A fully distributed node allocation scheme with partition protection for Mobile Ad Hoc Networks

Computer Communications 33 (2010) 1949–1960

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A fully distributed node allocation scheme with partition protection for Mobile Ad Hoc Networks q Ting Wang a,b,*, Chor Ping Low a,1 a b

School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Infocomm Research Lab, S2.1 B4-03, Nanyang Technological University, Singapore 639798, Singapore

a r t i c l e

i n f o

Article history: Received 20 October 2009 Received in revised form 12 June 2010 Accepted 15 June 2010 Available online 30 June 2010 Keywords: Mobile Ad Hoc Network Partition protections Node allocation Distributed schemes

a b s t r a c t In this paper, a fully distributed node allocation scheme, namely Distributed Allocation Scheme (DisAS), is proposed for the purpose of partition protection in Mobile Ad Hoc Networks (MANETs). The proposed scheme controls the locations of nodes in a MANET in a dynamic way. A theoretical upper bound on the probability of partitioning is derived in the scheme. An implementation of DisAS using normal distribution, which is referred to as Normally DisAS (N-DisAS), is also discussed as a case study in this paper. Extensive simulations are carried out to show the effectiveness of N-DisAS in protecting MANETs from partitioning. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction A Mobile Ad Hoc Network (MANET), as described in [1], is a kind of mobile wireless network formed by a collection of mobile hosts connected through wireless channels. The hosts of a MANET are usually called nodes while direct connections between the nodes are referred to as links. Nodes in a MANET function as network terminals as well as routers. They exchange packets with other nodes and forward packets as intermediate routers at the same time. The routing path of a packet in a MANET is formed by a series of mobile nodes. Packets are forwarded hop-by-hop. While mobility of nodes in a MANET enables the network to span over a large area, it also causes a highly dynamic network topology, which is a major challenge in the applications of MANETs. In particular, a MANET is connected by mobile devices without centralized connectivity coordination. Therefore, if two nodes in a MANET are not directly connected, the connection between them would depend on other intermediate nodes. A broken link may destroy the entire path. This possibility of link breakage may split nodes into a number of components among which there exists no paths that interconnect them. It in turn results in the inability to transmit packets from the source node in one component, to a destination node which resides

q

A preliminary version of this paper has been presented in IEEE WCNC 2009. * Corresponding author at: School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Tel.: +65 6790 6528. E-mail addresses: [email protected] (T. Wang), [email protected] (C.P. Low). 1 Tel.: +65 6790 6368. 0140-3664/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2010.06.021

in another component of the network. We refer this phenomenon as network partitioning. This problem makes a critical strike on Ad Hoc routing because most routing protocols typically assume that the network is always connected. To enhance the reliability of MANETs, network partitioning should be prevented. The probability of network partitioning needs to be minimized or eliminated. One way is to control the speed, directions or mobility patterns of the nodes. Some of the existing works in this direction include [2–4]. In this paper, we propose a scheme to evaluate suitable locations for node placement and move nodes toward such positions in a fully distributed manner. We refer to this scheme as the Distributed Allocation Scheme (DisAS). Our proposed scheme uses probability distribution to control the positions of the nodes in the MANET. A particular implementation of DisAS that is based on Normal Distribution is discussed in this paper. This scheme can minimize the probability of partitioning in MANETs. We refer to this scheme as the Normally(N)-Distributed Allocation Scheme (N-DisAS). Our proposed scheme is able to maintain good connectivity while keeping satisfactory coverage for exploration and surveillance purposes of MANETs. Triangular geometry and probabilistic principles are used to prove that the network connectivity can be controlled effectively when N-DisAS is used. The effectiveness of this scheme is verified through empirical studies. Our proposed scheme is also able to cooperate with any existing communication protocols and, consequently, is applicable to many real life applications. The remaining parts of this paper are organized as following: Section 2 introduces the application scenarios of DisAS; Section 3 gives an overview of existing allocation schemes; Section 4 describes how DisAS works and how the probability of partitioning

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can be estimated; N-DisAS is discussed in Section 5; Section 6 shows the results from our extensive simulations, while the plan of future work and the conclusion are given in Section 7. 2. Application scenarios One of the common applications of MANETs is for disaster relief assistance. Mobile sensors, as nodes, form a MANET which is used to explore the disaster area to search for survivors. The sensors are usually distributed from one spot, such as a truck or a helicopter which arrives at the disaster area. They need to be dispersed from this spot and start searching as quickly as possible. However, as the environment is not yet detected, it is not possible to place the sensors on some optimal locations. Instead, the sensors have to scatter in the area by themselves. We call this process the distribution of sensors. During distribution, we need to keep the sensors connected so that the sensed data or any useful information could be collected with minimum delay. Also, we want to minimize the traffic between the nodes so that bandwidth and energy can be conserved. Therefore, we propose DisAS to allocate the sensors autonomously while keeping the MANET connected. Another possible application is to disseminate armed robots to perform surveillance in a battle field. Similar to the previous scenario, robots need to undergo the distribution process before the surveillance commences. If the robots only stay at fixed positions, failure of a robot may produce a blind area in the region and it may become a potential vulnerable zone. Therefore, robots have to patrol around to perform the surveillance task. In such a case, DisAS can be used to distribute and maintain connectivity among the robots so that the system can response to intruders or attacks with shorter delay. We note that in these scenarios, the MANET is deployed in a wide open area (referred to as network area) with small amount of obstacles inside. We assume that the average size of the obstacles is much smaller than the size of the network area. If there are too many obstacles, or very large ones blocking the transmission of the radio signal, the connection can hardly be restored by any allocation scheme of the nodes. Instead, nodes with stronger communication and mobile capability should be deployed in these scenarios. In our scheme, all the necessary system information for the nodes will be either stored before they are distributed into the network area, or obtained locally. The assumptions we make on the nodes capabilities are: 1. Be aware of the direction they are heading to; 2. Be able to measure the distance they have moved. These assumptions can be easily satisfied with current technology. 3. Related works Different approaches have been used for nodes allocation purpose. The Facility Location Problem [5] in Operation Research has been adopted to find out the optimal layout of nodes in a MANET. However, even for the stationary nodes allocation, it has been proven to be NP-Hard for most of the formulations [6,7], and sub-optimal heuristics, such as [8,9], have been proposed. But in the before mentioned applications, deterministic deployment of sensors is very risky and/or infeasible due to the insufficient knowledge of the environment or high risk of placing nodes. Therefore, movement-assisted node deployment and dynamic allocation schemes are more practical. Wang et al. in [10] utilizes the mobility of sensors in order to distribute them as evenly as possible in a wireless sensor network,

