Wireless Ad Hoc and Sensor Networks

Predictive Congestion Control for Wireless Sensor Networks 437Backpressuretoward sources(i − 1) thnodeReduce incommingflow rateData flowi thnode(i +1) thnodeu i (k) = ∑u ij (k)f j − 1 (k + 1) = u j (k + 1)aggregated outgoing ratedictated by i th nodeu i (k + 1) - incoming ratecalculated foraggregated data flowOnset of congestion= > ui(k + 1) reducedFIGURE 9.3Rate selection overview.where T is the measurement interval, qi( k)is the buffer occupancy ofnode i at time instant k , ui( k) is a regulated (incoming) traffic rate, dk ( )is an unknown disturbance in traffic, f i() • represents an outgoing trafficthat is dictated by the next hop node i + 1 and is disturbed by changes inchannel state, and Sat p is the saturation function that represents the finitesizequeue behavior. The regulated incoming traffic rates ui( k)have to becalculated and propagated as a feedback to the node i − 1 located on thepath to the source, which is then used to estimate the outgoing traffic forthis upstream node f i− 1() ⋅ .Select the desired buffer occupancy at node i to be q id . Then, bufferoccupancy error defined as ebi( k) = qi( k)−qidcan be expressed using Equation9.2 as ebi( k+ 1) = qi( k) + T⋅ui( k) − fi( ui+1( k)) + d( k)−qid. Next, the controlleris introduced and its stability analysis is presented by using two differentscenarios.In the simple case, where the objective is to show that the scheme works,it is assumed that the outgoing traffic f i() ⋅ value is known. Theorem 9.3.1shows the asymptotic stability of the system. Consequently, the queuelevel, q i( ⋅), will closely track the ideal level, q id . Moreover, if the queue levelexceeds the ideal level at any time instance, the feedback controller willquickly force the queue level to the target value. The second case presentedin Theorem 9.3.2 relaxes the assumption of full knowledge aboutthe outgoing flow f i() ⋅ . The stability will hold even when the full knowledgeof the outgoing flow is unknown as long as the traffic flow estimationerror does not exceed the maximum value f M . On the other hand, Theorem9.3.3 shows that an adaptive scheme is capable of predicting the outgoingtraffic fiˆ () ⋅ with error bounded by the maximum value f M . In consequence,the proposed controller with adaptive scheme will ensure trackingof the ideal queue level even in the presence of bounded estimationerrors in traffic flow.