### Using event detection latency to evaluate the coverage

### of a wireless sensor network

### You-Chiun Wang, Kai-Yang Cheng, Yu-Chee Tseng

*Department of Computer Science, National Chiao-Tung University, Hsin-Chu 30010, Taiwan, ROC Available online 15 June 2007

Abstract

A wireless sensor network (WSN) consists of many tiny and low-power devices deployed in a sensing ﬁeld. One of the major tasks of a WSN is to monitor the surrounding environment and to detect events occurring in the sensing ﬁeld. Given an event appearing in a WSN, the event detection latency is to model the time that it takes for the WSN to be aware of the event. In this work, we analyze the latency using a probabilistic approach under an any-sensor-detection and a k-sensor-detection models, where k > 1 is an integer. Such an analysis can be used as an index to evaluate a WSN’s coverage and thus can help guide the deployment of a WSN. We also develop simulations to verify our analytical results.

2007 Elsevier B.V. All rights reserved.

Keywords: Ad hoc network; Network coverage; Pervasive computing; Ubiquitous computing; Wireless sensor network

1. Introduction and problem statement

Wireless sensor networks (WSN) have been intensively

studied recently [1]. A WSN consists of many tiny and

lower-power sensor nodes, each of which can collect sur-rounding environmental data and communicate with neighboring nodes. Communications in a WSN typically

takes place in an ad hoc manner[2]. Applications of WSNs

include surveillance and agriculture, habitat, traﬃc, and

civil infrastructure monitoring[3–7].

One of the major tasks of a WSN is to detect events occurring in the sensing ﬁeld. Given an event appearing in a WSN, the event detection latency is to model the time that it takes for the WSN to be aware of the event. Such latency is an important metric to measure the monitoring capability of a WSN’s deployment for real-time

applica-tions such as surveillance[8–10]or object tracking[11–13].

We propose our model to analyze the event detection latency. Speciﬁcally, we are given a sensing ﬁeld, on which

there are n homogeneous sensors. Each sensor has a sensing distance of r. Without loss of generality, we assume that these n sensors form a connected network. To simplify the analysis, we assume that the time axis is divided into ﬁxed-length slots and the working schedule of each sensor is modeled by a sequence of working cycles, each of length T slots. Each working cycle is led by an active phase fol-lowed by an idle phase. The former consists of the ﬁrst D

slots, and the latter the rest of the T D slots. Sensors only

conduct detection jobs in their active phases, and go to sleep in idle phases. However, sensors do not synchronize their clocks, so their working cycles are not necessarily

aligned. Fig. 1 shows an example. Note that this model

can be applied to most of the MAC/network protocols that are proposed for WSN recently. For example, for energy

conservation, the Zigbee/IEEE 802.15.4 standard [14]

allows a sensor node to wake up and sleep very similarly

to our working cycles inFig. 1. In fact, several other

pro-tocols (such as Bluetooth [15]and S-MAC[16]) also have

such an awake-sleep behavior.

Our objective is to evaluate the detection latency when an event occurs in the sensing ﬁeld. Note that we make no assumption on the locations of events. To take errors into account, we also assume that in an active slot, a sensor has 0140-3664/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.comcom.2007.05.009

*

Corresponding author. Tel.: +886 3 5131366; fax: +886 3 5724176. E-mail addresses: wangyc@csie.nctu.edu.tw (Y.-C. Wang),

kycheng@csie.nctu.edu.tw (K.-Y. Cheng), yctseng@csie.nctu.edu.tw

(Y.-C. Tseng).

a probability of p to successfully detect the occurrence of an event if the event is within this sensor’s sensing range. To sim-plify the analysis, we assume that p is a constant but not a function of the distance between a sensor and the event [17,18]. We consider two detection models in this work:

• Any-sensor-detection model: To capture the event, the network needs at least one sensor to successfully detect the event.

• k-sensor-detection model: To capture the event, the net-work needs at least k sensors to successfully detect the event, where k > 1. (The value of k is application-depen-dent. For example, positioning protocols using

triangu-lation[19–21]require at least three sensors.)

2. Analysis of event detection latency

To facilitate the calculation of the event detection latency, we establish a system clock, which starts at the instant when the event appears. The system time is also slotted and each set

of continuous T slots forms a system cycle, as shown inFig. 1.

