In this letter, we focus on a system where N sources send n ≤ N different packets to one destination, through M ≥ N relays. Each relay employs random linear network coding to encode the packets it received by randomly choosing coefficients in a finite field F_q, then forwards it to the destination. Owing to the inherent errorprone nature of erasure channels, data packets received by the relay and the destination nodes may not be correct. We analyze the optimal throughput with respect to n, given a series of parameters and derive the upper and lower bounds of throughput performance. We also analyze the impact of the number of relays and the erasure probability on the throughput performance. Simulation results are well matched with the theoretical analysis.