A Queuing Model for an Automated Workstation Receiving Jobs from an Automated Workstation

Transcription

1 Intenatonal Jounal of Opeatons Reseach Intenatonal Jounal of Opeatons Reseach Vol. 7, o. 4, 918 (1 A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton Davd S. Km School of Mechancal, Industal and Manufactung Engneeng, Oegon State Unvesty, Covalls, Oegon, USA Receved June 1; Revsed Octobe 1; Accepted ovembe 1 Abstact In ths pape we develop queung model esults fo a sngle automated wokstaton that eceves jobs fom anothe automated wokstaton. An automated wokstaton s a seve wth detemnstc pocessng tmes that expeences andom opeatng tmes between falues, and then subsequent andom epa tmes. We develop analytcal expessons fo the queue sze dstbuton, the aveage numbe n system and the vaance of the numbe n system usng a dscete model of ths queung system. KeywodsQueung, automated wokstaton, poducton systems, makov chan aggegaton. 1. ITRODUCTIO In the analyss of poducton systems, the pesence of automated wokstatons s vey common. An automated wokstaton s a seve wth detemnstc pocessng tmes that expeences andom opeatng tmes between falues, and then subsequent andom epa tmes. In ths pape we develop analytcal expessons fo the queue sze dstbuton, the aveage numbe n system, and the vaance of the numbe n system fo a sngle automated wokstaton that eceves nput fom anothe automated wokstaton. The actual wokstaton that s consdeed s assumed to have fxed job pocessng tmes and exponentally dstbuted opeatng tmes between falues, and exponentally dstbuted epa tmes. These have been shown to be easonable assumptons n pactce (Inman 1999, Dalley and Geshwn 199. Much s known about such automated wokstatons opeatng n solaton. In Km and Alden (1997 a mxed dscete/contnuous pobablty mass/densty functon fo the tme to poduce a fxed numbe of jobs on such a wokstaton s deved (whch also ncludes the specal case of a sngle job. Fo a dscete model of such a wokstaton a fomula fo the vaance of the numbe of jobs poduced n a fxed tme peod s deved n Geshwn (199. In Km and Alden (1997 and Hopp and Speaman (1 fomulas can be found fo the mean and vaance of the tme a job spends n such a wokstaton that ncludes pocess and epa tme. One addtonal step n the analyss of automated wokstatons s the analyss of a wokstaton and ts nput buffe. In ths eseach we develop an analytcal model fo a sngle automated wokstaton that eceves ts nput fom an upsteam automated wokstaton and compae ths aganst commonly known G/G/1 appoxmatons wth espect to estmatng the aveage numbe of jobs n the system. Analytcal expessons fo the dstbuton of the numbe of jobs n the system (the wokstaton and ts nput buffe ae also deved. In a elated pape (agaajan and Km 6, lnkng equatons have been developed so that the esults developed hee can be appled to the analytcal analyss of a sees of automated wokstatons usng a two-wokstaton decomposton appoxmaton. Ths pape pesents the detals of the modelng appoach and devaton of the esults fo an automated wokstaton queung system utlzed n agaajan and Km (6. Smplfed fomulas as well as the esult fo the vaance of the numbe n system ae new. A lage amount eseach addessng the modelng of seal automated poducton systems focuses on modelng Coespondng autho s emal: davd.km@ost.edu X Copyght 1 ORSTW

2 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 (1 1 two automated wokstatons n sees wth fnte buffe capactes between wokstatons. The goal of ths po eseach has focused on estmatng the thoughput of such systems, and ncludes contnuous and dscete appoxmatons of automated wokstatons. These models have seved as the foundaton fo models developed to analyze a longe sees of automated wokstatons (wth fnte buffes usng numecal two-wokstaton decomposton appoxmatons. A evew of such models can be found n Dalley and Geshwn (199. Thee has been less eseach dected at developng explct analytcal models of an automated wokstaton wth an nfnte nput buffe, wth a focus on pedctng the