# ARTICLE IN PRESS. Journal of Network and Computer Applications

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5 D. Wu et al. / Journal of Network and Computer Applications 32 (2009) The set of VSs hosted by node i: hi 1 ; i 2 ;...; i m i, where i k refers to the k-th VSs hosted by node i, k ¼ 1::m. The load vector of VSs hosted by node i: hl i1 ;...; L im i, where L ik refers to the load on VS i k, k ¼ 1::m. Let L i be the total load on the node i, L i ¼ P m k¼1 L i k. The node s IP address. For a small system with only one group G, the load information of each peer can be delivered quickly to all members within a short period. After that, each peer has the load information of all the other peers, and knows the total capacity C G and the total load L G of the whole system by C G ¼ P i2g C i and L G ¼ P i2g L i, respectively. Thus, the system utilization m can be easily calculated by m ¼ L G =C G. For a single node i, its utilization is given by m i ¼ L i =C i. Based on the ratio between m i and m, each node can be classified into three types: heavy node, if m i 4m; light node, if m i om and normal node, if m i ¼ m. In practical implementation, as m and m i are real numbers, we will round the value of m and m i before comparing their value. For example, if m equals to 0.67, it will be rounded to 0.7. Similarly, all the m i in the range of [0.65, 0.75) will be rounded to 0.7 and identified as normal nodes. Another possible approach is to use a range for the identification of heavy node, light node and normal node. For instance, when 0:9mom i o1:1m, that node is classified as normal node ; when m i p0:9m, it is classified as light node ; when m i X1:1m, it is classified as heavy node. In case of a large system, there exist multiple groups. Each node i should run at least two VSs (see Fig. 2). i 1 Primary Server of node i, which is with the identifier id 1 generated by hashing the node s IP address i 2 Secondary Server of node i, which is with the identifier id 2 generated by hashing id 1. Each node only publishes load information via its primary server. During load reassignment, the primary and secondary server cannot be moved to other nodes, but their size can be changed. Group T Group S i1 i2 Fig. 2. Primary server and secondary server.

9 D. Wu et al. / Journal of Network and Computer Applications 32 (2009) Table 1 Notations Symbol N g c W GMS T u N max g max g min l FðxÞ R p real Definition The system size, i.e., the total number of nodes The group size The ratio between the group size g and the number of groups N=g The message traffic that a node experiences per interval The load information update period The maximum possible system size The threshold of maximum group size The threshold of minimum group size The arrival rate of peers CDF of node lifetime distribution The residual lifetime of a node The probability that a peer is a real live peer 4. Performance analysis In this section, we perform an analytical study on the scalability and efficiency of GMS. For convenience, the related notations are summarized in Table Scalability The main concern is the scalability of GMS, which determines the applicability of GMS in real large systems. Define g to be the group size and N to be the system size. In case of a small system with a few thousands of peers, one group is formed and the group size g equals N. The message traffic that a node (including virtual node) experiences per interval is given by W GMS Oðg=T u Þ¼OðNÞ. For example, given the update period T u ¼ 60 s, N ¼ 1000 and message size ¼ 30 bytes, W GMS is about 0:5k bytes=s. If a node hosts multiple VSs, the traffic increases proportionally to the number of VSs. However, the traffic is still rather low and tolerable for normal users. Even for a powerful node with 20 VSs, the message traffic is about 10k bytes=s. To limit the traffic towards a node, each node can maintain a threshold to limit the maximum number of VSs. In a large system, multiple groups are formed. Define p c to be the ratio between the number of nodes in a group and the number of groups, we have g ¼ ffiffiffiffiffiffi p cn ffiffiffiffi. In case that c5n, the message traffic for a peer per interval is given by W GMS Oðg=T u Þ¼Oð N Þ. Given the update period Tu ¼ 60 s, N ¼ 250; 000, c ¼ 4ðc5NÞ and message size ¼ 30 bytes, W GMS is about 0:5k bytes=s, which is comparable to the small system. In our approach, c is an important tuning parameter. In order to guarantee efficient information dissemination between groups, c cannot be too small, and normally we set p cx5. As c ¼ g 2 =N, we can guarantee the above requirement by keeping the group size g bigger than ffiffiffiffiffiffi cn. Therefore, given the maximum possible system size N max, we can define two thresholds: the minimum group size g min, which is to guarantee a bigger c; and the maximum group size g max, which is to limit the message traffic towards a node. In the evolution process of the system, the groups are split only when the group size is beyond the threshold g max. The group split does not require global cooperation between nodes. As each node knows exactly the same number of group members due to information dissemination, when the group size is beyond g max, it will increase its GroupID by one bit locally. With the changes happening in each node, the system is reorganized into more groups. When the group size is below the threshold g min, the merging of groups is incurred by self-adaptation similarly. Each node decreases the length of GroupID by one bit. The process is stopped when the length of GroupID reaches zero, which means there exists only one group in the system.

