Approximations for a Fork/Join Station with Inputs from Finite Populations
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1 Approxmatons for a Fork/Jon Staton th Inputs from Fnte Populatons Ananth rshnamurthy epartment of ecson Scences ngneerng Systems Rensselaer Polytechnc Insttute 0 8 th Street Troy NY 80 USA Rajan Sur enter for Quck Response Manufacturng Unversty of Wsconsn-Mason 53 Unversty Avenue Mason WI USA Mary Vernon epartment of omputer Scences Unversty of Wsconsn-Mason 0 W ayton Street Mason WI USA Abstract Fork/jon statons moel synchronzaton constrants n queung netork moels of many manufacturng computer systems. We conser a fork/jon staton th to nput buffers general nputs from fnte populatons erve approxmate expressons for throughput mean queue lengths at the nput buffers. We assume that the arrvals to the fork/jon statons are reneal but our approxmatons only use nformaton about the frst to moments of the nter-reneal strbutons. Therefore the approxmatons can be use to prect performance for a varety of systems. We verfy the accuracy of these approxmatons aganst smulaton report sample results. eyors: Parametrc ecomposton to-moment approxmaton close queung netorks fork/jon statons.. Introucton Fork/jon statons are use to moel synchronzaton constrants beteen enttes n a queung netork. The fork/jon staton of nterest n ths paper conssts of a server th zero servce tmes to nput buffers. As soon as there s one entty n each buffer an entty from each of the buffers s remove jone together. The jone entty exts the fork/jon staton nstantaneously. Subsequent to ts eparture the jone entty forks back nto the component enttes hch then get route to other parts of the netork. Fork/jon statons fn many applcatons n queung moels of manufacturng computer systems. In queung moels of assembly systems the assembly operaton s typcally moele usng a fork/jon staton [][4][8]. Fork/jon statons are also use moel the synchronzaton constrants n moels of kanban control strateges []. They are also use to moel parallel processng atabase concurrency control n computer systems analyss [3]. As a startng pont for unerstng the behavor of queung netorks th fork/jon statons several researchers have analyze such statons n solaton. For the sake of analytcal tractablty a majorty of the prevous research efforts assume that the fork/jon statons have Posson nputs [][4][9][0]. Although these results are useful n many of the applcatons cte above the nput processes are not Posson. Most stues that assume arrval processes other than Posson such as those reporte n [] assume nfnte populatons for each arrval process. Hoever f the fork/jon staton s part of a close queung netork then once the queue length of an nput buffer equals the sze of the populaton that can arrve to the buffer the arrval process shuts on. The analyss of fork/jon statons th general arrval processes from fnte populatons can become very complex even hen the nter-arrval tmes are nepenent have a oxan strbuton [6]. Thus approxmatons for the performance of the fork/jon statons n partcular the netork n general can be hghly useful. In ths paper e erve approxmate expressons for throughput mean queue lengths at the nput buffers of a fork/jon staton th general nputs from fnte populatons. The approxmatons are base on the assumpton that the arrvals to the fork/jon statons are reneal but they only use the frst to moments of the nter-reneal
2 strbutons can therefore be use to prect performance for a e varety of systems. In the lterature such approxmatons are often referre to as to-moment approxmatons []. In aton to provng performance estmates for a fork/jon staton n solaton the approxmatons can also be use as bulng blocks n parametrc ecomposton approaches for solvng larger queung netorks. The outlne of ths paper s as follos. Secton escrbes the fork/jon staton uner conseraton an overve of our approach. We erve the general form for the approxmatons n Secton 3 n Secton 4 e erve the etale form of the approxmatons test ther accuracy aganst smulaton. Secton 5 proves the conclung remarks.. System escrpton Approach We escrbe our moel of the fork/jon staton next. As shon n the Fgure the fork/jon staton has to nput buffers B B. SN enotes the rest of the queung netork for enttes that arrve to buffer B. If an entty arrvng n buffer B (B fns buffer B (B empty t ats for the corresponng entty to arrve n nput buffer B (B. As soon as there s at least one entty n each queue one entty s remove from each buffer. The remove enttes jon together mmeately epart from the fork/jon staton. As a result the contents of both nput buffers reuce by one. Subsequent to eparture from the fork/jon staton the jone entty forks back nto to enttes that are route back to SN SN respectvely. In SN SN these enttes are subjecte to rom elays before they revst the fork/jon staton. There s a fnte populaton of sze for each entty. onsequently the number of enttes n nput buffer B queung netork SN alays sum up to. Atonally the arrval process to buffer B shuts on hen there are unts n buffer B. Snce the sub-netorks SN SN from hch enttes arrve to nput buffers can have fferent confguratons resultng n arbtrary elays the arrval processes to the fork/jon statons can have arbtrary characterstcs. Hoever analyss of fork/jon statons for general arrval processes can be qute complcate. To smplfy our analyss n keepng th other to-moment approxmaton methos e ll assume that the arrval processes are nepenent reneal processes that the nter-arrval tmes to the nput buffers are nepenent entcally strbute (.. havng means / / square coeffcents of varaton (SVs c c respectvely. Snce e assume that the arrval process to buffer B (B shuts on once t has ( unts the arrval processes are reneal beteen shut ons. Wth these assumptons our moel of ths fork/jon staton s completely characterze by the parameter 6-tuple ( c c. For a fork/jon staton characterze thus e obtan approxmatons for the throughput the mean queue length of each buffer L. In les n the nterval [0.33.0]. evelopng the approxmatons e assume that the rato of nput rates Ths s justfe for most practcal stuatons snce n a hgh performance system one oul not normally expect the arrvals rates at one nput buffer of a synchronzaton staton to be more than three tmes that of the other. We also assume that both c c le n the nterval [0.54.0]. These values capture the typcal SVs observe n actual manufacturng systems [5]. / / To evelop these approxmatons e frst stuy the mpact of the mean of the nter-arrval tmes ( populaton sze ( on the performance of the fork/jon staton usng the exact expressons reporte n [9] [0]. Although these expressons are exact only for the case of exponentally strbute nter-arrval tmes the nsghts about the mpact of arrval rates help us unerst behavor for more general arrval processes. Hoever to stuy the mpact of secon moments of the arrval strbutons (n partcular c c on the performance of the fork/jon staton a moel assumng Posson nputs s naequate. In [6] e analyze a fork/jon staton here the c c SN Subnetork B B SN Subnetork cˆ Fgure. Fork/Jon staton
3 nter-arrval tmes have a -phase oxan strbuton. Ths permts analyss for nput processes th a e range of means (0 SVs [0.5. Usng the results of ths analyss e stuy the mpact of both means SVs on the performance measures. Usng nsghts from all the above cases e evelop to-moment approxmatons for a c. fork/jon staton characterze by the 6-tuple ( c 3. General Form of Approxmatons In ths secton e use the nsghts from the exact analyss presente n [9] [0] [6] to evelop the general form of the to-moment approxmatons for the throughput the mean queue length of each buffer L. For the sake of clarty e mofy the notaton use a superscrpt hen the performance measures are base on the scussons presente n [9] [0] assumng exponental nputs a superscrpt hen the staton performance measures are base on the scussons presente n [6] assumng oxan nputs. Frst e conser the case th exponental nputs. Assumng thout loss of generalty that < Takahash et al. [0] erve the follong expressons for the throughput the mean queue lengths L at the fork/jon staton: ( + + L + + L Note that the above expressons are for the case here. The corresponng expressons for are obtane by takng the lmts as. Usng these expressons e stuy the mpact of the mean rates of the nput processes on the performance of the fork/jon staton. To stuy the mpact of hgher moments of the arrval strbutons on the performance of the fork/jon staton e conser the case here the nter-arrval tmes to the nput buffers have -Phase oxan strbutons th mean / SV c for respectvely. Usng the atonal constrant of balance means e erve unque - phase oxan strbutons to characterze the nter-arrval tmes at each nput buffer. Then base on the exact analyss presente n [6] e stuy the mpact of the SVs of the nput processes on staton performance measures. Snce the close form expressons for mean queue length throughput are substantally more complcate than those obtane for exponental nputs e obtan the requre nsghts from the numercal results obtane from the exact computatons. To obtan these nsghts e compute the throughput mean queue length L for over 900 nput parameter settngs th n the nterval [0.33.0] c n the nterval [0.54.0] n the nterval [0]. These numercal results prove the follong nsghts: ( For any value of SV the upper boun of the throughput from the fork/jon staton s mn (. Ths throughput s acheve as or tens to nfnty. ( For gven values of c c + the value of s nsenstve to the choce of. ( Substantal queues are observe at the buffers of the nput processes th hgher rates of arrvals.e. ( (3 L >> L hen >. (v When nput rates are equal.e. s qute senstve to c c. (v When > or the staton performance measures become ncreasngly less senstve to the SVs are more prmarly < epenent on. (v Let c enote 0.5( c + c. Then e observe that for a gven c > f 3
4 c < < f >0 then L < L c >. Atonally hen ( - ( c -<0 then L > L. L > L L < L hle ( -( c - To obtan quanttatve nsghts nto the mpact of the SVs of the nter-arrval tmes e stuy the relatve fferences of these measures for the oxan exponental cases. Thus e efne: We compute L L L L L plot ther values aganst (4 c - for the parameter settngs consere n the numercal stuy for oxan nputs. The etals are lste n [7]. From these graphs e nfer that vary roughly lnearly th c - th atonal (varaton epenng on or. Base on these observatons e propose the follong functons as cates for the approxmatons: Note that hen ( ( c + L L ( ( c + (5 L c c the approxmaton functons above yel L L. Ths mples that the approxmatons are exact for exponental arrvals. Next base on the nsghts from the stuy for oxan nputs e entfy the general form of. In partcular nees to satsfy the follong propertes:. (. We requre that ( ( L must be a sngle value functon of 3. ( 0 < Ths s because > f. ue to symmetry. c < 4. ( 0 for 0 5. We requre that ( ( 6. Snce mn ( e requre ate functons ( snce. ( ( c < f c >. hen or 0 + mn (. that satsfes propertes through 5 above are: ( -a 0 ( + ( + + (6. here a 0 s a postve constant or a postve functon of +. Further f 0 a 0 then property 6 s also satsfe. (For proof see [7]. Next e note that. L must satsfy the follong propertes: L must be a sngle value functon of hen 0. 0 L snce L hen 0 L 3. By symmetry ( (. for. 4
5 4. ( 5. For ( 0 ( ( 0 snce for gven values of have to be of opposte sgn. hle for ( 0 ( 0 ( 0 ( 0 L > L L 6. Snce e requre ate functons L L < L hle ( -( c ->0 then. Ths s because hen ( - ( ( c L ( + L < L. that satsfy propertes through 5 above are: L b b + b + b L + + L > L c - c -<0 then here b b are constants or postve functons of +. The value of these constants that ensures satsfes property 6 n aton s etermne usng results from the smulaton experments escrbe belo. L 4. etale Form of Approxmatons ther Accuracy In ths secton e etermne the etale form of the approxmatons test ther accuracy. To etermne the etale form of the approxmatons e only nee to etermne the best values of the constants a 0 n equaton (6 b b n equatons (7. We use smulatons to etermne the best values of these constants. In the smulaton experments e evaluate the approxmatons for