This is the post-print version of the following article: Kohei Watabe and Masaki Aida, ``Analysis on the Fluctuation Magnitude in Probe Interval for Active Measurement,'' in Proceedings of the 30th IEEE International Conference on Computer
Communication (INFOCOM 2011) Mini-Conference, 2011. The original publication is available at https://doi.org/10.1109/INFCOM.2011.5934954 (DOI: 10.1109/INFCOM.2011.5934954). (c) 2011 IEEE. Personal use of this material ispermitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of
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Analysis on the Fluctuation Magnitudein Probe Interval for Active Measurement
Kohei Watabe∗ and Masaki Aida∗∗ Graduate School of System Design,

Tokyo Metropolitan University, Asahigaoka 6–6, Hino-shi, 191–0065 Japan
Email: {watabe-kouhei, maida}@sd.tmu.ac.jp

Abstract—Active measurement, which can provide end-toend measurements of network performance, is critical since
the Internet is managed by multiple organizations. Recently,
on the active measurement of delay and loss, Baccelli et alreported that many probing policies can be used to provide
appropriate estimation in addition to the traditional policy basedon PASTA property if the volume of probe stream is negligible
compared to the traffic stream. Probing schemes with fixed probe
packet intervals suffer from the phase-lock phenomenon due
to synchronization against the network performance; they do,
however, provide superior accuracy. A remaining issue is how
to decide the optimal probing policy while taking the phase-lockphenomenon into consideration. In this paper, we propose theprobing policy that randomly fluctuates the probe packet interval
to avoid the phase-lock phenomenon. We start by clarifying therelationships among the fluctuation magnitude, the properties of
the target network, and estimation accuracy, and we discuss the
optimal probing policy with regard to the properties of the target
network.

I. I NTRODUCTION
The Internet has spread widely over the last several decades,
and it has come to play a key role as an infrastructure
supporting society and the economy. On the other hand,
Quality of Service (QoS) which is represented by delay andloss in addition to connectivity, is appropriately regarded asa key technology goal. Controlling network quality demands
an understanding of the internal state of the network, thekey metric of which is congestion. However, the Internet iscomposed of multiple networks that are managed by different
Internet Service Providers (ISP) or other organizations, and it
is inherently difficult to gather the information needed fromthe different managers of the networks. Therefore, we needa way of measuring, from outside the Internet, the QoS of
end-to-end connections.
Active measurement is an end-to-end measurement technique that can estimate the QoS of a network path. Active
measurement derives the QoS from information obtained byinjecting probe packets into the target network path [1], [2],[3]. From the the delay and loss experienced by probe packets,
active measurement can estimate one-way delay, round-trip
time (RTT), and loss rate on the network path. It can also
estimate link capacity [4] and available bandwidth [5] fromchanges in the delay or interval of the probe packets.
Active measurement is easy for the end user to perform
but it is not trouble free. The injection of probe packets
impact the path’s performance. We cannot estimate the true

performance since that state occurs only without the probestream. If we inject a lot of probe packets to increase sampling
rate, estimation accuracy suffers due to the extra traffic. The
problem of extra traffic is discussed in [6]. Therefore, it isimportant to achieve high accuracy with a limited number ofprobe packets.
Work by Baccelli et al [7], [8] gaves an important suggestion with respect to the accuracy of active measurement
in assessing delay and loss. The current favorite is to use
the PASTA (Poisson Arrivals See Time Averages) property
by issuing probe packets so that their arrival intervals follow
an exponential distribution, i.e. packet injection is a Poissonprocess [9]. Probing policy which is fixing the probe packet
arrival intervals (periodic-probing) is also used because itis easier to manipulate. PASTA-based and periodic-probing
was compared in [10]. However, recent work indicates that
there may be many other distributions that can provide higher
measurement accuracy if a non-intrusive context (the effectof probe packets is insignificant) can be assumed [7]. This
property is named NIMASTA (Non-Intrusive Mixing Arrivals
See Time Averages).
NIMASTA contains the following three assumptions.
1) The stochastic process that expresses the network state
we are interested in (e.g. virtual delay and loss/no-loss
indication) is stationary and ergodic. This process iscalled the ground truth process.
2) The point process of probe packets arrival {Ti } (i =1, 2, . . . , m) is stationary and mixing. Mixing is therequirement that guarantees joint ergodicity between theprobe and ground truth processes (see [7] for details).
3) The last assumption is the non-intrusive context, i.e. we
can ignore the impact of probe packets. Namely, the ratioof the probe stream to all streams is very small.
Under the above assumptions, Baccelli et al proved that thefollowing equation holds,
1 ∑f (X(Ti )) = E[f (X(0))]
m→∞ m
i=1
m

