Improving the Functional Reliability of an Urban Public Transport Line

Abstract

In this study we consider correlated and simultaneous interventions regarding: i—the physical infrastructure (by crossover lines between the two tracks of a tram line), ii—the characteristics of the trams (by bi-directional trams), as well as iii—tactical and operative decisions of the line manager. How these interventions are reflected in the functional reliability of the tram line service is demonstrated for both cases of the current operation and for the case of overloads, respectively, for the case of the temporary degradation of circulation caused by random disruptive events. The theoretical analysis, generalizing findings regarding the effectiveness of solutions to improve functional reliability, is supplemented with quantitative evaluations related to certain situations of disruptions. The proposed solutions aim to increase the attractiveness of urban public transport. Even if the evaluations are focused mainly on the interests of the service quality perceived by the user, the beneficial consequences for the line manager (in terms of technical and commercial efficiency) are also addressed.

1. Introduction

1.1. For a More Attractive Urban Public Transportation

Since the intensive use of cars by an increasingly large part of the population, the planning of the territory has been unable to integrate the new urban spatial mobility supported by cars. As a result, almost everywhere, large urban agglomerations are faced with the negative effects of excessive car traffic: decreasing the accessibility of places of interest (especially in central areas), loss of street functionality as public spaces for informal communication, air, and noise pollution, degradation of mobility conditions for those who do not use cars (pedestrians, cyclists, public transportation users), and others [1,2,3,4].

The responsibility of the authorities is only partially justified. Moreover, in the most frequent situations, in addition to other specific interventions and with less relevant effects at the global level, the authorities have already offered solutions for spatial movements consistent with the sustainability of life and urban activities by promoting efficient and qualitative public transport.

The question that needs to be answered is: “Why do these better transportation alternatives (which should generate significant migration to the public transport offers) have not always met the expected larger social adhesion?”.

Of course, a possible answer would be that the offer of public transportation did not prove attractive enough (even in the hypothesis of services with dedicated tracks that provide average travel speeds higher than their own car’s speed). In addition, this is the case even for those who seemed willing to give up their cars [5,6]. In short, the offered public transportation still does not meet the quality requirements.

Even if a synthetic definition of quality, as the totality of attributes that provide maximum satisfaction to the user, is suitable for both a product and a service, we must note that the differentiation between the quality of a service versus that of a product is necessary. If we examine the attributes of quality, then we find that in the case of service quality, they are more diversified and more sensitive. This involves specific and subtle actions to increase quality or to compensate for the lack of attributes specific to products [7,8].

All the attributes of the service—intangibility, inseparability, variability, perishability, lack of ownership—are at the origin of the actions through which it is also possible to increase the attractiveness of urban public transport [9]. Some attributes of the service quality are the consequence of strategic decisions regarding the compliance qualities of the resources related to the urban public transportation line (infrastructure, means of transport, materials, energy, human operators). Other attributes are the consequence of tactical decisions that are related to the operating technologies adopted according to the estimated demand. There are also attributes of service derived from operative decisions, those through which the transport operator aims to match the supply with the temporal and spatial variations of the demand or with the disruptive effects affecting the planned supply [10].

Regardless of the decision-making level (strategic, tactical, operative), the manager of a public transportation network of services faces in his actions the difficulty of solving the same problem—the harmonization between productive technical efficiency and commercial efficiency with direct implications in service quality.

Productive technical efficiency is defined as the ratio between the performance achieved (measured in vehicles. km, offered in-vehicle places. km, commercial speed, level of use of the active fleet, etc.) and the number of consumed resources (material, human, energetic, financial) that are the subject of maximization in the transport operator’s analyses. The commercial efficiency, as a ratio between the revenues achieved and the actual operation performed, provides the measure of the financial profitability of the operation of the services network, and both the transport operator and the public authority are interested in its value [11,12,13].

The difficulty of harmonization between the two kinds of efficiencies is because of the significant spatial and temporal variations of demand for which the service does not have, like industrial production, the possibilities of choosing between the manufacture of products, and storage.

Regardless of the complexity of the decision-making process and the ambiguity that the designer of the transport liner service is aware of, the public transportation operator (PTO) must opt for a specific offer and make it public. This involves a certain quality of presentation. This must be found as faithfully as possible in the achieved quality [14]. This achieved quality is the one perceived by the user.

Due to spatial and/or temporal variations of demand with large dispersions, but also due to endogenous or exogenous disruptive events that impact the planned performance of the offer, it is possible to register deviations between the quality of the service presented to the public and the one achieved or perceived by users [15,16,17]. This variability of service, which we noted among the attributes of public transport quality, is, in our opinion, essential for increasing the attractiveness of public transportation services whose different types of performance will attract the attention of potential new users. Therefore, the present paper proposes to identify actions to achieve the reduction of service variability. That is, what must be accomplished so that the quality of presentation of the urban public transport service (made public) can be found as faithfully as possible in the quality realized and perceived by the user.

