A) Utilization is defined as process time/arrival rate (make sure units are uniform). Therefore 10mins/5mins. Utilization rate 200% or 2.0. From this we can extrapolate that there should be at least another employee. At worst we want a utilization rate of 100%, meaning the employee has to be processing 100% of the time.
B) Exponential distribution describes the time in between events in process in which events occur continuously and independently at a constant average rate. I think we are missing some information here? Specifically how long is this server's shift. I will assume that this is an 8 hour work day. The mean of which would be 4 hours in. If we check in at 4.0 hrs and and 12 customers come per hour (60/5), a total of 48 customers will have come in. One customer is served every 10 minutes, 4hrs is 240 mins. 24 customers would have been served. Thusly 24 customers will be waiting at midday. 48 waiting at the end of the day and if we close exactly then we will have to turn them away even though the first at line at that point will have been waiting 5x48minutes or 4 hours.
C) The first customer waits 0 mins. The 2nd waits 5. The 3rd waits 10 (came in at min 10 and served at min 20. The 4th waits 15 (came in at 15, served at 30). The 5th waits 20. 0+5+10+15+20=50/5 the average wait time for the first 5 customers is 10 minutes. Of course, this all assumes superhuman efficiency from our one employee and no unforeseen events so it's not exactly realistic business math.
Do you see the trend here? If utilization rate is 2.0, we know wait time for customer x is x(y) where x is customer number and y is the difference between process rate and arrival rate. So customer 120 would be 120(5min) wait time. It is unlikely anyone would wait this long, so it's possible your test could include churn or balk rate. For example, if customers have a limit of 30 mins waiting and will leave after having waited that long. In this original problem if a customer balks at 30 mins, there would be a maximum of 6 customers in line at a time because service for customer b is always 5 minutes behind customer a and 30/5=6.
What if utilization rate is .5? For example process in 5 mins and arrive every 10 mins. Then we do, for customer 20 for example, 20(-5). -100 mins. This essentially means the worker has had 100 mins inactive by the 20th customer. To check our work we could say ok 20(10)=200mins. Utilization rate is .5 so 200(.5)=100 so yes that's 100 mins unutilized.
