Lower and Upper Bounds for Multicasting under Distance Dependent Forwarding Cost Functions

Assume a forwarding cost function which depends on the sender receiver separation, and assume further that noncooperative relaying is applied. What is the minimum total forwarding cost required for sending a message from source to one or more destinations when multicasting along optimal placed relaying nodes is applied? In this paper, I define and analyze cost function properties from which I derive general lower bound expressions on multicasting costs. I consider an MAC layer model which does not exploit the broadcast property of wireless communication and an MAC layer model which exploits it. For specific cost functions, I show further that in case of optimal relay positions, multicasts can be constructed whose cost always stays below the derived lower bound expression plus an additive constant depending on the number of destinations. For both, lower and upper bounds, I define a general procedure to check if-and if yes how-my findings can be used to derive the specific lower and upper bound expressions for a given cost function. I explain the procedure with three cost function examples: the euclidean distance, energy cost function, and the expected number of retransmissions under Rayleigh fading.