Time Dependent System State Probabilities of Single Server Queuing System with Infinite Queue

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1 Uglješa Bugarć Professor Uversy of Belgrade Faculy of Mechacal Egeerg Duša Perovć Assocae Professor Uversy of Belgrade Faculy of Mechacal Egeerg Mlca Gerasovć Advsor/Coordaor Isue for Iprovee of Educao Belgrade Zora Perovć Techcal drecor Teco Sse d.o.o. Belgrade Te Depede Syse Sae Probables of Sgle Server Queug Syse wh Ife Queue Aalycal expresso for e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/ s derved. Expresso s derved by fdg he l value of expresso for e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M//K, whe uber of places he queue es o fy, he case ha syse s epy a he begg. Oly eleeary aheacal operaos are used. Keywords: e depede probables, M/M/ queug syse, fe capacy.. INTRODUCTION. M/M//K QUEUING SYSTEM The aalyses of queug syses, based o Marov processes, are os cases resrced o seady sae behavor.e. o syses equlbru. The reaso for ha les he fac ha for obag e depede syse sae probables of hese queug syses, a syse of lear frs order dffereal equaos has o be solved. Uforuaely, aalycal soluos rarely exs ad f hey exs, her obag eds o be que dffcul ad coplcaed. Several auhors have obaed he resuls e depede syse sae probables (properes) of soe queug syses aalycal for. These aalycal expressos are usually obaed by use of geerag fucos ad rasfors such as Laplace rasfor, z-rasfor ec. The derved expressos are very coplcaed ad requre alerave copuaoal echques by he fac ha hey ofe refer o Bessel fucos. The rase aalyss of sgle server queug syses wh fe queue capacy, based o Marov processes.e. M/M/, ca be foud, for exaple, he wors of: Morse [,], Greeberg ad Greeberg [3], Heahcoe ad Wer [4], Gross ad Harrs [5], Kleroc [6], Cooper [7], Taacs [8]. The a dea of hs paper s o rasfor, usg eleeary aheacal operaos, he expresso for e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M//K, order o fd s l value whe he uber of places he queue () eds o fy. I aoher words, a dea s o oba aalycal expresso for e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/ drecly whou solvg adequae syse of lear dffereal equaos, he case ha he syse s epy a he begg. The correspodg syse of lear frs order dffereal equaos, defg he e depede sae probables of he M/M//K queug syse, s [6]: λ µ p p p λ ( λ µ ) µ p p p p λ µ pk pk pk,,..., K Receved: Sepeber 6, Acceped: Noveber 6 Correspodece o: Uglješa Bugarć Faculy of Mechacal Egeerg, Kraljce Marje 6, Belgrade 35, Serba,,,,...,. (3) E-al: ubugarc@as.bg.ac.rs, do:.5937/fe7463b Faculy of Mechacal Egeerg, Belgrade. All rghs reserved FME Trasacos (7) 45, where: λcos. s ea arrval rae, µcos. s ea servce rae, K s axal possble uber of us (cusoers) he syse, p () s e depede probably ha he syse a he e wll be he sae, ad p ( ) dp ( ) d s frs dervao of he / () p () per e,,,..., K,. Aalycal soluo of dffereal equaos syse () ay be foud wors by several auhors, such as: Morse [], Taacs [8], Shara ad Gupa [9], Taraba ad El-Baz [], Bugarc [] ec. Geeral aalycal soluo of syse (), whou egrao cosas expressos, ay be wre he followg for []: C p ( ) C ( ) s θ s ( ) θ s ( θ )} λ µ cos θ λ µ e () where:,,,..., K, λ/µ < s ulsao facor, θ () /( ), C are egrao cosas deered upo al values of syse sae probables ( ). The case whe he syse s epy a he begg eas ha here s o cusoers beg prese he syse ally.e. a. I such case he al values of he syse sae probables are:

