An Admission Control Algorithm in Cloud Computing Systems

Transcription

1 An Admsson Control Algorthm n Cloud Computng Systems Authors: Frank Yeong-Sung Ln Department of Informaton Management Natonal Tawan Unversty Tape, Tawan, R.O.C. ysln@m.ntu.edu.tw Yngje Lan Management Scence and Informaton Systems Guanghua School of Management Bejng, Chna ylan@gsm.pku.edu.cn Yu-Wen Chou Department of Informaton Management Natonal Tawan Unversty Tape, Tawan, R.O.C. r02029@m.ntu.edu.tw Yu-Shun Wang Department of Informaton Management Natonal Tawan Unversty Tape, Tawan, R.O.C. d98002@m.ntu.edu.tw Presenter: Yeong Sung Ln 1

2 Agenda I. Introducton II. Soluton Approach III. Experment Results IV. Conclusons 2

3 I. Introducton A. Descrpton and Scenaro B. Mathematcal Formulaton 3

4 A. Descrpton and Scenaro End users of cloud computng hope cloud servce provders can offer hgh qualty cloud envronments that satsfy ther ndvdual requrements. Cloud computng servce provders wsh to maxmze ther resource utlzaton, whle maxmzng ther profts. To make effcent use of ther cloud computng system resources whle ensurng those resources avalablty to the end users, servce provders need to adopt a sutable load balance technque. Hence, to provde an effcent cloud servce to users whle maxmzng proft margns at the same tme s mperatve to cloud servce provders. 4

5 A. Descrpton and Scenaro (Cont d) We proposed a generc mathematcal programmng model that can be used for developng an algorthm for allocatng resources and vrtual machnes. The algorthm smulates the roles of the cloud servce provder and users n a cloud computng system. The Lagrangean Relaxaton Method was adopted here to obtan the optmal soluton for the problem of allocaton. 5

6 A. Descrpton and Scenaro (Cont d) Notaton B s k B c P k M k H k V k Descrpton nternal communcaton bandwdth of server k S shared communcaton bandwdth wthn the cloud computng system total number of CPU cores n server k S total amount of RAM capacty n server k S total amount of HD capacty n server k S the maxmum number of VMs allowable on server k S 6

7 B. Mathematcal Formulaton Problem Assumptons Vrtual machnes leased by specfc users cannot be shared wth others. Rewards ganed from each user s satsfacton s not n complete accord. Some users vrtual machnes should not be n the same physcal machne due to prvacy consderatons. The CPU core capablty of each physcal machne vares. The number of vrtual machnes each user needs vares. 7

8 Gven Parameter Gven Parameter Notaton Descrpton the ndex set of physcal servers n the cloud computng system, whch s S equal to {1,2,3,,s} B k s B c P k k M k H k V k nternal communcaton bandwdth of server k S shared communcaton bandwdth wthn the cloud computng system total number of CPU cores n server k S processng capablty of each CPU core n server k S total amount of RAM capacty n server k S total amount of HD capacty n server k S the maxmum number of VMs allowable on server k S 8

9 Gven Parameter Notaton Descrpton D W the ndex set of demands, hereafter referred to as ether VPCs or users nterchangeably, onto the cloud computng system, whch s equal to {1,2,3,,d} the reward of admttng user D (user can be admtted only f hs/her demands on all types of resources be fully satsfed) the ndex set of VMs requred by user D, whch s equal to {1,2,3,,w } (w s also referred to as the total number of VMs requred by user D) p j m j h j c jk a jkl total amount of CPU processng capablty requred by user D on VM j W total amount of RAM capablty requred by user D on VM j W total amount of HD capablty requred by user D on VM j W total amount of communcaton channel capacty requred by user D between VMs j and k W the ndcator functon whch s 1 f VM j W and VM l W k are allowed to be allocated on the same physcal server, where users and k are n D, and 0 otherwse 9

10 Decson Varables Decson Varables Notaton Descrpton y 1 f user D s admtted to the cloud computng system, and 0 otherwse x jk t jk f jkl 1 f VM j W of user D s allocated to server k S, and 0 otherwse the number of CPU cores allocated to VM j of user on server k 1 f VM j W of user D and VM k W of user D are allocated to server l S, and 0 otherwse 10

