Call Now to Set Up Tutoring:

(888) 888-0446

Math Modeling : Stochastic Models

Study concepts, example questions & explanations for Math Modeling

Create An Account

All Math Modeling Resources

20 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Probability Models

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 X_t $\displaystyle X_t$ which represents the number of computers in the shop for repair at time $\displaystyle t$ months and calculate the EX_t $\displaystyle EX_t$.

Possible Answers:

$EX_t\approx 1.067$ $\displaystyle EX_t\approx 1.067$

$EX_t\approx 1.036$ $\displaystyle EX_t\approx 1.036$

$EX_t\approx 1.133$ $\displaystyle EX_t\approx 1.133$

$EX_t\approx 1.037$ $\displaystyle EX_t\approx 1.037$

$EX_t\approx 1.307$ $\displaystyle EX_t\approx 1.307$

Correct answer:

$EX_t\approx 1.037$ $\displaystyle EX_t\approx 1.037$

Explanation:

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

$X_t\epsilon\begin{Bmatrix} 1,2,3,4,...,m \end{Bmatrix}$ $\displaystyle X_t\epsilon\begin{Bmatrix} 1,2,3,4,...,m \end{Bmatrix}$

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

Let

$T_i=\text{time spent in state }i$ $\displaystyle T_i=\text{time spent in state }i$

$Pr{\begin{Bmatrix} T_i>t+s|T_i>s \end{Bmatrix}}=Pr\begin{Bmatrix} T_i>t \end{Bmatrix}$ $\displaystyle Pr{\begin{Bmatrix} T_i>t+s|T_i>s \end{Bmatrix}}=Pr\begin{Bmatrix} T_i>t \end{Bmatrix}$

Recall that the exponential distribution, and thus the T_i $\displaystyle T_i$ could have the density function,

$F_i(t)=\lambda_ie^{-\lambda_it}$ $\displaystyle F_i(t)=\lambda_ie^{-\lambda_it}$

Now formulate the Markov Process specifically for this problem.

$X_t\epsilon\begin{Bmatrix} 0,1,2,..., 29\end{Bmatrix}$ $\displaystyle X_t\epsilon\begin{Bmatrix} 0,1,2,..., 29\end{Bmatrix}$

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

X_t=i $\displaystyle X_t=i$ to X_t=i+1 $\displaystyle X_t=i+1$ or X_t=i-1 $\displaystyle X_t=i-1$

Now calculate the rates up and down.

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

$\\\lambda=5.6 \\\mu=11$ $\displaystyle \\\lambda=5.6 \\\mu=11$

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.

$[\text{Rate In}]=[\text{Rate Out}]$ $\displaystyle [\text{Rate In}]=[\text{Rate Out}]$

$\\\lambda P_0=\mu P_1 \$\mu +\lambda)P_1=\lambda P_0+\mu P_1 \\(\mu +\lambda)P_2=\lambda P_2+\mu P_3\\ \begin{matrix} .\\ .\\ . \end{matrix} \\\mu P_{29}=\lambda P_{28}$ \(\displaystyle \\\lambda P_0=\mu P_1 \\(\mu +\lambda)P_1=\lambda P_0+\mu P_1 \\(\mu +\lambda)P_2=\lambda P_2+\mu P_3\\ \begin{matrix} .\\ .\\ . \end{matrix} \\\mu P_{29}=\lambda P_{28}$

Solving with

$\sum P_i=1$ $\displaystyle \sum P_i=1$

$EX_t=\sum iP_i$ $\displaystyle EX_t=\sum iP_i$

Continuing in this fashion results in the following.

$\\P_n=(\lambda / \mu)P_{n-1} \\P_n=(\lambda / \mu)^nP_0 \\\\\sum_{n=0}^{29}P_n=P_0\sum_{n=0}^{29}\left(\frac{\lambda}{\mu} \right )^n=1 \\\\P_0=\frac{1-\rho }{1-\rho^{30}}$ $\displaystyle \\P_n=(\lambda / \mu)P_{n-1} \\P_n=(\lambda / \mu)^nP_0 \\\\\sum_{n=0}^{29}P_n=P_0\sum_{n=0}^{29}\left(\frac{\lambda}{\mu} \right )^n=1 \\\\P_0=\frac{1-\rho }{1-\rho^{30}}$

Now, writing the sum of a finite geometric series is

$P_n=\rho^nP_0=\frac{\rho^n(1-\rho)}{1-\rho^{30}}$ $\displaystyle P_n=\rho^nP_0=\frac{\rho^n(1-\rho)}{1-\rho^{30}}$

