ROUTING 137

5.4.1 Network and Trafﬁc Demand Models

The discussion of the distributed routing to maximize the network lifetime will consider a multi-hop

static wireless network with

Ν

= {1, 2, ..., n} nodes. Two nodes in the network, which are within the

transmission range, communicate directly and form a wireless link. Let L = {(i,k) | wireless link from

i to k} be the set of wireless links, where each link (i,k) has a weight d

ik

representing the distance

between the antennas of node i and k. Figure 5.12 shows an example network with 6 nodes N =

{1,2,3,4,5,6} and links L = {(2,1), (3,1), (1,4), (1,5), (5,6), (4,6)} that we will use to show some of

the concepts.

Let r

i

( j ) ≥ 0 be the end-to-end trafﬁc rate (in bits per second), generated at node i and des-

tined for node j. For example, in Figure 5.12, the end-to-end trafﬁc rate generated at node 2 and

destined to node 4 is r

2

(4) = 2kbps, the rate generated at node 3 and destined to node 6 is r

3

(6) =

3kbps, and the rate generated at node 1 and destined to node 6 is r

1

(6) = 1k.

Let t

i

( j ) be the total trafﬁc rate of node i destined for node j, including both r

i

( j ) and trafﬁc

from other nodes that is routed through i to destination j. For example, in Figure 5.12, the total trafﬁc

rate from node 1 to node 6 is t

1

(6) = r

3

(6) + r

1

(6) = 3 + 1 = 4kbps, total trafﬁc rate from node 1 to node 4

is t

1

(4) = 2kbps; the other total trafﬁc rates are t

2

(4) = 2kbps; t

3

(6) = 3kbps; t

4

(6) = 1kbps; and t

5

(6) =

3kbps. Let us consider routing variable

ik

j( )φ

to be the fraction of the total node ﬂow rate t

i

( j ) routed

over link (i,k). For example, in Figure 5.12, the fraction

14

(6)φ

of the total node ﬂow rate t

1

(6) =

4kbps routed over link (1,4) from node 1 through node 4 to node 6 is

14

1

(6)

4

φ =

and the fraction

15

(6)φ

of the total node rate t

1

(6) = 4kbps routed over link (1,5) from node 1 through node 5 to node

6 is

15

3

(6)

4

φ =

; the other fractions are

(4) (6) (4) (6) (6) 1

= = = = =

21 31 14 46 56

φ φ φ φ φ

. This rout-

ing variable is important and deﬁnes a routing solution with

FIGURE 5.12: Example of ﬁve-node network.

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