Waiting Time - Exponential distribution

Question

Question: I go to a grocery store and when I check out, there is one line with one customer waiting to be served while the other line has two customers. Naturally I choose to stand in the line with one customer. It seems as though many times the longer line would have been the better choice: those two customers get served before the person in front of me on my line is finished being served, which gets me annoyed. Is my getting upset reasonable, or should I expect this to happen sometimes? How often?

I felt kind of confused by this question. How could the waiting time of the line with one customer is longer than that with two customers? I would appreciate it if someone could give me some hints.

Answer 1

How could the waiting time of the line with one customer is longer than that with two customers?

it could be longer because you are dealing with random variables, and the serving time of the customers is not fixed...it is represented by an exponential density with a certain mean, i.e. 5 minutes. Now, it could happens that the only customer in line 1 takes 4 minutes to be served and the two customers in line 2 takes less than 1 minute each... thus the first line is longer in time w.r.t. the second one.

Possible solution

Given some reasonable assumptions, as exponential distribution for the service time, independence and same average time of serving, the problem can be formalized as follows:

• T1: time to get served in the first line is a rv

$$X\sim\exp(\theta)$$

• T2: time to get served in the second line is a rv

$$Y\sim\text{Gamma}(2;\theta)$$

and you have to calculate

$$\mathbb{P}[X<Y]$$

can you proceed and conclude?

After some calculation, you get that the line with one customer is convienient in 75% of the cases but, in 25% of the times, the line with 2 customer is quickly than the other one.