## 3 Traffic Model
Its advantage is the uncomplicated method of obtaining final results by summation of the mean values of message sojourn times in links and network elements, which is dedicated to engineering applications. Of course, the detailed formulas for \(E(CSD)\) and \(E(CDD)\) times in the service stratum resource virtualization case need to be modified to take into account the fact that message streams are split into multiple substreams at certain points in the network, served by multiple instances of individual IMS servers, and the resulting messages are then recombined into a single stream. The average time of all the abovementioned operations, taken into account during the calculation of the \(E(CSD)\) and \(E(CDD)\) times, can be computed as the weighted average of the sojourn time of individual message substreams in the appropriate links and instances of IMS servers. The weights of particular message substreams correspond to their share in the total stream of exchanged messages. To analyze various configurations of IMS/NGN service stratum virtualization, the described steps should be repeated to create a set of analytical traffic models, each of which includes different considered numbers of IMS server instances. It is very important that the analytical traffic model of the multidomain IMS/NGN described in this paper was successfully verified by simulations (service stratum and MPLS-based transport stratum separately; [23, 25, 26]). For the IMS/NGN service stratum and the MPLS-based transport stratum, simulation models were developed [23, 25], in which standardized communication scenarios were implemented. The simulators take into account the most important elements of the IMS/NGN network and communication links for exchanging messages between these elements. Network elements handle messages according to definable processing times depending on the element and message type. Messages waiting to be handled are stored in FIFO queues. Similarly, in the case of communication links, FIFO queues are used, and the time of message handling depends on its length and definable link bandwidth. The exchange of messages between the modeled network elements complies with the standards, and the introduced delays result from the operation of elements and links. In this way, the implemented simulation models reflect the phenomena taking place in the real IMS/NGN network and can be a reference for examining the quality of the analytical results. Our simulation studies described in Ref. [27] demonstrated that for exponential intervals between generated requests, message inter-arrival time distributions at the inputs of IMS/NGN network elements and links are generally not exponential but close to multi-modal. This results from the overlapping of many correlated messages sent in particular service scenarios. The nature of message service time distributions is similar, and it results from the assumed method of modeling the operation of network elements and links. Each network element and link processes its own set of messages (resulting from the implemented service scenarios), where the processing times of particular messages depend on their type and the type of element/link. The simulation model of the service stratum was used to evaluate the quality of the analytical results for overall IMS/NGN network responses (\(E(CSD)\) and \(E(CDD)\) times for all types of successful call scenarios) [26]. In the analytical research, various types of queuing models were used to calculate mean message waiting times for service in the IMS/NGN network elements and links. A wide range of queuing models were examined, regardless of how well they fit to message inter-arrival and service time distributionspreviously measured using simulations in Ref. [27]. The set of tested queuing models included M/M/1, M/G/1, G/G/1 approximations based on moments, PH/PH/1 and their special cases (PH/M/1 and M/PH/1) with algorithms for fitting phase-type distributions to arrival and service distributions based on moments or whole histograms.