In this paper, we study the performances of the registration protocol in Ethernet passive optical networks (EPONs). In each registration process, the newly-connected optical network units (ONUs) send their requests to the optical line terminal (OLT) without any scheduling, which may cause collisions and lower the registration success probability. Thus, a random-delay based registration protocol has been defined in the IEEE 802.3av standard to prevent collisions. Intuitively, the registration protocol of EPON possesses similar characteristics as medium access control (MAC) protocols. Based on this principle, we derive the closed-form expression of the registration throughput to evaluate the random-delay based protocol. We show that an optimal discovery-window size can be determined by maximizing the registration efficiency. Our analytical results indicate that the random delay introduced in the protocol is helpful to enhance the registration efficiency when the number of ONUs is large, or if all ONUs are clustered. However, the throughput improvement by random delay is marginal if there is only a small number of evenly distributed ONUs in the vicinity of EPON.