Modified Gamma and Hyper-Erlang Distribution Models for Group Broadcasting

There are numerous decentralized and distributed computing applications running across the Internet today. In some applications such as those consensus-related, senders may dispatch duplicated messages, i.e., one-to-many identical messages at the application layer, to reach multiple recipients simultaneously. To effectively characterize the performance of such systems, tractable and accurate traffic models are highly desirable. Although one-to-one traffic modeling, e.g., M/M/1 model, has been extensively studied, traffic characterization for one-to-many connectivity remains underdeveloped. To accurately understand the performance of applications that generate one-to-many traffic, closed-form solutions are always desirable. For example, the hyper-Erlang distribution accurately models the residence time and channel holding time of a user in a wireless cellular network. However, the relationship between the coefficients of the hyper-Erlang distribution and system parameters or control variables is often unclear. In this paper, we find the traffic distribution for one-to-many traffic delay distributions over a Selected Group Broadcasting (SGB) in a wide-area communication system. Given the system parameters of an SGB model, we obtain closed-form formulas for the density functions of both the first and second order statistics – mean and variance – of the latency density functions of a broadcast overlay. Through simulations, we verify that the derived hyper-Erlang and modified gamma processes align well with the mean and variance of the simulated delays, as confirmed by the Kolmogorov-Smirnov test.