We consider a random, irreversible, and sequential deposition of identical objects onto a substrate. In packing processes, if an attempted deposition event leads to overlap with the previously deposited object, such an attempt is rejected. In covering processes, only deposition events increasing the coverage are accepted. A finite system eventually reaches a congested state, and we investigate the statistics of congested configurations. One-dimensional systems are analytically tractable. We study the statistics of congested configurations, e.g., we compute the average number of deposited objects and all higher cumulants.