Markov model and delay analysis for group ALOHA systems

In the mobile environment, the power level difference between packets may arise due to Rayleigh fading (multipath propagation), shadowing (by building and hills) and the near/far effect (the geographic distribution of mobile users). In controlled grouped slotted ALOHA systems based on the backlog estimate, a steady state analysis is performed assuming that the backlog is equal to the estimate of the backlog S_k,i=n_k,i. It provides bounds on the performance of any control strategy of its type. Moreover under this assumption a multi-dimensional Markov chain become one-dimensional ergodic, time-homogeneous, irreducible Markov chains. A Markov chain is given to formulate the proposed system in the DFT mode in the genie model. A mean backlog number in each group is computed. The delay bound for the proposed grouped ALOHA is obtained under different new packet arrival rates joining backlog using Little´s theorem when the system is in stable state