1 Introduction
2 ENC/RED and SAP-LAW congestion control mechanisms
2.1 ECN/RED
2.2 SAP-LAW
3 Theoretical background
4 Models for ECN/RED and SAP/LAW congestion mechanisms
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The presence of UDP traffic as well as TCP. Specifically, we will be able to model both the case in which UDP traffic is smooth and the case in which it is bursty.
-
The case in which the TCP connections are not greedy, i.e. in case of an infinite capacity gateway, their windows would not tend to be at their maximum size. In practice, this is a frequent scenario which approximates the situation in which TCP connections are closed and new ones are newly opened maintaining the total number of active connections approximatively constant.
4.1 Modelling ECN/RED with UDP traffic and greedy TCP
p
(p
u
respectively) be the proportion of TCP (respectively UDP) connections, then we have
\(\phantom {\dot {i}\!}\mu _{\textit {\text {p,i}}}(t+1)=(1-q(\rho _{p}(t)))^{s_{p}(i-1)}\mu _{\textit {\text {p,i}}-1}(t)1_{\{i>1\}}\)
|
\(+\sum \limits _{j:d(j)=i}\left (1-(1-q(\rho _{p}(t)))^{s_{p}(j)}\right)\mu _{p,j}(t)\quad 1<i<I_{p}\)
|
\(\mu _{\textit {\text {p,I}}_{p}}(t+1)=(\!1\!-q(\rho _{p}(t)))^{s_{p}(I_{p}-1)}\!\mu _{\textit {\text {p,I}}_{p}-1}(t) +(1-q(\rho _{p}(t)))^{s_{p}(I_{p})}\mu _{\textit {\text {p,I}}_{p}}(t)\)
|
\(\mu _{\textit {\text {u,i}}}(t+1)=\sum \limits _{j=1}^{I_{u}}\kappa _{(\textit {\text {u,j}}),(\textit {\text {u,i}})}\mu _{\textit {\text {u,j}}}(t)\)
|
\(\sigma _{p}(t+1)=\sum \limits _{i=1}^{I_{p}}\mu _{\textit {\text {p,i}}}(t+1)s_{p}(i)\quad \quad \sigma _{u}(t+1)=\sum \limits _{i=1}^{I_{u}}\mu _{\textit {\text {u,i}}}(t+1)s_{u}(i)\)
|
ρ
c
(t+1)= max (ρ
p
(t) + σ
p
(t + 1) + σ
u
(t + 1) − C, 0) ρ
p
(t+1)=ρ
c
(t) |
4.2 Modelling SAP-LAW with UDP traffic and greedy TCP
\(\mu _{\textit {\text {p,i,j}}}(t+1)=\sum _{i^{\prime }=1}^{I_{p}} \sum _{j^{\prime }=0}^{i-1}\mu _{\textit {\text {p,i}}^{\prime },j^{\prime }}(t)1_{f(i^{\prime },j^{\prime },h(\vec \rho _{u}(t)))=(\textit {\text {i,j}})}\)
|
\(\mu _{\textit {\text {u,i}}}(t+1)=\sum _{j=1}^{I_{u}}\kappa _{(\textit {\text {u,j}}),(\textit {\text {u,i}})}\mu _{\textit {\text {u,j}}}(t)\)
|
\(\sigma _{p}(t+1)=\sum _{i=1}^{I_{p}}\sum _{j=1}^{i-1}\mu _{\textit {\text {p,i,j}}}(t+1)\min (s_{p}(i),s_{p}(h(\vec \rho _{u}(t))))\)
|
\(\quad \quad \quad \quad \quad \times \sigma _{u}(t+1)=\sum _{i=1}^{I_{u}}\mu _{\textit {\text {u,i}}}(t+1)s_{u}(i)\)
|
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
ui
(t+1)=ρ
u(i−1)(t),1≤i≤Y
ρ
u0(t+1)=σ
u
(t+1) |
4.3 Modelling ECN/RED with UDP and 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) |
4.4 Modelling SAP-LAW with UDP and temporary TCP connections
\(\tilde {\mu }_{p,1,1}(t+1)=\sum _{i^{\prime }=1}^{I_{p}}\sum _{j^{\prime }=1}^{i} \tilde {\mu }_{\textit {\text {p,i}}^{\prime },j^{\prime }}(t)\left (1-(1-w)^{\min (s_{p}(i^{\prime }),s_{p}(h(\vec \rho _{u}(t))))}\right) \)
|
\(\tilde {\mu }_{\textit {\text {p,i,j}}}(t+1)= \sum _{i^{\prime }=1}^{I_{p}}\sum _{j^{\prime }=1}^{i}\tilde {\mu }_{\textit {\text {p,i}}^{\prime },j^{\prime }}(t)1_{f(i^{\prime },j^{\prime },h(\vec \rho _{u}(t)))=(\textit {\text {i,j}})} (1-w)^{\min (s_{p}(i^{\prime }),s_{p}(h(\vec \rho _{u}(t))))} \)
|
\(\sigma _{p}(t+1)=\sum _{i=1}^{I_{p}} \tilde {\mu }_{\textit {\text {p,i,j}}}(t+1) \frac {1-(1-w)^{\min (s_{p}(i),s_{p}(h(\vec \rho _{u}(t))))}}{w} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(20) \)
|
\(\quad \qquad \times \sigma _{u}(t+1)=\sum _{i=1}^{I_{u}}\mu _{\textit {\text {u,i}}}(t+1)s_{u}(i) \)
|
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) |
5 Validation of the model
6 Performance evaluation
6.1 Greedy TCP connections and smooth UDP traffic
SAP-LAW | ECN γ=5E−6 | ECN γ=5E−7 | |
---|---|---|---|
T
| 591.27 | 585.69 | 594.23 |
Q
| 0 | 5.73 | 53.72 |
W
| 0 | 0.098 | 0.0901 |
6.2 Greedy TCP connections and bursty UDP traffic
SAP-LAW | ECN γ=5E−6 | ECN 5E−7 | |
---|---|---|---|
T
| n/a | 276.00 | 428.85 |
Q
| n/a | 64.52 | 155.34 |
W
| n/a | 0.234 | 0.3622 |
SAP-LAW | ECN γ=5·10−6 | ECN γ=5·10−7
| |
---|---|---|---|
T
| 713.27 | 333.29 | 529.4077 |
Q
| 153.95 | 35.20 | 83.79 |
W
| 0.2000 | 0.1056 | 0.1583 |
6.3 Temporary TCP connections
SAP-LAW | ECN γ=5E−6 | ECN γ=5E−7 | |
---|---|---|---|
T
| 219.17 | 300.06 | 295.50 |
Q
| 0.2160 | 361.1346 | 39.335 |
W
| 0.85E−4 | 1.2035 | 0.133 |