FAQ: Why does the effective service rate f(x) go to zero when x = 0 in Example 2.10 on queuing systems?
From FBSwiki
(Contributed by Richard Murray, 10 Oct 09)
Q: The formula for f(x) which scales μmax is x / (x + 1). This is zero when x = 0 (no queue) and 1 when x goes to infinity. Why is this the right model?
A: The model used by Agnew [5] is that the rate at which jobs are processed is linear in the queue length when the length is small, and saturates and the maximum service rate μmax. At the extreme where there are no jobs on the queue, there is no need to process incoming requests and Agnew's assumption was the more processing would be applied as the queue got longer, until it saturates for x very large.
Notice that the term x + 1 in the denominator means that the service rate very rapidly reaches it maximum as the queue increases. If there is 1 job on the queue then the service rate is 0.5μmax, 3 jobs gives a rate of 0.75μmax and 9 jobs gives a rate of 0.9μmax.
