Load Balancing in Distributed Web Server Systems with Partial Document Replication *

Transcription

1 Load Baancing in Distributed Web Server Systems with Partia Document Repication * Ling Zhuo Cho-Li Wang Francis C. M. Lau Department of Computer Science and Information Systems The University of Hong Kong {zhuo, cwang, fcmau}@csis.hku.hk Abstract How documents of a Web site are repicated and where they are paced among the server nodes have an important bearing on baance of oad in a geographicay Distributed Web Server (DWS) system. The traffic generated due to movements of documents at runtime coud aso affect the performance of the DWS system. In this paper, we prove that minimizing such traffic is NPhard. We propose a new document distribution scheme that periodicay performs partia repication of a site s documents at seected server ocations to maintain oad baancing. Severa approximation agorithms are used in it to minimize traffic generated. The simuation resuts show that this scheme can achieve better oad baancing than a dynamic scheme, whie the interna traffic it causes has a negigibe effect on the system s performance.. Introduction The increasing popuarity of the Word Wide Web has resuted in arge bandwidth demands which transate into high atencies perceived by Web users. To tacke this atency probem, mutipe copies of documents are distributed geographicay and paced in caches at optima ocations. Web caching attempts to reduce network atency by storing the commony requested documents as cose to cients as possibe. Simpe caches have no information on users access pattern, and so they woud habituay try to keep a copy of any document just requested. This may imit the performance of caches. For exampe, research in [] shows that the maximum cache hit rate achievabe by any caching agorithm is bounded under 0% to 50% A proactive Web server on the other hand can decide where to pace copies of a document in a distributed Web * This research was supported by CRCG Grant 009 and HKU 00/0 Large Items of Equipment Grants 000.0/0/00 server (DWS) system where the server nodes are distributed geographicay. In most existing geographicay DWS systems, each server node keep the entire set of Web documents managed by the system. Incoming requests are distributed to the server nodes via DNS servers [6,9,]. Athough such systems are simpe to impement, the caching of IP addresses on the cient side or in intermediate DNS servers coud easiy resut in uneven oad among the server nodes. Moreover, the fu repication eads to much waste of disk space due to those documents that are not frequenty requested. To achieve better oad baancing as we as to avoid disk wastage, one can repicate part of the documents on mutipe server nodes [5,,,5,8,9] and use contentaware distributor software to redirect a cient request to a server node that has the requested document [6]. Some rues are then needed in such a geographicay DWS system to determine each document s number of repicas and the distribution of these repicas. These rues constitute what we ca the document distribution scheme, and they shoud achieve the foowing goas. Load Baancing: Since requests tend to target at a sma part of the entire coection of documents [], frequenty requested documents shoud be repicated to avoid bottenecks. Documents and their repicas shoud be paced in such a manner that most of the time the oad of the participating server nodes is equaized. Reduced Traffic: To adapt to users access patterns, documents need to be re-dupicated and re-distributed among the server nodes dynamicay or periodicay. Communications caused by such actions shoud be kept to the minimum so that the performance of the geographicay DWS system woud not be adversey affected. Existing schemes mainy focus on baancing the oad, but not the traffic issues. In this paper, we propose a new document distribution scheme that can improve oad baancing performance of geographicay DWS systems,

