Minimisation of the Average Response Time in a Cluster of Servers

Transcription

1 Mnmsaton of the Average Response Tme n a Cluster of Servers Valery Naumov Abstract: In ths paper, we consder task assgnment problem n a cluster of servers. We show that optmal statc task assgnment s tantamount to equalzng an approprate cost functons assocated wth the servers. We also propose an mprovement of dynamc Shortest Expected Delay (SED) task assgnment polcy. Key words: Dstrbuted System, Task Assgnment, Average Response Tme.. INTRODUCTION Task assgnment polcy s an mportant factor affectng the performance of a dstrbuted system because t coordnates the use of processng capacty of servers. Hghspeed Web clusters [] and Internet routers [] mplement varous task assgnment polces. An mportant element of a task assgnment polcy s the nformaton t requres to operate. In general, dynamc polces operate under tme dependent nformaton, whereas statc polces operate under tme ndependent characterstcs of the system [3]. tasks dspatcher n servers Fgure. Task assgnment n a cluster of servers. We consder the task assgnment problem n a cluster of servers where dfferent servers exhbt dfferent task processng tmes as shown n Fgure. Tasks arrve to dspatcher, whch s responsble for dstrbutng them among servers accordng to a task assgnment polcy takng nto account avalablty of resources at servers. We are nterested n polces, whch mnmze the average response tme and dstngush two optmsaton problems. Optmzaton crteron n the Mnmum Average Response Tme (MART) problem s to mnmze the average response tme taken over all processed task. In the Mnmum of the Maxmum Response Tmes (MMRT) problem, the crteron of optmalty s to mnmze the maxmum of average task response tmes at the servers, to whch tasks are routed. Statc MART problem has been consdered n several papers. Tantaw and Towsley [4] studed an arbtrarly connected dstrbuted system. Tasks may arrve at any server and can be executed ether locally or be sent to another server. They derved an teratve algorthm that determned the optmal statc task assgnment polcy for a system wth general response tme and communcaton delay functons. Km and Kameda mproved ths algorthm n [5]. Buzen and Chen [6] derved equatons for optmal arrval rates at servers for a cluster of servers wth Posson task arrval process and generally dstrbuted task processng tmes. N and Hwang [7], and Tang and Chanson [8], found closed form soluton for partcular case when task-processng tmes are exponentally dstrbuted. Mnmzaton of the maxmum of average task response tmes may be unfar snce the average response tme at the slower servers can be much hgher than the average response tme at the faster ones. Georgads et al. n [3] derved soluton of the statc MMRT problem for a cluster of servers wth general response tme functons. Optmal polcy attans farness by equalzng the average response tmes at actve servers, at an ncrease n average response tme taken over all processed tasks. Dynamc task assgnment polces usually outperform statc polces [9], but optmal dynamc polcy s unknown because of analytcal dffcultes [0]. Chow and Kohler [] proposed smple dynamc Mnmum Response Tme polcy. In Mnmum Response Tme, s s s n µ µ µ n - II.- -

2 also known as Shortest Expected Delay (SED), an arrvng task s routed to the server wth the least expected response tme,.e. the server, for whch ( s + ) µ s mnmal, where s s the number of tasks at server ncludng the one n servce, and µ s ts task processng rate. In ths paper, we analyse optmal solutons for statc task assgnment polces. Based on ths analyss we propose new dynamc task assgnment polcy - Modfed SED, whch s as smple as SED but results n smaller average response tme.. OPTIMAL STATIC POLICIES Consder a cluster of n servers. We denote the rate of tasks arrvng at the system, the rate of tasks arrvng at server, θ the maxmum task arrval rate that server can sustan wthout becomng saturated, and Θ = θ + K+θn the maxmum throughput of the system. The mean processng tme needed to execute the task at server at the absence of other tasks arrved from dspatcher, s denoted by β. In what follows we assume that servers are numbered n nondecreasng order of the mean processng tmes β,.e. β β K β n, and t wll be convenent to defne an addtonal quantty, β n+ =. The th server wll be called faster than the j th server f β β j... Equalzaton Problem Assume that the th server has a cost functon c ( ) assocated wth t, and the followng condtons hold: ) c (x) s strctly ncreasng and contnuous for x ( 0, θ ), ) lm c =, x θ 3) lmc = β > 0. x 0 The load can be dstrbuted among the fastest servers, so that the values of cost functons c ( ) at all these servers are equalzed. In ths case the arrval rates form a soluton of the followng equalzaton problem: Equalzaton problem: For a gven, 0 < < Θ, fnd an nteger k, k n, and task arrval rates 0 < < θ, =, Kk, so that + + K + k = and c ( ) = c( ) = L = c k ( k ) βk +. Georgads et al. present n [3] the soluton of the equalzaton problem. Let g (x), x > β be the nverse of c (x), and set g = 0 for 0 x β. Server s called actvated f the arrval rate to ths server s nonzero. As the task arrval rate ncreases from 0 to Θ, the number of actvated servers ncreases from to n. Server s actvated when exceeds a threshold A, called the actvaton rate for the server. The server actvaton rates can be computed by k Ak = g ( βk ), k n, = and have the followng propertes: a) A = 0, b) A A K A n < An + = Θ, c) A = A j when β = β j. When Ak < < Ak +, then the arrval rates at servers satsfy c ( ) = c ( ) = L = ck ( k ) = E( ), where β k < E( ) < βk +, and = 0 for > k. Detals of the algorthm for calculaton of the arrval rates for arbtrary can be found n [3]. - II.- -

