Q$1.$ Cuts$-$R$-$Us provides low$-$cost haircuts at a shopping center in Boise, Idaho. During the day, customers arrive at an average rate of $9$ per hour following an exponential distribution. After a customer is in a cosmetician$$s chair, it takes an average of $18$ minutes for the haircut to be completed, with the actual service time following a Poisson distribution.

a$.$ If there are $3$ cosmeticians on duty, how long will customers have to wait on average before receiving service and how many customers are typically waiting for service?

b$.$ If there are $4$ cosmeticians on duty, how long will customers have to wait on average before receiving service and how many customers are typically waiting for service?

c$.$ If there are $5$ cosmeticians on duty, how long will customers have to wait on average before receiving service and how many customers are typically waiting for service?

d$.$ If you managed this store, how many cosmeticians would you employ and why?

Solve it in Excel using the queuing theory problem that can be modeled using the M$/$M$/$c queuing system.

Problem Breakdown:

Arrival rate $($$\backslash$lambda$)$: Customers arrive at a rate of $9$ per hour.

Service rate $($$\backslash$mu$)$: Each cosmetician can serve at a rate of $1$ haircut per $18$ minutes or $=60/18=3\mathrm{.}33$ per hour.

Number of servers $($c$)$: There are $3$ cosmeticians$.$

Calculate on Excel:

U $-$ Utilization factor, $\%$ of the time that all servers are busy.

P$(0)-$ Prob. that there are no units $($zero$)$ in the system.

Lq $-$ Avg number of units in line waiting for service.

L $-$ Avg number of units in the system $($inline & being served$).$

Wq $-$ Avg time a unit spends in line waiting for service.

W $-$ Avg time a unit spends in the system $($inline & being served$).$

Pw $-$ Prob. that an arriving unit has to wait for service.

Pn $-$ Prob. of n units in the system