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Effective Probabilistic Approach Protecting Sensor Traffic
Xiaoyan Hong*, Pu Wang†, Jiejun Kong‡, Qunwei Zheng*, Jun Liu*
*Computer Science Department, University of Alabama, Tuscaloosa, AL 35487
†Mathematics Department, University of Alabama, Tuscaloosa, AL 35487
‡Computer Science Department,University of California, Los Angeles, CA 90095
Abstract—Sensor networks are often deployed in environ-
ments where malicious nodes present. Among all possible forms
of the attacks threatening the sensor networks, in this work,
we focus on traffic analysis attacks. Typically, in performing
traffic analysis, an attacker will eavesdrop on-going wireless
transmissions and analyze contents and timing instances of the
transmissions so to infer critical events or to trace valuable assets
in the network (e.g, data sources or sinks). The paper presents a
probabilistic approach to shape the sensor network traffic to de-
correlate time instances in transmissions. The security properties
of the approach are studied both analytically and empirically,
showing strong protection in high probability.
I. INTRODUCTION
Rapid advances of wireless sensor network technologies
have enabled numerous applications in civilian, law enforce-
ment and military environments, from large scale environmen-
tal monitoring, structural defect monitoring, border intrusion
tracking, hostile terrain surveillance, to small scale personal
health monitoring. Many applications of wireless sensor net-
works require strong security protection over the network
and over the data collected and transmitted in the network.
Among various security threats, traffic analysis is one that
has been made easier in the wireless medium than the wired
lines, in that a malicious node can eavesdrop all the wireless
transmissions without physically attaching to a line or com-
promising a node. Though the wireless communications can
be protected by strong cryptographic methods at application
(end-to-end) or at link level, an eavesdropper can still obtain
information on traffic pattern, traffic change pattern, or end-
to-end traffic flow paths [8][9]. With such information, the
eavesdropper can infer the events happening in the network,
or trace the paths to the sources/destinations of the flows or
the locations of the events. More deadly consequences could
directly result from the information. Thus countermeasures
that prevent sensor traffic against traffic analysis have to be
studied as an important sensor security issue.
Traffic analysis [1] can be performed on both packet
contents (content analysis) and timing events of transmissions
(timing analysis). Typical countermeasures to content analysis
are padding data packet into a constant length and using per
hop key to encrypt packets. When source and destination
addresses in packet headers are concerned, per hop link level
encryption or link pseudonyme can be used [11]. With these
methods, every transmission looks different to an observer.
Though encryption and decryption is considered expensive
with regard to a sensor’s resource constraints, measurements
have shown that the overhead is tolerable, e.g., the times
that a Berkeley MICA1 Mote sensor takes to encrypt and
decrypt 30 bytes data using RC5 are 1.94 and 2.02 msec
respectively [6][7]. In addition, encryption serves a broad
range of security purposes like data integrity, content privacy,
access authentication, etc.
Timing analysis [15] can not be prevented with the en-
cryption methods. Successful correlation between two trans-
mission events reveals the path of a data flow. Solutions
studied for wired networks [2][4][14][17] use traffic mixing
techniques including sending messages in reordered batches,
sending decoy/dummy messages, and introducing random
delays. However, sensor networks have features different from
a wired network, requiring reevaluation of the existing work
and new designs of anti timing analysis solutions.
Features of wireless sensor networks include constraints
on the resources each sensor possesses, namely, computing,
storage, bandwidth and energy. Such constraints raise con-
cerns on a scheme that requires decoy packets due to the
extra usage of bandwidth and energy. On the other hand,
communications among sensors is multiplexed within a radio
range (depending on encryption method) in the wireless
medium, which helps hiding a particular transmission among
neighbors’ transmissions. Our study, thus, adopts the random
delay approach instead of using decoy packets and exploits
the multiplexing feature of the wireless medium to protect
sensor traffic from timing analysis.
We base our work on the assumption that a sender is able
to hide the identity of its intended receiver in transmission but
not its own. The rationales behind this assumption are that:
on one hand, the sender can either use link level encryption
or simply broadcast its messages to hide the receiver. The
intended receiver shall use the right key to decrypt the mes-
sages; on the other hand, an eavesdropper assisted with radio
detection technique can detect a wireless transmission from a
referable location and name it. By discovering the temporal
dependency of two transmissions at different locations, he
acquires the knowledge about the relationship of two consec-
utive nodes along a data path. When attackers have substantial
network-wide monitoring ability, or physical mobile tracking
ability, they will be able to integrate piecewise information
into global knowledge to finally infer the locations of the
events or the base stations in the sensor network.
In this paper, we study random delay strategies to prevent
temporal correlation of wireless transmissions within a neigh-
borhood. Specifically, our schemes introduce random delays
independently and distributively at each hop. Our strategies do
not use dummy messages but exploit the broadcast media in
improving the effectiveness and the efficiency of the schemes.
The key of such design is how to derive the delay and what
the anonymity properties will be. We introduce two schemes
and analyze their anonymity properties in the paper. The work
is presented as follows. We start with brief reviews on related
terminologies and previous work in Section II, follow by the
descriptions of the schemes in Section III. Section IV analyzes
the security properties of the schemes, and Section V validates
them through simulations. Section VI concludes the paper
with future work outlined.
II. BACKGROUND AND RELATED WORK
A. Terminology
The concept of anonymity is defined as ”the state of
being not identifiable within a set of subjects, the anonymity
set” [13]. The anonymity set is the set of the identities of
possible senders/recipients [5][16]. The larger the set is, the
better the anonymity protection will be. Further, anonymity is
defined in terms of the relationship of a sender and a receiver
not to be identified. If an algorithm always provides the size of
the anonymity set to be greater than 1, the algorithm provides
deterministic anonymity. Otherwise, if an algorithm has a
probability that the size of the set goes to 0 in an exponentially
decreasing rate given a linear increasing complexity parameter
of the algorithm, the algorithm is said to provide probabilistic
anonymity [10].
B. Related work
Strategies thwarting timing analysis are studied within
many Internet anonymity protocols, e.g., with various MIX-
Net designs [3][10][14]. The strategies include introducing
random delay; buffering messages in a pool, reordering mes-
sages and flushing the pool; injecting dummy/decoy packets;
and padding each packet to the same length or random lengths.
For example, a delay-and-playout technique (traffic mixing)
is used in [2][14]. Serjantov et al. [17] classify traffic mixing
strategies into two major categories—simple MIXes and pool
MIXes. Both categories use either one threshold or both: a
message pool size nand/or a time period tthat a message
stays in the pool. Decoy messages could be used when
necessary. These threshold based schemes are vulnerable to
either flooding attacks, where the adversary sends its own n-1
messages to flush a pool of size n; or trickle attacks, where the
adversary blocks all the incoming messages except the target
one until the mix node sends it out after t; or the two blended.
Various mixing schemes have been proposed to reduce the
success ratio of such attacks [17].
A close related work is Stop-and-Go-MIX (SG-MIX) [10].
SG-MIX does not use thresholds on a message pool or a
time window, nor decoy messages. Rather, it uses independent
random delays for each message. In SG-MIX, the source
pre-computes a route passing through a few MIX nodes
and values of random delays at each node. The values are
encrypted by the keys of each node en route and embed-
ded in the message header. SG-MIX achieves probabilistic
anonymity, i.e., two transmissions could be correlated with
an exponentially decreasing probability when traffic intensity
increases linearly. However, SG-MIX can not be used in
wireless sensor networks due to many reasons: (i) The ene-to-
end approach is not suitable for distributed sensor networks;
(ii) pre computed data paths can not be obtained in sensor
networks; (iii) cryptographic keys are more likely managed
locally in sensor networks; (iv) many applications use mobile
sensor networks; and moreover, (v) while per flow treatment
by SG-MIX is not a problem for a wired line because MIXes
usually have high traffic volume, inputs to a wireless sensor
node is not as high as those to a wired MIX. Our schemes use
a distributed approach and exploit the broadcast nature of the
wireless medium to facilitate anti-traffic analysis strategies.
For wireless multihop networks, preventing traffic analysis
has been studied as a routing problem. Jiang [9] presented
a routing algorithm that finds appropriate routing paths for
various end-to-end flows aiming at maintaining global link
traffic patterns as invariable as possible. Our work differs
from the work in that we model aggregated traffic in a local
transmission radius. For sensor networks, anti-traffic analysis
strategies have also been presented in [6], where window
based delay randomization and dummy packets are used. Our
approach does not use dummy packets, rather, longer delays
in trading for bandwidth. By exploiting the wireless media,
we are able to control the overall average delay.