3 .. Mnmzaton of the Average Response Tme The response tme functon R (x) specfes the average response tme of a task at server for a task arrval rate x to that server. It s well known that for server modelled as M/M/ queue R (x) s gven by smple formula R = ( µ x) []. If the servce tmes are not exponentally dstrbuted and/or the server processes addtonal dedcated arrval stream then the response tme functon would be more complex functon. We assume that the response tme functons satsfy the followng condtons: ) R (x) s strctly ncreasng and contnuous for x ( 0, θ ), ) lm R =, x θ 3) lm R = β > 0. x 0 Mnmum of the Maxmum Response Tme and Mnmum Average Response Tme problems for a cluster of servers can be formulated as follows. Mnmum of the Maxmum Response Tmes (MMRT) problem: For a gven, 0 < < Θ, fnd task arrval rates at servers 0 < θ, =, Kn, so that + + K + n = and the maxmum average response tme max { R ( )} s > 0 mnmzed. Mnmum Average Response Tme (MART) problem: For a gven, 0 < < Θ, fnd task arrval rates at servers 0 < θ, =,K n, so that the + + K + n = and n average overall response tme R( ) = R ( ) s mnmzed. = Both problems are tantamount to the equalzaton problem. Georgads et al. show n [3] that the soluton of the MMRT problem can be found as a soluton of the equalzaton problem wth the cost functon gven by c = R. Tantaw and Towsley studed n [4] the problem of mnmzaton of the average response tme for an arbtrarly connected dstrbuted system assumng that the response tme functons R (x) are dfferentable and convex for x ( 0, θ ). It follows from [4] that the soluton of the MART problem can be found d( xr x as a soluton of the equalzaton problem wth the cost functon gven by c x ( )) ( ) =. dx 3. MODIFIED SED POLICY For partcular case, when task arrval process s Posson and task processng tmes are exponentally dstrbuted, we have θ = µ, β = µ. The cost functons c (x) and ts nverse g (x) are gven by µ, >, c =, ( ) = xµ g x x for MMRT problem, µ x 0, 0 < xµ, µ µ c =, g ( ) =, >, x µ xµ x for MART problem. ( µ x) 0, 0 < xµ, From results presented n the prevous secton we may conclude that equalzaton of functons c ( x ),.e. the average response tmes, results n mnmzaton of the maxmum response tme ncurred on any actve server. Whle equalzaton of ether functons c ( x ) or c ( x ), whch are the average response tmes multpled by square roots of task - II.-3 -

