In this section we we analyze the performance of our derivations with the parameters listed below:
E Initial Energy of a sensor 1000 J
l packet size of each event 20 bits
r sensing range & transmitting range 1 m
Ai area of i-th region m2
Ni number of sensors deployed in i-th region N total number of sensors
λEi event occurring rate of the i-th region 1/m2/sec Table 2.1: Parameters used in the analyses.
From (2.13) and (2.14) we obtain the optimal value of N1 and N2 to maximize I(N1, N2).
We can verify this derivation through Fig.2.2. It clearly depicts the variation of the total information rate I(N1, N2) as a function of N1 with respect to different λE1 and λE2 when the areas of A1 and A2 are fixed and identical(A1 = A2 = π(102)). Those arrows indicate the optimal deployment of N1 and N2to obtain maximum information rate. We can observe that the total information rate increases as N1 increases from 0 towards the optimal value, where the slope of total information rate is positive and sharp initially. The total information rate would reach the maximum value where the corresponding value of N1 is the best deployment, which is exactly the value obtained from (2.13).
0 10 20 30 40 50 60 70 80 90 100
Figure 2.2: Variation of maximum information rate with respect to N1 under different λE, while (A1 = A2 = π(102)) and N = 100.
With the assumption of identical areas (A1 = A2 = π(102)), from Fig.2.2 we clearly obtain a sense of putting more sensors in the area with larger λE if maximum information rate is desired. This is consistent with our intuition since more sensors in the area with larger λE would receive more information. For example, if λE1 = 3 and λE2 = 2, the optimal value of N1 is 71, that is larger than N2 = 29. Notice that when λE1 = λE2 = 2, we get are identical. Fig.2.3 depicts the variation of I(N1, N2) as a function of N1 with respect to different area ratios A1/A2 = m. We assume λE1 = λE2 = 2‘ and A2 = π ∗ (102), and vary
the ratio m to see the variation of I(N1, N2). The result coincides with our intuition that we should put more sensors in the larger area to obtain more information. From (2.4) we know A1 and A2 are related to area coverage fa1 and fa2, respectively, through the parameters D1 and D2. It implies that by putting more sensors in the area, we could obtain more information while better area coverage fa is achieved.
0 10 20 30 40 50 60 70 80 90 100
Figure 2.3: Variation of maximum total information rate with respect to N1 for different areas of A1/A2 where λE1 = λE2 = 2, N = 100.
Comparing Fig.2.3 with Fig.2.2, we conclude that variation of λE is more critical than variation of area, since a slight change in λE would cause a significant change of N1 and N2. Notice that when A1 = A2 = π(102) and λE1 = λE2 = 2, the optimal deployment is N1 = N2 = 50, fa1 = fa2 = 0.3935 ; when A1 = 4 · A2 and λE1 = λE2 = 2, we get N1 = 80, fa1 = 0.1813 and N2 = 20, fa2 = 0.1813; when A1 = 9 · A2 and λE1 = λE2 = 2, we get N1 = 90, fa1 = 0.0952 and N2 = 10, fa2 = 0.0952, which indicates that if λE1 = λE2, the optimal deployment is the number of sensors that makes both fa1 and fa2 the same.
From the results of Fig.2.2 and Fig.2.3, we should have an appreciation of deploying more sensors in the region which either has a larger λE or a larger area, hence area and λE can be considered as two critical factors in determining the optimum value of N1 and N2. We keep some parameters in Fig. 2.3, i.e., A1/A2 = m, m = 1, 4, 9, but change λE1 to be 3 in Fig.
2.4. Since A1 and λ1 are larger than A2 and λ2, respectively, we would put more sensors in A1. In the case of m = 4 and m = 9, even if we put all the sensors N = 100 in A1, the slope of total information rate still positive, which means N1 has not reached the optimal value yet even if N1 = N.
Figure 2.4: Variation of maximum information rate with respect to N1 and with λE1 = 3, λE2 = 2, N = 100.
What if we let λE2 = 3 and λE1 = 2 while A1/A2 = 1, 4, 9? We can see the result in Fig.2.5. When λE2 = 3, λE1 = 2, A2 = π(102) and A1/A2 = 1, the optimal deployment is N1 = 30, fa1 = 0.2592 and N2 = 70, fa2 = 0.5034; when A1/A2 = 4, the optimal deployment is N1 = 48, fa1 = 0.1131 and N2 = 52, fa2 = 0.4055; when A1/A2 = 9, the
0 10 20 30 40 50 60 70 80 90 100
Figure 2.5: Variation of maximum information rate with respect to N1 and with λE2 = 3 and λE1 = 2.
optimal deployment is N1 = 54, fa1 = 0.0582 and N2 = 46, fa2 = 0.3687. We observe that there is only a slight change of N1 even when A1/A2 changes from 4 t0 9. Although more sensors should be put in the larger area while λE1 = λE2, here the numbers are less than those in Fig. 2.3. By observing the variation of fa1 and fa2 in both Fig. 2.3 and Fig. 2.5, we conclude that the effect of λE is critical when fa is relatively small, and is irrelevant when fa is relatively large. We can verify this conclusion in later discussion.
Now we examine the optimal value of N1 with respect to the ratio of λE when areas are identical and fixed. In Fig.2.6 we assume A1 = A2, and vary the ratio of λ1/λ2 = K while N could be 100, 200, 300 and 400. We see that the optimal value of N1 increases as the ratio λ1/λ2 = K increases, and eventually N1 = N at some critical value of K. Namely, even if we put all the N sensors in one area, it doesn’t achieve enough coverage of the area yet to obtain information. For example, when N = 100, we obtain N1 = 100 at λE1/λE2 = 2.9,
0 1 2 3 4 5 6 7 8 9 10
Figure 2.6: Variation of optimal N1 with respect to λE1/λE2 = K under different N and identical areas.
where the sensor density D = 0.6321 only.
In Fig. 2.6, N acts like a DC component for it shifts the curves upwards or downwards but does not change the shape, as we can see in the cases of N = 300, when we deploy N1 = 250 in A1 and the corresponding fa1 is 0.9179; when N = 400, we deploy N1 = 300 in A1 and the corresponding fa1 is 0.9502, which indicates an enough area coverage is achieved to obtain most of the information.
To examine the effect of λE in detail, we vary both λE and area to see the variation of optimal N1. In Fig.2.7 we assume N = 400, A1/A2 = m, m = 1, 4, 9, 16 and λE1/λE2 = K.
We observe that the curve reaches to the limitation of N quickly, thus the effect of λE is critical when fa is relatively small. Here we obtain the same conclusion again.
In Fig.2.8, we assume N = 200 is fixed and let A2/A1 = m and λE1/λE2 = K, which is a little bit different from the case in Fig. 2.7. Surprisingly we observe that all the curves
in-0 0.5 1 1.5 2 2.5 3 3.5
tersect at the same point. In order to generalize the form of intersection point, we formulate as follows. From (2.13) we obtain:
N1 = A1
which is independent of A2/A1 = m. Therefore the intersection point is determined once A1
is determined.
0 1 2 3 4 5 6 7 8 9 10
0 50 100 150 200 250
ratio of λ1/λ2=K
number of sensor N1
N = 200
A2/A 1 = 1 A2/A1 = 4 A2/A
1 = 9 A2/A
1 = 16
Figure 2.8: Variation of optimal N1 with respect to λE1/λE2 = K under A2/A1 = m.
Chapter 3
Maximum Information Capacity Deployment
In the previous chapter, we propose the “Maximum Information Rate Deployment”(MIRD) method for optimal sensor deployment in two areas. This method doesn’t consider the is-sue of energy constraint, for the objective is to obtain maximum information rate at some instant of time, thus energy consumption is completely ignored. Without lose of generality, now consider the situation : given two regions A1 and A2, each region has its own area and event occurring rate, then how to deploy sensors in each region to obtain the maximum information capacity within the network lifetime? To deal with this issue, we first introduce the formulation of energy consumption in each transmission. Next, we formulate the rela-tionship between energy consumption and network lifetime in a probability sense, thereby the total generated information capacity during the network lifetime period is determined.
In short, through the combination of network lifetime and the “Maximum Information Ca-pacity Deployment”(MICD), we can determine the optimal number of sensors to be deployed in each region.
3.1 Network lifetime and information capacity
The idea behind our analyses of network lifetime is quite simple. We assume every sensor has the same initial energy E and define one transmission of an event as “one round”. Each
round would consume the same energy Eevent, then the number of rounds that one sensor can last is simply determined by Rounds = E/Eevent. The detail analyses starts from this simple assumption.
In our work, we only discuss communication energy consumption. We do not include the energy loss of sensing here. The reason is that energy loss of sensing depends heavily on the specific application. Nevertheless, such energy loss can be easily integrated into the equations once the sensing energy model of specific application is defined. The first order model of energy consumption presented in [7] is used here. In this model Eelec = 50nJ/bit is the energy dissipated to active the transmitter or receiver circuitry and ²amp = 10pJ/bit/m2 for the transmit amplifier to deliver each bit. Thus we can get the energy consumption to transmit a k -bit packet to distance d, denoted as ET x(k, d), and the energy to receive the same packet, denoted as ERx(k, d), as follows :
ET x(k, d) = Eelec∗ k + ²amp∗ k ∗ d2 , (3.1)
ERx(k, d) = Eelec∗ k . (3.2)
We assume the transmission range of each sensor is r , which is the same as the sensing range.
Then, we can simplify 3.1 as :
ET x(k, r) = KT x∗ k ,
where KT x = Eelec+ ²amp∗ r2. Hence, the energy consumption of a sensor on receiving l bits from the distant d and transmit k bits to the distant r can be computed as follows:
ET x(k, r) + ERx(l, d) = Eelec∗ k + ²amp∗ k ∗ r2+ Eelec∗ l
= KT x∗ k + Eelec∗ l . (3.3)
Since we adopt the random deployment of Poisson Point Process under Boolean Sensing Model, it is reasonable and straightforward to assume all sensors are identical. Specifically, we further assume only one sink node exists to collect the information and the rest are all identical sensors with same sensor parameters and functionalities. In a wireless sensor network, the sensor located far away from the sink node would transmit data to the sink node by means of multi-hop communication. Therefore, the sensors closer to the sink node will have to transmit not only their own sensing data, which is defined as the “Originating traffic”, but also relayed data of the other nodes, being termed as the “Relaying traffic”. As a result, the initial energy of these sensors will be used up the earliest among all sensors, since their traffic load is heavier than the other sensors.
Thus, in such a random deployed wireless sensor network, we focus on the region R having area πr2 around the location of sink node, which is called the “inner circle”. If all the sensors deployed in the inner circle, called inner sensors, use up their initial energy, there will be no sensors to relay the information obtained from outer sensors to sink node. The time period when all the inner sensors use up their energy is therefore defined as the network lifetime for the wireless sensor network of concern.
Here are some implicit assumptions to simplify the network lifetime analysis. Firstly, the traffic load from outer sensors to be relayed to sink node is evenly divided by sensors deployed in the inner circle R, which implies some ideal routing protocol is in place. Secondly, we assume the packet size of each event l is small enough and sensors can use any level of power to complete the transmission. Finally, the sensed outer information to be relayed is proportional to the outer area (Ai− πr2) · fai and is independent of the number of sensors
deployed in the outer region.
To begin with, we consider the following situation : there are two regions to be monitored, which have area size A1, A2 and event occurring rate λE1, λE2, respectively. In general, we assume that there are N1 sensors in A1, N2 sensors in A2 and take s sensors in the inner circle of A1, w sensors in the inner circle of A2.
Without lose of generality, we discuss the case of A1 first, and then it is easy to obtain similar result of A2. Since there are s sensors in the inner circle of A1, hence (N1− s) sensors are in the outer region (A1− π · r2). From previous results, the averaged traffic to be relayed by s inner sensors to the sink node is expressed as follows :
Ireceive,s|A1 = λE1· l · (A1− πr2) · fa1,s,
where
fa1,s= 1 − e(−Douter,s·πr2), and
Douter,s= N1− s A1− πr2 .
The information to be transmitted by the s inner sensors per unit time can be denoted as :
Itransmit,s|A1 = [πr2+ (A1− πr2)fa1,s] · λE1· l . (3.4)
Therefore, the energy consumption of the s inner sensors can be computed as :
Ereceive,s|A1+Etransmit,s|A1 = λE1·l ·(A1−πr2)·fa1,s·Eelec+[πr2+(A1−πr2)fa1,s]·λE1·l ·Ktx , (3.5)
Assume that every sensor has identical initial energy E and the energy consumption can
be evenly divided by s inner sensors, we define a parameter named Rounds(s) which is the network lifetime time that s sensors can last before their total energy s · E be used up. Thus Rounds(s) is written as :
Rounds(s) = s · E
Ereceive,s+ Etransmit,s
= s · E
λE1· l · (A1− πr2) · fa1,s· Eelec+ [πr2+ (A1 − πr2)fa1,s] · λE1· l · Ktx .
(3.6)
The network lifetime of N1 is therefore determined by Rounds(s). From (3.4) and (3.6), the total information capacity that the sink node of A1 can receive from s inner sensors through the network lifetime is :
Itotal,s|A1 = Rounds(s) · Itransmit,s|A1
= [πr2+ (A − πr2)fa1,s] · λE1· l · s · E
λE1· l · (A − πr2) · fa1,s· Eelec+ [πr2+ (A − πr2)fa1,s] · λE1· l · Ktx .
(3.7)
Similarly, we define the network lifetime of A2 to be Rounds(w), expressed as :
Rounds(w) = w · E
Ereceive,w+ Etransmit,w
= w · E
λE2· l · (A2− πr2) · fa2,w· Eelec+ [πr2+ (A2− πr2)fa2,w] · λE2· l · Ktx ,
(3.8)
and the total information capacity that the sink node of A2 can receive from w inner sensors through the network lifetime is :
Itotal,w|A2 = Rounds(w) · Itransmit,w|A2
= [πr2+ (A2− πr2)fa2,w] · λE2· l · (w · E)
λE2 · l · (A2− πr2) · fa2,w· Eelec+ [πr2+ (A2− πr2)fa2,w] · λE2· l · Ktx .
(3.9)
Since we observe two regions to obtain desired information, we consider A1 and A2 as two sub-networks of the whole network. Thus it is straightforward to define the network lifetime of the whole network as :
Rounds = min{Rounds(s), Rounds(w)} , (3.10)
which means the shorter one of Rounds(s) and Rounds(w) would dominate the network lifetime. When either of these two sub-networks run out of its energy in the inner circle, it stops gathering data from the environment. Since information is the main concern of the
discussion, it doesn’t matter whether another sub-network is working or not, hence (3.10) is a reasonable definition of network lifetime of the whole network.
From the above discussion, the total information capacity of the whole network can be formulated as follows:
Itotal|(s,w) = Itotal,s|A1+ Itotal,w|A2
= Rounds · Itransmit,s|A1 + Rounds · Itransmit,w|A2 , (3.11)
where the Rounds is given as (3.10).
Thus, we obtain the total information capacity when there are s sensors in the inner circle of A1 and w sensors in the inner circle of A2. Recall that we assume all the sensors are deployed in the area by the Poisson Point Process, we can compute the probability of s sensors deployed in the inner circle of A1, and the probability of w sensors deployed in the inner circle of A2, and through the concept of probability we can obtain the average total information capacity.
From Sec.2.1.3, we can compute the probability of s sensors in the inner circle of A1 when there are N1 sensors deployed in the area A1, given as :
P (N(A) = s|N1) = e(−D1·A)· (D1· A)s
s! , (3.12)
where A = πr2 and D1 = N1/A1. Next, the probability of w sensors located in the inner circle of A2 is:
P (N(A) = w|N2) = e(−D2·A)· (D2· A)w
w! , (3.13)
where again A = πr2 while D2 = N2/A2.
As (3.12) and (3.13) are independent, the probability of obtaining Itotal(s,w) in (3.11) can be expressed as :
P (Itotal|(s,w)) = P (N(A) = s|N1) · P (N(A) = w|N2) . (3.14)
We observe that s and w are variables which can vary from 0 to N1 and from 0 to N2 respec-tively, thus if N1 and N2 are specified, the expected total information capacity is formulated as:
E[Itotal] = E[Itotal|(s,w)]
= X
s,w
Itotal|(s,w)· P (Itotal|(s,w))
=
N1
X
s=0 N2
X
w=0
P (N(A) = s|N1) · P (N(A) = w|N2) · Itotal|(s,w) . (3.15)
Consequently, we can apply (3.15) to estimate the total information capacity for a particular (N1, N2) and then look for the optimum deployment where E[Itotal] is maximum.
3.2 Simulation Results
Total Information Capacity : E[ I total ]A2 = π (52), λ1 = λ2 = 2
Figure 3.1: The effect of varying area ratio upon maximum information capacity where N = 1000.
Now we have the analytical form of E[Itotal], by MATLAB simulations we can observe the properties of E[Itotal]. First we assume N = 1000, let λE1 = λE2 = 2 and A2 = π(52), A1
could be π(52), π(7.52) and π(102), the resulting variation of E[Itotal] is shown in Fig. 3.1.
In order to demonstrate more clearly, in Fig. 3.2 shown next page, we let A1 be π(12.52), π(152) and π(202) and again show the variation of E[Itotal]. From the figures we can tell that the effect of area size is evident and significant. Since the network lifetime is determined by the number of sensors deployed in the inner circle, and the deployment follows the Poisson Point Process, we should put more sensors into the larger area to increase the probability of sensors be deployed in the inner circle. In this case, the number of sensors deployed in the inner circle can increase, so are the network lifetime and the information capacity. When we increase the number of senors in A1, which is N1, the number of sensors N2 is decreased, i.e.
we increase the network lifetime of A1 but decrease the network lifetime of A2. The total information capacity would reach an maximum value for some optimal values of N1 and N2, as we can see in the figures. There is an interesting observation that the effect of varying area ratio would saturate eventually, as depicted in Fig. 3.2. The reason is simple, we can’t put too few sensors in A2, that would cause the network lifetime of A2 be too short. Since very few sensors would be deployed in the inner circle of A2, the lifetime of the whole network will be short as well. Thus there exists a threshold of saturation. When optimal deployment of N1 reaches this threshold, it would remain the same but the total information capacity decreases. For example, for the case A1 = π(152) and A1 = π(202) in Fig. 3.2, the optimal N1 for both cases are the same, but the total informal capacity decreases as A1 increases.
0 100 200 300 400 500 600 700 800 900 1000
Figure 3.2: The effect of varying area ratio upon maximum information capacity where N = 1000.
After examining the effect of area upon total information capacity, now we fix the area A1, A2and vary λE to see the effect of λE upon total information capacity, the result is shown
0 100 200 300 400 500 600 700 800 900 1000 Total Information Capacity : E[ I total ]
A1 = A
Figure 3.3: The effect of varying λE upon maximum information capacity.
in Fig. 3.3. We can tell the difference between varying area ratio and varying λE ratio. The variation of total information capacity is irrelevant to the variation of λE ratio, which is due to the assumption that the traffic load is evenly divided by those sensors deployed in the inner circle. The total energy of those inner sensors implicitly imply the total information capacity that the sink node can receive during the network lifetime. When λE is larger, the energy consumption is larger, which leads to a shorter network lifetime, and vice versa. In either case the total information capacity is the same, although the optimal values of N1
and N2 varied with different λE ratio. Hence the conclusion here is that the maximum total information capacity is irrelevant to variation of λE ratio, only the area ratio would affect the total information capacity.
3.3 Searching Algorithm for Optimal Deployment
Here we introduce an algorithm to search for the optimal value of N1, and then N2is therefore determined by N2 = N − N1. The detailed MATLAB algorithm is listed in Appendix A. We start from the concept of expected value, and by Dichotomy we can converge our searching to the optimal value.
From the definition of network lifetime of (3.10), we have an idea of that the optimal value of N1 occurs when Rounds(s) = Rounds(w), that is because if either one of them is bigger, assume Rounds(s) is bigger than Rounds(w), it implies that we should take some sensors from A1 and deploy them into A2 to prolong the lifetime Rounds(w) in the probability sense.
From the idea described above, we first examine the initial deployment, which is the expected value of sensors to be deployed in the inner circle of each area. For example, in A1, the expected number of sensors to be deployed in the inner circle can be expressed as :
E[s] = X
A1 · πr2 . (3.16)
Next, the expected number of sensors to be deployed in the inner circle of A2 is :
E[w] = (N − X) A2
· πr2 . (3.17)
The initial deployment can be determined from (3.16) and (3.17), i.e., E[s] = E[w],given as :
X = A1 A1+ A2
· N . (3.18)
After obtaining X we can compute E[Itotal] for the cases of X − 1, X and X + 1, respectively.
This is because, from the observations of the simulation results of MICD, the maximum value of N1 occurs at the point that the slope changes from positive to negative. Since it is not possible to determine the derivative of E[Itotal] and X is a discrete value, hence we compute E[Itotal] for the cases of X − 1, X and X + 1 for obtaining the slope.
Here we use the simplest method of Dichotomy to approach the optimal value from the initial deployment X, i.e., if E[Itotal] of X + 1 is greater than E[Itotal] of X, and E[Itotal] of X is greater than E[Itotal] of X − 1, the slope of E[Itotal] at X is positive, which implies we should put more sensors to obtain a larger E[Itotal]. Thus, the optimal value certainly exists in the interval [X, N ]. We set an variable named of f set to be (N − X)/2, and by examining the slope at Y = X + (N − X)/2, we can determine whether the optimal value lies in the interval [X, Y ] or [Y, N ]. Namely, if the slope at Y is positive, then the optimal value exists in the interval [Y, N]; if it is negative, the optimal value appears in the interval [X, Y ].
Assuming that the optimal value exists in the interval [X, Y ], then we set the value of
Assuming that the optimal value exists in the interval [X, Y ], then we set the value of