Suppose that an event appears at location (x, y) in the sensing ﬁeld. Let M(x, y) be the number of sensors whose sensing ranges cover location (x, y). Consider the time slots that these M(x, y) sensors start their new working cycles in a system cycle. We can classify them into T groups such that the ith

group contains the misensors that start their working cycles

at the ith slot in a system cycle, i = 1, . . . , T. For example, in Fig. 1, sensor 1 belongs to group 2, sensor 2 belongs to group

1, and sensor n belongs to group T. Clearly, PT_{i¼1}mi¼

Mðx; yÞ. Taking all combinations of mi’s into consideration,

the event detection latency under this particular M(x, y) can be written as LatencyðMðx;yÞÞ ¼X Mðx;yÞ m1¼0 X Mðx;yÞm1 m2¼0 X Mðx;yÞðm1þþmT2Þ mT1¼0 Mðx; yÞ! m1! . . . mT! 1 T Mðx;yÞ! dðm1; . . . ; mTÞ;

where the ﬁrst term is the probability to observe a

particu-lar combination (m1, . . . , mT), and the second term

d(m1, . . . , mT) is the expected latency for this particular

combination.

As the event may appear in any location (x, y) inside the sensing ﬁeld, we have to consider all possible M(x, y). Thus, the overall expected latency of the WSN can be expressed as ET ;D¼

Z

x

Z

y

LatencyðMðx; yÞÞdx dy;

¼X

n

i¼0

Prob½Mðx; yÞ ¼ i LatencyðiÞ: ð1Þ

When the event appears in an i-covered region, it will be sensed by i sensors (i.e., M(x, y) = i). Therefore, Prob[M(x, y) = i] is the ratio of areas that are i-covered

in the sensing ﬁeld. So Eq.(1)can be simpliﬁed as

ET ;D¼

Xn

i¼0

Ai

A LatencyðiÞ;

where A is the area of the sensing ﬁeld, and Aiis the total

area in A in which each point is covered by exactly i sensors.

In Sections 2.1 and 2.2, we will show how to compute

d(m1, . . . , mT) under our two detection models, respectively.

Table 1summarizes the notations used in this work. 2.1. Any-sensor-detection model

Under this model, the event is considered to be captured by the network if any sensor successfully detects its existence.

Let xi be the number of active sensors at the ith slot,

i = 1, . . . , T. These xi sensors are composed of three types

of sensors: (1) sensors which turn into active at the ith slot, (2) sensors which turn into active between the ﬁrst and the

(i 1)th slots, and (3) sensors which turns into active before

the ﬁrst slot. Note that case (2) can be true if D > 1 and i > 1, while case (3) can only occur when i < D. This leads to xi¼ miþ X minðD1;i1Þ j¼1 mijþ X Di1 j¼0 mTj:

We also deﬁne xaT+bas the number of active sensors at the

(aT + b)th slot for any a P 1. Since cycles repeat every T slots, we have xaT+b= xb.

Time
**.**
**.**
**.**
*sensor n*
sensor 2
sensor 1
T

*system cycle i* *system cycle i +1*
slot
active phase
(D slots)
idle phase
(T-D slots)
. . . .
system cycle 1
. . . .
Time that an
event occurs
. . . .
. . . .
. . . .
working cycle
(T slots)

The probability that there is at least one sensor
success-fully detecting the event in the ﬁrst slot is ð1 ð1 pÞx1_{Þ.}

For i P 2, the probability that the event is not detected

in the ﬁrst (i 1) slots but is successfully detected in the

ith slot isð1 ð1 pÞxi_{Þð1 pÞ}x1þþxi1_{. Hence, as the time}

goes to inﬁnity, the expected detection latency under the
any-sensor-detection model is
dðm1; . . . ; mTÞ ¼
X1
a¼0
XT
b¼1
ðaT þ bÞ ð1 ð1 pÞxb_{Þ}
ð1 pÞaðx1þþxTÞþx1þþxb1_{:} _{ð2Þ}

Eq.(2)contains an inﬁnite number of expressions. The

fol-lowing theorem shows that it will converge.

Theorem 1. The expected delay d(m1, . . . , mT) under the

any-sensor-detection model is bounded by

dðm1; . . . ; mTÞ 6

T2 ð1 aÞ2; where a = (1 p)D·M(x, y).

Proof. Since ð1 ð1 pÞxb_{Þ 6 1 and (1 p) 6 1, we can}

obtain that
dðm1; . . . ; mTÞ ¼
X1
a¼0
XT
b¼1
ðaT þ bÞ ð1 ð1 pÞxb_{Þ}
ð1 pÞaðx1þþxTÞþx1þþxb1_{;}
6X
1
a¼0
XT
b¼1
ðaT þ bÞð1 pÞaðx1þþxTÞ_{;}
6X
1
a¼0
ð1 pÞaDMðx;yÞX
T
b¼1
ða þ 1Þ T
!
;
¼X
1
a¼0
aa_{ða þ 1ÞT}2_{;}
¼ T
2
ð1 aÞ2:
2.2. k-Sensor-detection model

Under this model, the event is considered to be captured by the network, once there are at least k sensors success-fully detecting its occurrence. Since the sequence x1, x2, . . .

has a period of T, the expected latency can be written as dðm1; . . . ; mTÞ ¼ X1 a¼0 XT b¼1 ðaT þ bÞ Pkðm1; . . . ; mT; aTþ bÞ; ð3Þ where Pk(m1, . . . , mT, aT + b) is the probability that there are

at least k sensors successfully detecting the event at the (aT + b)th slot, but not so before that slot. To ﬁnd Pk(m1, . . . , mT, aT + b), let Nebe the number of sensors that

have already succeeded in detecting the event before the

(aT + b)th slot, and Nfbe the number of sensors that succeed

in detecting this event at the (aT + b)th slot for the ﬁrst time. We ﬁrst categorize sensors according to their behaviors as

shown inFig. 2. There are xaT+b= xbactive sensors at the

(aT + b)th slot, and the rest of M(x, y) xbsensors are

inac-tive. The inactive sensors can be further divided into a set of

N1sensors which have ever succeeded in detecting this event

before the (aT + b)th slot, and a set of M(x, y) xb N1

sensors which have not. Similarly, the active sensors can be

divided into a set of N2sensors which succeed in detecting

this event at this slot, and a set of xb N2sensors which fail

to detect this event at this slot. From the latter set, we further

identify a set of N3 sensors which have ever succeeded in

detecting this event before the (aT + b)th slot, but fail to de-tect this event at the current slot.

Based on the above deﬁnitions, once the values of Ne, N1,

N2, and N3are given, the rest of the variables inFig. 2will all

be ﬁxed. Speciﬁcally, the number of sensors that successfully detect this event at the (aT + b)th slot and have also

suc-ceeded in doing that before is Ne N1 N3, and the number

of sensors that succeed in detecting this event for the ﬁrst time at the (aT + b)th slot is Nf= N2 (Ne N1 N3). In

Eq.(3), the latency is considered to be aT + b if Ne< k and

Table 1

Summary of notations used in this work

Notations Deﬁnition

n Number of sensors in the sensing ﬁeld

k Minimum number of sensors required to successfully detect the event

T Number of slots in a working or system cycle

D Number of slots that a sensor continues detecting the event

p Probability that a sensor successfully detects the event in an active slot

M(x, y) Number of sensors that can detect the event when the event occurs at location (x, y) mi Number of sensors in M(x, y) that repeat their working cycles at the ith slot in a system cycle

xi Number of sensors detect the event at the ith slot in a working cycle

Pk(m1, . . . , mT, aT + b) Probability that there are at least k sensors succeeding in detecting the event

Ne Number of sensors that have ever succeeded in detecting the event before the (aT + b)th slot

Nf Number of sensors that ﬁrst succeed in detecting the event at the (aT + b)th slot

N1 Number of sensors that have ever succeeded in detecting the event before but do not detect at the (aT + b)th slot

N2 Number of sensors that succeed in detecting the event at the (aT + b)th slot

N3 Number of sensors that have ever succeeded in detecting the event before but fail at the (aT + b)th slot

Si Number of sensors in the subset i

Nf= (N1+ N2+ N3) NeP k Ne. By enumerating all

combinations of Ne, N1, N2, and N3, we can derive that

Pkðm1; . . . ; mT; aTþ bÞ ¼ Xk1 Ne¼0 XNe h1¼0 Prob½N1¼ h1 Xxb h2¼0 Prob½N2¼ h2 X Neh1 h3¼0 Prob½N3¼ h3 Prob½NfP k Ne !!! : ð4Þ Depending on the value of b, we can further derive the four

terms Prob[N1= h1], Prob[N2= h2], Prob[N3= h3], and

Prob[NfP k Ne] with three cases.

Case (1): b < D. Consider the set of M(x, y) xbinactive

sensors at the (aT + b)th slot. We divide them into two subsets:

• S1: The set of sensors whose active phases do not cross

the boundaries of system cycles.

• S2: The set of sensors whose active phases cross the

boundaries of system cycles.

Clearly, jS1j = mb+1+ mb+2+ + mT(D1) and

jS2j = mT(D1)+1+ mT(D1)+2+ + MT(Db). For

example, when b = 2,Fig. 3 shows the above two subsets

in case 1. Recall the deﬁnition of N1. Among these N1

sen-sors, let R1be the number of sensors belonging to S1, and

R2 the number of sensors belonging to S2. Since

R1+ R2= N1, we can expand Eq.(4)as follows:

Pkðm1; . . . ; mT; aTþ bÞ ¼X k1 Ne¼0 XNe h1¼0 Xh1 r1¼0 Prob½R1¼ r1 Prob½R2¼ h1 r1 X xb h2¼0 Prob½N2¼ h2 X Neh1 h3¼0 Prob½N3¼ h3 Prob½Nf P k Ne !!! : ð5Þ

Given two integers x and y such that x P y and a probabil-ity value z, let us deﬁne

Binoðx; y; zÞ ¼ ðx_{y}Þzy_{ ð1 zÞ}xy_{:}

The probability functions in Eq. (5) are derived as

follows: Prob½R1¼ r1 ¼ BinoðjS1j; r1;1 ð1 pÞ aD Þ; Prob½R2¼ h1 r1 ¼ XD2 i¼1 mTðD1Þþi jS2j BinoðjS2j; h1 r1;1 ð1 pÞ aDþi Þ; Prob½N2¼ h2 ¼ Binoðxb; h2; pÞ; Prob½N3¼ h3 ¼ Binoðxb h2; h3; Xb1 i¼0 mbi xb ð1 ð1 pÞaDþbiÞ þ X Db1 i¼0 mTi xb ð1 ð1 pÞaDþbÞÞ; and

Succeed for the first time? Succeed at this slot?

Succeeded before?
Succeeded before?
Yes
No
Yes
Yes Yes
Yes
No No
No No
Conduct detection
at this slot?
*M(x,y)*
*M(x,y)-xb*
*M(x,y)-xb-N*1 *N*1
*xb-N*2*-N*3 *N*3 *Ne-N*1*-N*3
*N*2*-(Ne-N*1*-N*3*) = Nf*
*N*2
*xb-N*2
*xb*

Fig. 2. Classiﬁcation of sensors in the (aT + b)th slot. Numbers in ovals indicate numbers of sensors.

### {

### {

## {

Case (1) Case (2) Case (3) Slot Number*a T +*

*a T +*

*a T +*

*a T +*

*a T +*

*a T +*

*a T +*

*a T +*3 6 5 4 1 2 7 8 1 2 7 8

*(a+1)T +*

*(a+1)T +*

*(a 1)T +*

*(a 1)T +*

*S*2

*S*1

*S*3

*S*2

*S*1

*S*2

*S*1

*m*

2
*m*

1 *m*

3*m*

4*m*

5*m*

6*m*

7*m*

8
... ... ... ... ... ... ... ...
...
...
...
...
...
...
...
...
### . . .

### . . .

: active slotProb½Nf P k Ne ¼ Binoðh2; Nf; Xb1 i¼0 mbi xb ð1 pÞaDþbi þ X Db1 i¼0 mTi xb ð1 pÞaDþb1Þ:

Prob½R1¼ r1 is the probability that r1sensors in S1 have

ever succeeded in detecting this event before the (aT +

b)th slot, where 1 (1 p)aD is the probability that such

a sensor has ever successfully detected this event before the (aT + b)th slot. Prob½R2¼ h1 r1 is derived similarly,

except that we are concerned about sensors in S2 and,

among these sensors, there is a ratio ofmTðD1Þþi

jS2j of sensors

which have tried to detect this event for aD + i slots (and we take their average). Prob[N2= h2] is the probability that

there are h2sensors among xbsensors successfully detecting

the event at the (aT + b)th slot. Prob[N3= h3] is the

prob-ability that there are h3sensors among xb h2sensors that

have ever successfully detected the event before the (aT + b)th slot. Note that the third term in Bino(Æ) is to take care of those sensors whose active slots do not (the ﬁrst expression) and do (the second expression) cross the boundaries of system cycles, and we take their average.

Prob[NfP k Ne] is similar to the previous probability

ex-cept that these sensors succeed for the ﬁrst time at the (aT + b)th slot.

Case (2): D 6 b 6 T D + 1. In this case, we divide the

set of inactive M(x, y) xbsensors at the (aT + b)th slot

into three subsets according to whether their active slots cross the boundaries of system cycles:

• S1: The set of sensors which have ﬁnished their active

slots in the current system cycle and whose active slots do not cross the boundaries of system cycles.

• S2: The set of sensors which have not started their

active slots in the current system cycle and whose active slots do not cross the boundaries of system cycles.

• S3: The set of sensors whose active slots cross the

boundaries of system cycles.

We can obtain that jS1j ¼PbD_{i¼1}mi, jS2j ¼PTðD1Þ_{i¼bþ1} mi,

and jS3j ¼PT_{i¼TðD1Þþ1}mi. For example, when b = 4,

Fig. 3shows these subsets in case 2. Again, let R3be the

number of sensors belonging to S3. Since

R1+ R2+ R3= S1, we can expand Eq.(4) as follows:

Pkðm1; . . . ; mT; aTþ bÞ ¼X k1 Ne¼0 XNe h1¼0 Xh1 r1¼0 X h1r1 r2¼0 Prob½R1¼ r1 Prob½R2¼ r2 Prob½R3¼ h1 r1 r2 ! X xb h2¼0 Prob½N2¼ h2 X Neh1 h3¼0 Prob½N3¼ h3 Prob½Nf k Ne ; where

Prob½R1¼ r1 ¼ BinoðjS1j;r1;1 ð1 pÞðaþ1ÞDÞ;

Prob½R2¼ r2 ¼ BinoðjS2j;r2;1 ð1 pÞ aD Þ; Prob½R3¼ h1 r1 r2 ¼X D2 i¼0 mTðD1Þþ1þi jS3j BinoðjS3j; h1 r1 r2;1 ð1 pÞaDþiþ1Þ; Prob½N2¼ h2 ¼ Binoðxb; h2; pÞ; Prob½N3¼ h3 ¼ Binoðxb h2; h3; XD1 i¼0 mbi xb ð1 ð1 pÞaDþiÞÞ; and Prob½NfP k Ne ¼ Binoðh2; Nf; XD1 i¼0 mbi xb ð1 pÞaDþiÞ:

Again, Prob½R3¼ h1 r1 r2 is the probability that

h1 r1 r2sensors in S3 have ever succeeded in

detect-ing this event before the (aT + b)th slot, where

1 (1 p)aD+i+1 is the probability that such a sensor

have ever successfully detected this event before the

(aT + b)th slot. However, among these sensors in S3,

there is a ratio of mTðD1Þþ1þi

jS3j of sensors which have tried

to detect this event for aD + i slots, and thus we take their average.

Case (3): b > T D + 1. In this case, we divide the set of

inactive M(x, y) xbsensors at the (aT + b)th slot into two

subsets according to whether their active slots cross the boundaries of system cycles:

• S1: The set of sensors whose active slots do not cross the

boundaries of system cycles.

• S2: The set of sensors whose active slots cross the

boundaries of system cycles.

We have S1¼PbD_{i¼1}mi and S2¼PT_{i¼bþ1}mi. Fig. 3

gives an example when b = 7. We derive Eq. (4) as

follows: Pkðm1; . . . ; mT; aTþ bÞ ¼X k1 Ne¼0 XNe h1¼0 Xh1 r1¼0 Prob½R1¼ r1 Prob½R2¼ h1 r1 ! X xb h2¼0 Prob½N2¼ h2 X Neh1 h3¼0 Prob½N3¼ h3 Prob½Nf P k Ne ; where

Prob½R1¼ r1 ¼ BinoðjS1j; r1;1 ð1 pÞðaþ1ÞDÞ;
Prob½R2¼ h1 r1 ¼X
D2
i¼1
mTðD1Þþ1þi
jS2j
BinoðjS2j; h1 r1;1 ð1 pÞaDþiþ1Þ;
Prob½N2¼ h2 ¼ Binoðxb; h2; pÞ;
Prob½N3¼ h3 ¼ Binoðxb h2; h3;
XTb
i¼0
mTðD1Þi
xb ð1 ð1 pÞ
aDþiþ1
Þ
þ X
bTþðD1Þ1
i¼0
mTðD1Þþ1þi
xb ð1 ð1 pÞ
aDþiþ1_{ÞÞ;}
and

Prob½Nf P k Ne ¼ Binoðh2; Nf;X Tb i¼0 mTðD1Þi xb ð1 pÞ aDþiþ1 þ X bTþðD1Þ1 i¼0 mTðD1Þþ1þi xb ð1 pÞ aDþiþ1 Þ:

Finally, by replacing Pk(m1, . . . , mT, aT + b) in Eq. (3)

with one of the above three cases, we can obtain the expected latency d(m1, . . . , mT) under the

k-sensor-detec-tion model.

Table 2 summarizes the four terms Prob[N1= h1],

Prob[N2= h2], Prob[N3= h3], and Prob[Nf P k Ne] in

Eq.(4) under the three cases.

3. Using detection latency to guide deployment

Event detection latency can be used as an index to evaluate a WSN’s coverage and thus can help guide the deployment of a WSN. Below, we brieﬂy discuss how to improve the coverage of a WSN. First, we can partition the sensing ﬁeld into several subregions. Then, we can evaluate the event detection latency of each sub-region. If the expected latency of a region is larger than a tolerable threshold, it means that there are not enough sensors deployed in the region. For such regions, we can deploy more sensors to improve their expected detection latencies.

Beside, the event detection latency can also be used to measure the latency to detect a node newly joining a wireless personal area network (WPAN). We observe that for a device to join a WPAN, usually a network discovery procedure needs to be taken. To facilitate network discovery, coordinators in a WPAN normally need to send beacons periodically to announce their

presence (for example, Bluetooth, WiMedia [22], and

ZigBee follow this model). If we regard the beacon windows as our active phases, then the event detection latency under our any-sensor-detection model is the latency for a new node to discover the WPAN.

4. Simulation results

We have developed a simulator using C++ language to verify our analytical results. In the simulations, we set up a

sensing ﬁeld of size 10· 10, on which there are 50 sensors

randomly deployed. Each sensor has a sensing distance of 3 units. Events may appear in any location inside the sens-ing ﬁeld. Given a network conﬁguration, we evaluate the

event detection latency by both Eq.(1)and the simulations.

For each simulation, at least 1000 experiments are repeated, and we take their average.

Fig. 4shows the event detection latencies under diﬀerent values of detection probability p. The simulation results coincide well with the analytical results, except when p = 0.1 under the 5-sensor-detection model. This is because the simulator only simulates 1000 possible locations that an

event may occur, while the analysis (Eq. (1)) has to

con-sider all possible locations inside the sensing ﬁeld. Since the value of p is small, the network requires longer time to capture the event than we expect.

Fig. 5shows the event detection latencies under diﬀerent values of M(x, y). In the simulation, when an event occurs within i sensors’ sensing ranges, we record the detection latency in the corresponding M(x, y) = i statistics. From Fig. 5, we can observe that the simulation results coincide well with the analytical results, except when p = 0.1 and M(x, y) 6 5 under 3-sensor-detection model. This is because our analysis assumes a larger-scale network. It can be observed that a larger M(x, y), which implies a higher network density, can help reduce the detection latency. A larger detection probability p, which reﬂects the sensibility of sensors, can also reduce the detection latency. The result can be used to determine how sensors should be arranged at the deployment stage.

In both Figs. 4 and 5, we can observe that the event

detection latency can be greatly reduced when we increase the number of active slots D, especially when the detection probability p is small. Thus, we have interest in observing

Table 2

Summary of the four terms Prob[N1= h1], Prob[N2 = h2], Prob[N3= h3], and Prob[NfP k Ne] in Eq.(4)under diﬀerent cases, where q = 1 p

Cases Terms

b < D Prob½N1¼ h1 ¼Prh11¼0BinoðjS1j; r1;1 q

aD_{Þ }PD2

i¼1 mTDþiþ1jS2j BinoðjS2j; h1 r1;1 q

aDþi_{Þ}

Prob[N2= h2] = Bino(xb, h2, p)

Prob½N3¼ h3 ¼ Binoðxb h2; h3;Pb1i¼0mxbib ð1 q

aDþbi_{Þ þ}PDb1

i¼0 mxTib ð1 q

aDþb_{ÞÞ}

Prob½NfP k Ne ¼ Binoðh2; Nf;Pb1i¼0mxbib q

aDþbi_{þ}PDb1
i¼0 mxTib q
aDþb1_{Þ}
D 6 b 6 T D + 1 Prob½N1¼ h1 ¼Phr11¼0
Ph1r1
r2¼0BinoðjS1j; r1;1 q
ðaþ1ÞD_{Þ BinoðjS}

2j; r2;1 qaDÞ Pb2i¼0mTDþiþ2jS3j Binoðj S3j; h1 r1 r2;1 q

aDþiþ1_{Þ}

Prob[N2= h2] = Bino(xb, h2, p)

Prob½N3¼ h3 ¼ Binoðxb h2; h3;PD1i¼0 mxbib ð1 q

aDþi_{ÞÞ}

Prob½NfP k Ne ¼ Binoðh2; Nf;PD1i¼0 mxbib q

aDþi_{Þ}

b > T D + 1 Prob½N1¼ h1 ¼Prh11¼0BinoðjS1j; r1;1 q

ðaþ1ÞD_{Þ }PD2

i¼1 mTDþiþ2jS2j BinoðjS2j; h1 r1;1 q

aDþiþ1_{Þ}

Prob[N2= h2] = Bino(xb, h2, p)

Prob½N3¼ h3 ¼ Binoðxb h2; h3;PTbi¼0TDiþ1xb ð1 q

aDþiþ1_{Þ þ}PbþDT2

i¼0 mTDþiþ2xb ð1 q

aDþiþ1_{ÞÞ}

Prob½NfP k Ne ¼ Binoðh2; Nf;PTbi¼0TDiþ1xb q

aDþiþ1_{þ}PbþDT2
i¼0 mTDþiþ2xb q

the eﬀect of D on the event detection latency under diﬀerent

values of M(x, y) and p, as shown inFig. 6. To show the

eﬀect of D, we set the period T as a constant of 16 slots.

From Fig. 6, we can observe that the latencies drop as

the value of D increases, but this eﬀect becomes less signif-icant when D P 4. Since a sensor will consume more

(a) T = 5 and D = 1 (b) T = 5 and D = 2

(c) T = 5 and D = 1 (d) T = 5 and D = 2 0 5 10 15 20 25 30 35 40 45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability p

Detection latency (slots)

any-sensor model (analysis) any-sensor model (simulation) 5-sensor model (analysis) 5-sensor model (simulation)

0 5 10 15 20 25 30 35 40 45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability p

Detection latency (slots)

0 5 10 15 20 25 30 35 40 45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability p

Detection latency (slots)

0 5 10 15 20 25 30 35 40 45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability p

Detection latency (slots)

any-sensor model (analysis) any-sensor model (simulation) 5-sensor model (analysis) 5-sensor model (simulation)

any-sensor model (analysis) any-sensor model (simulation) 5-sensor model (analysis) 5-sensor model (simulation)

Fig. 4. The event detection latencies under diﬀerent values of probability p.

0 3 6 9 12 15 4 9 10 11 12 M (x,y)

Event detection latency (slots)

Event detection latency (slots)

p = 0.1 (simulation) p = 0.1 (analysis) p = 0.5 (simulation) p = 0.5 (analysis) p = 0.9 (simulation) p = 0.9 (analysis) 0 3 6 9 12 15 p = 0.1 (simulation) p = 0.1 (analysis) p = 0.5 (simulation) p = 0.5 (analysis) p = 0.9 (simulation) p = 0.9 (analysis) 0 10 20 30 40 50 60 p = 0.1 (simulation) p = 0.1 (analysis) p = 0.5 (simulation) p = 0.5 (analysis) p = 0.9 (simulation) p = 0.9 (analysis) 0 10 20 30 40 50 60 p = 0.1 (simulation) p = 0.1 (analysis) p = 0.5 (simulation) p = 0.5 (analysis) p = 0.9 (analysis) p = 0.9 (simulation)

(a) T = 5 and D = 1 (any-sensor model) (b) T = 5 and D = 2 (any-sensor model)

(c) T = 5 and D = 1 (3-sensor model) (d) T = 5 and D = 2 (3-sensor model)

Event detection latency (slots)

Event detection latency (slots)

5 6 7 8 4 9 10 11 12 M (x,y) 5 6 7 8 4 9 10 11 12 M (x,y) 5 6 7 8 4 9 10 11 12 M (x,y) 5 6 7 8

energy as the length of active slots D increases, this result can be used to decide the length of a sensor’s active phase to reduce both event detection latency and energy con-sumption of a WSN.

5. Conclusions

We have proposed a methodology to analyze the event detection latency of a WSN. Such a latency analysis can be used to measure the network coverage and the time that a new node needs to discover a network. We have adopted a probabilistic approach to analyze the latency under an

any-sensor-detection and a k-sensor-detection models. We have also developed a simulator to verify our analyses. Simulation results not only coincide well with the analyses, but also indicate the potential factors that aﬀect the latency.

Our analysis assumes that the detection probability p is a constant. It deserves to further study the same problem when the value of p is a function of the dis-tance between a sensor and the event. Also, our analysis models time in a discrete manner (by ﬁxed-length slots). It is also interesting to investigate the continuous time case.

Acknowledgements

Y.C. Tseng’s research is co-sponsored by Taiwan’s MOE ATU Program, by NSC under Grant Nos. 93-2752-E-007-001-PAE, 95-2623-7-009-010-ET,

95-2218-E-009-020, 95-2219-E-009-007, 94-2213-E-009-004, and

94-2219-E-007-009, by MOEA under Grant No.

94-EC-17-A-04-S1-044, by ITRI, Taiwan, and by Intel Inc.

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You-Chiun Wang received his B.S. and M.S. degrees in Computer Science and Information Engineering from the National Chung-Cheng University and the National Chiao-Tung Univer-sity, Taiwan, in 2001 and 2003, respectively. He is currently a Ph.D. candidate in the Department of Computer Science, National Chiao-Tung Univer-sity, Taiwan. His research interests include wireless communication and mobile computing, QoS management and wireless fair scheduling, mobile ad hoc network, and wireless sensor networks.

Kai-Yang Cheng received his B.S. and M.S. degrees in Computer Science from the National Tsing-Hua University and the Chiao-Tung Uni-versity in 2003 and 2005, Taiwan, respectively. His research interests include wireless communi-cation and sensor networks.

Yu-Chee Tseng received his B.S. and M.S. degrees in Computer Science from the National Taiwan University and the National Tsing-Hua University in 1985 and 1987, respectively. He obtained his Ph.D. in Computer and Information Science from the Ohio State University in January of 1994. He was an Associate Professor at the Chung-Hua University (1994–1996) and at the National Cen-tral University (1996–1999), and a Professor at the National Central University (1999–2000). Since 2000, he has been a Professor at the Department of Computer Science, National Chiao-Tung University, Taiwan, where he is currently the Chairman.

Dr. Tseng served as a Program Chair in the Wireless Networks and Mobile Computing Workshop, 2000 and 2001, as a Vice Program Chair in the International Conference on Distributed Computing Systems (ICDCS), 2004, as a Vice Program Chair in the IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS), 2004, as an Associate Editor for The Computer Journal, as a Guest Editor for ACM Wireless Networks special issue on ‘‘Advances in Mobile and Wireless Systems’’, as a Guest Editor for IEEE Transactions on Computers special on ‘‘Wireless Internet’’, as a Guest Editor for Journal of Internet Technology special issue on ‘‘Wireless Internet: Applications and Systems’’, as a Guest Editor for Wireless Communications and Mobile Computing special issue on ‘‘Research in Ad Hoc Networking, Smart Sensing, and Pervasive Com-puting’’, as an Editor for Journal of Information Science and Engineering, as a Guest Editor for Telecommunication Systems special issue on ‘‘Wireless Sensor Networks’’, and as a Guest Editor for Journal of Information Science and Engineering special issue on ‘‘Mobile Computing’’. Dr. Tseng received the Outstanding Research Award, by National Science Council, ROC, in both 2001–2002 and 2003–2005, the Best Paper Award, by International Conference on Parallel Processing, in 2003, the Elite I.T. Award in 2004, and the Distinguished Alumnus Award, by the Ohio State University, in 2005. His research interests include mobile computing, wireless communication, network security, and parallel and distributed computing. Dr. Tseng is a member of ACM and a Senior Member of IEEE.