11 Objectve Functon The problem s then formulated as the followng problem to maxmze the total revenue: max d 1 y (IP 1) 11

12 Constrants yw x, D jk j W k S (.1) c D j W hx H, k S j jk k (.7) c x p t, k S jk j jk k ( c. 2) D j W x V, k S jk k (.8) c x x a 1, D, j W, jm klm jkl k D, l W, m S D j W, k X, k j D j W D j W k c f B S jk jkl l t P, k S jk k mx M, k S j jk k (.3) c ( c. 4) (.5) c ( c. 6) D j W, k W, k j k S c c (1 f ) B, l S jk jkl x 1, D, j W jk y 0or 1, D x 0or 1, D, j W, k S jk t 0or 1, D, j W, k S jk f 0or 1, D, j W, k W, l S jkl ( c. 9) ( c.10) ( c.11) x x f +1, D, j W, k W, l S jl kl jkl ( c.12) 12

13 II. Soluton Approach A. Lagrangean Relaxaton B. Gettng Prmal Feasble Solutons 13

14 Lagrangean Relaxaton 14

15 Lagrangean Relaxaton (Cont d) Z d D(,,,,,,,,,, ) mn y 1 [ yw ( x )] [ x p t ] 1 2 jk jk jk j jk k D j W k S D j W k S [ x x a 1] D j W k D l W m S l S [( k 4 l jk jkl D j W, k W, k j 3 jklm jm klm jkl c f S ) B ] [( t ) P ] [( m x ) M ] 5 6 k jk k k j jk k k S D j W k S D j W [( hx ) H] [( x ) V] 7 8 k j jk k k jk k k S D j W k S D j W [( c (1 f ) B 9 l jk jkl l S D j W, k W, k j [( x ) 1] [ x x f 1] j jk jkl jl kl jkl D j W k S D j W k W l S l c ] (LR 1) 15

16 Lagrangean Relaxaton (Cont d) Subject to: y = 0 or 1, D (LR 1.1) t jk 0, D, j W, k S (LR 1.2) x jk = 0 or 1, D, j W, k S (LR 1.3) f jkl = 0 or 1, D, j W, k W, k j, l S (LR 1.4) 16

17 Lagrangean Relaxaton (Cont d) The Lagrangean multplers μ 1, μ 2, μ 3, μ 4, μ 5, μ 6, μ 7, μ 8, μ 9, μ 10, μ 11 3 are the vectors of {μ 1 }, {μ 2 jk }, {μ jklm }, {μ 4 l }, {μ 5 k }, {μ 6 k }, {μ 7 k }, {μ 8 k }, {μ 9 l }, {μ 10 j } and {μ 11 jkl }, respectvely, where μ 1, μ 2, μ 3, μ 4, μ 5, μ 6, μ 7, μ 8, μ 9, μ 10, μ 11 are non-negatve. In order to solve (LR 1), t s decomposed nto four ndependent and easly solvable subproblems, as shown below. 17

18 Subproblem 1.1 related to decson varable y : Z y w sub 1 1 sub1.1 ( ) mn ( ) ( 1.1) D Subject to: y 0or 1, D ( sub 1.1.1) 18

19 Lagrangean Relaxaton (Cont d) Subproblem 1.1 (sub 1.1) s further decomposed nto D ndependent mnmzaton subproblems. In Subproblem 1.1 (sub 1.1), decson varable y has two optons. The y can be decded by examnng the coeffcent γ + μ 1 w. When γ + μ 1 w s negatve or zero, we set y to 1. Otherwse, we set y to 0. The tme complexty of Subproblem 1.1 (sub 1.1) s O( D ). 19

20 Subproblem 1.2 related to decson varable t jk : Z t P sub sub1.2 (, ) mn jk ( k jk k ) k k ( 1.2) D j W k S k S Subject to: t 0, D, j W, k S ( sub 1.2.1) jk 20

21 Lagrangean Relaxaton (Cont d) Subproblem 1.2 (sub 1.2) can be further decomposed to D W S subproblems. Ths subproblem can be smply and optmally solved by examnng the coeffcent of decson varable t jk. If the coeffcent μ 5 k μ 2 jk π k s negatve or zero, the value of t jk s set the bggest value; conversely, f μ 5 k μ 2 jk π k s bgger than zero, t jk s set to zero. The tme complexty of Subproblem 1.2 (sub 1.2) s O( D W S ). 21

