TSR: Topology Reduction from Tree to Star Data Grids

Transcription

1 03 Seventh Internationa Conference on Innovative Mobie and Internet Services in biquitous Computing TSR: Topoogy Reduction from Tree to Star Data Grids Ming-Chang Lee #, Fang-Yie Leu *, Ying-ping Chen #3 # Department of Computer Science, Nationa Chiao Tung niversity, Taiwan * Department of Computer Science, TungHai niversity, Taiwan mingchang09@gmai.com; eufy@thu.edu.tw; 3 ypchen@cs.nctu.edu.tw Abstract To speed up data transmission of data grids, severa co-aocation schemes have been proposed. However, data grids are often arge in scae, heterogeneous in participating resources, and compicated in architecture and network topoogy, consequenty increasing the anaytica compexity of its data transmission behaviour. In other words, if we can reduce the data transmission topoogy for the grid, the anaysis wi be easier. Therefore, in this paper, we propose a topoogy reduction approach, caed the Tree-to-Star Reduction method (TSR for short), which can reduce a packet deivery tree topoogy to a star for a data grid so that the data transmission of a co-aocation scheme can be more convenienty anayzed. Here, a deivery tree topoogy, as a tree topoogy rooted at the destination node, is a network topoogy for deivering a fragments of a fie to the destination node. Keywords data grid, deivery tree, deivery star, topoogy reduction, co-aocation scheme I. INTRODCTION To shorten fie retrieva time and improve fie repication performance for data grids, severa co-aocation schemes have been proposed [][][3][4]. However, it is difficut for us to evauate these schemes' data transmission behaviour since a data grid is often arge in scae [5], heterogeneous in participating resources [6], and compicated in architecture and network topoogy. Coates et a. [7] ogicay transformed a compicated network topoogy into a simpe tree to simpify their performance anayses. Levitin et a. [8] introduced a concept in which the network topoogy used to transmit the data generated by different service sites for a grid job can be treated as a star topoogy in which a ogica ink connects a service site and the destination node. With this star topoogy, users can more easiy evauate the execution time of a task and data transmission reiabiity. However, these studies neither mention how to reduce a compicated network topoogy to a simpe one, nor discuss whether or not the performance of the two topoogies is identica. Hence, in this paper, we propose a topoogy reduction approach, caed the Tree-to-Star Reduction method (TSR for short), which can reduce a compicated packet deivery tree topoogy to a star in a data grid, where a deivery tree topoogy (or simpy a deivery tree), as a tree topoogy rooted at the destination node, comprises a set of service sites and network components for cooperativey deivering a fie or task resuts generated by the service sites to the destination node. Consequenty, the compexity of evauating a co-aocation scheme can be dramaticay reduced. II. BACKGROND A co-aocation scheme is an scheme invoked by the resource broker of a data grid to coordinate a set of service sites to cooperativey deiver a fragments of a fie to where,, is the site at which a user submits a request to access []. In fact, from 's viewpoint, the routing paths that transmit a fragments of to form a deivery tree rooted at. Each fragment may pass through severa routers and inks before arriving at. Due to containing so many network components (i.e., inks and routers) and parameters (e.g., ink bandwidths and the departure and arriva rates of routers), it may be hard for users to anayse the fie transmission time for the scheme in this deivery tree. III. THE TSR METHOD The process of reducing a deivery tree to a star is as foows. A deivery tree basicay comprises two types of structures. The first is a series structure (SS for short) in which a network components form a chain topoogy. If the topoogy contains nodes and inks,, we ca it an SS of ength. Given an SS of ength n-, if there does not exist a node that together with the SS forms an SS of ength, we ca the given SS a maximum series structure (MSS for short). The chain topoogy shown in Fig. a, i.e.,, is an exampe in which and are the head and tai nodes, respectivey. We wish to reduce the MSS to a simpe chain, which consists of the head node and the tai node connected by a ogica ink, denoted by (see Fig. b), impying that is simpified to. The reduction process is caed the series structure reduction process (the SS-reduction process for short). The other type is a parae structure (PS for short), which consists of upstream nodes and,, and a conjunction node, denoted by, in which is connected to through a direct ink, i.e., is a simpe chain which is an SS of ength,, where may be a physica or ogica ink. The structure iustrated in the dashed rectange shown in Fig. is an exampe. Given a PS, if there exists at east one sub-tree rooted at (see Fig. ) with as its eaf nodes, and /3 $ IEEE DOI 0.09/IMIS