which is also a type of MANETs. In [11], signal strength is used as ‘‘virtual force” by the nodes to compute new locations for better coverage. Yang et al. tries to minimize the distance of nodes movement in [12]. However, none of these works addresses the problem of connectivity or tries to prevent the network from the thread of partitioning, despite the fact that they are fundamental to the network performance. Winfield in his work [13] has proposed using mobile node to reconnect the partitioned MANETs, and his effort has been improved by Basu and Redi [14] with the Movement Control Algorithm (MCA), which constructs a fault-tolerance topology of the network by achieving bi-connectivity. By definition, a graph is bi-connected if it remains connected after removing any of its vertices. A MANET with bi-connected topology will not be partitioned even when one of its nodes fails. Fig. 1 demonstrates how bi-connectivity is achieved by moving node 1. Before node 1 moves, in Fig. 1(a), node 6 is a cutvertex or an articulation point, in the sense that the network will be partitioned to component {1, 2} and {3, 4, 5} if node 6 is removed or fails. After applying the MCA algorithm, we move node 1 into node 3’s communication range, as depicted in Fig. 1(b). The network becomes bi-connected, and will not be partitioned when any of the nodes fails. Despite the correctness, MCA and many other existing solutions have some drawbacks and limitations that we have identified in our study:  Similar to [10–12], in MCA, nodes move to ‘‘better” locations from their earlier positions. However, the distribution process is also important as described in the application scenarios, and we are interested in how the nodes move to their initial locations while remain connected. Yamauchi has addressed this problem in [15] for achieving better coverage in an indoor environment, but the connectivity issue is not discussed.  MCA is a centralized scheme, as many other topology control schemes such as [10,13]. The movement decision in these schemes is made based on other nodes’ positions and moving directions, or a global topology graph. Obtaining such information consumes much communication bandwidth, time and energy, and may not be available in MANETs, where the resources are limited. In many real life applications, due to the rapid change of nodes’ location and partitioning, it is infeasible to obtain a steady global topology at all. For example, in Fig. 1(b), it is possible that node 2 has moved towards left before node 1 arrives its ‘‘better” location calculated by MCA, and disconnects itself with node 6 and 1, resulting a partitioned network. Moreover, if the network is partitioned, many schemes, such as those using signal strength as ‘‘virtual force” to move the nodes would fail to work, since no signal can be received across partitions.  Nodes are located closer to each other to prevent partitioning in MCA and other partition protection schemes (e.g. [16]). The areas within the nodes’ sensing ranges (i.e. coverage) thus over-

3

1

6

3

1

6

4

4

2

2 5

Fig. 1. Using bi-connectivity to prevent partitioning.

5

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lap. The coverage provided by the entire network is therefore reduced. This phenomenon can be observed by comparing Fig. 1(a) and (b). Coverage is as vital as connectivity in many applications such as sensor network or area surveillance, and we do not wish to compromise one for another.  A practical bound on the probability that partitioning takes place has not been derived in the exiting schemes. It is desirable if one could know how well the network is connected with a particular number of nodes in a certain given area beforehand, so that the network can be constructed more efficiently. For example, if one knows that with 10 nodes in an area, the probability of partition will be high, he can increase the number of nodes to 20 or restricting them in a smaller area to achieve stronger connectivity. In summary, by computing the optimal layout or nodes locations, the existing works focus on answering the question that ‘‘where” to allocate the nodes in MANETs. We argue that there can be new approach that tells the nodes ‘‘how” to allocate themselves when they need to, or are forced to move. In other words, we aim to propose a rule, or a regulation, by which if each node locally abides, the global connectivity and coverage can be maintained. Moreover, the level of connectivity and coverage can be determined by the number of nodes, network area, and the rule itself. The Distributed Allocation Scheme (DisAS) is the rule that we have created to serve this purpose. It is a distributed scheme which is executed locally by each node, to prevent partitioning in MANETs. To the best of our knowledge, no similar scheme has been proposed with such philosophy. 4. Distributed Allocation Scheme (DisAS) 4.1. Notations We assume that a MANET consists of N nodes in a given region A and nodes are distributed from a point O inside A. All nodes are assumed to have the same communication and mobile capabilities. The communication range of the nodes is Rc and speed is v. Where the coverage of the nodes is concerned, we assume the sensing range Rs = Rc/2. Communication between two nodes is always successful when they are in each other’s range. Neighbors of a node are defined as the nodes with direct contact with this node, i.e., those nodes located within its communication range. For the purpose of convenience, we denote each node by using an ID from 1 to N. 4.2. States In DisAS, a node either remains stationary or moves towards a predefined destination which is referred to as a waypoint. It uses the similar concept defined in the Waypoint Mobility Model in [17]. Each node can be in one of the four states shown in Fig. 2, namely initialize, stay, move and terminate. Nodes are in the ‘‘initialize” state when they are still at the distribution spot, O. Each node generates a waypoint using the method described in the next section. The nodes move towards the corresponding waypoints as they are distributed in the area A. Upon reaching the waypoint, it enters the ‘‘stay” state. A new waypoint will be generated at the same time. The node stays stationary till certain events trigger it into the ‘‘move” state in which the node moves to the new waypoint. After arriving, the node enters the ‘‘stay” state again. When the task of the network is accomplished, nodes enter the ‘‘terminate” state. In this state, the nodes may wait at their positions till the battery runs out, or they may move back to O, where they were distributed from. In the following discus-

