The churn rate of a peer-to-peer system places direct limitations on the rate at which messages can be effectively communicated to a group of peers. These limitations are independent of the topology and message transmission latency. In this paper we consider a peer-to-peer network, based on the Engset model, where peers arrive and depart independently at random. We show how the arrival and departure rates directly limit the capacity for message streams to be broadcast to all other peers, by deriving mean field models that accurately describe the system behavior. Our models cover the unit and more general k buffer cases, i.e. where a peer can buffer at most k messages at any one time, and we give results for both single and multi-source message streams. We define coverage rate as peer-messages per unit time, i.e. the rate at which a number of peers receive messages, and show that the coverage rate is limited by the churn rate and buffer size. Our theory introduces an Instantaneous Message Exchange (IME) model and provides a template for further analysis of more complicated systems. Using the IME model, and assuming random processes, we have obtained very accurate equations of the system dynamics in a variety of interesting cases, that allow us to tune a peer-to-peer system. It remains to be seen if we can maintain this accuracy for general processes and when applying a non-instantaneous model.
The most closely related work is that of Yao, Leonard et. al. @cite_6 . They model heterogeneous user churn and local resilience of unstructured P2P networks. They also concede early that balancing model complexity and its fidelity is required to make advances in this area. They examine both the Poisson and Pareto distribution for user churn and provide a deep analysis on this front. Their work focuses on how churn affects connectivity in the network and we have separated this aspect from our work and concentrated on message throughput.
@cite_6
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Previous analytical results on the resilience of un-structured P2P systems have not explicitly modeled heterogeneity of user churn (i.e., difference in online behavior) or the impact of in-degree on system resilience. To overcome these limitations, we introduce a generic model of heterogeneous user churn, derive the distribution of the various metrics observed in prior experimental studies (e.g., lifetime distribution of joining users, joint distribution of session time of alive peers, and residual lifetime of a randomly selected user), derive several closed-form results on the transient behavior of in-degree, and eventually obtain the joint in out degree isolation probability as a simple extension of the out-degree model in [13].