A multiple access system with bursty data arrivals to the terminals is considered. The users are frame-synchronized, with variable sized packets independently arriving in each slot at every transmitter.Each packet needs to be delivered to a common receiver within a certain number of slots specified by a maximum delay constraint. The key assumption is that the terminals know only their own packet arrival process, i.e. the arrivals at the rest of the terminals are unknown to each transmitter, except for their statistics. For this interesting distributed multiple access model, we design novel online communication schemes which transport the arriving data without any outage, while ensuring the delay constraint. In particular, the transmit powers in each slot are chosen in a distributed manner, ensuring at the same time that the joint power vector is sufficient to support the distributed choice of data-rates employed in that slot. The proposed schemes not only are optimal for minimizing the average transmit sum-power, but they also considerably outperform conventional orthogonal multiple access techniques like TDMA.