Wait Time Analysis at a Restaurant

One more!

Burger Dome is a fast-food restaurant currently evaluating its customer service. In its current operation, an employee takes a customer’s order, tabulates the cost, receives payment from the customer, and then fills the order. Once the customer’s order is filled, the employee takes the order of the next customer waiting for service. Assume that time between each customer’s arrival is an exponential random variable with a mean of 1.35 minutes. Assume that the time for the employee to complete the customer’s service is an exponential random variable with a mean of 1 minute.

We have two exponential distributions at play here, and with them we are modelling two events: customer arrival (DISTRIBUTION_A) and customer service by the employee (DISTRIBUTION_B). Since both are exponential, I will define the customer’s waiting time as difference between two: DISTRIBUTION_A - DISTRIBUTION_B. With that logic, the $=0.35$.

1 Question A

What is the average wait time experienced by a customer?

The $=0.35$, as defined in the introduction.

2 Question B

What is the longest wait time experienced by a customer?

This one is tricky, since in theory, the waiting time has range between 0 and $\mathrm{\infty }$, that is $x\in [0,\mathrm{\infty }]$. But, simulating workday 20.000 times led me to conclusion that I can be 95% sure that the max waiting time will be in the range of 1.90 to 3.65 minutes. Do note that with more simulations (i.e. above 20k) I get greater range of maximum values (but the confidence interval stays roughly the same).

3 Question C

What is the probability that a customer waits more than 2 minutes?

Around 0.33%. If anybody knows this kind of restaurant, please let me know! :joy: