Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

212 T. Chu and I. Nikolaidisenergy demands. Clearly, in reality, not all nodes are expected to place the sameload burden on a wireless network. One cannot discount the value of nodes thatpotentially forward traffic but do not introduce their own traffic (or lots of theirown traffic). In this sense, even if the asymptotic diminishing returns of spatialreuse dominate, the scale of the network at which the introduced load dominatesover space reuse may be sufficiently large for most applications. What becomesevident at this point is that the allocation of source-destination demands onactual paths in a wireless network, specifically in an ad hoc network, has notbeen widely studied as a function of the source-destination loads. The conceptof a load matrix is certainly old and quite heavily used in circuit switchingnetworks [3]. We will use it in the context of path establishment in a packetswitchingwireless network. The idea is to assign all the source-destination pathsin an ad hoc network, such that their global (over all nodes) energy demandsare collectively minimized. How packet transmissions take place, at each intermediatehop, are left outside the scope of the paper. Suffice is to say that weconsider the existence of an ideal MAC protocol which properly synchronizes thetransmission of competing nodes without causing any undue reduction on theeffective capacity. The particular problem of implemented MAC protocols thatprovide coordinated access to the medium in order to capitalize on spatial reuseis left outside the scope of the paper, and it is a research topic on its own right,e.g., [4,5].In a wireless environment, each transmission is a local broadcast, that is, itis received by more than just the next hop along the path. As such, even thoughnodes outside the path from source to destination are uninterested in the specificpacket (they will neither forward it, nor they are the intended destination), theywill nevertheless “hear” it. Assuming the energy required to receive a packetis substantially small, at least compared to transmitting a packet, overhearingpackets can result in a small energy penalty but, more importantly, it results ina congestion penalty. That is, the nodes that overhear the packet transmissionshave to refrain from using the medium for their own purposes.Consider a specific node, i, that is within the range of several other nodes.Let us assume that these other nodes transmit traffic of τ i bits per second total.The specific node is liable for forwarding some of this received traffic, Θ i bitsper second (clearly Θ i ≤ τ i ). Finally, the node also originates some of the traffic(acts as a source) for a load of A i bits per second. From the perspective of thenode, the wireless medium provides a capacity of C bits per second. In orderfor the routes in the network to be feasible, the capacity constraint τ i + Θ i +A i ≤ C must be satisfied at each nodes in the network. We note however, thatany technique that attempts to minimize energy consumption, will attempt tominimize transmission radii. The result is that energy consumption reduction isexpected to force τ i to be reduced as well, except for the fraction that has tobe absolutely (because of the topology) forwarded by node i which implies thatthe least traffic possible as seen from node i is 2Θ + A i ≤C. Nevertheless, τ i alsoreflects the fact that nodes are assigned a particular transmission range to ensurethe connectivity of the network, or more specifically, the existence of paths from