**2.5.1 Modeling the retransmission of the INVITE request**

For the case of INVITE requests, the exponential retransmission behavior is used up to a timer called TimerB. That is a request is retransmitted at time points T1, 3T1, 7T1, 15T1 and up to TimerB. This can be represented as a series in the form of:

$$\{2^1 - 1\} \text{T1}, \{2^2 - 1\} \text{T1}, \{2^3 - 1\} \text{T1} \dots \dots \{2^{\aleph\_i} - 1\} \text{T1} \tag{1}$$

With (2�� − 1)T1 = TimerB . Thereby the maximum number of retransmitted INVITE requests (N�) is

$$\mathbf{N}\_{\mathbf{l}} = \left\lfloor \log\_2(\frac{\text{TimeB}}{\text{T1}} + 1) \right\rfloor \tag{2}$$

With a loss rate of l, out of r issued INVITE requests per T1 seconds (r x l) packets would be lost on average. These would be retransmitted T1 seconds later. The retransmitted packets would also suffer from a loss and will have to be retransmitted later. Hence, the call generation rate (Ri) can be depicted is shown in Table 1.


Table 1. Retransmission Behavior of Invite Requests Due To Network Losses in IMS Network (Sisalem et al. 2008)

At time point 0, r requests are sent per T1 seconds. After T1 seconds the senders will continue generating r new INVITE requests per T1 second and will retransmit the lost (l x r) requests, e.g, (r + (l x r)) will be sent. Out of those (l x (r + (l x r)) will be lost. These would be retransmitted at time 3T1.At time 2T1 r new requests will be sent plus the requests that were lost T1 seconds ago, e.g., (l x r) requests. Out of the sent request (l x (r + (l x r)) will be lost. These would be retransmitted at time 4T1 and so on.The number of INVITE requests (Ri) sent by the sender at any time point (n) can, hence, be determined as:

$$\mathbf{R}\_{\mathbf{l}}(\mathbf{l}, \mathbf{n}) = \mathbf{r} \times \left(\mathbf{1} + \sum\_{\mathbf{m}=\mathbf{l}}^{\mathbf{m}=\mathbf{k}} \mathbf{l}^{\mathbf{m}}\right) = \mathbf{r} \times \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{k}} \mathbf{l}^{\mathbf{m}} \tag{3}$$

$$\mathbf{l\_e} = \mathbf{1} - (\mathbf{1} - \mathbf{l})^2 \tag{4}$$

$$\{2^1 - 1\} \text{T1}, \{2^2 - 1\} \text{T1}, \{2^3 - 1\} \text{T} \dots \{2^{N\_n} - 1\} \text{T1}, \{2^{N\_n} - 1\} \text{T1} \dots \text{TimerF} \tag{5}$$

$$(2^{\aleph\_n^{\bullet}} - 1)\mathbf{T}\mathbf{1} = \mathbf{T}\mathbf{2}$$

$$\mathbf{N\_n^e} = \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + 1) \right| \tag{6}$$

$$\mathbf{N\_n} = \mathbf{N\_n^e} + \mathbf{N\_n^l} = \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + 1) \right| + \left| \frac{\mathbf{TimeF} - \mathbf{T2}}{\mathbf{T2}} \right|$$

$$= \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + 1) \right| + \left| \frac{\mathbf{TimeF} - \{2^{N\_n^e} - 1\} \times \mathbf{T1}}{\mathbf{T2}} \right| \tag{7}$$

$$\mathbf{R}\_{\mathbf{o}}(\mathbf{l},\mathbf{n}) = \begin{cases} \mathbf{r} \times \left( 1 + \sum\_{\mathbf{m=1}}^{\mathbf{m=k}} \mathbf{l}^{\mathbf{m}} \right) & \mathbf{n} \le \frac{\mathbf{T} \mathbf{2}}{\mathbf{T} \mathbf{1}} \\\\ \mathbf{r} \times \left( 1 + \sum\_{\mathbf{m=1}}^{\mathbf{m=k}} \mathbf{l}^{\mathbf{m}} + \sum\_{\mathbf{m=k+1}}^{\mathbf{m=q}} \mathbf{l}^{\mathbf{m}} \right) & \text{otherwise} \end{cases} \tag{8}$$

$$\mathbf{q} = \left| \frac{(\mathbf{n} - 2^{\mathbf{N}\_n^{\mathbf{e}}} - 1) \times \mathbf{T}1}{\mathbf{T}2} \right| \tag{9}$$


$$\mathbf{L} = \mathbf{1} - (\mathbf{1} - \mathbf{l})^\eta \tag{10}$$

$$\mathcal{L}\_{\mathbf{e}} = \mathbf{1} - (\mathbf{1} - \mathbf{L})^2 = \mathbf{1} - (\mathbf{1} - \mathbf{l})^{2\eta} \tag{11}$$

