Semi-Markov model of the availability of the means of municipal transport system

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1 Sem-Markov model of avalablty of the means of muncal transort system TRIBOLOGY 25 KLAUDIUSZ MIGAWA * SCIENTIFIC PROBLEMS OF MACHINES OPERATION AND MAINTENANCE 3 (159) 29 Sem-Markov model of the avalablty of the means of muncal transort system Keywords Transort system, muncal transort, sem-markov model, technologcal object avalablty. Słowa kluczowe System transortowy, komunkacja mejska, model sem-markowa, gotowość obektu techncznego. Summary The artcle resents a method of desgnng the avalablty of technologcal objects used n comlex oeratonal systems wth the assstance of a theory ertanng to sem-markov rocesses. The comlete consderaton was resented based on the chosen authentc system of transort means oeraton muncal bus transort system n a chosen urban comlex. Drect realsaton of transort goals of the muncal transort system s undertaken by a utlsaton subsystem comrsed of elementary subsystems of the oerator transort means (drver bus) tye, the avalablty of whch sgnfcantly nfluences the ossblty of arorate realsaton of these goals. In order to buld a model of the avalablty of transort means, sgnfcant oeratonal states of the oeratonal rocess were desgned and a dvson and reducton of the number of states was done takng nto consderaton the crteron avalablty for oeraton. Based on ths, both eventbased and mathematcal models of the rocess of transort means oeraton were bult, assumng that the mathematcal model of the rocess of transort means oeraton consttutes a homogenous * Unversty of Technology and Lfe Scences, Deartment of Machne Mantenance, ul. Prof. S. Kalskego 7, Bydgoszcz, Poland, e-mal: km@karor.com.l

2 26 K. Mgawa sem-markov rocess. Then, for the oeratonal data obtaned from the testng of authentc oeratonal rocess, lmt values of the avalablty factor for the delneated levels of transort means avalablty n muncal transort system were defned. Ths artcle resents a way to defne the avalablty of a sngle technologcal unt transort means, based on the constructed sem-markov model of oeratonal rocess used n a utlsaton subsystem of the tested transort means system. The resented model for desgnng the avalablty of transort means s art of a decson-makng model of controllng transort system avalablty created wthn the framework of a larger research roject. 1. Introducton Technologcal object (element or system) avalablty s the object s feature, whch s characterstc from the ont of vew of the ossblty of tmely obtanng or mantanng the state of effcency (facltatng the realsaton of goals) [6, 8, 1, 11]. The term avalablty ertans to the systems charactersed by the necessty of quck reacton n emergency stuatons; such systems nclude the mltary, olce, ambulance servce, fre deartment, as well as transort systems. In such systems, whenever there s a goal n an emergency stuaton, an ndvdual or a team of ndvduals wth ther assgned technologcal objects attemts ts mmedate realsaton. A unque tye of transort system s the muncal bus transort system wth the man goal of assenger transort. The frequency of rdes and a comlex number of ersons on board determne the sze of the transort goals, n an assgned erod of tme, on a defned route. In systems of ths tye, the drect mlementaton of assenger transort remans the resonsblty of the utlsaton subsystem comrsed of elementary subsystems of the oerator transort means (drver bus) tye. It s on the avalablty of these subsystems that the ossblty of arorate mlementaton of transort goals deends. Avalablty of transort means remans at an arorate level, as a result of the rear rocesses mlemented n the utlsaton subsystem by technologcal suort unts and n the effcency mlementaton n rear osts at the bus deot. The above results n the avalablty of transort system beng deendent on the ossblty of correct control of the rocess of oeraton s realsed both n the utlsaton and effcency mlementaton subsystems. Ths artcle resents a way of defnng avalablty of a sngle technologcal object means of transort, based on the bult sem-markov model of oeratonal rocess realsed n the utlsaton subsystem of the tested muncal transort system. The model of determnng the avalablty of transort means resented here s a art of a decson-makng model of controllng transort system avalablty created wthn a larger research roject. In the followng stages, a model for determnng utlsaton subsystem avalablty comrsed of N number of transort means combned wth an arorate structure wll be created, together wth a model of defnng and the evaluatng of the avalablty of an effcency mlementaton subsystem.

