Sampling cluster endurance for peer-to-peer based content ...unstructured P2P systems have been...

Multimedia Systems (2007) 13:19–33DOI 10.1007/s00530-007-0078-9

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Sampling cluster endurance for peer-to-peer based contentdistribution networks

Vasilios Darlagiannis · Andreas Mauthe ·Ralf Steinmetz

Published online: 10 March 2007© Springer-Verlag 2007

Abstract Several types of Content Distribution Net-works are being deployed over the Internet today, basedon different architectures to meet their requirements(e.g., scalability, efficiency and resiliency). Peer-to-peer(P2P) based Content Distribution Networks are promis-ing approaches that have several advantages. StructuredP2P networks, for instance, take a proactive approachand provide efficient routing mechanisms. Nevertheless,their maintenance can increase considerably in highlydynamic P2P environments. In order to address thisissue, a two-tier architecture called Omicron that com-bines a structured overlay network with a clusteringmechanism is suggested in a hybrid scheme.

In this paper, we examine several sampling algorithmsutilized in the aforementioned hybrid network that col-lect local information in order to apply a selective joinprocedure. Additionally, we apply the sampling algo-rithms on Chord in order to evaluate sampling as a gen-eral information gathering mechanism. The algorithms

The original paper with the same title has been published in theproceedings of MMCN’2006. This is an extended versionincluding further results invited for submission at ACM/SpringerMMSJ.

V. Darlagiannis (B)Swiss Federal Institute of Technology (EPFL),1010 Lausanne, Switzerlande-mail: [email protected]

A. MautheComputing Department, Lancaster University,Lancaster, LA1 4YR, UKe-mail: [email protected]

R. SteinmetzMultimedia Communications (KOM), Technische UniversitätDarmstadt, Merckstr. 25, 64283 Darmstadt, Germanye-mail: [email protected]

are based mostly on random walks inside the overlaynetworks. The aim of the selective join procedure is toprovide a well balanced and stable overlay infrastructurethat can easily overcome the unreliable behavior of theautonomous peers that constitute the network. The sam-pling algorithms are evaluated using simulation exper-iments as well as probabilistic analysis where severalproperties related to the graph structure are revealed.

1 Introduction

Content Distribution Networks (CDNs) are used todeliver content to potentially very large user popula-tions [1]. Within the Internet, different types of CDNsare being envisaged ranging from simple Web basedapplications to sophisticated multimedia entertainmentsystems, including interactive systems such as multiplay-er games and virtual environments. Whereas first gen-eration CDNs have mostly focused on Web content, thecurrent, second generation systems also deal with mediadelivery.

A special, very successful type of CDNs are peer-to-peer (P2P) file sharing systems. In 2001 for instance,Napster was the fastest growing application in theInternet’s history [2]. Since Napster, a number ofunstructured P2P systems have been developed such asGnutella [3], eDonkey [4], as well as structured approa-ches such as Chord [5], CAN [6], etc. Structured P2P net-works aim in maintaining a topology based on explicitrules (e.g., a hypercube) while unstructured are morefreely evolving network structures (e.g., power-law net-works). What these systems share is the idea to haveindependent, collaborating nodes that organize andshare information in a peer-to-peer fashion. Ideally for

20 V. Darlagiannis et al.

large expandability and freedom in interactivity, thereis no central instance that polices or governs the inter-action between the peers as is the case in client-serverinteraction. The P2P paradigm basically states that P2Psystems are self-organizing systems consisting of equal,autonomous entities where the interaction is governedby rules. However, in reality P2P systems are composedof peers with heterogeneous characteristics and userbehavior [7].

This paradigm can be applied to a multitude of struc-tures and systems. Apart from file sharing, P2P mecha-nisms are also proposed for media (mainly video)streaming [8–11]. Here, the focus is on the streamingof media from multiple senders to one receiver exploit-ing certain media properties (e.g., layered video coding).P2P structures are also being used for the transmission ofmedia to multiple receivers, as in the case of applicationlevel multicast. A single tree approach is for instancetaken in PeerCast [12] and SpreadIT [13]. In order toachieve better load balancing and improve resilience tonode failures, multiple multicast trees are employed inthe case of P2PCast [14] and SplitStream [15]. Otherquestions that are being addressed in the context of P2Pbased content distribution networks is the replication offiles on a large set of peers. This has been labeled Qual-ity of Availability (QoA) and defines a metric for theavailability of certain content items within the system[16]. BitTorrent [17] is one of the most popular systemsusing replication on a wider scale. Though it uses a cen-tral instance (i.e., a web-service to redirect the client tothe tracker) the exchange and organization of contentis essentially P2P. FastReplica [18] is another replica-tion system for large scale replication that uses a centralinstance, comparable to the control of surrogate serversin the case of Content Distribution Interworking (CDI).

The problem most P2P based CDNs encounter is themultitude of requirements placed on them. Hence, anumber of solutions are very restricted in their approachconcentrating on a (sub-)set of the critical requirements.However, this only provides a solution for specific casesand thus, they cannot be applied more widely. In orderto build more generic CDN infrastructures based on P2Pprinciples that maintain the advantages of P2P (such asflexibility and dynamicity), it is necessary to take certainrequirements into account. Such a system has to be, forinstance, able to cope with the inherent heterogeneityof peers since a common denominator approach wouldmake it very inefficient. Further, it has to provide scala-bility, be incrementally expandable and dependable inits service. It should also balance the load of requestsin a way that no hot-spots occur. The aim is to developprinciples and methods that can be used to build P2Pbased content infrastructures that can be used in a mul-

titude of environments and cases. Eventually, the goalis to create a generic infrastructure that can support allkinds of multimedia applications, including content pro-duction and delivery networks, interactive multimediaapplications, multiplayer games, etc.

