# Hadron production in d-Au, p-Pb, Pb-Pb and Xe-Xe collisions at RHIC and LHC:
onset of color glass condensate in light of an analytical solution of BK
equation

Pragyan Phukan<EMAIL_ADDRESS>HEP laboratory, Department of Physics,
Tezpur University, India  Madhurjya Lalung<EMAIL_ADDRESS>HEP
laboratory, Department of Physics, Tezpur University, India  Jayanta Kumar
Sarma<EMAIL_ADDRESS>HEP laboratory, Department of Physics, Tezpur
University, India

###### Abstract

An exact analytical solution to the Balitsky-Kovchegov equation is proposed
incorporating Mellin transformation with saddle point approximation. The
theoretical significance of this solution is its simplicity in form which
requires very less number of parameters. Using this solution along with the
idea of color glass condensate, a very good description of RHIC and LHC data
on differential yield is obtained for d-Au, p-p and p-Pb collisions.
Quantitative prediction for the nuclear modification factors for p-Pb, Pb-Pb
and Xe-Xe collisions relevant for very recent LHC run 2 is yielded.

Heavy ion collisions performed previously at RHIC BRAHMS collaboration et al.
(2004); STAR collaboration et al. (2006) and recently at LHC CMS collaboration
et al. (2018); ALICE collaboration et al. (2018a, b) probe the color glass
condensate (CGC) Weigert (2005) regime of QCD governed by the nonlinear
effects in the colliding nuclei wave functions as well as gluon saturation.
Pb-Pb collision at LHC opens up an excellent opportunity to study the hot-
dense hypothetical QCD medium arguably called quark gluon plasma (QGP).
Various theoretical frameworks Kharzeev and Levin (2001); Kharzeev et al.
(2005); Armesto et al. (2005a) that are modeled to study these QCD processes,
primarily based on the concept of saturation scale ($Q_{sA}$) that
characterizes the onset of nonlinear coherent phenomena. The growth of the
saturation scale with increasing energy is yielded by the Balitsky-Kovchegov
(BK) nonlinear evolution equation Balitsky (1996); Kovchegov (1999). This
portrays a direct link between the gluon saturation dynamics and the
experimentally measured hadron yields.

The BK equation in position space is characterized by the dipole scattering
amplitude, $\mathcal{N}(r,Y)$ on a target, where the target is specified by
the initial condition. For large and homogeneous targets, Fourier
transformation of the scattering amplitude to the momentum space yields

$\partial_{Y}\mathcal{N}=\bar{\alpha}_{s}\chi(k^{2})\mathcal{N}-\bar{\alpha}_{s}\mathcal{N}^{2},$
(1)

where $\bar{\alpha}_{s}=N_{c}\alpha_{s}/\pi$ and the quantity
$\mathcal{N}(k,Y)=\int_{0}^{\infty}\frac{d^{2}r}{2\pi
r^{2}}e^{ir.k}\mathcal{N}(r,Y)$ is related to unintegrated gluon distribution
(UGD) of the target. The operator $\chi$ is the well known LL BFKL kernel in
momentum space. The nonlinear term in (1) is not acted on by the kernel
operator and thereby posses a particularly simple representation.

To solve the BK equation (1) in Mellin space we recast (1) in terms of its
inverse function $\underline{\mathcal{N}}=\mathcal{N}^{-1}$ such that the
semilinear PDE transforms into a linear one and then diagonalize the same by
Mellin transformation

$\partial_{Y}\tilde{\underline{\mathcal{N}}}=-\bar{\alpha}_{s}\tilde{\chi}(\gamma)\tilde{\underline{\mathcal{N}}}-2\pi\bar{\alpha}_{s}\delta(i\gamma),$
(2)

where $\delta(i\gamma)$ is a Dirac-delta function. The Mellin transformation
of the scale invariant kernel $\chi$ in (2) can be formulated in terms of the
first order harmonic number $\mathcal{H}_{-\gamma}^{(1)}$ as follows,

$\displaystyle\tilde{\chi}(\gamma)$
$\displaystyle=\int_{0}^{\infty}\frac{du}{u}\left[\frac{u^{-\gamma+1}-1}{|u-1|}+\frac{1}{\sqrt{4u^{2}+1}}\right]$
(3) $\displaystyle=\pi\cot(\pi\gamma)-2\mathcal{H}_{-\gamma}^{(1)}$

implies,

$\tilde{\chi}^{\prime}(\gamma)=-\pi^{2}\csc(\pi\gamma)^{2}+\pi^{2}/3-2\mathcal{H}_{-\gamma}^{(2)}.$
(4)

Inverse Mellin transformation on the solution of (2) yields

$\displaystyle\underline{\mathcal{N}}$ $\displaystyle=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}d\gamma(k^{2}/k_{0}^{2})^{\gamma}\times$ (5)
$\displaystyle\left[\frac{2\pi\bar{\alpha}_{s}\delta(i\gamma)}{\tilde{\chi}(\gamma)}+e^{\tilde{\chi}(\gamma)(y_{0}-Y)}\left(\tilde{\underline{\mathcal{N}}}_{0}-\frac{2\pi\bar{\alpha}_{s}\delta(i\gamma)}{\tilde{\chi}(\gamma)}\right)\right],$

with
$\tilde{\underline{\mathcal{N}}}_{0}=\tilde{\underline{\mathcal{N}}}(k,y_{0})$
where the initial condition $y_{0}$ is chosen large enough to ensure the
validity of BK equation. An arbitrary scale $k_{0}$ is introduced for
dimensional purpose. Note that $\tilde{\chi}(\gamma)$ is symmetric around
$\gamma=1/2$ and along the contour $\gamma=1/2+i\nu$, with
$-\infty<\nu<\infty$ has its maximum at the ”saddle” point $\gamma=c=1/2$.
This indicates that the $\gamma\sim 1/2$ region dominates the small-$x$
behavior of the $N(k,Y)$ which seem to recover the familiar LL BFKL
characteristic function behavior. To produce the iterative solution of (2) we
therefore set $\gamma=1/2+i\nu$ in (5) and expand $\tilde{\chi}(\gamma)$ about
$\nu=0$ i.e.
$\tilde{\chi}(1/2+i\nu)=\lambda-1/2\Omega\nu^{2}+\mathcal{O}(\nu^{2})$ with
$\lambda=\bar{\alpha}_{s}4\ln 2$ and $\Omega=\bar{\alpha}_{s}28\zeta(3)$,
$\zeta$ being Reimann zeta function. We also need to expand
$\tilde{\underline{\mathcal{N}}}(1/2+i\nu,y_{0})$ around $\nu=0$,

$\tilde{\underline{\mathcal{N}}}(1/2+i\nu,y_{0})\approx\tilde{\underline{\mathcal{N}}}(1/2,y_{0})e^{\frac{\nu\partial\ln\tilde{\underline{\mathcal{N}}}(1/2+i\nu,y_{0})}{\partial\nu}|_{\nu=0}}.$
(6)

Substituting these expansions back to (5) and performing $\nu$ integration
along with various Dirac-delta operations one may arrive at

$\displaystyle\underline{\mathcal{N}}=$
$\displaystyle\frac{(k^{2}/k_{0}^{2})^{\frac{1}{2}}e^{\lambda(y_{0}-Y)}\tilde{\underline{\mathcal{N}}}(1/2,y_{0})}{(2\pi)^{\frac{1}{2}}\sqrt{\Omega|y_{o}-Y|}}e^{-\frac{\ln^{2}(k^{2}/\tilde{k}_{s}^{2})}{2\Omega\ln(y_{0}-Y)}}$
(7)
$\displaystyle+\frac{\bar{\alpha}_{s}}{\lambda+1/8\Omega}\\{1-e^{(\lambda+1/8\Omega)(y_{0}-Y)}\\},$

where we define an arbitrary scale
$\ln\tilde{k_{s}}^{2}=i\frac{d}{d\nu}[\ln\underline{\tilde{\mathcal{N}}}(y_{0},1/2+i\nu)]|_{\nu=0}$.
Above equation serves as the analytical solution of (2), however, the original
solution $\mathcal{N}$ of the BK equation (1) would be the immediate
reciprocal of (7) as we recall $\underline{\mathcal{N}}=\mathcal{N}^{-1}$ from
our assumption.