Now

$\rho=\frac{\lambda}{\mu}=\frac{5.6}{11}=0.509$ $\displaystyle \rho=\frac{\lambda}{\mu}=\frac{5.6}{11}=0.509$

therefore,

$1-\rho^{30}=1-0.509^{30}\approx 0.9999999984$ $\displaystyle 1-\rho^{30}=1-0.509^{30}\approx 0.9999999984$

Next,

$Pr\begin{Bmatrix} X_t>0 \end{Bmatrix}=1-P_0=\rho\approx 0.509$ $\displaystyle Pr\begin{Bmatrix} X_t>0 \end{Bmatrix}=1-P_0=\rho\approx 0.509$

$\\EX_t=\sum_{n=0}^{29}nP_n \\EX_t=\sum_{n=0}^{29}n\rho^n(1-\rho) \\EX_t\approx 1.037$ $\displaystyle \\EX_t=\sum_{n=0}^{29}nP_n \\EX_t=\sum_{n=0}^{29}n\rho^n(1-\rho) \\EX_t\approx 1.037$

Report an Error

Example Question #2 : Probability Models

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 X_t $\displaystyle X_t$ which represents the number of computers in the shop for repair at time $\displaystyle t$ months and calculate the EX_t $\displaystyle EX_t$.

Possible Answers:

$EX_t\approx 4.176$ $\displaystyle EX_t\approx 4.176$

$EX_t\approx 4.716$ $\displaystyle EX_t\approx 4.716$

$EX_t\approx 4.236$ $\displaystyle EX_t\approx 4.236$

$EX_t\approx 4.761$ $\displaystyle EX_t\approx 4.761$

$EX_t\approx 4.426$ $\displaystyle EX_t\approx 4.426$

Correct answer:

$EX_t\approx 4.176$ $\displaystyle EX_t\approx 4.176$

Explanation:

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

$X_t\epsilon\begin{Bmatrix} 1,2,3,4,...,m \end{Bmatrix}$ $\displaystyle X_t\epsilon\begin{Bmatrix} 1,2,3,4,...,m \end{Bmatrix}$

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

Let

$T_i=\text{time spent in state }i$ $\displaystyle T_i=\text{time spent in state }i$

$Pr{\begin{Bmatrix} T_i>t+s|T_i>s \end{Bmatrix}}=Pr\begin{Bmatrix} T_i>t \end{Bmatrix}$ $\displaystyle Pr{\begin{Bmatrix} T_i>t+s|T_i>s \end{Bmatrix}}=Pr\begin{Bmatrix} T_i>t \end{Bmatrix}$

Recall that the exponential distribution, and thus the T_i $\displaystyle T_i$ could have the density function,

$F_i(t)=\lambda_ie^{-\lambda_it}$ $\displaystyle F_i(t)=\lambda_ie^{-\lambda_it}$

Now formulate the Markov Process specifically for this problem.

$X_t\epsilon\begin{Bmatrix} 0,1,2,..., 32\end{Bmatrix}$ $\displaystyle X_t\epsilon\begin{Bmatrix} 0,1,2,..., 32\end{Bmatrix}$

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

X_t=i $\displaystyle X_t=i$ to X_t=i+1 $\displaystyle X_t=i+1$ or X_t=i-1 $\displaystyle X_t=i-1$

Now calculate the rates up and down.

$\\\lambda=5.9 \\\mu=7.3$ $\displaystyle \\\lambda=5.9 \\\mu=7.3$

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.

$[\text{Rate In}]=[\text{Rate Out}]$ $\displaystyle [\text{Rate In}]=[\text{Rate Out}]$

$\\\lambda P_0=\mu P_1 \$\mu +\lambda)P_1=\lambda P_0+\mu P_1 \\(\mu +\lambda)P_2=\lambda P_2+\mu P_3\\ \begin{matrix} .\\ .\\ . \end{matrix} \\\mu P_{32}=\lambda P_{31}$ \(\displaystyle \\\lambda P_0=\mu P_1 \\(\mu +\lambda)P_1=\lambda P_0+\mu P_1 \\(\mu +\lambda)P_2=\lambda P_2+\mu P_3\\ \begin{matrix} .\\ .\\ . \end{matrix} \\\mu P_{32}=\lambda P_{31}$

Solving with

$\sum P_i=1$ $\displaystyle \sum P_i=1$

$EX_t=\sum iP_i$ $\displaystyle EX_t=\sum iP_i$

Continuing in this fashion results in the following.