This paper elaborates on a particular issue in design-ing overlay networks, the network stability issue. Sincepeers are autonomous entities, they dynamically partic-ipate in the constructed network by joining and leaving.In fact, several empirical observations of the uptime dis-tribution (c.f. [19,20]) indicate that the majority of peersdo not stay connected for long time periods. Therefore,structured approaches utilizing proactive mechanisms inorder to provide efficient routing mechanisms, requiresignificant signaling to maintain the targeted topologyand update the indexing data on the advertised content.The required information exchanged in this process canbe further increased if the proactive design of the sys-tem replicates the content, too. Omicron (OrganizedMaintenance, Indexing, Caching and Routing for Over-lay Networks) [21] addresses this issue with a two-tierarchitecture combining a structured approach with aclustering mechanism. Omicron constructs clusters ofpeers that (as a set) form reliable components to developa stable structured overlay.

In order to do so, new peers perform a selective joinmechanism, so that the resulting joint reliability of eachcluster is above a minimum threshold. Generally defin-ing, selective join is the procedure of sampling a numberclusters gathering their properties and then selecting tojoin the cluster that meets better the predefined require-ments. The joint reliability of a cluster is called endur-ance to reflect the differences from single peer reliabilityand network stability. The selective join mechanism isbased on random sampling to select a subset of clus-ters in order to decide which one is the weakest fromthis subset. Over time, this has the effect of strengthenthe weakest. The sampling technique is implementedby simply initiating random walks into the network andcollecting the local properties as the messages are for-warded to their neighbors (messages increase in size oneach hop). Several algorithms have been investigated inorder to evaluate their performance on cluster coverageand the related properties.

Stable P2P networks provide the required infrastruc-ture for different kinds of CDNs to operate efficiently.Omicron’s design provides the additional aforemen-tioned requirements, such as being scalable, incremen-tally expandable and dependable, providing evenlydistributed workload properties, dealing adaptively withpotential hot-spots, supporting heterogeneous popula-tions, etc. However, sampling can be used as an effectivemechanism for gathering several types of information in

Sampling cluster endurance for peer-to-peer based content distribution networks 21

P2P systems. In fact, sampling can be effectively appliedin cases where probabilistic techniques can outperformtheir deterministic counterparts. Therefore, the evalu-ation of the proposed algorithms on other P2P net-works can provide generalized results of greater interest.Chord [5] is a well-known P2P network with appeal-ing scalability properties, which is used as a test-bedfor our algorithms in addition to the Omicron-basedexperiments.

This paper is organized as follows: In Sect. 2, an over-view of the Omicron network is provided focusing onthe graph structure, the clustering mechanism and theresulting architecture. The network management mech-anism dealing with peers joining the network is discussedin Sect. 3. The investigated sampling algorithms are pro-vided in Sect. 4, and the related simulation results aregiven in Sect. 5 together with probabilistic analysis ofthe most critical aspects. The related work is providedin Sect. 6. Finally, Sect. 7 summarizes the paper and givesan outlook for the future.

2 Omicron

Omicron is a P2P overlay network aiming to addressissues of heterogeneous, large-scale and dynamic P2Penvironments. Its hybrid, two-tier, DHT-based approachmakes it highly adaptable to a large range of appli-cations. Omicron deals with a number of conflictingrequirements, such as scalability, efficiency, robustness,heterogeneity and load balance. Issues to consider inthis context are:

Topology. The rational in Omicron’s approach is toreduce the high maintenance cost by having a smalland fixed node degree, thus, requiring small and fixedsize routing tables (at least for the majority of peers),while still performing lookup operations at low costs.For this reason, the usage of appropriate graph struc-tures (such as de Bruijn graphs [22], which are furtherdiscussed in Sect. 2.1) is suggested. However, while thesmall fixed node degree reduces the operational cost, itcauses robustness problems.

Clustering mechanism. To address the robustnessissue, clusters of peers are formed with certain require-ments on their endurance. The clustering mechanism isdescribed in deeper detail in Sect. 2.2.

Roles. A unique feature of Omicron is the inte-grated specialization mechanism that assigns particu-lar roles to peers based on their physical capabilitiesand user behavior. The specialization mechanism pro-vides the means to deal with peer heterogeneity. Thisscheme fits the contribution of each node to its resourcecapabilities and aims at the maximization of the cluster

efficiency by providing appropriate incentives to peersto take a certain role. As it can be observed fromOmicron’s name, four different core roles have beenidentified: Maintainers (M), Indexers (I), Cachers (C)

and Routers (R). Maintainers are responsible to main-tain the overlay network topology, while Indexers han-dle the relevant indexing structures. Routers forwardthe queries towards their logical destination, and Cach-ers reduce the overall routing workload by providingreplies to popular queries. Roles are additively assigned,meaning that peers do not remove their older roles asthey get new ones.

Identification scheme. Since Omicron is based onclusters, there is a need to identify both the clusters andthe peers. Therefore, a dual identification scheme is pro-posed to satisfy the identification requirements. Peersare distinguished by a (GUID) that is created usingsecure hash functions. Such peer GUIDs have constantlength. Clusters are distinguished by dynamically mod-ified GUIDs that follow a de Bruijn-like value assign-ment. The peculiarity of this scheme is that the length ofthe identifiers is adapted to the size of the graph. More-over, the length of de Bruijn identifiers may differ bymaximum one for neighbor nodes in order to fulfill therequirement of incremental expandability and of evenlydistributed workload. The benefits of this dual identi-fication scheme are multi-fold. Nodes can be uniquelyidentified and their actions may be traced when this isdesirable. Thus, peers are responsible for their actions. Inaddition, peers cannot “force” responsibilities for index-ing certain items since the mapping is not depending onthe peer GUID but on the cluster GUID. Further, neigh-bor selection is not strictly defined. Thereby, peers canselect their neighbors from neighbor clusters. This selec-tion may be based on network proximity, trust or otherspecific metrics.

