Estimating Packet Loss-Rate by Using Delay
Information and Combinated with
Change-of-Meaure Framework
Keisuke Ishibashi, Masaki Aida, and Shin-ichi Kuribayashi
NTT Information Sharing Platform Laboratories, NTT Corporation
3-9-11 Midori-cho, Musashino-shi, Tokyo 180-8585, Japan
E-mail: {ishibashi.keisuke, aida.masaki, kuribayashi.shinichi}@lab.ntt.co.jp

Abstract— We previously proposed a change-of-measure based
performance measurement method which combines active andpassive measurement to estimate user-experienced performance.
We also applied this method to packet-delay estimation. In thispaper, we apply this method to loss-rate estimation. Because
packet loss rarely occurs in current networks, its measurement
usually requires a huge number of probe packets, which imposesa non-negligible load on the networks. We propose a loss-rate
estimation method which requires significantly fewer number ofprobe packets. In our proposed method, the correlation between
delay and loss is measured in advance, and at the time of
measurement, the time-averaged loss rate is estimated by using
the delay of probe packets and the correlation. We also applied
our change-of-measure framework to estimating the loss rate inuser packets by using this time-averaged loss rate. We prove thatthe mean square error in our method is lower than that simple
loss measurement, which is estimated by dividing the number oflost packets by the total number of sent packets. We evaluatedour method through simulations and actual measurements andfound that it can estimate below 10−3 packet loss rate with only
900 probe packets.

I. I NTRODUCTION
We previously proposed a lightweight and scalable methodof measuring performance, CoMPACT Monitor, that combines
active and passive measurement to estimate performance experienced by users. We also used it to estimate the delay inuser packets [1], [2]. In this paper, we report the use of it toestimate the end-to-end loss rate for user packets.
Packet losses significantly degrade the QoS of UDP applications such as streaming video or voice over IP [3],[4].
TCP throughput also depends on a packet loss rate in a large
bandwidth-delay product environment [5]. Therefore, it isnecessary to measure the packet loss rate to manage the service
level of these applications. In terms of managing service level
management, we need to know not only the stationary packet
loss rate as an indicator of network performance, but also
the time-varying loss rates for relatively short time periods(e.g., duration of streaming videos) as service-level statistics.
However, there are two drawbacks with current active loss
measurement.
First, when loss rate is low, active methods send a huge
number of probe packets to detect packet losses that rarely
occur. This imposes a non-negligible load on the network,
especially when we measure the loss rate for a short period of

GLOBECOM 2003

time. While there are many tools for actively measuring packet
loss rate, they only make use of information on lost probe
packets and do not use other available packet information.
With those methods, the required number of probe packets
to measure the loss rate is roughly an order more than thereciprocal of the rate when losses occur independently. When
there is a correlation between loss events, which has beenreported in many Internet loss measurements [6]-[9], we do
not need as many probe packets. Yet, the required numberof probe packets is still large and those probe packets may
perturb the networks [10].
Second, the loss rate measured by probe packets may not
be the same as that experienced by user packets [10]. If
we assume that active monitoring measures the time averageof network performance and that user traffic is Poissonian,then the performance experienced by users and actively measured performance will be the same. This is a well-known
property called PASTA (which stands for “Poisson Arrivals
See Time Average”) [11]. It is known, however, that current
Internet traffic exhibits burstiness and is not Poissonian, in
general [12]. In that case, more user packets are transmitted
during congested periods, which means that more user packets
experience a high loss rate. Thus, the loss rate experienced
by users may actually be higher than that measured by those
active monitoring. We previously proposed a method that canestimate the performance experienced by users by combining
active and passive monitoring [1],[2]. We also applied thismethod to packet-delay estimation. However, if we simply
apply it to estimate loss rate without considering the firstdrawback, the estimated loss rate may deviate from the actual
loss rate when the number of probe packets is limited.
In this paper, we propose a method of estimating packet
loss-rate using delay information in probe packets to overcome
the first drawback. We use an intuitive expectation that when
the delay of a probe packet is large, then the loss probability ofuser packets sent near the probe packet is high. We propose an
estimator for the time-average loss rate that uses the correlation
between loss and delay. Then, combining the concept of the
CoMPACT monitor and this estimator, we propose an estimator
for the loss rate experienced by users packets. We provethat the mean square error of our method is smaller than