3 Sem-Markov model of avalablty of the means of muncal transort system Event-based oeraton rocess model An event-based model of an oeraton and mantenance rocess was bult on the bass of the analyss of the oeraton and mantenance states sace and the oeraton and mantenance events regardng the techncal objects beng oerated and mantaned wthn a real transort system under analyss. Each of the oerated and mantaned techncal object may, at any moment t, stay n only one of the dstngushed states, formng a fnte set of oeraton and mantenance states of an object. The followng sgnfcant oeraton and mantenance states have been dstngushed n the analysed oeraton and mantenance rocess model: S 1 stoover at bus deot arkng sace, S 2 rear at bus deot arkng sace, S 3 carryng out of the transort goal, S 4 fuel ntake between transort eak hours, S 5 rear by techncal suort unt wthout losng a tr, S 6 rear by techncal suort unt wth losng a tr, S 7 awatng the start of task realsaton after techncal suort rear, S 8 emergency ext, S 9 techncal object rear at the effcency mlementaton subsystem osts. Afterwards, the ossble transtons between the dstngushed oeraton and mantenance states were determned. It was the bass for makng a grah of the oeraton and mantenance state changes, as resented n the Fg. 1. S 7 S 6 S 3 S 5 S 8 S 2 S 4 S9 S 1 Fg. 1. Drected grah reresentng the transort means oeraton rocess Rys. 1. Graf skerowany odwzorowana rocesu eksloatacj środków transortu

4 28 K. Mgawa 3. Sem-Markov model of the avalablty of transort means In general, the avalablty of technologcal object to realse the assgned goal s descrbed as a technologcal object feature that characterses t from the ont of vew of the ossblty of tmely obtanng and mantanng the state of effcency (facltatng the realsaton of the goal) at moment t or durng tme range τ r of tme reserve for sules and/or object rear. In ths aer, the value of technologcal object avalablty s defned for three selected levels: Frst level avalablty G OT () 1 s determned for objects, whch at any gven moment t are effcent and suled; n case of the tested system, t ertans to technologcal objects whch realse the assgned goal or awat the begnnng of goal realsaton at the arkng lace of the bus deot or followng rear done by techncal suort unt. Second level avalablty G OT ( 2) s addtonally assgned for effcent technologcal objects whch are to be suled, e.g. n fuel; however, the suly rocess wll be realsed n T z τ z shorter than the erod of tme reserve for sulyng technologcal objects wthout losng a tr,.e. durng the breaks between trs determned by the schedule of transort goals realsatons (bus schedule). Thrd level avalablty G OT ( 3) s addtonally determned for neffcent technologcal objects whch were damaged at the arkng lace of the bus deot whle awatng the realsaton of the transort goal or en route whle realsng the transort assgnment but were reared by bus deot techncans or techncal suort unt n T u τ u tme shorter than the erod of tme reserve wthout losng a tr,.e. durng the breaks between trs determned by the schedule of transort goals (bus schedule). In the examned model, t s assumed that both the sules of the technologcal object durng tme reserve τ z and the rear durng tme reserve τ u do not sto the goal from beng realsed and, at the same tme, do not make t necessary to substtute the suled or reared object wth a dfferent one (reserve object) Mathematcal model of the oeraton rocess Usng the sem-markov rocesses n the mathematcal modellng of the oeraton rocess, the followng assumtons were ut forward: The modelled oeraton rocess has a fnte number of states S, 1,2,,9. The random rocess X(t) beng the mathematcal model of the oeraton rocess s a homogenous rocess. At moment t, the rocess fnds s n state S 3 (the ntal state s state S 3 ).

5 Sem-Markov model of avalablty of the means of muncal transort system 29 The homogenous sem-markov rocess s unequvocally defned when ntal dstrbuton and ts kernel are gven. From our assumtons and based on the drected grah shown n Fgure 1, the ntal dstrbuton ( ) P{ X ( ) }, 1,2,..., 9 takes the followng form: whereas, the kernel of rocess Q(t): () t Q13() t Q () t 1 when 3 () (1) when 3 Q12 23 Q () () () () () 34 t Q35 t Q36 t Q38 t Q39 t Q41() t Q() t Q () 53 t Q63() t Q67() t Q () () () 73 t Q89 t Q91 t (2) where: Q { },, 1,2,..., 9 () t P X ( ) j τ τ t X ( τ ) n+ 1, n+ 1 n n j τ (3) means that the state of sem-markovan rocess and the erod of ts duraton deends solely on the revous state, and does not deend on earler states and erods of ther duraton, where τ 1, τ 2,, τ n, are arbtrary moments n tme, so that τ 1 <τ 2 < <τ n < ; as well as Q () t F () t (4) where: () t lm (5) t means that the condtonal robablty of transfer from state S to state S j,

6 3 K. Mgawa as well as F { } () t P X () t j X ( ) () t P τ t X ( τ ) X ( τ ) (6) { j},, 1,2,..., 9 n+ 1 n n, n+ 1 j τ (7) s a dstrbuton functon of random varable Θ sgnfyng the erod of duraton of state S, under the condton that the next state wll be state S j Avalablty of transort means In general, avalablty of technologcal objects determned based on the sem-markov oeraton rocess model s defned as the sum of lmt robabltes * of beng n states belongng to the set of avalablty states G, dla S S G, 1,2,..., 9 (8) In order to assgn the values of lmt robabltes * of stayng n the states of sem-markovan model of transort means oeraton, based on the drected grah shown n Fgure 1, the followng matrx P was created of the states change robabltes n rocess X(t): P 1 (9) Lmt robablty * of stayng n states of sem-markov rocess were assgned on the bass of the lmt theorem for sem-markovan rocesses [2, 7]: If hdden Markov chan n sem-markovan rocess wth fnte state S set and contnuous tye kernel contans one class of ostve returnng states such that for each state S, f 1 and ostve exected values E( Θ ), S are fnte, lmt

7 Sem-Markov model of avalablty of the means of muncal transort system 31 * lm t () t π π S E ( Θ ) E( Θ ) (1) exst where robabltes π, S consttute a statonary dstrbuton of a hdden Markov chan, whch fulfls the smultaneous lnear equatons S π, j S, π 1 π. (11) In order to determne the avalablty of transort system objects (transort means) based on the sem-markov oeraton rocess model, the oeratonal states of the technologcal object should be dvded nto avalablty states S G and non-avalablty state S NG of the object to realse the assgnment. Technologcal object avalablty states are states durng whch the object, ncludng the oerator, remans n the oeraton system, and s effcent and suled or wll be reared and/or suled n a erod of tme shorter than the tme reserve whch s to serve the urose. Non-avalablty states are states n whch the object or the oerator remans outsde the oeraton system (effcent or neffcent), as well as when an neffcent and/or unsuled object remans n the oeraton system. In the resented model, for each level of avalablty the followng technologcal object avalablty states were defned: for the frst level: State S 1 stoover at bus deot arkng sace, State S 3 carryng out of the transort goal, State S 7 awatng the start of task realsaton after techncal suort rear, for the second level: State S 1 stoover at bus deot arkng sace, State S 3 carryng out of the transort goal, State S 7 awatng the start of task realsaton after techncal suort rear, State S 4 ntake between transort eak hours, for the thrd level: State S 1 stoover at bus deot arkng sace, State S 3 carryng out of the transort goal, State S 7 awatng the start of task realsaton after techncal suort rear, State S 4 ntake between transort eak hours, State S 2 rear at bus deot arkng sace, State S 5 rear by techncal suort unt wthout losng a tr. Then, wth the use of the MATHEMATICA software, the lmt robablty * of stayng n states of sem-markov rocess and the avalablty of technologcal objects of the transort system were determned, defned by the followng deendences: j S

8 32 K. Mgawa G OT () 1 G OT ( 2) G OT ( 3) ( + + ) G OT + (12) * * * ( ) Θ + Θ [( ) ( Θ Θ2 )] + Θ Θ Θ5 + [ 36 ( Θ Θ7 )] + 38 Θ8 + ( ) Θ9 Θ * * * * ( ) (13) G OT + (14) ( + + ) Θ + Θ + Θ [( ) ( Θ Θ2 )] + Θ Θ Θ5 + [ 36 ( Θ Θ7 )] + 38 Θ8 + ( ) 9 Θ * * * * * * ( ) Θ (15) G OT + (16) ( ) ( Θ Θ2 ) + Θ Θ Θ Θ7 [( ) ( Θ Θ2 )] + Θ Θ Θ5 + [ 36 ( Θ Θ7 )] + 38 Θ8 + ( ) Θ9 (17) Then, usng the above formulas, values of avalablty of transort means used n the tested muncal bus transort system were defned (Table 1). Table 1. Values of avalablty of transort means used n the tested muncal bus transort system Tablca 1. Wartośc gotowośc środków transortu eksloatowanych w systeme autobusowej komunkacj mejskej G OT () 1 G OT ( 2) G OT ( 3),8444,859, Concluson Usng sem-markov rocesses for the modellng of the oeratonal rocess of transort means facltates the determnaton of transort means avalablty n case of tme erods between changes of ndvdual rocess states havng arbtrary robablty dstrbuton and transfer to the followng state deends only on the current rocess state. Avalablty of technologcal objects (transort means) used n the muncal bus transort system, determned on the bass of sem-markovan oeratonal rocess model, deends drectly on the values of lmt robabltes

