Strategies evaluation on the attempts to gain access to a service system (the second problem of an impatient customer)

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Milton Singleton
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1 h ublcaton aard n Sostk R.: Stratgs valuaton on th attmts to gan accss to a vc systm th scond roblm of an matnt customr, Intrnatonal Journal of Elctroncs and lcommuncatons Quartrly 54, no, PN, Warsa 8, Stratgs valuaton on th attmts to gan accss to a vc systm th scond roblm of an matnt customr ROMN SOSE, Ph.D., Eng. Rsó Unvrsty of chnology Dartmnt of Quanttatv Mthods ul. Wncntgo Pola, Rsó rsostk@r.rso.l h ar ntroducs th soluton of th otmaton roblm consstng n choosng a mor ffcnt stratgy hl accssng a vc systm. o solv th roblm a mthod of uung systms analyss by ncludd Markov chans [][4] as usd. h xamnd systm s dscrbd by non-markovan modl. h rsntd ssu as calld th scond roblm of an matnt customr. h frst roblm of an matnt customr as dscussd n ar [5]. uthor s comrhnsv study formulats and solvs an orgnal task of th scond roblm of an matnt customr. yords: otmaton, objctv cost functon, systm aramtrs, robablstc mthods, Markov rocss, non-markovan rocss, ncludd Markov chan. Introducton h frst roblm of an matnt customr as ntroducd and solvd n ar [5]. h task as to dtrmn th otmal tm btn th succssv attmts to accss a vc systm. In ths ar th scond roblm of an matnt customr as formulatd and solvd. h objctv s to slct a mor ffcnt of th to dffrnt stratgs on th attmt to accss a vc systm.

2 o solv th abov mntond roblms t s ncssary to ntroduc th modls of th xamnd systms and an objctv functon. h modls ar nondtrmnstc and non- Markovan.. Problm formulaton h gvn confguraton of four satllts s vsuald n Fg.. Satllt s task s to snd a data st to satllts D or D. s th drct conncton btn thos satllts s not ossbl, th data transmsson has to b mad by satllt C, hch chargs a f for th tm durng hch ts rsourcs ar usd. Satllt C s alays dl and rady to coorat, hl satllts and C ar dl and busy ntrchangably. Whl on of thm s busy dong othr tasks, t cannot rcv data from satllt. Satllts D and D orat ndndntly and t dos not mattr to hch of thm satllt snds data. t a gvn tm nstant satllt can try to snd data to only on of th satllts D or D. C D D Fg.. Problm llustraton Satllt s task s, thrfor, to gan accss to th rsourcs of satllts D or D. Hovr, t has no nformaton about th stat of th othr satllts, untl t maks an attmt to snd data. If th attmt fals bcaus th satllt to hch an attmt to transmt data as mad s busy, thn aftr tm satllt maks anothr attmt to snd data to thr satllt. h follong attmts ar mad n th sam tm ntrvals, untl on of th satllts s rady to rcv data. Evry attmt to snd data by satllt C, vn f t s unsuccssful, rsults n costs amountng to c. It s assumd that th atng tm of satllt to obtan conncton also gnrats a unt cost of c. hus, th costs functon of obtanng conncton by satllt has th follong form: F c N c hr: N th man numbr of attmts mad to obtan conncton by satllt, th man tm of obtanng conncton by satllt, c R {} cost of on attmt to obtan conncton, c R {} cost of conncton atng tm unt. h MM- systm s assumd to b th modl dscrbng th stat of satllt D. h tm hn th satllt s dl and hn t s busy s, thrfor, subjctd to xonntal