A few comments on the solution (7) are in order. The first term of (7)
forecasts the characteristic $\sim e^{\lambda Y}$ behavior of $\mathcal{N}$,
so does UGD, similar to LL BFKL evolution. On the other hand, the second term
modulates the power growth arising from the linear contribution thereby
suppressing the gluon evolution. Interestingly, at pre asymptotic rapidities
($Y\sim y_{0}$), the nonlinear contribution vanishes since the driving term

$\displaystyle e^{(\lambda+1/8\Omega)(y_{0}-Y)}|_{Y\rightarrow
y_{0}}\rightarrow 1.$

Thus at not so large $Y$, the BK solution (7) recovers usual linear BFKL
behavior. Surprisingly the nonlinear contribution in the solution is
irrelevant of the momentum scale, $k^{2}$ and it depends particularly on the
rapidity only.

Equation (7) suggests the exact asymptotic solution of the BK equation and
thus geometrical scaling appears as a universal property of these kinds of
nonlinear equations. However, the detailed phenomenological studies on the
gluon evolution in context of geometrical scaling is beyond the scope of this
letter. Instead, in this work, we try to address a more realistic aspect of
the solution relating our theory to the ongoing collider physics experiments.

In high energy asymmetric collisions, there are at least two particular
approaches leading to hadron production. The first one is the $k_{t}$
factorization formalism, appropriate for central rapidity region, where both
the target and the projectile are characterized in terms of UGD. But at more
forward rapidities, $k_{t}$ factorization fails to grasp the dominant
contribution to the cross section coming from valence quarks of the
projectile. In this situation, so called hybrid formalism is most pronounced
where the large-$x$ degrees of freedom of the projectile is characterised by
usual collinear gluon distribution. In hybrid formalism, the differential
cross section for forward hadron production with transverse momentum, $p_{t}$
and pseudo rapidity $\eta$, in proton-nucleus collisions Dumitru et al.
(2006); Albacete and Marquet (2010) is given by

$\displaystyle\frac{dN_{h}^{pA\rightarrow hX}}{d\eta d^{2}p_{t}}=$
$\displaystyle\frac{C}{(2\pi)^{2}}\sum_{q}\int_{x_{F}}^{1}\frac{dz}{z^{2}}\bigg{[}\frac{x_{F}}{z}f_{q/p}(x_{F}/z,p_{t}^{2})$
(8)
$\displaystyle\times\tilde{N}_{F}(x^{\prime},p_{t}^{2})D_{h/q}(z,p_{t}^{2})$
$\displaystyle+\frac{x_{F}}{z}f_{g/p}(x_{F}/z,p_{t}^{2})\tilde{N}_{A}(x^{\prime},p_{t}/z)D_{h/g}(z,p_{t}^{2})\bigg{]}$

where the kinematics are $x_{F}=\sqrt{m_{h}^{2}+p_{t}^{2}}/\sqrt{S_{NN}}$ and
$x^{\prime}=x_{F}e^{-2y_{h}}/z$ with $\sqrt{S_{NN}}$ being the collision
energy per nucleon. For highly relativistic particles the difference between
rapidity $y_{h}$ and pseudorapidity $\eta$ can be neglected, $\eta\approx
y_{h}$. The projectile is characterized by the collinear PDFs $f_{i/p}$,
however its partons eventually acquire a large transverse momentum $p_{t}$
after interaction with the small-$x$ fields of the saturated nucleus causing
multiple scattering which is described by the UGDs in the adjoint or
fundamental representation $\tilde{N}_{A}(F)$. The term $D_{h/i}$ refer to the
final-state hadron fragmentation function which characterizes the
hadronization of the scattered parton. The lack of impact parameter dependence
in the calculation is accounted for by the normalization factor C.

Figure 1:  Charged hadrons $h^{-}$ / $h^{\pm}$ and $\pi^{0}$ yields in (a)
deuteron-gold collision at RHIC ($\sqrt{S_{NN}}=200~{}\text{GeV}$) and (b)
proton-proton, proton-lead collison at LHC
($\sqrt{S_{NN}}=5.02~{}\text{TeV}$). Solid lines are theoretical predictions.
Data by BRAHMS, STAR and ALICE collaboration.

The nuclear UGDs, $\tilde{N}_{F(A)}$ entering (8) are given by the two
dimensional Fourier transform of the imaginary part of the forward dipole
scattering amplitude in the fundamental (F) and adjoint (A) representation

$\displaystyle\tilde{N}_{F(A)}(x,k)=\int d^{2}\text{r}e^{-i\text{
k.r}}(1-\mathcal{N}_{F(A)}(r,Y)),$ (9)

where $r$ is the dipole size and rapidity $Y=\ln(x_{0}/x)$. The small-$x$
dynamics of the dipole amplitudes $\mathcal{N}_{F(A)}(r,Y)$ in (9) is given by
the solution of BK equation (7) (inverse) but in momentum space, however, it
is straightforward to switch between position and momentum domains via Fourier
transform. The initial conditions for the evolution of dipole-nucleus
amplitude are taken from the semiclassical Mclerran-Venugopalan (MV) model
McLerran and Venugopalan (1998),

$\tilde{\mathcal{N}}_{A}(r,Y=0)=1-\text{exp}\bigg{[}-\frac{r^{2}Q_{s0}}{4}\ln\left(\frac{1}{\Lambda
r}+e\right)\bigg{]}$ (10)

where $Q_{s0}$ is the initial saturation scale probed by quarks. The constant
$e$ serves as an infrared regulator and $\Lambda\approx 0.24$ GeV. The
scattering amplitude in fundamental representation, $\mathcal{N}_{F}$ can also
be parametrized as in (10) but with the replacement $Q_{s}^{2}\rightarrow
Q_{s}^{2}C_{F}/C_{A}=4/9Q_{s}^{2}$ Kharzeev et al. (2004).

Figure 2:  Neutral pion ($\pi^{0}$) and charged pions ($\pi^{+},\pi^{-}$)
nuclear modification factors for Pb-Pb collision at
$\sqrt{S_{NN}}=5.02~{}\text{TeV}$.