Initiaize Distribute

Trigger Move

Stay

Terminate

Fig. 2. States of nodes in DisAS.

sion, we will focus on how the waypoints are generated in ‘‘initialize” and ‘‘stay” states, and how the ‘‘move” state is triggered to achieve better connectivity and coverage. The task-oriented trigger of ‘‘move” state and the procedures in ‘‘terminate” state depend more on the specific real application for the network deployment and equipment in the node, and thus will not be discussed in this paper. When a node is being distributed or in the ‘‘move” state, it is possible that it collides with another node or be blocked by some obstacles. In this case, the node simply generates a new waypoint and moves towards it. Simulation shows that this method works effectively under the assumption that the size of the obstacles are relatively much smaller than the network area. 4.3. Waypoint generation Let’s take any node, say node i as an example. In the ‘‘initialize” state, node i generates a random variable ri,1 from a certain probability distribution function (pdf) fr(r). ri,1 is the distance between the 1st waypoint of node i (denoted as Pi,1) and the distribution point O. As we assumed, the nodes are able to measure the distance they have moved. So node i is able to choose a random direction and move to Pi,1. At Pi,1, node i enters the ‘‘stay” state and becomes stationary. Simultaneously, it will generate the next (the 2nd in this case) waypoint, Pi,2. At this time, ri,2 will be generated from the same pdf fr(r) together with another random variable xi,1. xi,1 is uniformly distributed in (0, ±p), and is the angle between OPi,1 and OPi,2 as shown in Fig. 3. Since node i is not moving from O this time round, it needs to compute the distance between Pi,1 and Pi,2, denoted as li,1. Using triangular geometry in MOPi,1Pi,2, li,1 can be computed as follows:

li;1 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2i;1 þ r 2i;2  2r i;1 r i;2 cos xi;1 :

ð1Þ

In addition, the angle \OPi,1Pi,2, denoted as hi,1, can also be computed using

hi;1 ¼ arcsin

ri;2 sin xi;1 : li;1

ð2Þ

In the same way, at any location Pi;nP , the next waypoint of node i, namely Pi;nP þ1 , can be generated by calculating li;nP and hi;nP , where nP denotes the number of waypoints which have been generated so far. As we also assumed that each node is able to control the

Pi,2 li,1

ri,2

θi,1 Pi,1

ωi,1 ri,1

Fig. 3. Waypoints generation.

O

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direction of its movement, node i can always go to P i;nP þ1 from Pi;nP after the ‘‘move” state is triggered. During this process of generating a waypoint, the only overhead is to produce the random numbers. Considering the nodes in MANETs such as sensors or mobile robots are usually equipped with micro controller with limited computational capability, we use the simplest pseudo random number generator, the linear congruential generator [18] to generate the uniformly distributed random variables. The time variable from embedded clock is used as the seed of the generator. Simulation shows that it provides very good randomness to ensure the effectiveness in our scheme. Since its linear, the overhead can be neglected.

Pi,n

P

P

Pi,n +1 P

4.4. Trigger of the ‘‘move” state In MANETs, a node may either move voluntarily or forcedly, depending on the application scenarios. In the disaster relief scenario we described in Section 2, a node, represented by a sensor, will only move to a new location when the area within its sensing range is sufficiently explored, i.e. only the completion of its task will trigger the ‘‘move” state, even though moving may disconnect it from others. This is because the exploration task, instead of communication, is the primary purpose of the network. We say the nodes move voluntarily in this case, since nothing can interrupt a node’s task and force it to move away. On the other hand, in the battle field surveillance scenario of Section 2, the primary objective of the MANET is to provide low latency communication and high coverage in the area. Thus the nodes need to be re-allocated to ensure better connectivity and coverage ratio, and we say this is the situation where nodes move forcedly, by the topology change of the network. To show the effectiveness of DisAS in both voluntary and forced movement cases, we adopt two methods as the triggers, namely using random number generation and using neighbor nodes count, respectively.

HIGH_Ω

ωi,n

Fig. 4. Trigger of the ‘‘move” state in DisAS(rn).

the current position of node i, which is denoted as P i;nP . In this case, node i will be forced to move to a new position that is further away from Pi;nP for better coverage. Therefore, by setting jxi;nP j > HIGH X, as shown in Fig. 4, node i’s next waypoint P i;nP þ1 will be allocated at the opposite side of the network, where there may be fewer nodes. The actual values of these parameters depend on N and the size of A. On the other hand, the ‘‘move” state will also be triggered when ni drops to 0. This is because when ni = 0, it is clear that node i is partitioned from other nodes and the ‘‘move” state should be triggered so that node i can reconnect itself with others. This helps to restore network connectivity when a certain node becomes isolated. We also note that by restricting the number of neighbors, we can also reduce the possibility of radio communication interference between the nodes due to being overly crowded at one place. DisAS with the adoption of this triggering mechanism is referred to as DisAS (nc), where ‘‘nc” refers to neighbor count. 4.5. Partition probability in DisAS

4.4.1. Voluntary movement-trigger by random number generation We assume that each node in the network spends an average duration of T seconds on its task, i.e. in the ‘‘stay” state. Random number generation is used to control the length of T. Every t seconds, each node in the ‘‘stay” state generates a uniform random number X in the interval (0, 1]. The ‘‘move” state will be triggered only if X is less than a predetermined threshold x 2 (0, 1], i.e. x is the probability that the node is triggered to the ‘‘move” state after every t seconds. Therefore, T will always be a multiple of t, denoted as T = kt. The factor k will be a geometric random variable with success rate x. We have P(k = K) = P(T = Kt) = (1  x)K1x, and the expected value of K, E[K] = 1/x. Therefore E[T] = E[Kt] = E[K]t = t/x. In this paper, DisAS with the adoption of this triggering mechanism is referred to as DisAS (rn), where ‘‘rn” refers to random number.

