0.6

b) Here, A⁵ =

0.7 0.4 0.3 0.6

i.e., A⁵ =

0.57247 0.57004 0.42753 0.42996

In first day, there are 808 employees at work and 202 employees absent.

Then, the predicted number of employees that will be at work five days from now is =

[(0.57247*808)+(0.57004*202)]

= 577.70384

$\approx$ 578

Therefore, 578 employees will be at work five days from now.

c) For steady-state vector, we must have AX = X

i.e.,
0.7 0.4 0.3 0.6
$\begin{bmatrix} x\\ y \end{bmatrix}$ = $\begin{bmatrix} x\\ y \end{bmatrix}$

i.e., 0.7x+0.4y = x

0.3x+0.6y = y

i.e., -0.3x+0.4y = 0

0.3x-0.4y = 0

i.e., 0.3x-0.4y = 0

i.e., 0.3x = 0.4y

i.e., 3x = 4y............(i)

We also have, x+y = 1010

Multiplying both sides by 3 we get,

3x+3y = 3030

i.e., 4y+3y = 3030

i.e., 7y = 3030

i.e., y = 3030/7

i.e., y $\approx$ 433

Then, x = 1010-y

i.e., x $\approx$ 1010-433

i.e., x $\approx$ 577

Therefore, the steady-state vector is = .

577 433