Bandwidth Modeling and Estimation in Peer to Peer Networks

Napster [2] was the first popular peer to peer service. This service has allowed hundreds of thousands of users to efficiently share MP3 formatted files. The success of Napster was a big motivation and several other peer to peer file sharing systems were introduced. These include KaZaA [3], Gnutella [4], and eDonkey [5] and BitTorrent [6].

According to Cache Logic [7] by the end of 2004, BitTorrent accounted for as much as 30% of all Internet traffic. Peer to peer represented 60% of Internet traffic at the end of 2004. Most dominant P2P * Mehdi Vasef is currently a graduate student at University of Duisburg-Essen, Duisburg, Germany systems are BitTorrent, eDonkey and Gnutella. The number of people that use peer to peer file sharing is growing. In January 2005, 2,975,477 online eDonkeyy2k users were reported. In January 2006, the number was increased to 3,351,754 [8].

The number of P2P users , the average file size transported in P2P file sharing systems and percentage of overall network traffic include P2P network traffic are growing, so bandwidth management plays an inevitable role in designing efficient computer networks nowadays. We still do not have accurate models for P2P bandwidth. In this paper, we propose a novel model for modeling P2P bandwidth. To achieve this goal, we should consider both the customers and the share of these customers. The customers are peers and the shares are their traffics in a given network. The second part was accomplished in the authors’ previous work. So our starting points deeming the behavior of peers and pondering it. We use OU type process to describe the peer behavior. We then proceed to model the P2P bandwidth. We use the stochastic calculus approach in our proposed model. Some statistical parameters of the bandwidth model are derived. We also present a model for the total bandwidth .It is shown that the total bandwidth asymptotically has a LRD property. We also derive the length of the buffer fed by the bandwidth. A method for generating synthetic bandwidth is presented. Estimation of bandwidth parameters is another contribution of this paper.

More accurately the contributions of this paper are:

• Modeling the P2P bandwidth and obtaining some parameters such as mean and variance • Modeling total P2P bandwidth and obtaining some parameters such as mean and variance • Study of autocovariance in bandwidth model both in individual and total case and Showing that the total bandwidth is asymptotically a Long Range Dependent process • Obtaining Length of queue fed by a bandwidth process • Synthetic generation of bandwidth process • Estimating bandwidth model parameters using MLE method

The rest of paper is organized as follows. In section 2, we review related works. In section 3, we introduce the mathematics required to understand the model. In section 4, we explain our proposed model. Queuing analysis of the proposed model, synthetic bandwidth generation and the estimation of bandwidth parameters are also studied in section 4. Finally section 5 concludes the paper.

To best of our knowledge the mathematical model for P2P bandwidth is an unexplored area in the literature. The next paragraphs illustrate the related works that have been done in P2P traffic modeling, modeling the behavior of peers and their impact on the P2P system properties. [9] proposes a stochastic differential equation approach for modeling the behavior of Peers. The steady state behavior of the peers is investigated. But the bandwidth model is not studied. [10] Proposes a fluid model that characterizes the number of peers in BitTorrent like networks. The arrival and departure processes for sinks and sources are Poisson processes. Another assumption is that all the peers have the same uploading/sinking bandwidth. [11] Proposes a fluid model for BitTorrent system but in comparison to [10], the peer arrival rate is exponential. Their finding is that the existing BitTorrent system provides poor service availability, fluctuating sinking performance, and unfair services to peers. Their model has revealed that these problems are due to the exponentially decreasing peer arrival rate. A stochastic fluid model [12] is proposed to study performance of peer to peer web cache (SQIRREL) and cache cluster by extending [10]. The [13,14] use Markovian model to describe P2P file sharing system .But they don’t work well in some P2P system such as BitTorrent. In addition the steady state behavior of peers is not studied.

Recently [15] have proposed a model for P2P traffic. The model is based on Alternating Fractal Renewal Process (AFRP) such that the ON/OFF periods are k

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