Question

The computer center at Rockbottom University has been experiencing computer downtime. Let us assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities: To From Running Down Running 0.70 0.30 Down 0.20 0.80 If the system is initially running, what is the probability of the system being down in the next hour of operation

Answer 1

The probability of the system being down in the next hour of operation is 0.3.

Explanation:

We have a transition matrix from one period to the next (one hour) that can be written as:

`T=\left[\begin{array}{ccc}&R&D\\R&0.7&0.3\\D&0.2&0.8\end{array}\right]`

We can represent the state that system is initially running with the vector:

`S_0=\left[\begin{array}{cc}1&0\end{array}\right]`

The probabilties of the states in the next period can be calculated using the matrix product of the actual state and the transition matrix:

`S_1=S_0\cdot T`

That is:

`S_1=S_0\cdot T= \left[\begin{array}{cc}1&0\end{array}\right]\cdot \left[\begin{array}{cc}0.7&0.3\\0.2&0.8\end{array}\right]= \left[\begin{array}{cc}0.7&0.3\end{array}\right]`

With the inital state as running, we have a probabilty of 0.7 that the system will be running in the next hour and a probability of 0.3 that it will be down.