9 Sem-Markov model of avalablty of the means of muncal transort system 33 * of beng at the states of the analysed rocess, and ndrectly on the values of robabltes of transfers between rocess states (values of the elements of matrx P) as well as values of condtonal duraton erods of rocess states Θ (values of matrx Θ). Change of the values of robabltes as well as duraton erods causes the change of the values of transort system technologcal objects avalablty G OT. Values of avalablty of transort system technologcal objects G OT deend on many factors: In case of the frst level: The relablty of the technologcal objects n use, The effcency of rear rocesses carred out at logstcs subsystem osts and by techncal suort unts, In case of the second level: The number of fuel suly osts n the subsystem, The effcency of the subsystem fuel ntake osts, The values of the tme reserve for suly of technologcal objects wthout losng a tr, In case of the thrd level: The number of techncal suort unts, Equng the techncal suort unts wth tools and devces used n technologcal object rear ensurng hgh effcency of the erformed rears, The servceablty and rear effcency of technologcal objects, The values of tme reserve for technologcal object rear wthout losng a tr. Values of transort means avalablty obtaned on the bass of oeratonal data resented n Table 1 are not hgh. However, one should take nto consderaton the fact that hgh avalablty of the utlsaton subsystem comrsed of N number of transort means may be obtaned as a result of usng the arorate structure whch ntegrates technologcal objects and numbers of reserve objects. The model of defnng the avalablty of the utlsaton subsystem comrsed of N transort means wll be reared at future stages of the work conducted. Θ References [1] Bobrowsk D.: Modele metody matematyczne w teor nezawodnośc. Warszawa, WNT, [2] Grabsk F.: Sem-markowske modele nezawodnośc eksloatacj, Polska Akadema Nauk, Instytut Badań Systemowych, sera: Badana systemowe, tom 3, Warszawa 22.

10 34 K. Mgawa [3] Grabsk F., Jaźwńsk J.: Funkcje o losowych argumentach w zagadnenach nezawodnośc, bezeczeństwa logstyk, WKŁ, Warszawa, 29. [4] Iosfescu M.: Skończone rocesy Markowa ch zastosowane, PWN, Warszawa, [5] Jaźwńsk J., Grabsk F.: Nektóre roblemy modelowana systemów transortowych, Instytut Technolog Eksloatacj, Warszawa Radom, 23. [6] Kołowrock K., Woroay M., Modelowane gotowośc systemu transortu autobusowego. Materały Szkoły Zmowej, Szczyrk, 21. [7] Koroluk V.S., Turbn A. F.: Sem-Markov rocesses and ther alcaton, Naukova Dumka, Kev, [8] Praca zborowa od redakcją J.M. Mgdalskego, Inżynera nezawodnośc. Poradnk. Warszawa, Wydawnctwo ZETOM, [9] Woroay M., Mgawa K.: Markov model of the oeratonal use rocess n an autonomous system. Polsh Journal of Envronmental Studes, Vol. 16, No 4A, 27. [1] Woroay M., Szubartowsk M., Mgawa K.: Model oceny kształtowana gotowośc oeracyjnej odsystemu wykonawczego w systeme transortowym. Wydawnctwo Zakład Polgraf Instytutu Technolog Eksloatacj, Radom, 23. [11] Żurek J., Problemy gotowośc technk lotnczej. Rozdzał 13, Praca zborowa: Problemy badań eksloatacj technk lotnczej, Tom 2, Warszawa, Wydawnctwo ITWL, Manuscrt receved by Edtoral Board, November 15 th, 29 Semmarkowsk model gotowośc środków transortu systemu komunkacj mejskej Streszczene W artykule rzedstawono metodę wyznaczana gotowośc obektów techncznych, użytkowanych w złożonych systemach eksloatacj, z wykorzystanem teor dotyczącej rocesów sem-markowa. Całość rozważań rzedstawono na rzykładze wybranego rzeczywstego systemu eksloatacj środków transortu systemu autobusowej komunkacj mejskej w wybranej aglomeracj. Bezośredną realzacją zadań rzewozowych systemu komunkacj mejskej zajmuje sę odsystem wykonawczy złożony z odsystemów elementarnych tyu oerator środek transortu (kerowca autobus), których gotowość w stotny sosób wływa na możlwość rawdłowej realzacj tych zadań. W celu zbudowana modelu gotowośc środków transortu wyznaczono stotne stany eksloatacyjne rocesu eksloatacj oraz dokonano odzału redukcj lczby stanów ze względu na kryterum gotowośc do dzałana. Na tej odstawe zbudowano zdarzenowy oraz matematyczny model rocesu eksloatacj środków transortu, zakładając, że matematycznym modelem badanego rocesu eksloatacj jest jednorodny roces sem-markowa. Nastęne dla danych eksloatacyjnych, uzyskanych z badań rzeczywstego rocesu eksloatacj, wyznaczono granczne wartośc wsółczynnka gotowośc dla wyróżnonych ozomów gotowośc środków transortu eksloatowanych w systeme komunkacj mejskej. W rezentowanej racy rzedstawono sosób wyznaczana gotowośc ojedynczego obektu techncznego środka transortu, na odstawe zbudowanego semmarkowskego modelu rocesu eksloatacj, realzowanego w odsysteme wykonawczym badanego systemu transortu mejskego. Przedstawony w racy model wyznaczana gotowośc środków transortu jest częścą składową oracowywanego w ramach szerszego rojektu badawczego, decyzyjnego modelu sterowana gotowoścą systemu transortowego.