ECE $5371--$ Engineering Analysis Assgn. #$6$ Due: April $25,2024.$

$1.$ Question: Parts arrive at station #$1$ from a conveyor belt with a Poisson distribution at a

mean rate of $1-$per$-$second. Here they mesh with parts arriving from another conveyor belt

at a steady and constant rate of $1$ part in $1\mathrm{.}5$ seconds, to form a more complicated assembly

and emerge from Station #$1.$ Assume the complicated assembly can be completely by

Station #$1$ instantaneously. Station #$1$ is known to have a failure rate of $0\mathrm{.}01$ per second

that is characterized by the exponential distribution and stays $$down$$ for $5$ seconds. Based

on the above, write a Monte Carlo simulator of this scenario that can then yield the discrete

event average output rate for the assembled parts from Station #$1.$

Hints: $($a$)$ Poisson distribution is given as: f$($k$)=\backslash$mu k

exp$(-\backslash$mu $)/$k$!,$ where $\backslash$mu is the mean. You

could generate a histogram that mimics the Poisson distribution by using either of the two

methods given next. $($a$)$ The Sequential Search Algorithm, or the $($b$)$ Exponential Random

Number Algorithm are given below.

The two algorithms will yield near$-$identical results if you take a lot of sample points. Use

either algorithm to generate number of arrivals at Station #$1$ in given time period $\backslash$Delta T$.$

Sequential Search Algorithm

Step $1->$ Generate a random number $"$r$"$ and set i $=1.$

Step $2->$ If $"$r$"<$ Pi $,$ return N$=$ i$-1.$

Otherwise increase i by $1$ and repeat step $2.$

Exponential Random Number Algorithm

Step $1->$ Set r $=$ e $-\backslash$mu

$,$ N $=0$ and s $=1.$

Step $2->1.$ Generate $"$r$"($random number$)$ and put s $=$ s r$.$

Step $3->$ If s $>$ r $,$ set N $=$ N$+1$ and go to $1.$ Otherwise return N$.$

Results for the distribution comparing the Poisson formula with the Algorithms $($both

yielded the same results, so only $1$ figure is shown$)$ are given below:

Histogram from the Algorithms $(\backslash$mu $=15$ and $\backslash$mu $=6).$

Plot for the analytic formula for the Poisson Distribution Histogram $(\backslash$mu $=15$ and $\backslash$mu $=6).$

$($a$)$ For the $$exponential distribution of the failure rate$,$ simply obtain the relation

between the $$time$-$to$-$failure$$ and a random number.