11 D. Wu et al. / Journal of Network and Computer Applications 32 (2009) ti -Tu New Arrival Peers X t i Tu R T R L Y1 Update Time Y2 Arrival Time Y3 Fig. 3. Peer arrival and departure. For Pareto lifetime distribution, we have F R ðxþ ¼1 ð1 þðx=bþþ 1 a. Due to the randomness, the update time T of a peer since t i T u can be assumed to follow a uniform distribution in ½0; T u Š, with the PDF given by f ðxþ ¼1=T u. Define the probability that the peer s residual lifetime R is less than its first update time T as p Y1, then p Y1 can be calculated by p Y1 ¼ PðRoTÞ ¼ Z Tu 0 PðRoxÞf ðxþ dx ¼ 1 Z Tu F R ðxþ dx T u Under Pareto lifetime distribution, we have " b p Y1 ¼ 1 1 þ T # 2 a u 1 ð2 aþt u b 0 From the above, we can deduce jy 1 j¼m p Y1 and obtain M 0 accordingly. Thus, we have p real ¼ M 1 1 M 0 ¼ ¼ & 1 þ lt u p Y1 ðða 1Þ=bÞT u þðb=ð2 aþþt u ðð1 þðt u =bþþ 2 a 1Þ To get a better understanding, we take a real DHT-based Peer-to-Peer file-sharing system Kad (Emule project, 2006) as a case for study. According to the measurement results in Stutzbach and Rejaie (2005), the peer lifetime distribution in Kad follows a heavy-tailed distribution with the expectation of 2.71 h, which can be approximated by a Pareto distribution with a ¼ 2:1; b ¼ 3:0. Based on Theorem 4.1, we get the numerical results of p real under different update periods (as shown in Table 2). It can be observed that, by tuning the update period to be smaller, we can keep the value of p real very high. In a group with 1000 nodes, by setting T u ¼ 60 s, there are less than 4 faked live peers in the balancing set on average. In the real environment, if the update period is properly configured, the impact of churn will not be a big issue for GMS. 5. Experimental evaluation 5.1. Methodology We evaluate the performance of GMS on top of a discrete-event Chord simulator, which is enhanced to support VSs. In our experiments, we simulate a system with up to 8192 nodes, which is bigger than that in the previous simulation settings (e.g., 4096 nodes in Godfrey et al., 2004). The

14 58 D. Wu et al. / Journal of Network and Computer Applications 32 (2009) intra-group load balancing, GMS also needs to perform inter-group balancing in order to balance the whole system Impact of system size Although the scalability of GMS has been analyzed in Section 4, we further validate our results by simulation. In the experiment, the system size is varied from 500 to We measure the average message overhead per VS. From Fig. 7, we can observe that the message overhead per VS is less than 0:5k bytes=s most of the time, even with the increasing of the system size. It is because that, when the group size reaches a threshold, the group will be split to keep the traffic low. The effect of group splitting can be found in the figure. The message overhead increases continuously with the increasing of system size, but it drops down when reaching a threshold. Percentage of ill requests 100% 90% 80% 70% 60% 50% 40% 30% 20% Proportion ISS RDS GMS 10% 0% Time (minutes) Fig. 6. Percentage of ill requests under workload shift. 3 Message overhead per node (kbps) System Size Fig. 7. Message overhead per virtual server under different system size.

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