2 Iegrao cosas, deered upo he al values of he syse sae probables (3), are []: C ( ) /( ), (4) C λ s θ / ( ) (5) λ µ cos θ λ µ } ;,,..., By subsug (4) ad (5) o () he fal expresso for e depede syse sae probables, whose l has o be foud whe, has followg he for: ( ) ( λ µ ) p ( ) λ ( ) e s θ λ µ cos θ λ µ (6) s ( ) θ s ( θ )} λ µ cos θ e ;,,,..., 3. LIMIT OF THE SYSTEM STATE PROBABILITIES WHEN Applyg ow rgooerc forulas, expresso (6) ca be rasfored as: ( ) ( λ µ ) p ( ) λ ( ) e λ µ cos θ λ µ cos ( ) cos ( ) } ( ) θ ( ) θ } θ θ cos cos λ µ cos θ e ;,,,..., (7) O he oher sde, par of expresso (7) /( λ µ cosθ λ µ ) s soluo of defe egral: e e ( λ µ ) d Prevous defe egral, usg he followg forulas: θ ( θ e e ) (8) cosθ, [] θ ( e θ e λ µ ) e I ( λ µ ) e ( λ µ ) I ( λ ) θ [7] I µ [3] ca be rasfored o he followg for: ( ) e ( λ µ ) λ µ I cos d ( ) I ( ) e ( λ µ ) θ λ µ d (9) where I ( λ µ ) s odfed Bessel fuco of he frs d of order ( - eger uber). [4] Soluo of defe egral: () ( ) e ( λ µ ) λ µ d I,,,..., volves Hypergeoe rcpfq,, fuco ad for 4 µ /( λ µ ) λ<µ) s: 4λ µ }, λ µ λ < (λ>, µ> ad r Γ r r Γ! r Γ Γ ( λ µ ) r! where ( x) Γ s Gaa fuco. ( r)! r λ µ λ µ Prevous expresso depeds oly fro (,,...) ad furher ex wll be deoed as R(). Aalycal values of R() depedece of λ ad µ wll be calculaed laer. Fally, he soluo of defe egrals gve by expresso (9).e. (8) s: ( ) R( ) R() cos θ. () Replacg /( cosθ λ µ ) wh expresso (), expresso (7) obas he followg for: ( ) ( λ µ ) p ( ) λ ( ) e R() cos ( θ ) R cos ( ) cos ( ) } ( ) θ ( ) θ } θ θ () cos cos λ µ cos θ e ;,,,..., For furher aalyss, prevous expresso should be wre expaded for.e. decoposed o dvdual sus per. Afer ulplyg ad dvdg expresso () by, eleeary aheacal rasforaos ad applcao of ow rgooerc forulas, a ew covee for for fdg he l of expresso () s followg: p ( ) ( ) ( λ µ ) λ R R e ( ) cos( θ ) cos e [( ) θ ] e ( ) cos[ ( ) θ ] e FME Trasacos VOL. 45, No 4, 7 63

3 R ( ) cos[ ( ) θ ] R cos e [( ) θ ] e cos R cos [( ) θ ] e [( ) θ ] e cos R cos [( ) θ ] e [( ) θ ] e cos R cos [( ) θ ] e [( ) θ ] e,,,...,. cos[ ( ) θ ] e (3) Accordg o defo of defe egral, egral su becoes defe egral f he followg l exss [5]: b l f ( ζ ) x f ( x) dx ; (4) a ax x x ζ < x, x x x ;,,..., (). Each of he sus expresso (3) accordg o (4) ca be rasfored o defe egral: θ J f ( θ ) dθ whe he followg way: θ ( ) θ θ ( ) θ, θ,,,...,. Values for θ () are depede fro ad es o fy whe. The upper boud θ ( ) of egral J s ( ) deered wh: l ; θ, The lower boud θ ( ) of egral J s deered wh: l ; θ. Accordg o he prevous ex, l whe ad θ of frs su per expresso (3) s: l cos( θ ) e θ θ cos( θ ) e dθ. Prevous defe egral s, fac, a odfed Bessel fuco of he frs d of order of argue λ µ.e. [4, 6]: cosθ cos( θ ) e dθ I ( λ µ ). The l of all sus per expresso (3) whe ad θ ca be deered he sae way. The l of he frs added expresso (3) for < whe s: ( ) l ~ ( ),,,...,. (5) Replacg all sus per (3) wh s l ad frs added of (3) wh expresso (5), he l of e depede syse sae probables whe, s obaed as: ( λ µ ) ( ) λ ( ) e p ( ) ( λ µ ) ( λ µ ) ( λ µ ) ( λ µ )} R I I I I R I ( λ µ ) I ( λ µ ) ( λ µ ) ( λ µ ) ( λ µ ) ( λ µ ) ( λ µ ) λ µ I I I I I I,,..., }} (6) Prevous expresso, usg ow relao bewee odfed Bessel fucos of frs d I ( x) I ( x) I ( x), ca be reduced o [3]: x p ( ) ( λ µ ) λ e ( ) I ( λ ) R( ) µ λ µ ( λ ) I µ λ µ 63 VOL. 45, No 4, 7 FME Trasacos

4 ( ) I ( λ ) R µ (7) λ µ ( ) I ( ) λ µ λ µ ( ) I ( ) λ µ λ µ ( ) I ( λ ) µ,,,...,. λ µ Aalycal values of R() ca be deered drecly fro expresso (7). Frs, s ecessary o replace al codos (3) o (7).e. for, p (), p (),,,...,. I order o deere he values of expresso (7) for, for each syse sae probably he L'Hospal- Beroull rule has o be appled o solve udefed fors of ype / whch wll appear. Also, he forula for dffereao of odfed Bessel fucos of he frs di ( x) d: I ( x) I ( x) dx as well as he fac ha I () ad I (),,,..., have o be appled. Applyg prevous o expresso (7) for varous