Queuing Network Approximation Technique for Evaluating Performance of Computer Systems with Input to Terminals

Transcription

1 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance ueung Netwo Appoxmaton Technque fo Evaluatng Pefomance of Compute ytem wth Input to Temnal Ha Yoh Nozom Myamoto Mohamad Zan Nuhafqah Bnt Daue Myae Itau oe and Tohyu nohta chool of Compute cence Toyo Unvety of Technology Hacho Toyo Japan Abtact: ueung netwo technque ae effectve fo evaluatng the pefomance of compute ytem. We dcu a queung netwo technque fo a compute ytem wth nput to temnal. The fnte numbe of temnal ext n the netwo and a ob ave andomly fom outde of the temnal. Afte a thn-tme at the temnal the ob move to the eve and t acque ome pat of memoy and execute CPU and I/O poceng n the eve. Afte the ob complete CPU and I/O poceng t eleae the memoy and goe bac to t own temnal. Howeve becaue the temnal and the memoy eouce can be condeed a a econday eouce fo the CPU and I/O equpment the queung netwo model ha no poduct fom olu-ton and cannot be calculated the exact oluton. We popoed hee an appoxmaton queung netwo technque fo calculatng the pefomance meaue of a compute ytem wth nput to temnal on whch multple type of ob ext. Th technque nvolve dvdng the netwo nto two level; one nne level n whch a ob execute CPU and I/O poceng and the othe oute level that nclude temnal and communcaton lne. By dvdng the netwo nto two level we can pevent the numbe of tate of the netwo fom nceang and appoxmate the pefomance meaue of the netwo. We evaluated the popoed appoxmaton technque by ung numecal expement and clafed the chaactetc of the ytem epone tme and the mean numbe of ob n both level. eywod: pefomance evaluaton queung netwo cental eve model a compute ytem wth nput to temnal. Intoducton ueung netwo technque ae effectve fo evaluatng the pefomance of compute ytem. In compute ytem two o moe ob ae geneally executed at the ame tme whch caue delay due to conflct n acceng hadwae o oftwae eouce uch a the CPU I/O equpment o data fle. We can evaluate how th delay affect the compute ytem pefomance by ung a queung netwo technque. ome queung netwo have an explct exact oluton whch called a poduct fom oluton []. Wth th oluton we can ealy calculate the pefomance meaue of compute ytem fo example the buy ato of hadwae and the ob epone tme. Howeve when the excluon contol ae actve o when a memoy eouce ext the queung netwo doe not have a poduct fom oluton. To calculate an exact oluton of a queung netwo that doe not have a poduct fom oluton we have to contuct a Maov chan that decbe the tochatc chaactetc of the queung netwo and numecally olve t equlbum equaton. When the numbe of ob o the amount of hadwae n the netwo nceae the numbe of tate of the queung netwo datcally nceae. nce the numbe of tate of the queung netwo the ame a the numbe of unnown quantte n the equlbum equaton the numbe of unnown quantte n the equlbum equaton alo datcally nceae. Theefoe we cannot numecally calculate the exact oluton of the queung netwo. Moeove when the queung netwo an open model whee ob ave fom o depat fo the outde of the netwo the numbe of tate of the netwo can become nfnte the numbe of ob can be nfnte and we cannot actually calculate an exact oluton. Hee we dcu the queung netwo model fo compute ytem wth nput to temnal Fgue.. In the model the ob ave andomly fom the outde to the netwo and acque a temnal. If all temnal ae occuped the 06