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those of a simple loss measurement. The method was also
evaluated through simulations and actual measurements andthose showed that it can estimate below 10−3 packet loss ratewith only 900 probe packets.
The rest of this paper is organized as follows. Section II
presents the proposed loss measurement method. Simulations
and actual measurement experiments are described in section
III. Finally, we give summary in section IV.
II. P ROPOSED MEASUREMENT METHOD
Our method of estimating loss rate in user packets involves
the following two steps:
Step 1: Estimating the time-average loss rate by using actively measured delay and loss.
Step 2: Converting the time-average loss rate to the loss rate
experienced by user packets by using passively measured
traffic intensity.
These steps are explained more fully in the next two subsections.
A. Time average loss rate
The main difficulty in this step is measuring rare events withfewer samples (probe packets). We overcome this difficulty byusing delay information about probe packets that are not lost.
The objective of this step, time-average loss rate during(t1 ,t2 ] LR(t1 ,t2 ), is defined as 
t2
1
1
dt,
(1)
LR(t1 ,t2 ) :=t2 − t1 t1 {V (t)=Dl }where 1{·} is the indicator function, V (t) is the virtual delay ofthe packet sent at t, Dl is a value set larger than the maximum
delay, and V (t) = Dl indicates that the packet is lost. Here, we
use the word “virtual” because there may not be any actualuser packets at time t [11].
A simple estimator of LR(t1 ,t2 ) with n probe packets sent
during (t1 ,t2 ], SELR(n), is
1 n
SELR(n) := ∑ 1{Y =D } ,(2)
n i=1 i lwhere Yi is the delay for i-th probe packet and Yi = Dl indicatesthe packet is lost. We also define ti as the transmission time forthe i-th probe packet, so V (ti ) = Yi . Actually, if probe packets
are sent independently of V (t), i.e. in a Poisson manner, (2)is a consistent estimator of (1). However, when seeing an
individual estimation for a measurement of a short time period,
it may be significantly different from LR(t1 ,t2 ) when the loss
rate is low and the number of probe packets is limited. Now,
we can intuitively expect that even if a probe packet is not
lost, when the delay for the probe is large, then the loss ratearound the time of the probe packet is high as reported in [8].
Therefore, we propose a method of estimation that uses thecorrelation between the delay in the packet and the loss ofneighboring packets.
First, let us define the conditional loss probability lc (t, τ , x)
as follows:(3)
lc (t, τ , x) := Pr [V (t + τ ) = Dl V (t) = x].

Loss probability,τ, x)

l_c(t

τ
δ−

δ+

lc (t, τ , x)

Fig. 1.

(independent of t), and denote this as lc (τ , x). Then, given the
delay of the i-th probe packet, Yi , we obtain the unconditionalloss probability for time ti + τ as lc (τ ,Yi ). Therefore, the loss
rate in the neighborhood of ti , (ti − δ− ,ti + δ+ ] (δ− , δ+ ≥ 0),is estimated by δ+
1lc (τ ,Yi ) d τ .
(4)
δ+ + δ− −δ−While 1{Y =D } in (2) only takes 0 or 1, (4) takes value fromil0 to 1 even if Yi = Dl . The expectation of (4) in terms of Yi
agrees with E[LR(ti − δ− ,ti + δ+ )] because δ δ  D++lE
lc (τ ,Yi ) d τ =lc (τ , x) dFi (x) d τ
−δ−

−δ− 0

 δ

+

Pr[V (ti + τ ) = Dl ] d τ t +δ+i= E
1{V (τ )=D } d τ ,=−δ−ti −δ−(5)lwhere Fi (x) = Pr[V (ti ) ≤ x]. Therefore, using conditional
probability lc (τ , x) and n measurements of probe packets{Y1 ,Y2 , . . . ,Yn } sent in (t1 ,t2 ], we obtain another estimator forLR(t1 ,t2 ) by changing 1{Y =D } in SELR to (4) as
i

l

1
1 n
ELR(n, δ+ , δ− ) := ∑n i=1 δ+ + δ−

 δ

+

−δ−

lc (τ ,Yi ) d τ .