With this setup, we obtain a very good description of RHIC and LHC run 2 data
for hadron yields of different hadron species in p-p, d-Au and p-Pb
collisions. The invariant differential yields in d-Au collision at RHIC energy
$\sqrt{S_{NN}}=200$ GeV and pseudo-rapidity $\eta$ = 2.2 and 3.2 for
negatively charged hadrons ($h^{-}$) (BRAHMS collaboration BRAHMS
collaboration et al. (2004)) and $\eta$ = 4 for neutral pions ($\pi^{0}$)
(STAR collaboration STAR collaboration et al. (2006)) is shown in Fig. 1. In
analogy to Armesto et al. (2005b), we obtain a quark saturation scale
$Q_{s0}\approx 0.43~{}\text{GeV}^{2}$ for gold nucleus at $x_{0}=0.01$.
Corresponding gluon saturation scale $Q_{s0}^{2}$ is found to be
$Q_{s0}^{2}\approx 0.97~{}\text{GeV}^{2}$. For entire calculations performed
with hybrid formalism, we adopt the DSS NLO fragmentation functions de Florian
et al. (2007) and CT14 NLO collinear PDFs Dulat et al. (2016) as default. Our
results show a good agreement with RHIC data. However, for the better
description of neutral pion data, a normalisation factor $C=0.6$ is
multiplied, while charged hadron yields are sketched without any normalisation
(i.e. C=1). Considering the sizable uncertainty in the data, the precise value
of the normalisation constant does not have much meaning. In Fig. 1, the
$\pi^{0}$ and $h^{\pm}$ yields for p-Pb collisions and $h^{\pm}$ yield for p-p
collisions measured in ALICE at $\sqrt{S_{NN}}=5.02~{}\text{TeV}$ ALICE
collaboration et al. (2018a) are also shown. Corresponding pseudo rapidity
windows for $h^{\pm}$ and $\pi^{0}$ are $|\eta|<0.8$ and $|\eta|<0.4$
respectively. Here for all calculation the normalisation is set to 1. The
theoretical measurement are clearly consistent within the uncertainties over
the entire $p_{t}$ range.

The modification of forward particle production is quantified with the nuclear
modification factor,

$R_{pA}=\frac{d^{2}N^{pA}/d\eta dp_{t}}{\braket{T_{pA}}d^{2}\sigma^{pp}/d\eta
dp_{t}},$ (11)

where $N^{pA}$ is the charged particle yield in pA collisions and
$\sigma^{pp}$ is the particle production cross-section in p-p collision. The
average nuclear overlap function,
$\braket{T_{pA}}=\braket{N_{coll}}/\sigma_{inel}^{NN}$ depends on the
collision centrality determined from the Glauber model Loizides et al. (2018).
For instance, at $\sqrt{S_{NN}}=5.02$ TeV, $\braket{T_{pPb}}\approx
0.098\quad\text{mb}^{-1}$ for minimum bias. Note that computing the ratio (11)
removes sensitivity to the normalization factor C. Our $R_{pPb}$ calculations
scaled by $\braket{T_{pPb}}$ are displayed in Fig. 2 at
$\sqrt{S_{NN}}=5.02~{}\text{TeV}$ subject to the pseudo rapidity windows
$-0.8<\eta<0.8$ for neutral pions ($\pi^{0}$) and $-1.265<\eta<0.335$ for
charged pions ($\pi^{+}+\pi^{-}$) agree well with the ALICE data within the
uncertainties over the complete $p_{t}$ domain. Both theory and data indicate
a sizable suppression around 30% at $p_{t}=1~{}\text{GeV}$, whereas the ratio
$R_{pPb}$ is seen tending to unity for $p_{t}$ above 2 GeV for both neutral
pions and charged pions.

Figure 3:  Charged hadron $R_{AA}$ for (a) Pb-Pb collisions at
$\sqrt{S_{NN}}=5.02~{}\text{TeV}$ for centrality bins 0-100%, 50-70% and (b)
Xe-Xe collision at $\sqrt{S_{NN}}=5.44~{}\text{TeV}$ for centrality bins
10-30%, 0-5%.

We have also checked our BK analytical solution + CGC framework to testify
charged hadron production particularly at mid rapidities using $k_{t}$
factorization formalism. At mid rapidities, $x_{F}/z$ becomes too small to
valid hybrid formalism in (8). The $k_{t}$ factorization formula is valid for
small values of $x$, for both the projectile and targets. But since it
includes only gluonic degrees of freedom, this formalism is insufficient at
very forward rapidities or large $p_{t}$. Within $k_{t}$ factorization
approach Albacete (2007) the pseudo rapidity density of charged particle can
be calculated via.

$\frac{dN^{ch}}{dydp_{t}}=\frac{2\pi K}{C_{F}}\frac{\alpha_{s}}{p_{t}^{2}}\int
dk_{T}^{2}\phi_{A_{1}}(x_{1},k_{t}^{2})\phi_{A_{2}}(x_{2},(p-k)_{t}^{2}),$
(12)

where the variables $x_{1,2}=p_{t}/\sqrt{S_{NN}}e^{\pm y}$ and $K$ is
accounted for the normalisation purpose. The UGD $\phi$, in (12) is related to
collinear gluon distribution, $\phi(x,k)\propto\frac{d(xG(x,k^{2}))}{d^{2}k}$
and is given in terms of the dipole scattering amplitude

$\phi(y,k)=\int\frac{d^{2}r}{2\pi r^{2}}e^{ir.k}\mathcal{N}(y,r).$ (13)

The CMS and ALICE data CMS collaboration et al. (2018); ALICE collaboration et
al. (2018b) for the nuclear modification factor $\text{R}_{\text{AA}}$ for
primary charged particles in Pb-Pb ($\sqrt{S_{NN}}=5.02~{}\text{TeV}$) and Xe-
Xe ($\sqrt{S_{NN}}=5.44~{}\text{TeV}$) collisions are shown in Fig. 3 for the
centrality bins 0-100%, 50-70%, 10-30% and 0-5%. For minimum bias (0-100%),
the ratio $R_{pA}$ is also included in Fig. 3 to sketch a quantitative
comparison with $R_{AA}$. The pronounced suppression in our theoretical
prediction is seen at $p_{t}<2~{}\text{GeV}$ for p-Pb system and at
$8<p_{t}<100~{}\text{GeV}$ for Pb-Pb system which is consistent with CMS data.
In the peripheral (50-70%) bin, $R_{AA}$ measurements at
$\sqrt{S_{NN}}=2.76~{}\text{TeV}$ is also shown to examine the sensitivity of
the ratio towards probe energy. The measured distribution $R_{AA}$ at 2.76 TeV
and 5.02 TeV are found to be quantitatively similar at intermediate $p_{t}$,
but above about 10 GeV, $R_{AA}$ at 5.02 TeV tend to be slightly smaller than
the same at 2.76 TeV. In Fig. 3, $R_{AA}$ for Xe-Xe collision at
$\sqrt{S_{NN}}=5.44~{}\text{TeV}$ is also sketched for two centrality bins
0-10% and 10-30%. The theoretical prediction seem to be in rough agreement
with CMS data at moderate $p_{t}$, however, towards high $p_{t}$, our theory
tend to predict much stronger suppression compared to the data. From this
analysis it is clear that our theory holds good for minimum bias measurements,
while it might be inappropriate for most central events. This is expected as
our theory is impact parameter $b$ independent and thus one have to improve
the theory for the $b$ dependence of the UGD to study different centrality
bins which is beyond the scope of this letter.

In summary, we have proposed an analytical solution to BK equation
incorporating Mellin transformation with saddle point approximation. The main
key feature of this solution is its simplicity in form with relatively very
less number of parameters, which might be handy for investigation of various
characteristics of such parton evolution processes. At pre asymptotic $y$, the
solution is found to reconstruct the characteristics linear BFKL solution. The
feasibility of the solution is tested at both RHIC and LHC energies. As per
the good consistency between the theory and data is concerned, the analytical
solution of BK equation could be a very reliable framework for exploring quark
gluon plasma at LHC as well as future colliders.

