# Measurements, Analysis, and Modeling of BitTorrent-like Systems

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11 number of number of raw data linear fit Probability of T sd > t Probability of T sl > t peers (a) For all peers in the trace 1 4 peers (b) For peers born in the middle of trace collection time seeding time (hour) (a) The probability distribution of seeding time sleeping time (hour) (b) The probability distribution of sleeping time Figure 12: Torrent involvement of peers (x-axis is in log scale) Figure 13: The seeding time and sleeping time of peers (y-axis is in log scale) peer joins is: m = k=1 kp k 1 (1 p) = 1 1 p. (4.13) Figure 12(a) shows the distribution of number of torrent files that each peer downloads in the trace. The curve in the figure is a little convex, deviating from what Equation 4.12 predicts (a straight line when x-axis is in log scale). The reason is that the number of joined by peers born before the trace collection is under-estimated, since some of these requests cannot be recorded in the trace. A similar situation exists for peers that are not dead before the end of trace collection. Figure 12(b) shows the distribution of number of joined by each peer that was born in the middle of the trace collection duration (indicated by raw data) and our linear fit. The curve fits Equation 4.12 very well, and we estimate from the analysis that p.8551, while the average number of each peer joins is about To verify the probability model we use in the above analysis, we estimate p in another way as follows. Assuming the peer birth rate is λ p and the torrent request 1 rate is λ q, since each peer joins 1 p during its lifetime in average, we have λ q = 1 1 p λ p. (4.14) Based on the peer request arrival rate and the peer birth rate we derived before (see Figure 1(b) and 1(c)), we have p = 1 λp λ q = This is very close to the value we got from Equation 4.12,.8551, meaning that there are more than 85% peers joining multiple. Having characterized the torrent request pattern of peers, finally we consider the distribution of the seeding time and the sleeping time of peers. According to our fluid model, 1 γ represents the average seeding time. Figure 13(a) and 13(b) show the probability distribution functions of the peer seeding time and the peer sleeping time in the system. Note that the y-axis is in log scale. Both the peer seeding time and sleeping time roughly follow the exponential distribution with probability den- τ sd, and f sl (t) = 1 sity function f sd (t) = 1 respectively. Based on the trace analysis, we estimate τ sd = 1 γ = 8.42 hours, and τ sl = hours. τ sd e t τ sl e t τ sl, 4.2 Characterizing the Inter-Torrent Relation In this part we study how different are connected through peers that download multiple files, based on our previously verified assumptions. For simplification, we consider a homogeneous multitorrent environment where all and peers have the same λ, τ, µ, c, η, γ, and average sleeping time. We denote each torrent in the system as torrent i with birth time t i (1 i < ). For any two that are born successively, torrent j first born and torrent i born next, we have i = j + 1 and t i > t j. Assume the probability that a peer selects torrent i at time t as its k-th torrent is Pi k(t ), Pi k(t ) = when t i > t. We also denote Pi 1(t ) as P i (t ) for simplicity. Without loss of generality, we assume that the most recently born torrent by time t is torrent 1, and P i (t ) satisfies e t t i τ P i (t ) =, (4.15) t j=1 e t j τ where t j = t j λ t, 1 j <. Thus, we have P i (t ) = e i λ t τ j j=1 e λ t τ = (e 1 λ t τ 1)e i λ t τ = (e 1 λ t τ 1)e t t i τ. (4.16) For a peer that requests its k-th torrent file, the peer does not select the that it has requested. Assuming P k i (t ) = α k P i (t ), (4.17)

12 out degree (trace) out degree (model) torrent size in degree (trace) in degree (model) torrent size weighted out degree torrent size weighted in degree torrent size (a) Weighted out-degree (b) Weighted in-degree Figure 14: The inter-torrent relation (y-axis is in log scale) the peer arrival rate of a torrent can be expressed as λ(t) = αλ q P i (t ) = α 1 p λ p(e 1 λ t τ 1)e t t i τ, (4.18) where α = k=1 α kp k 1 (1 p). When λ t r, we have α k 1 and α 1. Comparing Equation 3.1 with 4.18, we have λ = α 1 p λ p(e 1 λ tτ 1). Considering that a peer in a torrent may have downloaded files from other, we can model the relationship among different in the P2P system as a directed graph. Each node in the graph represents a torrent. A directed edge from torrent i to torrent j denotes that some peers in torrent i have downloaded the file from torrent j, and thus have the potential to provide service to peers in torrent j, even though they are not in torrent j currently. The weight of the directed edge W i,j represents the number of such peers. For simplicity, we define W i,i =. The graph changes dynamically over time. Now let us consider the graph at time t. During time [t, t + dt], t j t < t, there are λ(t)dt peers who joined torrent j. Let k(t) = r(t t). During time [t, t ], these peers can download up to k(t) 1 completely in addition to torrent j and may request (or be requesting) the next torrent at time t. If torrent i is not requested before the last requests during time [t, t ], the probability of such events is k(t) 1 Q i (t) = p p (1 α l P i (t + l )). (4.19) r l=1 When i j, we have W i,j = t t j λ(t)dt Q i (t) α k(t) P i (t+ k(t) ). (4.2) r Therefore, the weighted out-degree of torrent i represents the total potential capability its peers can provide to peers in other, denoted as SP i, where SP i = W i,j. (4.21) j=1 Correspondingly, the weighted in-degree of torrent i represents the total potentials its peers can get from peers in other, denoted as SG j, where SG j = W i,j. (4.22) i=1 Figure 14(a) and 14(b) show the weighted out-degree and weighted in-degree at a time instant based on trace analysis and our probability model, respectively. In the figures, each point in the x-axis denotes a torrent, sorted in non-ascending order of weighted out-degree or weighted in-degree. The right y-axis in the figures denotes torrent size, the number of peers in the torrent at this time instant. In general, with more peers tend to have large out-degree and in-degree. The weighted out-degree and in-degree distribution according to our trace analysis follows power law rules roughly. It deviates from our model somewhat because of the heterogeneity of in the real system. In the multi-torrent environment, old peers that had downloaded the file from a torrent may come back for other torrent files, and the lifespan of this torrent can be extended if these old peers are willing to provide service. Assume the request arrival rate of this torrent is λ(t) and λ(t) = when t <. If we consider both new requesting peers and old returning peers, the peer arrival rate of the torrent is λ (t) = k(t) l= pl λ(t l r ) = k(t) l= pl λ e t l r τ = λ e t τ qk(t)+1 1 q 1, where k(t) = rt and q = pe 1 rτ trace analysis). (4.23) (q > 1 based on our

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