$$\mathbf{R}\_{\mathbf{l}}(\mathbf{l}\_{\mathbf{e}}) = \mathbf{r} \times \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{\mathbf{l}}} \mathbf{l}\_{\mathbf{e}}^{\mathbf{m}} = \mathbf{r} \times \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{\mathbf{l}}} (1 - (\mathbf{1} - \mathbf{l})^2)^{\mathbf{m}} \tag{12}$$

$$\mathbf{R}\_{100}(\mathbf{l}) = \mathbf{r} \times \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{\parallel}} \mathbf{l}^{\mathbf{m}} \tag{13}$$

$$\mathbf{N}\_{\mathbf{i}} = \left\| \log\_2(\frac{\mathbf{TimeB}}{\mathbf{T1}} + 1) \right\|$$

$$\mathbf{R}\_{183} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{183}} \mathbf{L}^{\mathbf{m}} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{183}} (\mathbf{1} - (\mathbf{1} - \mathbf{l})^{\eta})^{\mathbf{m}} \tag{14}$$

$$\mathbf{N}\_{183} = \left| \log\_2(\frac{\mathbf{T}\_{183} + \mathbf{T}\_{\text{PRACK}}}{\mathbf{T} \mathbf{1}} + \mathbf{1}) \right| \tag{15}$$

$$\mathbf{T\_{163}} = \mathbf{T1} \times \{ 2^{\aleph\_l} - \mathbf{1} \} \tag{16}$$

$$\mathbf{N}\_{\rm l} = \left| \log\_2(\frac{\text{TimeB}}{\text{T1}} + 1) \right| \tag{17}$$

$$\mathbf{T\_{PRACK}} = \mathbf{T1} \times \left\{ 2^{\mathbf{N\_0^e}} - 1 \right\} + \text{ Max (0, T2 \times N\_n^l)}\tag{18}$$

$$\mathbf{N\_n = N\_n^e + N\_n^l} = \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + 1) \right| + \left| \frac{\mathbf{TimerF} - \mathbf{T2}}{\mathbf{T2}} \right| \tag{19}$$

$$\mathbf{R\_{PRACK}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} \mathbf{L\_e}^{\mathbf{m}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} (1 - (1 - l)^{2\eta})^{\mathbf{m}} \tag{20}$$

$$\mathbf{R\_{200}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} \mathbf{L^m} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} (1 - (1 - \mathbf{l})^\eta)^\mathbf{m} \tag{21}$$

$$\mathbf{N\_n = N\_n^e + N\_n^l} = \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + 1) \right| + \left| \frac{\mathbf{TimeF} - \mathbf{T2}}{\mathbf{T2}} \right| \tag{22}$$

$$\mathbf{L} = \mathbf{1} - (\mathbf{1} - \mathbf{l})^{\eta} \tag{23}$$

$$\mathbf{L\_e} = \mathbf{1} - (\mathbf{1} - \mathbf{L})^2 = \mathbf{1} - (\mathbf{1} - \mathbf{l})^{2\eta} \tag{24}$$

$$\mathbf{R\_{UPDATE}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_o}} \mathbf{L\_e}^{\mathbf{m}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_o}} (\mathbf{1} - (\mathbf{1} - \mathbf{l})^{2\eta})^{\mathbf{m}} \tag{25}$$

$$R\_{200} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_0} \mathbf{L}^{\mathbf{m}} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_0} (1 - (1 - \mathbf{l})^\eta)^{\mathbf{m}} \tag{26}$$

$$\mathbf{N\_{o}} = \mathbf{N\_{o}^{e}} + \mathbf{N\_{o}^{l}} = \left| \log\_{2}(\frac{\mathbf{T2}}{\mathbf{T1}} + 1) \right| + \left| \frac{\mathbf{TimeF} - \mathbf{T2}}{\mathbf{T2}} \right| \tag{27}$$

$$\mathbf{L} = \mathbf{1} - (\mathbf{1} - \mathbf{l})^{\eta} \tag{28}$$

$$\mathcal{L}\_{\mathbf{e}} = 1 - (\mathbf{1} - \mathbf{L})^2 = 1 - (\mathbf{1} - \mathbf{l})^{2\eta} \tag{29}$$

$$\mathbf{R}\_{180} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{180}} \mathbf{L}^{\mathbf{m}} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{180}} (1 - (1 - l)^{\eta})^{\mathbf{m}} \tag{30}$$

$$N\_{180} = \left| \log\_2(\frac{T\_{180} + T\_{\text{PRACK}}}{T1} + 1) \right| \tag{31}$$

$$\mathbf{T}\_{1\\$0} = \mathbf{T}1 \times (\mathbf{2}^{\mathbf{N}\_{\parallel}} - \mathbf{1})\tag{32}$$

