# Markov jump Processes: finding the generator matrix

• May 17th 2012, 05:00 AM
amul28
Markov jump Processes: finding the generator matrix
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:

$\dots\dots$ $\begin{matrix} W & A & M & S & H \end{matrix}$
$\begin{matrix} W \\ A \\ M \\ S \\ H \end{matrix}\left[ \begin{matrix} \frac{-1}{15} & \frac{1}{15} & 0 & 0 & 0 \\ 0 & \frac{-1}{20} & & & \frac{1}{25} \\ 0& 0 & \frac{-1}{30} & 0 & \frac{1}{30} \\ 0 & 0 & 0 & \frac{-1}{180} & \frac{1}{180} \\ 0 & 0 & 0 & 0 & 0 \end{matrix} \right]$

but i dint know how to find the 2 empty spaces in second row. Any help would be appreciated.
• May 18th 2012, 06:43 AM
harish21
Re: Markov jump Processes: finding the generator matrix
how did you get to $p_{AH}=\frac{1}{25}$?
• May 18th 2012, 06:54 AM
amul28
Re: Markov jump Processes: finding the generator matrix
As 80% of them drive home directly their only waiting time will be 1/20. So, I calculated (80%)/20 = 1/25
• May 18th 2012, 06:58 AM
harish21
Re: Markov jump Processes: finding the generator matrix
likewise.. $0.05 \times \frac{1}{20}=p_{AS}$

$0.15 \times \frac{1}{20}=p_{AM}$
• May 18th 2012, 07:03 AM
amul28
Re: Markov jump Processes: finding the generator matrix
I was calculating taking into account the waiting time in H also. As you take waiting time in H when you enter that state. But actually it has reached state H from A, so the waiting time will be only 1/20. This will agree with your post, right.
• May 18th 2012, 07:10 AM
harish21
Re: Markov jump Processes: finding the generator matrix
Yes. After assessment, it can go to any of the other progressive states. and your average time for the assessment state is 20 minutes and the calculation follows.