Queuing models - Solving problem with different service time 1/μ

#1
Hi everyone
I'm struggling with a problem because the service time is different depending on the number of tickets purchased.
Following is the case:

"The owner wants to reduce the waiting and serving time for customers to buy tickets knowing that this has a negative effect on the time available to them to consume at the food concessions. He knows that the configuration of the store must take into account the needs of the customers. The manager has compiled data from his other branches and is ready to evaluate different options. The peak period is Friday evening between 6:00 p.m. and 10:00 p.m. On average, 520 customers are served during this period. It is therefore necessary to evaluate the options so as to be able to serve these customers when the traffic is maximum. The options under consideration are as follows: (

  1. Have a single queue and ask the first person in line to go to the next available counter. The manager knows that this option would require five counters to be open.
  2. Have two queues: one for a counter clerk serving customers needing more than 2 tickets and the other line of four counter clerks for customers needing less than 2 tickets.
  3. Have a single queue for five automated teller machines to be used by all customers. According to the study, 10% of customers buy more than 2 tickets on Friday evening.
The average service time for customers requiring 2 tickets or less and using the services of a counter clerk is 90 seconds, while this average time is 3 minutes and 30 seconds for customers requiring more than 2 tickets. As for the use of an automated counter, the average duration is 96 seconds for those who buy 2 tickets or less and 2 minutes and 30 seconds for the others.

If you were the manager, which option would you choose? Why ? What assumptions did you make in your analysis? Explain the disparity between the results of the various options. To do this, specify all the calculations and all the elements used in your analysis. To make sure the owner understands your recommendation, present your comparison time in minutes for each of the options. part 2 The owner also needs your advice for another branch where 80% of customers buy more than 2 tickets. He absolutely wants to set up a queue for a certain number of counters serving customers who need 2 tickets or less and another queue for other counters who would take care of other customers (those with more than 2 tickets). He knows that the average length of service is the same as that of the study. This branch also has the same peak period and the same traffic. Unfortunately, the space is too limited to install automated counters, but it allows for a total of a maximum of ten counters. Each category of customers must have at its disposal at least one teller. How many tellers of each type do you suggest? Present him with various options to justify your recommendation.