Question

A new restaurant just opened-up in a town near you. The manager of this establishment has asked you to evaluate their drive-thru system queue. More specifically, the restaurant owner wishes to better understand the relationship between service rates and the resulting average amount of time a customer spends in the system. The manager has a choice in the number of servers working ranging from a minimum of 8 servers to a maximum of 15 servers (only consider integers values of servers). The restaurant's average service rate is a function of the number of servers working such that (number_of_servers*5 = average_customers_serviced_per_hour). Assuming that the restaurant meets M/M/1 model assumptions and has an average arrival rate of 37 customers/hour, what is the minimum number of servers required to meet the threshold maximum time of 5 minutes for average time a customer spends in the system?

How to solve this

Answer 1

Given,

Arrival Rate () = 37 customers per hour

Let service Rate =

It is mentioned that threshold of waiting time of system should maximum be 5 minutes

Average waiting time in the system ≤ 5 minutes

It is given that the restaurant follows M/M/1 assumptions.

Average waiting time in system, W

Therefore, Service Rate () ≥ 49 customers / hour

Given that

Number of servers × 5 = Service Rate

Number of servers × 5 ≥ 49

Number of servers ≥ 9.8

Therefore, minimum number of servers needed = 10

Therefore, 10 servers are needed to meet the threshold maximum time of 5 minutes for average time a customer spends in the system.