An Investigation into Transport Protocols and Data Transport ...

7.1. Metrics 139
assuming equilibrium such that w i (k) = w i (k + 1) and T i (k) = T i (k + 1):
w i (k) = β i w i (k) + α i T i (k) (7.6)
Assuming that queuing delay is negligible and where α i and β i are constant
values, define that the expectation values of w i and T i are E(w i ) and
E(T i ), Equation 7.6:
E(w i ) = α iE(T i )
λ i (1 − β i )
(7.7)
where λ i is the probability that a congested system will result in immediate
back-off of flow i. Similar to [CJ89], define that cwnd fairness as the
ratio of cwnd between the two AIMD TCP flows, i and j, such that:
w i
w j
= min
( αi
λ i (1−β i )
α j
,
λ j (1−β j )
α j
)
λ j (1−β j )
α i
λ i (1−β i )
(7.8)
where the quantity λ j
λ i
defines the synchronicity between the two flows as
defined by the ratio of drops experienced between the two competing flows.
Using the BDP (Equation 4.1) to relate goodput and latency for a single
flow, the fairness, F i,j between two flows, i and j with α i = α j and β i = β j
(i.e. both flows have the same AIMD parameters) is:
F i,j = ¯X
( ( ) 2
i λ j RT Tj
= min
, λ ( ) ) 2
i RT Ti
¯X j λ i RT T i λ j RT T j
(7.9)
Therefore, the fairness between two competing Standard TCP flows depends
upon the synchronicity and the square of the ratio between the latencies
of the flows. As the growth rate of cwnd of the short RTT flow is greater
than that of the long RTT flow, the increased growth rate in time results in