The probability of partition, Ppar, reflects the likelihood of network partitioning taking place in a MANET. The lower its value is, the stronger the network is connected. It is reasonable to assume that nodes in a MANET spend more time performing tasks in the ‘‘stay” state rather than moving around in the ‘‘move” state. Therefore, in this section we will prove that when all the nodes are in the ‘‘stay” state, Ppar of a MANET which is deployed using DisAS can be bounded by a value calculated from fr(r) and Rc. A node is placed on its waypoint while it is in the ‘‘stay” state. For nodes i and j, we denote their positions as I and J respectively. Let ri be the distance between I and O while d is the distance between I and J. / is the angle between OI and OJ as shown in Fig. 5. In a similar way which we obtained Eq. (1), we can see in MOJI, 2

4.4.2. Forced movement-trigger by neighbor nodes count When the nodes are only forced to move to preserve network connection and coverage, we use xi;nP and the number of neighbors of each node to trigger the ‘‘move” state. Similar method can be found in [19,20]. Node i’s number of neighbors, denoted as ni, can be detected using neighbor-discovery algorithms in [21]. They work on the MAC layer or the link layer of network; therefore no application data package needs to be transmitted. In addition, we define two parameters in DisAS, namely HIGH_N and HIGH_X. HIGH_N is an upper bound of ni. HIGH_X is used to control xi;nP . When ni > HIGH_N, it means that there are too many nodes crowded around

d ¼ r 2i þ r 2j  2r i r j cos /:

ð3Þ

This equation shows that d2 is a new random variable generated from two independent and identically distributed random variables ri and rj. The cumulative distribution function (cdf) of d2, as given in [22], will be 2

F d2 ðD2 Þ ¼ Pðd  D2 Þ Z Z ¼

r2i þr2j 2r i rj cos /D2

ð4Þ fri rj ðr i ; rj Þdr1 dr 2 :

Since ri and rj are independently generated,

ð5Þ

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T. Wang, C.P. Low / Computer Communications 33 (2010) 1949–1960

N M I

J

N M

O φ

rj

ri

Communication Range

I

L

d

J

Communication Range

Link between Connected Nodes

O

Link between Connected Nodes Fig. 6. A special case of event E.

Fig. 5. Adjacent nodes i and j.

F d2 ðD2 Þ ¼

¼

Z Z r 2i þr 2j 2ri r j cos /D2

Z

D2 sin /

fr ðr j Þ

r j ¼0

¼

Z

Z

ri ¼cos /r j 

D2 sin /

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 4

r i ¼cos /rj þ

D r 2j sin /

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi fr ðr i Þdr i dr j 2 4 2 D r j sin /

 fr ðr i ÞF R cos /r j þ

r j ¼0



fr ðri Þfr ðr j Þdr i dr j

D2 sin /

r j ¼0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 D4  r 2j sin / dr j

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 fr ðr i ÞF R cos /r j  D4  r 2j sin / dr j ;

ð6Þ

in which FR(R) is the corresponding cdf of fr(r). 2 Since d  0 and Rc > 0, the event d  R2c is equivalent to the event d  Rc. Therefore, by replacing the variable in Eq. (6) by R2c , we have







2



F d2 R2c ¼ P d  R2c ¼ Pðd  Rc Þ:

ð7Þ

We can see nodes i and j are not directly  connected if d > Rc, which takes place with a probability 1  F D2 R2c . We say nodes i and j are adjacent if there is no other nodes that lie in the region bounded by lines extending from OI and OJ. In Fig. 5, nodes pair (i, j) and (m, n), located at (I, J) and (M, N) respectively, are two pairs of adjacent nodes. Obviously, when the distance between every pair of adjacent nodes in the MANET is less than Rc, the MANET must be connected. On the other hand, if the MANET is partitioned, there must exist at least two pairs of adjacent nodes whereby the distance between each pair is larger than Rc. As shown in Fig. 5, the network is partitioned into two connected components, represented by the dashed area. It means both adjacent nodes pairs (i, j) and (m, n) are disconnected. In this case, we say (i, j) and (m, n) are disconnected adjacent nodes pairs. We also denote the event that there are at least two pairs of disconnected adjacent nodes as event E. The probability of E taking place can be given as follows:

PðEÞ ¼ P2 ðd > Rc Þ:

ð8Þ

We note that the occurrence of event E is only a necessary but not sufficient condition for partitioning to take place. Because although the disconnected adjacent nodes pairs are partitioned from each other, it is possible that they are connected through other nodes. In particular, we consider a special case shown in Fig. 6, where the network is still connected although event E takes place.

