In this section, we propose an energy-efficient construction of the proposed proximate graph for mobile nodes. For simplicity, we first discussion the (r, α)-neighborhood graph for mobile nodes, where r is identical among nodes . This construction provides the skeleton of the mobile protocol in the next Chapter.
The basic idea is borrowed from a distributed protocol of the primitive enclosed graph [20, 24], though we confront more challenges when designing for our structure, discussed below. This mechanism is based on the characteristic the (r, α)-enclosed graph. Recall that this graph has been shown to be equivalent to the (r, α)-neighborhood graph. Hence the following the two structures will be used interchanged when necessary.
Consider a node u. Let denote the set of information collected by u from its neighboring nodes during a period of time According to Definition 3.7, we define
as the enclosed region of u based on the set of collected information , i.e.
broadcast using the least radius that covers the all points in . We denote such radius as λu, i.e.
ERu
} ,
|
max{ ∈ ∈ℵ
= ux x ERu x
λu . (4.3)
Note that in (4.3) only considers points in deployment region ℵ. The transmission radius in (4.3) ensures that each node will aware its neighbors in rα(V). Hence, all
. NG links in NG rα can be preserved using possibly less construction power
(a)
(b)
Figure 4.2: (a) the shrunk power λw; (b) the enlarge power (r = 1, α = 2).
In the a ufficient to inclu
used, w
bovementioned, we discuss how reduce the power that is s
de all necessary neighbors. To construct the desired graph, we need to further ensure that all non-neighbor nodes will be blocked. However, the shrunk power may prevent nodes from be aware of some non-neighbor nodes that are necessary to block other non-neighbor nodes. See the example in Figure 4.2, where r = 1 and α = 2. transmission radius of w shorter than its distances to u and v. Thus, both u and v will be no longer being able to find w. In other words, Consequently, both uv and vu exists. It however are not allowed in the de
||uw|| ≤ ||uv|| and ||uw|| ≤ ||uv||,
which mean that the transmission radiuses of both u r w. So, w must be )
To fix this problem, a simplest way is to enlarge w’s power such that both u and v can be aware of w, while the prerequisite is that w should be able to be aware of both u and v first. Fortunately, this prerequisite can be self-contained, since ifw∈ERα( vu, ), by definition 3.3,
r
and v will cove able to overhear the existences of u and v. In other word, there must be
S
u∈ and u v∈ . Su
From this observation, for each node u, to ensure that all links which should be blocked by u will not exist, it is sufficient for u to transmit to all nodes that have been received by u. We denote such transmission radius as χu', i.e.
} ,
' max{
u = uv v∈
χ Su . (4.5)
The transmission radius in (4.5) ensures that all links that are belong to
s is now
ermined by u. Consider a node w. By )
α(V NGr will be removed. Nevertheless, the readers may node notice that the radiu
enlarged, which may counteract the original benefit from (4.3). Therefore, below we attempt to further shrink the radius of (4.5).
Let Nu denote the set of neighbors det So, from (4.5), the following radius χu is sufficient to covering all nodes in Bu
} correctly using less using less power, it is sufficient to broadcast using the following transmission radius
{
, '}
max u u
Tu = λ χ . (4.9)
.3 Neighborhood Graph based Topology Control Protocol 4
Based on above discussion, a distributed protocol that constructs the(r,
α )-n
orrection is directly followed from the meaning of each variable. We omit the p
u is
ses, by (3.12), the region will be smaller, which leads to a
ations on the NGT proto
n see that
neighborhood graph for mobile nodes is now presented here. In this protocol, each ode will periodically broadcast a message in every T time interval using the shrunken radius Tu. Each message will consist of the current position and the radius ηu. As a message is received from a node v, it will include v in the collecting set Su. In addition, if the node observes that the distance to v is shorter than the received radius ηu, it includes v into Bu such that χu will be sufficiently large whenever there are some links of v that should be block by u. On the other hand, if u do not received from v over an beacon interval, then u discard v’s information. Upon a message is either received or expired, the reconstruction process will proceed. Then, before sending the next beacon, related variables mentioned above will be recalculated based upon the information collected in the previous interval. The protocol is summarized below. We named it the Neighborhood Graph based Topology Control Protocol, abbreviated as NGTC.
The c
roof. Now we show an interest feature of this protocol below.
PROPERTY 4.1: For any set of nodes, the construction radius Tu of each node decreased by r.
