THE ARCHIVES OF TRANSPORT VOL. XXV-XXVI NO 1-2 213 Sem-Markov Model of the Operato ad Mateace Process of Cty Buses Mrosław Szubartowsk Receved December 212 Abstract Ths paper presets a model of the operato ad mateace process of the techcal objects wth the urba trasport bus system. All the vestgatos have bee preseted o the bass of the selected real operato ad mateace system of the cty buses. Basg o the detfcato of the system uder vestgato ad o the process beg realzed wth t a model of the operato ad mateace process of the cty buses was bult, assumg that the model of ths process s the homogeeous sem-markov process X(t). For ths purpose, crucal states of operato ad mateace process selected trasport system were determed as well as possble trasfers betwee those states. Based o ths, a evet-based model of the operato ad mateace process of the cty buses was bult, assumg that ts model s the homogeous sem-markov process. For operato data obtaed after research coducted a authetc trasport system, values of ucodtoal perods of durato of process states, values of statoary dstrbuto cluded the Markov cha as well as values of probabltes of lmt dstrbuto of the sem-markov process were determed. Based o ths, a aalyss of the cty buses operato ad mateace process questo was performed. Preseted ths artcle the sem-markov model operato ad mateace process of cty buses s the frst stage of the creato of the avalablty model of the urba trasport bus system. 1. Itroducto Oe of the trasport system examples, whch the cotrolled processes are beg carred out, are urba trasport bus systems. The urba trasport bus systems are such systems whose task s to satsfy trasport eeds as a result of the trasportg performed over the specfc routs. The operato ad mateace process of the cty buses may be dvded geeral to the operatoal use process ad the servceablty assurace process. I the operato process the basc task of trasport system s obtaed, at the same tme geeratg profts due to the performace of the task. The actvtes performed sde the Wloclawek College of Eterprse ad Techology, Łęgska Street 2, 87-8 Włocławek, Polad, e-mal: km@karor.com.pl
11 Mrosław Szubartowsk servceablty assurace process are amed at restorg the task servceablty of the operated ad mataed cty buses. The realzato of the process descrbed s to brg back servceablty of the requred umber of the cty buses wth the specfed perod of tme. The realzato effectveess of the operato ad mateace processes has fluece o the possblty to accomplsh a trasportg task, as well as o the operato effectveess of the system descrbed here. That s why t s so ecessary to detfy properly the system ad process uder vestgato as well as to aalyse ad evaluate t order to assure hgh effectveess level of the system operato [6, 8, 9]. The aalyss ad evaluato of the cotrolled processes are performed o the bass of studyg ther models. The most frequetly used represetatos of the operato ad mateace process model are mateace graphs, cludg the drected graphs, whose peaks represet operato ad mateace states, whle the arcs of the graph represet possble trastos betwee the states. Due to a radom ature of the factors affectg the course ad effectveess of the operato ad mateace process beg realzed wth a complex system, the stochastc process, wth extesvely appled Markov ad sem- Markov processes are most frequetly appled for mathematcal modellg of the operato ad mateace process [1, 3, 5, 6, 7]. Implemetg sem-markov processes makes t possble to create ad aalyze the mathematcal model of the operato ad mateace process the case of the varables characterstc of the defed process states beg dstrbutos other tha expoetal. Realzato of the vestgatos by meas of the Markov ad sem-markov models of the operato ad mateace process makes t possble to aalyse the detaled problems related to the operato ad mateace of a techcal object o the oe had, ad o the other had t eables to aalyse the terrelatos betwee the set umber of the model parameters. It has bee attempted ths paper to descrbe mathematcally ad aalyse the operato ad mateace process of the cty buses, beg performed a selected real trasport system, assumg that the model of ths process s the homogeous sem-markov process X(t). The urba bus trasport operato ad mateace system formg oe of the subsystem of a urba trasportato system wth a selected courbato was chose as the vestgato object. 2. Evet-based Model of the Operato ad Mateace Process of Cty Buses The model of operato ad mateace process was created o the bass of the aalyss of state space as well as operatoal evets pertag to cty buses used the aalyzed authetc trasport system. Due to the detfcato of the mult-state operato ad mateace process of cty buses, crucal operato states of the process as well as possble trasfers betwee the defed states were desgated. Based o ths, a graph was created, depctg the chages of operato ad mateace process states, show fgure 1.
