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Polyanskiy [1] proposed a framework for the MAC problem with a large number of users, where users employ a common codebook in the finite blocklength regime. In this work, we extend [1] to the case when the number of active users is random and there is also a delay constraint. We first define a random-access channel and derive the general converse bound. Our bound captures the basic tradeoff between the required energy and the delay constraint. Then we propose an achievable bound for block transmission. In this case, all packets are transmitted in the second half of the block to avoid interference. We then study treating interference as noise (TIN) with both single user and multiple users. Last, we derive an achievable bound for the packet splitting model, which allows users to split each packet into two parts with different blocklengths. Our numerical results indicate that, when the delay is large, TIN is effective; on the other hand, packet splitting outperforms as the delay decreases.