An Investigation into Transport Protocols and Data Transport ...

5.4. Analysis of AIMD Congestion Control 105
packets traversing along the network, plus the packets buffered at the queue.
Given this, the time taken to just fill the pipe after the k th epoch can
be determined by considering the rate at which packets are put onto the
network:
(βP ) + α T (t 1(k) − t 0 (k)) = BT
t 1 (k) − t 0 (k) = T α (BT (1 − β) − βq) (5.2)
At period t 1 (k) to t 2 (k), all subsequent packets are buffered at the queue
such that:
α
T (t 2(k) − t 1 (k)) = q
t 2 (k) − t 1 (k) = T q
α
(5.3)
And the time between congestion epochs is defined as,
α
T (t 2(k) − t 0 (k)) = P − βP
t 2 (k) − t 0 (k) = T α (P − βP ) (5.4)
Defining the provision parameter γ = q max /BT and that 1/1+γ ≤ β, i.e.
the queue empties on backoff, the throughput, y, can be calculated as the
area under the graph between t 0 (k) and t 2 (k) where the area of the triangle
before the queue begins to fill, (t 1 (k) − t 0 (k) × βB, is subtracted from the
area for the entire epoch, (t 2 (k) − t 0 (k)) × B.
y = B2 T 2 (β − 1)(1 + γ)
α
− B2 T 2 (βγ + β − 1)
2α
(5.5)
Normalising by the total number of packets sent out (i.e. the total area