3 dstrbuton. It s not ossbl to uu to non of satllts D rsourcs. h grah of satllts stats s shon n Fg.. In stat W satllt s dl, hl n stat satllt s busy. W Fg.. Grah of stats for th MM- systm In ordr to obtan conncton satllt can rocd dffrnt stratgs, to of hch ar comard n ths ar. h frst stratgy conssts n slctng on of th satllts D or D and makng attmts to connct to ths satllt only n tm ntrvals. In th scond stratgy th attmts to connct ar mad th both satllts D and D ntrchangably. m ntrval btn th attmts s as ll. In ths stratgy both th frst and th scond satllt ar chckd n tm ntrvals. In th follong sctons t ll b rovd hch of th to roosd stratgs s mor ffcnt, that s, for hch stratgy th cost functon has th smallr valu.. nalyss of th frst stratgy h attmts to obtan conncton by satllt can b rgardd as an addtonal dtrmnd stram of arrvals to satllt D rsourcs. h systm gvn n Fg. togthr th an addtonal dtrmnd stram ll b calld th xtndd systm. In Fg. grah of th xtndd systm stats s shon. h xtndd systm s non-markovan. W Fg.. Grah of stats for MM- systm xtndd by dtrmnd stram of arrvals rrsntng an addtonal customr h addtonal absorbng nod rrsnts th stuaton n hch satllt obtans conncton. ranston to stat s ossbl only from stat W dl systm. h addtonal transton from stat to rrsnts th stuaton hn th attmt to obtan conncton by satllt has fald.

4 4 h xtndd systm has a nonxonntal vc nod and can b analysd by ncludd Markov chans... ranston rocsss for satllt D hs aragrah dmonstrats th transton rocsss n th systm shon n Fg.. hr knoldg s ncssary for furthr stratgy analyss. ssum th follong symbols: vnt that satllt obtand conncton n th attmt,,,, vnt that satllt dd not obtan conncton n th attmt,,,. h follong attmts ar mad n tm ntrvals. h attmt,,, s mad at tm. t a tm nstant of ro-tst satllt D s n statonary stat. hus, obtan: hat s: If th frst attmt to obtan conncton has fald, thn aftr tm satllt trs agan. ssumng that th systm gvn n Fg. s th Markov systm mmorylss thn obtan:,,,, :,,,, : 4 h follong symbols ar ntroducd: robablty that th satllt D s dl at a tm nstant, assumng that at a tm nstant t as busy, that s, robablty that th satllt D s busy at a tm nstant, assumng that at a tm nstant t as busy, that s. For th calculaton of th abov valus t s ncssary to solv th systm of dffrntal uatons hch dscrbs th transton rocsss n th systm from Fg.. & & 5 h soluton of th systm of uatons 5 has th follong form:

5 hr:,,, 6 h grahs of and functons ar shon n Fg Fg. 4. Grahs of functons and.. Contructon of th ncludd marko chan h ncludd Markov chan for th systm gvn n Fg. has to stats. hat s so bcaus th systm at a tm nstant just aftr th attmt to gan accss to a vc systm can b n stat satllt obtand conncton or n stat satllt dd not obtan conncton. In Fg. 5 grah of stats of ncludd Markov chan s shon. Fg. 5. Grah of stats of ncludd Markov chan ssum th follong symbols,,, s th attmt numbr: th robablty that th Markov chan s n stat, th robablty that th Markov chan s n stat. h uatons dscrbng th ncludd Markov chan hav th follong form: 5

6 6 [ ] [ ],, 7 Whr:,,, h follong ntal condtons ar also fulflld on th bass of ], [ ], [ 8 h uatons 7 and 8 dfn Markov chan dscrbng transton rocsss n th xamnd xtndd systm. Lt us dtrmn th robablty ˆ that satllt obtans conncton xactly n th attmt,,,. It can b don n to ays: Frst mthod drct,,,, ˆ ˆ 9 Scond mthod us 7,,,, ˆ ˆ In both th frst and th scond cas obtan:,,,, ˆ ˆ h valus of ˆ dscrb th dstrbuton of atng tm of satllt for th conncton to satllt D. h follong rlaton occurs: ˆ.. h xtndd systm aramtrs h man numbr of attmts to obtan conncton s:

7 ˆ ˆ ˆ ˆ N h man atng tm to obtan conncton s gvn by: ˆ ˆ ˆ ˆ 4 ftr takng nto account 6 and obtan dd aramtrs: 5 N 6 In Fg. 6 grah of functon N s vsuald. 8 N Fg. 6. Grah of functon N Grah of functon s shon n Fg. 7. 7

8 Fg. 7. Grah of functon 4. nalyss of th scond stratgy For th scond stratgy th xtndd systm conssts of to MM- systms and dtrmnd stram of arrvals rrsntng th addtonal satllt attmtng to obtan conncton. 4.. ranston rocsss n th vc systms ttmts to obtan connctons ar mad to satllts D and D ntrchangably. y th numbrs,, 4, dnot th attmts to connct to satllt D, hl by th numbrs,, 5, th attmts to connct to satllt D. ssum th follong symbols: th vnt that a conncton to satllt D n th attmt s obtand th vnt that a conncton to satllt D n th attmt n not obtand th vnt that a conncton to satllt D n th attmt s obtand th vnt that a conncton to satllt D n th attmt s not obtand hr:,,, Durng th ro-tst and th frst attmt th satllts D and D ar n a stady stat. hus, smlarly to cas hav: 7 8