III. PROBABILISTIC TRAFFIC SHAPING
In this section, we describe two random delay strategies
to shape data traffic so that direct correlation of a previous
transmission (by the upstream node) with the current one
is impossible with high probability. Our strategies do not
use dummy messages but exploit the broadcast media in
improving the effectiveness and the efficiency of the schemes.
Analysis of the anonymity properties will be given in the next
section.
A. Network Model
In our targeted sensor network scenario, all the sensors
independently generate data reports and relay reports for some
of its neighbors. The packets are padded to a constant size.
We assume all the sensors have a radio range of r. Sensors
within the range rcan either send to or receiver from each
other. Our assumptions on sensor security follow the ones
used by [6][12], i.e., sensors have established pair-wise keys
to encrypt transmissions at each hop and also to decrypt the
packets and process them at the intended receivers within the
transmission range. A further relay of the message will be
re-encrypted to generate a different payload pattern. Note this
network security model is vulnerable to localized DOS attacks
(energy depletion) but the damage will not propagate further.
Using current radio detection technique, an eavesdropper
can easily detect a wireless transmission from a referable
location. It can identify this transmission using its own coor-
dination system and name convention. When it intercepts two
consecutive transmissions from two sensors that are within
the range r, the eavesdropper can perform traffic analysis on
the two packets for content correlation and timing correlation.
With the above sensor security assumption, content correlation
is impossible (is hard or resource consuming). Because a
relay node will re-encrypt the content with a different key,
the same data payload appears differently at every relay
transmission. However timing correlation is possible if one
packet is sent after a reasonable delay counting the processing
and media access control. More over, a compromised node is
more dangerous in that it can act protocol-compliantly but
to break the privacy and security system. For example, it
can generate its own payload and monitor who will relay
the packet. When an adversary detects the relation between
upstream and down stream nodes, it can trace the routing path
towards either the data source or the destination (base station)
by physical movements or by feeding information to its global
adversarial information center for integration with other data.
These actions could lead to deadly consequences.
B. Distributed Random Delay Strategies
Communications in wireless sensor networks typically lack
global centralized control and ene-to-end knowledge. MIX-
net techniques, including SG-MIX, can not be used directly.
Our attempt is to apply the end-to-end SG-MIX scheme in
a distributed way, i.e., each node along a path independently
introduces a random delay for the packet it relays. Thus we
are able to avoid the many drawbacks of original SG-MIX.
Our first scheme thus works as follows. In the scheme, a
sensor node independently generates data packets and emits
them immediately after generation. A sensor also relays
packets for its neighbors. When a packet is received, it will
be delayed for a while before being emitted again. The value
of the delay draws from an exponential distribution with
parameter µ. When a transmission is delayed, the packet is
put into a queue until the delay time expires. During the delay
period or even before receiving the packet, the neighbors
of the node may transmit or have transmitted packets, or
the node itself may have originated packets. Thus when
this node transmits the delayed packet, an eavesdropper may
not be able to tell whether the current transmission is an
originated data or a relay of an early transmission; or, to
which early transmission this packet might relate. Thus a
relay transmission is hidden among its own data, neighbors’s
transmissions or its other relay traffic.
The scheme presents the worst case when a packet is
emitted after delay but with no mixing traffic. When this
happens, an eavesdropper has the highest probability (50%) in
guessing whether this transmission is a relay of the previous
one or an emission by the node itself. If an eavesdropper is
able to identify idle periods of the system, especially, when
traffic load is not high, he will have a higher probability in
correlating events (we defer analysis to the next section). To
be more specific, let’s define the following events.
•Event A: the queue is empty when the packet first arrivals
at the node;
•Event B: During the delay period, there is no other
transmissions in the neighborhood.
•Event C: During the delay period, there is no packets
originated by the node itself.
Thus at the time of an expiration, if at least one of the
three events does not occur, immediate transmission of the
current packet is considered safe because it is mixed with
other transmissions when observed by another node in the
neighborhood. However, when all of the events A, B and
C have happened, transmission of the packet is less safe
because this is the only transmission following the previous
transmission by a neighbor. We will show that such probability
is very low.
We extend the above scheme to further eliminate the
worst case by introducing a second or more delay(s) if each
expiration of delay returns finding event A happened at the
first place and all the later delays have both events B and C
happened. There are several issues relate to this extension.
