Operations Management - Queue calculations

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** Please see the attached Word & Excel files. **

Thanks
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1. The check-in line at Alpha Airlines has two agents who serve all passengers during peak hours. Arrivals to the check-in counter are Poisson with a mean of 50 passengers per hour. Approximately, thirty percent of the arrivals are Priority Class (e.g., First Class, Gold members, etc) while the remainder holds coach tickets. The time that a passenger spends at the ticket check-in counter with an agent is exponentially distributed with a mean of 1.44 minutes. Alpha promises that if a passenger waits in line for more than 5 minutes, a coach passenger will be awarded 500 frequent flyer miles while a Priority Class passenger will be awarded 1000 miles. Currently, all passengers go through the same queue for check-in.

a) How many total miles are awarded per hour on average currently?
This is M/M/m with l = 50/hour and 1/m = 1.44 minutes= 1.44/60 hours. Therefore, we use the "MMm" spreadsheet and set cells E2-D4 are 50, 41.667 and 2.
From spreadsheet, we get
Utilization =60.00%
p0 =Probability that the system is empty=0.25
Lq = Expected queue length =0.675
Ls = Expected number in system =1.875
Wq = Expected time in queue =0.0135 hours
Ws = Expected total time in system =0.0375 hours

Now we calculate the probability of waiting in queue for more than 5 minutes i.e. 5/60 hours. Change the value in cell E22 to =0.0833 hour
Prob(waiting in queue > t) = Prob(wait) * e-(mm-l)t
P(Wait>5 minutes) = 0.02798
Since 30% of arrivals are priority, the total miles awarded per hour on average is
=30%*50*0.02798*1000+70%*50*0.02798*500
=909.33 miles

b) Management is considering the creation of two separate lines, one for each of the two classes of passengers. In this situation, each check-in line will have one agent who will serve their respective passengers; line switching will not be allowed. Find the average total miles awarded hourly under this configuration.

Here, we will have two separate systems for the two types of customers.

For priority customers:
The system is M/M/1 with l = 15/hour and 1/m = 1.44 minutes= 1.44/60 hours. Therefore, we use the "MMm" spreadsheet and set cells E2-D4 are 50, 41.667 and 1.
From spreadsheet, we get
Utilization =36.00%
p0 =Probability that the system is empty=0.64
Lq = Expected queue length =0.2025
Ls = Expected number in system =0.5625
Wq = Expected time in queue =0.0135 hours
Ws = Expected total time in system =0.0375 hours

Now we calculate the probability of waiting in queue for more than 5 minutes i.e. 5/60 hours. Change the value in cell E22 to =0.0833 hour
Prob(waiting in queue > t) = Prob(wait) * e-(mm-l)t
P(Wait>5 minutes) = 0.03901
Total miles ...