2.1 de Bruijn Digraphs

Directed graphs (digraphs) have been extensively usedin interconnection networks for parallel and distributedsystems design (cf. [23,24]). Digraphs received specialattention from the research community aiming to solvethe problem of the so-called (k, D) digraph problem [25],where the goal is to maximize the number of vertices(order) N in a digraph of maximum out-degree k anddiameter D. Some general bounds relating the order,the degree and the diameter of a graph are providedby the well-known Moore bound [26]. Assume a graphwith node degree k and diameter D; then the maximumnumber of nodes (graph order) that may populate thisgraph is given by Equation 1:


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N ≤ 1 + k + k2 + · · · + kD = kD+1 − 1k − 1

. (1)

Interestingly, the Moore bound is not achievable forany non-trivial graph [26]. Nevertheless, in the con-text of P2P networks, it is more useful to reformulateEquation 1 in a way that provides a lower bound forthe graph diameter (DM), given the node degree and thegraph order [27]:

DM = �logk(N(k − 1) + 1)� − 1 ≤ D. (2)

The average distance (µD) among the nodes of a graphmay also be bounded by the following inequality [28](which is approximated by Loguinov et al. [29]):

DM − k(kDM − 1)

N(k − 1)2 + DM

N(k − 1)≈ DM − 1

k − 1≤ µD. (3)

An interesting class of digraphs is the so-called lexico-graphic digraph class [30], which includes the de Bruijnand Kautz digraphs.1 de Bruijn digraphs have asymp-totically optimal graph diameter and average node dis-tance [29]. Thereby, they are employed in the designof our work. de Bruijn graphs have been suggested tomodel the topology of several P2P systems, however,we exploited them in an innovative way. Considerableexamples of P2P systems that use de Bruijn graphs areKoorde [31], D2B [32] and Optimal Diameter RoutingInfrastructure (ODRI) [29].

Figure 1 shows a directed de Bruijn(2, 3) graph denot-ing a graph with a maximum out-degree of 2, where thediameter length is 3 and the graph order is 8. For graphswith fixed out-degree of 2, the maximum number ofnodes2 is always limited by 2D. The graph contains 2D+1

directed edges in this case. Each node is represented by

1 de Bruijn graphs are less dense than Kautz graphs but theyare more flexible since they do not have any limitations on thesequence of the represented symbols in every node.2 The Moore bound determines always maximum upper boundson the size of the graphs that are not reachable for non-trivialcases.

string of length D (D = 3 in this example). Every char-acter of the string can take k different values (2 in thisexample). In the general case, each node is representedby a string such as u1u2...uD. The connections betweenthe nodes follow a simple left shift operation from nodeu1(u2...uD) to node (u2...uD)ux, where ux can take oneof the possible values of the characters (0, k − 1). Theshifted-in character determines the selected neighbor tofollow in the routing procedure. The solid lines in thefigure denote links where the ‘0’ character is shiftedin, while the dotted lines denote links where the ‘1’character is shifted in.

2.2 Clustering

Clusters have been introduced into the design of P2Psystems in a variety of approaches. JXTA defines theconcept of PeerGroups [33] to provide service compat-ibility and to decompose the large number of peers intomore manageable groups. Further, SHARK [34] clus-ter peers based on the common interests of users. Also,Considine [35] proposes multiple cluster-based overlaysfor Chord. The cluster construction in the latter proposalis based on network proximity metrics aiming to reducethe end-to-end latency. Furthermore, even hierarchicalapproaches like eDonkey and KaZaA might be consid-ered as clustering approaches to a certain extent, wherenormal peers are clustered around the super-peers. Thepurpose of this “clustering” is to transform the costlyall-to-all communication pattern into a more efficientscheme. However, by doing so it introduces additionalload-balancing concerns. In fact, this is a more generalissue that appears in every acyclic hierarchical organiza-tion (i.e., tree-like organization). Thus, the cluster orga-nization must be restricted to non acyclic structures inorder to provide even distribution of responsibilities.

A desirable property of each overlay network isacquiring a topology that remains as stable as possibleover time and minimizes the related required communi-cation cost to maintain the targeted structure. However,in highly dynamic P2P systems that consist of unreli-able peers, perfect stability cannot be attained. This isthe most crucial motivating factor for introducing theconcept of clusters in the architectural design of theOmicron overlay network. Clusters can be consideredas an essential abstraction, which can be used to absorbthe high peer attrition rate and accomplish high net-work stability. They can be considered as an equivalentmechanism to the suspensions used in vehicles to absorbshocks from the terrain. In order to make more clear theinvolved concepts, it is required to define peer reliabil-ity, network stability and cluster endurance. We definenetwork stability as follows:


Definition 2.1 Network stability SN(t) is the probabilitythat the topology of the network remains unmodifiedfor some time t.

A definition for peer reliability is given below.

Definition 2.2 Peer reliability RP(t) is the probabilitythat the peer remains connected for some time t.

Assuming that the lifespan of a peer is modeled withthe random variable X, then the reliability of the peeris given by:

RP(t) = Pr{X > t} = 1 − F(t). (4)

where Pr{·} denotes the probability statement and F(t) isthe Cumulative Distribution Function (CDF) of peer’slifetime over the set of peers active in the system at anygiven moment. This distribution is different from theone used to describe the peers’ lifetime as they arrive inthe system This is due to the fact that longer-lived peersspend more time in the system and therefore make up alarger fraction of (currently) active peers.