2 whie minimizing the communication cost needed. We assume that each document has approximatey the same popuarity in a the server ocations. Therefore, we wi not consider network proximity to cients in repicating and distributing the documents. The performance of our scheme is evauated through simuation using rea access og data. The resuts show that this scheme can baance the oad in the DWS system during run-time efficienty, and the interna traffic generated due to these agorithms is reasonaby minima. The rest of the paper is organized as foows. Section formuates the document distribution probem in the DWS systems and gives a proof of its NP-hardness. Section presents our document distribution agorithms. In Section, we describe our simuation methodoogy and present the performance resuts. Section 5 surveys reated work, and Section 6 concudes the paper and discusses future work.. Probem Formuation In this section, we formuate the document distribution probem in DWS systems. Chen [7] proved that minimizing the maximum oad over a server nodes is NP-compete. We wi prove that even when the oad baancing constraint is removed, the probem of minimizing the communication cost of moving the documents is NP-hard. Suppose there are N documents and M server nodes in the system. Each server node has storage capacity C. Each document has size of s i and number of repicas c i (In this paper, if we don t state otherwise, we assume i =, N and j =, M). A cost ink is constructed between each document and each server: p, associated with the number of bytes to be transferred if document i is assigned to server j. We aso have variabes t ( =, c i ), which is if th repica of ith document is paced on jth server; otherwise, it is 0. The determination of c i is under the imitation of tota N storage, i.e., ( s c ) M * C. i= i i After c i is determined, a the documents and their repicas are paced on the server nodes under these constraints: () each server can ony hod repicas whose tota size does not exceed its disk space; () each server can hod at most one repica of a document; () no document is eft unassigned to any server node; () oad is equaized among the server nodes. As we stated at the beginning of the section, we won t incude constraint () in the formuation. The repica pacement probem formuation is therefore as beow: minimize z M N ci = j= i= = t p N c i subject to ts C () i= = c i = i t, () M c i t = c () i j= = t = 0 or, =,... c A repica pacement that fufis a the above constraints is a feasibe pacement. Our discussion is under the assumption that a feasibe pacement aways exists. We ca this optimization probem the Repica Pacement Probem (RPP). When c i =, the probem is 0- RPP. Lemma 0- RPP is NP-hard Proof: We reduce the bin-packing probem, which is NPhard [], to the 0- RPP. For the bin-packing probem, s i denotes the size of object i and the bin s size is C. We assume that, in any feasibe soution, the owest indexed bins are used. This means that if there are two bins with the same avaiabe storage, the object wi be paced in the one with the ower index. Given the bin-packing probem, we can construct a 0- RPP with costs p as foows., i {,... N}, j = p = ( pi, j )* N +, i {,... N}, j {,... M} With such costs, the tota cost of any set of repicas assigned to {s, s j } is ower than the tota cost of any set of repicas assigned to {s, s j+ }. It is then obvious that the bin-packing probem gets the minima number of bins used if and ony if the 0- RPP gets the minima tota communication cost. Since the 0- RPP is a specia case of the RPP, our document pacement probem is NP-hard.. Document Distribution Scheme In this section, we propose our document distribution scheme, which periodicay re-repicates and re-distributes the documents based on the access pattern in the past period. We first describe an agorithm for determining the number of repicas for each document. Next, we present severa heuristics that use avaiabe information in different ways in order to achieve minima communication cost... Density Agorithm Intuitivey, we shoud prefer to dupicate documents that require more work on the part of the DWS system as i

3 we as the sma-size ones. We use the concept of density to represent the workoad per unit storage of the document. The arger a document s density is, the more repicas it wi have. Input: d i, s i, C, M, N, Variabes: S, tota size of document S_disk, avaiabe disk space d min, minima density temp_s, tota size of temporary repicas temp_c i, temporary number of repicas Output: c i (i =, N).compute S, S_disk = M * C - S.sort documents by decreasing density d i, find d min.for i = to N { temp_c i = d i / d min } compute temp_s.for i = to N { c i = temp_c i * S_disk / temp_s if (c i >= M-){ c i = M- temp_s = temp_s temp_c i * s i S_disk = S_disk c i * s i }} 5.finay decide c i (i =, N) Figure. Density agorithm To compute a document s density, we associate document i with weight w i, which represents the workoad it brings to the server node hoding it. In our agorithm, w i is computed as s i * r i, where s i is the document s size and r i is its access rate in the past period. The density of a document d i, therefore, equas to w i / s i. If a document is dupicated, we assume that the workoad is divided eveny among its repicas (true if we assign requests to the repicas in a round-robin manner). Therefore, a repica of document i has weight of w i / c i and density of d i / c i. The Density Agorithm is shown in Figure. First, the space equa to the tota size of documents is reserved to guarantee that each document has at east one copy in the system. Step sorts the documents by their densities decreasingy. In Step, each document gets a temporary repica number. The densities of the temporary repicas are neary equa to the minima density. Step adjusts the temporary repica numbers under the storage imitation. Repica numbers are computed according to the ratio between avaiabe disk space and the tota size of the temporary repicas, thus the resuting repicas sti maintain simiar densities. In Step 5, each repica number is finay decided as an integer not arger than M. This agorithm repicates the documents according to their densities under the storage imitation. The time compexity of it is Θ ( N og N + N). From Step, we know that for any two documents u and v, if < cu, c v M and d u > d v, d u / (c u ) d v / (c v ). Thus, we can assume if d u > d v, d u / c u > d v / c v, for any two documents u and v. Since we have proved that the minimization probem of communication cost is NP-hard, in the rest of this section, severa approximation agorithms for distributing the repicas to the server nodes are proposed. Before we begin the discussion, we need to introduce a new variabe, W j. It denotes the weight of server j and is computed as the sum of the weights of a repicas aocated to it. Aso, in the foowing discussion, we ca document i and its repicas repica set i"... Greedy-cost Agorithm Input: c i, s i, p, C, M, N Output: t (i =, N, j =, M, =, c i ).sort (i, j) pairs by increasing cost, p.for each (i, j) in the sorted ist{ if (c i > 0) { aocate a repica to server j if it has enough space and t = 0 ( =, c i ). c i = c i }} Figure. Greedy-cost agorithm Greedy-cost agorithm aims to minimize traffic by keeping as many as documents as they are in the system, without caring if the oad of the servers is baanced. This agorithm is shown in Figure. To minimize the cost, it first sorts the (document, server) pairs by the communication cost between the document and the server increasingy. Then in this order, a repica of document i is aocated to server j if it has enough storage space and has not been assigned the same repica in this period. The tota time compexity is Θ ( MN og MN + MN )... Greedy-oad/cost Agorithm Unike Greedy-cost agorithm, Greedy-oad/cost agorithm considers baancing the oad among the server nodes as we as minimizing the cost caused in distributing the documents.