2 , then we ca the PS an expandabe PS. If no such sub-tree exists, we ca the PS a maximum parae structure (MPS for short), denoted by MA. The one contained in the dashed rectange shown in Fig. 3a, i.e.,, is an MPS. Given a MA, if there exists an MSS with as its head node, e.g., the one shown in Fig. 3a, i.e.,, then after appying the SS-reduction process on the MSS, the resuting structure as depicted in Fig. 3b is an MPS foowed by a simpe chain. We ca this structure a parae-chain structure, which wi be reduced to a PS (see Fig. 3c). The corresponding reduction process is caed the parae-chain reduction process (the PC-reduction process for short). In the foowing, we assume that ) a packet deivery tree is drop-free, impying that no packet wi be ost and retransmitted during the packet transmission; ) the queuing mode of a node, either a router or a service site, for sending and receiving packets is M/M/ [9], i.e., the arriva and departure rates foow a Poisson distribution; 3) due to appying the TSR to a data grid, each node 's packet service rate (service rate for short), denoted by, and packet departure rate (departure rate for short), denoted by, are a arger than its packet arriva rate (arriva rate for short), named. Here, we separate service rate and departure rate since service rate means the maximum capacity in pkts/sec that a node can transmit, whereas departure rate is the number of packets that physicay sends. For a node in a drop free deivery tree, the capacity of is either or. Let be the function used to cacuate the transmission time of a packet through a ink at time point. Then, () where is the size of, and is the bandwidth of at. Let be the function empoyed to cacuate the expected queueing deay of in a node at. Then, () where is 's arriva rate and is 's service rate. Let be a function used to cacuate 's departure rate. Then, (3) where is the service rate of at. Let be the function empoyed to cacuate 's arriva rate where is the set of 's immediate upstream nodes, in which may be a service site or a router transmitting packets to,. Then, k Fig. The topoogy before and after the SS-reduction process Fig. An expandabe parae structure A. The SS-reduction process To describe the SS-reduction process, we assume that the packet generation rate, i.e., departure rate (rather than service rate), of a service site, denoted by, is known and depoy the topoogy shown in Fig. a as an exampe, in which and are respectivey the head node and tai node of the MSS, and the set of routers are the intermediate nodes and the set of inks are the intermediate inks between and. The resuting topoogy is. The SSreduction process consisting of three phases is as foows.. The first phase uses the function shown in Eq. () to cacuate the transmission time of fowing through a ink at, named, where,. When,.. The second phase empoys the functions shown in Eqs. ()~(4) to cacuate the expected queueing deay of in router at, named, where, in which k MSS n n n (a) S PS conj conj m 'conj m k ' conj (4),, and ' k (b) m+ when,, and when,. n m+ m+ m+ : SS-reduction process : receiving site or router, 679

3 MPS conj conj m MSS m m m S MPS ' conj conj ' S ' ' m m S n n n n (a) Topoogy T (b)topoogy T' (c) Topoogy T'' : receiving site or router : SS-reduction process : PC-reduction process Fig. 3 The topoogy before and after the SS-reduction and PC-reduction processes. (a) A topoogy T in which an MPS is foowed by an MSS; (b) The topoogy T' after reducing the MSS in T to a simpe chain; (c) The topoogy T'' is a PS structure, which is aso an MPS after reducing the parae-chain structure shown in T' with the PC-reduction process 3. The third phase cacuates the accumuated transmission time of which is aso the transmission time of on, denoted by, where and the equivaent bandwidth of at, denoted by, is (6) B. The PC-reduction process In the foowing, we use the topoogy shown in Fig. 3b as an exampe, i.e., MA foowed by a simpe chain, to describe the PC-reduction process which consists of four phases. Aso assume that 's packet generation rate, i.e.,, is known,.. The first phase aso uses Eq. () to cacuate the transmission time of a ink at, denoted by, where,.. The second phase empoys Eq. () to cacuate the queueing deay of in at, denoted by,, in which (5) is the service rate of 's outink at,, and. 3. The third phase aso invokes Eq. () to cacuate the transmission time of through the ink at (see Fig. 3b), denoted by, where. 4. In the fourth phase, the accumuated deay of on traveing through,, and at, denoted by, is cacuated as (7) where is a constant for a,, since and are the average vaues of a the packets fowing through and, respectivey. The bandwidth of at is aso cacuated by invoking Eq. (6),. C. Tree-to-Star Reduction (TSR) agorithm In the foowing, a deivery tree topoogy rooted at with a set of service sites shown in Fig. 4a is given as an exampe to describe the TSR agorithm iustrated in Fig. 5, in which between ines and 3, the TSR first checks to see whether a service site and its foowing MSS is a simpe chain or not (assume that the conjunction node of the MSS is ), where and. If not, the TSR utiizes the SS-reduction process to reduce the MSS to a ogica ink connecting to. The resut is shown in Fig. 4b. Now each service site is the head node of a simpe chain. Between ines 4 and 0, on identifying a new MPS, e.g., MA, the TSR checks to see whether the MSS with and as its head and tai nodes, respectivey, is a simpe chain or not,. If yes, the MPS and the simpe chain form a parae-chain structure. If not, the TSR reduces the MSS to by using the SS-reduction process. After that, the resuting topoogy is aso a parae-chain structure. Next, the TSR utiizes the PCreduction process to reduce the parae-chain structure to a PS. 680