## References

  * BRAHMS collaboration et al. (2004) BRAHMS collaboration et al., Physical review letters 93, 242303 (2004).
  * STAR collaboration et al. (2006) STAR collaboration et al., Physical review letters 97, 152302 (2006).
  * CMS collaboration et al. (2018) CMS collaboration et al., Journal of High Energy Physics 2018, 138 (2018).
  * ALICE collaboration et al. (2018a) ALICE collaboration et al., The European Physical Journal C 78, 624 (2018a).
  * ALICE collaboration et al. (2018b) ALICE collaboration et al., Journal of High Energy Physics 2018, 13 (2018b).
  * Weigert (2005) H. Weigert, Progress in Particle and Nuclear Physics 55, 461 (2005).
  * Kharzeev and Levin (2001) D. Kharzeev and E. Levin, Physics Letters B 523, 79 (2001).
  * Kharzeev et al. (2005) D. Kharzeev, E. Levin, and M. Nardi, Nuclear Physics A 747, 609 (2005).
  * Armesto et al. (2005a) N. Armesto, C. A. Salgado, and U. A. Wiedemann, Physical review letters 94, 022002 (2005a).
  * Balitsky (1996) I. Balitsky, Nuclear Physics B 463, 99 (1996).
  * Kovchegov (1999) Y. V. Kovchegov, Physical Review D 60, 034008 (1999).
  * Dumitru et al. (2006) A. Dumitru, A. Hayashigaki, and J. Jalilian-Marian, Nuclear Physics A 765, 464 (2006).
  * Albacete and Marquet (2010) J. L. Albacete and C. Marquet, Physics Letters B 687, 174 (2010).
  * McLerran and Venugopalan (1998) L. McLerran and R. Venugopalan, Physics Letters B 424, 15 (1998).
  * Kharzeev et al. (2004) D. Kharzeev, Y. V. Kovchegov, and K. Tuchin, Physics Letters B 599, 23 (2004).
  * Armesto et al. (2005b) N. Armesto, C. A. Salgado, and U. A. Wiedemann, Physical review letters 94, 022002 (2005b).
  * de Florian et al. (2007) D. de Florian, R. Sassot, and M. Stratmann, Physical Review D 75, 114010 (2007).
  * Dulat et al. (2016) S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C.-P. Yuan, Physical Review D 93, 033006 (2016).
  * Loizides et al. (2018) C. Loizides, J. Kamin, and D. d’Enterria, Physical Review C 97, 054910 (2018).
  * Albacete (2007) J. L. Albacete, Physical review letters 99, 262301 (2007).


# FaiRTT: An Empirical Approach for Enhanced RTT Fairness and Bottleneck
Throughput in BBR

Akshita Abrol1, Purnima Murali Mohan1, Tram Truong-Huu12 Email:
{akshita.abrol, purnima.mohan<EMAIL_ADDRESS>1Singapore Institute of Technology (SIT), Singapore
2Agency for Science, Technology and Research (A*STAR), Singapore

###### Abstract

In next-generation networks, achieving Round-trip Time (RTT) fairness is
essential for ensuring fair bandwidth distribution among diverse flow types,
enhancing overall network utilization. The TCP congestion control algorithm —
BBR, was proposed by Google to dynamically adjust sending rates in response to
changing network conditions. While BBRv2 was implemented to overcome the
unfairness limitation of BBRv1, it still faces intra-protocol fairness
challenges in balancing the demands of high-bandwidth, long-RTT elephant flows
and more frequent short-RTT mice flows. These issues lead to throughput
imbalances and queue buildup, resulting in elephant flow dominance and mice
flow starvation. In this paper, we first investigate the limitations of
Google’s BBR algorithm, specifically in the context of intra-protocol RTT
fairness in beyond 5G (B5G) networks. While existing works address this
limitation by adjusting the pacing rate, it eventually leads to low
throughput. We hence develop the FaiRTT algorithm to resolve the problem by
dynamically estimating the Bandwidth Delay Product (BDP) sending rate based on
RTT measurements, focusing on equitable bandwidth allocation. By modeling the
$Inflight$ dependency on the BDP, bottleneck bandwidth, and packet departure
time after every ACK, we can resolve the intra-protocol fairness while not
compromising the throughput on the bottleneck link. Through extensive
simulations on NS-3 and comprehensive performance evaluations, FaiRTT is shown
to significantly improve the fairness index and network throughput,
significantly outperforming BBRv2, for diverse flow types. FaiRTT achieves an
average throughput ratio of $1.08$ between elephant and mice flows, an average
fairness index of $0.98$, and an average utilization of the bottleneck link of
$98.78\%$.

###### Index Terms:

BBR, Traffic Congestion Control, Flow Fairness, Network Throughput, Elephant
and Mice Flows.

##  I Introduction

The Bottleneck Bandwidth and Round-trip propagation time (BBR) algorithm,
introduced by Google in 2016 [1], stands out as a pivotal mechanism,
especially considering its capability to dynamically adjust sending rates in
response to real-time network conditions. The versatility of BBR makes it an
indispensable tool, not merely for enhancing throughput and reducing latency
in the ever-evolving and complex network landscapes but also for effectively
navigating the challenges brought forth by the continuous advancement in
network technologies and their accompanying protocols. In next-generation
networks (NGN) such as beyond 5G (B5G) or 6G, the analysis of flow size
statistics showcases a consistent pattern of heavy-tail behavior for
applications ranging from bandwidth-demanding long flows (e.g., video-
surveillance drone applications, AR/VR) to the delay-sensitive short flows
[2]. Such heavy-tail flow distribution in NGN is similar to that of the
Internet’s familiar mice-elephant phenomenon.

The high-bandwidth, high Round-Trip Time (RTT) applications, which account for
a smaller fraction of the overall network traffic, are referred to as the
“elephant” flows. This contrasts sharply with the plethora of smaller “mice”
flows, such as tactile internet signaling traffic, HTTP requests, etc., which,
despite their higher frequency, low-RTT constitute a lesser segment of the
total traffic volume. In the B5G network landscape, the distinct roles of
high-bandwidth, long-RTT elephant flows, and frequent short-RTT mice flows are
even more pronounced with their varying QoS and RTT requirements. To model the
network flows, BBR works on Kleinrock’s optimal operating point [3] by
estimating the Maximum Delivery Rate referred to as the Bottleneck Bandwidth
(BtlBw) and the Minimum Round-Trip Time known as RTprop. Based on the
Bandwidth Delay Product (BDP), measured as the product of BtlBw and RTprop,
the algorithm calculates three control parameters: (i) sending quantum, (ii)
pacing rate, and (iii) congestion window. BBR estimates bottleneck bandwidth
to set sending rates — aiming for high throughput without increasing the
waiting time of the packets in the queue [4].