4 Input: c i, s i, p, C, M, N Output: t (i =, N, j =, M, =, c i ) Variabe: D j, density of server j (j =, M).sort repica sets by increasing density, d i / c i.for i = to N { sort servers by increasing communication cost, p. Servers having the same p are sorted by decreasing density, D j aocate repica set i update D j (j =, M) } Figure. Greedy-oad / cost agorithm To achieve oad baancing, we expect seeing that after document distribution, the weights of the server nodes are approximatey the same. Since the density of a server node D j equas to W j / (amount of used disk space in server j), in a homogeneous DWS system, this means that after document distribution, the server nodes have simiar densities. Therefore, in Greedy-oad/cost agorithm, the repica sets are sorted by decreasing density and are aocated in this order. When choosing server nodes for repica set i, the server nodes are sorted by increasing communication cost p. If two server nodes have the same cost, the one with the arger density D j is chosen. The time compexity of Greedy-oad agorithm is Θ ( N ogn + NM og M). To simpify it, we can use the sorting resut of the Density agorithm, based on the assumption that if du > d v, d u / c u > d v / c v. Thus the agorithm ony takes Θ(NM og M ) time... Greedy-penaty Agorithm It is possibe that aocating a document at different times generates different traffic. For exampe, if we aocate document i immediatey, we can assign it to server x with p ix = 0; if we deay aocating it for a whie, however, server x may have become fu and the document has to be paced on server y with p iy = s i. In this case, we say we are punished and use f i to refer to the vaue of penaty. A penaty-based agorithm hopes to decrease cost by pacing documents in certain order. It has been used to sove the Genera Assignment Probem []. We say a pacement is better if it incurs ess communication cost. In Greedy-penaty agorithm, f i is computed as the difference in the costs of repica set i's best and second best pacements, according to the current status of the server nodes. This agorithm iterativey paces the repica sets unti they are a aocated. Each time it computes f i for a unassigned repica sets, and the one yieding the argest penaty are paced with its best pacement. The time compexity of this agorithm is Θ ( N og N + NM ogm). Input: c i, p, s i, C, M, N Output: t (i =, N, j =, M, =, c i ) Variabes: f j, penaty for document i (i =, N) whie there are unassigned repica sets { for each unassigned repica set i{ if ony c i server nodes have enough storage to hod document i{ aocate repica set i goto whie } ese { sort servers by increasing cost with document i, p. compute f i }} sort repica sets in decreasing penaty, f i aocate the repica set with minima f i in its best pacement} Figure. Greedy-penaty agorithm If there are ony c i server nodes having enough storage to hod document i, we need to aocate repica set i immediatey. Otherwise, we might eave a repica unassigned and vioate constraint (). In this case, we set f i to. If there are mutipe repica sets with infinite penaty, they are paced in the order of decreasing densities. To do this, we can use the sorting resuts of the Density agorithm.. Simuation Resuts.. Experimenta Setup We use the CSIM 8 package [] for our simuation. In our simuation mode, requests are redirected to the server nodes that have the requested documents using HTTP redirection. When a document has copies in mutipe server nodes, the requests for it are assigned in round-robin fashion. Initiay, Web documents are randomy paced on the server nodes without repication. Afterwards, documents are repicated and distributed among the server nodes every hours. In our simuation, processing a web request comprises () redirection (if necessary), () waiting in the queue of the serving server node, () reading the fie from disk. The connection estabishment time and teardown time is negected. The round-trip time of redirection is 00 ms [6]. The disk access time is about 9 ms and the disk transfer time about MB/s []. We use two rea traces of Web access. One is from a website mainy used for hosting persona homepages, caed Data Set. Another, caed Data Set, is obtained from The Internet Traffic Archive [0]. For simpicity, the documents in the same directory are grouped and these groups are used as basic units of repication and distribution.

