Synchronization of TCP Flows in Networks with Small DropTail Buffers
ABSTRACT A recent fluidmodel formulation of an Internet congestioncontrol problem considers routers with small Droptail buffers. This paper is interested in the oscillatory regime of such networks and considers a topology where two TCPcontrolled flows (each regulated by separate (edge) routers) merge to compete for bandwidth at a common (core) router. We describe this dynamic using a weaklycoupled oscillator model and analyze the coherence of oscillation as a function of coupling strength. We show that increased coupling leads to increased coherence and to larger variations in the arrival flow at the core router. The coupling strength can be expressed in terms of network parameters.
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Conference Paper: Nonlinear oscillations in TCP networks with DropTail buffers
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Page 1
Synchronization of TCP Flows
in Networks with Small DropTail Buffers
H. Han, C.V. Hollot, Don Towsley and Y. Chait
Abstract—A recent fluidmodel formulation of an Internet
congestioncontrol problem considers routers with small Drop
tail buffers. This paper is interested in the oscillatory regime
of such networks and considers a topology where two TCP
controlled flows (each regulated by separate (edge) routers)
merge to compete for bandwidth at a common (core) router.
We describe this dynamic using a weaklycoupled oscillator
model and analyze the coherence of oscillation as a function
of coupling strength. We show that increased coupling leads to
increased coherence and to larger variations in the arrival flow
at the core router. The coupling strength can be expressed in
terms of network parameters.
I. INTRODUCTION
The TCP (Transmission Control Protocol) is widelyused
for reliable data transmission through the Internet. TCP
sources have adjustable sending rates, and on route to
destinations, this data is buffered and directed to different
links. At the beginning of a session, TCP sources send data
at an exponential rate which is called the slowstart phase. As
traffic rates approach a link’s capacity, packets are queued at
the link buffers. Some packets may be dropped or marked at
the routers by mechanisms called Active Queue Management
(AQM)s, and such packets will trigger TCP receivers to issue
negative acknowledgements back to the source. Upon receipt
of a negative acknowledgement, a TCP source halves its
sending rate and enters the socalled congestion avoidance
phase. At this stage, a TCP source increases its rate linearly
on receiving a positive acknowledgement and decreases its
rate by half after receipt of a negative acknowledgement.
The previously mentioned AQM mechanisms were intro
duced to anticipate and improve performance such as net
work throughput. Different types of AQMs include DropTail,
Random Early Detection (RED) [1], proportionalintegral
(PI) [3], Adaptive Virtual Queue (AVQ) [2], Random Early
Marking (REM) [4], and their variants. The most widely
implemented AQM in the current Internet is the Droptail
queue which is simple; i.e., if the buffer is full, all incoming
This work is supported in part by the National Science Foundation under
Grant ANI0238299,
H. Han is with the Electrical and Computer Engineering De
partment, University of Massachusetts, Amherst, MA 01003, USA
hhan@ecs.umass.edu
C.V. Hollot is with the Electrical and Computer Engineering De
partment, University of Massachusetts, Amherst, MA 01003, USA
hollot@ecs.umass.edu
DonTowsley iswiththe
University of Massachusetts,
towsley@cs.umass.edu
Y. Chait is with the Mechanichal and Industrial Engineering De
partment, University of Massachusetts, Amherst, MA 01003, USA
chait@ecs.umass.edu
Computer
Amherst,
Science
MA
Department,
01003,USA
packets are dropped. Stability issues of TCP/AQM systems
are of recent interest, and readers are referred to [5] – [12]
and the references contained therein. However, there has been
little research devoted to the analysis of Droptail routers,
which is mostlikely attributable to the discontinuous nature
of its feedback. Recently, a model [13] was proposed for
the case of small Droptail buffers where it’s argued that,
for large numbers of flows, the blocking probability of an
M/M/1 queue is a suitable model for packet loss. As a result,
a fluid model is adopted for analysis where it is demonstrated
that limitcycling occurs when the system becomes linearly
unstable.
The present paper is interested in the oscillatory regime
of small Droptail buffer networks, and our approach is
inspired by the research on oscillator synchronization; e.g.,
see [14] – [17]. This research has exposed the interesting and
rich phenomenon arising when many independent oscillators,
with differing intrinsic frequencies, are coupled together. The
signature behavior of such networks is characterized by the
oscillators spontaneously locking to a common frequency.