We consider 5 nodes, namely i, j, l, m and n in the MANETs. They are located at I, J, L, M and N respectively. The disks represent the communication ranges of the nodes. We can see that nodes i and j are adjacent nodes of node m, and they are located outside node m’s communication range. In this case, (m, i) and (m, j) are two pairs of disconnected adjacent nodes, thus the event E takes place. However, in the figure, node n is inside both i’s and m’s communication ranges, and node i and m can thus be connected though n. Similarly, although node m is not directly connected to node j, there may exist a node l that connects to both of m and j. Therefore, although event E has occurred, the five nodes are not partitioned. Hence the probability of partitioning is bounded from above by the probability of event E. We have

  2 Ppar  PðEÞ ¼ P2 ðd > Rc Þ ¼ 1  F D2 R2c ¼ P0par :

ð9Þ

Eq. (9) shows that an upper bound for partition probability in DisAS can be calculated from fr(r), the pdf that is used to generate r for a waypoint. However, this result only applies with the assumption that all the nodes are on their corresponding waypoints. When nodes are moving, the partition probability will be less predictable. However, our simulation shows that P 0par can still give a good indication to the overall connectivity. In the next section, we use normal distribution2 as a case study of our proposed node allocation scheme. As normal distribution can be represented by letter N, we name DisAS with normal distribution as N-DisAS. There are basically two reasons that we use normal distribution instead of other choices: 1. The integral result of P 0par is relatively easy to obtain when r follows normal distribution. This will be illustrated in Section 5. 2. The simulation results show that normally distributed waypoints can maintain a reasonably high coverage while keeping the nodes connected, which is a desired feature in the application scenarios mentioned in Section 2. This will be discussed in Section 6.

5. N -DisAS In N-DisAS, ri and rj are generated from Nðm; r2 Þ, where m and r are the mean and standard deviation of the normal distribution, respectively. We define 2

Normal distribution is also referred to as Gaussian Distribution

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T. Wang, C.P. Low / Computer Communications 33 (2010) 1949–1960

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 d ¼ r2i þ r2j :

ð10Þ

It is worth to mention that normally distributed variables can be generated from the uniform random numbers. In N-DisAS, we use the polar form of the Box-Muller transform [23,24] to generate ri and rj. All the instructions in this generator are available from a micro controller with floating-point instructions and the complexity can be considered as linear. Using the result fromp[25], we know that d0 is distributed as a ffiffiffi Rician Distribution Rðr; 2mÞ, of which the pdf is

pffiffiffi !   0 0  pffiffiffi  d0 ðd þ 2m2 Þ d 2m 0 I0 fd0 d j 2m; r ¼ 2 exp ; 2r2 r r2

where I0(z) is the modified Bessel function of the first kind with order zero [26]. The corresponding cdf of the Rician Distribution is 0

0

0

F d0 ðD Þ ¼ Pðd  D Þ ¼ 1  Q 1

! pffiffiffi 2m D0 ; ;

r

r

ð12Þ

where Q1 is the Marcum Q-Function [27]. 0

If we define a ¼ dd , i.e., d0 = a d, the probability of d > Rc is given by 0

0

Pðd > Rc Þ ¼ Pðd > aRc Þ ¼ 1  Pðd  aRc Þ ! pffiffiffi 2m aRc ; : ¼ Q1

r

r

ð13Þ

Based on Eq. (9), we have 2

Ppar  P ðd > Rc Þ ¼

Q 21

! pffiffiffi 2m aRc ¼ P0par : ;

r

r

Fig. 7. The layout of waypoints with different r value.

ð11Þ

ð14Þ

This shows that, in N-DisAS, the actual value of P 0par depends on the parameters m, Rc, a and r, which will be discussed in the following subsections.

5.2. Determine m and r One of the potential drawbacks of N-DisAS is the difficulty in distributing the waypoints as evenly as possible over the area. However, we found that this can be overcome by setting the values of m and r. We assume that the MANET is applied in a circular area with a radius M. Since normal distribution is symmetric with respect of the mean value m, we set m = M/2. Moreover, in normal distribution the pdf peaks at the mean value, which makes the waypoints in N-DisAS being more likely to be generated around the circumference of the circle centered at O with radius m. If r is small, the nodes will be restricted in the ring area which is so close to the circumference of this circle that the freedom of the nodes is limited (as shown in Fig. 7(a)3). On the other hand, a very large r will in turn result in a large variance in the random numbers generated. Our estimation of Ppar will in turn become less accurate as the random numbers becomes less predictable. Therefore, we need to set r to a moderate value. From extensive simulations, we found that r = 2 m is a balanced setting for connectivity and coverage, as shown in Fig. 7(b). 5.3. Estimate Ppar Given r = 2 m and a0N ¼ 1:1 þ 0:03N, we can rewrite Eq. (16) as

5.1. Determine a 0

As defined, a ¼ dd

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2i þr 2j ¼ r2 þr2 2r . Because ri and rj are both ranr cos / i

j

0

0

Pðd > a0 Rc Þ  Pðd > a0 djd > Rc Þ ¼ Pðd > a0 dÞ  Pðd > Rc Þ  Pðd > Rc Þ:

ð15Þ

It then follows from Eqs. (13) and (14) that

P0par  Q 21

! pffiffiffi 2m a0 Rc : ;

r

r

ð16Þ

Clearly, a0 is inversely proportional to /, which in turn is related to N, the number of nodes in the network. In this work, the value of a0 is determined from simulations. For different N value, we randomly generate waypoints according to N-DisAS, and compute the corresponding a0 that satisfies P(d0  a0 d)  1. From this result, we use function regression with respect to N to find the following relationship between a0 and N:

a0N ¼ 1:1 þ 0:03N:



Q 21

! pffiffiffi 2 ð1:1 þ 0:03NÞRc : ; 2m 2

ð18Þ

i j

dom numbers, the exact value of a varies for each pair of points (i,j). As long as we can find some a0 such that P(d0  a0 d)  1, we will have 0

P0par

ð17Þ

Using this value of a0 , we have P(d0  aNd0 )  95% in the simulations. The effectiveness of this relationship can also be confirmed by the results illustrated in Section 6.

