Network Calculus

From fidlerGuideStochasticNetwork2015 we have the following definition of the backlog $B(t)\le {\max }_{\tau \in [0,t]}\{A(\tau ,t)-S(\tau ,t)\}$Where $B(t)$ is the length of the backlog at time $t$. $A(\tau ,t)=A(t)-A(\tau )$ & $S(\tau ,t)=S(t)-S(\tau )$ where $A(t)$ is the total amount of bits arrived in $[0,t]$ and $S(t)$ is the service at time $t$

Alternatively we can use the following definition: $B(t)=A(t)-D(t)$ where $D(t)={\min }_{\tau \in [0,t]}\{A(\tau )+S(\tau ,t)\}$Putting it together we get

$B(t)=A(t)-{\min }_{\tau \in [0,t]}\{A(\tau )+S(t)-S(\tau )\}$