Application of cost models over traffic dimensioning with QoS restrictions

Abstract

Network operators and Internet service providers are offering “Triple Play” products integrating services with different quality of service (QoS) requirements. The provision of QoS guarantees implies the revision of current dimensioning methods and consequences for costing and pricing. This paper proposes a cost model which considers QoS parameters, based on the Total Element Long Run Incremental Cost (TELRIC) model, calculating the cost of a network element and distributing it over the different services whose traffic uses it, taking into account the QoS requirements of each service. For this purpose, three traffic engineering methods are analyzed: complete traffic aggregation by “Over-engineering,” complete traffic segregation by separated virtual tunnels, and partial traffic aggregation by priority queuing. As an example, the cost model is applied to the connection in a Next Generation Network aggregation network for estimating the influence of QoS and traffic engineering on the cost estimation under the TELRIC model.

Appendices

Annex A: approximation toward Gi/Gi/1 model

Kingman’s formula, given in [8] provides:

$$ \overline n = \rho \left[ {1 + \frac{{\rho \left[ {{C^2}\left( {{T_s}} \right) + {C^2}\left( {{T_a}} \right)} \right]}}{{2\left( {1 - \rho } \right)}}} \right] $$

(21)

Note that $\rho = \lambda \cdot {\overline T_s}$ and Little’s formula gives $\tau = {{\overline n } \mathord{\left/{\vphantom {{\overline n } \lambda }} \right.} \lambda }$.Note that from $\tau > {\overline T_s}$, a unique value is always obtained for λ under $C\left( {{T_s}} \right) + C\left( {{T_a}} \right) > 0$, while the special case of a D/D/1 system with $C\left( {{T_s}} \right) = C\left( {{T_a}} \right) = 0$ results any λ with ${1 \mathord{\left/{\vphantom {1 \lambda }} \right.} \lambda } > {\overline T_s}$.

For T _s , a quadratic equation is obtained where:

$$ \lambda \left[ {1 - \frac{{{C^2}\left( {{T_s}} \right) + {C^2}\left( {{T_a}} \right)}}{2}} \right]\overline T_s^2 - \left( {1 + \lambda \tau } \right){\overline T_s} + \tau = 0 $$

(22)

from which Eq. 1 is obtained.

Annex B: approximation toward hyperexponential model

H ₂ (T_ak) is a proposed distribution for modeling the interarrival time T_ak for each service k = 1...K, see [17]. The PDF of T_ak from the arrival stream λ _k is:

$$ {f_{\rm{Tak}}}(t) = \alpha \cdot {\eta_k} \cdot {e^{ - {\eta_k} \cdot t}} + \left( {1 - \alpha } \right) \cdot {x_k} \cdot {\eta_k} \cdot {e^{ - {x_k} \cdot {\eta_k} \cdot t}} $$

(23)

The corresponding parameters x _k and η _k are calculated by the given first and second moment of T_ak, results:

$$ {x_k} = \sqrt {{\frac{{2 \cdot \left( {1 - \alpha } \right)}}{{\overline {T_{\rm{ak}}^2} \cdot \eta_k^2 - 2 \cdot \alpha }}}} $$

(24)

$$ {\eta_k} = \frac{{\alpha \cdot \overline {{T_{\rm{ak}}}} - \sqrt {{\alpha \cdot \left( {1 - \alpha } \right) \cdot \left( {\frac{{\overline {T_{\rm{ak}}^2} }}{2} - \overline {T_{\rm{ak}}^2} } \right)}} }}{{\overline {T_{\rm{ak}}^2} - \frac{{\left( {1 - \alpha } \right)}}{2} \cdot \overline {T_{\rm{ak}}^2} }} $$

(25)

where we tuned α to 0.1 to avoid a negative value in the root of Eq. 24.

For calculating the PDF of T _a as expressed in Eq. 4, we consider K H ₂ (T _ak) distributions as Fig. 8 shows, see [17]. Applying a Laplace transformation results:

$$ {F_{\rm{Ta}}}(s) = \frac{\alpha }{{{\lambda_0}}}\sum\limits_{k = 1}^K {{\lambda_k}} \frac{{{\eta_k}}}{{{\eta_k} + s}} + \frac{{\left( {1 - \alpha } \right)}}{{{\lambda_0}}}\sum\limits_{k = 1}^K {{\lambda_k}\frac{{{x_k} \cdot {\eta_k}}}{{{x_k} \cdot {\eta_k} + s}}} $$

(26)

from where Eq. 5, 6, and 7 results.

Fig. 8

Unit cost (in monetary units) per user class under real cost function (Segg segregation, Over-eng over-engineering, P.Q. Priority Queuing)