wok-n-pocess o tme-n-system. In Altok (1997 M/G/1 queung esults wee appled to a system consstng of a sngle (possbly automated wokstaton wth Posson avals. The dstbuton of the numbe n system fo a wokstaton wth Posson avals, exponentally dstbuted tme between falues (both opeatng tme and elapsed tme, and phase-type epa tme dstbutons s examned n Altok (1997 usng contnuous tme Makov chan analyss. Anothe analyss appoach fo a sngle automated wokstaton wth an nfnte capacty nput buffe s to apply exstng geneal queung appoxmaton methods. Such methods ae employed n Hopp and Speaman (1. In Hopp and Speaman (1 two-moment G/G/1 queung model appoxmatons ae appled. G/G/1 appoxmatons have also been used as pat of softwae packages that estmate the pefomance of netwoks of wokstatons wth nfnte buffe capactes (Whtt, Thee ae multple two-moment G/G/1 appoxmaton models n use, as well as lnkng equatons that estmate the paametes of the nput pocess to a wokstaton as a functon of the po wokstatons paametes. Vaous G/G/1 appoxmatons and lnkng equatons ae summazed n Shanthkuma and Buzacott (198 and Buzzacott and Shanthkuma (1993. Because these models do not explctly model the opeaton of an automated wokstaton t s expected that they may not always pefom accuately, n patcula when the coeffcent of vaaton (CV of the nteaval and/o sevce pocess s hgh (Buzzacott and Shanthkuma, The emande of ths pape s oganzed as follows. In secton we pesent a Makov chan model fo an automated wokstaton wth nfnte nput buffe capacty ecevng ts jobs fom anothe automated wokstaton (assumed to always have wok. In secton 3, the expessons fo queue sze dstbuton, the aveage numbe n system, and the vaance of the numbe n system ae deved. In secton 4, the accuacy of the model s examned by compasons to smulatons and exstng G/G/1 appoxmatons.. MARKOV CHAI MODEL OF TWO AUTOMATED WORKSTATIOS I SERIES We develop a Makov chan model of an automated wokstaton wth nfnte nput buffe space, ecevng jobs fom anothe automated wokstaton wth an nfnte supply of unpocessed jobs. By assumng that the fst wokstaton always has jobs to pocess, the output pocess fom the fst wokstaton epesents output fom an automated wokstaton wth no nfluence of a andom aval pocess. Modelng the output pocess n the pesence of vaable nput s addessed n agaajan and Km (6, whee a sees of automated wokstatons s analyzed. The Makov chan model s a dscete tme model whee the fxed wokstaton pocessng tme t, seves as the dscete tme unt (t s assumed that both wokstatons poduce at the same speed when up. By the dscete natue of the model, the opeatng tmes between falues, and epa tmes wll follow geometc dstbutons as appoxmatons to exponental dstbutons. In most automated wokstatons, ths type of dscete appoxmaton s suffcently accuate snce the fxed pocessng tmes ae nomally much smalle than the tme between falues and epa tmes. The mechancs of the dscete tme Makov chan ae as follows: State tanstons occu at the end of each tme step. Any wokstaton that s down at the begnnng of a tme step may be epaed even f the wokstaton s empty at the begnnng of the tme step. A wokstaton that s up and not empty at the begnnng of a tme step wll complete ts job even f t moves to a down state at the end of the tme step. Any jobs completed at the end of a tme step ae moved out of the wokstatons and new jobs ae moved nto the wokstatons even f a wokstaton moves to a down state. ote that a job may be moved out of both wokstatons, and a job moved nto both wokstatons at the end of a tme step. Wokstatons that ae up at the begnnng of a tme step but dle because they ae staved, cannot change to a down state at the end of a tme step. We assume that the long-un pocessng capacty of the fst wokstaton s stctly less than that of the second wokstaton. Ths ensues that the numbe of jobs n the second wokstatons nput buffe wll not steadly ncease ove tme. The objectve of the model s to analyze the behavo of the second wokstaton and ts nput buffe, whch we wll efe to as the system. We let the state of the Makov chan at tme unt n, X =( n x,, 1 x, whee x = status of wokstaton, =1,

