This paper aims at analyzing the performance of an anonymous communication system 3-Mode Net with respect to the number of relay nodes required for communication and sender anonymity. As for the number of relay nodes, we give explicit formulas of the probability distribution, the expectation, and the variance. Considering sender anonymity, we quantify the degree of sender anonymity under a situation where some relay nodes collude with each other. The above analyses use random walk theory, a probability generating function, and their properties. From obtained formulas, we show several conditions for avoiding a situation where the number of relay nodes becomes large, and for providing high sender anonymity. Furthermore, we investigate the relationship between the number of relay nodes and sender anonymity, and give a condition for providing a better performance of 3 MN.