The Mathis equation states that, for a small loss rate (less than 1%), t= he maximum achievable throughput of a TCP connection is limited by:

Rate <=3D (MSS/RTT)*(1 / sqrt{p})

where MSS is Maximum Segement Size

RTT is Round Trip Time as measured by TCP

p is the probability of packet loss

(Extract from Les Cottrell's '= Throughput versus loss')

An improved form of the above formula that takes into account the TCP in= itial retransmit timer and the Maximum TCP window size, and is generally mo= re accurate for larger (> 2%) packet losses, can be found in: . The form= ula is given below (derived from eqn 31 of Padhye et. al.):.

if w(p) < wmax

Rate =3D MSS * [((1-p)/p) + w(p) + Q{p,w{p}}/(1-p)] /

(RTT * [(w{p}<=
span style=3D"text-decoration: underline; ">1)](Q{p,w{p}}*G{p}*T0)/(=
1-p))

otherwise:

Rate =3D MSS * [((1-p)/p)+ wmax+Q{p,wmax}/(1-p)] /

(RTT * [0.25*wmax=
+((1-p)/(p*wmax)+2)] + (Q{p,wmax}*G{p}*T0)/(1-p)])

Where:

We have assumed the number of packets acknowledged by a received ACK is =
2 (this is b in the Padhye et. al. formula 31)

wmax is the maximum conge=
stion window size

w{p} =3D (2/3)(1 + sqrt{3*((1-p)/p) + 1} from eqn. 13 =
of Padhye et. al. substituting b=3D2

Q{p,w} =3D min{1,[(1-(1-p)3)**(1+(1-p)3)**(1-(1-p)(w-3))] /=

[(1-(1-p)w)]}

G{p} =3D 1+p+2*p2+4*p3+8*p4+16*p5+32*p6 from eqn 28 of Padhye et. al.

T0 =3D Initial retransmit timeout (typically this is suggested by RFCs 793=
and 1123 to be 3 seconds).

Wmax =3D Maximum TCP window size (typical d=
efault for Solaris 2.6 is 8192 bytes)

If you are tuning your hosts for best performance then also read Enablin= g High Performance Data Transfers on Hosts and TCP Tuning Guide for Distrib= uted Application on Wide Area Networks. Also The TCP-Friendly Website summa= rizes some recent work on congestion control for non-TCP based applications= in particular for congestion control schemes that maintain the arrival rat= e to at most some constant over the square root of the packet loss rate.

Recently, some of the authors of the initial TCP model have proposed aba= ndoning these closed-form "macroscopic" models. Reasons include that some o= f the original assumptions, such as sufficient buffer space in routers, are no longer tenable, and tha= t the models don't fit for promising new congestion control approaches such= as BBR.

*The macroscopic behavior of the TCP co=
ngestion avoidance algorithm,* Mathis, Semke, Mahdavi & Ot=
t in Computer Communication Review, 27(3), July 1997

*Modelling TCP throughput: A sim=
ple model and its empirical validation,* J. Padhye, V. Firoiu,=
D. Townsley and J. Kurose, in Proc. SIGCOMM Symp. Communications Architect=
ures and Protocols Aug. 1998, pp. 304-314

*Deprecating the T=
CP Macroscopic Model, *M. Mathis, J. Mahdavi, in Computer Comm=
unications Review, 49(5), October 2019

=E2=80=93 TobyRodwell - 12 Aug 2006

=E2=80=93 Simon Leinen=E2=80=9414=
Oct 2019