The WTFast's Gamers Private Network (GPN R) is a client/server solution that makes online games faster. GPN R connects online video-game players with a common game service across a wide-area network. Online games are interactive competitions by individual players who compete in a virtual environment. Response time, latency and its predictability are keys to GPN R success and runs against the vast complexity of internet-wide systems. We have built an experimental network of virtualized GPN R components so as to carefully measure the statistics of latency for distributed Minecraft games and to do so in a controlled laboratory environment. This has led to a better understanding of the coupling between parameters such as: the number of players, the subset of players that are idle or active, the volume of packets exchanged, the size of packets, latency to and from the game servers, and time-series for most of those parameters. In this paper we present a mathematical model of those system game network parameters and show how it leads to: (1) realistic simulation of each of those network or game parameters, without relying on the experimental setup; (2) very large-scale numerical simulation of the game setup so as to explore various internet-wide performance scenarios that: (a) are impossible to isolate from internet "noise" in their real environment and; (b) would require vast supercomputing resources if they were to be simulated exhaustively. We motivate all elements of our mathematical model and estimate the savings in computational costs they will bring for very large-scale simulation of the GPN R. Such simulations will improve quality of service for GPN R systems and their reliability.