On the other hand, a cluster is a virtual entity com-posed by several peers. We define the endurance of acluster as follows:

Definition 2.3 Cluster endurance EC(t) is the probabil-ity that at least one peer of the cluster will remain con-nected for some time t.

The endurance of clusters is calculated by the follow-ing equation.

EC(t) = 1 −K∏

i=1

(1 − Fi(t)), (5)

where K is the size of the cluster and Fi(t) is the CDF ofthe ith peer in the cluster.

Clustering algorithms aim mainly at “partitioningitems into dissimilar groups of similar items”. Theyrequire the definition of a metric to estimate the sim-ilarity of the items in order to perform the partitioningprocedure. Since an overlay network is a virtual net-work, there is a lot of freedom in defining the optimalpartitioning metric. In the context of P2P overlay net-works, the similarity of the peers forming an individualcluster is that all the members are responsible for thesame part of the address space, which does not neces-sarily define a meaningful metric for assigning peers toclusters. Therefore, the proposed clustering algorithmrequires criteria other than the usual similarity metrics.The key factors that motivate the construction/decon-struction of clusters are the following:

• Clusters should fulfill the endurance requirements.• Clusters should have the smallest possible size in

order to reduce the intra-cluster communicationcomplexity.

• Clusters should be divided when it is possible tocreate d other endurable clusters, where d is thedegree of the employed de Bruijn graph. Selectivedivision should be applied to maximize the endur-ance of each new cluster.

• Clusters should be merged when their estimatedendurance is lower than a predefined endurancethreshold.

Moreover, a hysteresis-based mechanism is requiredto avoid oscillations in splitting and merging clusters.In order to describe the membership of peers in theclusters, the ClusterMap concept has been used. A Clus-terMap includes the peers participating in a cluster. Clus-terMaps may be realized as tables collecting entriesfor each member peer. Every entry may hold infor-mation about peers’ GUID, their role in the system,the observed reliability and other useful informationthat could be used by every peer of a cluster to effec-tively construct its local routing table. In fact, Clus-terMaps are supersets of Routing Tables, including thepotential peers of clusters that may become neighborsof a particular peer. ClusterMaps are periodically dis-seminated to neighbor clusters as well as the clusteritself.

2.3 Two-tier network architecture

The suggested two-tier network architecture is a majorstep towards accomplishing the fulfillment of thetargeted requirements. It enables the effective usageof de Bruijn graphs by successfully addressing theirshortcomings, such as inflexible network expandabil-ity and low network resilience for low node degree. Infact, the successful “marriage” of two different topologydesign techniques (in a combination of a tightly struc-tured macro level and a loosely structured micro level)provides a hybrid architecture with several advantages.

1. Tightly structured macro level. Adopting the topo-logical characteristics of de Bruijn graphs, the macrolevel is highly symmetrical enabling simple routingmechanisms. Composed of endurable components,it results in a relatively stable topology with smalldiameter and fixed node degree.

2. Loosely structured micro level. On the other hand,the micro level provides the desirable characteris-tics to the macro level by following a more looselystructured topology with a great degree of freedom


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Fig. 2 Omicron overlay network

in the neighbor selection. This freedom may beinvested on regulating and achieving a finer loadbalance, offering an effective mechanism to handlepotential hot spots in the network traffic. Moreover,locality-aware neighbor selection may be used tomaximize the matching of the virtual overlay net-work to the underlying physical network. Finally,redundancy may be developed in this micro levelsupplying seamlessly fault-tolerance to the macrolevel.

An example of the hybrid topology is illustrated inFig. 2. The structured macro level is a de Bruijn(2, 3)

digraph. Two nodes (representing peer clusters) are“magnified” to expose the micro level connectivity pat-tern between them. Two different connection types areshown: inter-cluster connections and intra-cluster con-nections. Further information (e.g., on the relevant rout-ing mechanism) can be found at [7].

3 Overlay network management

In the resulting two-tier architecture, two entities areused to construct the network topology: individual peersand clusters of peers. Thereby, a set of efficient proce-dures need to be defined to handle the dynamic par-ticipation of the peers in the system and the resultingconsequences in both the endurance and the mainte-nance cost of clusters. Three crucial requirements aredriving the developed solutions: dependability, load-balance and efficiency.

When new peers request to join an Omicron-basedP2P system, Maintainers perform a number of oper-ations in order to place the new peers in the network.The purpose of their actions is to achieve a well balanced

topology where clusters have sufficient endurance andthe total workload is minimized and well distributed.

Obviously, the optimal selection can be made whenthe endurance of every cluster of the network is globallyknown (or at a particular central entity). However, sucha solution raises scalability issues as the size of the net-work increases considerably. Therefore, an alternativeapproach has been investigated where Maintainers per-form a random walk collecting the endurance of eachcluster in the path in order to decide which one is thebest selection to direct the new peer to. In addition to therandom walk based solution, we have considered a pub-lish/subscribe mechanism where low endurance clustersbecome publishers. In this case, all the clusters of thenetwork should subscribe to these events, which makesthe solution unscalable. An advanced source to destina-tion assignment will increase significantly the complex-ity of the solution in contrast to the elegant probabilisticapproach of random walks.

A variety of bootstrap phases may be assumed, pro-viding an initial online peer PY that triggers the mech-anism to accept the newly joining peer PX . Withoutloss of generality it can be assumed that PY has beenassigned the Maintainer role (otherwise the request hasto be simply redirected to another peer PY of the samecluster that has been assigned the Maintainer role).