Sem-Markov Model of the Operato ad Mateace Process of Cty Buses 111 7 4 6 8 5 3 2 9 1 Fg. 1. Drected graph represetg the operato ad mateace process of cty buses 1 - stopover at the bus depot, 2 - obta the trasport task, 3 - damage to the bus at the bus depot, 4 - carryg out of the trasport task, 5 - damage to the bus o route, 6 - repar by techcal support ut wthout losg a rde, 7 - repar by techcal support ut wth losg a rde, 8 - emergecy ext, 9 - repar at the effcecy mplemetato subsystem posts 3. Mathematcal Model of the Operato ad Mateace Process of Cty Buses The mathematcal model of the operato ad mateace process of cty buses was bult wth the use of the sem-markov processes theory. The sem-markov X(t) process s oe, where perods of tme betwee the chages of cosecutve process states have arbtrary probablty dstrbutos ad a trasfer to the cosecutve state depeds o the curret process state. Usg the sem-markov processes mathematcal modelg of the operato ad mateace process, the followg assumptos were put forward: the radom process X(t) beg the mathematcal model of the operato ad mateace process s a homogeous process, the modeled operato ad mateace process has a fte umber of states = 1,2,,9, f techologcal object (cty bus) at momet t s state, the X(t) =, where = 1,2,,9,
112 Mrosław Szubartowsk at momet t =, the process fds s state = 1,.e. the tal dstrbuto takes up the followg form: p 1 whe 1. (1) whe 1, 1, 2,..., 9 I order to assg the values of o-codtoal durato perods for process states, values of lmt probabltes π of Markov cha hdde the process ad values of lmt probabltes p of sem-markov process, based o the drected graph show fgure 1, the followg were created matrxes P of the states chage probabltes ad matrx Θ of codtoal perods of durato of the states process X(t): where: p12 p23 p24 p 39 p41 p45 p49 P p 56 p57 p58, (2) p64 p 74 p79 p89 p91 p the codtoal probablty of trasfer from state to state j: p lm p t t lm P X t j X t ; (3)
Sem-Markov Model of the Operato ad Mateace Process of Cty Buses 113 T 12 T 23 T 24 T 39 T 41 T 45 T 49 T T 56 T 57 T 58, (4) T 64 T 74 T 79 T 89 T 91 where: the perod of durato of techologcal object -th state, uder the codto that the ext state wll be j-th state. T The homogeous sem-markov process s uequvocally defed whe tal dstrbuto (1) ad ts kerel Q(t) are gve [4, 6, 7]: where: Q t Q41 Q91 Q Q12t Q23t Q24t Q39t t Q45t Q49t Q56t Q57t Q58t Q64t Q74t Q79t Q89t t,, j 12, 9 t P X j, t X 1, (5) 1,...,, (6) meas that the state of sem-markov process ad the perod of ts durato depeds solely o the prevous state, ad does ot deped o earler states ad perods of ther durato, where 1... are arbtrary momets tme, ad: 2 Q t t p F, (7)
114 Mrosław Szubartowsk where: F j,, j 12, 9 t P t X,X s a dstrbuto fucto of radom varable 1,...,, (8) uder the codto that the ext state wll be state j. T 1 sgfyg perod of durato of state, The lmt theorem for sem-markov processes [4, 7]: If hdde Markov cha sem-markov process wth fte state S set ad cotuous type kerel cotas oe class of postve returg states such that for each state S, f 1 ad postve expected values ET, S are fte, lmt probabltes: S ET p E T P lm P t, (9) t p where probabltes p, S costtute a statoary dstrbuto of a hdde Markov cha, whch fulflls the smultaeous lear equatos: S 4. Examples of Test Results S p p p, j S, p 1. (1) j Based o source data obtaed from operato research performed authetc mucpal bus trasport system, the values of the elemets of matrx P (2) as well as matrx T (4) were estmated. The matrces P ad T are thus defed:. 297 P 1 1. 556. 9444 1. 8333. 891. 6522. 269. 869 1. 622,(11). 1667 1
Sem-Markov Model of the Operato ad Mateace Process of Cty Buses 115 T 3. 154 3. 153 5. 788. 199. 145. 92 1. 29 3. 79. 49. 297. 444. 197 12. 27,(12) 1. 626 1. 169 where the values of codtoal durato perods of process T states were show [h]. The, wth the use of the MATHEMATICA software, for the data show matrces (11) ad (12) values of o-codtoal durato perods for process states, values of statoary dstrbuto hdde Markov cha process as well as values of lmt dstrbuto of sem-markov process. Results were show tables 1 to 3. Tab. 1. Values of probabltes p of statoary dstrbuto of hdde Markov cha p 1. 2523 p 2523 p 3. 14 2. p 4. 2582 p 5. 23 p 6. 15 p 7. 6 p 8. 2 p 9. 1772 Tab. 2. Values of o-codtoal durato perods T [h] of stayg the states of process X(t) T 1 5. 788 T 2. 148 T 3. 197 T 4 8. 714 T 5. 148 T 6. 92 T 7 1. 346 T 8 1. 169 T 9 3. 153 Tab. 3. Values of probabltes P of lmt dstrbuto of the sem-markov process P 1. 3377 P 86 P 3. 6 2. P 4. 524 P 5. 8 P 6. 3 P 7. 19 P 5 P 9. 1292 8.
116 Mrosław Szubartowsk 5. Summary Based o the aalyss of the values of matrx P (2) ad defed values of lmt probabltes of the hdde Markov cha t may be determed, that almost 9% of cty buses mplemetg the task udergo damage e route. At the same tme t may be stated that most of them are dagosed ad repared (91.31%) o the rout by the techcal servce uts. The remag cty buses, damaged durg the mplemetato of the trasport task, are moved to the bus depot (emergecy ext o ther ow or towed). Out of the repars carred out by the techcal servce uts, more tha 71% are so called mor repars, maly such as adjustmet ad replacemet of small parts, performed wthout loss of a rde. The remag repars (28.57%) are doe over a loger perod of tme tha breaks betwee rdes makg t ecessary to substtute cty buses damaged durg the performace of the trasport task by reserve cty buses (about 3% of the cty buses realzg the task). Almost 5.5% cty buses awatg the start of trasport task udergo damage at the bus depot parkg space before the performace of the trasport task. Based o the o-codtoal value of tme perods as well as lmt probabltes p defed for the sem-markov process oe may coclude that the states of the process whch a statstcal cty bus stays the logest are states: 4 - carryg out of the trasport task (more tha 52% of the operato ad mateace tme), 1 - stopover at the bus depot (about 33.8% of the operato ad mateace tme), 9 - repar at the effcecy mplemetato subsystem posts (almost 13% of the operato ad mateace tme). The overall percetage of tme spet o reparg cty buses by techcal emergecy uts the total tme of operato ad mateace amouts to oly a lttle more tha.3%. However, t s sgfcat because of the possblty to realze the task of the trasport system as well as of the sgfcat costs suffered whe realzg these processes. Refereces 1. Cox D.R., Mller H.D.: The theory of stochastc processes. Chapma ad Hall, Lodo, 1965. 2. Dyk E.B., Juskevc A.A.: Cotrolled Markov processes. Sprger Verlag, Berl, 1979. 3. Flemg W.H., Soer H.M.: Cotrolled Markov processes ad vscosty solutos. Sprger Verlag, New York, 1993. 4. Grabsk F.: Sem-markowske modele ezawodośc eksploatacj. Polska Akadema Nauk, Istytut Badań Systemowych, Badaa systemowe, tom 3, Warszawa 22. 5. Iosfescu M.: Skończoe procesy Markowa ch zastosowae. PWN, Warszawa, 1988. 6. Jaźwńsk J., Grabsk F.: Nektóre problemy modelowaa systemów trasportowych. Istytut Techolog Eksploatacj, Warszawa-Radom, 23. 7. Koroluk V.S., Turb A.F.: Sem-Markov processes ad ther applcato. Naukova Dumka, Kev, 1976. 8. Leszczyńsk J.: Modelowae systemów procesów trasportowych. Wydawctwo Poltechk Warszawskej, Warszawa, 1994. 9. Woropay M., Szubartowsk M., Mgawa K.: Model ocey kształtowaa gotowośc operacyjej podsystemu wykoawczego w systeme trasportowym. Istytut Techolog Eksploatacj, Radom, 23 T P