The contribution of this paper is a network problem
formulation for small Droptail buffers that admits a model
of weaklycoupled oscillators. More specifically, we consider
a situation where two TCPcontrolled flows (each regulated
by separate (edge) routers) merge to compete for bandwidth
at a common (core) router; see Figure 2. Presumption that
the bandwidth of the core router is larger than the edge
router implies that the two TCP flows are weakly coupled. If
each of the two TCP flows are intrinsically oscillatory (as a
result of interaction with their Droptail edge routers), then we
have an instantiation of weaklycoupled oscillators. The edge
routers induce intrinsic oscillations that are weakly coupled
by the core. Our ensuing analysis shows that as coupling
strength increases, the synchronizing frequency converges to
a constant which is decided only by the roundtrip time
delays, regardless of the intrinsic frequencies. When the
coupling strength exceeds a bound, the synchronizing state
becomes unstable.
The rest of this paper is outlined as follows. In Section
II, the TCP dynamics at the edge routers are described and
the intrinsic frequencies computed. In Section III, a core
router is considered and a model for the weeklycoupled
oscillators is derived. In Section IV we provide analysis and
describe the synchronized state, and its stability, in terms of
network parameters. In Section V we provide simulations
of both the fluid and discreteevent (ns) network models.
Proceedings of the
44th IEEE Conference on Decision and Control, and
the European Control Conference 2005
Seville, Spain, December 1215, 2005
ThA14.5
0780395689/05/$20.00 ©2005 IEEE
6762
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II. TWO UNCOUPLED TCP FLOWS
In this section we consider two uncoupled TCP flows
that experience congestion at smallsized Droptail buffers.
We assume that each of these loops are linearly unstable
and approximate their ensuing limitcycling with harmonic
oscillations. We express their intrinsic frequencies in terms
of the Droptail buffer lengths and roundtrip times.
A. TCP dynamics with small Droptail buffer
Consider two TCPcontrolled flows each passing through
a congested edge router as shown in Figure 1. From [7], a
1c
2c
2b
1b
Fig. 1.
edge router
Two distinct groups of TCPcontrolled flows traversing a common
fluid model for the TCP window dynamics is
˙ wk(t) =1
τk
−wk(t)wk(t −τk)
2τk
pk(t −τk); k = 1,2 (1)
where wk, pk and τk are the window size, edgebuffer
packetloss probability and roundtrip time (RTT) for the
kth TCP flow respectively. In [13], it was argued that for
large numbers of flows, the blocking probability of an M/M/1
queue is a suitable model for the packet loss incurred by a
small Droptail buffer. Thus, our edge routers will be assumed
to have the packet loss model
?nkwk(t)
By setting ˙ wkin (1) equal to zero, we obtain the equilibrium
window size w∗
?
nk
pk(t) =
ckτk
?bk
; k = 1,2.
(2)
k:
w∗
k=
2
?ckτk
?bk?
1
bk+2
; k = 1,2.
(3)
Linearizing (1) about this equilibrium point then gives
∆ ˙ wk(t) = −
1
kτk(∆w+(bk+1)∆w(t −τk))
k+∆wk.
w∗
(4)
where wk= w∗
B. Intrinsic frequencies
Assume that each of the TCP dynamics described in (1)
and 2 are linearly unstable. We approximate their ensuing
limitcycle oscillations with harmonic oscillations. To obtain
their frequency, we substitute ∆wk= rkeiθk(t)into (4) and
obtain
1
w∗
rkeiθk(t) ˙θk= −
kτk(rkeiθk(t)+(bk+1)rkeiθk(t−τk))
(5)
where θk(t) = ωkt +φkand where ωkdenotes the intrinsic
frequency of the kth TCP flow and φkits phase. From (5):
1+(bk+1)cos(ωkτk) = 0
and
ωk−bk+1
w∗
kτk
sin(ωkτk) = 0.
so that
ωk=
?bk(bk+2)
w∗
kτk
.
(6)
Remark: If the edge router buffer sizes bk are small,
then the intrinsic frequencies in (6) are approximately
inversely proportional to their link capacities ck. Therefore,
as technology advances and the ck increase, the intrinsic
frequencies decrease.
III. WEAKLY COUPLED TCP FLOWS
In the previous section, we considered uncoupled TCP
flows, each passing through distinct edge routers having
small Droptail buffers. We assumed that these TCP flows
were in limitcycle oscillation and we approximated their
intrinsic frequencies ωk as in (6). Now, we modify the
network configuration in Figure 1 and allow the two edge
regulated flows to pass through a common Droptail (core)
router having link capacity C and small buffer size B; see
Figure 2.