lim

a.s. ,(1)

where X(t) and f are ground truth process and an arbitrary
positive function, respectively. If we can obtain X(Ti ) (e.g.
the delay of the probe packet or loss/no-loss of probe packet)
from a probe packet injected at time Ti , (1) means that we
can estimate E[f (X(0))] by injection of m probe packets.

The only requirements for the point process of probe packet
arrival {Ti } (i = 1, 2, . . . ) are that it be stationary and mixing.
Therefore, there are a lot of point processes of probe packet
arrival that can achieve appropriate estimation of ensemble
mean E[f (X(0))] besides PASTA-based probing. Such mixing
point process include those whose intervals follow the Gamma
and the uniform distribution. Note that periodic-probing with
fixed interval is not a mixing process, and does not satisfy (1).
Moreover, Baccelli et al investigated how to select an
optimal probing process [8]. We can select an optimal probing
process under a specific assumption by using the inter-probe
time given by the parameterized Gamma distribution.
If we estimate the mean of X(0)∑by using active measurem
ment, we can use estimator P̂ = i=1 X(Ti )/m. Thus thevariance of P̂ ism m ∫1 ∑∑ ∞r(τ )fi−j (τ )dτ ,(2)
Var[P̂ ] = 2
m i=1 j=1 −∞measurements with regard to delay and loss (we don’t use
the approach with Gamma-probing). To avoid the phase-lockphenomenon, we add a random fluctuation to probe packet
intervals. We clarify the relationships among the fluctuation
magnitude imposed on probe packet interval, the property ofthe target network and accuracy of estimation, and we discuss
the optimal probing policy corresponding to the property ofthe target network.
The rest of the paper is organized as follows. In Section II,
we verify the cause of the phase-lock phenomenon and describe a method to assess it. Section III proposes our probing
policy with random fluctuations. In Section IV, we analyze
the relationships between the fluctuation magnitude and theaccuracy of active measurement, and confirm the validityof the proposal by simulations. We conclude the paper insection V.

where fi−j is probability density function (pdf) of Ti − Tj ,r(τ ) = Cov (X(t), X(t + τ )) is the autocovariance function(ACF) of the ground truth process X(t) (we can express r(τ )by τ alone, because X(t) is stationary).
If r(τ ) is convex on interval [0, ∞) and the average of theinter-probe time is µ, the following equation can be proven byusing Jensen’s inequality [11].
∫ ∞∫ ∞r(τ )fk (τ )dτ ≥r(τ )δ(τ − kµ)dτ ,(3)

Reference [8] did not mention the factor of phase-lockphenomenon in detail because it was not main topic of [8].
This study examines the phase-lock phenomenon in detailbecause our goal is to specify the optimal probing process.
Accordingly, this section investigates the cause and effectsof the phase-lock phenomenon and introduces an evaluation
function that can assess this phenomenon appropriately.
The convex ACF r(τ ) of the ground truth process X(t)means X(t) has no special periodicity. The phase-lock phenomenon occurs when the cycle of the ground truth processsynchronizes to that of the probing process. Hence, phase-lockphenomenon will not occur if the ACF is strictly convex. This
fact was proven by (3). However, the experiments of Baccelli
et al demonstrated that there are multiple instances in which
the accuracy of the estimator of periodic-probing was worse
than that of other probing processes (periodic-probing is notalways bad).
We consider that phase-lock phenomenon is caused by
accidental periodicity which is realized by finite measurement periods. Even if there is no special periodicity of
the ground truth process (in terms of long-time average),
accidental periodicity can be generated. Thus, there is apossibility that a specific frequency component will be present
by chance if the measurement period is limited. Even if ACFr(τ ) = E[X(t)X(t + τ )] − {E[X(t)]}2 in terms of ensemble
mean
average) is∫ strictly convex, ACF∫ l−τ (namely long-time∫ l−τl−τ
X(t)X(t+τ)dt−
X(t)dt 0 X(t + τ )dt is, in
00
terms of a time average on finite period (0, l], not necessarily
convex. In Fig. 1, we display the ACF of a simple ON-OFF
process. The line of time average in Fig. 1 represents the time
average for a single sample path generated by simulation.
Therefore, the accuracy will become bad if the groundtruth process contains a lot of frequency component thatcan synchronize with the probing process. Conversely, if it
contains few such frequency component so much, the accuracy
will be quite good. The accuracy of periodic-probing may
become extremely bad for the specific sample path though
it is better on average.