4 m,, m,, m,,, n, m, n, m p m, z p, m, z p, a q, ' m ' p p, p, a, ' m m m m m ' ' p ' p, ' m ' ' ' ', ' m ' ' ' ' ' ' m ' ' ' ' ' ' u ' ' m u u u (a) u u u (b) u u u u u u (c) (d) (e) (f) u : receiving site : SS-reduction process : PC-reduction process Fig. 4 A deivery tree is reduced to a deivery star by invoking the TSR agorithm TSR agorithm: Input: A deivery tree with as its root (eaf nodes); Output: The equivaent deivery star; Procedure: : For each service site, { : If there is an MSS with and as the head node and tai node, respectivey{ /* may be a router or */ 3: Reduce the MSS by using the SS-reduction process; }} /* by invoking Eqs. ()~(6) */ 4: Whie there is a MA newy identified where { 5: If there is an MSS connecting and a node, and its ength is onger than { /* is a router or */ 6: Appy the SS-reduction process to reduce the MSS to a simpe chain ;} 7: Appy the PC-reduction process to reduce the MA and to a PS ; 8: /* by invoking Eqs. ()~(7)*/ 9: Expand the to a PS ; } /*,..., are those service sites currenty and directy connected 0: to by a physica or ogica ink.*/ Fig. 5 The TSR agorithm If the PS is an expandabe one, the TSR expands the PS to a MA by reducing each sub-tree rooted at to an MPS by using the SS-reduction and PCreduction processes interchangeaby where are those service sites that are now directy connected to by a physica or ogica ink. This procedure continues unti is and no MPS can be newy identified. Now a deivery star topoogy (or simpy a deivery star), in which a service sites are connected to the center node, is derived. Fig. 4f is the exampe resut. Figs. 4b, 4c, 4d, and 4e are the intermediate resuts of reducing Fig. 4a to Fig. 4f. IV. A DEMONSTRATION EAMPLE In this section, we woud ike to evauate the transmission time of a packet, KB in ength, in a deivery tree and its corresponding deivery star. We use the tree topoogy iustrated in Fig. 6a as an exampe, which has six service sites that send packets of a fie to, to cacuate 's deivery deay. The bandwidths of a the inks are isted in TABLE I, and the service and departure rates of a service sites and routers are accordingy isted in TABLE II. We assume that service site 's packet generation rate is pkts/sec,. TABLE I THE NETWORK BANDWIDTHS OF ALL THE LINKS SHOWN IN FIG. 6A Network inks The bandwidth of a ink (Mbps) TABLE II THE SERVICE AND DEPARTRE RATES OF ALL SERVICE SITES AND ROTERS SHOWN IN FIG. 6A Network inks The bandwidth of a ink (Mbps) A. The transmission time in a deivery tree The routing path of a packet issued by, denoted by, consists of,,,,,,,,,, and, and hence the tota transmission time of at, denoted by, is (8) 68

5 where based on TABLE I,,,, and. Based on Eq. (), TABLE II, and Fig. 6a, since pkts/sec, and has two incoming inks and, impying pkts/sec. Simiary, in which pkts/sec, pkts/sec, and because pkts/sec and pkts/sec. Furthermore, since pkts/sec and pkts/sec. Based on Eq. (8) and the vaues cacuated above,. The deay of sending a packet by to, denoted by, can be cacuated by the same method,. B. The transmission time in a deivery star We first reduce the deivery tree shown in Fig. 6a to a deivery star by empoying the TSR agorithm. A network components on the paths from and to, and from and to form two parae-chain structures. After appying the PC-reduction process to them, the resut is shown in Fig. 6b, in which PS is an MPS and the path from to is reduced to a ogica ink,. Based on Eq. (7), (9) (0) Hence, the bandwidth of, i.e.,,, can be cacuated by invoking Eq. (6). Now we can further reduce MA and its foowing simpe chain, i.e., forming a parae-chain structure, to a, in which foowed by and is reduced to,. The resuting structure is depicted in Fig. 6c. Based on Eq. (7), () Therefore, the bandwidth of can be obtained by invoking Eq. (6), and then the TSR expands the PS to MA, in which and are individuay connected to by a physica ink, and which as shown in Fig. 6c is a newy identified MPS, and is an MSS which can be reduced by appying the SS-reduction process. The resuting structure is depicted in Fig. 6d, in which the path from to is reduced to a ogica ink. Based on Eq. (5), () where the bandwidths of can be cacuated by empoying Eq. (6). Now MA and is a parae-chain structure which can be reduced to an MPS, i.e., MA, in which,, and become,, and,, and becomes,. p to this point, a deivery star centered at as depicted in Fig. 6e has been derived. Each service site is directy connected to with either a ogica ink or physica ink. Based on Eq. (7) (3) and (4) Among the six service sites, the deay time of a packet sent by, and, to is the ongest. Hence, just as in cacuating the transmission time of a packet in a deivery tree, we cacuate the transmission time of a packet sent by to as an exampe. Let be the transmission deay of from to. Then, (5) Based on Eqs. (5), (3), (), (9), and (), which is equa to Eq. (8), impying that and. The deay of transmitting a packet from to,, can be cacuated by the same method. The deays are,,,, and, respectivey, consequenty demonstrating that the transmission deay of deivery star is equivaent to that of deivery tree. Additionay, if the deivery tree can be identified and the bandwidths of and,,, are cacuated beforehand, the cacuation of, i.e., Eq. (5), is simper than the cacuation of, i.e., Eq. (8). Therefore, it is easier for users to evauate the data transmission deays of a co-aocation scheme. 68

