Symbol | Represents |
Q(t) | Queue length at time t |
A(t) | Cumulative arrivals to the queue at time t |
D(t) | Cumulative departures from the queue at time t |
The maximum service rate | |
(t) | The service rate at time t |
(t) | Arrival rate at time t |
Time label | Significance |
t0 | The first point at which (t) equals the maximum service rate, |
t1 | The point where (t) is a maximum |
t2 | The point where (t) has decreased back down to |
t3 | The point where the queue has gone, and so (t) = (t) |
The above details tend to imply that a graph of (t) is quadratic, and so would be differentiable twice at or near t1. This means that (t) has a taylor expansion about the point t1:
The length of the queue is the area between the curves of (t) and (t), i.e.:
By the using similar substitutions to those used in the previous derivation we can simplify this to: