Ethernet passive optical access network (EPONs) have become widespread optical access network. In EPON uplink communication, communication is reserved by polling from an optical network unit (ONU) to an optical line terminal (OLT) using multi-point control protocol (MPCP). A method of delaying the transmission timing of REPORT messages in MPCP, called delayed REPORT messages-MPCP (DR-MPCP) has been proposed as a way of reducing the mean packet delay time of EPON uplink communication . However, this method cannot limit the transmission window size of each ONU (making it gated service). In real networks, the transmission window size should be limited to be fair to each ONU (i.e. it should be limited service). This paper derives the upper limit and theoretical expression of the mean packet delay time in DR-MPCP limited service using the queueing theory M/G/1 model. It also analyzes the characteristics of mean packet delay.