>Intuitively, this result doesn't seem obvious
It isn't obvious because human arrival times at checkout counters follow a Poisson distribution and their service times follow an Exponential distribution. The stated results follow immediately if you look at the cumulative density function for the exponential.
Stated another way, suppose service times followed a Uniform distribution. Then none of this would hold. But because they are Exponential, these results come into play. Intuitively, we think in terms of Uniform distribution. So your mind is saying, wait a minute, if there are 1000 people in a queue, they probably average a 10 to 15 service minute per person. But that's like saying if there are 1000 people in an office, they probably make 100k on average because that's about what you (might) make. In reality incomes follow a Pareto, so your janitors will take home 30k and your managers will pull in a couple mil. A similar sort of dynamic applies here with the exponential distribution. The key takeaway is: Service times are not uniform but exponential.
>Intuitively, this result doesn't seem obvious It isn't obvious because human arrival times at checkout counters follow a Poisson distribution and their service times follow an Exponential distribution. The stated results follow immediately if you look at the cumulative density function for the exponential.
Stated another way, suppose service times followed a Uniform distribution. Then none of this would hold. But because they are Exponential, these results come into play. Intuitively, we think in terms of Uniform distribution. So your mind is saying, wait a minute, if there are 1000 people in a queue, they probably average a 10 to 15 service minute per person. But that's like saying if there are 1000 people in an office, they probably make 100k on average because that's about what you (might) make. In reality incomes follow a Pareto, so your janitors will take home 30k and your managers will pull in a couple mil. A similar sort of dynamic applies here with the exponential distribution. The key takeaway is: Service times are not uniform but exponential.