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

On the Interaction of Bandwidth Constraints and Energy Efficiency 217that its bandwidth violation is the most. Then, we will fix the transmission radiiof the nodes in the vicinity of the node found in violation of the capacity constraint.If a path being overheard by the node with the capacity violation turnsout to consist of a two-hop sub-path and two corresponding transmission radiicover that constrained node, we will reduce it to one-hop path if possible. Theidea is taken from the triangulation relaxation of the shortest path construction.Note that when we compute the minimum energy path, the relaxation step byadding an edge to any existing path can reduce the cost of a specific node. Tothis end, we have computed the minimum energy path but we would like tosatisfy the bandwidth constraint by sacrificing energy consumption. Hence, wewill take the reverse procedure of the relaxation step. We will select the firsttwo-hop sub-path with such property, and after we replace it by a single relay,we will remove the load from these two hops and recalculate the load for thenew single transmission. This examination process will continue until no nodehandling or overhearing the traffic or no path re-construction has been made.This approach differs from the previous one in the process of repairing path.The previous algorithm considers the reconstruction of the entire path whilethis approach considers the subsection of a path that causes the bandwidth constraintviolation. Again, it is totally within reason to end up with an infeasibleconfiguration.3 Simulation StudyThe cost we computed captures the global energy consumption. After we constructthe paths from all source-destination pairs, we compute the total transmissionenergy consumption. If a node is involved the communication of differentsource-destination pair, its transmission power may not be the same in these differentpaths. As a result, we consider the total transmission power Energy[i ][j]for a particular path, and using T i,j as the weight for each paths’ transmissionpower, the average global energy consumption is equal to the sum of T i,j x Energy[i][j].That is, furthering the concept of an ideal MAC, the cost calculationassumes that a node can vary its transmission power depending on the destinationof the packet being handled each time. The simulations were conducted withrandomly placed nodes within a 1500x500 rectangular area and without nodemobility. The capacity of each node, C, was the unit of bandwidth, hence C=1throughout the simulations. The traffic load matrix, T i,j , was produced in a randomfashion. Specifically, each element of T i,j , is uniformly randomly generatedto be a demand between 0 and L/2(N-1) (where L is a parameter controllingthe relative load over all nodes and N is the number of nodes). The particularformula guarantees that the sum of traffic originating from source i is less thanwhich guarantees that the load of another nodes due to forwarding the load ofthis source-destination pair is going to be less than 1 (i.e. A i =y:x=i∧ j ∈V ∧ y= ∑ T x,j ≤ 1 ). Note however the restriction of the traffic matrix andcapacity generation is not sufficient to avoid infeasible solution scenarios. Therelative load is a predefined parameter while the absolute load is defined as the