First, this could lead to long total delay depending on traffic
load. We argue that since a source always sends its packets
without delaying, any long delay will eventually end when
any of its neighbors or itself sends a packet. The occurrence
of more than one delays has decreasing smaller probabilities.
Second, we introduce a delay upper bound Dmax at each
hop. The upper bound enables the control over the overall
end-to-end delay to meet QoS requirements. Especially, if
the sensor network is delivering time critical data, the upper
bound provides a tradeoff for optimal anonymity control and
performance. Finally, this method creates more chances for
a malicious node to perform trickle (or blocking) and flood
attack - only one packet is enough to trigger an emission from
its neighborhood. When this happens, this scheme reverts to
the previous one. However, a successful trickle and flood
attack has also very small probability in success, given that
sensor sources are uniformly distributed and each source
generates packets independently.
We name the basic scheme probabilistic reshaping
(PRESH) and the extended one extended probabilistic re-
shaping (exPRESH). In the next section, we will provide
analysis on the security properties of the two schemes. The
distributions enable a better understanding, especially about
the trade-offs between delay and protection.
IV. SECURITY ANALYSIS
A. Attacks
An eavesdropper in the system running PRESH or
exPRESH has no predictable clues in correlating two trans-
missions since each packet is delayed independently and
randomly. Transmission events {e1, e2, ..., eh}that the eaves-
dropper has collected during a time window Thdo not
help him in guessing which event the next transmission will
most likely relate to. Each event has the same probability
of 1
h. However, if an eavesdropper is able to monitor the
neighborhood for a long time, he might be able to identify
the busy period and the idle period of the system, especially,
when traffic load is not high. Thus, when a packet enters the
system with an empty queue, and when it is emitted again
but there is no other transmissions in the neighborhood up to
the time, the eavesdropper can correlate the two events most
successfully. This presents the worst case for the two schemes.
In the wireless medium, a single compromised node can
only block traffic passing through itself. To successfully
break the mixing scheme supported by the neighborhood,
the adversary has to compromise a number of nodes in the
neighborhood. These nodes then act coordinately to block the
transmissions in the neighborhood except the target packet
for a period in order to empty the queue and to wait for
the target packet to come out. If the adversary could obtain
Dmax, he can block the neighborhood for the same period to
increase blocking success probability. Then exPRESH reverts
to a threshold-based scheme. The success rate of such attack
depends on the time period the adversary spends in blocking
the traffic. After that the adversary only needs to send one
additional packet to convince the target packet to come out
after its delay period. In this case, exPRESH reverts to
PRESH. After all, launching such attacks requires very strong
adversary.
B. Assumptions
For the convenience of the analysis, we assume transmis-
sion delay and processing delay are negligible compared to
the random delay we introduced. Benchmark measurements
of Berkeley Mote MICA sensors show that transmission of
a sensor data packet uses 40-50 ms, encryption/decryption
requires about 2 ms using RC5 for 30 bytes data, and sensors
generate data at a rate of 60s [6]. Thus it is safe to assume a
mean of random delay at 60s or tens seconds. The aggregated
transmissions over the neighborhood, provide enough mixing
traffic.
We base our analysis on the uniformity of node distribution
and traffic distribution. In a neighborhood with mnodes,
each node originates data independently and each has equal
probability in relaying data for its neighbors. We simplify
the problem by assuming a packet flow will be relayed in
a neighborhood only once. We also simplify the model by
not differentiating a specific node from its neighbors since
we study the worst case. Thus, we model the traffic that
each node generates (both originating and relaying) as a
Poisson process with parameter λ. The aggregated traffic in
a neighborhood is a Poisson distribution with parameter mλ
(including the node itself). At a specific node, a portion of
the arrivals (at rate of λ) will be processed (here, delayed and
then re-emitted) with a service time exponentially distributed
with parameter µimmediately at arrival. From the view
point of an eavesdropper, the system we are modelling is
the neighborhood around it. So we redefine the events that
lead to the worst case, i.e., a transmission of a packet directly
following its previous transmission, to the following:
•Event A0: the queue is empty when the packet arrivals
at a node;
•Event B0: during one delay period, there is no transmis-
sions in the neighborhood.
Based on this aggregated neighborhood traffic model, we ana-
lyze the anonymity properties for both PRESH and exPRESH
schemes.
C. PRESH
For PRESH, a packet will be delayed only once. We model
the scheme as a m/m/∞queue system, where the arrivals
are in a Poisson distribution with parameter mλ, and infinite
number of servers are able to serve each arrival immediately
with a service time exponential distributed with parameter µ.
