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

\(\hspace {-16pt}\tilde {\mu }_{p,1}(t+1)=\sum _{j:d(j)=1}\left (1-(1-q(\rho _{p}(t)))^{s_{p}(j)}\right)\tilde {\mu }_{j}(t) (1-w)^{s_{p}(j)}\)

\(\qquad \qquad + \sum _{j=1}^{I_{p}} \tilde {\mu }_{p,j}(t) (1-(1-w)^{s_{p}(j)})~~~~~~~~~~~~~~~(18) \)

\(\tilde {\mu }_{\textit {\text {p,i}}}(t+1)=\sum _{j:d(j)=i}\left (1-(1-q(\rho _{p}(t)))^{s_{p}(j)}\right)\tilde {\mu }_{j}(t) (1-w)^{s_{p}(j)} \)

\(~~~~~~~~~~+(1-q(\rho _{p}(t)))^{s_{p}(i-1)}\tilde {\mu }_{\textit {\text {p,i}}-1}(t) (1-w)^{s_{p}(i-1)} \quad 1<i<I_{p} \)

\(\tilde {\mu }_{\textit {\text {p,I}}_{p}}(t+1)=(1-q(\rho _{p}(t)))^{s_{p}(I_{p}-1)}\tilde {\mu }_{\textit {\text {p,I}}_{p}-1}(t) (1-w)^{s_{p}(I_{p}-1)} \)

\(\quad \qquad \quad + (1-q(\rho _{p}(t)))^{s_{p}(I_{p})}\tilde {\mu }_{\textit {\text {p,I}}_{p}}(t) (1-w)^{s_{p}(I_{p})} \)

\(\mu _{\textit {\text {u,i}}}(t+1)=\sum _{j=1}^{I_{u}}\kappa _{(\textit {\text {u,j}}),(\textit {\text {u,i}})}\mu _{\textit {\text {u,j}}}(t) \)

\(\sigma _{p}(t+1)=\sum _{i=1}^{I_{p}} \tilde {\mu }_{\textit {\text {p,i}}}(t+1) \frac {1\,-\,(1\,-\,w)^{s_{p}(i)}}{w} \quad \quad \sigma _{u}(t+1)\,=\,\sum _{i=1}^{I_{u}}\mu _{\textit {\text {u,i}}}(t+1)s_{u}(i) \)

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)