However, the first version of BBR (BBRv1) overestimates the sending rate by
constantly filling the BDP leading to queue buildup, bandwidth, and RTT
unfairness issues [5, 6, 7]. BBRv2 was developed to overcome these limitations
by introducing mechanisms to better estimate and adapt to the actual available
bandwidth and minimize queue buildup. While BBRv2 adapts better to network
changes than BBRv1, it still struggles in NGN networks with highly varying QoS
requirements. Ensuring complete fairness among multiple BBRv2 flows is
challenging, especially in NGN scenarios of the coexistence of elephant and
mice flows due to the overestimation (elephant flows) or underestimation (mice
flows) of the optimal operating point. For example, the elephant flows tend to
dominate, causing unfairness and queue build-up at the router while the mice
flows face starvation due to the unintended shift in the operating point due
to the overestimation. Consequently, this unintended shift in the operating
point leads to slow, unstable convergence, RTT unfairness, and throughput
underutilization by different flow RTTs [8, 9]. Some efforts have been made to
improve the intra-protocol fairness for BBRv1 in [8] and BBRv2 in [10]. They
utilize queue size estimation to adjust the pacing gain to improve fairness.
However, there is no emphasis on changing the BDP itself depending on the
incoming flow.

In this paper, we develop a novel algorithm (named FaiRTT) that can
dynamically estimate the BDP based on flow-centric RTT estimate to mitigate
the observed fairness challenges. The primary objective is to ensure equal
opportunity for both elephant and mice flows within the bottleneck link,
addressing the dominance and starvation issues, respectively, prevalent in
existing BBR implementations. We use the standardized Jain’s fairness index as
one of the key performance metrics to demonstrate the effectiveness of FaiRTT,
combined with other network performance metrics such as throughput, and link
utilization. Through the performance evaluation on NS-3, we show that FaiRTT
has a better average throughput ratio of 1.08 between elephant and mice flows
and an average fairness index of up to 0.98 compared to that of BBRv2 while
improving the overall bottleneck link bandwidth utilization to $98.78\%$.

The rest of the paper is organized as follows. In Section II, we discuss the
related work. We present the FaiRTT algorithm for improving RTT fairness in
Section III. Section IV presents our experiments and analysis of results
before we conclude the paper in Section V.

##  II Related Work

Since the introduction of Google’s BBR congestion control algorithm, numerous
experiments and analyses have been performed to evaluate its performance in
various network scenarios. In this section, we review several works and
discuss potential issues that can happen to traffic flows.

###  II-A BBR Operational Intricacies

In contrast to the initial promising results reported by Google on the
effectiveness of BBRv1 (BBR-version 1), subsequent studies uncovered several
issues in its behavior. Scholz et al. [5] identified the creation of long-
standing queues during the startup phase, suppressing existing flows. Hock et
al. [6] observed bandwidth discrepancies between elephant and mice flows
sharing a bottleneck link due to the higher BDP of elephant flows. Scherrer et
al. [7] found that in the presence of multiple flows managed by BBRv1,
overestimated delivery rates lead to standing queues exceeding 1.5 times BDP,
causing packet loss and unfair bandwidth shared among flows, particularly for
small buffer sizes. To address the above concerns, Google introduced BBRv2
(BBR-version 2) [11] in 2018, incorporating explicit congestion notification
(ECN) and packet loss rate. Zhang et al. [12] while evaluating various BBR
versions, highlighted BBRv2’s improvements in flow fairness and enhanced
coexistence with CUBIC and Reno. However, suboptimal channel utilization was
noted with a 5% loss rate. Song et al. [13] reported fairness enhancements and
reduced re-transmissions with BBRv2, emphasizing its effectiveness in limited
buffer scenarios but noting challenges in fair bandwidth convergence with two
flows entering a bottleneck link at different times. Nandagiri et al. [14]
conducted an experimental evaluation comparing BBRv1 and BBRv2, revealing
BBRv2’s ability to overcome inflight capacity limitations in networks with
small buffers. However, in networks with large buffers, long RTT flows still
consume more bandwidth.

###  II-B BBR Enhancing Fairness

Ma et al. [15] introduced the BBQ algorithm to enhance RTT fairness by
minimizing probing periods for flows with extended round-trip times. While
effective, BBQ’s performance degrades with an increasing number of competing
flows. Kim et al. proposed solutions in [16] and [17] to limit BBRv1’s
inflight capacity during the ProbeBW phase, addressing bias towards long RTT
flows and preventing excessive data transmission. However, it causes a
decrement in total throughput due to limiting inflight capacity to 1 BDP to
prevent congestion. In [18], the authors presented the Delay-Aware BBR (DA-
BBR) algorithm to improve RTT fairness and throughput using the relation of
each flow’s RTT with RTprop. However, this algorithm was noted for slow and
unstable convergence. Njogu et al. [8] introduced BBR-With Enhanced Fairness
(BBR-EFRA), dynamically adjusting congestion window (CWND) based on buffer
queue status for equitable competition among different RTT flows. Pan et al.
[9] addressed the fairness and excessive re-transmissions in BBRv2 with the
introduction of BBRv2+, utilizing flow-aware explicit congestion notification.
However, these algorithms rely on accurate real-time queue estimation, which
is challenging in dynamic network conditions.

Addressing these issues remains crucial for achieving equitable bandwidth
allocation among different RTT flows without compromising on the throughput
through a stable convergence. With the continuous updating of the BBR
algorithm, BBRv2 can solve some fairness problems and limitations of BBRv1.
BBRv3 (BBR-version 3) [19] has recently been released, aiming to rectify bugs
and optimize performance parameters observed in BBRv2. While BBRv2 and BBRv3
present updates and improvements, experiments reveal persistent RTT
unfairness. Thus, continued research is essential to enhance RTT fairness and
intra-protocol fairness, providing potential refinements for the final version
of BBR.

##  III FaiRTT Algorithm

###  III-A Overview of BBRv2

Given a network, the main objective of BBR is to maximize the utilization of
the bottleneck link with a minimal delay. BBR estimates the bandwidth of the
bottleneck link as the maximum observed delivery rate and the propagation
delay as the minimum observed RTT over periodic intervals. The bandwidth of
the bottleneck link is denoted as BtlBw, and the propagation delay is denoted
as RTprop. However, sending more traffic (probing more bandwidth) to estimate
BtlBw may cause congestion at the bottleneck link, thus increasing RTprop and
vice-versa. BBR, therefore, estimates these values separately in its different
execution phases.

Figure 1: BBRv2 Phases and Flow Diagram. TABLE I: Mathematical Notations Notation | Definition
---|---
$BDP_{t}$ |  Bandwidth delay product at time window $t$
$\texttt{RTprop}_{t}$ |  Minimum RTT captured at time window $t$
$\texttt{BtlBw}_{t}$ |  Max. bottleneck bandwidth at time window $t$
$lastRTT_{t}$ |  RTT of the last ACK at time window $t$
${minRTT}_{t}$ |  Minimum value of RTT at time window $t$
${dRate}_{t}$ | Delivery rate of the bottleneck link
at time window $t$
$maxBw_{t}$ |  Maximum delivery rate at time window $t$
$Inflight_{t}$ | Estimated volume of in-flight data to utilize
available bottleneck bandwidth at time window $t$
$\alpha_{t}$ |  RTT fairness threshold at time window $t$
$Wfcount_{t}$ |  Number of flows at time window $t$
$WminRTT_{t}$ |  RTT estimate at time window $t$
$\beta$ | Balance factor set to 0.8
$\gamma$ | Discount factor set to 0.99
PROBE_INT_EXP | Expiration flag of the probe bandwidth phase
RATE_LIM_APP | Rate limiting flag