$\\P_n=(\lambda / \mu)P_{n-1} \\P_n=(\lambda / \mu)^nP_0 \\\\\sum_{n=0}^{32}P_n=P_0\sum_{n=0}^{32}\left(\frac{\lambda}{\mu} \right )^n=1 \\\\P_0=\frac{1-\rho }{1-\rho^{33}}$ $\displaystyle \\P_n=(\lambda / \mu)P_{n-1} \\P_n=(\lambda / \mu)^nP_0 \\\\\sum_{n=0}^{32}P_n=P_0\sum_{n=0}^{32}\left(\frac{\lambda}{\mu} \right )^n=1 \\\\P_0=\frac{1-\rho }{1-\rho^{33}}$

Now, writing the sum of a finite geometric series is

$P_n=\rho^nP_0=\frac{\rho^n(1-\rho)}{1-\rho^{30}}$ $\displaystyle P_n=\rho^nP_0=\frac{\rho^n(1-\rho)}{1-\rho^{30}}$

Now

$\rho=\frac{\lambda}{\mu}=\frac{5.9}{7.3}=0.808$ $\displaystyle \rho=\frac{\lambda}{\mu}=\frac{5.9}{7.3}=0.808$

therefore,

$1-\rho^{33}=1-0.808^{33}\approx 0.9991198131$ $\displaystyle 1-\rho^{33}=1-0.808^{33}\approx 0.9991198131$

Next,

$Pr\begin{Bmatrix} X_t>0 \end{Bmatrix}=1-P_0=\rho\approx 0.808$ $\displaystyle Pr\begin{Bmatrix} X_t>0 \end{Bmatrix}=1-P_0=\rho\approx 0.808$

$\\EX_t=\sum_{n=0}^{32}nP_n \\EX_t=\sum_{n=0}^{32}n\rho^n(1-\rho) \\EX_t\approx 4.176$ $\displaystyle \\EX_t=\sum_{n=0}^{32}nP_n \\EX_t=\sum_{n=0}^{32}n\rho^n(1-\rho) \\EX_t\approx 4.176$

Report an Error

Example Question #3 : Probability Models

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 X_t $\displaystyle X_t$ which represents the number of computers in the shop for repair at time $\displaystyle t$ months and calculate the EX_t $\displaystyle EX_t$.

Possible Answers:

$EX_t\approx 0.999$ $\displaystyle EX_t\approx 0.999$

$EX_t\approx 0.989$ $\displaystyle EX_t\approx 0.989$

$EX_t\approx 0.997$ $\displaystyle EX_t\approx 0.997$

$EX_t\approx 1.989$ $\displaystyle EX_t\approx 1.989$

$EX_t\approx 1.999$ $\displaystyle EX_t\approx 1.999$

Correct answer:

$EX_t\approx 0.999$ $\displaystyle EX_t\approx 0.999$

Explanation:

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

$X_t\epsilon\begin{Bmatrix} 1,2,3,4,...,m \end{Bmatrix}$ $\displaystyle X_t\epsilon\begin{Bmatrix} 1,2,3,4,...,m \end{Bmatrix}$

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

Let

$T_i=\text{time spent in state }i$ $\displaystyle T_i=\text{time spent in state }i$

$Pr{\begin{Bmatrix} T_i>t+s|T_i>s \end{Bmatrix}}=Pr\begin{Bmatrix} T_i>t \end{Bmatrix}$ $\displaystyle Pr{\begin{Bmatrix} T_i>t+s|T_i>s \end{Bmatrix}}=Pr\begin{Bmatrix} T_i>t \end{Bmatrix}$

Recall that the exponential distribution, and thus the T_i $\displaystyle T_i$ could have the density function,

$F_i(t)=\lambda_ie^{-\lambda_it}$ $\displaystyle F_i(t)=\lambda_ie^{-\lambda_it}$

Now formulate the Markov Process specifically for this problem.

$X_t\epsilon\begin{Bmatrix} 0,1,2,..., 23\end{Bmatrix}$ $\displaystyle X_t\epsilon\begin{Bmatrix} 0,1,2,..., 23\end{Bmatrix}$

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

X_t=i $\displaystyle X_t=i$ to X_t=i+1 $\displaystyle X_t=i+1$ or X_t=i-1 $\displaystyle X_t=i-1$

Now calculate the rates up and down.

$\\\lambda=2.2 \\\mu=4.4$ $\displaystyle \\\lambda=2.2 \\\mu=4.4$

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.