−∞−∞

where δ(·) denotes Dirac δ function. (3) means that no otherprobing process with an average interval µ has a variance thatis lower than that of periodic-probing. Estimator variance isassociated with accuracy, lower is better, so periodic-probing
is the best probing process if we focus only on asymptotic
variance.
On the other hand, periodic-probing does not satisfy theassumptions of (1) due to non-mixing, so periodic-probing
is not necessarily the best. This is because the phase-lockphenomenon may occur and the estimator may converge on afalse value when a frequency component of the ground truth
process is synchronized with the cycle of the probing process.
Namely, periodic-probing has a bias. Therefore, we cannot
categorize periodic-probing as the optimal probing process.
To tune the tradeoff between traditional probing process(which obeys Poisson arrival) and periodic-probing (which has
a bias but has the advantage in terms of asymptotic variance),
Baccelli et al proposed a probing process that gives an interprobe time that obeys the parameterized Gamma distribution.
Gamma-probing provides multiple selections lying between traditional PASTA-based probing and periodic-probing
through parameter, and it is a great advance in network
measurement techniques. However, since Baccelli et al did
not indicate how to decide upon the optimal parameter, it hasremained a problem with no solution. Therefore, we cannot
decide upon probing policy which we should use.
In this paper, we introduce a method to specify the optimal
probing policy so as to maximize the accuracy of active

II. E VALUATION OF P HASE - LOCK P HENOMENON0.25

Tim e average
Ensem ble m ean

0.2

III. P ROPOSED P ROBING M ETHOD WITH F LUCTUATIONS

ACF

0.15
0.1
0.05
0
-0.05 0Fig. 1.

5

Tim e lag

10

15

ACF in terms of time average and ensemble mean

Accordingly, to assess the phase-lock phenomenon appropriately we must consider the accuracy of the target sample
path. We assume the∫ metric that we want to measure is theltime average X = 0 X(t)/l dt on the measurement period(0, l] and E[P̂ ] = X (namely bias-free) holds. By using a
conditional variance, we can express the accuracy of the target
sample path as follows.
Var[P̂ |X(t)] .

(4)

According to the property of conditional variance, we get thefollowing equation.
E[Var[P̂ |X(t)]] = Var[P̂ ] − Var[X] .

(5)

Baccelli et al proved that periodic-probing gives Var[P̂ ], theminimal value (we introduced this in the Introduction). (5)means that the expectation of (4) is minimized with periodicprobing. Therefore, by using (5), we can assess the average
accuracy (in which periodic-probing has a superiority). On theother hand, the effect of phase-lock phenomenon is that (4) isvaried with the intensity of the frequency component (which issynchronized with the probing process). The effect of phaselock phenomenon can be assessed by the following:
Var[Var[P̂ |X(t)]] .

(6)

This paper looks for the probing process which can avoid
extremely bad accuracy due to phase-lock phenomenon (extremely bad accuracy occurs when the ground truth processcontains high intensity of the frequency component which
is synchronized with the probing process.). In addition, theprobing process must be accurate in terms of average accuracy.
To satisfy the two above requirements (which are assessedby (5) and (6), respectively), we introduce the following
evaluation function:√
E[Var[P̂ |X(t)]] + Var[Var[P̂ |X(t)]] .
(7)
We define the probing process that minimizes evaluation
function (7) as the optimal probing process, and we investigate
the optimal probing process as determined by the property ofthe target network.
(7) is one of the simplest evaluation functions which canassess the phase-lock phenomenon. We can consider othervarious patterns of evaluation function. For example, we canuse the square root of (6) multiplied by a constant instead of
the second term of (7). Our evaluation function is simple andintuitive one though we cannot show that it is the best one.

