Vehicles in a certain country are required to be assessed every year for road-worthiness. At one vehicle assessment center, drivers wait for an average of 15 minutes before the road-worthiness assessment of their vehicle commences. The assessment takes on average 20 minutes to complete. Following the assessment, 80% of vehicles are passed as road-worthy allowing the driver to drive home. A further 15% of vehicles are categorized as a “minor fail”; these vehicles require on average 30 minutes of repair work before the driver is allowed to drive home. The remaining 5% of vehicles are categorized as a “significant fail”; these vehicles require on average three hours of repair work before the driver can go home.
A continuous-time Markov model is to be used to model the operation of the vehicle assessment centre, with states W (waiting for assessment), A (assessment taking place), M (minor repair taking place), S (significant repair taking place) and H (travelling home).
generator matrix i got:
$\space\space\space\space\space\space\space $ $\begin{matrix} W & \space\space A & M & \space\space S &\space\space\space H \end{matrix}$
$\begin{matrix} W \\ A \\ M \\ S \\ H \end{matrix}
\left[ \begin{matrix} \frac{-1}{15} & \frac{1}{15} & 0 & 0 & 0 \\ 0 & \frac{-1}{20} & & & \frac{1}{25} \\ 0& 0 & \frac{-1}{30} & 0 & \frac{1}{30} \\ 0 & 0 & 0 & \frac{-1}{180} & \frac{1}{180} \\ 0 & 0 & 0 & 0 & 0 \end{matrix} \right]$
But I don't know how to find the 2 empty spaces in second row. Any help would be appreciated.
W, A, M, S, H are labels of rows and columns.
Best Answer
Think of the entries as rates of flow. Drivers flow out of state $A$ at a rate of 1/20 minutes (the - sign means "out"), and therefore must flow into states $M$, $S$, and $H$ combined at a rate of << fill in the blank. >> Now, given that total flow rate into $M$, $S$, and $H$, why is the number for $A \to H$ 1/25? Obviously it must relate to a number given in the text somewhere...
Once you figure that out, the rest will be clear.