Upon the reception of the joining request PY trig-gers a sampling procedure using an inter-cluster randomwalk. It contacts a Maintainer PZ of a randomly selectedneighbor cluster, which is recursively repeating this stepmaking a random walk of length w = α · log(CS), whereCS is the number of clusters in the system and α is aweight. It should be noted that w is asymptotically equalto the diameter of the inter-cluster overlay network. Thegoal of the procedure is to equally distribute the newpeers in the deployed clusters considering the internalstate of each cluster, i.e., its endurance and its size. Byperforming a random walk of length at least equal to thediameter of the network, every cluster has a non-zeroprobability of being included in the sampling procedureof each join request. This probability is related to thecluster location in the overlay network. Moreover, hav-ing a logarithmic number of samples provides a fairlygood approximation of collecting the state of all clus-ters, which otherwise, it would have been very costlyin terms of communication traffic to obtain accurately.Thus, the selected approach can provide a well-balancedoutcome with a low cost. Finally, as it will become moreclear in Sect. 5.3, such sampling based on random walksof logarithmic length has statistical properties similar toindependent sampling.

After performing the sampling procedure with therandom walk, a suitable cluster is selected for joining. In


PXPY

Join

PZ...

(Sampling procedure via random walk)

Accept

ConnectUpdateClusterMap

Fig. 3 Join sequence diagram

this phase, a newly joined peer is considered unreliable,and it is merely assigned the Router (and optionally theCacher) role. Thus, the selected cluster is the one withthe least number of unreliable peers so that the load forthe Maintainers of the clusters is fairly equal. The Main-tainer of the last visited cluster included in the randomwalk indicates to the newly joined peer the selected clus-ter of which it should become a member of (i.e., via anAccept message). Afterwards, PX asks the providedMaintainer of the target cluster to connect and becomea cluster member. As a reply, the Maintainer providesupdated ClusterMap structures of the cluster itself andthe neighbor clusters so that the PX can correctly buildits routing table.

The whole process is illustrated in Fig. 3 by a sequencediagram. Peer PZ represents the Maintainers thatparticipate in the sampling random walk. The selectedMaintainer receives the Connect message and replies byproviding the necessary ClusterMap structures. A fur-ther issue related to the network management is the waythe structured macro level expands or shrinks in order tofit to the network size. For this purpose, a decentralizedalgorithm to split and merge the clusters is described in[21]. Moreover, a similar random walk mechanism maybe applied when peers are becoming reliable enoughto be assigned more critical roles (i.e., Indexers andMaintainers). In this case, a migration phase takes placeensuring the better distribution of the reliable peersamong the clusters.

4 Investigated sampling algorithms

In this section, we describe four different algorithmsto accomplish effective cluster coverage at low cost.Three of them are probabilistic and one is determin-istic. All algorithms start randomly at any peer, assum-ing that new peers randomly select their first contact tosend the requests to. Even if the employed bootstrap

phase does not comply with this assumption, it can beeasily achieved by performing an additional randomwalk before the sampling phase begins. Sampling is per-formed by initiating the submission of a message to arandomly (or deterministically) chosen neighbor clus-ter. The GUIDs and the relevant endurance values ofthe visited clusters are collected in the body of the mes-sage itself. This random walk terminates following therules defined by the particular algorithms.

4.1 Probabilistic algorithms

The probabilistic algorithms differ both in length andneighbor selection policy. Their description is providedin the following list.

1. Random destination. This algorithm starts fromany random peer, which randomly selects the finaldestination. This has the advantage of simplicitysince it does not differ from a typical query routingprocedure. However, the number of covered clus-ters is equal to the average query length. There-fore, the average random walk length is given byEquation 3, which is shorter than the networkdiameter. The achieved cluster coverage is verysimilar to the assigned routing workload assuminguniform query distribution [7].

2. Short random walk. This algorithm starts from anyrandom peer by randomly selecting only the nextpeer to follow among the neighbors found in therouting table. The procedure is recursively applieduntil a random walk of length equal to the diameterD of the network is reached. This algorithm has theadvantages of (i) equal length random walks and(ii) better coverage distribution than the previousalgorithm since seldom reached clusters are morelikely to be visited. However, its implementationis more complex. It requires a non-oblivious rout-ing mechanism to avoid cycles in the random walk.Clusters should be visited only once for efficiency,however, this rule can be ignored for particulartopologies where it will not allow visiting enoughnodes.

3. Long random walk. This algorithm is very similarto the previous one. The only difference is that therequired length of the random walk must be twicethe length of the diameter (2D). It is expected thatthe longer random walk combined with the cycleavoidance restriction will provide a much bettercluster coverage. The disadvantage of this algo-rithm is that it costs twice as much as the shortrandom walk.


Fig. 4 Deterministic R-shiftalgorithm

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4.2 Deterministic R-shift algorithm

The aforementioned algorithms perform random walksin order to sample the endurance of the clusters. In thissection, a deterministic algorithm is investigated in orderto evaluate such an alternative. It is assumed that thedeterministic walk begins randomly at any peer (simi-larly to the random alternatives).

There are certain restrictions and guidelines in design-ing an effective deterministic walk appropriate to effec-tively sample the endurance of clusters.

1. The length of each walk must be as close as possi-ble to its maximum value (i.e., the diameter of thenetwork).

2. The length of each walk should not differ consid-erably (independently of the position of the initialcluster).

3. The cluster coverage should be as wide and asevenly distributed as possible.

There are several algorithms that can fulfill the afore-mentioned requirements. We have designed one thatis as simple as possible. It is called “R-Shift” algorithm.Basically, each peer deterministically select the final des-tination by applying a right-shift operation at the GUIDof its cluster (note that the conventional routing in Omi-cron utilizes left-shift operations). The new symbol atthe left end of the GUID must be different than thesymbol at the right end before the right-shift operation.Formally, this operation can be expressed as follows.

r shift(u1u2...uD) = uDu1u2...uD−1, (6)

where the uD is an operation that provides a differentsymbol from the available alphabet (deterministically).For example, for binary de Bruijn graphs, it holds that

Table 1 Sampling algorithms routing cost

Sampling algorithm Expected walk length

Random destination DDB − 1k−1

Short random walk DDBLong random walk 2 · DDBR-shift ≈DDB

0 = 1 and 1 = 0. It is guaranteed that using the R-Shiftalgorithm all the clusters will be included in the samplingsince Equation 6 provides a direct and unique mappingof the input cluster to the output cluster.