(6)Compared to (2) which only uses the loss rate in timing theprobe packet transmission, the estimator (6) is expected to be
more accurate because it uses the loss rate of the neighborhoodof the probe packets by using the delay in probe packets andthe correlation between the delay and loss probability. Actually, we can prove that the mean square error in the proposed
method is smaller than that in simple loss rate estimation (2).
Here, we compare the error of the estimation when using
one probe packet; E[(ELR(1, δ+ , δ− )−LR(−δ− , δ+ ))2 ]. Errors
for multiple packets can roughly be obtained by dividing
the MSE by the number of probe packets if we can assume
that error between probe packets is independent. Let R(x) beR(x) := δ+ lc (t, x)dt/(δ+ + δ− ). Then,

For large delay x, the conditional loss probability of a packet
sent near t is expected to be high and to decrease as |τ |increases (Fig. 1). Here, we assume that lc (t, τ , x) is stationary

GLOBECOM 2003t

- 3879 -−δ−E[(ELR(1, δ+ , δ− ) − LR(−δ− , δ+ ))2 ]

2 
δ
+ 1
 D
dt
l 
−δ− {V (t)=Dl |V (0)=x} ER(x) −=dF1 (x)
δ
+
δ
0
+
−
=

 D

l

0


VarR(x) −

δ

+

1

−δ− {V (t)=Dl |V (0)=x}

δ+ + δ−dtdF (x)10-7803-7974-8/03/$17.00 © 2003 IEEE

 D

+

l



δ

+

1

−δ− {V (t)=Dl |V (0)=x}

ER(x) −dt

2
 dF (x)δ+ + δ−

 δ
+ 1
 D
dt
l
−δ− {V (t)=Dl |V (0)=x}
dF (x).
=Var
1
δ+ + δ−
0
0

1

(7)

On the other hand, the mean square error for the SELR
is obtained as follows. Let f1 (Dl ) as the stationary loss
probability as f1 (Dl ) := E[1{V (0)=D } ] = 1 − F1 (Dl −). Then,l

E[(SELR(1) − LR(−δ− , δ+ ))2 ]

 δ
+ 1
 D
dt
l
−δ− {V (t)=Dl |V (0)=x}
dF (x)=Var
1
δ+ + δ−0
 D−+ f1 (Dl ) +

E

l

+

1{V (t)=D |V (0)=x} dt−δ−lδ+ + δ−0

 δ−2 f1 (Dl )E 

=

 δ

+

1{V (t)=D |V (0)=D } dt−δ−lδ+ + δ−l

2
 dF (x)
1





1
dt
{V
(t)=D
|V
(0)=x}l−δl
dF (x)Var −1
δ+ + δ−
0

 D

 δ

+ f1 (Dl ) +

+

 D−
l

R(x)2 dF1 (x) − 2 f1 (Dl )R(Dl ).

(8)

Therefore, the difference between (8) and (7) is2  Dl −f1 (Dl ) 1 − R(Dl ) +R(x)2 dF1 (x) ≥ 0,

(9)

