Jitter in IP networks: A cauchy approach

Abstract

Jitter is recognized as an important phenomenon that degrades the communication performance. Particularly, in real time services such as voice and video over the Internet, there is evidence that jitter departs from already proposed Laplacian models and that it has a heavy tail behavior. In this paper, we show that an Alpha-Stable jitter model is adequate, and that in some cases the Cauchy distribution provides a satisfactory approximation. Furthermore, this work shows how the jitter dispersion increases with the number of hops in the path, following a power law with scaling exponent dependent on the index of stability ¿. This allows us to predict the expected QoS in terms of the number of nodes and traffic parameters.

Full text

IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 2, FEBRUARY 2010 1
Jitter in IP Networks: A Cauchy Approach
L. Rizo-Dominguez, D. Torres-Roman, D. Munoz-Rodriguez, and C. Vargas-Rosales, Senior Member, IEEE
Abstract—Jitter is recognized as an important phenomenon
that degrades the communication performance. Particularly, in
real time services such as voice and video over the Internet, there
is evidence that jitter departs from already proposed Laplacian
models and that it has a heavy tail behavior. In this paper, we
show that an Alpha-Stable jitter model is adequate, and that
in some cases the Cauchy distribution prov ides a satisfactory
approximation. Furthermore, this work shows how the jitter
dispersion increases with the number of h ops in the path,
following a power law with scaling exponent dependent on the
index of stability 𝛼. This allows us to predict the expected QoS
in terms of the number of nodes and traffic parameters.
Index Terms—Alpha-stable model, jitter, QoS.
I. INTRODUCTION
A
S mistiming is a major concern in telecommu nication
systems; it has been addressed, in the literature, from
different perspectives. Since jitter impair s severely real-time
applications such as videoconferencing, network g aming, VoIP
(Voice over IP) and VIP (Video over IP), among others, delay
and packet loss have been studied extensively. For instance,
Fulton and Li, [1], deal with delay in ATM networks, while
Qiong and Mills, [2], consider the jitter-bound estimation
problem from the TCP perspective; and Daniel, et. al., [3],
consider the Round Trip Time (RTT) in the Internet environ-
ment and suggest a Laplacian model.
For this study an ample data set of network delay measure-
ments was obtained and examined. Conducted observations
show that jitter has a behavior that departs from the Laplacian
distribution, [3], thus a jitter model that matches the heavy
tail behavior exhibited by packets traveling in the network
is proposed. The model helps to determine the maximum
number of allowable hops in an end-to-end path maintaining
a specified QoS. This information is relevant for routing
purposes, and for resource assignment and reservation. We
describe the heavy tail jitter observations by a general alpha-
stable representation, and show a description based on the
Cauchy distribution that provides an accurate approximation.
Applicability to QoS is also presented, and r esults comparing
against network measurements, show strong agreement. The
proposed jitter model is described in Section II. The evaluation
scenarios are presented in Section III. The jitter accumulation
law and its validation are introduced in Section IV. QoS based
Manuscript received March 24, 2009. The associate editor coordinating the
re view of this letter and approving it for publication was N. Nikolaou.
This work was partially sponsored by CONACyT
L. Rizo and D. Torres are with Research Center and Advanced
Studies, CINVESTAV, Guadalajara, Jal., Mxico (e-mail: {lrizo, dtor-
res}@gdl.cinvestav.mx).
D. Munoz and C. Vargas-Rosales are with ITESM-Campus Monterrey,
Monterrey, N.L., 64849, Mexico (e-mail: {dmunoz, cvar gas}@itesm.mx).
Digital Object Identifier 10.1109/LCOMM.2010.02.090702
on the proposed model is discussed in Section V. Concluding
remarks are given in Section VI.
II. J
ITTER MODEL
Jitter is defined as the difference of the trip d elays of
consecutive packets in an end-to-end connection. Under ideal
conditions, all packets should undergo the same delay. How-
ever, due to traffic queueing, processing time variations in the
nodes and even route changes, packets experience jitter, which
can be expressed as
𝐽
𝑁
(𝑘)=𝐷
𝑁
(𝑘) − 𝐷
𝑁
(𝑘 − 1), (1)
where 𝐷
𝑁
(𝑘) is the delay of the 𝑘-th packet as observed in the
𝑁-th node. A negative jitter is known as a packet clustering
phenomenon, and a positive is known as packet spreading. Let
𝜉
𝑖
(𝑘) be the 𝑖-th stage deterministic delay (i.e., propagation,
and processing times), and 𝑊
𝑖
(𝑘) be the 𝑖-th stage random
delay (i.e., queueing and route change phenomena), then the
end-to-end delay through the 𝑁 hops of the path is given by
𝐷
𝑁
(𝑘)=
𝑁

