## 3 Traffic Model
The delays related to sending messages through links (\(T_{link-i}\)) can be further decomposed: \[T_{link-i}=T_{link-w-i}+T_{link-t-i}+T_{link-p-i},\] (2) where:
* \(T_{link-w-i}\) represents mean message waiting time in a communication queue storing messages when links are busy (to obtain the value of this delay, the M/M/1 or M/G/1 queuing model is applied);
* \(T_{link-t-i}\) is a message transmission time calculated by dividing the message length by the link bandwidth (message lengths are set according to the values measured experimentally in Ref. [22]; link bandwidths are taken from the **b** vector);
* \(T_{link-p-i}\) is a propagation time proportional to link length (taken from the **d** vector) and equal to \(5\mu\)s/km for optical links. For network elements whose response times are modeled as random variables, message sojourn times (\(T_{NE-j}\)) are simply included in calculations of \(E(CSD)\) and \(E(CDD)\) times by taking mean values of these random variables (**ETR, EXA** and **EY** vectors). Message sojourn times in the network elements containing queues can be calculated as follows:  
\[T_{NE-j}=T_{NE-w-j}+T_{NE-p-j}, \tag{3}\]  
where:  
* \(T_{NE-w-j}\) represents mean message waiting time in the CPU queue buffering messages when the CPU is busy (to obtain the value of this delay, the M/M/1 or M/G/1 queuing model is applied);
* \(T_{NE-p-j}\) is a message processing time. For each CSCF server, which is the \(j\)-th network element in (1), message processing times are calculated in the following way: \[T_{NE-p-j}=a_{msg}\cdot T_{INV},\] (4) where:
* \(a_{msg}\) is a factor determining the time of processing a message \(msg\) in relation to the time of processing the SIP INVITE message (this parameter depends only on the message type and does not depend on the CSCF server type);
* \(T_{INV}\) is the SIP INVITE message processing time by a particular CSCF server, and values of this time for all CSCF servers are stored in the **TINV** vector. The message processing times (\(T_{NE-p-j}\)) of RACF C1 and RACF C2 are obtained by appropriately summing elements of the **TA, Tproc** and **Tresp** vectors. Components of these sums are dependent on the type of resource operation and the necessity of adjusting LSP bandwidth. Detailed calculations are described in Ref. [23]. It is worth noting that the method of constructing our analytical model is universal. Therefore, it can be used to develop new analytical models, which would allow analysis of other aspects of the IMS/NGN network operation or even analysis of other network types. For example, IMS/NGN resource virtualization can be investigated in which new instances of individual IMS servers are launched in order to guarantee quality (defined by values of \(E(CSD)\) and \(E(CDD)\)) as the load increases. Then, new virtual machines for a specific type of server can be added, and when the load decreases, they can be removed, as suggested in Ref. [24]. In this way, the use of resources allocated for service control in the service stratum can be optimized. Implementation of the analytical traffic model for the IMS/NGN network with service stratum resource virtualization requires modification of the previously presented network model (Figure 1) and the set of service scenarios (Table 1). They must take into account the fact that there are many instances of individual IMS servers in the network working in parallel, to which messages generated in particular service scenarios are distributed. Then, according to the service scenarios, messages handled by these servers are sent to subsequent network elements that may be implemented in one or more instances. Thus, in addition to changing the number of network elements, the number of links and the way of handling messages in the network also change. The general calculation methodology presented in this section (Formulas (1)-(4)) is suitable for computing mean values of \(CSD\) and \(CDD\) times in the modified IMS/NGN network structure involving virtualization of service stratum resources.