values of paraeer (dffere syse sae probables), he syse of recurre forulas suable for deerg aalycal values of coeffces R(),,,,..., s obaed as: R λ R R ( 3) ( ) R ( 4) ( ) R ( 4) λ λ R λ 3 λ R λ 3 5 (8) λ ec. I order o solve he syse (8) s ecessary o ow frs wo values of coeffces R().e. R() ad R(). Those values ca be obaed by solvg defe egral () for ad. Geeral aalycal soluo of he syse (8) s: R,,,..., (9) µ λ ( ) Fally, by replacg (9) (7), aalycal expresso for he e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/, obaed as he l value of he syse sae probables of sgle server queug syse wh fe queue capacy M/M//K whe he uber of places he queue () eds o fy, has he followg for: p ( λ µ ) e ( ) ( ) µ λ I ( λ µ ) ( ) I ( λ µ ) ( ) ( ) I ( λ µ ) [ () ( ) I ( λ ) µ ( ) I ( λ ) µ ( ) I ( λ )]} µ,,,..., 4. SIMULATION MODEL OF M/M/ QUEUING SYSTEM Valdao of he expresso for he e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/ () wll be doe by usg adequae sulao odel (dscree eve sulao). The reaso for ha s he fac ha he syse of fe dffereal equaos ca o be exacly solved wh ow uercal ehods. S T A R T λ, µ, No,, d s s, Nos al syse sae X ss, s arrval servce, Nos al syse sae f (, Nws) X ss [ ] Xss [ ] X j - hsogra of absolue frequeces of sgle syse saes depedece of gve erval legh d, cosderg all sulao experes; p j - hsogra of relave frequeces (probables) of sgle syse saes, p X / ( No. ); j j s s E N D Fgure. Algorh of sulao odel. X j p j ar Sae Wa N ws N w p j FME Trasacos VOL. 45, No 4, 7 633

5 arrval servce ar o sc o Sae Wa ser (-/ µ ). l( RN) sc ser Sae Wo Nws Nws o ar (-/ λ). l( RN) Nws Nws - N w > o Sae Wa Nws Nws Nw Nw ser (-/ µ ).l( RN) sc ser s - durao of sulao expere, sc - oe of servce copleo.e. chage of servcg chael sae, ser - servcg durao, d - erval legh for syse sae absolue ad relave hsogras, X ss - e depede syse sae vecor for each sulao expere, X j - e depede absolue frequeces of j-h syse sae, p j - e depede relave frequeces (proba bles) of j-h syse sae, RN - rado uber geeraed accordg o u for dsrbuo erval fro o. Durao of each sulao expere s 6 e us, he uber of execued sulao experes s 4 for oe par of values for λ ad µ.e. ulsao facor. 5. COMPARISON OF ANALYTICAL AND SIMULA TION RESULTS Nw Nw - Fgure. Algorh of sulao odel. (coue) The sulao odel used for valdao of expresso () as oupu has oly chage of syse sae probables e depedece of λ - ea arrval rae ad µ - ea servce rae.e. ulsao facor. The algorh of he developed sulao odel s show fgure. Ial codos for sulao odel (expere) are: uber of us he syse as well as he queue equal o zero, servcg chael s free ( he sae "wag"), ad frs u coes o he syse a. Ier arrval es of us o he syse are geeraed accordg o expoeal dsrbuo wh paraeer λ, whle u servcg es are geeraed accordg o expoeal dsrbuo wh paraeer µ. For every e u durg sulao e, each sulao expere, e ad syse sae are wre separae daabase (depedg o syse sae). Whe gve uber of sulao experes are fshed, fro creaed daabases, absolue frequeces of syse saes gve e ervals are calculaed (hsogra of absolue frequeces). Afer ha relave frequeces hsogra.e. syse sae probables, gve e ervals, are calculaed. Overvew of labels used he sulao odel: Sae - sae of servcg chael: "Wo" - wor, "Wa" - wag, No s - uber of sulao experes, N w - curre uber of us he queue, N ws - curre uber of us he syse, - curre syse e, ar - oe of arrval of ew u o he syse, Dagras pcures, 3, 4 ad 5 show chage e of he frs four syse sae probables (p, p, p ad p 3 ) of sgle server queug syse wh fe queue capacy M/M/, for dffere values of ulsao facor, such as.35,.5,.65 ad.8 respecvely. Te depede syse sae probables obaed as a resul of sulao are ared wh sybols (*,,, ), whle e depede syse sae probables obaed usg expresso () are o ared. Fgure. Te depede syse sae probables (.35). Fgure 3. Te depede syse sae probables (.5). 634 VOL. 45, No 4, 7 FME Trasacos