5 LBM.5.5 /6 /8 / / C / S DC Figure 5. Load baancing performance with Data Set (6 server nodes) LBM no. of servers Figure 7. Load baancing performance with Data Set (C / S = /8) LBM 5 /6 /8 / / C / S DC Figure 6. Load baancing performance with Data Set (6 server nodes) We simuated the agorithms presented in Section. Density agorithm is combined with Greedy-cost (), Greedy-oad/cost (), Greedy-penaty () respectivey. For the purpose of comparison, we added a Dynamic scheme (DS). In this scheme, each server node owns a part of documents. Dynamicay it examines the other servers oad and determines if they are underoaded or overoaded or. It then repicates one of its documents to the under-oaded node or revokes one repica of its documents from the overoaded node. This scheme is simiar to the one used in DC-Apache. In our simuation, the servers check oad status every 0 minutes... Load Baancing Anayses The Load Baance Metric (LMB) [] is used as a performance metric for measuring oad baancing resuts. We record the peak-to-mean ratio of server utiization every samping period (0 minutes) during the simuation. The LBM vaue is obtained by cacuating the weighted average of the peak-to-mean ratios measured, using the tota server utiization at the samping point as the weight. A smaer LBM vaue indicates better oad baancing performance. Figure 5 and Figure 6 present the oad baancing performance of our scheme when the number of servers is LBM no. of servers Figure 8. Load baancing performance with Data Set (C / S = / 8) fixed as 6. The y-axis is LBM vaue. The x-axis is C / S, where C is the storage capacity of each server node and S is the tota size of the documents. We can see that DS doesn t improve oad baancing much as the storage capacity increases. This is because in DS, each server node can ony repicate one document once a time so that the avaiabe disk space is not utiized efficienty to remove hot spots. On the contrary, the oad baancing performance of our scheme increases as storage capacity increases because the Density Agorithm fuy utiizes the disk space. Among the document distribution agorithms, performs worst in oad baancing whie and s performance is simiar. Next, we fix the storage capacity, and increase the number of server nodes from 6 to 56. The resuts are shown in Figure 7 and Figure 8. We notice that s and s performance is sti cose when the node number is not very arge. When there are more than 8 nodes, however, appears to deteriorate faster than... Traffic Anayses We record the tota number of bytes transferred inside the system each period (except the first period, as documents are randomy distributed without dupication initiay). At the end of the simuation, the ratio between

6 Average Traffic / S Average Traffic / S /8 / / 5/8 5/6 C / S Figure 9. Average traffic with Data Set (6 server nodes) no. of servers Figure 0. Average traffic with Data Set (C / S = /8) the average traffic each period and tota size of Web documents S is computed. In the figures, y-axis represents this ratio. We can see from Figure 9 and Figure 0 that when number of server nodes is fixed, the traffic caused by the agorithms first increases as the storage capacity C increases, and then decreases. This is because when there is more avaiabe disk space, more documents are repicated and the numbers of repicas of popuar documents are arger. Once the access pattern changes, therefore, more repicas of the past period are revoked and more new repicas of this period need to be distributed. It is easy to understand that, which cares most about communication cost, incurs the east cost. We find that when the storage capacity is arge, the traffic caused by and is amost the same. As the number of nodes increases, the tota storage space increases, therefore, the traffic in the system increases. From Figure 0 and Figure, we can see that sti causes east traffic, and the traffic caused by and get coser as the number of nodes increases. The actua time needed to move the documents t tota bytes /( B* M) where M is the number of servers and B is the bandwidth between any two server nodes. Therefore, if we assume that the bandwidth is MB/s and tota size of the DC Average Traffic / S Average Traffic / S /8 / / 5/8 5/6 C / S Figure. Average traffic with Data Set (6 server nodes) DC no. of servers Figure. Average traffic with Data Set (C / S = /8) documents is G, moving documents woud take no more than severa minutes. Since during this period, the DWS system can continue to serve requests with documents not in the move, its performance woud not be substantiay affected. In the figures, DS s average traffic may be smaer than that of our scheme. But since its period is much shorter, its tota traffic is actuay arger than ours. This may be because that it frequenty repicates a document and then revokes it. From the simuation resuts, we see that our document distribution scheme can achieve better oad baancing in a geographicay DWS system and generate ess traffic than the dynamic scheme. Among the document distribution agorithms, s oad baancing performance is not as good as that of and. However, generates the east interna traffic. needs shortest computing time. Its oad baancing performance is best in most cases and ony generates a itte more traffic than. When the number of server nodes is arge, however, performs much worse than. baances the oad we but it requires more computation than the others. A suitabe agorithm can be chosen according to the practica situation of a geographicay DWS system.