𝑖=1
[𝜉
𝑖
(𝑘)+𝑊
𝑖
(𝑘)]. (2)
Several studies have shown that network traffic exhibits
long range dependence, [4]; this implies, according to [5],
that the waiting time 𝑊
𝑖
(𝑘) in the queue is he avy tailed.
This has been revealed through delay measurements whose
distribution exhibits a Paretian behavior, [7]; and in 1925 L´evy
showed, [8], that Pareto laws belong to the so-called stable-
Paretian or stable non-Gaussian distribu tions. This implies that
𝑊
𝑖
(𝑘) can be modeled by an alpha-stable distribution
1
[6], and
then 𝐷
𝑁
(𝑘), in (2), converges to an alpha-stable distribution
[6] as well when 𝑊
𝑖
(𝑘) are independent. This independence
assumption is discussed in Section IV.
It is known that if two alpha-stable random variables are
independent, then their difference is also alpha-stable, [6].
Thus, jitter becomes alph a -stable with characteristic function
given by a symmetrical distribution with 𝜇 =1,𝛽=0,as
𝐶
𝛾,𝜇
𝛼,𝛽
(𝜁)
𝐽
𝑁
(𝑘)
= 𝑒𝑥𝑝(−𝛾
𝛼
∣𝜁∣
𝛼
). (3)
1
𝐷
𝑁
(𝑘) has an alpha-stable distribution if its characteristic function is, [6],
𝐶
𝛾,𝜇
𝛼,𝛽
(𝜁)
𝐷
𝑁
(𝑘)
= exp(𝑗𝜇𝜁 − 𝛾
𝛼
∣𝜁∣
𝛼
[1 − 𝑗𝛽𝑠𝑖𝑔𝑛(𝜁)𝜔(𝜁, 𝛼)]),
𝜔(𝜁, 𝛼)=
{
𝑡𝑎𝑛
(
𝛼𝜋
2
)
,𝛼∕=1,
2
𝜋
𝑙𝑜𝑔∣𝜁∣,𝛼=1,
where 𝛼 is the index of stability, 𝛾 the dispersion parameter, 𝛽 the skewness
parameter and 𝜇 the shift parameter . There are three closed forms of alpha-
stable distributions: the Gaussian distribution when 𝛼 =2, the Cauchy dis-
tribution when 𝛼 =1,𝛽=0, and Levy distribution when 𝛼 =0.5,𝛽=1.
1089-7798/10$25.00
c
⃝ 2010 IEEE

