Consider the following model describing the required energy EAB to send a packet from node A to node B: EAB = kd (A,B) . Here, d(A,B) is the distance between node A and B, is a system parameter with > 2 and k is a positive constant. Assume that we are allowed to place a number of equidistant relay nodes between source node A and destination node B. Here, relay nodes serve as intermediate nodes to route packets from A to B. For instance, if A and B would use relay nodes R1 and R2, the message would be sent from A to R1, from R1 to R2 and finally from R2 to B.

(a) (3 points) What is the optimal number of relay nodes in order to send a message from A to B with minimum energy consumption?

(b) (3 points) How much energy would be consumed in the optimal case of part (a)?

(c) (3 points) Assume now a modified energy model which determines the energy required to send a message from node A to node B: EAB = kd (A,B) + c, with c > 0. Argue why this energy model is more realistic.

(d) (3 points) Show that under the modified energy model introduced in part (c) there exists an optimal number n of equidistant intermediate nodes between A and B that minimizes the overall energy consumption when using these intermediate nodes in order to route a packet from A to B.

(e) (3 points) Derive a closed-form expression on how much energy will be consumed when using this optimal number n of relay nodes.