Proof: As r increa ERrα(u)
lower λu. The same observation is on χu. By Property 3.2 (ii), the node degree of each node v will be strictly decreased as r goes up. So, by (4.6), a large r leads to a smaller ηv, which in turn declines the size of Bu for any u received v. As a sequel, by (4.8),
'
χu can be lower. Combining these two facts, we provide this.
The rest of this subsection, we discuss additional observ C col.
1) We ca Tu ≥λu and Tu ≥χu' are the sufficient conditions for u to
preserve all necessary links and block all unnecessary links, respectively. However, we have to admit that there is a potential wasting on Tu. It occurs when Tu > λu but there is no v ∈ Bu such that u ∈ NR(v, s) for some s ∈ Nv. In other words, u enlarges its r to cover Bu but there is no additional blocked by u, see Figure 4.2 (b).
So, the radius Tu is not the minimum.
NGTC Protocol
1 Nu = {}; Su = {}; Bu = {};
2 For every T time
3 u
{
x∈ERu,x∈ℵ}
; 4=max ux | λ
{
u}
u =max uv |v∈B
χ ;
5 Tu =max
{
λu,χu}
;6 ηu = max
{
uv |v∈Nu}
;roadcast (Loc ius Tu;
a node v,
= Bu + {v}, otherwise, Bu = Bu – {v};
r T time),
pired 7 B (u), ηu) in rad
8 Upon received a message (Loc(v), ηv) from 9 Su = Su + {v};
10 If ||uv|| ≤ ηv, Bu
11 Upon a message received from some v in Su is expired (ove 12 Su = Su – {v}; Bu = Bu – {v};
13 Upon a message is received or ex
14
I ( )
Su
w r
u D uT RR u
ER = ∈ ℵ∩ ( , max)− α( ,w) ; 15 Nu ={u∈Su|u∈ERu};
Since λu ≥ ηu, by sending instead of λu ηu in the message, it is also sufficient to lock all unnecessary neighbors. However, whether sending λu or ηu is better? See the foll
As nodes placement changes, each node u would maintain a new set of neighbor us Tu. However, due to the recursive dependency among node
t – T, t), i.e. ∃v ∈ V , and there is no
chan +
hronous (time slot lay and computatio
b
owing discussion: if u sends λu, since λu ≥ ηu, a node v covered by u would include u into Bv even if ||uv|| > ηu. So, the radius χu' which supports blocking other links increases. But by receiving information from some node farther than the farthest neighbor in Nu, the coverage of ER(u) is more possibly to be shrunk down and thus lead to a lower χu. In other words, if u sends λu, the radius Tu of u itself could be lower (at least one larger), while its neighbor’s radiuses would be increased, and vice versa. So, there is a tradeoff between sending λu or ηu. We will give our suggestion in the later part.
4.4 Convergency
Nu and recalculate the radi
s, such as χu depends on Bu and Bu further depends on ηv, some variable (radius or set) may require several iterations to recover from the change. In this section, we show that the topology as well as construction power of NGTC can converge in a constant time.
Consider a node u. Suppose the current timer of u is at t and some topological change occurs during [ , )Loct−1(v)≠Loct(v
ge after t. i.e. ∀v ∈ V, c ∈ Z , )Loct(v)=Loct+cT(v (note: the change may be caused by u itself). Assume the network is sync is aligned among each node), and the propagation de n time are negligible with respect to the interval T.
Table 4.1: The sufficient status of each variable. We define three statuses for each variable in NGTC:
Stale: it is neither of the following two statuses;
z 4.1;
means the radius can not be
z
Sufficient: it is sufficient for its functionality, see Table z Converge: it will change any more (For Tu, it
es with updated positions is gathered. By Definition 3.6, a point cons
PERTY 4.2: In a synchronous network, if each node u sends (Loc(u), λu) every T cover all logical neighbors. At time t+1: λt+1 is now sufficient due to the sufficiency of ER . In turn, t St+1 is sufficient, since
Nt . Accordingly, and B are sufficient since a node v should be blocked only if
∈NG
uv r
v
covers all nodes that should be blocked due to the suff
v
wil
The statuses hese variables are summ d
+
Tu converge at the beginning of the fourth and fifth intervals after the change occurred.
Figure 4.3:
For the alternative where sending
Th ues of eac ble o col can converge even faster.
Property 4.3: In a synchronous net
time, the neighborhood set Nu can converge in 3T and the radius Tu can converge in and 5T.
Chapter 5
Mobile Topology Control Protocol
In this Chapter, we present our mobile topology control protocol. The protocol is based the most general version, where each node is allowed having its own r. So, first we extend the shrink power mechanism to the (fr, α)-neighborhood graph. Then, the protocol is presented. We will discuss how automatically configure parameter to adapt to changing. Lastly, we discuss how efficiently perform the protocol and the corresponding time complexity.
5.1 Extending on Shrinking Power mechanism
Now we extend the shrink power mechanism such that all links of will be preserved and all links not belong to will be blocked. Before that, we should extend the (r, α)-enclosed region to the (fr, α)-neighborhood graph. different is the identical r which is not replaced by ruv. Institutively, it seems that the mechanism can be applied to based on directly. However, there is one difficulty: The enclosed region of u is now depending on not only ru but also rv. It means that the least radius where λu covers a possible neighbor v would be variant
)
by rv, which however is uncertain before acquiring a message from v.
Fortunately, there is an upper bound that can be calculated using a node’s own r.
Recall that in (3.11), for any 0≤r1≤r2 ≤1, ( ) ( ). Since ruv ≤ ru, we get
Further, similar to (4.3), we redefine
( )
So the following radiusru
λ , redefined from (4.3), is sufficient to cover all possible neighbors. λu is sufficiently large to cover all points in , which means that u can still receive from all w’s that block uv. So, ηu, Bu, χu, Tu are still corrected here.
5.2 Adaptive Mobile Topology Control Protocol
The main idea of this protocol is based on adjusting the parameter ru for each u.
Thus we start with a series of analyses how the parameter ru of each node u influence the overall energy-efficiency from the following three dimensions:
1) Energy efficiency of routes vs. operation time of individual node:
Consider a node u. Given a ratio r0, 0 ≤ r0 ≤ 1, we denote NGαf|r=r (V) to be the (fr,
α)-neighborhood graph where ru is fixed on r0. We have the following observations.
Based pm these properties, we can observe that for each node u, no matter what the parameters of other nodes are taken, a smaller ru will strictly lead to an overall better energy efficiency communication routes (at least on worse), and on the other hands, a smaller ru can reduce the adjacency of u to its neighboring nodes. A smaller node degree can help prolong the operation time of an individual node in two reasons:
Broadcasting power: Since the relationship in (5.4), a smaller degree implies that the farther selected neighbor is closer. So, for broadcasting operation, the node can spend less power to cover all neighbors.
Traffic Load: A node with more links will let more traffic flow (both flooding and unicasting) pass through it, which may draw out its energy rapidly for those transmissions. Thus a smeller degree can help release the node’ traffic.
2) High mobility vs. low mobility:
Consider how node mobility effects the energy consumption. As a node has high mobility. It will cause its surrounding nodes changing the links status (establish or remove a link) to itself frequently, which will in turn triggers more route reconstruction at the upper layer. More reconstruction implies extra energy wasting on
flooding route discovery packets. To alleviate such undesirable circumstance, a highly moving node can reduce the adjacency to its neighboring. In other words, a large ru which leads to a lower node degree on u is preferable as u is in high mobility.
3) Topology maintenance Power:
Addition consideration is from the topology maintenance power. Recall in Property 3.4, a larger r will cause a smaller Tu. It means the energy consumption of u can be reduced as a large ru is used, which is surprisingly consist with the tendency of ru toward the residual energy in the first consideration.
Combining the above considerations, a configuration rule for the parameter ru is characterized as follows.
⎟⎟⎠
where Energyu and Mobilityu are the current residual energy and mobility level of node u, and EnergyFull and MobilityMax are the full power level and the maximum node mobility. The formulation in (5.7) can completely consist with all observation and anticipation in above considerations. We can see that the rule is extremely simple and can be carried out automatically by each node relied on only inherent statuses of itself.
In addition, the configuration can be conducted independently by each node without additional control message to negotiate the symmetric, connectivity and planarity, since theoretically all these properties are preserved, see properties 3.4 and 3.5. For these reasons, the protocol will be very practical. More importantly, by reducing node dependency according mobility, the drawback of using nodes position in proximate graph can elegantly alleviated, since a node with lower degree will now trigger less reconstruction. The overall conceptions are depicted in Figure 5.1.
In practice, each node u can set its ru = 0 at the initial stage, and then configure ru
periodically according to several distinct energy levels and mobility. The node mobility can be measured by node speed or remaining pause time, i.e. as a node stops moving and anticipates that it will stay on the place for a relatively long period of time, it can turn up its ru to allow more neighbors accessing to it.
Figure 5.1: The relationship among the considerations, effects, and the configure process.
The final version of the mobile topology control protocol is given below. We named it the Adaptive NGTC Protocol, abbreviated as ANGTC. To save page space, we only highlight the different parts, in comparison with NGTC. The other part encapsulated from line 1 to 13 here is the same except that ERu is no replaced by .
ru
ER
ANGTC Protocol
1 Nu = {}, Su = {}; Bu = {}, ru = 0.
The procedure from lines 2 to 13 are the same of the NGTC protocol;
14 ⎟⎟
5.3 Efficient Calculation and Time Complexity
In the rest part of this section, we discuss the complexity issues and suggest some efficient way for calculation the related variables.
1) Calculation on Nu:
If there are relatively smaller number of point on ℵ, each node requires only O(|V|) to compute its neighbors in by set operation. However, if the there are infinite number of points on ℵ, each node can turn to determine its neighbors in in
In the following, we just consider α = 2 for two reasons: First, there is no simple root function for ||ux|| when α > 2. Second, the radius λu covering is sufficiently ERr2(u)
large for any α > 2, since ERr2(u)⊇ERrα(u). and another one of the equations from
ε
Consider four subcases:
v), then
Otherwise, if x satisfies (iii) and (vi), we get
a
ux = ; otherwise, if x satisfies either (ii) and (iii), the farthest point will finally be crossed by another w’ ∈ Su. As a sequel, it is sufficient to
In addition, in the second and third subcases,
⎟⎟
and in the fourth subcase
⎟⎟ Since x is itself a point in ER(u), the longest distance can be obtained by
{
u u}
u =max ux |w∈S ,andv∉NR(u,w),∀v∈S
λ . (5.16)
Chapter 6
Experiments
In this chapter, a series experiments will be conducted to evaluate the observations on theoretic results as well as the protocol designs.
6.1 Evaluations on Graph Structures
First of all, we evaluate the theoretic properties shown in Chapter 4. Recall that for any α ≥ 2, the power stretch of the r-neighborhood graph is bounded from above by an increasing function of r and conversely, the upper bound of the maximum node degree is decreased by r. Figure 6.1 draws the two theoretic functions for n = 100 and α = 2. We can see that NGr(V) indeed has the flexibility to be adjusted between the two metrics through the parameter r.
Power stretch factor
0 20 40 60 80 100 120
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r
Maximum node degree
0 10 20 30 40 50 60 70
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
r
Figure 6.1: The upper bounds on the power stretch factor and maximum node degree of NGr.
The topologies of NGr(V) of 3 different levels r are depicted in Figure 6.2. We can see that NGr(V) can construct any immediate structure between RNG and GG. A sparser topology can be constructed using a larger r, and contrarily, more routes will be preserved as a smaller r is applied.
(a) NG1(V) ≡ RNG(V) (b) NG0.5(V)
(c) NG0 ≡ GG(V)
Figure 6.2: The topologies for 3 different levels of r.
The results shown in Theorem 3.4 and Theorem 3.5 are the worst upper bound.
Actually, the average values will be much better. See figures 6.3 and 6.4. The results
ration d, 0 ≤ d ≤ 1 it has the following means: As the nodes placements are created, we sort every nodes pair (u, v) according to their distance ||uv|| in non-decreasing order. Then we set the maximum transmission range Tmax as the d×100% percent shortest distance. It means that given a density ratio d, in the underlying UDG(V), there will be at least d×100% of nodes can transmit to their neighbor using the directly a directly transmission.
Figure 6.3: The power stretch factor of the
We can observe that when n is 100, the maximum value of ρ(NGr(V)) is still within two times to the optimal value 1 in the most case, and the minimal relaying power among any nodes pair is almost closed to the optimal. The same observation is also on the node degree. The average and maximum node degrees are all limited within 4 and 8 respectively.
Figure 6.3: The power stretch factor and maximum node degree
graph. The curves indicate the when the environment antennae factor α become worse (larger), both matrices decline significantly. This confirms our argument that a generalized structure of the r-neighborhood can gain better quantity results. The observation also tells use that our structure can adaptive well in a highly interference or obstacle environment. Therefore, the generalization is worth.
