Homework Statement
A repairman fixes broken televisions. The repair time is exponentially distributed with a mean of 20 minutes. Broken television sets arrive at his shop according to a Poisson process with arrival rate 12 sets per working day. (8 hours).
(i) What is the fraction of time that...
Well, I'm trying to find a value for A(P(t)) - A is the matrix that I found at the very begining, so I was thinking I should be subbing in the birth and death rates into that matrix to find P(t)/P(t+1)?
I'd imagine I start off by subbing the values into the given matrix:
\ P = \left(\begin{array}{cc}P_0 \\P_1\\P_2\\P_3\end{array}\right) = \left(\begin{array}{cc}3\\2\\2\\0\end{arra y}\right)
I also have P(t + 1) = AP(t).
So maybe sub in values for t = 0, 1 and 2 (but sub them into...
That's actually part of the next question, which I've only just seen:
Consider the evolution of the species over the next three years, where the initial population is P_0 = 3, P_1 = 2, P_2 = 2, P_3 = 0. Using the following birth and death rates:
d_0 = 0.15,
b_1 = 0.5, d_1 = 0.2
b_2 =...
My head is fried... sorry for all the obvious questions!
f(1) is negative, f(2) is positive... so somewhere in between there is an f(x) that is equal to zero. I was concentrating on the magnitude instead of the signs!
Thanks for your help.
I only spotted my mistake when I took the time to type it all out in latex code - and it was a small mistake with mixing up a zero in a place where it shouldn't have been! So I did manage to get out my eigenvalue equation, but now I am stuck on another section:
If d_0 = 0.15, d_1 = 0.2, d_2...
Ok, so here's my full work on finding the determinant so far:
\left(\begin{array}{cccc}-\lambda&b_1&b_2&0\\1-d_0&-\lambda&0&0\\0&1-d_1&-\lambda&0\\0&0&1-d_2&-\lambda\end{array}\right)
-\lambda \left(\begin{array}{ccc}-\lambda&0&0\\1-d_1&-\lambda&0\\0&1-d_2&-\lambda\end{array}\right)...
\left(\begin{array}{cccc}-\lambda&b_1&b_2&0\\1-d_0&-\lambda&0&0\\0&1-d_1&-\lambda&0\\0&0&1-d_2&-\lambda\end{array}\right)
That's the (A-\lambdaI) matrix that I'm using
Then I'm trying to find the determinant by taking each element of the first row as follows:
-\lambda...