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Table 3 Iteration system for ECN/RED model with temporary TCP connections

From: Formal modelling of TCP congestion control mechanisms ECN/RED and SAP-LAW in the presence of UDP traffic

\(\hspace {-16pt}\tilde {\mu }_{p,1}(t+1)=\sum _{j:d(j)=1}\left (1-(1-q(\rho _{p}(t)))^{s_{p}(j)}\right)\tilde {\mu }_{j}(t) (1-w)^{s_{p}(j)}\)
\(\qquad \qquad + \sum _{j=1}^{I_{p}} \tilde {\mu }_{p,j}(t) (1-(1-w)^{s_{p}(j)})~~~~~~~~~~~~~~~(18) \)
\(\tilde {\mu }_{\textit {\text {p,i}}}(t+1)=\sum _{j:d(j)=i}\left (1-(1-q(\rho _{p}(t)))^{s_{p}(j)}\right)\tilde {\mu }_{j}(t) (1-w)^{s_{p}(j)} \)
\(~~~~~~~~~~+(1-q(\rho _{p}(t)))^{s_{p}(i-1)}\tilde {\mu }_{\textit {\text {p,i}}-1}(t) (1-w)^{s_{p}(i-1)} \quad 1<i<I_{p} \)
\(\tilde {\mu }_{\textit {\text {p,I}}_{p}}(t+1)=(1-q(\rho _{p}(t)))^{s_{p}(I_{p}-1)}\tilde {\mu }_{\textit {\text {p,I}}_{p}-1}(t) (1-w)^{s_{p}(I_{p}-1)} \)
\(\quad \qquad \quad + (1-q(\rho _{p}(t)))^{s_{p}(I_{p})}\tilde {\mu }_{\textit {\text {p,I}}_{p}}(t) (1-w)^{s_{p}(I_{p})} \)
\(\mu _{\textit {\text {u,i}}}(t+1)=\sum _{j=1}^{I_{u}}\kappa _{(\textit {\text {u,j}}),(\textit {\text {u,i}})}\mu _{\textit {\text {u,j}}}(t) \)
\(\sigma _{p}(t+1)=\sum _{i=1}^{I_{p}} \tilde {\mu }_{\textit {\text {p,i}}}(t+1) \frac {1\,-\,(1\,-\,w)^{s_{p}(i)}}{w} \quad \quad \sigma _{u}(t+1)\,=\,\sum _{i=1}^{I_{u}}\mu _{\textit {\text {u,i}}}(t+1)s_{u}(i) \)
ρ c (t+1)= max(ρ c (t)+σ p (t+1)+σ u (t+1)−C,0)
ρ p (t+1)=ρ c (t) ρ up (t+1)=ρ uc (t),i>0 ρ uc (t+1)=σ u (t+1)