$$\mathbf{T\_{PRACK}} = \mathbf{T1} \times \left(2^{\aleph\_{\rm n}} - 1\right) + \text{ Max (0, T2} \times \mathrm{N}\_0^1\text{)}\tag{33}$$

$$\mathbf{R\_{PRACK}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} \mathbf{L\_e}^{\mathbf{m}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} (1 - (1 - l)^{2\eta})^{\mathbf{m}} \tag{34}$$

$$\mathcal{R}\_{200} = \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{\mathbf{n}}} \mathbf{L}^{\mathbf{m}} \sum\_{\mathbf{m}=\mathbf{0}}^{\mathbf{m}=\mathbf{N}\_{\mathbf{n}}} \left\{ 1 - (\mathbf{1} - \mathbf{l})^{\boldsymbol{\eta}} \right\}^{\mathbf{m}} \tag{35}$$

$$\mathbf{N\_n = N\_n^e + N\_n^l} = \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + \mathbf{1}) \right| + \left| \frac{\mathbf{TimerF} - \mathbf{T2}}{\mathbf{T2}} \right| \tag{36}$$

$$\mathbf{L} = \mathbf{1} - (\mathbf{1} - \mathbf{l})^{\eta} \tag{37}$$

$$\mathcal{L}\_{\mathbf{e}} = 1 - (\mathbf{1} - \mathbf{L})^2 = 1 - (\mathbf{1} - \mathbf{l})^{2\eta} \tag{38}$$

$$\mathbf{R\_{200}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} \mathbf{L\_e}^{\mathbf{m}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} (1 - (1 - l)^{2\eta})^{\mathbf{m}} \tag{39}$$

$$\mathbf{R\_{ACK}} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} \mathbf{L^m} = \sum\_{\mathbf{m=0}}^{\mathbf{m=N\_n}} (\mathbf{1} - (\mathbf{1} - \mathbf{l})^\eta)^\mathbf{m} \tag{40}$$

$$\mathbf{N\_n = N\_n^e + N\_n^l} = \left| \log\_2(\frac{\mathbf{T2}}{\mathbf{T1}} + \mathbf{1}) \right| + \left| \frac{\mathbf{TimeF} - \mathbf{T2}}{\mathbf{T2}} \right| \tag{41}$$


Design and Analysis of IP-Multimedia Subsystem (IMS) 81

3. Find voice coder and background link utilization level given link bandwidth and packet

The E-Model, (ITU-T Rec. G.107 2005) is extremely complex with 18 inputs that feed interrelated components. These components feed each other and recombine to form an output (R). The recommendation (ITU-T Rec. G.108 1999) gives a thorough description on

Due to the complexity of the E-Model, the approach used here is to try to identify which E-Model parameters are fixed and which parameters are not. In the context of this research the

Where (T) is the mean one way delay of the echo path, (Ta) is the absolute delay in echo free

Next, the relationship between these parameters is identified. Since, we are making the assumption that the echo cancellers on the end are very good, we can say that T = Ta and Ie

According to the above assumption, R-Factor equation can be reduced to the following

The factor Id is the delay impairment factor that can be calculated as follow (ITU-T Rec.

<sup>6</sup> <sup>6</sup> <sup>X</sup> Id 25 1 X 1 <sup>2</sup>

Ta log <sup>100</sup> <sup>X</sup>

log2 

6 6 1

3

R = 93.2 – Id (Ta) – Ie (codec, packet loss) (43)

1

(44)

conditions. In addition, parameters that affect delay Id and Ie are introduced:

is directly related to a particular coding scheme and the packet loss ratio.

OPNET and MATLAB are the optimization tools that are used in this chapter.

how to carry out an E-model QoS calculation within VoIP networks.

only parameters of the E-Model that are not fixed are:

loss level

**3.1 Assumptions for E-Model** 

T and Ta – Delay variables

 PL - Packet Loss % ρ- Link Utilization Coder Type

G.107 2005):

With:

 Ie – Equipment Impairment Factor Id – Delay Impairment Factor

expression (ITU-T Rec. G.107 2005):

**3.1.1 Calculation of the delay impairment Id** 

B= η × (R� ×S� + R��� × S���)

+( R��� × S��� + R����� × S����� + R��� × S���) + (R������ × S������ + R��� × S���) (42)

+ μ × (R��� × S��� + R����� × S����� + R��� × S���) + (R��� × S��� + R��� × S���)

l : is the probability of losses between two hops (Assume l is the constant).

L : is the one way End to End Losses.

L�: is the two ways End to End Losses.

S: is the SIP Message Size.

η : is the number of hops.

μ: is the number of ringing.

r : is the number of Calls or Sessions per Second.

B: The Bandwidth needed for IMS Sessions Establishment.
