Question $2.(45$ marks$)$ At an assembly line, jobs arrive to the facility with an inter$-$arrival time exponential $(2)$ minutes. There are two types of jobs, each job is type I with probability $0\mathrm{.}4$ and type II with probability $0\mathrm{.}6.$ Each job must pass through two production stages: stage $1$ and stage $2.$ Stage $1$ has a single queue and takes an exponential $(2)$ minutes to complete and $2$ workers are available to perform stage $1.$ After completing stage $1,$ the job immediately passes to stage $2.$ Stage $2$ has two machines, machine $1$ and machine $2,$ working on the jobs. Each machine at stage $2$ takes a triangular $(0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}5)$ minutes to complete a job. The machines have separate lines, and the jobs will be routed to the line with the fewest number of jobs waiting in queue, ignoring whether any job is in service. If there is a tier, it will be routed to machine $1.$ In both stages $1$ and $2,$ type II jobs have a higher priority over type I jobs. After completing stage $2,$ each job is inspected. Inspection takes an exponential $(1\mathrm{.}2)$ minutes, and $1$ repair worker is available to perform inspection. After inspection, $5\%$ of the jobs must be returned to stage $1$ for reprocessing. The rest $95\%$ of the jobs are finished products and leave the system. The assembly line is running $12$ hours a day.

a$.(15$ marks$)$ Run the simulation for $60$ days and replication number $10.$ Report the following performance indicators, both mean and $95\%$ confidence interval: the cycle time $($or$)$ total time for each type of job. $($Save your model as q$2$a$.$doe$)$ Interpret the meaning of $95\%$ confidence interval.

b$.(5$ marks$)$ What percentage of time the worker at stage $1$ is busy $($or utilization$)?$

c$.(10$ marks$)$ Now suppose that a consultant recommends the assembly line to use a single line for the two machines in stage $2.$ Re$-$run the simulation. What is the average cycle for each type of jobs for this setting? $($Save your model as q$2$c$.$doe$)$

d$.(10$ marks$)$ Now that at stage $1,$ due to the space constraint the queue length for stage $1$ cannot exceed $12$ jobs. If there is already $12$ jobs in the queue at stage $1,$ the arriving job will be discarded. Under this modification, re$-$run. Report the following performance indicators, both mean and $95\%$ confidence interval: the cycle time $($or$)$ total time for each type of jobs. $($Save your model as q$2$d$.$doe$)$

e$.(5$ marks$)$ What if in stage $1$ and the inspection stage, they share the common pool of three workers while keep the service time distributions the same as before? Re$-$run the simulation. What is the average cycle for each type of jobs for this setting? $($Save your model as q$2$e$.$doe$)$

$($ the questions in c$.,$ d$.,$ e$.$ are based on a$.)$