With this model, the analysis for PRESH follows the work
presented in [10] directly. We give the main results below
with a definition on traffic intensity ρ=mλ/µ.
1) Probability of worst case: The worst case happens when
the two events A0and B0occur. The probability is
P(worstC ase) = P(A0∩B0) = P(A0)P(B0)
For event A0,P(A0)is simply the probability that the
system is empty at an arrival, which is e−mλ
µ. For event B0,
the probability that during a service time there is no arrival is
equal to the probability that an sample drawn from Exp(µ)is
larger than a sample from Exp(mλ), which is µ
mλ+µ. Thus,
we have
P(worstC ase) = e−ρ
1 + ρ(1)
2) Size of anonymity set: When a packet is emitted, the
packet is mixed within the anonymity set. The anonymity set
of a packet Xconsists of the packets in the queue when X
arrivals, Xitself and the packets arriving during the service
time of X. The expectation of the number in the queue when
a packet arrives is mλ
µ. And the expectation of the number
arrivals during the busy period of a packet is e
mλ
µ+1
2. Thus
the expectation of the size of the anonymity set U0is:
U0=ρ+eρ+ 1
2(2)
Note that limρ→0U0= 1, i.e, when traffic load is very
low, the only transmission would be the packet itself. At that
time, the scheme fails to protect Xby mixing it with others:
limρ→0P(worstC ase) = 1.
3) Success rate of blocking attack: Suppose a time interval
τis used by the adversary to block the traffic and then to send
its own one packet. The blocking attack will success if all the
packets in the system leave within time τ. The memoryless
property gives us the probability that a packet in the system
leaving within τas P[X≤τ]=1−e−µτ . At the time the
blocking attack starts, the system could have arbitrary number
of packets, say i, with a probability of P[X=i] = ρi
i!e−ρ.
The probability that all of them leave the system within time
τis V[allLeave|X=i] = (P[X≤τ])i= (1−e−µτ )i. Thus,
the expectation of success probability of blocking attack is
V0(τ) =
∞
X
i=0
P[X=i](P[X≤τ])i=
∞
X
i=0
ρie−ρ(1 −e−µτ )i
i!
=exp(−mλe−µτ
µ) (3)
Equation (3) shows that if a blocking interval τis set,
the success probability decreases exponentially with the linear
decrease of the delay parameter µ.
D. exPRESH
For exPRESH, additional delays will be used when
PRESH’s worst case occurs with a probability of
P(exP RES H) = P(worstCase). When that happens,
an independent exponential delay starts. The procedure
repeats thereafter if the message terminates its service but
finds there is no arrival to the system during its last service
time. The procedure stops until at least one new message
has arrived or the maximum delay bound Dmax has been
reached, then the packet is emitted. When the scheme is
under blocking attack, the best strategy the attacker will use
is to force a packet to come out after the first delay. In that
case, exPRESH reverts to PRESH. The key performance
issue for exPRESH is the trade off between a longer delay
and a zero probability of worst case. The total delay time a
packet experiences becomes a critical factor. We model the
system using the state transition diagram shown in Figure 1.
In this model, the bound is not considered.
In the diagram, n, n ≥0,denotes the states of queue
length n. States 1Fand 1Lare the steady states for the
first and last message during a busy period in the system.
The corresponding probabilities of queue length in steady-
states are xn.x1Land x1Fare for steady states 1Fand 1L
respectively. We have the following balance equations:
mλx0=µx1L
mλx1F=mλx0
(mλ +µ)x1L= 2µx2
(mλ + 2µ)x2=mλx1F+mλx1L+ 3µx3
···
(mλ +nµ)xn=mλxn−1+ (n+ 1)µxn+1, n > 2
The derivation of xnis given in the Appendix I. An explicit
expression for x0is
x0=e−ρρ
(1 + ρ)−e−ρ(4)
Note that limρ→0x0=1
2. Based on the probabilities, we
are able to give the following properties.
