# Math Modeling : Markov Chains & Processes

## Example Questions

### Example Question #1 : Markov Chains & Processes

A computer company has one service repair man and has space for 29 computers in the shop at one time. Last year the shop repaired 67 computers with an average repair time of 2 days per computer. Formulate a Markov process model for  which represents the number of computers in the shop for repair at time  months and calculate the .

Explanation:

Using Markov Process for this question assume that the space is finite.

The stochastic process which determines the future of the process from a probabilistic standpoint is defined as follows.

Let

Recall that the exponential distribution, and thus the  could have the density function,

Now formulate the Markov Process specifically for this problem.

The transitions of computers in and out of the shop are,

to  or

Now calculate the rates up and down.

To calculate  take the total number of computers fixed last year and divide it by the total months in the year. To calculate  assume there are 22 working days in a month and it takes on average, two days to fix a computer.

Its key to know that at zero we cannot move down a state and like wise when at 29, we cannot move up a state.

Solving with

Continuing in this fashion results in the following.

Now, writing the sum of a finite geometric series is

Now

therefore,

Next,

### Example Question #2 : Markov Chains & Processes

A computer company has one service repair man and has space for 32 computers in the shop at one time. Last year the shop repaired 71 computers with an average repair time of 3 days per computer. Formulate a Markov process model for  which represents the number of computers in the shop for repair at time  months and calculate the .

Explanation:

Using Markov Process for this question assume that the space is finite.

The stochastic process which determines the future of the process from a probabilistic standpoint is defined as follows.

Let

Recall that the exponential distribution, and thus the  could have the density function,

Now formulate the Markov Process specifically for this problem.

The transitions of computers in and out of the shop are,

to  or

Now calculate the rates up and down.

To calculate  take the total number of computers fixed last year and divide it by the total months in the year. To calculate  assume there are 22 working days in a month and it takes on average, two days to fix a computer.

Its key to know that at zero we cannot move down a state and like wise when at 29, we cannot move up a state.

Solving with

Continuing in this fashion results in the following.

Now, writing the sum of a finite geometric series is

Now

therefore,

Next,

### Example Question #3 : Markov Chains & Processes

A computer company has one service repair man and has space for 23 computers in the shop at one time. Last year the shop repaired 51 computers with an average repair time of 5 days per computer. Formulate a Markov process model for  which represents the number of computers in the shop for repair at time  months and calculate the .

Explanation:

Using Markov Process for this question assume that the space is finite.

The stochastic process which determines the future of the process from a probabilistic standpoint is defined as follows.

Let

Recall that the exponential distribution, and thus the  could have the density function,

Now formulate the Markov Process specifically for this problem.

The transitions of computers in and out of the shop are,

to  or

Now calculate the rates up and down.

To calculate  take the total number of computers fixed last year and divide it by the total months in the year. To calculate  assume there are 22 working days in a month and it takes on average, two days to fix a computer.

Its key to know that at zero we cannot move down a state and like wise when at 29, we cannot move up a state.

Solving with

Continuing in this fashion results in the following.

Now, writing the sum of a finite geometric series is

Now

therefore,

Next,