This is a function of Rc, N and m. As all these three parameters are usually known before we setup the MANET, this result is extremely useful when we design a MANET. We are able to control Ppar by adjusting these values with N-DisAS. This in turn enables the network designer to evaluate the probability of network partitioning for different values of Rc, N and m. Fig. 8 plots P0 par versus m with different N with the assumption that Rc = 10. In each figure, the x axis shows the value of m and y axis refers to Ppar. m varies from 1 to 50. To estimate the actual partition probability in the network, we generate N waypoints by using the method in N-DisAS simultaneously. Then Depth First Search (DFS) is used to evaluate the connectivity of the network which is formed by these N waypoints. In particular, DFS will return the number of connected components of the network, together with the number of nodes in the largest component. Network partition would have taken place if the number of components is more than 1. This process is repeated for 1000 iterations for each N value. We divide the number of iterations in which partitioned is detected by 1000 to estimate Ppar. The results are shown in the corresponding figures. It is easy to see from Fig. 8 that the empirical values of the network partition probability are bounded from above by our theoretical bound P0par . This verifies the validity of the theoretical 3

In Fig. 7, M = 20, N = 100.

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T. Wang, C.P. Low / Computer Communications 33 (2010) 1949–1960

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bound that we have derived. We can also see that as m increases, the nodes will be scattered in a larger area. This in turn increases the probability of network partitioning. Hence we could see from Fig. 8 that both Ppar and P0 par increase towards 1 for both the theoretical bound and the empirical results when m increases. From the four figures, we can also observe that when the number of nodes in a MANET increases, partition becomes less likely to take place with the same m value. It is also true that with more nodes, the network is able to span over a larger area while keeping connected. On the other hand, we can also observe that for one or two values of m (e.g. N = 50, m = 20), the theoretical prediction exhibits large deviation from the simulation result and the bound seems to be less effective than expected. This is first of all because of the random variation in the simulations, since fluctuating simulation results can be observed in all four figures of Fig. 8 and most of them are close to the theoretical bound. Secondly, it is partially due to the fact that a is determined empirically in N-DisAS, and may not be the most accurate value. With more sophisticated mathematical procedures, a can be evaluated analytically to propose an even tighter bound to the partition probability. However, since we would like to focus on the more important parts of DisAS and N-DisAS schemes, such as how the waypoints are generated and how is normal distribution related to the Q function, determining the exact value of a has to be left as a future research topic for further study. More importantly, it should be noted that this is the first time such kind of bound on partition probability is analytically derived

by a node allocation scheme. We believe it is a more flexible and practical solution than the existing deterministic schemes. It demonstrates a new dynamic way of controlling MANETs topology, and is one of our major contributions. 6. Simulation results Extensive simulations are carried out to verify the effectiveness of N-DisAS. Two mobility models, namely the Random Walk model and the Random Waypoint model, and the Movement Control Algorithm (MCA) from [14] are used to compare with N-DisAS (rn) and N-DisAS (nc). In the Random Waypoint model [17], upon reaching a waypoint, the node pauses for a random period and changes its direction to another uniformly random generated waypoint. This model is similar to N-DisAS (rn), where nodes only move voluntarily after completing a task with random time length. However, in the Random Waypoint Model, there is no rule to guide the generation of waypoints, and thus will have weaker connectivity than N-DisAS (rn). With MCA, the nodes move closer to each other to construct a bi-connected network topology graph. It is similar to N-DisAS (nc) as the nodes are forced to move to new locations for better connectivity, whereas coverage is not concerned in MCA. We note that MCA is a centralized algorithm which guarantees that all nodes are eventually connected, and offers an optimal solution in the concern of connectivity. Since the distribution of nodes are

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not discussed in MCA, we have to assume that the nodes are initially allocated at random locations inside the area. In the Random Walk model [28], each node changes its direction and speed randomly, i.e. nodes movement is purely randomized, as Brownian motion. We also show it in the comparison because in some works the nodes movement in a MANET is assumed to be Random Walk, or Brownian. We use three different measures to evaluate the connectivity and coverage of MANETs:

network is already partitioned, the fewer the number of the connected components there are, the less severe the partitioning is. The method of measuring coverage ratio can be found in [29]. Section 6.1 discusses the overall performance of a network with and without N-DisAS. From Sections 6.2–6.5. We demonstrate that how the performance level varies if some of the parameters change in N-DisAS.

1. The number of nodes in the largest connected component (i.e. partition), which is referred to as partition size or PS, 2. The number of connected components, which is referred to as number of partitions or NoP, 3. The ratio between the size of area within the sensing range of the nodes and the total network area size, which is referred to as coverage ratio or CR.

We assume that the MANET consists of 100 nodes. The network area (A) is a circular area with diameter 100 m. 10 smaller circular area with random radius from the interval [1,5]m are randomly placed in A to represent the obstacles. Rc is set to 10 m. Rs is 5 m. The speed of the nodes is set to 2 m/s. Similar setup can be found in [30] and many other existing works on MANETs. We set m = 25 m and r = 50 in N-DisAS. For N-DisAS (rn), x is 0.2, and t = 3 s so that the expected task length T = t/x = 15 s. In Random Waypoint Mobility Model, the pause time is also set to a random variable up to 15 s so that it is similar to N-DisAS (rn). In N-DisAS (nc), HIGH_N is 8 and HIGH_X = p/2. Although the values

The first two measures listed above can be obtained using DFS on the network topology graph. The more nodes there are in a connected component, the stronger the MANET is connected. If the