22 Subproblem 1.3 related to decson varable x jk : Z sub1.3(,,,,,,, ) mn ( xjk jk xjk pj k mj xjk k hj xjk k xjk j xjk j ) D j W k S ( x x ) ( x x a 1) 11 3 jkl jl kl jklm D j W k W l S D j W k D l W m S k jm klm jkl ( km k kh k kvk) ( sub 1.3) k S Subject to: x jk = 0 or 1, D, j W, k S ( sub 1.3.1) 22

23 Lagrangean Relaxaton (Cont d) In order to solve the subproblem 1.3 (sub 1.3), we frst reformulate (sub 1.3) by applyng [1], as shown below. [1] Frank Yeong-Sung Ln, Quas-statc Channel Assgnment Algorthms for Wreless Communcatons Networks, Informaton Networkng, (ICOIN-12) Proceedngs, Twelfth Internatonal Conference, pp , Jan

24 Subproblem 1.3 related to decson varable x jk : Z sub1.3(,,,,,,, ) mn ( xjk jk xjk pj k mj xjk k hj xjk k xjk j xjk j ) D j W k S x ( x x a 11 3 jkl jl jklm jm klm jkl j W D l S D j W k D l W m S k ( km k kh k kvk) ( sub 1.3') k S 1) Subject to: x jk = 0 or 1, D, j W, k S ( sub 1.3.1') 24

25 Lagrangean Relaxaton (Cont d) In Subproblem 1.3 (sub 1.3 ), decson varable x jk has two optons. As a result, the value of x jk can be determned by applyng exhaustve search. The tme complexty of Subproblem 1.3 (sub 1.3 ) s O( D W S ). 25

26 Subproblem 1.4 related to decson varable f jkl : Z c f c f sub1.4(,, ) mn ( l jk jkl l jk (1 jkl )) l S D j W, k W, k j c ( f 1) ( B B ) ( sub 1.4) 11 4 S 9 jkl jkl l k l D j W k W l S l S Subject to: f 0or 1, D, j W, k W, l S ( sub 1.4.1) jkl 26

27 Lagrangean Relaxaton (Cont d) Subproblem 1.4 (sub 1.4) can be optmally solved through analyzng the composton of (sub 1.4). When σ l S σ D σ j W,k W,j k(μ l 4 c jk f jkl + μ l 9 c jk (1 f jkl )) of subproblem 1.4 (sub 1.4) has mnmum value and σ D σ j W σ k W σ l S μ 11 jkl (f jkl + 1) σ l S (μ 4 l B S k + μ 9 l B c ) of subproblem 1.4 (sub 1.4) has maxmum value, (sub 1.4) s mnmzed. The tme complexty of Subproblem 1.4 (sub 1.4) s O( D W W S ). 27

28 Gettng Prmal Feasble Solutons By applyng Lagrangean Relaxaton method, a theoretcal lower bound on prmal objectve functon can be found. Moreover, t provdes some suggestons for obtanng prmal feasble solutons. However, the result of the dual problem may be nvald to the orgnal problem snce some mportant and complex constrants are relaxed. Therefore, a heurstc s needed here to tune nfeasble solutons feasble. In order to obtan prmal feasble solutons and an upper bound of (IP 1), the outcome of (LR 1) and Lagrangean multplers are used as hnts for dervng solutons. The concept of the proposed heurstc s descrbed below. 28

29 Gettng Prmal Feasble Solutons (Cont d) Recall that subproblem 1.1 s related to decson varable y, whch determnes whether the user s admtted by cloud servce provder or not. The concept used to develop a Drop and Add Heurstc to obtan prmal feasble solutons by droppng and addng users to cloud envronment. Intally, we check whether the outcome of (LR 1) can satsfy all constrants of (IP 1). If one of the constrants of (IP 1) s not satsfed, Drop and Add Heurstc wll be used. There are two steps n Drop and Add Heurstc and the frst step s droppng step. 29