7 5. Reated Work Much research work has been done on ways to keep a baanced oad in geographicay DWS systems. Various DNS based scheduing techniques have been proposed. The NCSA scaabe web server depends on round-robin DNS to dispatch requests [], whie [9] found that the DNS poicies combined with a simpe feedback aarm mechanism coud effectivey avoid overoading the server nodes. Adaptive TTL agorithm [8] was proposed to address the uneven cient request distribution and heterogeneity of server capacities. The main probem with these techniques is that DNS ony has a imited contro on the requests reaching the Web servers, due to the caching of IP address in intermediate DNS servers and cient caches. The content-aware requests distribution strategy LARD [6] makes it possibe to baance the oad among the server nodes through partitioning the Web documents among the server nodes, with or without repication. DCWS [5] makes use of a graph-based Web documentpartitioning agorithm. Each document resides on its home server at first and can be migrated to a co-op server for oad baancing reason. To redirect cient requests from the home server to the co-op server, a hyperinks pointing to the document are modified. However, if the system happens to contain many hot spots (i.e., popuar Web pages with extremey high request rates), to equaize the oad is absoutey non-trivia. DC-Apache [] is simiar to DCWS, except that documents are repicated instead of migrated among the server nodes. Each document has a home server that keeps its origina copy. Every time the number or ocations of copies of a document change, the document s home server needs to regenerate a the hyperinks pointing to this document based on goba oad information. This operation requires substantia computation. Riska et. a. observed that directing tasks of simiar size to the same server reduces the sowdown in a web server and proposed a oad baancing poicy ADAPTLOAD [8] which partitions the documents among the server nodes according to their sizes. How to effectivey choose parameters of the poicy sti needs more work. In RobustWeb [], each document has the same number of repicas. The repicas are paced on the server nodes based on past access pattern to equaize the servers oad. Mutipe copies of a document may have different weights of redirection, and the requests are assigned to them in a weighted round-robin way. Instead of moving documents ike we do, in RobustWeb, ony the weights of the copies are computed periodicay. When the access pattern change dramaticay, however, it s difficut to maintain oad baancing using this method. Ng at e. [5] incuded the prefetching feature in their EWS system. In this system, documents that are aways accessed together are grouped and paced on the same server node. Ony the first request of a session has to go through the redirection server, thus cutting down on the redirection overhead. Load baancing is achieved by using a revised document pacement agorithm of the one used in RobustWeb. Our work can be considered a derivative from theirs by taking disk utiization and communication cost into account. The agorithms we propose in this paper can be depoyed in EWS. Recenty there has been an increase in interest in repica pacement in Content Deivery Networks (CDN) that offer hosting services to Web content providers [,0,7]. Athough the probem formuation in CDN is very simiar to ours, it mainy focuses on minimizing cients atency or tota bandwidth consumption, and not baancing the oad among the servers. 6. Concusion and Future Work In this paper, we study how to repicate and distribute the documents in a geographicay DWS system to achieve oad baancing. In contrast with existing work, we aso take the communication cost caused by distributing the repicas into consideration. We prove that even without oad baancing constraint, minimizing this cost in homogeneous DWS systems is NP-hard. We propose a document distribution scheme which periodicay repicates the documents and distributes the repicas. In this scheme, we utiize the concept of density of a document to decide number of repicas for each document. Severa distribution agorithms are proposed and they use the avaiabe information from different perspectives to reduce interna traffic of the geographicay DWS system. Our scheme is compared with a dynamic scheme using rea og fies. The resuts show that our scheme coud baance the oad in a DWS system more efficienty during run-time and causes ess traffic. We aso discuss the difference between the distribution agorithms and the situations for which they are suitabe. Our next step is to incorporate geographica information into our document distribution scheme. We aim at a geographicay DWS system which woud automaticay copy a document to a ocation where it is in most demand, whie maintaining oad baancing and minimizing communication cost. Such a geographicay DWS system woud reduce access atencies and be most suitabe for Web sites where different parts of the content are of interest to peope from different geographica ocations.

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