Wireless Ad Hoc and Sensor Networks

Optimized Energy and Delay-Based Routing 375PROOF This statement means that, using a minimum-cost spanning tree,a node can compute optimal routes to all the destinations in the network.In OEDR, each node knows the links between the MPR nodes and theirselectors, along with the costs of the links between them, from the TCmessages. According to Lemma 1, the intermediate nodes on the optimalpath are selected as MPRs by the previous nodes on the path. As a result,for a given node, the optimal paths to all the destinations are covered bythe network topology known through the MPR nodes. Therefore, the nodecan correctly compute the optimal path for the whole network topology.Theorem 8.4.2 directly follows from Lemma 8.4.1 and Lemma 8.4.2.THEOREM 8.4.2The OEDR protocol results in an optimal-route (the path with the minimumenergy-delay cost) between any source–destination pair.THEOREM 8.4.3For all pairs of nodes s and d, s generating and transmitting a broadcast packet P,d receives a copy of P .PROOF The proof follows on similar lines to (Jacquet et al. 2001). Let kbe the number of hops to d from which a copy of packet P has retransmitted.We shall prove that there exists a minimum k = 1, that is, a onehopneighbor of d eventually forwards the packet.Let n k be the first forwarder at distance k ( k ≥ 2)from d , which hasretransmitted P . There exists a MPR n′ k−1of n k which is at distance k − 1from d . To be convinced, let us imagine a path of length k from n kto d : n k −> n k−1 −> n k−2 −> … −> n 1 −> d and consider for n′ k−1asthe MPR of n k , which covers n k−2(on the optimal route calculated usingOEDR n k−1will be the MPR n′ k−1).Because n′ k−1received a copy of P the first time from n k (the prior transmittersare necessarily two hops away from n′ k−1), n′ k−1will automaticallyforward P : packet P will be retransmitted at distance k − 1 from d andwill reach n k−2 . Similarly, packet P will be retransmitted at distance k − 2,k − 3 , …, 2, 1 until the packet reaches d . Hence, the theorem is proved.8.5 Performance EvaluationThe proposed OEDR algorithm is implemented in the NS-2 simulator asa new routing protocol. Simulations were performed by varying nodemobility and with variable numbers of nodes in the network. The OEDR,OLSR, and AODV routing protocols and IEEE 802.11 MAC protocol areemployed.