2 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance ob on the ytem watng queue and wat untl a temnal become avalable. Afte a thn-tme at the temnal the ob move to the eve and acque ome pat of the memoy and exe-cute CPU and I/O poceng. When the ob complete CPU and I/O poceng at the eve t eleae the memoy and goe bac to t own temnal. Th model eemble that a cutome ente and leave an ATM Automated Telle Machne temnal. nce a ob execute CPU and I/O poceng occupyng the temnal and the memoy they can be condeed a a econday eouce fo the CPU and I/O equpment. Geneally when a queung netwo nclude a econday e-ouce t doe not have poduct fom oluton To get the tct oluton of the Model we have to contuct a Maov chan whch decbe the ente model and have to numecally olve t equlbum equaton. In ode to pevent the numbe of tate of the Maov chan fom nceang we dvde the model nto two level one oute level that nclude the temnal and communcaton lne and the othe nne level that nclude CPU I/O equpment and memoy eouce Fgue. Fg. Cental eve model wth nput to temnal Fg. Concept of appoxmaton In the ame way a [8][9] thee ae multple type of ob cla ext n the netwo. Each ob cla behave dffeently n the oute level and the nne level. Although both the nne level and the oute level ha a poduct fom oluton when the model ha oluton when the model ha a ngle ob cla the both level doe not have a poduct fom oluton when the model ha multple ob clae. Theefoe an appoxmaton technque fo the both level needed to analye t pefomance meaue. In th pape we have popoed an appoxmaton technque fo calculatng the pefomance meaue of a compute ytem wth nput to temnal. We pevouly epoted multple ob cla ncludng memoy eouce model aved andomly fom the outde [8] and a model n whch a ob move bac and foth between a temnal and a netwo [9]. In th eeach we epot a model n whch a ob ave va a temnal fom the outde ncludng the memoy eouce. Dvdng the model nto two level one of two-laye queung netwo technque [3]. Ou popoed technque alo a two-laye technque fo a compute ytem wth nput to temnal. In ou pevou tudy [4] we epoted an appoxmaton technque fo evaluatng pefomance of compute ytem wth fle eouce. Meanwhle heteoge- 07

3 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance neou paallel compute ytem wth dtbuted memoy wa eeached n [6] and the Maov chan nvolvng two dmenonal tate tanton mla to ou popoed model wa dcued n [7].. Model Decpton The CPU and I/O model n the nne level equvalent to the odnay cental eve model wth multple ob type each of whch called a ob cla. In th model R ob clae ext and each of them numbeed = R by affxng. We denote n a the numbe of ob of ob cla n the cental eve model and n a the total numbe of ob n the cental eve model. We alo denote a the numbe of ob of ob cla n the nne level the dffeence of n and the numbe of ob n the ytem watng queue. The nne level cont of ngle CPU node and multple I/O node. We denote M a the numbe of I/O node. The I/O node ae numbeed m = M by affxng m and the CPU node numbeed m = 0 by alo affxng m. The evce ate of ob cla at the CPU node η 0 and the evce ate of ob cla at an I/O node m η m. The evce tme at each node a mutually ndependent andom vaable ubect to common exponental dtbuton. Job ae cheduled on a ft come ft eved FCF pncple at all node. At the end of CPU poceng a ob pobabltcally elect an I/O node and move to t o complete CPU and I/O poceng and goe bac to t own temnal. The electon pobab lty of I/O node m of ob cla M p m m M; R and the completon pobablty of ob cla p 0. Theefoe p m m0... R. In the oute level thee ae temnal ext and the ob ave andomly fom the outde to the netwo and acque a temnal. If all temnal ae occuped the ob on the ytem watng queue and wat untl a temnal become avalable. When the ob complete the CPU and I/O poceng n the cental eve model t etun to t own temnal. We denote a the numbe of ob of the ob cla =... R acqung the temnal. R Thee ae not contant and hold. 08

4 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance The ob tay n the temnal fo hot whle. The tayng tme called thn-tme. The thn-tme mutually ndependent andom vaable ubect to common exponental dtbuton wth paamete ν of ob cla ν ob depatue ate fom the temnal. Memoy eouce ae added to th cental eve model Fgue. We denote a the numbe of the unt of the memoy acqued by a ob of ob cla and a the total numbe of the unt of the memoy. Afte the thn-tme a ob of ob cla move to the nne level and equet and acque unt of the memoy befoe enteng the cental eve model. If uffcent unt of the memoy do not avalable the ob on the ytem watng queue and wat fo the memoy to be eleaed by anothe ob. When the ob complete CPU and I/O poceng t eleae the memoy and leave the nne level and goe bac to t own temnal n the oute level. nce the ob ha to acque the memoy befoe enteng the cental eve model the total numbe of unt of the R occuped memoy n the cental eve model ha to be le than o equal to.e. n. By eplacng CPU oute level tanton wth CPU CPU tanton the cental eve model modfed to a cloed model n whch the numbe of ob contant Fgue. 0049n th modfed model when CPU CPU tanton occu we conde a the ob temnate and a new ob bon. Theefoe the mean ob epone tme the mean tme between two ucceve CPU CPU tanton. Th mean that the ob epone tme can be condeed a the ob lfetme. 3. Appoxmaton Model We ue the followng notaton. t : mean thn-tme of ob cla ν : depatue ate fom the temnal of ob cla τ m : mean total evce tme of ob cla at node-m n the cental eve model : numbe of unt of memoy acqued by a ob of ob cla : total numbe of unt of memoy eoce n m : numbe of ob of ob cla at node-m n the cental eve model =... R; m=0... M n : numbe of ob of ob cla n the cental eve model n = n n... n R : vecto of numbe of ob n the cental eve model n =0... : numbe of ob of ob cla n the nne level : numbe of ob of ob cla n the netwo = numbe of temnal of ob cla n * = n 0 n... n M n 0 n... n M... n R0 n R... n RM : tate vecto of the cental eve model M * F n { n nm n nm 0 m 0 M } n n nr m0 R : et of all feable tate of the cental eve model when the numbe of ob of ob cla n P n * : teady-tate pobablty of tate n * T n : mean ob epone tme of the cental eve model when the vecto of numbe of ob n μ n : completon ate fom the cental eve model of ob cla T : ytem epone tme of ob cla = lfetme of ob cla 3.. Inne level nce the cental eve model n the nne level equvalent to the odnay cental eve model wth multple ob clae t ha the poduct fom oluton. Then the teady-tate pobablty Pn* epeented by the followng fomula [][]. 09

