Table 9 Notation used in the model

O.f. calculation

$W_{Q\left(B\right)}=\sum _{s\in {\mathcal{S}}_{Q\left(B\right)}}{A}_s^c{w}_s$

Total expected network revenue associated with QoS(BE)

traffic flows

$B_{\mathit{\text{Mm}}|Q}=\underset{s\in {\mathcal{S}}_Q}{max}\left\{B_{\mathit{\text{ms}}}\right\}$

Maximal average blocking probability among all QoS service

types

$B_{\mathit{\text{ms}}|Q}=\frac{1}{A_s^o}\sum _{f_s\in {\mathcal{F}}_s}A\left(f_s\right)B\left(f_s\right)$

Mean blocking probabilities for flows of type $s\in {\mathcal{S}}_Q$

$B_{\mathit{\text{Ms}}|Q}=\underset{f_s\in {\mathcal{F}}_s}{max}\left\{B\right(f_s\left)\right\}$

Maximal blocking probability defined over all flows of

type $s\in {\mathcal{S}}_Q$

Blocking probabilities calculation

B(f _s )

Node-to-node blocking probability for all flows $f_s\in {\mathcal{F}}_s$

$B_{\mathit{\text{ks}}}={\mathcal{ℬ}}_s\left(\overline{d_k},\overline{{\rho }_k},C_k\right)$

Blocking probabilities for micro-flows of service type s in

link l _k

${\mathcal{ℬ}}_s$

Basic function (implicit in the teletraffic analytical model)

to calculate B _{k s}

Decision variables

$\overline{R}={\cup }_{s=1}^{\left|\mathcal{S}\right|}R\left(s\right)$

Network routing plans

$R\left(s\right)={\cup }_{f_s\in {\mathcal{F}}_s}R\left(f_s\right),s\in {\mathcal{S}}_Q\cup {\mathcal{S}}_B$

Set of all the feasible routes for the traffic flows of type s

R(f _s )=(r^p(f _s )), p=1,⋯, M

First, second, ⋯, M-th choice route for flow f _s

Path metrics and auxiliary parameters - MMRA-S2

$m_{\mathit{\text{ks}}}^1=c_{\mathit{\text{ks}}}^{Q\left(B\right)}$

Marginal implied costs

$m_{\mathit{\text{ks}}}^2=-log(1-B_{\mathit{\text{ks}}})$

Marginal blocking probabilities

$\mathcal{D}\left(f_s\right)$

Set of all feasible loopless paths for flow f _s

Simulation parameters

T= [ T _{i j} ]

Base matrix with offered bandwidth values

from node i to node j (Mbps)

α

Compensation parameter

t ₀

Duration of the first stage of the initialization phase,

where only periodical updates of the estimate

of the offered traffic are performed

t ₁

Duration of the second stage of the initialization phase,

where periodical updates of the estimate of the offered

traffic and of the routing plan are performed

t_warm-up=t₀+t₁

Duration of the initialization phase

τ

Update period of the estimates of the offered traffic

and of the network routing plans

${\tilde{x}}_n\left(f_s\right)=(1-b){\tilde{x}}_{n-1}\left(f_s\right)+b{\tilde{X}}_{n-1}\left(f_s\right)$

Estimate of the average traffic offered to the network

by the flow f _s in the time interval [ n τ;(n+1)τ[

${\tilde{X}}_{n-1}\left(f_s\right)$

Estimator of the average value of the traffic offered

by f _s to the network in the previous interval [ (n−1)τ;n τ[

b

Compromise value between the need to obtain a quick

response of the estimator to rapid fluctuations in $\tilde{X}\left(f_s\right)$

and the stability of the long-run variations

Miscellany of auxiliary parameters

f _s

Flow of service type s

${\mathcal{S}}_{Q\left(B\right)}$

Set of QoS(BE) service types

$A_s^o$

Total traffic offered by flows of type s

$A_s^c$

Carried traffic for service type s

A(f _s )

Mean traffic offered associated with $f_s\in {\mathcal{F}}_s$

w _s

Expected revenue per call of service type s

ρ _{k s}

Reduced traffic loads offered by flows of type s to l _k

$\overline{{\rho }_k}=\left({\rho }_{k1},\cdots ,{\rho }_{k\left|\mathcal{S}\right|}\right)$

Vector of reduced traffic loads

d _{k s}

Equivalent effective bandwidths for flows of type s in l _k

$\overline{d_k}=\left(d_{k1},\cdots ,d_{k\left|\mathcal{S}\right|}\right)$

Vector of equivalent effective bandwidths

$d_s^{\prime }$

Required bandwidth for service s (kbps)

$d_s=\frac{d_s^{\prime }}{u_0}$

Required effective bandwidth for service s (channels)

$\mathcal{ℳ},\mathcal{E},\mathcal{G},\mathcal{\mathscr{H}}$

Test networks

$\left|\mathcal{N}\right|$

Number of nodes in the network

$\left|\mathcal{ℒ}\right|$

Number of unidirectional links in the network

$C_k^{\prime }$

Link bandwidth (Mbps)

$C_k=⌈\frac{C_k^{\prime }}{u_0}⌉$

Link capacity (channels)

u ₀

Basic unit capacity

h _s

Average duration of a type s call

D _s

Maximum number of arcs for a type s call

ξ(f _s )

Function for choosing candidate paths for flow f _s

for possible routing improvement

δ

Average node degree of a network