Random vector approach to the calculation of the number of railway vehicles to hold in reserve

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1 Origina Artice Random vector approach to the cacuation of the number of raiway vehices to hod in reserve Proc IMechE Part F: J Rai and Rapid ransit 216, Vo. 23(1) ! IMechE 214 Reprints and permissions: sagepub.co.uk/journaspermissions.nav DOI: / pif.sagepub.com Jan Famfuík, Jana Míková, Micha Richtář and Radim Haama Abstract he goa of every pubic transport operator is to not ony provide high-quaity service, but aso to minimize investment and operating costs. A significant proportion of the costs are associated with the acquisition, operation and maintenance of reserve vehices. If the number of vehices hed in reserve is too high, an operator wi incur economic osses because vehices are underused. Conversey, if the number of reserve vehices is too ow, the quaity of service wi be reduced due to disruptions in the timetabe. o determine the number of vehices to hod in reserve, a coefficient of avaiabiity is commony used. he coefficient of avaiabiity is determined by two parameters: reiabiity and maintainabiity. Both parameters have the same physica dimensions, the mean time between faiures (MBF) and mean time to repair, and they are expressed in hours. Vehice wear, which resuts during the emergence of faiures, is not very dependent on the time of operation but is strongy dependent on the distance traveed. It is therefore appropriate to use mean distance between faiures (MDBF) instead of MBF, because it better describes the reiabiity of vehices. Using MDBF, however, means that the coefficient of avaiabiity is not considered, because MDBF and MR have different physica dimensions. his probem is soved by using a random vector that makes it is possibe to determine the number of vehices to hod in reserve based on the distance traveed and maintenance time. his origina approach aows the acquisition of better and more authentic data necessary for an operator s decision-making process. herefore, the number of required reserve vehices can be much better panned. Utimatey, this positivey affects the quaity of services and aso the investment and operating costs of the operator. Keywords Raiway vehice, reiabiity, maintenance, random vector, stochastic mode, operationa mode Date received: 14 August 213; accepted: 4 Apri 214 Introduction he maintenance of raiway equipment is commony based on preventive and corrective maintenance actions. 1 he preventive maintenance of raiway vehices is aready we-managed. 2 Preventive maintenance performed on the basis of operationa data can be reativey accuratey panned. However, corrective maintenance is probematic, because random faiure is a random variabe. hus, there is an extra random variabe to account for in addition to the maintenance time. he occurrence of faiures is represented by the mean time between faiures (MBF) and the maintenance time is represented by mean time to make a repair (MR). 3 he standard reiabiity mathematica modes contain random variabes that have identica units, MBF and the MR, as described in CSN IEC 5(191). 4 A unit conversion is required if the random variabes have different measurement units. Because of the actua situation in raiway stock, a unit conversion can be a compicated procedure. he reiabiity of a raiway vehice changes as a function of the distance traveed. he repair times of the components of a raiway vehice is recorded in hours. he proposed random vector is a suitabe method to sove these conversion difficuties. Mode assemby procedure he mathematica mode wi be assembed in two phases. A stochastic operation mode of the vehice is created in the first phase. he input variabes are the probabiity of the occurrence of a faiure, which Facuty of Mechanica Engineering, VSB echnica University of Ostrava, Czech Repubic Corresponding author: Jan Famfuik, Facuty of Mechanica Engineering, VSB echnica University of Ostrava, 17. Listopadu 15/2172, Ostrava Poruba, Czech Repubic. Emai: jan.famfuik@vsb.cz