This algorithm is graphically illustrated in Fig. 4, whichdisplays a de Bruijn(2, 4) digraph. Two different deter-ministic walks are shown. Assume that the two walksstart at nodes (0011) and (0101), respectively. Thereby,the R-shift algorithm produces the sequence (0011) �(0110) � (1100) � (1000) � (0001) for the first case,which is traversed by Msg1. Similarly, the sequence(0101) � (1010) � (0100) � (1001) � (0010) is tra-versed by Msg2. However, in certain cases the lengthof the path is smaller than the diameter, e.g., (1101) �(1011) � (0110).

The described algorithms have different inter-clustercommunication cost that determines the sampling walklength. Table 1 summarizes this cost.

5 Evaluation and analysis

The evaluation of the sampling algorithms has been per-formed by simulation experiments using the generalpurpose discrete event simulator for P2P overlay net-works described in [36]. In most of the experiments,the constructed overlay network forms an Omicron net-work, however, in several cases we additionally observethe sampling properties in Chord [5] overlay networks.


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Chord and Omicron differ in the node degree and thesymmetry properties. Both networks have been stabi-lized before the sampling procedures. Moreover, a prob-abilistic analysis is engaged in some aspects to verifyto observed simulation results. During the simulatedexperiments, message delivery considers the physicaldistance between the nodes enhanced by a statisticalqueue model. However, connection bandwidth limita-tions have been ignored, assuming that the involved sig-naling protocol is light enough and does not generatelarge traffic.

5.1 Cluster Coverage

The most critical aspect we need to evaluate is the abil-ity of the four sampling techniques to visit evenly eachcluster of the network. This case is more interesting forthe de Bruijn digraph based network that lacks nodesymmetry. Figure 5 provides the results for a specificOmicron configuration where the structured de Bruijnnetwork is composed of 2,048 clusters and the inter-cluster degree is k = 2. Figure 5a describes the cov-erage distribution (the number of times each cluster is

sampled) for the random destination algorithm. Similarresults are provided in Fig. 5b–d for the R-shift algo-rithm, the short random walk algorithm and the longrandom walk algorithm, respectively. Furthermore, itcan be observed that the results are symmetrical amongthe left half and the right half of the figures (patternscan be observed). This is the result of the symmetryproperties of the de Bruijn digraphs combined with theeven, deterministic distribution of the sampling initia-tion among the clusters.

As expected, the long random walk algorithmprovides the most evenly distributed sampling wherethe majority (in most of the experiments over 90% of thepopulation) of samples differ less than 10% from themean value. The reason for this lies on the fact thatmore clusters are sampled at every join, which is com-bined with the cycle-removal mechanism (non-obliviousrouting mechanism). Therefore, clusters that are seldomsampled by other algorithms have a higher probabilityof sampling by this algorithm.

Aiming at providing additional results on the abil-ity of each algorithm to evenly sample the networkclusters, further experiments have been performed. In


Fig. 6 Cluster sampling in deBruijn networks 450

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these experiments, the size of the structured macrolevel (de Bruijn network) is modified between 64 and16, 384 clusters. For each experiment, a number of joinrequests is generated that is related to the size of thenetwork and the utilized algorithm. The target is togenerate approximately equal workload for all of thealgorithms.

The quantity of interest in these experiments is theevaluation of the standard deviation of the cluster sam-pling distribution for each algorithm. Figure 6a summa-rizes the results of the experiments. It should be notedthat the x-axis scales logarithmically in order to pro-vide a more comprehensive view. As it can be observed,the long random walk algorithm achieves the smalleststandard deviation for the complete range of the evalu-ated network sizes. Moreover, the short walk algorithmachieves better performance compared to the randomdestination algorithm as the network grows. The stan-dard deviation of the R-shift algorithm grows consider-ably more compared to the three probabilisticalternatives.

However, the standard deviation provides an abso-lute value for the effect. In many cases, it is more impor-tant to observe a relative metric that relates the standarddeviation with the mean value. Such a metric is the coeffi-cient of variation, which is defined as CV = σ/µ, whereσ is the standard deviation and µ is the mean value. Itshould be noted that the mean value of the determin-istic algorithm differed from the provided mean valuesset. Therefore, it is not included in the provided results.Figure 6b summarizes the results on the coefficient ofvariation. As it can be observed, the long random walkalgorithm accomplishes always a coefficient of variationless than 10%. Also, it is interesting to notice the decreas-ing rate of CV for the short random walk algorithm. Itcan be stated that for very large network size (as networksize approaches infinity), the performance of the shortrandom walk algorithm approaches asymptotically theperformance of the long random walk algorithm.

5.2 Cluster sampling inter-arrival distribution

The aggregated behavior of the cluster sampling algo-rithms can be observed with the experiments performedin the previous sections. However, it is interesting toevaluate how often each particular cluster is revisited insubsequent random walks.

Self-connected clusters (i.e., with GUID 11...1 or00...0) are the least frequently involved clusters in therouting procedure [7], which can also be observed inFig. 5. Further, by examining the details of the col-lected results, it has been noticed that clusters withGUID lacking of “patterns” in their digit sequence,e.g., (011001001) or (0110010011) or (01100100110), areamong the most frequently visited clusters for eachsampling algorithm.