9 9 s assumd that satllt D and D ar oratng n th sam ay, th follong condtonal robablts ar dntcal for both systms. hs robablts r calculatd n scton. s E. 6. For satllt D hav: : : 8 hr:,, 4, and for satllt D : : : 9 hr:,, 5, 4.. Constructon of th ncludd markov chan h ncludd Markov chan for th consdrd systm has to stats and t s htrognous. ftr th attmt to obtan conncton th systm can b n stat satllt gand accss to a vc systm or n stat satllt dd not gan accss to a vc systm. In ths cas th uatons dscrbng th ncludd Markov chan hav th follong form: [ ] [ ] [ ] [ ],,,,,,,, h follong ntal condtons ar also fulflld on th bass of 7: ], [ ], [ h uatons and dscrb th transton rocsss n th xamnd xtndd systm. W calculat th robablts that satllt obtans conncton xactly n th th attmt,,,. nalogously to th frst stratgy t can b don n to ays: Fst mthod drct:

10 ,,,, ] [ ] [,,,, ] [ ] [ Scond mthod us :,,, hus, n both cass, aftr takng nto account 7, 8 and 9 obtan:,,4,, 4 h valus of dscrb th dstrbuton of atng tm for a vc. h follong rlaton occurs: h xtndd systm aramtrs h man numbr of attmts mad tll th tm nstant n hch a conncton s obtand: N 6 h man atng tm for obtanng conncton s:

11 7 ftr takng nto account 8, 9 and 4 obtan dd aramtrs: 8 N 9 5. Stratgs comarson For th frst stratgy th cost functon F gvn n th xrsson, aftr takng nto account 5 and 6, has th follong form: c c F c h grah of functon F C s shon n Fg. 8 hn c, c >. 8 F C 6 F C OP 4 c c c OP.5.5 Fg. 8. Grah of th objctv functon F C For th scond stratgy aftr takng nto account 8 and 9 obtan:

12 F c c c In ordr to comar both stratgs calculat th valu of th follong xrsson: F c F c N N < h grah of rlaton n tm functon s dctd n Fg Fg. 9. Grah of functon of E. F h man costs for th frst stratgy ar hghr than thos for th scond stratgy. hus, n trms of assumd crtra, th scond stratgy sms to b mor ffcnt. h advantags rsultng from choosng th scond stratgy ar th hghr th smallr th xrsson valu s tm ntrval btn th succssv attmts s consdrably small. 6. Concluson Prsntd n th ar th mthod of solvng th consdrd roblm can b asly ald n cass of mor comlx modls of vc systms as ll. h only condton of ts mlmntaton s th ossblty of transton rocsss dtrmnaton n vc systms satllt D. For mor comlx systms ths transton rocsss can b calculatd by numrcal mthods. s th assumd modls of vc systms r Markovan t as ossbl to solv th roblm by analytcal mthods. Invstgatd n th ar th xtndd systms ar non-markovan. hr analyss as mad usng ncludd Markov chans. For th soluton of th statd roblm t as ncssary to analys transton rocsss n ths systms.

13 7. Rfrncs. G.. rtamono, O. M. rcho: naltcskj rojatnostnyj modl funkconroanja EWM. Moska, Wyd. Enrgja, Cachórsk: Modl koljko systmó komutroych. Glc, Poltchnka Śląska, skryty uclnan nr 844, Floc: Modloan otymalacja systmó koljkoych c.i. Systmy markosk. rakó, Wyd.. Rudkosk, Floc: Modloan otymalacja systmó koljkoych c.ii. Systmy nmarkosk. rakó, Wyd. FHU Poldx, 5. R. Sostk: Systmy koljko nykładncym ęłm obsług. rakó, Wydancta GH, Elktrotchnka Elktronka,, om 9, str F. tk: Stracony cas. Elmnty tor obsług masoj. Ponań, PWN, 977

COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP

ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng