Wireless Ad Hoc and Sensor Networks

162 Wireless Ad Hoc and Sensor NetworksHere, the congestion controller operates at the packet level, whereas theadmission controller works at the source level. These two controllers operateat two time scales and their interaction has to be carefully studied to avoidany instability and performance deterioration. Here, we propose a hybridsystem theoretic approach to arrive at the overall network stability.Because the packets arrive at discrete intervals, their transmission overthe network is modeled as a continuous fluid flow behavior, and thecommunication among the end systems is via messaging, which is adiscrete-event system; a hybrid system theory is required to analyze theoverall stability of the congestion and admission control schemes. In theproposal, the admission controller design, expressed as a matrix-baseddiscrete-event controller (DE), provides a framework for rigorous analysisof the DE system including its structure, protocols, and overall stability.It is described by the following equations, where k stands for the kth eventand z(k) is the state of the admission controller, ( z(k) is the negation ofz(k)), which is written as:where,zk ( + 1) = J v ( k) + J v ( k) + J v ( k) + J u ( k), (4.30)s s r r c c u svand the task complete equation isvvsrc= Szk ( ),s= S zk ( ),r= Szk ( ),c(4.31)yk ( ) = Syzk( ). (4.32)Equation 4.30 through Equation 4.32 are logical equations. Input u s (k)represents the sources waiting to get admitted and y represents the sourcesthat ended transmission. The controller state equation (Equation 4.30) is a setof rules, so that it is formally a rule base. The coefficient matrices, J s , J r , J c , andJ u are referred to as bandwidth availability, resource availability, congestionprediction, and admission request matrices, which are sparse so that realtimecomputations are easy for such a large interconnected DE system. Therules can be fired using an efficient Rete algorithm. The overbar in Equation4.30 denotes logical negation (e.g., sources that have completed transmissionsare denoted by ‘0’). The other matrices, Ss, Sr,Sc, are bandwidth, resourcerelease, and congestion prediction matrices, whereas S y represents the sourcecompletion matrix. Finally, all the matrix operations are defined in the ‘maxmin,’‘or,’ and ‘and’ algebra. These matrices are created based on the initialstate of the network and updated with time and are not discussed in detailbecause of space limitations .