In Fig. 1, we present the flow diagram of BBRv2. We provide the meaning of
mathematical notations used in the BBRv2 algorithm in Table I. The BBRv2
algorithm is composed of four phases: Startup (BBR_START_UP), Drain
(BBR_DRAIN), Probe Bandwidth (BBR_PROBE_BW), and Probe RTT (BBR_PROBE_RTT).
The first phase (BBR_START_UP) adapts the exponential Startup behavior from
CUBIC by doubling the sending rate with each RTT. The pacing_gain is set to
$2.89$. If the increase in the sending rate does not exceed 25% for three
consecutive attempts or if packet loss or an ECN-marked rate surpasses a
predefined threshold (e.g., 2%), BBRv2 assumes to have reached the bottleneck
bandwidth. Since this observation is delayed by one RTT, a queue was already
created at the bottleneck. In the second phase (BBR_DRAIN), BBRv2 tries to
drain the queue by temporarily reducing the pacing_gain to approximately
$0.34$. Afterwards, BBRv2 enters the third phase (BBR_PROBE_BW), in which it
probes for more available bandwidth. BBRv2 specifies exit conditions for each
stage of the third phase (ProbeBW Refill, ProbeBW Up, ProbeBW Down, and
ProbeBW Cruise). BBRv2 initiates bandwidth probing by linearly augmenting the
amount of inflight data over one $RTprop$ in the ProbeBW Refill stage,
effectively filling the network pipe. Subsequently, BBRv2 engages in the
ProbeBW Up phase, during which it increases the inflight data swiftly to probe
for more bandwidth. This increase of inflight data ceases when the volume
reaches 1.25 times the BDP, or when packet loss or the ECN rate surpasses a
predetermined threshold. BBRv2 establishes inflight_hi when confronted with a
higher loss/ECN rate than the threshold, ensuring that it does not surpass the
operational point where excessive packet loss may occur. Upon concluding the
ProbeBW Up phase, BBRv2 transitions to the ProbeBW Down phase to offset the
queue. It exits this phase once the volume of inflight data becomes less than
BDP and maintains a constant delivery rate in ProbeBW Cruise until the
initiation of the next bandwidth probing cycle. If BBRv2 experiences packet
loss in the ProbeBW Cruise, it uses inflight_lo to cope with the temporary
packet loss. BBRv2 continuously samples the bandwidth for BDP estimation using
a time window basis. At the time window $t$, $\texttt{BtlBw}_{t}$ is defined
as follows:

$\displaystyle\texttt{BtlBw}_{t}$
$\displaystyle=\begin{cases}maxBw_{t},&dRate_{t}\geqslant\texttt{BtlBw}_{t-1},\\\
maxBw_{t},&\texttt{RATE\\_LIM\\_APP}=\texttt{FALSE},\\\
\texttt{BtlBw}_{t-1},&\text{otherwise}\end{cases}$ (1)

where $maxBw_{t}$ is the maximum delivery rate captured from the socket within
time window $t$, normally set to two times the probe bandwidth phase. It is to
be noted that $\texttt{BtlBw}_{t}$ is measured only for the applications with
no rate limit, indicated by the flag RATE_LIM_APP = FALSE.

After not measuring a new RTprop value for 5 seconds, BBRv2 stops probing for
bandwidth and enters the last phase (Probe RTT). During this phase, the
bandwidth is reduced to half of BDP to drain any possible queue and get a real
estimation of the RTT. This phase is kept for 200 ms, controlled by the flag
PROBE_INT_EXP. If a new minimum value is measured, $\texttt{RTprop}_{t}$ is
updated as follows:

$\displaystyle\texttt{RTprop}_{t}$
$\displaystyle=\begin{cases}{lastRTT}_{t},&0\leqslant{lastRTT}_{t}\leqslant\texttt{RTprop}_{t-1},\\\
{lastRTT}_{t},&\texttt{PROBE\\_INT\\_EXP}=\texttt{TRUE},\\\
\texttt{RTprop}_{t-1},&\text{otherwise}\end{cases}$ (2)

where $lastRTT_{t}$ is captured from the network whenever an ACK packet is
received.

The BBRv2 algorithm estimates the amount of data to be transmitted based on
the current BDP. The BDP is in turn calculated by the product of the
bottleneck bandwidth at time window $t$ and the estimated RTT of the flows
propagating in that time window, $\texttt{RTprop}_{t}$.

$BDP_{t}=\texttt{BtlBw}_{t}\times\texttt{RTprop}_{t}.$ (3)

###  III-B Drawback of BBR

When multiple flows with diverse RTT coexist over a shared bottleneck link,
BBR tends to transmit excess data, approximately $1.5$ times the intended
amount [5]. Consequently, BBR deviates from its typical operating point,
leading to a total transmission rate exceeding the available bandwidth of the
bottleneck link. This results in a persistent queue on the bottleneck link
which is shared among all flows — predominantly favoring the elephant flows,
causing bandwidth unfairness.

Elephant flows, with higher BDP values, transmit more packets and utilize more
bottleneck buffers. Our experimental analysis reveals that the BDP value of
mice flows becomes smaller than that of elephant flows. Consequently, mice
flows inject fewer packets into the pipeline, reducing the delivery rate. The
decrease in delivery rates amplifies the dominance of elephant flows, leading
to increased throughput imbalance, higher packet loss, re-transmission rates,
and elevated latency. In BBRv1, mice flows were found to be constrained by the
congestion window, thereby reducing their transmission rates. BBRv2 aimed to
mitigate this issue by using ECN and setting of inflight upper and lower
bound. However, in [13], the authors reported that BBRv2 still experienced
significant issues with coexistence between different RTT flows particularly
with large buffers. Furthermore, in [20], the authors revealed that BBRv3
continues to exhibit RTT unfairness. Malicious actors can exploit this
vulnerability, deliberately increasing delay to gain a disproportionately
higher bandwidth share.

###  III-C FaiRTT Algorithm

Algorithm 1 FaiRTT: RTT Fairness Improvement Algorithm

1:Input: $\texttt{BtlBw}_{t}$, $\texttt{RTprop}_{t}$, $cwnd\\_gain$,
$lastRTT_{t}$

2:Input: $minRTT_{t}$, $\gamma$

3:Output: $Inflight_{t}$

4:$Wfcount_{t}\leftarrow\texttt{length}(\texttt{unique}(minRTT)\text{ in
}\text{window }t)$

5:Calculate $\alpha_{t}$ using Eq. (5)

6:if $\text{BBR}\rightarrow\text{mode}==\text{BBR\\_PROBE\\_BW}$ then

7: if $lastRTT_{t}>\alpha_{t}$ then

8:
$BDP_{t}\leftarrow\texttt{BtlBw}_{t}\times\texttt{RTprop}_{t}\times\displaystyle\left(\frac{minRTT_{t}\times\gamma}{lastRTT_{t}}\right)$

9: else

10: $BDP_{t}\leftarrow\texttt{BtlBw}_{t}\times\texttt{RTprop}_{t}$

11: end if

12:else

13: $BDP_{t}\leftarrow\texttt{BtlBw}_{t}\times\texttt{RTprop}_{t}$

14:end if

15:$Inflight_{t}\leftarrow BDP_{t}\times cwnd\\_gain$

16:return $Inflight_{t}$

To address the inherent design issue of BBR, we develop Algorithm 1, which
dynamically captures the changes in RTT and adaptively adjusts inflight for
each BBR flow request. The proposed algorithm aims to improve RTT fairness
while maintaining the overall throughput of the network. We leverage the
calculated RTT from each acknowledgment packet (lastRTT) and minimum RTT
(minRTT) to calculate an adjustment coefficient for curtailing the amount of
inflight data for each flow. The inflight is defined as follows:

