Lindley recursion ($W_{k+1}=(W_k+Z_{k+1}{)}^{+}$)

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.

Consider two work-conserving slotted queues in tandem, with upstream service $c$ and downstream service $d<c$, where upstream departures can be served downstream in the same slot. Put $D_t=\min \{c,Q_{t-1}+a_t\}$. Then $Q_t=Q_{t-1}+a_t-D_t$ and $R_t=(R_{t-1}+D_t-d{)}^{+}$. If $R_{t-1}+D_t\ge d$, addition gives $(Q_{t-1}+R_{t-1}+a_t-d{)}^{+}$. Otherwise $D_t<d<c$, so the upstream queue empties and both sides of that identity are zero. Thus total workload obeys the Lindley recursion with service $d$. Starting empty in the remote past yields the displayed identity whenever the upstream workload is finite. Both workloads, and hence their difference, are continuous in the weighted cumulative-input topology for a slotted queue under strictly subcritical mean input $\lambda <d<c$.