0

0

and the error of our estimator is smaller or equal to that of asimple loss estimator. The first term equals zero if the mean
packet loss probability near the lost probe packet is one. The
second term is zero if the mean loss probability near the probe
packet is zero even when the delay of the probe packet is large.
Although packet loss on the Internet exhibits a high degree of
burstiness [9], a situation where both terms equal zero canhardly be expected, especially for a relatively large δ .
Our estimator (6) requires conditional loss probabilitylc (τ , x). This can be obtained through either preprocessing orin an online way. In preprocessing, probe packets are sent withshort intervals, and we can take the loss and delay pairs of
the interval. The interval should be sufficiently short that the
integral in (6) can be approximated by the sum of the intervals.
Then, lc (τ , x) can be calculated using the loss and delay pairs.
We expect this probability remains almost unchanged in a
network path during a period of time that can be considered
as stationary (such as a busy period). Thus, once the lc (τ , x) ofa period of time has been obtained, we can use this for othermeasurements done in the same period of time. However, tocope with gradual changes in network conditions, introducing
an online update of lc (τ , x) may be better. In that case thedelay and loss pairs for different intervals can be obtained bysending probe packets with different intervals (e.g. Poissonprocess), while simultaneously measuring the loss rate. Then,lc (τ , x) is updated with these delay and loss pairs, for example,
by taking a moving average.
B. Estimation of user packet loss rate
This step uses our previously proposed method to estimate
the delay in user packets [1], [2]. In this subsection, we
present a brief review of it and its combination with the results
obtained in Step 1. The mathematical foundation of the method
can be found in [1], [2].
Our proposed change-of-measure based network performance measurement method, CoMPACT Monitor, was aimed
at estimating the network performance experienced by users.
To do this, it actively measures the performance through probe
packets, and passively measures the number of user packets
sent near the probe packets. Then, by changing the measurefor the performance of probe packets to that of user packets,
we obtain an estimator of network performance experienced
by users.
The objective of this step is to estimate the loss rate
experienced by user packetssent in (t1 ,t2 ], ULR(t1 ,t2 ):t2
t1 1{V (t)=Dl } dA(t)ULR(t1 ,t2 ) =
,
(10)
t
2
t1 dA(t)where A(t) is the arrival process for user packets. Here, thedenominator represents the number of user packets sent in(t1 ,t2 ] and the numerator represents the number of lost user
packets.
Modeling A(t) as a fluid, we demonstrated that an empirical
distribution of the delay for any user fluid A(t) could beobtained using actively and passively measured values [1].
Let a(i) be user traffic intensity at ti , i.e., dA(t)/dt at thei-th active measurement timing. The a(i) is obtained throughpassive measurement by counting the number of user packets.
Assuming that measurement timing is stationary, it can beproved for t any D ∈ R+ that2∑ni=1 1{Y >D} a(i)t 1{V (t)>D} dA(t)ia.s. (11)lim 1  t= limn
2 dA(t)t2 →∞n→∞∑i=0 a(i)t1
Therefore, by setting D to Dl −, a(i) as the number of user
packets sent in (ti − δ− ,ti + δ+ ], and u as ∑ni=1 a(i), we obtainan estimator of (10) by simply applying the change-of-measuremethod as
1 n(12)SEULR(n) := ∑ 1{Y =D } a(i).
u i=1 i lCompared with (2), the estimator (12) weights the event
Yi = Dl by a(i) to convert the time-average loss rate touser-experienced loss rate. SELR has been proved to agreewith ULR if the measurement lasts long enough. However,
because it assumes that user packets sent near the lost probe
packet will all be lost, otherwise no user packets are lost, it
shares the same problem as (2). Therefore, by combining theresults of Step 1 and the change-of-measure based method,
we propose an estimator to determine the loss rate for user
packets EULR(n, δ+ , δ− ) as
 δ+
1 n a(i)EULR(n, δ+ , δ− ) := ∑lc (τ ,Yi ) d τ . (13)u i=1 δ+ + δ− −δ−This estimator is obtained by changing 1{Y =D } in (12) toδ1+
δ+ +δ− −δ− lc (τ ,Yi ) d τil

the same as the relationship between(2) and (6).

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Bottleneck router

10 Mbps

Exponential
On-Offror
r

10 Mbps
10 Mbps

Fig. 3.

Probe sink

Fig. 2. Network configuration for simulation.
1.E+00x=20 (ms)
1.E-01
1.E-02),x( 1.E-03c_l1.E-04x=0(ms)1.E-05
1.E-060200400τ(ms)600
800 1000
Conditional loss probability (lc (τ , x)) for x = 0 and 20 (ms).

III. E VALUATION

io
t
a
r
s
s
o
L

Fig. 4.