Multiple aspects of the variability of public transportation can be discussed. However, we limit ourselves only to those aspects of service variability that have quantitative measurements and are devoid of subjectivism. Such as respecting the time of departure and arrival (journey duration) and comfort (ensuring the place and ambient conditions throughout the journey in a vehicle). These are the ones for which regional and interregional transport passengers can incriminate the operators in case of non-compliance with the conditions assumed when they purchased the travel ticket [18]. In the case of urban public transportation, with trams on an exclusive way (for which the duration of the journey has a good constancy), we retain as objectives for the offered services’ stability: (a) respecting the planned tram headways, and (b) loading degree of tram capacity (as a measure of the service comfort).

Of course, the study of service variability is related to a time characteristic of a working day or holiday according to which the PTO plans the operating schedule.

The variability/uncertainty of the transport service quality impinges on the attractiveness of service, especially for the discerning and non-captive user (car owner). The variability of the service, like any obligation assumed but not achieved, is troublesome. It causes stress and is kept in the user’s memory [19,20]. The repeated underlining of the service variability of a public transport network points to the need to establish quality norms for the services. Compliance with these quality standards inevitably has a stochastic feature. Therefore, the average values are used. But, even using average values, they contain the necessary elements to guide the actions aimed at reducing the unpredictability of the quality of the offered service.

Based on these standards of quality, a designed public transportation service can be rated as reliable or non-reliable. Of course, it remains to be determined how the reliability of a services network, in general, and of a public transportation line, in particular, can be defined.

Analogies with the reliability of a product or a physical traffic network are not useful [21,22,23].

1.2. Literature Review

The literature offers a wide range of references that deal with the issue of the quality of public transport services and the reliability of services as an important part of this quality. In the following, we present some articles that are representative of a category of works on the topic of reliability that are closer to the approach of the present work.

By investigating approximately one hundred works, the authors of the review article [24] identify the most important aspects that measure the reliability of transport services. Most often these are related to time: travel time, waiting time, boarding/disembarking time, etc., or the availability of seats in the vehicle, missing connections, or compliance with the schedule.

By considering the total generalized cost of the urban public transport trip, the authors of the paper [25] try to take into account the users’ perception in estimating the reliability of the service; this being the most important merit of the paper. The model considers users separated into three categories depending on their generalized cost and their level of acceptability to make the trip under the conditions of a capacity reduction of the transport network links. The characteristics of the category that does not accept the trip when the transport capacity is reduced by a certain value represent the measure of reliability.

The reliability of the infrastructure network, just one of the components of the transport service, is addressed in the paper [26], which proposes an evaluation model by decomposing the network into several series and parallel configurations of the component elements using graph theory. This category of papers is widely spread, with graph theory addressing both the reliability and the vulnerability but also the resilience of transport networks, for example, paper [27], which, however, does not deal with the problem of service dysfunction in the network.

The problem of introducing the reliability of the service as an element in its design is brought into discussion by the paper [28]. The author surveys the international level and another at the level of the network of Dutch operators and highlights the inconsistency in defining the reliability of the public transport service.

Inspirational works [29,30] that rigorously describe the utility and the mode of operation of bi-directional trams. Assessments are made of the advantages (including those of land saving) of introducing this type of flexible-use tram. However, their advantage in increasing the reliability of the service is not addressed.

An important class of papers deals with the solution of using the short turning pattern. In the paper [31], the authors demonstrated that the short-turning solution leads to a more efficient management of resources in the case of some over congested metro lines on certain sections. The degree of loading of metro services is the main tool in determining the congested section. The authors of the paper [32] study different short-turning solutions for transit services (differentiated by length, frequency, and position concerning the full service) to reduce the waiting time and identify an acceptable level of vehicle occupancy. An interesting model for predicting the length of dysfunctionality [33] for the conventional railway (with optimistic and pessimistic assumptions) is proposed together with a short-turning solution for train rescheduling.

The present paper novelty consists in defining the functional reliability concept by considering all the three main components of a transport system, i.e., infrastructure, vehicle, and operational management. The framework for the evaluation of functional reliability is set considering the user’s point of view. We analyze the impact of specific technical interventions (crossing connections between the two lines of a generic tram line with a dedicated way, using bi-directional trams and appropriate operational management) on the functional reliability of tram services in two cases of disruptions. We found that the proposed investments and management actions improve the reliability of the tram service line. A rather constant headway of the trams according to the flow of users, together with maintaining a certain level of loading of the trams (from the point of view of the user) and a better utilization of the tram fleet (from the point of view of the urban transport operator), are achievable by comparison with the “no intervention” case.

In the remainder of the paper, the method of analysis of functional reliability and the way of its improvement is presented in Section 2. Two types of disruptions are analyzed in Section 3, one for anticipated occasional overloads (including a numerical experiment) and another for the random disturbances from scheduled circulation, both for the two specific technical interventions. Section 4 concludes with the main results and practical implications and requirements.

2. Materials and Methods

2.1. Functional Reliability of Urban Public Transport Network Services

The reliability of public, commercial, technological, or touristic transport, but also that of individual car trips, to a certain extent, stands out as one of the primary factors that determine the levels reached in the quality of transport or traffic.

Related to the quality of public transport, the reliability of the network cannot be limited to the reliability of the physical network/support. It means that the definition of the reliability of a product, or the reliability of the traffic/transport networks, does not sufficiently reflect the quality of the service provided by the network.