In this section, we will propose the probing process withfluctuations to specify the probing process that minimizes theevaluation function (7). To avoid the phase-lock phenomenon,
fluctuated probe intervals is effective. In other words, we force
the probing process to exhibit multiple phases (i.e. variableintervals). Actually, the Gamma-probing proposed by Baccelli
et al in [8] is one approach to adding fluctuations. In this study,
we propose a new method that takes three parameters into
consideration: measurement period, number of probe packets,and the fluctuation magnitude. It is expected that measurement
period has a critical affect on the intensity of accidental
periodicity.
We add fluctuations that obey a normal distribution to
the timing of probe packet arrivals, while specifying themeasurement period. We assume that interval (0, l] is themeasurement period and m is the number of probe packets
sent in the measurement period. Our probing method gives thepoint process of probe packet arrival {Ti } (i = 1, 2, . . . , m)
as follows.

S + GiTi = (S + Gi ) − l,(8)l
where S and Gi denote the random variables that follow uniform distribution U (0, l/m) and normal distribution
N ((i − 1)l/m, σ 2 ), respectively, b·c denotes a floor functionand σ denotes the magnitude parameter of the fluctuations.
To prevent {Ti } from taking a value on the outside of themeasurement period (0, l], we added the second term. {Ti }given by (8) occurs at an equal probability at an arbitrary
point ∑
in the measurement period (0, l]. Therefore, estimatormP̂ = i=1 X(Ti )/m is an unbiased estimator for X.
Our probing process provides multiple selections lying between traditional PASTA-based probing and periodic-probing
as well as Gamma-probing. If σ = 0 (namely there is nofluctuation), {Ti } corresponds to periodic-probing. On theother hand, if σ → ∞, Ti follows a uniform distribution
U (0, l). Therefore, intervals between Ti follow an exponential
distribution for sufficiently large l.
IV. F LUCTUATION M AGNITUDE AND ACCURACY
In this section, we clarify the relationship between fluctuation magnitude, which is given by the probe timing, r(τ ) (aproperty of the target network) and the evaluation function (7).
We then detail a method that specifies the optimal fluctuation
magnitude, theoretically. As we mentioned in Section II,
periodic-probing is the optimal probing process if we consider only average accuracy. It is important to specify theminimal fluctuation magnitude that can avoid the phase-lockphenomenon, because large fluctuations cause the probing
process to deviate too far from periodic-probing.
Our evaluation function (7) is composed of the averageand variance of (4). Therefore, we first address the stochastic
behavior of (4).
We define X̃(t) = X(t − lbt/lc) which depends onstochastic process X(t) in measurement period (0, l], andT̃i = S + Gi which is composed of a simple uniform
and normal random variable. Since X(t) observed at thetiming of {T∑i }m is equal to X̃(t) observed at the timing of{T̃i }, P̂ = i=1 X̃(T̃i )/m holds. Therefore, by using ACF∫l∫lR̃(τ ) = 0 X̃(t)X̃(t + τ )/l dt − { 0 X̃(t)/l dt}2 in terms oftime average, (4) can be expressed as follows (as well as (2)).
m m ∫1 ∑∑ ∞
Var[P̂ |X(t)] = 2
R̃(τ )fi−j (τ )dτ .
m i=1 j=1 −∞Note that R̃(τ ) is a stochastic process that depends on X(t).
Furthermore, when we expand R̃(τ ) in Fourier series, wefind that
Var[P̂ |X(t)]



)
(
∞
m ∑
m ∫ ∞


∑
∑
1
2πn
= 2
τ fi−j (τ )dτ ,
Kn
cos


m n=1
l
i=1 j=1 −∞

Kn =

2
l

(

∫ l
cos
0

2πn
τ
l

(9)

)
R̃(τ )dτ .