6.2 Evaluations on Shrinking Power Mechanisms
Next, we evaluate the shrink power mechanism for the (r, α)-neighborhood graph. The results of 100 test cases for 50 and 100 nodes are summarized below, where dist and pwr denote the remaining percent of radius and power of Tu in compared with the maximum radius Tmax. We can see the both dist and pwr can be strictly declined as r goes large. Such tendency does consist with the results proven in Property 5.1.
Table 6.1: The shrunken radius and power. (n = 50)
r = 0 r = 0.25 r = 0.5 r = 0.75 r = 1.0 α density dist pwr dist pwr dist Pwr dist pwr dist pwr
0.1 93.61% 87.68% 92.37% 85.37% 89.08% 79.42% 85.90% 73.87% 84.69% 71.80%
0.2 80.42% 64.76% 77.11% 59.55% 70.63% 49.95% 65.46% 42.91% 63.90% 40.89%
2
0.3 67.15% 45.18% 63.55% 40.45% 57.31% 32.90% 52.71% 27.84% 51.38% 26.46%
0.1 89.86% 72.71% 89.60% 72.10% 88.11% 68.59% 85.78% 63.34% 84.69% 60.95%
0.2 72.06% 37.57% 71.59% 36.84% 68.88% 32.81% 65.29% 27.96% 63.90% 26.20%
3
0.3 58.62% 20.25% 58.20% 19.82% 55.73% 17.41% 52.57% 14.62% 51.38% 13.66%
0.1 87.96% 60.19% 87.93% 60.10% 87.33% 58.51% 85.66% 54.23% 84.69% 51.81%
0.2 68.62% 22.36% 68.55% 22.26% 67.53% 20.96% 65.13% 18.15% 63.90% 16.82%
4
0.3 55.51% 9.60% 55.45% 9.56% 54.52% 8.94% 52.44% 7.65% 51.38% 7.07%
On the other hand, as the network density, network size, or attenuate factor increase, this mechanism can perform even better. This phenomenon is due the fact that both influences will cause each node u confronting to more neighboring nodes, which in turn means that the (α, r)-enclosed region of u will be smaller. A smaller
α
For this reason, the shrink power mechanism can perform well in a large scale as well as worse condition network.
Table 6.2: The shrunken radius and power. (n = 200)
r = 0 r = 0.25 r = 0.5 r = 0.75 r = 1.0 α density dist pwr dist pwr dist pwr dist pwr dist pwr
0.1 61.84% 38.27% 58.07% 33.75% 52.47% 27.55% 48.56% 23.59% 47.42% 22.50%
0.2 42.43% 18.02% 39.77% 15.83% 35.87% 12.87% 33.17% 11.01% 32.38% 10.49%
2
0.3 33.40% 11.17% 31.31% 9.81% 28.24% 7.98% 26.12% 6.83% 25.49% 6.50%
0.1 53.49% 15.33% 53.17% 15.06% 51.00% 13.29% 48.41% 11.37% 47.42% 10.68%
0.2 36.57% 4.90% 36.35% 4.81% 34.85% 4.24% 33.07% 3.62% 32.38% 3.40%
3
0.3 28.79% 2.39% 28.62% 2.35% 27.44% 2.07% 26.04% 1.77% 25.49% 1.66%
0.1 50.71% 6.63% 50.65% 6.60% 49.99% 6.26% 48.32% 5.47% 47.42% 5.07%
0.2 34.64% 1.45% 34.61% 1.44% 34.15% 1.37% 33.00% 1.19% 32.38% 1.10%
4
0.3 27.27% 0.56% 27.24% 0.55% 26.89% 0.52% 25.98% 0.46% 25.49% 0.42%
6.3 Evaluations on the Mobile Protocol
In the last section, we conduct simulation study to emulate the really performance. This experiment was conducted by ns2 simulator [41]. The IEEE 802.11 distributed coordination function has been implemented in ns2 kernel. It uses RTS/CTS/DATA/ACK pattern for all unicast packets and simply sends out DATA for all broadcast packets. The implementation uses both physical and virtual carrier sense.
The two-ray ground reflection model is chosen as radio propagation model. The initial energy of each node is 0.5 joules. Each node can choose a power level to transmit a packet according to distance to the next hop. We modified the route protocol DSDV [42] to find the least-energy path instead of the shortest path. That is, the transmission
The two-ray ground reflection model is chosen as radio propagation model. The initial energy of each node is 0.5 joules. Each node can choose a power level to transmit a packet according to distance to the next hop. We modified the route protocol DSDV [42] to find the least-energy path instead of the shortest path. That is, the transmission