Let us define each random walk experiment as a“round”. The event of interest EC is “how many randomwalks (rounds) are necessary until a particular clusterC is sampled”. Such a quantity can provide vital infor-mation on whether the sampling algorithm is adequatefor its need. Therefore, further experiments have beenperformed aiming to evaluate EC. Collecting the exper-iment results for these clusters, the diagrams of Fig. 7have been drawn to show the probability that the par-ticular cluster will be sampled after a certain number ofsampling walks. The different location in the de Bruijngraph of nodes (0000000000) and (01100100110) (pro-vided in Fig. 7a,b, respectively) results in different rates,following the same shape though. In order to generatethese measurements, the long random walk algorithmhas been employed.

It is interesting to observe that the cluster samplinginter-arrival distribution can be closely approximatedby an exponential distribution of the form f (x) = λx−λx,x ≥ 0. The reason for such behavior can be explainedas follows. Let us call VC the event of interest, which isvisiting a particular cluster C during a random walk. VCis a Bernoulli random variable:


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g(x) ={

g(0) = Pr{VC = 0} = 1 − p,g(1) = Pr{VC = 1} = p,

(7)

where p is the probability of success that depends onthe cluster position in the digraph. Each random walk isan independent event. If we let X be the number of theperformed events until a success occurs, then X is saidto be a geometric random variable with parameter p. ItsPMF is given by:

h(n) = Pr{X = n} = (1 − p)n−1p, n = 1, 2, . . . (8)

The geometric distribution is the discrete equivalent ofthe exponential distribution. Therefore, the cluster sam-pling inter-arrival distribution can be approximated wellwith the exponential distribution.

As it can be seen from the approximated rates ofFig. 7, seldom visited clusters have a lower rate than fre-quently visited clusters. Also, as the size of the networkgets larger, the approximated rates are getting smaller,which is expected since more clusters are available forsampling. However, the peer join rate is getting higher,providing the necessary lower bound for the samplingrate. In theory, there is a probability that a cluster mightbe under-sampled for a certain period, however, this is aworst-case scenario that does not occur often in practice.

The basic reason of the different inter-arrival rates inthe aforementioned nodes is the asymmetry of the deBruijn networks. On the contrast, Chord has a node-symmetric graph (the graph looks the same indepen-dently of the observation node). Therefore, we wouldexpect to observe similar sampling inter-arrival rate foreach Chord node. We have constructed such simula-tion experiments performing random walks in a Chordlike-network consisting of 2,048 nodes. The network has

been stabilized before the sampling procedure. At eachnode, a finger3 is randomly selected and followed unlessit has been sampled already during this walk (a newselection is made in this case). The results are shown inFig. 8 where two different nodes have been randomlyselected. It can be observed that similar to the Omicroncase, inter-arrival distribution while sampling in Chordnetworks has similar exponential distribution proper-ties. Moreover, all of the nodes have the same exponen-tial factor as a result of the higher symmetry found inthe Chord networks.

5.3 Cover-time

In this section, we examine the time it takes to sampleeach cluster at least once, the so-called cover-time. Wemeasure the cover-time in number of sampling randomwalks required, as each one is triggered, i.e., when a newpeer joins the network or a peer migration is necessary.

In order to model the aforementioned problem andprovide a probabilistic evaluation, a useful result foundin the related literature is utilized. Gkantsidis et al. [37]proves that as long as the random walk length is O(lg N)

and the graph is an expander, the samples taken from theconsecutive steps can achieve statistical properties simi-lar to independent sampling. Since de Bruijn graphs arewell-known expander graphs [38], the aforementionedresults holds on them too.

Moreover, Feller [39, p. 225] provides a probabilis-tic analysis of Coupon Collection Problem. The CouponCollection Problem is the following: Suppose that thereare N distinct types of coupons, uniformly distributed.

3 A finger is shortcut connection as it is defined for Chordnetworks [5].


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At each experiment step, we draw a coupon. Let Sr bethe time by which r distinct coupon types have beenencountered. It holds that4:

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Equation 9 provides the expected number of singlesamples to cover each different sample type based onindependent sampling. In our problem, we need to eval-uate the number of necessary random walks of length kthat suffice to sample each cluster of the network. There-fore, if we group k consecutive samples as a randomwalk, Equation 10 provides an approximation of theexpected number of random walks (we ignore terms oforder O(N)):

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We need to empirically verify the suitability ofEquation 10 in predicting the expected number of ran-dom walks. Therefore, we have performed simulationexperiments both on Omicron-based as well as onChord-based overlay networks. In both overlays, we

4 In this problem, a drawing event is successful if it results insampling a new node. Sr is the number of sampling iterationsuntil r successes. Let Xk = Sk+1 − Sk. Then, Xk − 1 is the num-ber of unsuccessful drawings between the kth and (k + 1)st suc-cess. During these drawings, the population contains N − k nodesthat have not yet been sampled. Therefore, Xk − 1 is the num-ber of failures preceding the first success in Bernoulli trials withp = (N − k)/N, alas, it holds that E[Xk] = 1 + (1 − p)/p =N/(N − k). Since Sr = 1 + X1 + · · · + Xr, the expectation isE[Sr] = N{ 1

N + 1N−1 +· · · 1

N−r+1 }. A useful approximation is given

by E[Sr] = N · lg N+1/2N−r+1/2 = N · lg N

N−r + O(N) [39, p. 225].

have considered networks of 2,048 nodes to be sam-pled. The number of samples collected in each randomwalk are k = s+1, where s is the random walk size, sincethe initiator of the random walk is also encountered inthe sampling process.