A. Weakly coupled TCP dynamics
1c
2c
2b
C
B
1b
Fig. 2.
regulated TCP flows traverse a core router with capacity C and small
Droptail buffer size B.
Modification to the network in Figure 1 wherein the two edge
If C > c1+ c2, the core link will not be a bottleneck
but will nevertheless affect synchronization of the oscillating
TCP flows because of the core router’s coupling effect. The
window dynamics for these coupled TCP window dynamics
now becomes
τk−wk(t)wk(t −τk)
2τk
˙ wk(t)=1
(pk(t−τk)+q(t,τ1,τ2)); k=1,2
(7)
where pkis as in (2) and
q(t,τ1,τ2) =
⎛
⎝
n1w1(t−τ1)
τ1
+n2w2(t−τ2)
C
τ2
⎞
⎠
B
(8)
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where roundtrip time delays are assumed to occur in forward
path. Substituting (2) and (8) into (7), and then linearizing
gives
1
? w∗
2τk
C
⎛
C
∆ ˙ wk
=
−
kτk(∆wk+(bk+1)∆wk(t −τk))
⎛
−B(? w∗
−bk? w∗
kis the new equilibrium window length. After some
manipulations (see Appendix A for details) we transform (9)
into the following phase form:
⎛
?njrj
⎛
C
k)2
⎝∑2
⎝∑2
j=1
nj? w∗
j
τj
⎞
⎠
B
B−1
2
∑
j=1
?nj∆wj(t −τj)
Cτj
?
k
2τk
j=1
nj? w∗
j
τj
⎞
⎠
∆wk(t −τk)
(9)
where ? w∗
˙θk
=
ωk−B(w∗
k)2
2τk
⎝∑2
j=1
njw∗
τj
j
C
⎞
⎠
B−1
·
2
∑
j=1
Crkτjsin(θj(t −τj)−θk(t))
⎞
?
−bkw∗
2τk
k
⎝∑2
j=1
njw∗
τj
j
⎠
B
sin(θk(t −τk)−θk(t)). (10)
Remark: Recall that the rkare the oscillation amplitudes.
They can computed using the techniques followed in [20].
For simplicity, we now focus on the case when the
flows experience equal coupling from the core router.
B. Equallycoupled TCP flows
Suppose n1= n2= n, c1= c2= c, b1= b2= b and that
τ1≈ τ2
Consequently, (10) can be expressed as the coupled oscilla
tors:
˙θk
=
ωk−Kbsin(θk(t −τk)−θk(t))−
BK
2
∑
j=1
.= τ. Then, w∗
k
.= w∗=
?
2?cτ
n
?b? 1
b+2, and r1≈ r2.
2
(sin(θj(t −τj)−θk(t))); k = 1,2 (11)
where the coupling strength K is given by
K =nB
2cB
0
?w∗
τ
?B+1
and where c0=C
2.
Remark: The model (11) differs from the standard
coupled oscillators model in that the coupling term
sin(θk(t − τk) − θk(t)) contains the roundtrip time delay
τ. The work in [16] considers such a timedelay coupled
oscillator model and, in the next section, we extend some
of [16]’s results to the particular situation in (11).
IV. MAIN RESULT
Our next result gives conditions under which the coupled
oscillators (11) synchronize.
Theorem
in (11) and assume that the difference in roundtrip time
delays is small; i.e., τ1≈ τ2(ω1≈ ω2). Suppose there exists
nonnegative numbers Ω and φ0satisfying:
1: Consider the coupledoscillators described
KBsinφ0cos(Ωτ) = ω2−ω1
(12)
and
Ω = ω +K(b+B
2)sin(Ωτ)+KB
2
sin(Ωτ +φ0).
(13)
Then, the coupled oscillators (11) synchronize at frequency
Ω with phase difference φ0. The synchronized state is locally
stable iff
K(b+B)cos(Ωτ) < 0.
(14)
Proof: From (11) we form the differential equation in
the phase difference to give:
˙θ1(t)−˙θ2(t)
K(b+B
=
ω1−ω2−
2)(sin(θ1(t −τ1)−θ1(t))−sin(θ2(t −τ2)−θ2(t)))−
(sin(θ2(t −τ2)−θ1(t))−sin(θ1(t −τ1)−θ2(t)))
Let φ(t).= θ1(t)−θ2(t) define phase difference between the
oscillators and let φ0 denote its steadystate value (defined
by˙φ = 0). Then, since τ1≈ τ2:
sin(θ1(t −τ1)−θ1(t))−sin(θ2(t −τ2)−θ2(t)) ≈ 0,
and
KB
2
(15)
θ1(t −τ1)−θ2(t) ≈ θ2(t −τ2)−θ1(t)+2φ0.
