Asymptotic Analysis for Reliable Data Dissemination in Shared loss Multicast Trees

Abstract

The completion time for the dissemination (or alternatively, aggregation) of information from all nodes in a network plays a critical role in the design and analysis of communication systems, especially in real time applications for which delay is critical. In this work, we analyse the completion time of data dissemination in a shared loss (i.e., unreliable links) multicast tree, at the limit of large number of nodes. Specifically, analytic expressions for upper and lower bounds on the expected completion time are provided, and, in particular, it is shown that both these bounds scale as $\alpha \log n$. For example, on a full binary tree with $n$ end users, and packet loss probability of $0.1$, we bound the expected completion time for disseminating one packet from below by $1.41 \log_2 n+o \left( \log n \right)$ and from above by $1.78 \log_2 n+o \left( \log n \right)$. Clearly, the completion time is determined by the last end user who receives the message, that is, a maximum over all arrival times. Hence, Extreme Value Theory (EVT) is an appropriate tool to explore this problem. However, since arrival times are correlated, non-stationary, and furthermore, time slots are discrete, a thorough study of EVT for Non-Stationary Integer Valued (NSIV) sequences is required. To the best of our knowledge, such processes were not studied before in the framework of EVT. Consequently, we derive the asymptotic distribution of the maxima of NSIV sequences satisfying certain conditions, and give EVT results which are applicable also beyond the scope of this work. These result are then used to derive tight bounds on the completion time. Finally, the results are validated by extensive simulations and numerical analysis.