Fig. 1. State Transition Diagram
1) Delay Distribution: For exPRESH, a packet may be
delayed for many terms till there is at least one arrival (packet
M) during its last delay term. Thus the total period Da
packet is delayed (the busy period) can be divided into two
sub periods T1and T2, where T1is the time for the first
packet Mto arrival in a busy period, and T2is the residual
service time after Marrivals. D=T1+T2. Due to the
memoryless property of the arrival and delay processes, T1
has the exponential interarrival distribution with parameter
mλ, and T2has the exponential service time distribution with
parameter µ. The probability density function of Dis:
fD(t) = Zt
0
mλe−mλs ·µe−µ(t−s)ds
=(mλµe−µt
mλ−µ(1 −e−(mλ −µ)t), if mλ 6=µ(5)
mλµte−µt , if mλ =µ(6)
Note that when mλ =µ,fD(t)becomes a Gamma
distribution with parameters mλ and 2. Figure 2 illustrates the
distribution of the total delay needed for a successful mixing
with µ= 1 packet/second. A reference curve of Exp(µ)
for PRESH is given too. The figure shows that when ρ= 5,
exPRESH and PRESH have very close probabilities at long
delay times. Owing to neighborhood multiplexing, ρ= 5 is
easy to reach. For example, for a neighborhood with 10 active
members, the mean delay could set to only a half of the mean
packet interarrival time.
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Probability
Time (at µ = 1 packet/second)
ρ = 0.5
ρ = 1.0
ρ = 5.0
reference, Exp(µ)
Fig. 2. Distribution of Delay
2) Mean total delay time: Now we compute the mean of
the total delay for exPRESH. The mean total delay time for
any message can be computed by conditioning on the system
is empty or not when the message arrives at the system. When
the system is empty, the massage has to wait until one arrives.
Thus,
D1=x0µ1
mλ +1
µ¶+ (1 −x0)1
µ=1
µ+x0
mλ
Substituting x0yields the mean total delay time
D1=1
µµ1 + e−ρ
(1 + ρ)−e−ρ¶(7)
One can see that when the traffic intensity ρis large, the
mean total delay is just the mean service time µ−1. When ρ
is small, however, the total delay increases because the first
message in the busy period has to wait for a new one to come.
The additional delay x0/mλ could be substantially large. An
illustration of the function is given in Figure 3, where the
mean total delay is normalized to the mean service time. In
order to demonstrate the convergence to 1, X axis starts at
0.25 in the figure.
V. SIMULATIONS
We investigate the influence of aggregated traffic on the
mixing schemes PRESH and exPRESH through simulation.
We use GlomoSim[18], a packet level simulator for wireless
and wired networks. In the simulation, 225 nodes are uni-
formly distributed in a square field at an average nodal density
of 16 nodes per transmission radius. In an attempt to collect
data from uniform traffic pattern, we set sinks on one side of
the field and all the data flow towards the same direction. The
statistics are only collected from the nodes at the other side of
the field. Those nodes also have their neighborhood within the
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
1 2 3 4 5 6
Normalized Ratio
ρ = mλ/µ
Normalized Mean Total Delay Time
Fig. 3. Mean Total Delay Time Normalized to µ
simulation area. Even though, for the simulations conducted,
we can not enforce that one flow being relayed only once in
a node’s neighborhood. We run the simulations over multiple
seeds for the random number generator. The results should
reflect accuracy statistically. In simulations, we set the mean
date generating interval to be fixed at one report every 60
seconds [6]. The mean delay time varies from one second to
60 seconds. At short delay intervals like one second, a node
does not have enough aggregated transmissions to have at least
one packet on average in order to hide its own traffic. The X
axis is the relay traffic intensity ρ0for one node, i.e, λ/µ. Thus
when each node generates packets every 60s on average in a
neighborhood of 15 nodes, a node has to wait for 4 seconds
(ρ0= 0.067) on average to hear another transmission. If a
node delays a packet longer, it has better chance for mixing.
We collect statistics to calculate average P(worstCase)for
PRESH and the size of the anonymity set for both schemes.
For exPRESH scheme, the worst case occurs when the total of
extended delays exceeds the bound Dmax. At that time, even
without an arrival, the packet has to be emitted. We count the
occurrences of this event to calculate the P(worstCase)for
exPRESH. Obviously the probability is affected by the value
of Dmax. In the simulation, we vary Dmax to be one, two or
three times of the chosen mean delay for exPRESH.
Figure 4 gives the probabilities for the worst cases. The
possibilities approach zero quickly when the traffic intensity
increases. Clearly exPRESH schemes decrease faster than
PRESH. In addition, the larger the delay bound, the smaller
the probabilities. The result suggests that we can trade delay
for stronger security protection. Figure 5 gives the sizes of
the anonymity sets corresponding to the above configured
parameters. The sizes increase when load increases. The two
schemes demonstrate little differences due to the fact that
most packets experienced the same neighborhood activities.
