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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