Note that {Kn } (n = 1, 2, · · · ) are random variables that
depend on X(t); they represent the intensity of each frequency
component of X(t).
Moreover, the summation in (9) reduces as follows.
(
)
m ∑
m ∫ ∞
∑
2πn
cos
τ fi−j (τ )dτl
i=1 j=1 −∞
)
(
m−1
∑∫ ∞
2πn
=m
τ fi (τ )dτcosl
i=0 −∞
) m−1
(
∫ ∞
∑ 1
(τ −il/m)22πn√ e− 4σ2 dτ
τ
=m+m
cos
l
2σ π
−∞
i=1

) 1 − τ2
(
∫
∞
2πn
2
2


m + (m − m) 0 cos l τ σ√π e 4σ dτ,=n = mj (j = 1, 2, . . . )τ2m − m ∫ ∞ cos ( 2πn τ ) √
− 4σ
1
2
dτ,
otherwise
e
l
0
σ π
{
2πn 2 2m + (m2 − m)e−( l ) σ , n = mj (j = 1, 2, . . . )=.
2πn 2 2m − m e −( l ) σ ,otherwise(10)
Substituting (10) for (9), the accuracy achieved for target
sample path Var[P̂ |X(t)] is expressed as follows using theintensity of each frequency component {Kn } (n = 1, 2, · · · ).
Var[P̂ |X(t)] =∞∑wi Ki ,


2πi 2 2
 1+(m−1)e−( l ) σ

(11)

i=1

wi =

m

2
2πi

 1−e−( l ) σ ,, i = mj (j = 1, 2, . . . )

2

m

.

otherwise∫lNote that K0 = 0 because 0 R̃(τ )dτ = 0.
To clarify the average and standard deviation of
Var[P̂ |X(t)], we must investigate the stochastic behavior

of {Kn } (n = 1, 2, · · · ). Since Fourier transformation of2
autocorrelation function R̃(t) + X of the stochastic process
yields a power spectrum (Wiener-Khintchin theorem [12]),2F[R̃(t) + X ] = |F[X̃(t)]|2 holds.
∫ l Consequently, Kn relates
to Fourier coefficient Cn = 2 0 cos(2πnt/l)X(t)dt/l and∫l
Sn = 2 0 sin(2πnt/l)X(t)dt/l as follows.
Cn 2 + Sn 2.
(12)2
Note that Cn and Sn are random variables because they
depend on X(t).
Hence, by using (11) and (12), (5) which composes ourevaluation function is given as follows.
∞[] 1∑
E Var[P̂ |X(t)] =wi {Var[Ci ] + Var[Si ]} . (13)
2 i=1
Kn =

Similarly, if we assume Ci and Si follow normal distributions,the other component (6) composing the evaluation function is
as follows.
∞ ∑
∞
[
] 1∑
{
(
)
Var Var[P̂ |X(t)] =wi wj Cov Ci 2 , Cj 2
4 i=1 j=1()}()
+ 2Cov Ci 2 , Sj 2 + Cov Si 2 , Sj 2
∞ ∞
{
1 ∑∑
2
wi wj 2 {Cov (Ci , Cj )}=4 i=1 j=1}
2
2
+ 4 {Cov (Ci , Sj )} + 2 {Cov (Si , Sj )} ,(14)
where the second equality follows from property of the moment of bivariate normal distributions.
Finally, if we can relate Cov(Ci , Sj ), Cov(Ci , Cj ) andCov(Si , Sj ) to r(τ ), we can evaluate any probing method by
our evaluation function. Calculating Cov(Ci , Sj ), we have
Cov(Ci , Sj )
(
)
(
)
∫ ∫
4 l l
2πi
2πj
= 2
cos
t sin
s r(s − t)dsdt
l 0 0
l
l
)
(
)
(
∫ l ∫ l−t
4
2πj
2πi
= 2
t sin
(τ − t) r(τ )dτ dtcos
l 0 −tll()∫ l ∫ −τ {2π(j + i)
2πj
2
sin
t+
τ
= 2
l 0 −lll)}(2πj2π(j − i)t+τdt r(τ )dτ
+ sin
l
l
(
)
∫ 0∫ l {22π(j + i)
2πj
+ 2
sin
t+
τ
l −l −τ
l
l
)}
(
2πj
2π(j − i)t+τdt r(τ )dτ .
+ sin
l
l
Moreover, by creating two cases i = j and i 6= j andintegrating the result, we get Cov(Ci , Sj ) = 0. Similarly, we
can calculate Cov(Ci , Cj ) and Cov(Si , Sj ) as follows.
{
2
− ilπ
rS,i + 4l rC,i ,(i = j)Cov(Ci , Cj ) =,4lπ(j 2 −i2 ) (irS,i − jrS,j ) , (i 6= j)1stterm