(15) now becomes
KBsinφ0cos(θ2(t −τ2)−θ1(t)+φ0) ≈ ω2−ω1
which implies that θ2(t − τ2) − θ1(t), θ2(t − τ2) − θ2(t),
θ1(t−τ1)−θ1(t), and θ1(t−τ1)−θ2(t) are constant. There
fore,
(16)
˙θ1
=
˙θ2
ω1−K(b+B
KB
2
ω2−KB
2
K(b+B
Ω.
=
2)sin(θ1(t −τ1)−θ1(t))−
sin(θ2(t −τ2)−θ1(t))
sin(θ1(t −τ1)−θ2(t))−
2)sin(θ2(t −τ2)−θ2(t))
=
.=
Consequently,
θ1(t) = Ωt,
θ2(t) = Ωt −φ0,
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Page 4
Ω = ω +K(b+B
2)sin(Ωτ)+KB
2
sin(Ωτ +φ0),
and
KBsinφ0cos(Ωτ) = ω2−ω1.
As K increases, φ0approaches zero and (13) becomes
Ω = ω +K(b+B)sin(Ωτ).
(14) then follows directly from [17].
?
Remark 1: If τ = 0, then (12) and (13) are satisfied
with sinφ0=ω2−ω1
KB
usual synchronization results for zero time delays.
Remark 2: Let K ∈ (Kc,Ku) denote the range of coupling
strengths K for which (12) – (14) are feasible. When
K < Kc, no synchronization occurs between two flows.
When Ku>K >Kc, the phase difference between oscillators
is locked. As we increase K, the phase difference decreases,
similarto case of nondelayed
However, thesynchronizing
when K → ∞. Indeed, for large K, φ0 is very small,
and Ω = ω + K(b + B)sin(Ωτ) from (13). To make
K(b+B)sin(Ωτ) small for large K, we will have Ωτ ≈ π,
so that Ω ≈π
do not affect the synchronizing frequency. Finally, when
K > Ku, the synchronized state becomes unstable.
Remark 3: Let reiψ(t).=1
the coupled oscillators are synchronized, r = cos(φ0) which
increases with coupling strength K. When synchronized, r
measures the coherence of the coupled oscillators and is a
measure of the variation in TCP traffic at the core router.
and Ω =ω1+ω2
2
, which recovers the
coupled oscillators.
approachesfrequency
π
τ
τ. Under this situation, the intrinsic frequencies
2(sin(θ1(t))+sin(θ2(t))). When
V. SIMULATIONS
A. Model Simulations
In this section, we present simulations of the coupled
oscillators model (11). In these simulations, τ1= 0.09s,
τ2= 0.11s, ω1= 99rps, ω2= 101rps, b = B = 20pkts. We
increase the coupling strength K from Kc= 0.03 to 40 and
plot the synchronizing frequency Ω against coupling strength
K in Figure 3. When K ≥ 50, the synchronized state is
unstable. From Figure 3, we observe that the synchronizing
frequency Ω converges toπ
nondelayed case; see Remarks 1 above. We also plot the
coherence r with respect to the coupling strength in Figure 4.
We see that flows quickly become coherent.
τ. This is quite different from the
B. ns simulations
In addition to fluidmodel simulations, we also conductted
ns (discreteevent) simulations of the TCP network de
scribed in Figure 1. The parameters are c1= c2= 0.4Mbps,
b1=b2=10pkts, τ1=40ms and τ2=44ms. For C =1Mbps
and B = 10pkts, the TCP window lengths behave as unsyn
chronized oscillators as shown in the top plot of Figure 5.
For C = 0.801Mbps, B = 2,pkts the two flows become syn
chronized as shown in the bottom plot of Figure 5. We also
plot the total and average window sizes for the synchronized
and unsynchronized cases in Figure 6. The dashed traces
10
−2
10
−1
10
0
10
1
10
2
30
40
50
60
70
80
90
100
Log(K)
Synchronized oscillating frequency (Ω)
Synchronized oscillating frequency for 2 flows
Fig. 3.
strength K.
Synchronizing frequency approachesπ
τwith increasing coupling
10
−2
10
−1
10
0
10
1
10
2
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Coherence of synchronized 2 flows
Log(K)
Coherence(r)
Fig. 4. Coherence r as a function of coupling strength K.
