A production system is as shown below, with the capacity rates as given in units per hour, potential. An average of $150$ customers arrive at a university Starbucks coffee shop every day between $7$ am and $7$ pm $.$ It takes, on average,

$6$ minutes to serve a customer. Treat the service times as exponentially distributed, and the arrivals as

Poisson distributed. There are two baristas working at all times.

Which model best matches the situation?

Model used:

arrival rate:

per

$($units$)$

$($units$)$ population:

$($units$)$ servers:

population:

$("$infinite$"$ or a number$)$

service rate: $$

per

b$.$ What is the average number of customers in the waiting line, not counting the one being served?

c$.$ What is the average time spent in the system by a customer?

$($units$)$

d$.$ What is the probability that there will be less than $3$ customers in the system?

e$.$ What would the service rate have to be for the time waiting in the line to be less than $3$ minutes on average? $($Hint: use Goal seek$).$ Kira Zrust is irritated by staffing issues. She doesn't want to increase her labor costs, but she is starting to think

that she is losing customers because of too much waiting. So the question is$,$ how many lanes should she have open?

Right now customers arrive at a rate of $93$ per hour. Each member of the staff can serve $16$ people per hour.

A staff member costs $$24$ per hour. She estimates cost of waiting before service per customer per hour to be $$200.$

Model used: $$

arrival rate:

per hour

service rate: $$

per hour

cost per server per hour:

cost of waiting per customer per hour:

a$.$ What is the minimum number of servers needed to make this system stable?

b$.$ What is the system cost with the number of servers in question a$?$

c$.$ What is the optimal number of staff members?

d$.$ what is the total system cost with the optimal number of servers?

Which station will control the system rate?

What will be the effective capacity of station A$?$

If the system runs at a rate of $6\mathrm{.}5$ per hour,

What is utilization of station D$?$

What is the efficiency of station D$?$

What is the average processing time per unit in minutes at each of the stations?