0.002

2nd term + 3rd term

Evaluation function

Variance

3.E-082.E-08
1.E-080.E+000

0.1

0.2

0.3 0.4 0.5 0.6 0.7
M agnitude ofßuctuations

0.8

Cov(Si , Sj ) =2
4ilπ rS,i + l rC,i ,4lπ(j 2 −i2 ) (jrS,i − irS,j ) ,()∫ lTheoreticalvalue
Sim ulation based value

0.00050(i = j),(i 6= j)2πi
τ r(τ )dτ ,
l
0
(
)
∫ l(
τ)
2πi
rC,i =1−cos
τ r(τ )dτ .
ll
0rS,i =

0.001

0

0.9

Fig. 2. Comparison of first and second terms on the right-hand side of (15)?D{

0.0015sin

Substituting them into (13), we have
∞
[
] ∑
4
E Var[P̂ |X(t)] =wi rC,i ,l
i=1[]
Var Var[P̂ |X(t)]
}2 ∑
}2
∞ {
∞ {
∑
4
2
=
wi rC,i +wi rS,i
(15)
l
liπ
i=1
i=1
{
}
∑
(i2 + j 2 ) rS,i 2 + rS,j 2 − 4ijrS,i rS,j+
16wi wj.
l2 π 2 (i2 − j 2 )2
i6=j

However, the second and third terms on the right-hand sideof (15) are trivial compared with the first term if we give actual
parameters to r(τ ), l and m. Hence, (15) is approximated as
}2
∞ {
[
] ∑
4
Var Var[P̂ |X(t)] 'wi rC,i
.
(16)
l
i=1
Now, if X(t) denotes a simple ON-OFF process whose ON
and OFF periods obey exponential distribution with parameters λ1 and λ2 , r(τ ) = (λ1 λ2 )/(λ1 + λ2 )2 · e−(λ1 +λ2 )τ
is easily verified. In Fig. 2, we show the first term andother terms on the right-hand side of (15) for the case of(λ1 , λ2 , l, m) = (2, 1, 100, 100). Obviously, we can judge thatthe approximation is appropriate according to Fig. 2.
Thus evaluation function e(σ) is given by the following:
v
u∞ {
}2
∞
∑
u∑ 4
4
e(σ) =
wi rC,i + t
wi rC,i .
(17)
l
l
i=1
i=1
We can plot evaluation function e(σ) if we specify thefollowing: measurement period l, the number of probe packets
m, the ACF r(τ ) of X(t), and the fluctuation magnitude σ.
This means that we can obtain the optimal probing process as
per the properties of the target network.
Finally, we conduct simulations to assess the validity of(17). We executed Monte-Carlo simulations that assumed
above mentioned ON-OFF processes, and we calculated (7)directly. In Fig. 3, we display comparisons between theFig. 3.

0.25
0.5
M agnitude ofßuctuations

0.75

Comparison of the simulation-based evaluation function and (17).

simulation-based value and (17). We can confirm that (17)
is appropriate, and we can judge that the optimal σ are 0.2.
V. C ONCLUSION
In this paper, we proposed a new probing policy that can
avoid the phase-lock phenomenon by randomly perturbing thetiming of probe packets. Moreover, we defined an evaluation
function which can well assess the phase-lock phenomenon,
and we provided a method that can specify the optimal
fluctuation magnitude. Consequently, given the properties of
the target network, we can identify the optimal probing policy.
We confirmed the validity of our method through simulations. In the future, we will validate the method in detailincluding verification on an actual network.
ACKNOWLEDGMENT
This work was partly supported by Grant-in-Aid for Scientific Research (B) No. 21300027 (2009-2011) from theJapan Society for the Promotion of Science, and Early-concept
Grants for Exploratory Research on New-generation Networkfrom NICT.
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