 Lindley recursion

For arrivals added before the slot's offered service, $Z_k$ is the arriving work minus that service. With independent identically distributed increments of negative mean, the stationary queue workload has the law ${\sup }_{m\ge 0}{\sum }_{j=1}^m{Z}_j$. Iterating the recursion from the past gives this supremum; the strong law makes it finite. A different arrival/service ordering changes slot-boundary workload but requires its convention to be stated.