As it can be observed from Fig. 9a,b where randomwalks of length lg N and 2 · lg N are, respectively, eval-uated, the expected probabilistic cost of independentsampling fits accurately with the random walk basedmechanism.5 However, in the case where the lengthof the random walk is getting significantly less thanthe diameter of the graphs, the difference between theexpected probabilistic cover time (based on indepen-dent sampling) and the simulation experiments basedon random walks is becoming apparent. The results areshown in Fig. 9c,d demonstrating the random walks oflength 2 and 3, respectively.6 Nevertheless, despite theirdifferences in the degree and symmetry, both Chordand Omicron perform similarly well with respect tothe expected cover time (almost indistinguishable). Thegreat majority of nodes is expected to be often sampled,which leads to high network stability.

Finally, some further experiments were contactedexamining the performance of randomly selecting thenext hop of a random walk in comparison with an alter-native adaptive mechanism where less frequentlyselected neighbors have higher probability to be fol-lowed in future walks (history log is maintained).Surprisingly, no improvement was noticed both forChord and Omicron with respect to cover-timeperformance.

5 Note that since the confidence interval bounds are very small,we have omitted them to increase the readability of the images.6 Consider that the network diameter in this case is lg N = 11.


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6 Related work

Churn, the continuous process of nodes arrival anddeparture, is one of the basic issues that should be han-dled by DHTs, since it determines the network main-tenance cost. A number of measurements reports (i.e.,[19,20,40–42]) have investigated deployed P2P systems,most of them being file sharing applications. Despite thefact that the results reveal slightly different uptime nodedistributions, they all agree that the majority of the peersstay online for a relatively short time resulting in unsta-ble networks. Therefore, proactive approaches such asDHTs require high maintenance cost to provide effi-cient routing of the lookup queries. This issue has beeninvestigated by several other researchers, too. Here wesummarize some of the most important related work.

Lam et al. [43] considered this problem and addressedit with two protocol extensions for hypercube-basedapproaches. The aim is to maintain K-consistency for thenetwork, meaning that K alternative paths are availablefor each node to reach each other node of the network.While this approach provides a reliable routing schemefor scenarios with high churn rates, the maintenance cost

is high. The solution does not consider the heterogeneityof the uptime distribution and in the evaluation of theapproach a uniform failure rate is assumed. In contrast,Omicron has been design in a way that capitalizes onthe existence of the most stable nodes to greatly reducethe workload generated by highly unstable peers.

Rhea et al. [44] address the problem considering threefactors: reactive versus periodic failure recovery, mes-sage timeout calculation and proximity neighbor selec-tion. For this context, Bamboo is used as the utilizedDHT infrastructure. By emulating the system, it is pos-sible to extract critical information on the networkingeffects. For example, how the message timeout selectioninfluences the process. Moreover, this work incorporatessampling as a mechanism to obtain information andselect physically close neighbors. They samplings algo-rithms are different from these suggested for Omicronsince they serve different purposes. Moreover, hetero-geneity of uptime distribution is not capitalized in theway Omicron dos, since it aims in addressing approacheswhere each peer has been assigned equivalent roles.

The advantage of the clustering mechanism of Omi-cron combined with the ability of selecting the assigned


roles provide the means to utilize and benefit from theheterogeneity of the node behavior in a novel way ascompared to the existing approaches. Clustering hasbeen suggested in other pieces of work. Yang et al.[45] clusters peers as a subnetwork around super-peers.While one might consider an Omicron Maintainer as asuper-peer, in reality Omicron differs significantly fromsuch an approach. Omicron routes queries over a prefix-based DHT achieving efficient routing and even work-load distribution. Considine [35]) suggests the usage ofclusters in DHTs (i.e., Chord) to reduce the routing costof a flat approach.

7 Conclusions

Omicron is a hybrid overlay network that has beendesigned to meet several critical requirements for P2Psystems, which fits adequately to the needs of a greatmultitude of P2P based CDNs. In this paper, we havefocused on the sampling mechanism of Omicron. Thismechanism has been employed in providing a well bal-anced and stable network, though the evaluation f thestability if is not the focus of this paper. The maintenanceoverhead that is required to ensure network stabilitythrough cluster endurance has been evaluated with mul-tiple sampling algorithms. Both deterministic and prob-abilistic algorithms have been employed where the lattershowed better cluster coverage capabilities. In addition,the cluster sampling inter-arrival distributions have beenestimated revealing an interesting exponential distribu-tion property that can be further exploited by analyticalmeans to obtain a stochastically described P2P network.The different rates of these exponential distributionsreveal properties of the topology and can be used toidentify potential traffic hot-spots. Moreover, the cover-time by applying random walks has been investigatedand approximated with that of independent samplingalgorithms. The graph characteristics and properties ofboth Omicron and Chord allow such an approximation.

The set of mechanisms proposed in Omicron coop-erate harmonically to provide a P2P overlay networkmeeting the large majority of the relevant requirements.For networks of small size, the long random walk mecha-nism will provide the best trade-off between theachieved coverage and the generated sampling traffic.As the network size grows, the short random walkalgorithm is an interesting alternative.

Sampling techniques are of a more general inter-est, and they can be applied to other quantities thancluster endurance. In fact, they are interesting candi-dates in many load-balancing problems. They fit well inthe distributed nature of P2P systems where no central

component exists. Their exploration is vital to developefficient systems and frameworks that can be deployedin the context of decentralized CDN infrastructures. Ourfuture research in the area includes mechanisms thatcombine caching allowing to reuse sampled informationfor a limited amount of time, thus, considering the peeruptime distribution. This may be combined with adapt-ing the length of random walks in order to achieve thesame coverage with less traffic.

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