Differences exist when load is low.
The distribution of the total delays that packets experienced
at each node is illustrated in Figure 6 and 7 with mean delay
setting to one second and 60 seconds respectively. For both
figures, the bound Dmax is enforced in such a way that a
delay drawn from the Exp() must be finished even if Dmax
is met. This allows us to show the full distribution smoothly.
Otherwise, one can expect the curves end with a peak at the
corresponding Dmax values. When calculating probability, the
placement of bins affects the shape of the curves slightly.
Here, each figure takes 20 bins. The corresponding value is
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Probability
Traffic Intensity
exPRESH, 3*MeanDelay
exPRESH, 2*MeanDelay
exPRESH, 1*MeanDelay
PRESH
Fig. 4. Probability of Worst Case
0
5
10
15
20
25
30
35
40
45
50
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Mean Set Size
Traffic Intensity
exPRESH, 3*MeanDelay
exPRESH, 2*MeanDelay
exPRESH, 1*MeanDelay
PRESH
Fig. 5. Size of Anonymity Set
drawn at the right side of a bin. For example, a point drawn
at ”Per Hop Delay = 1s” in Figure 6, counts data fallen in
the bin of (0.5,1]. This particular case actually counts for the
leverage of a possible higher peak at ”x = 1s”.
In Figure 6, PRESH shows exponential decay trend of the
probability with regard to the delay time, which is expected.
For exPRESH, more terms of delays will be used since the
probability of the worst case occurring after the first delay
is relatively high at one second as the mean delay time.
The figure shows that exPRESH exhibits higher probabilities
of longer overall delays than those of PRESH. The curves
shift towards the right side of PRESH. Especially, larger
Dmax shows more shifting corresponding to longer delays.
When the mean delay reaches 60 second (Figure 7), all the
schemes reveal the same distribution. This is because that the
aggregated traffic intensity is so high that all the packets will
be transmitted after one delay no matter what scheme is in
use. The distribution converges to the exponential distribution
with µ.
VI. CONCLUSION AND FUTURE WORK
The paper has presented traffic shaping schemes to coun-
termeasure traffic analysis over sensor data transmissions.
PRESH and exPRESH both use random delays to de-correlate
transmission events without introducing extra bandwidth over-
head for sensors. The approaches also exploit the traffic
multiplexing ability of the wireless medium to reduce the
overall delay and to reduce the probability of the worst case.
exPRESH presents an improvement over PRESH to eliminate
the occurrence of the worst case at the cost of longer ene-to-
end delays. Analysis on the anonymity properties and simu-
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6 7 8 9 10
Probability
Per Hop Delay (s)
exPRESH, 3*MeanDelay
exPRESH, 2*MeanDelay
exPRESH, 1*MeanDelay
PRESH
Fig. 6. Distribution of Delay at Mean = 1 second
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 50 100 150 200 250 300
Probability
Per Hop Delay (s)
exPRESH, 3*MeanDelay
exPRESH, 2*MeanDelay
exPRESH, 1*MeanDelay
PRESH
Fig. 7. Distribution of Delay at Mean = 60 seconds
lations are given in the paper. The main results are that with
very high probability, the schemes successfully mix sensor
data transmissions among neighborhood radio activities; and
the insecure facts of the schemes decrease exponentially with
linear increase of mean delay time.
Some applications of sensor networks do not demonstrate
uniform traffic patterns, e.g, applications with only a few base
stations will show traffic concentration near the base stations.
Our future work will study traffic shaping strategies for these
traffic patterns. We will also analyze the impact of capacity
on neighborhood mixing technique.
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APPENDIX I
STEADY-STATE PROBABILITY xn
The balance equations can be written as:
mλxn= (n+ 1)µxn+1, n ≥2
Let ρ=mλ/µ, we have xnas functions of x2:
xn= 2ρn−2
n!x2, n ≥2
We then obtain the following:
x2=ρ(1 + ρ)
2x0
x1L=ρx0, x1F=x0
Using the normalization condition yields
x0+x1L+x1F+
∞
X
n=2
xn= 1
x0"2 + ρ+ρ(1 + ρ)
∞
X
n=2
ρn−2
n!#= 1
So
x0=ρ
(1 + ρ)eρ−1