nter-arrval tmes that have -stage rlang Shfte exponental Lognormal Hyper-exponental strbutons. In these experments ( take values of ( (0.3 (0.8.5 respectvely hle take several values n the range [0] (7 c take values n the range [ ]. From these experments e observe that settng a n our approxmaton for settng b b 4 n our approxmatons for L gves the best performance. For atonal etals see [7]. Wth ths choce the fnal expressons for the to-moment approxmatons for throughput the mean queue length of each buffer L L L are as follos: + + c ( + + ( The corresponng equatons for ( (8 c (9 ( c (0 quatons 8 to 0 are for the case here are obtane by takng the lmts as. Fnally e test the performance of the approxmatons by computng the percentage fference n the estmates gven by the approxmatons estmates from smulaton. Sample results are prove n Table. In ths table L correspon to performance measures from smulaton experments hle the performance measures etermne by the approxmatons are gven by L respectvely. In these experments nter-arrval tmes th SV of 0.5 ere assume to have a shfte exponental strbuton hle nterarrval tmes th SV of ere assume to have a lognormal strbuton. 5
6 Table. Percentage fference beteen approxmatons smulaton (Sample results from over 40 test cases L L L L ε ( 00 ε ( L 00 ε ( L 00 Maxmum Average Maxmum Average Maxmum Average onclusons xtensons In ths paper e have propose approxmatons for the throughput mean queue lengths at the nput buffers of a fork/jon staton th general arrvals from a fnte populaton. From the sample results reporte e observe that the maxmum fference n the estmates prove by the approxmatons compare th smulaton as 6% for staton throughput % for mean queue lengths. Although these approxmatons have been evelope for a fork/jon staton n solaton a prncpal applcaton s n evelopng parametrc methos for analyss of larger close queung netorks th fork/jon statons. In such applcatons e also nee to-moment approxmatons for the varablty parameter of the eparture process c ˆ. Stues such as [3] reports several ssues that nee to be aresse hen etermnng the varablty parameter n the context of smple queues. We are currently orkng on aressng these ssues n the context of fork/jon statons ervng the necessary to-moment approxmatons. Usng these approxmatons as bulng blocks e nten to evelop ne methos to analyze close queung netork moels of sngle mult-stage kanban systems. References. Mascolo M. Fren Y. allery Y. 996 An Analytcal Metho for Performance valuaton of anban ontrolle Proucton Systems Operatons Research 44( Harsson J.M. 973 Assembly-lke Queues Journal of Apple Probablty 0( Heelberger P. Trve. S. 983 Queung Netork Moels for Parallel Processng th Asynchronous Tasks I Transactons on omputers -3 ( Hopp W. Smon J.T. 989 Bouns An Heurstcs For Assembly-Lke Queues Queung Systems amath M. Sur R. Sers J. L. 988 Analytcal Performance Moels for lose Loop Flexble Assembly Systems The Internatonal Journal of Flexble Manufacturng Systems rshnamurthy A. Sur R. Vernon M. 00 Analyss of a Fork/Jon Synchronzaton Staton th oxan Inputs From a Fnte Populaton submtte to Annals of Operatons Research. 7. rshnamurthy A. Sur R. Vernon M. 00 To-Moment Approxmatons for a Fork/Jon Staton th General Inputs Techncal Report epartment of Inustral ngneerng UW -Mason. 8. Rao P.. Sur R. 000 Performance Analyss Of An Assembly Staton Wth Input From Multple Fabrcaton Lnes Proucton Operatons Management 9( Som P. Wlhelm P.. sney R.L. 994 ttng Process n a Stochastc Assembly System Queung Systems Takahash M. Osaa H. Fujsaa T. 996 A Stochastc Assembly System Wth Resume Levels Asa-Pacfc Journal of Operatons Research Takahash M. Takahash Y. 000 Synchronzaton Queue Wth To MAP Inputs An Fnte Buffers Proceengs of the Thr Internatonal onference on Matrx Analytcal Methos n Stochastc Moels Leuven Belgum.. Whtt W. 983 The Queung Netork Analyzer Bell Systems Techncal Journal Whtt W. 995 Varablty Functons for Parametrc ecomposton Approxmatons of Queung Netorks Management Scence
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