2 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 2, FEBRUARY 2010
-20
-15
-10
-5
0
i
tter CCDF
M easurem ent
lh tbl (Ch)
-8 -6 -4 -2 0 2 4 6
-40
-35
-30
-25
log jitter (ms
)
log J
i
a
l
p
h
a-s
t
a
bl
e
(C
auc
h
y
)
G aussian
E xponential
Fig. 1. Jitter survival fitting.
III. EVA L UAT I O N SCENARIOS
In order to present a realistic jitter model, extensive delay
measurements were conducted for several hops and p aths. The
observation setup involved international destinations located
at six countries in different continents: Argentina, Australia,
Japan, Mexico, France and USA. All packets traveled through
the USA. A set of some 7.2 million measur ements taken
along a 24-day period was examined. A description o f the
experiment and r ecorded data are available in [9]. The survival
tail was studied, and typical results are presented in Figure 1;
it can be seen that jitter does not fit the Laplacian nor Gaussian
models, but tail exhibits a slow decay.
From a practical perspective, system performance forecast
based on alpha-stable modeling can be cumbersome due to
the inverse transform of the characteristic function, which
does not have a close expression, but for very limited values
of the stability indexes. However, observations show that the
parameter can be close to one, thus Cauchy distribution can be
considered, [10]. Figure 2 shows an example of a normalized
histogram of the parameter alpha in a 21-node path, where the
mean stability index is 𝐸(𝛼)=0.9716
IV. J
ITTER ACCUMULATION LAW
The packet delay experienced in a hop is dependent upon
the previous unprocessed traffic in the node. However, only a
proportion of the arriving packets will have that node as final
destination, while the remaining packets will be forwarded to
other routes [11]. Since, only a fraction of the packets travels
to the next node of the path, and due to the process of cross
traffic sources, the delay from node to node exhibits a small
correlation .
The multiplexing process at the nodes allows rebuilding an
independence assumption, [12], so that the delays in the nodes
tend to be independent as traffic and connectivity increase.
Then, according to (2) and (3), the jitter accumulated in an
𝑁 hop path will have a dispersion 𝛾
𝑝
= 𝛾𝑁
1/𝛼
,where𝛾 is
the jitter d ispersion in a single node, in a homogeneous jitter
scenario, i.e., 𝛾
𝑖
= 𝛾. The experimental observations show that
although, the network environment may not be homogeneous,
0.08
0.1
0.12
0.14
e
Frecuenc
y
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.02
0.04
0.06
α
Relativ
e
Fig. 2. Normalized histogram of alpha.
0.15
0.2
0.25
γ
M easurem ents M ex-USA-Australia
pow er law
γ
p
=E(
γ
i
)N
1/E(
α
)
M easurem ents M ex-USA-Japan
pow er law
γ
p
=E(
γ
i
)N
1/E(
α
)
0 5 10 15 20 25
0
0.05
0.1
hop
Fig. 3. Jitter dispersion (𝛾) vs. number of hops (𝑁).
jitter disp ersion grows as a power law function of the number
of nodes in the path. Figure 3 shows the comparison of the
observed jitter dispersion growth along a 21-node path and
the proposed model, note that 𝛾 ≈ Σ𝛾
𝑖
/𝑁 ,and𝛼 ≈ Σ𝛼
𝑖
/𝑁 .
This is 𝛾
𝑝
= 𝐸(𝛾
𝑖
)𝑁
1/𝐸(𝛼)
.
V. J
ITTER-QOSBASED ON PROPOSED MODEL
To illustrate the use of the model, we consider as QoS,
among other criteria, [13], that the mean jitter be less than 30
ms for VoIP, and be kept under a maximum value 𝐽
𝑚𝑎𝑥
for
at least 99% of the transmitted packets. 𝐽
𝑚𝑎𝑥
is set at 30 ms
for VIP services. However, in heavy tail environments mean
and variance may diverg e and constraint 𝐽
𝑚𝑎𝑥
may be more
appropriately described in terms of distribution percentiles.
This is 𝑄
𝑜
𝑆 ≤ 𝑃 (∣𝐽
𝑁
∣≤𝐽
𝑚𝑎𝑥
). For alpha-stable jitter
distributions, this can be expressed as the infinite series, [14],
𝑃 (∣𝐽
𝑁
∣≤𝐽
𝑚𝑎𝑥
)=
2
𝜋𝛼
∞

𝑘=1
Γ(1+𝜓(𝛼,𝑘))(𝐽
𝑚𝑎𝑥
𝛾
𝑝
(𝑛))
−𝛼𝑘
𝑘⋅𝑘!
⋅ sin

−𝛼𝜋𝑘
2

,
(4)

RIZO-DOMINGUEZ et al.: JITTER IN IP NETWORKS: A CAUCHY APPROACH 3
0.97
0.98
0.99
1
e
r
|
<=30m s
)
M easurem ents Mex-USA-France
C auchy Approach
0 5 10 15 20 2
5
0.94
0.95
0.96
hop (N)
P(
|
jitt
e
Fig. 4. 𝑃 (∣𝐽
𝑁
∣≤𝐽
𝑚𝑎𝑥
) vs Number of nodes.
Fig. 5. Regions of jitter-QoS, for 𝐽
𝑚𝑎𝑥
=30ms.
𝜓(𝛼, 𝑘)=

𝛼𝑘, 0 <𝛼<1,
𝑘/𝛼, 1 ≤ 𝛼<2,
(5)
where Γ is the gamma function. It has been shown that when
𝛼 ≃ 1, a Cauchy distribution is an adequate approximation
and the QoS requirement can be expressed as
𝑄
𝑜
𝑆 ≤ 𝑃 (∣𝐽
𝑁
∣≤𝐽
𝑚
𝑎𝑥)=
2arctan(𝐽
𝑚𝑎𝑥
/𝛾
𝑝
(𝑁))
𝜋
. (6)
Figure 4 sh ows the percent of observed packets with a jitter
below 30 ms for a given hop length in path Mex-USA-France;
we see that (6) provides a good QoS approximation.
Also, substituting in (6) the cumulative dispersion 𝛾
𝑝
(𝑁)=
𝐸(𝛾
𝑖
)𝑁
1/𝐸(𝛼)
, the maximum number of hops 𝑁 to guarantee
a jitter-QoS level is given by
𝑁<