30 Gettng Prmal Feasble Solutons (Cont d) The man purpose of the droppng step s to drop users one by one through adjustng the number of users or partcular users we admt. By usng the hnt provded by subproblem 1.1, we propose three strateges. In the frst strategy, subproblem 1.1 has the mnmum value when μ 1 w s mnmzed. Hence, we drop the bggest μ 1 w one after another. In the second strategy, once γ has the smallest value, the subproblem 1.1 can obtan the optmal soluton. Therefore, we drop the bggest γ one by one. The thrd strategy s smlar to the frst and second strategy. When γ + μ 1 w s mnmzed, subproblem 1.1 s also mnmzed. As a result, we drop the bggest γ + μ 1 w one after another. 30

31 Gettng Prmal Feasble Solutons (Cont d) The droppng process keeps droppng users sequentally accordng the gven strateges mentoned above, and t stops whle all the decson varables meet the constrans of (IP 1). The process n droppng step reallocates the remanng users VMs as much centralzed as possble (on the same physcal machne) under the premse that all constrans of (IP 1) are met. Ths reduces the severty of nternal fragmentaton, and we named such process as compresson. 31

32 Gettng Prmal Feasble Solutons (Cont d) When we drop users sequentally, the degree of volaton of constrants are also reduced. However, when the number of users s decreased, the rewards are also reduced. So, when we reduce the number of users that satsfy all constrants of (IP 1), we should consder about admttng more users to gan more rewards. 32

33 Gettng Prmal Feasble Solutons (Cont d) The second step of Drop and Add Heurstc s addng step. There are also three strateges. The frst strategy consders the rewards that users gve to cloud servce provder. We add users wth the bggest reward one by one. In the second strategy, the cloud servce provder has to spend resources for admttng users nto cloud envronment. Consequently, we admt users who has the smallest requrement one after another. At last, we use the net proft to decde the order of admttng users. The users wth the bggest net profts are added nto our system one by one. 33

34 Gettng Prmal Feasble Solutons (Cont d) The addng process consders the resource lmts of physcal machnes and the upper lmt of VM nstances. When the gven constrans are met, the cloud servce provder allocates the VM j of user on physcal machne k, and check over all constrans of (IP 1). Then executes these actons repeatedly untl all constrans of (IP 1) are satsfed. Every user requests wll perform the compresson process to mnmze nternal fragmentaton. 34

35 Gettng Prmal Feasble Solutons (Cont d) 35

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38 III. Experment Results 38

39 Experment Results To prove the LR algorthm and the proposed heurstc are effectve, we used the optmal software Lngo for comparson purposes. Before that, we compared the dfferent drop and add strategy combnatons n the LR algorthm to determne whch drop and add combnaton s more effectve. 39

40 Experment Results (Cont d) Parameter settng User demand (ncludng CPU, RAM, HD respectvely) Level 1: Hghest demand (150) Level 2: Medum demand (80) Level 3: Lowest demand (60) Cloud servce provder Total number of VM: 50 Total number of physcal server: 50 Total number of CPU cores, computng capablty, RAM and HD: 3,000 40

41 Experment Results (Cont d) Case 1: drop users by μ 1 w Case2: drop users by γ Case3: drop users by γ + μ 1 w Case 1: add users by ther rewards (γ ) Case 2: add users by ther costs (μ 1 w ) Case 3: add users by ther net profts ( γ + μ 1 w ) Reward 7,100 Reward 6,800 Reward 7,100 Tme (m) Tme (m) Tme (m) Reward 7,100 Reward 6,800 Reward 7,100 Tme (m) Tme (m) Tme (m) Reward 7,400 Reward 7,100 Reward 7,400 Tme (m) Tme (m) Tme (m)

42 Experment Results (Cont d) Reward of Dfferent Algorthm 42

43 Experment Results (Cont d) Tme Consumpton of Dfferent Algorthm 43

44 Experment Results (Cont d) Gap of Dfferent Algorthms (20, 50, 15) (30, 50, 15) (40, 50, 15) Lagrangean Relaxaton 0% 4.17% 5.71% Lngo (Local Soluton) 0% 10.34% 10.67% Lngo (Global Soluton) 0% 0% 0% 44

45 IV. Conclusons 45

46 Conclusons The man contrbuton of ths research s the producton of a generc mathematcal model that can develop an algorthm for allocatng resources and vrtual machnes. From the experment result, we found that our proposed Lagrangean relaxaton-based algorthm and drop and add heurstc show excellent performance n effectveness and effcency n comparson wth Lngo. From the outcomes of these experments, we conclude that our research can be effectvely appled n real-world stuatons. 46

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