Unclear about Sakasegawa formula

Question

I'm referring to Sakasegawa's forumla for calculating average line length in a queuing system.

I don't understand the result intuitively. For example, our server is able to serve 10 customers per hour, and we have 8 arriving every hour. So we have 1 queue and 1 server, and a utilization of 0.8:

• I think to myself, "we're underutilized, and not running at full capacity, so there should be no line". I.e. average line length = 0
• But the actual answer from the formula is 3.2

How can we have 3 people lined up if we are able to process them faster than they come in?? I just don't see it intuitively.

Answer 1

Disclaimer: I first heard of queueing theory today :)

I think the key is that 8 customers per hour do not arrive at a steady rate of 1 every 7.5 minutes but are randomly distributed. Taking an extreme example say you have 16 people appear at the start of the day. Nobody else arrives for the next two hours.

You obviously average 8 customers per hour for the first two hours and you have at least 6 people standing in line for 1 full hour. So for the first two hours your average queue is bigger than 3 people.

The key intuition is that there are times where your server sits around doing nothing.

Answer 2

This formula is for an M/M/m queue which has exponential interarrival times and exponential service times and is an approximation for $m>1$.