$.$ Consider a loan office processing applications. A loan application consists of $4$ stages: Credit Check,

Preparation of the Loan, Pricing and Disbursement. Applications arrive with exponential interarrival

times that has a mean of $1\mathrm{.}25$ hours, and each stage of the application has an exponential process time

with mean of $1$ hour. There are $4$ employees in the office. We assume the work is serial, each task is

assigned a single employee and an applicant has to go through all employees.

a$)$ Simulate the loan office for $120$ hours and collect statistics on the total time it takes to process a

loan application.

b$)$ The expected total time it takes to process an application is $20$ hours. $($This can be computed

using Queueing Theory for serial exponential queues.$)$ This means that if we run the simulation

for an infinitely long amount of time, the average time spend in the system should be $20$ hours.

When you ran your simulations for Question $1,$ you must have observed a lower total wait time.

Why is the result lower than $20?$

c$)$ The difference between the theoretical and simulation results could be due to random fluctuations.

Increase the number of replications to $100$ to account for the effect of randomness in the input

data. Does this improve the accuracy of your results?

d$)$ The results may also be biased because $120$ hours is not long enough. Increase the length of the

simulation to $12000$ hours and run a single replication. Is the problem resolved?

e$)$ The problem could be the way we initialized our simulation. $($Our model starts running with an

empty queue and idle servers.$)$ Try warming up your model, by assigning the

first $30$ hours as the warm$-$up period $($in the Run Setup$)$ and running for a replication length of

$120$ hours $($the statistics will be collected for the last $90$ hours$).$ How does the warm$-$up period

affect the results?