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of these parameters though have not been proven to achieve optimal results, they have already shown that such an implementation of NDisAS is able to improve network connectivity effectively. All the nodes are distributed from the center of A except in MCA, where they are randomly placed at the beginning. Every 3 s, the three measures are taken from the snapshot of the MANET topology. We refer to each 3 s interval as a step, and 500 steps are taken in the simulation. However, we only show the first 200 steps in Fig. 9, since it is already enough to illustrate how network connectivity and coverage varies over time. In Fig. 9, the horizontal axis represents the number of steps. Fig. 9(a) and (b) show that the connectivity is maintained at a high level by N-DisAS (rn) and N-DisAS (nc) as compared to the other schemes. We can see that N-DisAS (nc) and MCA eventually reaches a steady state where all nodes are connected as one partition. In particular, when every node’s number of neighbors is more than 0 and less than HIGH_N = 8, all nodes will remain in the ‘‘stay” state. According to our estimation in the previous sections, it is highly likely that they are connected and partitioning will not take place. Since in N-DisAS (nc) and N-DisAS (rn) the nodes move out of the same point, they maintain a strong connectivity from the beginning. Meanwhile in MCA, the algorithm reconnects different partitions together by moving them closer from randomly allocated places, therefore the number of connected nodes (i.e. partition size) increases gradually towards 100 and the number of partitions reduces slowly to 1 at the same time. We can see that

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it needs around 150 steps to connect all the nodes, while only about 120 steps are spent for N-DisAS (nc), showing that N-DisAS (nc) is a better solution. Although due to the voluntary movement of nodes, N-DisAS (rn) does not guarantee a full connectivity, it keeps a much stronger connection in the network than the Random Waypoint and Random Walk models, by having a larger partition size and a significantly smaller number of partition. We also note that the partition size sometimes drops to a relatively smaller value in N-DisAS (rn). This is because we bound the probability of partitioning from above, instead of the partition size in DisAS. Partitioning is still possible, but with a lower probability, to take place. When the network is partitioned, it becomes possible that a large number of nodes are disconnected from others, resulting a small number in the partition size. However, we can see that this situation does not last for a long duration. The network recovers from partitioning quickly using N-DisAS (rn). This is useful in practice because a short period of partitioning can be tolerated easily by buffering the packets in the storage of the nodes. Besides, even when partitioning takes place with NDisAS (rn) and N-DisAS (nc), the number of partitions remains as low as 2 or 3, which is much smaller than the Random Waypoint and Random Walk models. This shows that the negative effect of partitioning is significantly reduced by using N-DisAS (rn). In terms of coverage, Fig. 9(c) depicts that nodes in the Random Walk model cover more area than in all the other schemes after about 100 steps, because the Brownian motion of nodes eventually lead to a near-uniform node distribution in A. Similar value of cov-

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tion 6.1. We note that this experiment is equivalent to the case where network area size is changed while keeping the same number of nodes in the network. We run the simulations for 31,000 s for each N value. The average value of largest partition size, number of partitions and coverage ratio are measured every 3 s (as 1 step) after the first 1000 s, during which the network is initialized. The average values of these 10,000 samples are plotted in Fig. 10. The horizontal axis in each figure represents N. We note since MCA eventually enters a steady state where all nodes are connected, as we have shown in the previous section, it represents an optimal solution in the concern of connectivity in Figs. 10 and 11. As increasing the number of nodes causes higher density of nodes in the network area A, the connectivity of the MANET is enhanced. All the four schemes are able to connect almost all the nodes in the network as N increases to a large value, as shown in Fig. 10(a). To demonstrate that the increasing size of the largest partition is not only a result of the increasing total number of nodes, but also a proof for better connectivity, we divide the partition size by N to obtain a normalized partition size, which is plotted in Fig. 10(b). It can be interpreted as the ratio between the number of nodes connected in the largest partition and the total number of nodes. A value near 1 shows that almost all the nodes are connected in a single partition. We can see that N-DisAS (nc) always connects more nodes than other schemes with a much faster growth rate for 20 6 N 6 40, and reaches the optimal level represented by MCA. This shows that N-DisAS (nc) is significantly more

erage ration also presents for MCA when it has just been initialized, i.e. at the beginning steps of the simulation, because uniform node placement is used in the simulation for MCA. However, as we discussed in Section 2, MCA compromises coverage for connectivity as it tries to move nodes closer to others. This is justified by a considerable drop in its coverage ratio in Fig. 9(c). In its steady state, the coverage ration is much lower than N-DisAS (nc) and N-DisAS (rn). In particular, we note that by force the nodes move to places with fewer neighbors, N-DisAS (nc) improves the coverage ratio of N-DisAS (rn) by around 10%, almost as high as with the random Waypoint Mobility Model. This shows that the goal of using HIGH_N and HIGH_X to guide the generation of waypoints for better coverage is successfully achieved. We can see that N-DisAS (nc) performs better than N-DisAS (rn) in both connectivity and coverage, but we also should note that N-DisAS (rn) offers much more freedom to the nodes since it does not force the nodes on ‘‘when” to move and has less restriction on ‘‘where” to move as well. In many practical scenario such as the disaster relief example of Section 2, N-DisAS (rn) provides a more practical solution and will be more suitable. 6.2. Varying N One of the parameters which directly influences the performance of MANETs is the density of nodes in the network. In this section, we vary N from 10 to 150 with an incremental size of 5 while other parameters are kept to the same values as those in Sec-