𝐽
𝑚𝑎𝑥
𝐸(𝛾
𝑖
)𝑡𝑎𝑛(𝜋𝑄
𝑜
𝑆/2)

𝐸(𝛼)
. (7)
Since we consider the Cauchy distribution, 𝐸(𝛼)=1.
In practice, routing protocols must consider the maximum
number of hops 𝑁 permitted in a path as a QoS criterion.
This relationship is illustrated in Figure 5 fo r a QoS = 0.99,
0.96, 0.98 and 0.94 as a f unction of the mean jitter dispersion.
The presented model captures the heavy tail behavior and the
dispersion of jitter for different nodes in a path, and describes
as well the jitter-QoS for N nodes.
VI. C
ONCLUSIONS
In this paper, an IP network alpha-stable jitter model that
exhibits a better fit than that of the exponential formulation
was p resented. It was shown through measurements that jitter
is better described with our alpha-stable model by comparing
it to network measurements.
When the stability index has a mean value close to one, a
simplified model based on the Cauchy distribution is adequate.
Jitter dispersion follows a power law of the number of nodes
in the path with scaling exponent given by the stability index.
The proposed models permit the estimation of jitter-QoS as
a function of the number of nodes in the path, the stability
index and the jitter dispersion.
R
EFERENCES
[1] C. A. Fulton, and S. Li, “Delay jitter first-order and second-order
statistical functions of general traffic on high-speed multimedia networks,
IEEE/ACM Trans. Networking, vol. 6, pp. 150-163, Apr. 1998.
[2] L. Qiong and D. Mills, “Jitter-based delay-boundary prediction of wide-
area networks, IEEE/ACM Trans. Networking, vol. 9, pp. 578590, Oct.
2001.
[3] E. J. Daniel, C. M. White, and K.A. T eague, “An inter-arrival delay jitter
model using multi-structure network delay characteristics for packet net-
works, in Proc. Conf. on Signals, Systems and Computers, pp. 17381742,
2003
[4] W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-
similar nature of Ethernet traffic (extended version), IEEE/ACM Trans.
Networking, vol. 2, pp. 1-15, Feb. 1994.
[5] S. Resnick and G. Samorodnitsky, “Performance decay in a single server
exponential queueing model with long range dependence, Operations
Research, pp. 235-243, Mar.Apr. 1997.
[6] G. Samorodnistsky and M. S. Taqqu, Stable Non-Gaussian Random Pr o-
cesses: Stochastic Models with Infinite Variance. Chapman and Hall/CRC,
1994.
[7] SLAC (July 2009), Round Trip Delay Distribution between SLAC and
CERN. [Online]. Ava ilable: http://www.slac.stanford.edu/comp/net/wan-
mon/resp-jitter.html
[8] N. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Dis-
tributions, vol. 1. John Wiley and Sons, pp. 56-60, 1994.
[9] L. Rizo-Dominguez, “Jitter modeling for wide area networks, Technical
Report CVSTV 0809, Guadalajara, Mexico, CINVESTAV, Aug. 2008.
[10] L. Rizo, D. Torres, J. Dehesa, and D. Muoz, “Cauchy distribution
for jitter in IP networks, in Proc. 18th International Conference on
Electronics, Communications and Computers (CONIELECOMP ’08), pp.
35-40, 2008.
[11] R. Pastor-Satorras, A. Vzquez, and A. Vespignani, “Dynamical and
correlation properties of the Internet, Physical Review Lett.,vol.87,pp.
1-4, Dec. 2001.
[12] M. Schwartz, BroadBand Integr ated Networks. Prentice Hall, 1996.
[13] The Cisco website (Oct. 2007), Enterprise QoS Solution
Reference Network Design Guide. [Online]. Available:
http://www.cisco.com/univercd/cc/td/doc/solution/esm/qossrnd.pdf
[14] W. Feller, An Introduction to Probability Theory and its Applications,
vol. 2. New York: Wiley, pp. 581-583, 1970.