3 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 (1 11 and x {,1}. x f the wokstaton s down a the begnnng of a tme step, and x 1 f the wokstaton s up a the begnnng of a tme step. s the numbe of jobs n wokstaton plus the numbe n ts nput buffe. We let p p kj denote the tanston pobablty matx fo ths Makov chan. If wokstaton s up and opeatng (an unpocessed job s n the wokstaton at the begnnng of the tme step, t emans up dung the tme step and may tanston to a down state wth pobablty f at the end of the tme step. The job beng pocessed n ths cycle wll be completed and moved out of the wokstaton. If wokstaton s down and unde epa at the begnnng of the tme step, t emans down dung the tme step and s epaed wth pobablty at the end of the tme step. If an unpocessed job was pesent n ths wokstaton at the begnnng of the tme step, the job emans unpocessed and stays n the wokstaton untl the wokstaton s epaed befoe beng pocessed. A state tanston dagam of the Makov chan model s shown n Fgue 1. umbe In System 1 3 (1,1 Machne Status (1, (,1... (, Fgue 1. State tanston dagam fo the Makov chan model. State = ( x1, x, whee s a column label and ( x1, x ae ow label. Dffeent maked tanstons (e.g., dashed epesent tanston fom states wth the same ( x1, x values. The tanston pobabltes between any two states n fgue 1 ae functons of the wokstaton status at tme n and n+1. Fo example, p (1,1,,(1,, the tanston pobablty fom state (1,1, to (1,, equals(1 f1 f. When thee ae no customes n the system (.e., wokstaton s staved, the tanston pobabltes eflect the assumpton that wokstaton cannot fal f t s staved. Snce the actual wokstatons ae assumed to have exponentally dstbuted opeatng tmes between falues, and epa tmes, the pobabltes f and ae computed as a functon of the fxed pocessng tme t, MTBF, and MTTR, MTBF s the mean opeatng tme between falues fo wokstaton, and MTTR s the mean epa tme fo wokstaton (both paametes of exponental dstbutons. The pobabltes f and ae computed such that the mean tme to pocess a job, and the vaance of the tme to complete a job (whch ncludes wokstaton downtme n the dscete model match that fo the actual wokstaton. Wthout loss of genealty let t 1, and let T the total tme spent by a job n wokstaton. In Km and Alden (1997 and Hopp and Speaman (1 t s shown that, MTBF MTTR ET [ ] (1 MTBF

4 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 (1 1 * MTTR Va[ T ] ( MTBF Fo the dscete model t s staghtfowad to show that f. ET [ ] 1 To deve Va[ T ] as a functon of f and, let I 1 f a job just moves nto a wokstaton that has just faled, and I = othewse. By condtonng on the ndcato vaable I we get (agaajan and Km 6, E[ T ] E[ T I 1]* p( I 1 E[ T I ]* p( I k1 f ( E[ T ] ( k 1 (1 * f (1 *(1 f. k1 f ( f Va[ T ]. f and as functons of MTBF and MTTR f MTBF MTTR * MTBF * MTTR MTBF MTBF MTTR * MTBF * MTTR MTTR can then be expessed as: 3. DERIVATIOS OF THE AVERAGE UMBER I SYSTEM AD THE DISTRIBUTIO OF THE UMBER I SYSTEM To deve analytcal expessons fo the aveage numbe n system and the dstbuton of the numbe of jobs n the system, we take advantage of the tanston stuctue of the Makov chan, and Makov chan aggegaton/dsaggegaton esults (Fenbeg and Chu, 1987, Km and Smth, Fgue 1 was dawn n such a way that the numbe n system defnes a natual pattonng of the system states. Followng the temnology defned n Km and Smth (1995, a set of fou states n the Makov chan that epesent the same numbe n system wll fom a macostate. A Makov chan s n a patcula macostate wheneve t s n any state contaned n the macostate. The tanstons fom macostate to macostate also consttute a Makov chan (Km and Smth, The soluton to ths macostate Makov chan epesents the soluton to the queung model snce the macostates epesent the numbe n system, and the macostate Makov chan steady state pobabltes wll equal the sum of the steady state pobabltes of all states contaned n the macostate (Km and Smth, A dagam of the macostate Makov chan s shown n fgue Fgue. Macostate Makov chan model. 3.1 Mcostate Makov Chan Tanston Pobabltes One method to compute the tanston pobabltes of the macostate Makov chan (tanston pobablty matx denoted by P, s to examne the states wthn each ndvdual macostate n solaton whee all tanstons afte a macostate s left ae gnoed. The states wthn a macostate ae efeed to as mcostates, and the pocess ealzed by vewng the mcostates n solaton consttutes a Makov chan (Km and Smth, These Makov

5 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 (1 13 chans ae efeed to as mcostate Makov chans. If the steady state pobabltes of the mcostate Makov chan ae known, then they can be used to calculate the macostate Makov chan tanston pobabltes (Km and Smth, To fnd the tanston pobabltes of the mcostate Makov chans (denoted by p fo the mcostate chan assocated wth customes n system we take advantage of the tanston stuctue. Consde the mcostate chan that coesponds to zeo customes/jobs n the system. All tanstons leavng the set of states contaned n ths chan must eventually etun to these states (due to egodcty, and e-ente the set of states fom only a sngle state. Thus the tanston pobabltes fo those tanstons leavng the mcostate Makov chan ae known. Ths s shown n Fgue 3. umbe In System (1,1 (1, 1 ( 1 f 1 f 1 Tanstons leavng the mcostate chan Machne Status (,1 ( 1 (1 f 1 (, ( 1 f 1 Fgue 3. Fndng the mcostate Makov chan tanston pobabltes. The tanston pobablty matx fo the mcostate Makov chan fo zeo customes/jobs n the system s, p (1,1 (1, (,1 (, (1,1 1 (1 f 1 f (1 1 (1 f (1 1 f (1, 1 (1 f 1 f (1 1 (1 f (1 1 f (,1 1 (1 1 (, 1 1 (1 (1 1 (1 1 (1. We next addess the mcostate Makov chan tanston pobabltes when thee ae two o moe customes n the system. In fgue 1 t can be seen that all tanstons that ncease the numbe n system occu when wokstaton 1 s up and wokstaton s down. Smlaly all tanstons that decease the numbe n system occu when wokstaton 1 s down and wokstaton s up. Futhemoe the Makov chan stuctue (tanstons nto, wthn, leavng fo the mcostate Makov chans fo a numbe n system of two o geate s the same. Theefoe I J p p fo I, J. Usng smla easonng as used to fnd p, when tanstons leave the states n a mcostate chan to the left, they etun to the mcostate chan va a sngle state. Also when tanstons leave the states n a mcostate chan to the ght, they also etun to the mcostate chan va a sngle state. Theefoe we get,

6 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 (1 14 p (1,1 (1, (,1 (, (1,1 (1 f1(1 f (1 f1 f f1(1 f f1 f (1, 1 (1 f 1 f (1 1 (1 f (1 1 f (,1 (1 f1 (1 f1(1 f1 f1(1 (, 1 1 (1 (1 1 (1 1 (1 f fo. I To fnd p eques knowledge of the steady state pobabltes of p snce tanstons leavng the mcostate chan (fo one n the system to the left n fgue 1 may etun to the mcostate chan va two dffeent states (as shown n fgues 1 and 3. Let [ 1,1, 1,,,1,,] epesent the steady state pobabltes of the mcostate Makov chan fo n the system, whee epesentng wokstaton 1 n state, and wokstaton n state j. Let Then, s the steady state pobablty fo the mcostate, j c 1,1. ( 1,1 1, (1,1 (1, (,1 (1,1 (1,1 (1 f1(1 f (1 f1 f f1(1 f f1 f 1 (1, 1 (1 f 1 f (1 1 (1 f (1 1 f p. (,1 c(1 f1 (1 c(1 f1 (1 c(1 f1(1 cf1 (1 c f1 (1 c f1(1 (, 1 1 (1 (1 1 (1 1 (1 3.. Mcostate Makov Chan Solutons It s possble to deve manageable analytcal solutons fo the mcostate Makov chans as functons of the wokstaton falue and epa pobabltes, snce they ae only fou-state Makov chans. The solutons fo the mcostate chans satsfy p,, 1,1 1,,1, 1, and thus epesent the unque steady state soluton to the mcostate Makov chans. Although we ae pmaly nteested n the steady state pobabltes fo those mcostates that have tanstons out of the set of mcostates, all mcostate steady state pobabltes have been deved. These steady state solutons ae pesented next. Let a f1 f f1 f and b 1 1 Mcostate Makov chan steady state pobabltes fo. f 1 1,1 1 b f 1 1, b ( f1 f1,1 1 b f1 f1, b (1 Mcostate Makov chan steady state pobabltes fo 1. (( a 1 ( 1 bf f f ( b f ,1 bf 1 b f 1 b 1 f1 f b f a f1 f1 ( 1 ( 1( ( (3 1

7 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 ( f( b f1 1, bf 1 b f 1 b 1 f1 f b f a f1 f1 ( 1 ( 1( ( (3 1 b( b f ( a f 1 1 1,1 bf 1 b f 1 b 1 f1 f b f a f1 f1 ( 1 ( 1( ( (3 1 f ( b f ( a f 1 1 1, bf 1 b f 1 b 1 f1 f b f a f1 f1 ( 1 ( 1( ( (3 1 Mcostate Makov chan steady state pobabltes fo. ( b f *( b f 1 1 1,1 ( a b f1 f1 ( b f *( a f 1 1 1, ( a b f1 f1 ( a f *( b f 1 1,1 ( a b f1 f1 ( a f *( a f 1 1, ( a b f1 f Macostate Makov Chan Soluton and Queung Model Results Let P [ P IJ ] be the tanston pobablty matx fo the macostate Makov chan. The macostate Makov chan depcted n Fgue has the followng tanston pobablty matx stuctue and values fo P IJ. In ths secton s always ( ( P 1 1 ( ( The tanston pobabltes ae computed fom the esults of the mcostate Makov chan analyss. Tanston pobabltes fo a macostate ae computed as weghted aveages usng the mcostate chan steady pobabltes as weghts. Fo example, a tanston pobablty fo a tanston to the ght s the weghted sum ove the states wthn the mcostate chan, multpled by the pobablty of a tanston to the ght. That s, I I, I 1 (, j * p(, j, I,( m, n, I 1 (, j ( m, n P whee (, j,( m, n {(1,1,(1,,(,1,(,}. Let [, 1,, ] epesent the steady state pobabltes of P. Because of the smple stuctue (tme evesble of the macostate Makov chan, t s staghtfowad to show that,

8 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 ( K * 1 11 (3 1 K * ( K whee fo (5 1 ( K ( ( (6 K can be expessed as a functon of f and, f ( f b ( f f K ( f a( f ( f b (7 Equatons 3-7 epesent the pobablty dstbuton of the numbe of jobs n the second wokstaton and ts nput buffe. Expessons fo the aveage and the vaance of the numbe n system can then be obtaned fom equatons 3-7. Lettng q 1, and s 1 to smplfy the notaton, and lettng C = the numbe n system gves, 3 K 1 s s s Aveage umbe n System E[ C] K 3K 4K 5K q 1 q q q Smplfyng we get, 1 q s Aveage umbe n System K 1 1 ( q s The aveage numbe n system can be expessed as a functon of f and, Aveage umbe n System 1( f f b f ( f f ( f The vaance of the numbe n system s found by fst fndng E[C ] and then subtactng E[C], 3 K 1 3 s 4 s 5 s E[ C ] K 3 K 4 K 5 K q 1 q q q q ( q s K. 1 3 s s 1 ( q s Subtactng EC gves, 1 1 q s 4q 3qs s Vaance of the umbe n System K K ( q s ( q s 4. COMPARISO WITH TWO-MOMET G/G/1 APPROXIMATIOS AD SIMULATIOS As mentoned n secton 1, one appoach fo analyzng automated wokstatons ecevng nput fom automated wokstatons s the applcaton of geneal two-moment G/G/1 appoxmatons. Anothe moe tme consumng appoach s to use smulaton. In ths secton we show how the cuent model pefoms wth espect to pedctng the aveage numbe n system when compaed aganst smulaton esults and two popula G/G/1 appoxmatons. The system used n these compasons can be thought of as two automated wokstatons n sees, whee the fst wokstaton always has jobs to pocess. Unde ths system descpton the mean and vaance of the job ntedepatue tmes fom the fst wokstaton may be found usng equatons (1 and (. Smlaly the mean and vaance of the pocessng tmes (actual wok tme plus downtme at the second wokstaton can be calculated. Wth these values computed a numbe of two-moment G/G/1 appoxmatons may be appled to estmate the numbe of jobs n the second wokstaton and ts queue. The two G/G/1 appoxmatons used hee ae appoxmatons,. (1 (8 (9

9 Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 (1 17 fo the aveage numbe n queue fom Sakasegawa (1977 and Yu (1977, and Kame and Lagenbach-Belz (1976. The appoxmatons n Sakasegawa (1977 and Yu (1977 ae the same as that used n Hopp and Speaman (1. Whtt (1983 uses an appoxmaton that s the appoxmaton n Kame and Lagenbach-Belz (1976 when the squaed coeffcent of vaaton of the nteaval tmes s less than one, and s the appoxmaton n Sakasegawa (1977 and Yu (1977 when the squae coeffcent of vaaton of the nteaval tmes s geate than one. By addng the wokstaton utlzaton to these appoxmatons we get an estmate fo the aveage numbe n system. The cuent model s compaed to smulaton esults of a contnuous tme system wth exponentally dstbuted opeatng tmes between falues and epa tmes, to estmate the mpact of the dscete model appoxmaton. The smulaton esults wll also seve as the best estmate of the aveage numbe n system. A vaety of systems wee smulated to epesent dffeent nteaval and sevce tme coeffcents of vaaton, and dffeent wokstaton-two utlzatons. Wthn each sevce tme coeffcent of vaaton and utlzaton ange, dffeent systems wee smulated. Each smulaton stated wth a 1, tme unt wam-up (the fxed pocessng tme fo a sngle job s one tme unt. Afte the wam-up peod, the smulatons wee un untl 1,, jobs wee completed on the second wokstaton. 3 eplcatons wee conducted and the best estmate fo a systems aveage numbe n system was taken as the aveage of the 3 eplcatons. A summay of the smulaton esults s pesented n Table 1. In Table 1, the smulaton esults ae sepaated nto test sets. Wthn each set the ange fo the wokstaton pocessng tme (wok tme plus downtme and nteaval tme coeffcents of vaaton, and utlzatons fo the second wokstaton ae shown. The esults pesented ae the aveage, maxmum, and mnmum (ove the numbe of systems wthn a test set absolute pecent dffeence fom smulaton. The aveage pecent dffeences n Table 1 ae plotted ove the test sets n Fgue 4. As can be seen, the pefomance of the model s vey consstent ove a ange of wokstaton coeffcents of vaaton and utlzatons. Also, the aveage eos when usng G/G/1 appoxmatons can be vey lage. In geneal the appoxmaton of Kame and Lagenbach-Belz (1976 outpefoms the appoxmaton n Sakasegawa (1977 and Yu (1977 although they ae vey smla n pefomance. Ths s not supsng snce they ae vey smla n functonal fom. Addtonally the esults confm that the G/G/1 appoxmatons do pefom bette fo hghe utlzatons. Both appoxmatons can be vewed as modfcatons of the uppe bound n Kngman (196 fo a G/G/1 queue, whch becomes tghte as utlzaton appoaches one. Table 1. Summay of compasons the model wth smulaton and two-moment G/G/1 appoxmatons Absolute Pecent Dffeence Fom Smulaton Test Set Wokstaton Wokstaton o. of ew Model Sakasegawa-Yu KLB umbe CV Range Utl. Range Systems Avg. Max Mn Avg. Max Mn Avg. Max Mn 1 to 1 8%-9% 3.14% 9.81%.6% 6.6% 59.4%.71%.8% 58.97%.77% 1 to 8%-9% 1.49% 5.35%.3% 18.1% 61.76%.6% 16.14% 6.99%.65% 3 to 3 8%-9%.1% 6.1%.3% 3.1% 91.44% 5.93% 8.31% 83.76% 3.% 4 3 to 4 8%-9%.37% 4.88%.8% 3.7% 78.73% 4.75%.16% 69.99% 3.49% 5 4 to 5 8%-9% 1.86% 5.84%.6%.8% 56.49%.1% 16.95% 5.96%.54% 6 to 1 9%-95% 1.83% 5.88%.1% 16.74% 6.95% 1.43% 1.67% 51.13%.87% 7 1 to 9%-95% 1.3%.6%.% 16.5% 69.88%.3% 15.3% 67.41%.46% 8 to 3 9%-95% 1.77% 4.4%.8% 1.% 69.39%.44% 1.98% 66.4%.8% 9 3 to 4 9%-95% 1.18% 3.89%.1% 1.56% 76.66%.15% 11.9% 7.36%.7% 1 4 to 5 9%-95%.44% 5.85%.% 13.18% 35.5% 1.97% 11.66% 33.6% 3.34% 11 to 5 8%-95% 1.7% 6.1%.4%.59% 19.74%.9% 18.4% 93.79%.4% 5. SUMMARY In ths eseach we have developed an analytcal model fo the aveage numbe n queue, the vaance of the numbe n queue, and queue sze dstbuton fo an automated seve, whch eceves nputs fom an automated wokstaton. The analytcal soluton s deved fom a dscete tme Makov chan model of two automated wokstatons n sees wth nfnte buffe capacty. We show that the analytcal model pefoms much bette on aveage than two-moment G/G/1 appoxmatons appled to such systems. In such automated systems t s vey possble to have lage coeffcents of vaaton fo the tme jobs spend n the wokstaton. The analytcal expessons deved ae moe complcated than exstng two-moment G/G/1 appoxmatons, yet they ae vey easly mplemented n speadsheets. The next step n ths eseach s to examne moe than two wokstatons n sees. Ths has been conducted n agaajan and Km (6, whee two-moment lnkng equatons ae developed to estmate the mean and vaance of wokstaton ntedepatue tmes. Addtonal extensons ae to wokstatons that can pocess multple jobs n paallel, and to sees of wokstatons that have dffeent fxed job pocessng tmes.

10 Abs. % Dffeence n Avg. # n System Km: A Queung Model fo an Automated Wokstaton Recevng Jobs fom an Automated Wokstaton IJOR Vol. 7, o. 4, 9 18 ( % Aveage % Dffeence Fom Smulaton ( Systems Pe Test Set 3.% 5.%.% 15.% 1.% ew Model Saka.- Yu KL-Belz 5.%.% Test Set umbe Fgue 4. Aveage pecent dffeence n the tme aveage numbe n system. The aveage pecent dffeence s taken ove systems wthn a test set. REFERECE 1. Altok, T. (1997. Pefomance Analyss of Manufactung Systems, Spnge-Velag, ew Yok... Buzacott, J. A. and Shanthkuma, J. G. (1993. Stochastc Models of Manufactung Systems, Pentce-Hall Intenatonal Sees n Industal Engneeng. 3. Dalley, Y. and Geshwn, S.B. (199. Manufactung Flow Lne Systems: A Revew of Models and Analytcal Results, Queueng Systems: Theoy and Applcatons, 1: Fenbeg, B.. and Chu, S.S. (1987. A Method to Calculate Steady State Dstbutons of Lage Makov Chans, Opeatons Reseach, 35(: Geshwn, S.B. (199. Vaance of Output of a Tandem Poducton System, Poceedngs of The Second Intenatonal Wokshop on Queung etwoks wth Fnte Capacty, IBM-Reseach Tangle Pak, oth Caolna, May Hopp, W.J. and Speaman, M.L. (1. Factoy Physcs, nd Edton, Iwn McGaw-Hll. 7. Inman, R.R. (1999. Empcal evaluaton of exponental and ndependence assumptons n queueng models of manufactung systems, Poducton and Opeatons Management, 8(4: Km, D.S. and Alden, J.M. (1997. Estmatng the Tme to Poduce a Fxed Lot Sze on a Wokstaton wth Detemnstc Pocessng Tmes and Random Falues and Repas, Intenatonal Jounal of Poducton Reseach, 35(1: Km, D.S. and Smth, R.L. (1995. An Exact Aggegaton/Dsaggegaton Algothm fo Lage Scale Makov Chans, aval Reseach Logstcs, 4: Kngman, J.F.C. (196. Some Inequaltes fo the Queue GI/G/1, Bometca, 49: Kame, W. and Lagenbach-Belz, M. (1976. Appoxmate Fomulae fo the Delay n the Queueng system GI/G/1, Poceedngs of the Eghth Intenatonal Teletaffc Congess, Melboune, 1-17 ovembe, 35: agaajan, R.D. (3. Lnkng equatons fo the analyss of a seal automated wokstaton system, Maste of Scence Thess, Depatment of Industal and Manufactung Engneeng, Oegon State Unvesty, ovembe agaajan, R.D.. and Km, D.S. (6. Two-Moment Appoxmatons fo Analyzng a Sees of Automated Wokstatons, Intenatonal Jounal of Poducton Reseach, 44(6: Sakasegawa, H. (1977. An Appoxmaton Fomula / (1, Ann. Inst Statstcal Math., Tokyo, 9, A. L q 15. Shanthkuma, J. G. and Buzacott, J. A., (198. On the Appoxmatons to the Sngle Seve Queue, Intenatonal Jounal of Poducton Reseach, 18(6: Whtt, W., (1983. The Queueng etwok Analyze, The Bell System Techncal Jounal, 6(9, Yu, P.S. (1977. On Accuacy Impovement and Applcablty Condtons of Dffuson Appoxmaton wth Applcatons to Modelng of Compute Systems, TR-19 (Dgtal Systems Laboatoy, Stanfod Unvesty.