The Stochastic Model of Reliability for City Public Transport Operation

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 [2].

Suppose there is thetransport system S. The event A_S is normal operation of the system S within a certain period of time Dt_S. We may determine reliability of the system S operation as the probability P(A_S) of A_S. Boundaries of the reliability numerical measurements are determined by the two-sided inequality

. (1)

A risk for the system S is the probability that the event A_S does not take place. Hereinafter a line means an opposite event.

As events A_S and generate a full system of events, there is the following correlation between reliability and risk:

(2)

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

(3)

or boundaries of risk, which, according to (2) and (3), are given in correlation

(4)

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 [1]). 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 A_S, A_S1, A_S2 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:

(5)

Provided the events A_S1 and A_S2 are independent, then according to (5) the reliability of the system S is the product of reliability of systems S₁ and S₂.

(6)

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 R_2, — will begin to function according to the subsystems S₁and S_2, 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:

(7)

Uptime of the decision-makers activities Dt_R and uptime of the standby systems Dt_D must ensure to take all necessary measures to eliminate danger, related to events and during this period.

Calculate the reliability P(A_SRD) 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 (A_SRD) is failure-free operation of the system SRD. Therewith, the reliability (correctness of decisions) made by the decision-makers Р(А_R1), Р(А_R2) and the reliability of standby vehicles P(A_D1), Р(А_D2) are supposed to be independent and known. Here the events А_R1 and А_R2 are failure-free operation of the decision-makers during , while the events A_D1 and А_D2 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 S_1.

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:

(8)

Provided that all systems operate independently, the subsystem SRD1 reliability, basing on (8), is

(9)

Similarly we will have for the reliability of the subsystem SRD2 (supporting the subsystem S₂, including R₂ and the standby system D₂) as follows:

(10)

The event A_SRD (the system SRD operation), evidently, equals

(11)

Based on correlations (2), (9), (10) and (11) we will have the following for the SRD systems reliability:

(12)

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 A_SRD 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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