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reliable in enhancing the connectivity as compared to other schemes. We can see from Fig. 10(c), that the number of partitions falls to a small value in all the schemes. This phenomenon is consistent with the fact that network connectivity improves as the number of nodes increases. But we observe that the number of partitions increases with the number of nodes when there are few nodes. This trend persists and peaks when N is around 40. The reason is that when the number of nodes is few, say less than 40, any new nodes added to the network may fall on some areas which are disconnected from other nodes, thus forming new partitions. This in turn results in an increase in the number of partitions. When the number of nodes is further increased, new nodes may reconnect the partitions and reduce the number of partitions. Therefore, we conclude that increasing the node number is not always effective in term of reducing number of partitions in a MANET. When more nodes are in the network, the coverage increases. The results are similar as in Section 6.1. Random Walk has the best coverage, followed by N-DisAS (nc). Random Waypoint has a slightly poorer coverage than N-DisAS (nc), while MCA covers the least area of all. It is the drawback of MCA which we have already discussed before. 6.3. Varying Rc The communication range (Rc) of the nodes also changes the topology and coverage of the network. In this section, the simulation is repeated with varying Rc from 5 m to 15 m with an increment of 1 m. Rs also changes from 2.5 m to 7.5 m with Rc, since we always set Rs = Rc/2. Other parameters remain the same as those in Section 6.1. Each average value is obtained from 10,000 steps of simulation after the first 1000 s initialization duration (refer to Fig. 11). Again, MCA is used to illustrate the optimal case. We can see that increasing the communication range enhances both connectivity and coverage significantly. In all the schemes, the nodes are connected as one single partition when the communication range is larger than 13. The average coverage ratio for NDisAS (rn) when Rc = 15 is at least two times of the ratio when Rc = 5. In the Random Walk model, the average coverage ratio increases from 0.2 to 0.8. Comparing Fig. 11 with Fig. 10, we can see increasing Rc and N are both effective in enhancing the connectivity and coverage of the MANET. However, increasing Rc will result in additional energy consumption by the nodes while increasing N will require additional cost to deploy the additional nodes. 6.4. Varying HIGH_N in N-DisAS (nc) In N-DisAS (nc), HIGH_N controls the number of neighbors of a node in the MANET. This indirectly changes the frequency at which the ‘‘move” state is triggered. We have shown that N-DisAS (nc) reaches a steady state, and it is useful to find that how many steps are needed to reach the steady state, i.e. no more ‘‘move” is triggered by all the nodes, with different values of HIGH_N. When HIGH_N = 1, the nodes will always be triggered to the ‘‘move” state and N-DisAS (nc) never enters the steady state. On the other hand, when HIGH_N = 8, 127 steps are required, while it is 73 steps when HIGH_N = 15. It is because that when HIGH_N is larger, it is easier to reach the steady state. The results are shown in Table 1. We also measure the average partition size, number of partitions and coverage ratio before the steady state is reached. For HIGH_N = 1, it is the average value from 10,000 steps. It is interesting to see that although the steady state is reached sooner when HIGH_N = 15 the coverage ratio is lower than the case when HIGH_N = 8. This shows an interesting dilemma in using N-DisAS

Table 1 Results for varying HIGH_N in N-DisAS (nc). HIGH_N

1

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– 95.8900 1.8040 0.4222

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73 93.6960 2.0520 0.4980

Table 2 Results for varying x in N-DisAS (rn). x

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97.4860 3.9205 1.4060 0.4133

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rPS NoP CR

(nc): a higher upper bound for neighbor counts will stabilize the network sooner but also may cause a lower coverage, since more nodes may be allocated in a small area. Moreover, the coverage ratio is the lowest when HIGH_N = 1. This is because when HIGH_N = 1, the ‘‘move” state is triggered more often than higher HIGH_N values. Moreover, as HIGH_X = p/ 2, the new waypoint will be generated on the opposite side of the network. In order to reach this new waypoint, the node needs to move across the center area of the network. It becomes possible that several nodes cross through the region that is close to the center, resulting a lower coverage. 6.5. Varying x N-DisAS (rn) The value of x changes the average time duration that a node stays in the ‘‘stay” state in N-DisAS (rn). As T = t/x and we fixed t to be 5 s, x is inversely proportional to T, the expected length of the tasks. We repeated the simulations in previous sections for different values of x to see what is the impact of this variable. The results for different x values are shown in Table 2. We can see that the partition size, number of partitions and coverage ratio do not vary much, but the standard deviation (denoted as rPS) of the partition size, which is a measure of the dispersion among the sampled data (shown in the second row of Table 2), changes with x. For a smaller x, the ‘‘move” state is triggered more frequently. As x increases, nodes have a higher tendency to stay in their position and thus cause a lower standard deviation in the partition size. In conclusion, comparing to some of the existing mobility models, both N-DisAS (rn) and N-DisAS (nc) can effectively enhance the connectivity in MANETs while maintaining reasonably good coverage. 7. Future work and conclusion In this paper, we proposed a fully distributed node allocation scheme namely DisAS for Mobile Ad Hoc Networks. It uses random number generation to control the locations of the nodes in a MANET so that the probability of partitioning taking place can be estimated and controlled. This scheme is adaptive to different applications and scenarios. A case study of DisAS using normal distribution, namely N-DisAS, is carried out. Simulation shows that N-DisAS is able to maintain the connectivity at a high level while maintaining a good degree of coverage. Moreover, the probability of network partition in MANETs which is constructed using N-DisAS can be estimated using a Q-Function. This enables the degree of connectivity to be evaluated when the network is being designed.

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DisAS is adaptive to any existing and future communication protocols, which makes it a practical solution for area detection, surveillance, exploration and other applications in real life. Moreover, there are also several directions in which this work can be extended. First of all, other probability distribution functions may be used instead of normal distribution to control the waypoints generation. The functions may be chosen according to different purposes besides partition protection, such as coverage enhancement or faster exploration to the area. For example, if the application is to achieve higher degree of coverage in the center of area A, exponential distribution can be used so that the nodes will be allocated closer to the center. On the other hand, although this paper only addresses the twodimensional (2-D) case, the work can be extended to the 3-D case. The discussion in Section 4.5 uses triangular geometry in MOJI, which is applicable in both 2-D and 3-D. However, the a value will be different as the expected value of / will no longer be 2Np and the empirical study need to be modified to fit the 3-D environment. 3D DisAS will be extremely useful in various fields such as underwater robot network and space communication. As stated in this paper, we have not prove the optimality of the parameters that were used in the simulations. This is another possible future research area. The parameters of DisAS should be determined according to the environment (such as the area size, degree of coverage of connectivity required and number of nodes etc.) and the capabilities of the nodes (communication range, speed, energy restrictions etc.). An optimized solution will provide an even better partition protection mechanism for MANETs.

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