6 Fgure 4. Te depede syse sae probables (.65). Fgure 5. Te depede syse sae probables (.8). Aalyss of resuls preseed fgures 5, shows ha he values of he syse sae probables obaed as a resul of sulao ach values of he syse sae probables calculaed usg expresso (). Ths leads o he cocluso ha expresso () for e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/ s correc. 6. CONCLUSION I hs paper, aalycal expresso for e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/ s derved as a l value of expresso for he e depede sys e sae probables of sgle server queug syse wh fe queue capacy M/M//K. The l value s foud he case whe he uber of places he queue eds o f y ad he case ha he syse s epy a he begg. Valdao of derved expresso for he e depede syse sae probables of sgle server queug syse wh fe queue capacy M/M/ s doe by developed sulao odel. The applcao of derved expresso ca be foud he aalyss of o-saoary worg reges of ras porao devces dusry. For exaple, he source of rasporao us (palles) s a oupu fro produco (fal goods), whle servce cosss of sorg palles o warehouse usg oe AS/RS devce. All palles have o be sored he warehouse, so he lao of pu zoe (queue) of he warehouse, heorecally, does o exss. REFERENCES [] Morse, P.M.: Sochasc Properes of Wag Les, Joural of he Operaos Research Socey of Aerca, Vol. 3, No. 3, pp. 55-6, 955. [] Morse, P.M.: Queues, Iveores ad Maeace: The Aalyss of Operaoal Syses wh Varable Dead ad Supply, Joh Wley & Sos Ic., New Yor, 958. [3] Greeberg, H. ad Greeberg, I.: The Nuber Served a Queue, Operaos Research, Vol. 4, Iss., pp , 966. [4] Heacoe, C.R. ad Wer, P.: A Approxao for he Moes of Wag Tes, Operaos Research, Vol. 7, Iss., pp , 969. [5] Gross, D. ad Harrs, C.M.: Fudaeals of Queueg Theory, Joh Wley & Sos, New Yor, 974. [6] Kleroc, L.: Queueg Syses, Volue I: Theory, Joh Wley & Sos, New Yor, 975. [7] Cooper, B.R.: Iroduco o Queueg Theory, Elsever Norh Hollad, New Yor, 98. [8] Taacs, L.: Iroduco o he Theory of Queues, Oxford Uversy Press, Lodo, 96. [9] Shara, O.P. ad Gupa, U.C.: Trase behavour of a M/M//N queue, Sochasc Processes ad her Applcaos, Vol. 3, Iss. 3, pp , 98. [] Tababa, A.M.K. ad El-Baz, A.H.: Exac Trase Soluos of Noepy Marova Queues, Copuers ad Maheacs wh applcaos, Vol. 5, Iss. 6-7, pp , 6. [] Bugarc, U.: Modellg of rasporao syses o-saoary operag reges by applyg he queug heory, PhD hess, Faculy of Mechacal Egeerg Belgrade, Belgrade,. ( Serba) [] Kaša, R.: Vša aeaa I, Zavod za zdavaje uđbea SR BH, Sarajevo, 969. ( Serba) [3] ÀNKE, E., ÈMDE, F., LËÃ, F., Specalêìe fucforlì, graf, ablcì, zdae reêe, Izdaeêsvo Naua, Mosva, 977. ( Russa) [4] Kaša, R.: Vša aeaa II (jga druga), Nauča jga, Beograd, 95. ( Serba) [5] Baraeov, G.S., Dedovč, B.P. dr.: Zadac rješe prjer z vše aeae s prjeo a ehče aue, Tehča jga, Zagreb, 978. [6] Šor, J.B., Sasče eode aalze orole valea pouzdaos, SMEITS, Beograd, 975. ВЕРОВАТНОЋЕ СТАЊА У ЗАВИСНОСТИ ОД ВРЕМЕНА ЈЕДНОКАНАЛНОГ СИСТЕМА МАСОВНОГ ОПСЛУЖИВАЊА СА НЕОГРАНИЧЕНИМ РЕДОМ У. Бугарић, Д. Петровић, М. Герасимовић, З. Петровић Аналитички израз за вероватноће стања у зависности од времена, једноканалног система масовног опслуживања са неограниченим редом М/М/, је изведен. Израз је изведен налажењем граничне вредности израза за вероватноће стања у зависности од времена једноканалног система масовног опслуживања са ограниченим редом М/М//К, када број места у реду тежи бесконачности, у случају када је систем на почетку рада празан. При извођењу коришћене су само елементарне математичке операције. FME Trasacos VOL. 45, No 4, 7 635