RTT Prediction in Heavy Tailed Networks

Abstract

TCP is the most used transport protocol in the Internet and it relies on RTT (Round Trip Time) predictions for the retransmission control algorithm. Most of the algorithms reported in the literature consider memoryless traffic characteristics and do not study the performance under heavy tailed scenarios present in the Internet. In this paper, an algorithm for RTT prediction in a heavy-tailed environment is introduced, and it is shown to follow closely and accurately the actual RTT. The proposed algorithm is simple and permits online implementations. Results are compared with those obtained with other methodologies for real trace sets. It is shown that the proposed algorithm leads to a lower prediction error.

Full text

700 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 4, APRIL 2014
RTT Prediction in Heavy Tailed Networks
L. Rizo-Dominguez, D. Munoz-Rodriguez, C. Vargas-Rosales, D. Torres-Roman, and J. Ramirez-Pacheco
Abstract—TCP is the most used transport protocol in the
Internet and it relies on RTT (Round Trip Time) predictions
for the retransmission control algorithm. Most of the algorithms
reported in the literature consider memoryless traffic charac-
teristics and do not study the performance under heavy tailed
scenarios present in the Internet. In this paper, an algorithm
for RTT prediction in a heavy-tailed environment is introduced,
and it is shown to follow closely and accurately the actual
RTT. The proposed algorithm is simple and permits online
implementations. Results are compared with those obtained with
other methodologies for real trace sets. It is shown that the
proposed algorithm leads to a lower prediction error.
Index Terms—Heavy tail jitter distribution, RTO, RTT pre-
diction and Internet packet delay.
I. INTRODUCTION
FORECASTING is a common practice in engineering as it
permits anticipation of the best parameter settings to oper-
ate a system optimally. Nevertheless, predictions often depart
from true values due to the presence of unforeseen variations
that upset the system performance. A common forecasting
practice is to rely on the expected value since it optimizes
the mean square error criterion between the estimated and
the true value. This approach is valid when the assessment is
conducted over a large number of results and large time scales.
However, several system operations demand a short-term pre-
diction and must be based on recent observations of the current
realization. Also, network nodes carry out prediction of diverse
parameters in different protocols so that resources are used
as efficient as possible. A dominant transport protocol in the
Internet is TCP (Transmission Control Protocol), [1], where
the packet retransmissions are directly related to the current
RT T (Round Trip Time) parameter. However, as the packets
travel along the route they will be exposed to noise, queueing-
delays and packet losses that cause errors that render in RT T
variations. Therefore, it is necessary to forecast the RT T
parameter value in order to regulate the packet retransmission
process. For instance, if the predicted value (also known as
Retransmission Time Out -RT O) is lower than the actual
RT T , a packet loss is assumed, and the packet is retransmitted
reducing the effective bandwidth. On the other hand, an
RT O overestimation leads to a delayed packet loss detection
upsetting the congestion control process, [2], [3]. Thus, an
Manuscript received December 2, 2013. The associate editor coordinating
the review of this letter and approving it for publication was B. Bellalta.
L. Rizo is with ITESO (e-mail: lrizo@iteso.mx).
David Munoz and Cesar Vargas are with Tecnologico de Monterrey,
Campus Monterrey, N.L., 64849 (e-mail: {dmunoz, cvargas}@itesm.mx).
D. Torres is with CINVESTAV, Guadalajara (e-mail: dtor-
res@gdl.cinvestav.mx).
J. Ramirez is with the Universidad del Caribe (e-mail:
jramirez@ucaribe.edu.mx).
Digital Object Identifier 10.1109/LCOMM.2014.013114.132668
RT T prediction with good accuracy is fundamental to avoid
situations with congestion and delays. Packet delay prediction
often depends on the correlation characteristics of the data.
Some approaches propose simplified estimators that rely both,
on past observation and on previous predictions. A common
estimator proposed by Jacobson, [4], behaves adequately in
environments with Gaussian distributed delay with a small cor-
relation. However, for heavy-tailed scenarios, the predictions
depart substantially from the true value. Alternative methods,
[3], suggest an estimator where past observations are window
weighted for a set of realizations. In this paper, we present,
under a bound error criterion, a new simplified RT T predictor
suitable for heavy-tailed scenarios that exhibits a reduced
forecast error. It is also shown that small prediction errors
are seldomly achievable for big jitter dispersion scenarios and
that there is a tradeoff between a desirable small prediction
error and the jitter dispersion. Feasible regions according to
traffic characteristics and QoS requirement are also reported.
II. THE PROPOSED PREDICTOR
In this section, we introduce an RT T prediction algorithm
under an error bound probability criterion in a heavy tailed
environment. Let RT T (k)denote the round trip delay experi-
enced by the (k−1)-th packet, and let the next packet delay
prediction RT O(k)be of the form
RT O(k)=RT T (k−1) + ξ, (1)
where ξstands for a travel time variation that will be set
to keep probabilistically the error between the forecasted and
the true value, e(k)=RT T (k)−RT O(k), within specified
constraints.
It has already been said that some values of e(k)impact
adversely network performance. For instance, a positive value
of error e(k)indicates an RT T underestimation, leading to
unnecessary retransmissions and throughput reduction; while a
negative value will slow down recovery of network congestion
control.
The prediction error e(k)can be considered a random
variable that it is desirable to keep within certain limits. A
first approach could be to use the Chebyshev inequality that
takes the form
P{|e(k)|>}≤ E{(RT T (k)−RT O(k))2}
2=q. (2)
Stating that the absolute error |e(k)|is mean to exceed a value
with a probability smaller than a value q. This formulation is
not applicable to heavy tailed environments as the probability
of |e(k)|becoming large is not negligible and its variance may
not be defined. This may lead to a meaningless inequality. An
1089-7798/14$31.00 c
2014 IEEE

RIZO-DOMINGUEZ et al.: RTT PREDICTION IN HEAVY TAILED NETWORKS 701
alternative probabilistic performance criterion can be written
as
P{|e(k)|≤}≥ψ, (3)
where ψis a definable quality of service (QoS) parameter,
denoting the minimum proportion of time that the prediction
error is below the allowance error . The value of this error
allowance tends to be small, i.e., →0, thus the criterion
shown in (3). Now, considering the definition of error e(k)=
RT T (k)−RT O(k)and substituting RT O(k)from (1), then
(3) can be rewritten as
P{−+ξ≤RT T (k)−RT T (k−1) ≤+ξ}≥ψ, (4)
The difference RT T (k)−RT T (k−1) = J(k)is known as
the Internet delay-jitter and it has been reported to exhibit a
heavy tailed behavior [12], [13]. In particular J(k)follows a
Cauchy distribution, [10], i.e., the distribution of J(k)is
P{J(k)≤x}=1
2+1
πtan−1x
γ≥ψ, (5)
Therefore, using (5) in (4), we get
P{−+ξ≤J(k)≤+ξ}=1
πtan−1(+ξ
γ)
−1
πtan−1(−+ξ
γ)≥ψ(6)
Equation (6) is further reduce to
P(−+ξ≤J(k)≤+ξ)= 1
πtan−12γ
γ2−2+ξ2≥ψ
(7)
where parameter γis the jitter dispersion induced by the
Internet queueing process. Taking the tangent on both sides in
inequality (7), and taking advantage of the monotonic behavior
of the tangent function, for a given specified quality of service
parameter ψ(QoS) and error allowance the estimated travel
time variation can be given by
ξ2≤2γ
tan(πψ)+2−γ2(8)
Note that ξ2is upper bounded by a quadratic form of ,and
it can be verified that ξ2is real-valued for >γ{csc(πψ)−
cot(πψ)}.
Thus, for instance if the QoS is set to be of ψ=0.9, i.e.,
at least 90% of the time the delay-jitter is within the limits
desired, the error requirement cannot be smaller than 8.61γ
Figure 1 shows the normalized /γ feasible allowance region
for a given dispersion γand a probability requested criterion
ψ. As jitter dispersion increases, as in the case of heavy tailed
scenarios, the ratio /γ decreases, compromising QoS ψ.
Finally, according to (1) and (8) the value of the RT T
predictor is determined by the QoS parameter ψ, the desired
allowance , and the jitter-delay γdispersion. This is
RT O(k)=RT T (k−1) + 2γ
tan(πψ)+2−γ2.(9)
ψ
ε / γ
0 0.2 0.4 0.6 0.8 1
0
5
10
15
Feasibility region
Fig. 1. /γ feasibility region for a given probability ψ.
30 40 50 60 70 80 90 100
0
200
400
600
800
1000
1200
1400
1600
1800
2000
k − packet number
RTO(k) − ms
True RTT
Jacobson´s
RWM
CRTTP
Fig. 2. RT O(k)predictor comparison.
III. RESULTS AND DISCUSSION
In order to evaluate the performance of the proposed algo-
rithm, the predictor was assessed using reported traffic traces.
We present results that have been obtained by means of the
available data in those traces and using a random sampling
rate stated in the corresponding references [7], [8], [9]. We
present results that compare the performance of the proposed
predictor to that of other methods [9] by showing values of
the RT T predicted by all these methods and those true values
obtained analyzing the reported traffic traces. We also show
performance in terms of the error between the predicted and
the true value, e(k)=RT T (k)−RT O(k).
It can be seen in equations (8) and (9), that knowledge
of the jitter dispersion γparameter is required. This can be
obtained either from quartiles and interactions, [6]. Since it
is expected that the γestimation be in real-time, it must be
simple and swift. Thus, use of the quartile based estimation
is recommended. For the purpose of this work, Mc. Culloch
estimation, [5], is used to find the dispersion parameter.

702 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 4, APRIL 2014
0 20 40 60 80 100
−15000
−10000
−5000
0
5000
k − packet number
e(k)−ms
Jacobson´s
RWM
CRTTP
Fig. 3. Error of predictors.
Figure 2 presents the values of the predicted RT T , i.e.,
RT O(k), obtained by using the Jacobson estimator (estimator
mostly used in current TCP implementations), the Recursive
Weighted Median (RWM), [3], which is the algorithm de-
signed for general heavy tailed environment, and the proposed
Cauchy Round Trip Time Predictor (CRTTP). These predictors
are compared to the true RT T values obtained from the traffic
trace.
The results show that Jacobsons algorithm is very sensitive
to RT T sudden changes (large excursions from the mean
RT T values), and exhibits quick response to changes, but
a very slow recovery. We can also see that most of the
time Jacobsons algorithm provides overestimates of the true
RT T values. On the other hand, also in Figure 2, we can
see that performance of the predictors RWM and CRTTP
lead to significant accuracy and good recovery timing of the
predictions of RTT which result in closer predictions to the
true RTT values.
Figure 3 compares the error performance criterion for the
different predictors compared already in Figure 2. It can be
observed that the proposed predictor CRTTP has a smaller
error than any of the other methods for a heavy tailed delay
environment. It can also be seen that for some periods of
time, the RWM predictor is better than the CRTTP, but for
other periods of time, RWM does not have small values of
error. For all the time period, it can be seen that Jacobsons
algorithm gives error values larger than those of the other
methods.
The root mean square error (RMSE) in the observation
windows is also compared for different data sets and presented
in Table I. It can be seen that the proposed algorithm predictor
offers a better accuracy while keeping lower complexity for
online implementation.
In Figure 4 it can be seen that the proposed predictor takes
advantage with a reduced expected error and the error variance
of CRTTP is the lowest. The negative error is related to
overestimation, and produces a slowly packet loss detection.
Otherwise, the positive error damages the TCP performance
TAB L E I
ROOT MEAN SQUARE PREDICTION ERROR
Prediction
Algorithm
TIME AVERAGED
SQUARE ERROR
Jacobson 19.5ms(a) 136.21ms(b) 4680ms(c)
RWM 4.36ms(a) 67.77ms(b) 969ms(c)
CRTTP 3.16ms(a) 20.1ms(b) 768.9ms(c)
(a) RMSE value for data traces in reference [9]
(b) RMSE value for data traces in reference [7]
(c) RMSE value for data traces in reference [8]
−2000 −1500 −1000 −500 0 500 1000 1500 2000
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
e(k) − ms
Probability Density
Jacobson E[e(k)]=−2,263 ,E[e2(k)]= 1.6e7
RWM E[e(k)]=−153 ,E(e2(k))= 9.26e5
CRTTP E(e(k))=−87.5 ,E(e2(k))= 5.83e5
Overestimation Error Subestimation Error
Fig. 4. Probability density of prediction error.
with needless retransmissions.
Another comparison of the algorithms is through their
execution times in a common computer platform. All the
algorithms were scripted in Matlab using an i5-Core 3.1Ghz
processor with 4GB of RAM.
The mean execution time was 4.33μs for the Jacobson
algorithm that is a reduced value due to the low computational
complexity. However, a better accuracy was obtained by WM
and CRTTP algorithms that present an execution time of
230μsand207μs, respectively. Hence, the algorithm proposed
provides better accuracy and shorter execution times than
those of the Jacobsons algorithm. It is also important to point
out that the proposed predictor, CRTTP, requires only some
20 RT T observations for an adequate performance.
IV. CONCLUSIONS
In this paper, we addressed the problem of RT T prediction
(Retransmission Time Out-RTO). A simple predictor suitable
for a heavy tailed jitter environment has been presented and
compared to other predictors in the open literature. The per-
formance of the proposed predictor is assessed using different
Internet RT T measurement sets. Results show that the Cauchy
predictor offers advantages both in terms of low complexity

RIZO-DOMINGUEZ et al.: RTT PREDICTION IN HEAVY TAILED NETWORKS 703
and lower prediction error. As a consequence of the smaller
values of error, CRTTP, results in less overestimation and
subestimation instances than the other algorithms.
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