2 254 Proc IMechE Part F: J Rai and Rapid ransit 23(1) depends on the distance traveed, and the probabiity of repair of the faiure, which depends on the maintenance time. A mode of the vehice feet is created in the second phase. he modes are combined in the third phase. he fina formua for determining the number of vehices to be hed in reserve is obtained using a combination of both modes. Stochastic mode of vehice operation During operation, the distance a vehice traves is L.If a faiure occurs and is repaired in the aowed time, then the vehice s operationa task is sti accompished. However, if a faiure occurs and it is not rectified in the aowed time, then the vehice is not abe to compete its operationa task and must be repaced by a reserve vehice. he foowing conditions are used in the assemby of the mode:. faiures are mutuay independent random events;. faiures can be corrected ony in the aowed time ;. the process of the occurrence of faiures in the feet has its own random probabiity distribution;. the process of the correction of the faiures in the feet has its own random probabiity distribution. his situation is described using the probabiity density of the random vector f(,t), see Figure 1. Under these assumptions there can be three different situations. 1. Situation A. he raiway vehice is repaired in the aowed interva after the faiure within the traveed distance L. his means that the operationa task wi be successfuy fufied and this situation can be expressed by a two-dimensiona stochastic function whose probabiity is given by equation (1) PðAÞ ¼ Z L Z ð1þ 2. Situation B. he raiway vehice does not need to be repaired in the aowed interva because no faiure occurs within the traveed distance L. his means that the operationa task wi be fufied with a probabiity given by equation (2) PðBÞ ¼ Z 1 Z 1 L ð2þ 3. Situation C. he raiway vehice is not repaired in the aowed interva after a faiure within the traveed distance L. his means that the operationa task wi be not be successfuy fufied and the vehice must be repaced by a reserve vehice; the probabiity of this event is given by equation (3) PðCÞ ¼ Z L Z 1 Situations A, B and C represent the condition of a compete set of a possibe disjoint occurrences; the sum of the probabiities of the possibiities of occurrences is therefore PðAÞþPðBÞþPðCÞ ¼1 We can define the random event F, which represents the competion of the raiway vehice s operationa task. Situations A and B represent the defined random event F. We can use the probabiities of situation A and B and equation (5) to describe their forma sum ð3þ ð4þ f(,t) PðFÞ ¼PðAÞþPðBÞ ð5þ L SituationA Substituting equation (5) into equation (4), it is possibe to obtain equation (6) that describes the probabiity of competing the vehice s operationa task PðFÞ ¼1 PðCÞ ð6þ Situation B Situation C t Using equation (3), that describes the probabiity of situation C, it is possibe to express the probabiity of competing the vehice s operationa task using equation (7) Figure 1. Probabiity density distribution of the stochastic mode of the operation of the vehice. PðFÞ ¼1 Z L Z 1 ð7þ

3 Famfuík et a. 255 Assuming the statistica independence of the margina probabiity density functions f() and f(t) the formuas of the probabiity of competing the vehice s operationa task is given by equation (8) and equation (9). Statistica independence of the margina probabiity density functions f() and f(t) can be confirmed using hypothesis testing PðFÞ ¼Pð, tþ ¼1 Z L f ð Þ d Pð, tþ ¼1 Fð ÞRðtÞ Pð, tþ ¼1 ½Fð Þð1 FðtÞÞŠ Z 1 f ðtþ dt ð8þ ð9þ where P(, t) is the probabiity of the vehice competing its operationa task, F() is the distribution function of the distance between faiures, F(t) is the distribution function of the maintenance time and R(t) is the compement to one of the distribution function of the maintenance time. Ony probabiities appear in equation (9), i.e. dimensioness variabes. he mode eiminates the need for the same physica dimensions of the two random variabes that is the determinative factor for the avaiabiity coefficient of the standard utiization, as described in CSN IEC 5(191). 4 Mode of the vehice feet o ensure the provision of services based on a pubished timetabe it is necessary to have a minimum number Nt of faut-free vehices. If a vehice faiure occurs, faiure can be corrected in the aowed time t in accordance with the stochastic mode. For maintenance actions it is possibe to utiize a panned break or when a vehice is waiting for another train and thus the timetabe is not disrupted. However, if maintenance is not competed in the aowed time t, it is necessary to have a reserve vehice so that once again the number of operationa vehices is Nt. he tota number of vehices in the feet Nc can be expressed by equation (1) Nc ¼ Nt þ Nz ð1þ where Nc is the tota number of vehices in the feet, Nt is the minimum number of faut-free vehices and Nz is the number of reserve vehices. he number of reserve vehices can be defined using equation (9). hat equation describes the probabiity of competing the vehice s operationa task. he probabiity that the vehice s operationa task is not competed is a suppement to the one given in equation (9). he tota number of reserve vehices in the feet can be expressed using equation (11) or after modifications by equation (12) Nz ¼ Nt ð1 Pð, tþþ ð11þ Nz ¼ Nt ½Fð Þð1 FðtÞÞŠ ð12þ For further appication of equation (12) it is necessary to determine a probabiity distribution for both random variabes and estimate their parameters. Cacuation of the number of vehices needing to be hed in reserve he probem of estimating the parameters of the probabiity distribution is quite difficut and goes beyond the scope of this artice. Different methods of estimating the probabiity of distribution parameters have been described in the iterature, e.g. the maximum ikeihood estimation method or using the properties of order statistics. 5 We possess operationa data on the occurrence of faiures and the maintenance time for train units operated in the Czech Repubic; however, the operator did not give consent for their pubication. herefore, the number of reserve vehices is modeed using a bathtub-shaped curve with a constant faiure rate and constant repair rate. 6 It represents the standard usefu ife period of a vehice s ife cyce using a bathtub-shaped curve; this curve is appicabe after the run-in period to remove eary faiures or after a significant upgrade of a vehice. he mode assumes that the steady operation period is a homogeneous Poisson process, where both reiabiity and maintainabiity have an exponentia distribution as shown in equation (13) and equation (14). he mode cosey corresponds with our resuts obtained by anaysing the operationa data. he mode can be made suitabe for the wear-out period of a vehice by incorporating a Weibu distribution into the bathtub-shaped curve. 7 Concrete exampes of the utiization of a Weibu distribution in the bathtub-shaped curve to represent for the wear out period can be found in Famfuik 8 Fð Þ¼1 expð Þ 4, 5 ð13þ FðtÞ ¼1 expð t tþ t 4, t5 ð14þ where F() is the probabiity of the occurrence of a vehice faiure, is the vehice s faiure rate, is the distance traveed by the vehice, F(t) is the probabiity of the vehice s maintenance time, t is the vehice s repair rate and t is the maintenance time. he exponentia distribution can be expressed as ¼ t ¼ 1 MDBF 1 MR ð15þ ð16þ where is the vehice s faiure rate, t is the vehice s repair rate, MDBF is the mean distance between faiures and MR is the mean time to repairs.

4 256 Proc IMechE Part F: J Rai and Rapid ransit 23(1) Figure 2. he ratio Nz/Nt as a function of distance traveed and maintenance time. Figure 3. he ratio t/mr as a function of distance traveed and the number of reserve vehices. Substituting equations (13), (14), (15) and (16) into equation (12), it is possibe to obtain equation (17), which describes the number of vehices needed to be hed in reserve N z ¼ N t 1 exp exp t MDBF MR ð17þ N z ¼ 1 exp exp t N t MDBF MR ð18þ Equation (17) aows operators to determine the number of vehices required to be hed in reserve. o be abe to make genera concusions, the ratio Nz/Nt is defined in equation (18). his ratio indicates how many reserve vehices shoud be avaiabe with respect to the minimum number of faut-free vehices. Genera considerations can be carried out using the ratio Nz/Nt, because the ratio eiminates the dependence on a particuar situation. hus, our concusions wi not be dependent on the specific number of vehices Nt, which changes according to a specific timetabe. Equation (18) can be obtained by modifying equation (17). A diagram of the function in equation (18) is depicted in Figure 2. he foowing points arise from an anaysis of Figure If the distance a vehice traves is severa times onger than the MDBF (ratio /MDBF ¼ 6), the number of reserve vehices ony depends on the aocated maintenance time. his is because the function converges to a steady state and higher numbers of reserve vehices are not required. 2. he number of reserve vehices Nz has a noninear trend and depends on the shortening of the aocated maintenance time.

5 Famfuík et a An exampe of graph utiization can be as foows: if the ratio 1/MDBF ¼.6 and the ratio t/mr ¼ 1.5 (green curve in Figure 2.) then there must be for 1 faut-free vehices (Nt ¼ 1) ony one reserve vehice (Nz ¼ 1), because the ratio Nz/Nt ¼.1. Other probems for operationa practice can be addressed using the proposed procedure. For exampe, the transport company has a finite number of vehices Nc and in order to ensure the timetabe, Nt vehices are needed and thus are defined by the ratio Nz/Nt. It is therefore appropriate to make a graph showing the time required for maintenance as a function of the distance traveed and a fixed ratio Nz/Nt. he graph in Figure 3 aows the panning of maintenance time in the situation where there are either reativey few or a arge number of reserve vehices. he foowing points arise from an anaysis of Figure he various curves correspond to the different numbers of reserve vehices. For exampe, Nz/ Nt ¼.1, which means that when 1 vehices are necessary to ensure the timetabe there is one reserve vehice. 2. If we reduce the number of reserve vehices, we must extend the maintenance time. he extension of the maintenance time is tied to the shape of the curve. 3. For a given number of reserve vehices, it is efficient to extend the maintenance time t up to a certain imit vaue. Further extension of the maintenance time is ineffective, because the function converges to a steady vaue. Concusions he contribution of this work is to create a mathematica mode of the number of vehices that need to be hed in reserve that is a function of the distance traveed and the maintenance time. he use of a stochastic approach brings a new, better ook at operationa practice. he proposed mode aows for the panning of we-managed preventive maintenance actions aong with the panning of corrective maintenance. he proposed mode, based on an exponentia probabiity distribution, is suitabe for representing the standard usefu ife period of a vehice s ife cyce as modeed by a bathtub-shaped curve. his is especiay true for the case where a vehice has competed its run-in period (or it has been upgraded) and is not in the wear-out period. A mode that utiizes a Weibu distribution can be used to represent the run-in and wear-out periods when they can be represented by a bathtub-shaped curve, thus the fu ife of a vehice can be represented using the proposed mode. Working with distance traveed is another benefit of the proposed mode. his is because the wear of important vehice components such as the chassis, mechanica brakes and pantographs, which resuts in the emergence of faiures, is not very dependent on the time of operation, but is strongy dependent on the distance traveed. he use of distance traveed aows for better and more reaistic panning of vehice operation. his origina approach aows for the acquisition of better and more authentic data necessary for an operator s decision-making process. herefore the number of required reserve vehices can be much better panned. Utimatey, this positivey affects the quaity of services and aso the investment and operating costs of the operator. Funding his research was supported by the echnoogy Agency of the Czech Repubic (Project A Active safety system for object spatia ocaization between raiway vehices and ahead of it front ends). he authors gratefuy acknowedge the financia support. References 1. Kumar S, Esping U and Kumar U. Hoistic procedure for rai maintenance in Sweden. Proc IMechE, Part F: J Rai Rapid ransit 28; 222(4): Vintr Z and Vintr M. Preventive maintenance optimization of pantographs of raiway vehice. In: ransport Means 211. Kaunas: Kaunas University of echnoogy Press, 211, pp Grencik J and Stuchy V. Organizacia udrzby a udrzbove systemy: tutoria 1: tutoria pre ceozivotne vzdeavanie Manazer udrzby Katedry dopravnej a manipuacnej techniky (K DM a Sovenskej spoocnosti udrzby (SSU), Report, 212. Ziina, Sovakia: University of Ziina. 4. Cˇ SN IEC 5(191):199. Dependabiity and quaity of service. 5. Famfuik J, Hranos V, Krzyzanek R and Gavasova Z. Spoehivost pozemni dopravy. Poradove statistiky. Virtuani vzdeavni v deprave, (213, accessed 15 June 213). 6. Mencik J. Faiure rate, bathtub curve and reiabiity of components and systems. In: he internationa conference on engineering mechanics, Praha, Czech Repubic, 26, pp Moubray J. RCM: reiabiity centred maitenance. Oxford, UK: Butteworth Heinemann, Famfuik J. Modernizace hnacích vozide a stanovení pozˇ adavku na jejich spoehivost, hande/1195/544 (21, accessed 15 June 213).