LR
ELR
SELR

SELR

0

200

400

600

800

1000

1200

No. of pr obe pac kets

Fig. 5. Mean square errors for time-average loss rate
0.025
ULR
EULR
0.02
SEULR
o
ti 0.015
a
r
s
s
o
L

0.01

0.005
0

A. Simulation results
We evaluated our method through simulations for the network in Fig. 2. User sources generate exponential On-Off
traffic, whose means are 1 s and 14 s, respectively. During theOn period, 1,000-byte packets were sent with an exponentiallydistributed interval with a mean of 10.6 ms. Thus, each source
sent packets at 50 kbps on average. The number of sources was
140, so that the utilization rate of the intermediate link was
0.7. The buffer capacity of the router was set to 50 packets.
To obtain conditional loss probability, lc (τ , x), we first ran
a 50,000-s simulation in advance where 64-byte probe packets
were sent with an exponential interval with a mean of 10
ms. The lc (τ , x) were calculated using the loss and delay
information of these probe packets. After calculating lc (τ , x),
we ran 10 simulations each lasting 1, 000 s and calculated ELR
and EULR. In each run, probe and user packets were sent in
the same way as in the first 50,000-s simulation. Time-averageloss rate, LR, was calculated using all probe packets, and useraverage loss rate, ULR, was calculated using all 1,000-byte
packets. To estimate loss rate, we only used one probe packet
per 100 probe packets. In other words, we only sent one probe
packet every second on average during actual measurements.
We set (δ+ , δ− ) = (100, 100) (ms) in this paper. The maximum
delay was 29 ms. Figure 1 shows lc (τ , 0) and lc (τ , 20) when
the number of hosts is 140. As expected, lc (τ , 20) decreasedas τ increased. We can also see that lc (τ , 0) increased as τ
increased and converged to a constant, which is expected tobe the time-average loss rate (unconditional loss probability).
First, we tested ELR, the estimator for time-average loss
rate. Figure 4 shows LR (time-average loss rate), SELR (estimates using only the loss information of probe packets), andELR (estimates using both delay and loss information of probe
packets) for ten simulations. We see that while SELRs deviatedfrom LRs, ELR could estimate LR with high accuracy. Note
0.012
0.01
0.008
0.006
0.004
0.002
0

ELR

3.0E- 05eer
a 2.0E- 05
uqs 1.0E- 05nae 0.0E+00
M

Sinks

sources

Probe source

4.0E- 05

0
2
4
6
8
10
Simulation run
Fig. 6.
Loss rate for user packets and its estimations results for ten
simulations.

that while we used the same lc (τ , x) for all estimates, ELR
could estimate different values for LR for ten simulations, byreflecting difference in probe delays. We can also see that noprobe packets were lost in simulations 2, 3, 5 and 6, while
ELR could estimate loss rate using by delay information.
We also compared mean square errors, E[(ELR − LR)2 ] andE[(SELR − LR)2 ] by varying t2 from 200 to 1, 000 s (we fixed
t1 at 0). Figure 5 shows the results. The accuracy of our method
was higher than that for the simple loss rate estimation asproved in (9).
Next, we tested EULR, the estimator of the loss rate in user
packets. Figure 6 shows the ULR (user-experienced loss rate),
SEULR (estimates of ULR using only the loss informationof probe packets and number of user packets), and EULR
(estimates of ULR using both delay and loss information ofprobe packets and number of user packets). We can also seefrom this figure that our estimation could follow the loss rate
for user packets, which was about 1%, i.e. higher than thetime-average loss rate shown in Fig. 4. This was due to thecorrelation between the number of user packets and the loss
rate, which was discussed in Section I. By weighting the loss
rate with the number of user packets sent near probe packets,
our method could convert the time-average loss rate to an useraverage loss-rate. Figure 7 shows the mean square error of the
EULERs and SEULRs to ULRs. Our estimator achieved alower mean square error compared with the simple changeof-measure method.
B. Actual measurement results
We did end-to-end loss and delay measurements on an
actual network to evaluate our method. We measured oneway delay and loss from an asymmetric digital subscriberline (ADSL) customer to a company LAN during office hours(10:00-18:00) in February 2003. The path consisted of 15 hops
r 1.5E- 04
o
r
r

SEULR

e
e1.0E- 04rauqs5.0E- 05

EULR

n
a
e
0.0E+00
M

0

2

4
6
8
10
Simulation run
Loss rate and its estimation for ten simulations.