$[\text{Rate In}]=[\text{Rate Out}]$ $\displaystyle [\text{Rate In}]=[\text{Rate Out}]$

$\\\lambda P_0=\mu P_1 \$\mu +\lambda)P_1=\lambda P_0+\mu P_1 \\(\mu +\lambda)P_2=\lambda P_2+\mu P_3\\ \begin{matrix} .\\ .\\ . \end{matrix} \\\mu P_{23}=\lambda P_{22}$ \(\displaystyle \\\lambda P_0=\mu P_1 \\(\mu +\lambda)P_1=\lambda P_0+\mu P_1 \\(\mu +\lambda)P_2=\lambda P_2+\mu P_3\\ \begin{matrix} .\\ .\\ . \end{matrix} \\\mu P_{23}=\lambda P_{22}$

Solving with

$\sum P_i=1$ $\displaystyle \sum P_i=1$

$EX_t=\sum iP_i$ $\displaystyle EX_t=\sum iP_i$

Continuing in this fashion results in the following.

$\\P_n=(\lambda / \mu)P_{n-1} \\P_n=(\lambda / \mu)^nP_0 \\\\\sum_{n=0}^{23}P_n=P_0\sum_{n=0}^{23}\left(\frac{\lambda}{\mu} \right )^n=1 \\\\P_0=\frac{1-\rho }{1-\rho^{24}}$ $\displaystyle \\P_n=(\lambda / \mu)P_{n-1} \\P_n=(\lambda / \mu)^nP_0 \\\\\sum_{n=0}^{23}P_n=P_0\sum_{n=0}^{23}\left(\frac{\lambda}{\mu} \right )^n=1 \\\\P_0=\frac{1-\rho }{1-\rho^{24}}$

Now, writing the sum of a finite geometric series is

$P_n=\rho^nP_0=\frac{\rho^n(1-\rho)}{1-\rho^{30}}$ $\displaystyle P_n=\rho^nP_0=\frac{\rho^n(1-\rho)}{1-\rho^{30}}$

Now

$\rho=\frac{\lambda}{\mu}=\frac{2.2}{4.4}=0.5$ $\displaystyle \rho=\frac{\lambda}{\mu}=\frac{2.2}{4.4}=0.5$

therefore,

$1-\rho^{24}=1-0.5^{24}\approx 0.9999999404$ $\displaystyle 1-\rho^{24}=1-0.5^{24}\approx 0.9999999404$

Next,

$Pr\begin{Bmatrix} X_t>0 \end{Bmatrix}=1-P_0=\rho\approx 0.5$ $\displaystyle Pr\begin{Bmatrix} X_t>0 \end{Bmatrix}=1-P_0=\rho\approx 0.5$

$\\EX_t=\sum_{n=0}^{23}nP_n \\EX_t=\sum_{n=0}^{23}n\rho^n(1-\rho) \\EX_t\approx 0.999$ $\displaystyle \\EX_t=\sum_{n=0}^{23}nP_n \\EX_t=\sum_{n=0}^{23}n\rho^n(1-\rho) \\EX_t\approx 0.999$

Report an Error

View Tutors

Angela
Certified Tutor

American University, Bachelor of Science, Political Science and Government. Harvard University, Master's/Graduate, Sustainabi...

View Tutors

Molly
Certified Tutor

University of Southern Indiana, Master's/Graduate, Administration and Leadership .

View Tutors

Asha
Certified Tutor

Durham Technical Community College, Associate in Arts, Early Childhood Education. Wake Technical Community College, Bachelor ...

All Math Modeling Resources

20 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

English Tutors in Los Angeles, Math Tutors in San Francisco-Bay Area, Reading Tutors in San Diego, Biology Tutors in Chicago, ISEE Tutors in Atlanta, Reading Tutors in Philadelphia, SAT Tutors in Houston, Chemistry Tutors in Dallas Fort Worth, Spanish Tutors in Seattle, Math Tutors in Philadelphia

Popular Courses & Classes

SAT Courses & Classes in San Diego, SSAT Courses & Classes in Denver, SAT Courses & Classes in Phoenix, SSAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Philadelphia, ACT Courses & Classes in Washington DC, MCAT Courses & Classes in Washington DC, SAT Courses & Classes in Atlanta, MCAT Courses & Classes in Atlanta, ACT Courses & Classes in Miami

Popular Test Prep

SAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Seattle, ACT Test Prep in Washington DC, GRE Test Prep in Phoenix, GRE Test Prep in Philadelphia, MCAT Test Prep in Philadelphia, SSAT Test Prep in San Diego, ISEE Test Prep in Houston, SSAT Test Prep in New York City, ACT Test Prep in Miami

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Math Modeling : Stochastic Models

Study concepts, example questions & explanations for Math Modeling

All Math Modeling Resources

Example Questions

Example Question #1 : Probability Models

Example Question #2 : Probability Models

Example Question #3 : Probability Models

All Math Modeling Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: