= Slower-server tandem workload identity
{title2=$R(a,c,d)=Q(a,d)-Q(a,c),\quad d<c$}
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 $R_{t-1}+D_t\geq d$, 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 $\lambda<d<c$.
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