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Flashcards: Math Modeling

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 which represents the number of computers in the shop for repair at time months and calculate the EX_t .

$EX_t\approx 4.236$

$EX_t\approx 4.176$

$EX_t\approx 4.761$

$EX_t\approx 4.426$

$EX_t\approx 4.716$

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Flashcards: Math Modeling

Your answer:

$EX_t\approx 4.236$

Your answer:

$EX_t\approx 4.761$

Your answer:

$EX_t\approx 4.426$

Your answer:

$EX_t\approx 4.716$

Correct answer:

$EX_t\approx 4.176$

Explanation:

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

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

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

Let

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

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

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

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

Now formulate the Markov Process specifically for this problem.

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

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

X_t=i to X_t=i+1 or X_t=i-1

Now calculate the rates up and down.

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

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

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.

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

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

Solving with

$\sum P_i=1$

$EX_t=\sum iP_i$

Continuing in this fashion results in the following.

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

Now, writing the sum of a finite geometric series is

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

Now

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

therefore,

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

Next,

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

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

Flashcards: Math Modeling

In a meadow there are two types of wild flowers growing, a green floor crawler and yellow daisies. The more desirable flower is the yellow daisy as they can be picked and sold for bouquets. The yellow daisies are also the more slow growing flower of the two. The green floor crawlers grow more rapidly and consume more of the meadow land. Yellow daisies compete with the green flower crawlers by growing taller and shading the crawlers new growth. The yellow daisies are also more resistant to certain types of bugs. Can these two types of flowers coexist on one portion of the meadow, or will one over run the other, driving it to extinction?

Both flowers will thrive.

The Green Floor Crawler will go extinct.

The Yellow Daisies will go extinct.

The steady state analysis cannot answer the question.

Both flower will coexist.

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Flashcards: Math Modeling

Your answer:

Both flowers will thrive.

Your answer:

The Green Floor Crawler will go extinct.

Your answer:

The Yellow Daisies will go extinct.

Your answer:

Both flower will coexist.

Correct answer:

The steady state analysis cannot answer the question.

Explanation:

Since there is competition between the two differing species of flowers, the growth rate function is,

g(P)=rP-aP^2

where is the intrinsic growth rate and a<<r measures the strength of the limitations of resources.

Now identify the question as a dynamic model in steady state.

In other words,

$\\f_1(x_1,...,x_n) \\f_2(x_1,...,x_n)$

From here, identify what is known about the problem in mathematical terms.

$\\D=\text{Yellow Daisy Population} \\C=\text{Green Floor Crawler Population} \\g_D=\text{growth rate of yellow daisies} \\g_C=\text{growth rate of green floor crawler population} \\c_D=\text{loss due to competition for yellow daisies} \\c_C=\text{loss due to competition for green floor crawler}$

$\\g_D=r_1D-a_1D^2 \\g_C=r_2C-a_2C^2 \\c_D=b_1CD \\c_C=b_2CD \\D\geq0, C\geq0 \\r_1,r_2,a_1,a_2,b_1, b_2 \text{are positive reals}$

To answer the question determine whether $C\rightarrow0$ or $D\rightarrow0$ .

To formulate the model let

$\\x_1=D \\x_2=C \\\begin{Bmatrix} (x_1,x_2):x_1\geq0, x_2\geq0 \end{Bmatrix}$

where the steady state equations are as follows.

$\\\\r_1x_1-a_1x_1^2-b_1x_1x_2=0 \\r_2x_2-a_2x_2^2-b_2x_1x_2=0$

For the particular problem, the points on interests are at those where the two functions intersect.

$\\(0,0) \\\left( 0,\frac{r_2}{a_2}\right) \\\left(\frac{r_1}{a_1},0\right)$

and the intersection of

$\\a_1x_1+b_1x_2=r_1 \\b_2x_1+a_2x_2=r_2$

Using Cramer's rule to solve results in the following.

$\\x_1=\frac{r_1a_2-r_2b_1}{a_1a_2-b_1b_2} \\\\\\x_2=\frac{a1_r2-b_2r_1}{a_1a_2-b_1b_2}$

and when the question of coexistence is ask it is found that,

$\frac{r_2}{a_2}<\frac{r_1}{b_1}$ and $\frac{r_1}{a_1}<\frac{r_2}{b_2}$

Which after further examination learns to the conclusion that the steady state analysis cannot answer the question.

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Flashcards: Math Modeling

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