150 160170 180190 200
10
20
30
non−synchronization (B=10pkts, c=1.0Mbps)
time (s)
Window size (pkts)
150 160 170180 190 200
10
20
30
synchronization (B=2pkts C=0.801Mbps)
time (s)
Window size (Pkts)
Fig. 5.
ns simulation of two weaklycoupled TCP flows.
6765
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mark the unsynchronized TCP flows, while solid traces are
for the synchronized flows. The bolder lines denote the
averages. We observe that the average TCP windows for the
unsynchronized TCP flows is larger than the synchronized
case. Synchronization adversely affects throughput through
the core router.
150 160170 180190200
25
30
35
40
45
50
55
60
65
time (s)
window sizes (pkts)
Synchronized and dys−synchronized 2 TCP flows
Sum of dys−synchronized window sizes
Average dys−synchronized window sizes
Sum of synchronized window sizes
Average of syncrhonized window sizes
Fig. 6.
throughput.
ns simulation showing that synchronized TCP flows have lower
VI. CONCLUSION
Typical stability analysis of the congestionavoidance
phase of TCP networks has been primarily concerned with
motion about the equilibrium and, for the most part, has
been confined to local behavior. In contrast, this paper was
interested in the oscillatory regime of DropTail networks,
and in the synchronization and coherence of TCP flows. One
application of such results is towards the sizing of DropTail
buffers.
APPENDIX
Since the coupling is weak, i.e., q defined as (8) is small.
So ? w∗
∆ ˙ wk
=
−
w∗
⎛
C
k≈ w∗
k, where w∗
kis defined as (3). Therefore, (9)
becomes
1
kτk(∆wk+(bk+1)∆wk(t −τk))−
njw∗
j
τj
⎠
bkw∗
k
2τk
C
B(w∗
2τk
k)2
⎝∑2
⎝∑2
j=1
⎞
B−1
2
∑
j=1
?nj∆wj(t −τj)
Cτj
?
−
⎛
j=1
njw∗
τj
j
⎞
⎠
B
∆wk(t −τk)
Next, substitute ∆wk= rkeiθk(t)into above equation and
obtain
?
B(w∗
2τk
C
⎛
C
irkeiθk(t)˙θk= −
1
kτk
njw∗
τj
w∗
rkeiθk(t)+(bk+1)rkeiθk(t−τk)?
⎞
j=1
−
k)2
⎛
⎝∑2
⎝∑2
j=1
j
⎠
B
B−1
2
∑
?
njrjeiθj(t−τj)
Cτj
?
−
bkw∗
2τk
k
j=1
njw∗
τj
j
⎞
⎠
rkeiθk(t−τk).
Then,
i˙θk= −
1
kτk
w∗
⎛
?
1+(bk+1)eiθk(t−τk)−iθk(t)?
njw∗
j
τj
C
∑
j=1
⎞
−
B(w∗
2τk
k)2
⎝∑2
⎝∑2
j=1
⎞
⎠
B
B−1
2
?
njrjeiθj(t−τj)−iθk(t)
Crkτj
?
−
bkw∗
2τk
k
⎛
j=1
njw∗
τj
j
C
⎠
eiθk(t−τk)−iθk(t).
Since the coupling is weak, at the fast time scale, the
oscillators behaves like independently. So, as for the first
term, let θk≈ θk. Therefore, the above equation becomes
i˙θk= −
w∗
⎛
C
1
kτk
?
1+(bk+1)eiθk(t−τk)−iθk(t)?
njw∗
j
τj
⎠
njw∗
j
τj
⎠
−
B(w∗
2τk
k)2
⎝∑N
⎝∑N
j=1
⎞
B−1
N
∑
j=1
?
njrjeiθj(t−τj)−iθk(t)
Crkτj
?
−
bkw∗
2τk
k
⎛
j=1
C
⎞
B
eiθk(t−τk)−iθk(t).
After substitution we obtain
i˙θk= iωk−
B(w∗
2τk
⎛
k)2
⎛
⎝∑2
⎝∑2
j=1
njw∗
τj
j
C
⎞
⎠
B
B−1
2
∑
j=1
?
njrjeiθj(t−τj)−iθk(t)
Crkτj
?
−
bkw∗
2τk
k
j=1
njw∗
τj
j
C
⎞
⎠
eiθk(t−τk)−iθk(t).
By equating the imaginary part, we obtain the coupled
oscillators model (10)
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Page 6
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