We consider a multiple access system (MAC) with bursty arrivals. The transmissions are grouped into slots and the users are frame-synchronized. At the start of each time slot, variable sized packets independently arrive at each of the transmitting terminals. The packets are to be delivered to a common receiver by the end of the slot. Each terminal knows only its own arrival process, i.e. the packet-sizes at the rest of the terminals are unknown to each transmitter. The respective link gains from the transmitters to the receiver are assumed to be fixed and known to all. In this random access system, we propose schemes which will deliver the arriving data without any outage, under strict delay constraints. Our schemes are optimal in minimizing the total average power spent in data transport.