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0Fig. 7.

- 3881 -

200

400

600

800

1000

1200

No. of pr obe pac kets

Mean square errors for user-experienced loss rate.

0-7803-7974-8/03/$17.00 © 2003 IEEE)s 300m
05 2502=
ss 200
lo(
150
s]
[m
ya 100
le
D 50
2000
2100
2200
2300
2400
2500
Packet num.
Fig. 8. Sample path of delay and loss in measurement (Losses are shownas 250 ms delay)
1.E+00x=170 (ms)
1.E-01)
1.E-02,xt(c
1.E-03l_
1.E-04x=70 (ms)
1.E-05
0
200
400
600
800
1000
tau (ms)
Fig. 9. Conditional loss probability (lc (τ , x)) for x = 70 and 170 (ms).

and the narrowest link along the path was the ADSL up-link,with a bandwidth of about 400 Kbps1 . Here, we only evaluatedthe estimator for time-average loss rate because there were nouser packets on the link.
We sent 64-byte UDP packets between two GPSsynchronized PCs in both LANs as a Poisson process where
the mean interval was 20 ms. We ran thirty measurements
each lasting 900 s and used the first twenty to calculateconditional loss probability, and the last ten to estimate loss
rate. The maximum, mean and minimum delay were 207, 60,
and 20 ms, respectively. The time-average loss rate for thewhole measurement was 0.07 %.
Figure 8 shows a sample path for the delay and loss of theprobe packets (Here, loss is shown as 250 ms delay). We cansee the fluctuations in delay, and packet losses during somepeaks of fluctuation. This indicates that there is a correlation
between the loss and delay even in the actual network. To
provide more direct evidence of this correlation, Fig. 9 hasconditional loss probability for delays of 70 ms and 170 ms.
There is a clear correlation between loss and delay the sameas in simulation.
Figure 10 shows LR, SELR, and ELR for 10 measurements
where LR was calculated using all probe packets, and ELR
and SELR was estimated by using one probe packets per 50
packets. In these simulations, as the loss rate varied from 10−5
to 10−3 , we plotted the loss rate semi-logarithmically. Except
for measurements 2 and 3, no probe packets were lost, andthe SELR was zero, which cannot be shown in the figure.
Even so, our method could estimate time-average loss rateaccurately especially for loss rates over 10−4 . Figure 11 shows
the mean square errors of proposed estimator ELR and simple
loss estimator SELR. The errors for ELR are smaller than thosefor SELR for every number of probe packets.
IV. C ONCLUSION
We proposed a two-step method of estimating the loss
rate in user packets that involved: 1) estimating the time1 During measurements, because no packets except active probe packets
were sent to the link, it could not cause queueing delays or losses for probe
packets on the link.

GLOBECOM 2003

1.0E-02

LR
ELR
SELR

1.0E-03

io
t
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s
s
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1.0E-04
1.0E-05

Fig. 10.

0

2

4

6
8
10
12
Simulation run
Loss rate and its estimations for ten measurements.

1.4E- 06
r 1.2E- 06orr
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q 6.0E- 07s
n 4.0E- 07ae
2.0E- 07
M

ELR

SELR

0.0E+00
0

Fig. 11.

200

400

600

800

1000

No. of pr obe pac kets

Mean square errors for time-average loss rate.

average loss rate and 2) converting that rate to the userexperienced loss rate. We used the delay in probe packets
and the conditional loss probability given by their delay in
Step 1. In Step 2, the time-average loss rate estimated in Step 1
was converted to the user-experienced loss rate through ourpreviously proposed change-of-measure based method. Our
method can be used to estimate both the time-average anduser-experienced loss rate accurately with a limited numberof probe packets. It can be used to determine the average loss
rate over short periods of time such as duration of a streaming
videos or phone calls.
ACKNOWLEDGEMENTS
We with to thank Ms. Kyoko Ashitagawa for her help
with the simulations and Mr. Takashi G. Inoue with themeasurement in our experiment.
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