Analytical Model

formal analysis of broadcast probability for the DIS_

RAD algorithm. In [7], Tracy et al. proposed a predictive probability model of the counter-based scheme. Since the proposed scheme adds the concept of distance concept to the counter-based scheme, the analysis becomes more complicated. Two rebroadcast probabilities for the border and interior nodes need to be deduced.

Because the RAD of the border nodes is shorter than that of the interior nodes, the rebroadcast probability of border nodes should be higher than that of interior nodes

As in [7], several assumptions were made to simplify the analysis process. Fir ize of an area can represent the number of nodes located in that area. Second, each node in the network is independent, and moves unaffected by any other nodes.

That is, the topology is regarded as uniformly distributed at any time. Third, the broadcast requests are generated randomly from all nodes in the network. The DIS_RAD analysis is likely to be workable in a network with these properties.

As mentioned earlier the broadcast probability is divided into two comp

dcast probability of interior nodes (Pi) and of border nodes (Pb). The probability of an individual node cannot be precisely predicted. However, this analysis gives a general trend of the rebroadcast probability under DIS_RAD.

In a DIS_RAD scheme, each node initiates a RAD as so

dcast packet. The length of a node’s RAD is determined by its relative distance from the source node. When the RAD expires, each node rebroadcasts the packet only if its counter is less than the counter threshold. The counter of each node is increased

by 1 for each duplicated rebroadcast packet received before its RAD expires.

Therefore, the probability that a node increases its counter must be deduced. The analysis starts as follows [7].

When node v receives a duplicated packet from node u, three conditions must apply

Node u must be a neighbor of node v.

than v.

can be neglected, nodes u and v are assum

hat these three events are independent, the probability Q that node v’s coun

: A.

B. Node u must transmit the packet.

C. Node u must have a shorter RAD

Since the broadcast signal propagation delay

ed to have received the original broadcast request from the source node simultaneously.

Supposing t

ter increases by 1 can be obtained as follows:

)

The parameters P(A), P(B) and P(C ately as the followings. Parameter P(A) deno

) separ

tes the probability that node u locates within node v’s transmission range, and is calculated as:

ase I: Given a distance threshold Dth, node v is located at the border annulus if its notes the transmission radius and Anet den s the entire network r area. In DIS_RAD, the distance concept divides the rebroadcast probability into Pi and Pb. When node v receives a packet from an imaginary source, its position is determined according to two cases.

C

distance from the source is greater than Dth. Therefore the broadcast probability of v is given by Pb. When node v receives a duplicate packet from node u, in counter-based

rebroadcast model proposed by Tracy et al., the location of node u is insignificant as long as it falls into node v’s transmission radius. In the proposed algorithm, however, the position of node u affects the probability of increment of the v counter. Therefore, two situations are considered:

S1: Node u is located at interior circle of source node, as shown in Fig. 4(a). The

probability S1 is given as:

2

Since the border nodes are assumed to have the same rebroadcast probability equal to

(2)

Under S1, node is est d ca a e rde

Pb, and the interior nodes have the same rebroadcast probability equal to Pi, the rebroadcast probability of node u is given by:

i

S P

B

P( ₁)=

v d ine an SRAD since it is lo ted t th bo r annulus, while node u is destined an LRAD since it is located at the interior circle. Since node v’s RAD expires earlier than that of node u, node v does not receive a duplicated packet from node u before its RAD expires. Therefore, P(C) under S1 (referred as P(CS1)) equals zero. probability S2 is given as:

)

Since node u is now located at the bo t probability of node

(4)

destined SRADs. Since the two nodes have the same range of RAD (SRAD), the

probability that node u‘s RAD expires first is 1/2, because RADs are chosen randomly, and nodes v and u are assumed to receive the same broadcast request simultaneously (the signal propagation delay is negligible). Thus, P(CS2) is given as:

2

With (1)、(2) 、(3)、(4) and (5), the probab e v’s counter increases by

2

1 when node v is located at the border annulus, is computed by summing S1 and S2:

)

Equation (6) describes the probability that the counter of node v is increased by 1 by any other node in the network. Therefore, Pb can be computed by summing all possible scenarios when 0 to Cth−1 duplicated packets are received before node v’s RAD expires. Since the imaginary source definitely increases the counter value of node v from 0 to 1, only 0 to Cth −2 cases need to be considered. This is leading to:

ase II: Given a distance threshold Dth, node v is located at the interior circle if its

circle of the source node, as shown in Fig. 5(a).

C

distance from the source is less than Dth. Likewise, two situations are discussed according to the position of node u:

S3: Node u is located at the interior The probability of S3 is given as:

2

The rebroadcast probability of node u is given by:

i

S P

B

P( ₃)= (8) Under S3, both nodes v and u are located at the interior circle, and share the same RAD range (LRAD). As with S2, the probability that node u‘s RAD expires first is given by 1/2. Thus, P(CS3) is represented as:

2

S4: Node u is located at the border annulus of the source node, as shown in Fig. 5(b).

The probability of S4 is given by:

)

The rebroadcast probability of node u is given by:

(10)

Under S4, node u D, while

S Pb

B

P( ₄)=

is located at the border annulus and is attached with SRA

node v is attached with LRAD according to the DIS_RAD scheme. Therefore the node u’s RAD expires prior than node v. Thus the probability that node u causes node v’s counter to be increased is 1.

) that node v’s counter increases by 1 when node v is located at the interior circle is given by:

From (12), Pi can be calculated by summing all possible Qi values as the same method when calculating Pb. Therefore,

i

From the analysis above, Pb and Pi can be compared with the broadcast probability in the counter-based scheme analyzed in [3], given by Pc by observing the curve of the analytical results. Figures 6, 7, 8 and 9 show various L×L maps analyzed using MATLAB, where L denotes a multiplier of the length of the communication radius R, which was set to 250 meters. Hence, the area of a network Anet equals 250×250×L×L m². The number of nodes N was set to 100, and counter threshold Cth was set to 3.

The X-axis denotes the value of Dth, and the Y-axis denotes the probabilities calculated by the analytical models. Clearly, Pb approximates Pc when Dth approaches 0, while Pi approximates Pc when Dth is set to 250 (equals R). In both cases, DIS_RAD is degenerated into the counter-based scheme.

Figure 4. The cases where node v locates at the border annulus.

Figure 5. The cases where node v locates at the interior circle.

Figure 6. Pb, Pi and Pc vs. Dth with Cth=3 in 3×3 map

Figure 7. Pb, Pi and Pc vs. Dth with Cth=3 in 5×5 map

Figure 8. Pb, Pi and Pc vs. Dth with Cth=3 in 7×7 map

Figure 9. Pb, Pi and Pc vs. Dth with Cth=3 in 9×9 map