5 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance R M nm m whee R M P n * = m 0 nm n n nr M the nomalzng contant of teadytate pobablte when the numbe of ob of ob cla n the cental eve model n =0 ; =... m n n nr M nf n m0 R. Fom thee teady-tate pobablte we can calculate the mean ob epone tme T n of ob cla a n n n nr M Tn when the numbe of ob n the cental eve model n. n n nr M The memoy eouce n nne level can be condeed a an M/M/ queung model wth eve. In an odnay M/M/ queung model the evce ate at a eve contant egadle of the numbe of guet n the evce. In the memoy eouce of ou model howeve the evce ate change dependng on the numbe of occuped memoe. The mean ob epone tme T n of ob cla =... R when the vecto of numbe of ob n = n n... n R equal to the mean tme whle the memoy occuped. nce the completon ate μ n of ob cla fom the cental eve model denoted a μ n μ n alo depend on n = n n... n R that the numbe of ob n the cental T n eve model. The tate tanton of the M/M/ queung model wth two ob clae hown n Fgue 3 whee the completon ate fom the cental eve model change dependng on the numbe of ob n the cental eve model. Th a two dmenonal bth-death poce. The equlbum equaton wth the teady-tate pobablty when the total numbe of the unt of the memoy and the numbe of ob n the nne level ae a follow mla to the cae wth hghe dmenon. Whee the maxmum ntege uch a =0 =0 ν + ν 0 0 = μ μ { - ν + ν + } 0 0 = - + ν { - ν + ν + } 0 0 = - + ν ν = ν { ν + - ν + } 0 0.e. / = - + ν { ν + - ν + } 0 0 = - + ν ν = ν { - ν + - ν + + } = and { - ν + - ν + + } = and { - ν + - ν + + } =

6 0 0 denote the et of the hotet oute fom 0 0 to and thee the lattce pont on a oute u uch a and o. When we denote the teady-tate pobablty along the oute u a the followng equlbum equaton hold. } { can be epeented a. onu u Fo the tate of the Maov chan when all ob ae n the cental eve model and executng CPU and I/O poceng and when ome ob ae n the ytem watng queue and watng fo a pat of the memoy to be eleaed. The tanton dagam of the two dmenonal bth-death poce hown n Fgue 3. Howeve the equlbum equaton do not have the poduct fom oluton. Theefoe ome appoxmaton equed to olve t. When the model ha a ngle ob cla t can be decbed wth a one dmenonal bth-death poce. It tanton dagam hown n Fgue 4 and the equlbum equaton ae a follow: ν 0=μ { - ν + } = - + ν < 3 { - ν + } = - + ν < < 4 = ν - The oluton fo the equlbum equaton ae decbed n the followng poduct fom. 0 0 In th fomula fo the tate tanton at =... - multply by facto whle fo the tate tanton at = +... multply by facto. Fo two dmenon cae we conde a hotet oute fom lattce pont 0 0 to hown n Fgue 5 and fo the hozontal tate tanton at the lattce pont uch a on the oute multply by facto and multply by facto fo the hozontal tate tanton When the lattce pont uch a 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance

7 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance fo the tate tanton outde of the lattce pont uch a and o on the oute between and multply by facto fo the hozon- t a l t a t e t a n t o n o fo the vet- cal tate tanton Thu the coeffcent of elated to 0 0 epeented a the ummaton of the poduct along a l l t h e o u t e f o m 0 0 to. Fo example fo the oute fom 0 0 to when =4 = = and =5 =4 whch the cae of 4 the poduct along the oute of boen lne n Fgue Fo the oute fom 0 0 to 3 whch the cae of the oute the multplcaton along nce thee ae multple oute fom 0 0 to the coeffcent of elated to 0 0 appoxmately epeented a the total of the poduct along all oute fom 0 0 to. mla to the cae above we can appoxmate the tate pobablty of a queung netwo wth multple ob clae when R >. 3.. Oute level Fgue 6 how a tate tanton dagam of the oute level. The oute level alo expeed n a twodmenonal bth-death poce. In the dahed-lne tangle of Fgue 6 all ob ae n a tate of acqung a temnal. Unle the nne level t an open queung netwo whee the numbe of ob n each cla can be nfnte. The equlbum equaton wth the teady-tate pobablty U * = U when the numbe of temnal and the numbe of occuped temnal * = ae a follow mla to the cae wth hghe dmenon. =0 =0 λ +λ U 0 0 = ν 0 U 0 +ν 0 U 0 =... =0

8 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance λ +λ + U 0 0 = λ U U U 3 = +... =0 λ +λ + U 0 0 = λ U 0 U U 4 =0 =... λ +λ = λ U U =0 = +... λ +λ + U 0 0 =λ U 0 U + U =.. =.. λ +λ + + U =λ U λ U + U U = =... =... λ +λ + + U =λ U λ U U + + U > =... =... When the lattce pont l l uch a l +l = oluton the hotet oute l fom 0 0 to and U l the tate pobablty along the oute l of the tate λ +λ +l l l + l U l l l = λ U l 0 λ U l 0 l l l U l + U l. l l l a + > =... =... U b + > = =... U c + > =... = U d + > = = U 4. Numecal Expement l U l l U l l U l0 l U n n l0 We evaluated the popoed appoxmaton technque though numecal expement. We ued the followng paamete.. Numbe of temnal: = 0 5. Thn-tme and aval ate:. Numbe of memoy eouce: = 5 Fgue Numbe of I/O node: M = t t = Total evce tme at each node λ λ = τ 0 =.0 τ =τ =.0 τ 0 =.0 τ =τ =0.5 Fgue 9 0 whee τ m the total evce tme of ob cla t t = at node m. λ λ = whee t the mean thn-tme of ob cla and λ the aval ate of ob cla = 3

9 6th Intenatonal Confeence on Chemcal Agcultual Envonmental and Bologcal cence CAEB-7 Dec Pa Fance Fgue 7 ~ 0 how the mean ytem epone tme and mean numbe of ob a n the nne level of ob clae and epectvely when the aval ate λ fxed and λ change fom 0.05 to 0. by 0.0 and the thn-tme t fxed at.0 and t change fom.0 to.5 by 0.0. The mean ytem epone tme the mean tme fom ob aval to depatue fom the netwo that the mean tme of movng between temnal and the cental eve model plu the thn-tme at the temnal. mla to the cae of a ngle ob cla the mean ytem epone tme fo both ob cla nceae monotoncally n a convex cuve. When the thn-tme of ob cla nceae the mean numbe of ob n the nne level nceae lnealy along nceang of the thn-tme of ob cla. 5. Concluon We popoed an appoxmaton technque fo evaluatng the pefomance of a compute ytem wth nput to temnal ung a queung netwo technque and analyzed t pefomance meaue though numecal expement. The concept of the appoxmaton baed on epaately analyzng the nne level CPU I/O equpment and memoy and the oute level temnal and communcaton lne. The numecal expement clafed the chaactetc of the ytem epone tme. In the futue we plan to examne the accuacy of the popoed appoxmaton technque by compang t wth exact oluton o mulaton eult. Fg. 7: Mean numbe of ob n the cental eve modal Fg.8: Mean ytem epone tme Fg. 9: Mean numbe of ob n the cental eve modal Fg.0: Mean ytem epone tme 6. Refeence [] F. Baett. M. Chandy R. R. Muntz and F. G. Palacou Open Cloed and Mxed Net-wo of ueue wth Dffeent Clae of Cutome'' J. ACM Vol. No. pp Apl [] H. obayah Modelng and Analy'' Addon-Weley Publhng Company Inc

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