4 processng rates, results n mnmzaton of the average response tme taken over all processed tasks. In SED polcy, an arrvng task s sent to the server, for whch ( s + ) µ s mnmal, where s s the number of tasks at server. In other words, SED polcy equalzes response tmes at actve servers, that leads to mnmzaton of the maxmum response tme ncurred on any server. We modfy SED so, that n Modfed SED (MSED), an arrvng task s sent to the server, for whch ( s + ) µ s mnmal. MSED polcy equalzes response tmes multpled by square roots of task processng rates, and thus, accordng to prevous consderatons, would mnmze the average response tme taken over all processed tasks. We smulate three dfferent systems, whch were used n [9], and compare SED and MSED. The frst system, System, has 0 nodes. The task processng rates of the nodes are µ = 6, µ = µ 3 = K = µ 0 =. The second system, System, also has 0 nodes. The task processng rates of the nodes form an arthmetc seres, that s µ = 3, =, K0. Each of these two systems has an aggregate processng rate of 5. The thrd system, System 3, has 8 nodes, and the task processng rates of the nodes form the geometrc seres µ = 0, =, K0. In all cases, the arrval process s assumed to be Posson. Fgures and 3 show the average response tmes taken over all processed tasks under SED (sold lnes) and MSED (dash lnes) versus task arrval rate. System System System 3 Fgure. Average response tmes, when task processng tmes are exponentally dstrbuted. System System System 3 Fgure 3. Average response tmes, when task processng tmes are constant. Results shown n Fgure demonstrate that MSED polcy outperforms SED when task-processng tmes have exponental dstrbutons. Fgure 3 shows that MSED polcy also outperforms SED for systems wth constant task processng tmes. In ths case, the dfferences n average response tmes are even hgher then n the case of exponental dstrbutons. - II.-4 -

5 3. CONCLUSION We show that the statc polcy, that equalzes task response tmes at actve servers, does not mnmze average overall response tme. Well-known dynamc Shortest Expected Delay assgnment polcy tres to mnmse the average overall response tme by sendng new task to a server wth mnmal expected response tme. We propose modfcaton to SED polcy. In Modfed SED, an arrvng task s sent to the server, for whch expected response tmes multpled by square roots of task processng rates s mnmal. Modfed SED s as smple as SED but results n smaller the average overall response tme at hgh load. REFERENCES [] Cardelln V., E. Casalccho. The State of the Art n Locally Dstrbuted Web- Server Systems, ACM Computng Surveys, Vol. 34, No., June 00, pp [] Youns O., S. Fahmy. Constrant-Based Assgnment n the Internet: Basc Prncples and Recent Research, IEEE Communcatons Surveys, Vol. 5, No., 003, pp. -3. [3] Casavant T.L., J.G. Kuhl. A Taxonomy of Schedulng n General-Purpose Dstrbuted Computng Systems, IEEE Transactons on Software Engneerng, Vol. 4, No., February 988, pp [4] Tantaw A.N., D. Towsley. Optmal Statc Load Balancng n Dstrbuted Computer Systems, Journal of the ACM, Vol. 3, No., Apr. 985, pp [5] Km C., H. Kameda. An Algorthm for Optmal Statc Load Balancng n Dstrbuted Computer Systems, IEEE Transactons on Computers, Vol. 4, No. 3, March 99, pp [6] Buzen J.P., P.P.-S. Chen. Optmal Load Balancng n Memory Herarches, Proc. IFIP 974 (J.L. Rosenfeld ed.), North-Holland, Amsterdam, 974, pp [7] N L.M., K. Hwang. Optmal Load Balancng n a Multple Processor System wth Many Task Classes. IEEE Transactons on Software Engneerng, Vol., No. 5, 985, pp [8] Tang X., S.T. Chanson. Optmzng Statc Task Schedulng n a Network of Heterogeneous Computers, Proc. of ICPP 000, Aug. 000, pp [9] Banawan S.A., N.M. Zedat. Comparatve Study of Load Sharng n Heterogeneous Multcomputer Systems, Proc. 5th Annual Smulaton Symposum, Orlando, Florda, USA, Apr. 6-9, 99, pp [0] Boxma O., G. Koole, Z. Lu. Queueng-Theoretc Soluton Methods for Models of Parallel and Dstrbuted Systems, Performance Evaluaton of Parallel and Dstrbuted Systems - Soluton Methods, (O.J. Boxma and G.M. Koole eds.), CWI, Amsterdam, 994, pp. -4. [] Chow Y.C., W.H. Kohler. Models of dynamc load balancng n a heterogeneous multple processor system. IEEE Transactons on Computers, Vol. C-8, No. 5, May 979, pp [] Klenrock L., Queueng Systems, Vol., Theory, Wley Interscence, 975. [3] Georgads L., C. Nkolaou, A. Thomasan. A Far Workload Allocaton Polcy for Heterogeneous Systems, J. Parallel Dstrb. Comput., Vol. 64, No. 4, 004, pp ABOUT THE AUTHOR Prof. Valery Naumov, Laboratory of Communcaton Engneerng, Lappeenranta Unversty of Technology, Fnland, Phone: , E-mal: valery.naumov@lut.f. - II.-5 -