Queuing theory: $(25$ Points$)$

Use JaamSim to create a simulation of the process described in example $6\mathrm{.}7$ in the book $$Business Process Modeling, Simulation and Design$.$ This is an example of a M$/$M$/$c$/\backslash$infty $/$N queue. Your job is to compare the analytical approach with the simulation approach. We are particularly interested in the remark made just before the example $6\mathrm{.}7$ is presented, which says:

Remark: The M$/$M$/$c$/\backslash$infty $/$N model is based on the assumption that the time an individual job or customer spends in the calling population outside the queuing system is exponentially distributed. However, it has been shown $($see Bunday and Scraton, $1980)$ that the expressions for P$0$ and Pn $($and consequently, those for L$,$ Lq$,$ W$,$ and Wq$)$ also hold in more general situations. More precisely, the time that a job or customer spends outside the system is allowed to have any probability distribution as long as this distribution is the same for each job or customer and the average time is $1/\backslash$lambda $.$ Note that these situations fall outside the class of birth$-$and$-$death processes into a G$/$M$/$c$/\backslash$infty $/$N model.

Use your Jaamsim simulation model to verify this remark with the data of example $6\mathrm{.}7.$ You must do it at least with three general distributions for the interarrival times. You must clearly explain and justify your results, just numerical values without any explanation will be considered as an empty answer. Also, you must document your implementation in JaamSim $($Example $6\mathrm{.}7$: The Machine Repair Process at PaperCo Inc.

PaperCo Inc. is a manufacturer of fine paper. Its most important product is a special, environmentally friendly type of bleached paper used for laser printers and copiers. PaperCo has one production facility consisting of three large paper$-$milling machines. These machines are expensive and need to be kept operational to the largest extent possible. At the same time, their complexity makes them sensitive to malfunctions. Currently, PaperCo has one repair team on call $24$ hours a day, $7$ days a week. However, management is contemplating increasing this to two teams and would like a perfor$-$ mance analysis of this double$-$team scenario. They already know the performance of the current process with one team. Each team is configured so that all competencies needed are available within the team, and having two teams working on the same machine provides no advantages. Therefore, it can be assumed that the two teams work inde$-$ pendently of one another on different machines. Furthermore, their service times are assumed to be identical and exponentially distributed with a mean of $4$ hours. Empirical data show that the time between malfunctions in each of the three machines when they are operating is exponential with a mean of $8$ hours. The machines are equally impor$-$ tant$,$ so broken$-$down machines are serviced in a FIFO sequence.

PaperCo wants to know the steady$-$state probabilities that zero, one, two, or three machines are nonoperational. They also want to determine the average time before a machine that breaks down is operational again and how long on average a broken machine has to wait before a service team starts to work on the problem.

The first step in analyzing this problem is to recognize that the PaperCo repair pro$-$ cess can be modeled as an M$/$M$/$c$/\backslash$infty $/$N system with c $=2$ servers $($the service teams$)$ and N$=3$ jobs in the calling population $($the machines$).$ Furthermore, the mean ser$-$ vice rate for each of the two teams is $\backslash$mu $=1/4=0\mathrm{.}25$ jobs per hour, and the failure rate for each machine in operation is $\backslash$lambda $=1/8=0\mathrm{.}125$ machines per hour. The rate diagram for the repair process is depicted in Figure $6\mathrm{.}16.($See also Figure $6\mathrm{.}15$ for the genera structure.$)$ It should be noted that this problem can be analyzed using the general approach for birth$-$and$-$death processes discussed in Section $6\mathrm{.}2\mathrm{.}3,$ although a differ$-$ ent method is used here.