Wireless Ad Hoc and Sensor Networks

86 Wireless Ad Hoc and Sensor Networkswhere the cell losses are given byck ( ) = ek ( ), if ek ( ) > 0,= 0Otherwise.(3.7)Equation 3.6 can be expressed as:ek ( + 1) = xk ( + 1) −x d ,(3.8)where ek ( + 1)and xk ( + 1)denote the error in buffer occupancy and thestate at the instant k + 1, respectively. Using Equation 3.5 in Equation 3.8,the dynamics of the nonlinear system described in Equation 3.5 can bewritten in terms of the buffer-length errors as:ek ( + 1) = f( xk ( )) − x + Tuk ( ) + dk ( )d(3.9)In high-speed networks, each source should be able to use the availablebandwidth up to its peak cell rate (PCR) or, at least, should meetthe minimum-rate requirement for ensuring fairness. The goal of a trafficratecontroller then, is to fully utilize the available network bandwidthwhile maintaining a good QoS, where the QoS parameters are the CLR,fairness, and transmission/cell-transfer delay or latency. This objective isachieved by selecting a suitable traffic rate using feedback, uk ( ), via minimizingthe difference between the actual and desired buffer length usinga predicted congestion level obtained via buffer occupancy errors, its rateof change, cell losses, and so on.Define the traffic rate input calculated using feedback,uk ( ),as1uk ( ) = (T x d − f ˆ( x ( k )) + k v e ( k )),(3.10)where f ˆ( x( k))is the traffic estimate of the unknown nonlinear traffic accumulationfunction, f(x(k)), at the network switch and k v is a diagonal gainmatrix. Then, the closed-loop error in buffer occupancy system is givenby Equation 3.9. One can also select uk ( ) as1uk ( ) = (T x d − f ˆ( x ( k )) + k ve1( k ))(3.11)where f ˆ( x( k))is the traffic-input estimate of the unknown nonlineartraffic function f( x( k)),at the network switch, a diagonal gain matrix