The paper describes unstable provision of electric public transport in contemporary Ukrainian cities, under using the fleet with excess depreciable life, decreased vehicle production and consequently unpredicted failures. The research suggests the stochastic design of reliability of the system provided with a reserve fleet at minimal expenses for maintenance.
Keywords: transport system, vehicles, reserve fleet, system reliability, standby systems.
Introduction. Topicality. Nowadays population in large cities increases asmore and more people are migrating to cities in search of a job. Rapid growth of urban population necessitates rigid structuring of the public transportation system, and this undoubtedly makes this research urgent.
Assignment formulation. The problem of insufficient public transportation, and in some cases of stagnation, is especially urgent for all large CIS cities. This issue has been considered in numerous recent scientific and publicist papers [5, 9, 11].
One of the scientific objectives, leading to solving the specified problem, is to improve efficiency of public transportation service. Similar problems have recently often solved by the methods of probability theory [4, 6, 8].
Theoretical and practical principles of operation and development of thetransport complex. The task is to ensure uninterrupted public transportation by an electric transport fleet. To achieve the purpose a number of vehicles considering possible failures of several units while running are to be calculated. Vehicles quantitative reserve is to be provided, so that, on the one hand, the transport system operation remains stable via shifts, and on the other hand, it is ensured with minimal expenses for its maintenance .
Suppose there is thetransport system S. The event AS is normal operation of the system S within a certain period of time DtS. We may determine reliability of the system S operation as the probability P(AS) of AS. Boundaries of the reliability numerical measurements are determined by the two-sided inequality
A risk for the system S is the probability that the event AS does not take place. Hereinafter a line means an opposite event.
As events AS and generate a full system of events, there is the following correlation between reliability and risk:
As a rule, in practice technical capabilities do not provide sufficient confidence to make reliability equal to 1 (that means to reduce a risk to zero). In this case for each specific system, boundaries of reliability are to be integrated via the inequality
or boundaries of risk, which, according to (2) and (3), are given in correlation
The numeric value is determined for each specific system S considering technical and economic capabilities, and also degree of damage and loss due to the event.
In this problem, basing on the above definitions, we calculate the system reliability increase through the connection of «standby» systems to it (transport reserve fleet ). Therewith, the overall task, allowing to use its solution for different systems, will be combined with specific numeric results.
Consider the system S containing a few subsystems. We will discuss only a case with two subsystems S1 and S2. As it’s evident from the calculations below, extension by any number of subsystems is easy.
The events AS, AS1, AS2 will represent normal operations of the whole system S as well as normal operations of its two subsystems S1 and S2. Basing on the determination of the product of two events, we will have:
Provided the events AS1 and AS2 are independent, then according to (5) the reliability of the system S is the product of reliability of systems S1 and S2.
Suppose the current situation is that the system S reliability, calculated under the formula (6), is lower than the allowable reliability limit, determined by the inequality (3). Thus there are no technical or economic opportunities to improve the reliability (6) via enhancement of the subsystems S1 and S2.
The solution of this situation may be connection of the standby systems D1 and D2 to the subsystems S1 and S2, respectively. These subsystems via automatic control stations — decision-makers R1 and R2, — will begin to function according to the subsystems S1and S2, respectively, if their operation does not meet the standard requirements [3, 7, 10].
To create the standby systems may be easier than to improve the subsystems S1 and S2, since proper actions by the decision-makers and the standby systems may be designed to operate within relatively short period of time:
Uptime of the decision-makers activities DtR and uptime of the standby systems DtD must ensure to take all necessary measures to eliminate danger, related to events and during this period.
Calculate the reliability P(ASRD) of the system SRD, which along with the original system S contains two decision-makers R1 and R2 and two standby vehicles D1 and D2. Here the event (ASRD) is failure-free operation of the system SRD. Therewith, the reliability (correctness of decisions) made by the decision-makers Р(АR1), Р(АR2) and the reliability of standby vehicles P(AD1), Р(АD2) are supposed to be independent and known. Here the events АR1 and АR2 are failure-free operation of the decision-makers during , while the events AD1 and АD2 are failure-free operation of the standby vehicles over .
The subsystem S1, implying R1 and the standby system D1 is convenient to be considered as a single subsystem, SRD1.
The reliability of this system is determined by the event , which is the subsystem SRD1 functioning with each possibility. In this case the event is split into two variants:
1. Normal operation of the subsystem S1.
2. The S1subsystemdoes not operate, but the decision-maker R1was activated resulted in the standby system D1 operation.
The aforesaid resulted in the following inequality:
Provided that all systems operate independently, the subsystem SRD1 reliability, basing on (8), is
Similarly we will have for the reliability of the subsystem SRD2 (supporting the subsystem S2, including R2 and the standby system D2) as follows:
The event ASRD (the system SRD operation), evidently, equals
Based on correlations (2), (9), (10) and (11) we will have the following for the SRD systems reliability:
Formula (12) solves the specified problem. The first summand in the right part of equation (12), according to (6), provides the reliability of the system S, if standby vehicles are unavailable. The second, third and fourth summands in (12) determine the increased reliability of the system S, with available standby systems. The risk of the event ASRD is determined by formulas (2) and (12).
According to (12), the reliability of the system SRD tends to one, whereas the risk tends to zero, when the decision-makers and the standby systems tend to one. It should be noted that achieving the high reliability value of the latter items may be reached by inequality (7), which admits short uptime period for the decision-makers and the standby systems.
Conclusion. The stochastic modeling described in the paper enables to ensure sufficient public transportation at minimum costs to maintain the reserve fleet. The model is designed to be used in the transport structure in a large city with any passenger flow.
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