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During off-peak hours of a day, when incoming and outgoing traffic arrival rate used to be low, a Mobile Terminal (MT) can stay in idle mode for long time. For tracking the idle MTs, a wireless access network invokes them to conduct location update whenever they enter into a new location. This allows the network to route a call successfully. To provide any IP packet based service, Mobile IP (MIP) is very important protocol undeniably. However, Proxy Mobile IP was developed by IETF without considering idle mode condition of MTs. Consequently, an idle MT needs to conduct MIP binding (Layer 3 location update) whenever it moves to a new area of a Mobile Access Gateway (MAG) although it does not have any incoming or outgoing packets during idle periods. This phenomenon unnecessarily increases location update signaling cost. In our previous work, we proposed a mechanism using which only Layer 2 location update is needed when an MT is in idle mode and the Layer 3 location update is conducted after a call arrives for an idle MT in Optical and Wireless Converged Access Networks (OWCAN). This allows saving energy in the mobility management nodes (e.g. MAG, Base Stations) significantly. Based on our previous work, in this paper we propose a novel algorithm for the Optical Network Units (ONUs) of an OWCAN. This algorithm decides when to interrupt modules of a sleeping ONU to wake up and which frames should be forwarded for uplink transmission. We have found that adopting this algorithm an ONU of an OWCAN can save energy significantly. Therefore, our solutions mainly contribute in two-fold to make a green OWCAN. First, it minimizes energy consumption in the mobility management nodes. Second, our solution also reduces energy consumption and improves bandwidth utilization in the optical backhaul.