The paper considers random multiple access in a network where only a small portion of users have data to forward and transmit packets in each time slot because user activity ratio is not high in practice. For this reason, the AP has to not only identify the users who transmitted but also decode the received data codewords. Exploiting the sparsity of transmitting users, Lasso, a well-known practical compressed sensing algorithm, is applied for efficient user identification. The compressed sensing algorithm enables the AP to handle more users than the conventional random multiple access schemes do. We develop distributed scheduling methods for maximizing system sum throughput and analyze the corresponding optimal throughput for three different cases of channel knowledgechannel state information at transmitter (CSIT), channel state information at receiver (CSIR), and imperfect channel state information at receiver (ImCSIR). We also derive closed form expressions of asymptotically optimal scheduling parameters and the corresponding maximum sum throughput for each CSI assumption. The results show the effects of system parameters on sum throughput, and it provides useful insights on using compressed sensing for throughput maximization in random multiple access schemes.