An efficient numerical method for performance analysis of contention MAC protocols: a case study (PRMA++)

Several multiple access control (MAC) protocols of industrial interest can be modeled by discrete-state discrete-time Markov chains, with finite lower block Hessenberg probability transition matrices P_N, where the diagonal blocks have a size which decreases while passing from the top left block (A_0,0) down to the right bottom block (A_N,N). Such matrices have been identified in the literature as funnel matrices. In general, for meaningful systems of engineering interest, the size of P_N is so large that the computation of the P_N stationary probabilities (π) with common numerical methods cannot be performed due to the computational cost involved. To calculate the stationary probabilities of P_N we have developed an innovative computational method which fully exploits the block Hessenberg structure of the matrix I-P_N . In this way, we drastically reduce the overall computational cost with respect to the customarily used LU factorization, while still keeping the strong numerical stability of Gaussian elimination with diagonal adjustment. The potential of the new method is exploited to assess the performance of an access protocol for third-generation mobile systems, called PRMA++ (packet reservation multiple access protocol). This was designed within the European project RACE and has so far been studied, to the best of our knowledge, via simulative analysis. PRMA++ has received a lot of attention from manufacturers in the ongoing fifth framework program of the European Community