${Inflight}_{t}={BDP}_{t}\times cwnd\\_gain$ (4)

where $cwnd\\_gain$ is fixed at $0.5$ for the BBR_PROBE_RTT phase and at $2$
for all other phases of BBRv2. While all BBR flows exceed their intended
inflight data transmission, the magnitude differs, particularly with elephant
flows transmitting significantly larger volumes. To account for this, we
examine the relationship between lastRTT and minRTT, introducing an adjustment
coefficient with a value below $1$. This coefficient will approach $1$ for
mice flows, while for elephant flows it would have a smaller value. The BDP
calculation incorporates the adjustment coefficient, dynamically adjusting
inflight data based on RTT. The adjustment coefficient is multiplied with the
BDP only when the value of lastRTT is more than the RTT fairness threshold
$\alpha_{t}$, calculated as:

$\vspace{-0.5ex}\alpha_{t}={Wfcount}_{t}\cdot\sum\nolimits_{j\in
W_{t}}{WminRTT}_{j}/W_{t}$ (5)

where

${WminRTT}_{t}=\beta\cdot\sum_{i={0}}^{t-1}\frac{{minRTT}_{i}}{t}+(1-\beta)\cdot{minRTT}_{t}$
(6)

and

$W_{t}=\min(|\mathbb{R}\texttt{x}|,|\mathbb{T}\texttt{x}|)$ (7)

where $|\mathbb{R}\texttt{x}|$ and $|\mathbb{T}\texttt{x}|$ are the size of
the receiving window and sending window , respectively. $Wfcount_{t}$ is the
estimated number of BBR flows in the time window $t$ calculated by counting
the number of unique $minRTT$’s in that time window, and $\beta$ is a constant
fixed at 0.8 to balance the estimate. Further, we introduce a discount factor
$\gamma$ of $0.99$ to stabilize BDP and promote equitable bandwidth-sharing.
Elephant flows exhibit a lower adjustment coefficient than mice flows,
resulting in a more substantial reduction in elephant flow’s BDP compared to
that of mice flows. Consequently, by transmitting less data than the original
BBR, elephant flows experience a significant reduction in buffer occupancy,
thereby enhancing RTT fairness.

##  IV Performance Evaluation

###  IV-A Experimental Setup

We used a dumbbell topology setup in the NS-3 simulation framework, featuring
multiple flows sharing a common bottleneck link, as illustrated in Fig. 2. We
applied the default active queue management as the Drop-Tail policy and every
packet size as 1 kB. Each simulation lasted 120 seconds and was repeated 5
times with different random seeds for the error rate in packet delivery. The
obtained results were plotted with 95% confidence interval.

Figure 2: Experimental network topology.

####  IV-A1 Performance Metrics

We used the following three performance metrics to evaluate the proposed
algorithm.

  * •

Throughput: We used average throughput to assess the equal distribution of
bandwidth among competing flows. We defined the throughput ratio as the ratio
of the throughput of elephant flows and that of mice flows.

  * •

Fairness Index: We used Jain’s fairness index [21] to provide a more
comprehensive evaluation of fairness in the allocation of bandwidth among
competing flows. The Jain’s fairness index, $F$ is calculated using the
formula:

$F=\left(\sum\nolimits_{i=1}^{n}T_{i}\right)^{2}/\left(n\cdot\sum\nolimits_{i=1}^{n}T_{i}^{2}\right)$
(8)

Here, $n$ represents the number of competing flows, and $T_{i}$ denotes the
throughput of flow $i$. A value of $F$ close to 1 signifies fairness in the
allocation of bandwidth sharing among competing flows.

  * •

Bottleneck link utilization: Link utilization is vital for assessing network
efficiency, measuring how effectively the bottleneck link handles data
transmission. The percentage bottleneck link utilization $U$ is computed as:

$U(\%)=\frac{\sum_{i}\text{T}_{i}}{C_{\text{Btl}}}\cdot 100$ (9)

where $T_{i}$ is the throughput of flow $i$ traversing the bottleneck link and
$C_{\text{Btl}}$ is the bandwidth of the bottleneck link set to 10 Mbps for
all experiments. Higher link utilization indicates better use of the network
resources.

####  IV-A2 Comparisons

We compare FaiRTT with BBRv2 using above performance metrics in different
experimental scenarios:

  * •

Throughput with Time

  * •

Fairness Index with Time

  * •

Bottleneck link utilization with Time

  * •

Throughput with Queue size

  * •

Throughput with RTT

###  IV-B Analysis of Results

####  IV-B1 Throughput

(a) FaiRTT vs. BBRv2.

(b) With different Queue Size Values.

(c) With different RTT Values.

Figure 3: Throughput Comparison.

In this experiment, we set a bottleneck bandwidth of 10 Mbps, a delay of 10
ms, and a bottleneck buffer size of 10 BDP. The RTT of the elephant and mice
flows was set to 15ms and 5ms, respectively. From Fig. 3a, we can observe that
regardless of the algorithms, elephant flows occupy a larger portion of
throughput. BBRv2 presents highly suppressed mice flows by elephant flows with
an average throughput ratio of 1.44 between the two competing flows in
contrast to 1.05 in FaiRTT while maintaining the total throughput of the
network. The average total throughput increased by 1% for FaiRTT as compared
to BBRv2, respectively. Hence, FaiRTT is shown to be much-needed fairness when
elephant and mice flows compete for the bottleneck bandwidth.

To evaluate the effect of bottleneck queue size on fairness, we configured the
test topology having 10 Mbps bottleneck bandwidth and 10 ms bottleneck delay.
The RTT of the elephant and mice flows was set to 15ms and 5ms, respectively.
The bottleneck queue size was varied from 0.5 BDP to 100 BDP. Fig. 3b shows
the throughputs of both elephant and mice flows for BBRv2 and FaiRTT with
respect to the queue size. It is observed that fairness tends to decrease for
larger queue sizes for BBRv2. The elephant flows occupy a much larger portion
of the bottleneck bandwidth compared to mice flows particularly as the queue
size increases from 10 BDP. For FaiRTT, both flows maintain almost stable and
equitable throughput share for queue sizes more than 10 BDP. The algorithm
performs best in terms of equitable throughput share at 2 BDP. For values less
than 2 BDP, the performance of both algorithms is similar. The average
throughput ratio between the competing flows is 1.46 for BBRv2 and 1.04 for
FaiRTT.

We further evaluated the performance of FaiRTT for different RTTs. The RTT of
elephant flows was varied from 5 to 30 ms and that of the mice flows was fixed
to 5 ms. Fig. 3c displays throughput changes of both algorithms when mice
flows compete with elephant flows with different RTTs for a 10 Mbps bottleneck
bandwidth, 10 ms delay on a 10 BDP buffer size. It can be observed that as the
RTT difference between the flows increases, the fairness between the flows
deteriorates. The minimum and maximum throughput ratio between the competing
flows 1.02 and 1.18 for FaiRTT considerably reduces as compared to 1.32 and
1.82 for BBRv2. The overall average throughput ratio is 1.11 for FaiRTT and
1.44 for BBRv2.

####  IV-B2 Jain’s Fairness Index

(a) FaiRTT vs. BBRv2.