6 * : disk queue * 6 : repica site ' 5 ' 6 u ' '' 4 ' '' 3 ' '' ' '' * : disk queue * : repica site u (a) A deivery tree data grid u (b) A deivery tree reduced from the one shown in Fig. 6a u (c) A deivery tree reduced from the one shown in Fig. 6b 6 u (d) A deivery tree reduced from the one shown in Fig. 6c 4 ' 4 4 ' ' 4 ' ' ' ' * : disk queue * : repica site 3 ' ' 3 3 ' ' 4 ' ' 3 ' ' ' ' ' ' * : disk queue * : repica site ' ' * : disk queue * : repica site (e) A deivery tree reduced from the one shown in Fig. 6d Fig. 6 An exampe of reducing a deivery tree data grid to a deivery star V. CONCLSIONS In this artice, we propose a topoogy reduction approach, caed the TSR, to reduce a compicated deivery tree to a star in a data grid without impacting the evauation accuracy and correctness of data transmission deays for a co-aocation scheme. The deivery star simpifies the evauation and anaytica compexities of the network deay time of a coaocation scheme. Nevertheess, the TSR is aso appicabe to reducing the compexity of a deivery tree in other distribution and parae systems, e.g., a grid computing, coud computing, and PP system. ACKNOWLEDGMENT: The work was partiay supported by TungHai niversity under the project GREENs and the Nationa Science Counci, Taiwan under Grants NSC 0- -E MY3, NSC 00--E and NSC 0-68-E MY3. REFERENCES [] S. Vazhkudai, Enabing the co-aocation of grid data transfers, Proceedings of Internationa Workshop on Grid Computing, pp. 44 5, Phoenix, Arizona, SA, 7 November 003. [] R.S. Chang, C.M. Wang, and P.H. Chen, Repica seection on coaocation data grids, Proceedings of Second Internationa Symposium on Parae and Distributed Processing and Appications, vo. 3358, pp Springer, Heideberg, 004. [3] R.S. Bhuvaneswaran, Y. Katayama, and N. Takahashi, Dynamic coaocation scheme for parae data transfer in grid environment, Proceedings of First Internationa Conference on Semantics, Knowedge, and Grid, Beijing, China, vo. 7, pp. 8 9, Nov [4] C.T. Yang, I.H. Yang, S.Y. Wang, K.C. Li, and C.H. Hsu, A recursivey-adjusting co-aocation scheme with a cyber-transformer in data grids, Future Generation Computer Systems, vo. 5(7). Esevier B.V: pp , Amsterdam, Juy 009. [5] M.C. Lee, F.Y. Leu, and Y.-p. Chen, PFRF: An Adaptive Data Repication Agorithm based on Star-Topoogy Data Grids, Future Generation Computer systems, vo. 8, issue 7, pp , Juy 0. [6] I. Foster, C. Kesseman, and S. Tuecke, "The anatomy of the grid: enabing scaabe virtua organizations," Internationa Journa of High Performance Computing Appications, vo. 5, pp. 00, 00. [7] M. Coates, A. Hero, R. Nowak, and B. Yu, Internet tomography, IEEE Signa Processing Magazine, vo. 9, issue 3, Jan. 00. [8] G. Levitin, Y.S. Dai, and B.H. Hanoch, Reiabiity and performance of star topoogy grid service with precedence constraints on subtask execution, IEEE Transactions on Reiabiity, vo. 55, no. 3, pp , Sept [9] L. Keinrock, Queuing Systems. In: Theory vo., Wiey, New York (975). 683