Stationary workload supremum ($Q\stackrel{d}{=}{\sup }_{n\ge 0}({\sum }_{j=1}^n{A}_j-cn)$)

For independent and identically distributed arrivals with finite mean below service $c$, the strong law of large numbers makes the net cumulative input tend to minus infinity, so the supremum is finite almost surely. Applying the same formula to the past input gives a stationary solution of the Lindley recursion. This representation turns queue workload overflow into an event involving some lookback horizon.