(b) With different RTT Values.

(c) With different Queue Size Values.

Figure 4: Fairness Index.

Fig. 4a illustrates Jain’s fairness index evaluation for both algorithms. We
set bottleneck bandwidth at 10 Mbps and 10ms of delay. Elephant and mice flows
have an RTT of 15 ms and 5 ms, respectively and they coexist on a 10 BDP
bottleneck buffer size. For BBRv2, we recorded a minimum fairness index of
about 0.88 and an average of 0.95 as compared to FaiRTT with 0.91 and 0.99.
Fig. 4c shows the effect of varying bottleneck queue size from 0.5 BDP to 100
BDP on the fairness index for both algorithms. The fairness index exhibits the
maximum value at 2 BDP for FaiRTT at 0.99 as compared to 0.95 for BBRv2. Both
algorithms exhibit the same fairness for values less than 1 BDP. The average
fairness index of FaiRTT and BBRv2 are 0.99 and 0.95, respectively. Fig. 4b
illustrates the fairness index amongst competing elephant and mice flows with
varying RTTs. Despite the fairness indexes’ slight decrement when competing
with elephant flows with increasing RTT, FaiRTT outperforms BBRv2. For
competing flows of 5 ms and 30 ms, FaiRTT has a fairness index of 0.99 versus
0.93 for BBRv2. The maximum value of the fairness index obtained for FaiRTT is
0.99 as against 0.97 for BBRv2 at 5 ms and 10 ms competing flows. The average
fairness index for FaiRTT and BBRv2 are 0.98 and 0.94, respectively. This
demonstrates that FaiRTT enhances Jain’s fairness index and can guarantee
improved fairness amongst elephant and mice flows in different scenarios.

####  IV-B3 Bottleneck link utilization

Fig. 5 shows the bottleneck bandwidth utilization with the bandwidth set to 10
Mbps, delay at 10 ms, and a bottleneck buffer size of 10 BDP. FaiRTT achieves
an average link utilization of 98.78% as compared to 97.21% in BBRv2. Hence,
our proposed algorithm demonstrated its ability to optimize and utilize the
available bottleneck bandwidth efficiently as compared to BBRv2.

Figure 5: Bottleneck Link Utilization of FaiRTT vs. BBRv2.

##  V Conclusion

In this paper, we developed FaiRTT, a novel algorithm that dynamically
estimates the BDP sending rate based on RTT measurements to achieve equitable
bandwidth allocation between elephant and mice flows, addressing the drawback
of BBR. In FaiRTT, we modeled the in-flight dependency on BDP, bottleneck
bandwidth, and packet departure time after every ACK. We evaluated the
performance of FaiRTT through extensive simulation experiments on multiple
flows with different RTTs and buffer sizes using NS-3. The results show that
FaiRTT has a better average throughput ratio of 1.08 between elephant and mice
flows and an average fairness index of up to 0.98 compared to that of BBRv2
while improving the overall bottleneck link bandwidth utilization to
$98.78\%$. By effectively resolving intra-protocol RTT fairness without
compromising throughput on the bottleneck link, FaiRTT emerges as a promising
solution to enhance overall network utilization and address the challenges
posed by varying flow characteristics in next-generation networks. The
findings of this study contribute valuable insights to the ongoing efforts in
optimizing BBR for the evolving landscape of communication networks.

## Acknowledgment and Disclaimer

This research is supported by the Ministry of Education, Singapore, under its
Academic Research Tier 1 (Grant WBS number: R-R12-A405-0003).

Any opinions, findings, conclusions, or recommendations expressed in this
material are those of the author(s) and do not reflect the views of the
Ministry of Education, Singapore.

## References

  * [1] N. Cardwell _et al._ , “BBR: Congestion-based Congestion Control,” _ACM Queue_ , vol. 14, no. 5, 2016.
  * [2] C. Chaccour _et al._ , “Seven defining features of terahertz (THz) wireless systems: A fellowship of communication and sensing,” _IEEE Communications Surveys & Tutorials_, vol. 24, no. 2, pp. 967–993, 2022.
  * [3] L. Kleinrock, “Power and Deterministic Rules of Thumb for Probabilistic Problems in Computer Communications,” in _IEEE ICC_ , 1979.
  * [4] N. Cardwell _et al._ , “BBR Congestion Control,” in _IETF 97th Meeting_ , 2016.
  * [5] D. Scholz _et al._ , “Toward a Deeper Understanding of TCP BBR Congestion Control,” in _IFIP Networking_ , 2018.
  * [6] M. Hock _et al._ , “Experimental Evaluation of BBR Congestion Control,” in _Proc. International Conference on Network Protocols (ICNP)_ , 2017.
  * [7] S. Scherrer _et al._ , “Model-based Insights on the Performance, Fairness, and Stability of BBR,” in _ACM IMC 2022_ , 2022.
  * [8] C. K. Njogu _et al._ , “BBR-With Enhanced Fairness (BBR-EFRA): A New Enhanced RTT Fairness for BBR Congestion Control Algorithm,” _Computer Communications_ , vol. 200, pp. 95–103, 2023.
  * [9] W. Pan _et al._ , “Improvement of BBRv2 Congestion Control Algorithm Based on Flow-aware ECN,” _Sec. and Commun. Netw._ , Jan. 2022.
  * [10] ——, “Improved RTT Fairness of BBR Congestion Control Algorithm Based on Adaptive Congestion Window,” _Electronics_ , vol. 10, no. 5, 2021.
  * [11] N. Cardwell _et al._ , “BBRv2: A Model-based Congestion Control,” in _Proc. IETF 102th Meeting_ , 2018.
  * [12] S. Zhang, “An evaluation of BBR and its variants,” _arXiv preprint arXiv:1909.03673_ , 2019.
  * [13] Y.-J. Song _et al._ , “Intra-protocol Convergence Problem in BBRv2’s Bandwidth Probing,” in _ICTC 2020_ , 2020, pp. 1016–1018.
  * [14] A. Nandagiri _et al._ , “BBRv1 vs BBRv2: Examining Performance Differences through Experimental Evaluation,” in _IEEE LANMAN 2020_ , 2020.
  * [15] S. Ma _et al._ , “Fairness of congestion-based congestion control: Experimental evaluation and analysis,” _arXiv preprint arXiv:1706.09115_ , 2017.
  * [16] G.-H. Kim _et al._ , “Fairness Improvement of BBR Congestion Control Algorithm for Different RTT Flows,” in _ICEIC 2019_ , 2019.
  * [17] ——, “Enhanced BBR Congestion Control Algorithm for Improving RTT Fairness,” in _ICUFN 2019_ , 2019, pp. 358–360.
  * [18] G.-H. Kim and Y.-Z. Cho, “Delay-aware BBR Congestion Control Algorithm for RTT Fairness Improvement,” _IEEE Access_ , vol. 8, 2019.
  * [19] N. Cardwell _et al._ , “BBRv3: Algorithm Bug Fixes and Public Internet Deployment,” in _Proc. IETF 117th Meeting_ , 2023.
  * [20] J. A. Gomez Gaona, E. Kfoury, J. Crichigno, and G. Srivastava, “Evaluating TCP BBRv3 performance in wired broadband networks,” 2023.
  * [21] R. Jain _et al._ , “A quantitative measure of fairness and discrimination,” Digital Equipment Corporation, Hudson, MA, USA, Tech. Rep., 1984.