A public transport network with extremely good reliability of the underlying physical network (expressed by a probability tending to the maximum value) is not equivalent to a service network with similar reliability. The reliability of the service implies another way of definition, related to the performance of the operation as users perceive it. The reliability of the service is conditioned not only by the reliability of the physical network but is defined by the probability of maintaining connectivity between two network points. It is also conditioned by the reliability of means of transport, control, and command equipment, but also by the actions of human operators involved in the development of traffic. The technologies adopted to ensure the stability and continuity of movement contained in the presentation quality of the offered service (in terms of reliability) summarize the reliability of the above components. It follows that in the case of the service network, another definition of reliability must be used, more comprehensive, and as consistent as possible with how it is reflected in the quality of services provided by the traffic network, i.e., functional reliability [14], as a benchmark of the attractiveness of the offered service.

As a result, it is necessary to return to a broader understanding of the notion of reliability. This needs to be understood as the ability of the system to ensure the success of the entrusted mission.

We note certain difficulties in defining the functional reliability of the service network by reference to “mission success”. Specifying the “success of the mission” requires a collaboration between the public authority, the infrastructure manager, the user, the designer, and, possibly, a specialized third institution (e.g., a quality certification entity). Divergent points of view are required to be harmonized, referring to the preliminary specifications of the public authority, the operator, the user, and the information on the services provided by similar competing networks (if any).

As a result, two ways of defining reliability for transportation service networks are possible.

The first one reduces the mission of the network service to the keeping/preserving of certain parameters or functional characteristics (e.g., compliance with traffic schedule, comfort, duration, technical efficiency, commercial efficiency, etc.) within certain specified boundaries. Under these conditions, functional reliability represents the probability that the service network will preserve its functional indicators/indices/metrics during a given time interval, within certain boundaries.

$$P\left(S\right)=P\left(X_1\le X\le X_2\right)$$

(1)

where $P\left(S\right)$ is the probability of mission success S, which means maintaining the functional metrics X inside the range of boundaries; X—a random value representing the operational indicators/indices/metrics of the system; $X_1,X_2$—accepted boundaries of random indicators X.

The second way is related to the effectiveness of the system.

The occurrence of “defects” in the system has the main effect of worsening the effectiveness indicator, whose value could fall below a certain critical value (which is set according to the purpose pursued).

Under these conditions, functional reliability is defined as the ability of the service network to keep its effectiveness above the critical level, a property to which an appropriate probabilistic indicator can be associated.

For example, for the functional reliability of the service of an urban public transportation network, a metric/indicator α can be considered and probabilistically defined by the relation

$$\alpha =P(E\ge E_C)$$

(2)

where $E_C$ is the critical value of the effectiveness indicator, $E$—current value of the effectiveness indicator, which is influenced by the defects during the operation of the services’ network.

For example, if this objective is that of sustainable mobility, i.e., that of attracting as many users as possible to public transportation, the indicator of effectiveness could be defined by the average loading degree (calculated as the total number of passengers × km divided by the total offered capacity of transport × km). However, the total number of actual circulated passengers × km is difficult to register without specific equipment on trams, or in tram stops.

That is why using one or another functional reliability function needs a careful decision on the adopted metric. Anyway, the choice of the most suitable functional reliability indicator (X or E) must consider simplicity, stability in a statistical sense, and its importance in defining the system’s functionality.

In any of the two ways of defining the functional reliability of the transportation network services, it is obvious that there is a direct link between the functional reliability and the quality of mobility for individual journeys, respectively, for public transportation of different types.

Regarding the calculation of the reliability indicators for the physical network components (infrastructure, means of transport, equipment), there are no methodological difficulties [26,27]. On the contrary, regarding the functional reliability of the services’ network (which includes the planned operating technology/method for service adaptation to the size of the estimated demand as well as to the multitude of endogenous and exogenous disruptive phenomena producing deviations from the planned offer), the difficulties are major.

Defining one of the mentioned indicators (X or E) of the functional reliability of the transportation services’ network, the evaluation must be based on the types of technological disturbances and the correct estimation of their consequences. Technological irregularities in the operation of the services’ network mean complex events that cause deviations (discontinuities) in the programmed sequence of operations that ensure the continuity and stability of the entire process. In the case of the occurrence of these technological irregularities, in addition to the causes attributed to the technical elements (e.g., breakdowns, insufficient capacities for transport demand at rush hours, lack of correlations between the capacities of the subsystems, etc.), exogenous causes (coming from the natural environment or riparian man-made systems, etc.) must also be taken into account, as well as consequences of some non-compliant actions of the operators involved (due to imprecision, insufficiency or lack of information, information overload, insufficient time for the elaboration of management decisions, non-compliance with the regulations related to a certain department, etc.).

We consider functional reliability as a defining characteristic in user assessment/perception of the performance of a public transportation service network. We appreciate that there is no need to assign a norm/standard value of functional reliability, as generally valid for any public transport network. This is because functional reliability varies by project, location, user, reason for travel, and time.

In establishing a level of functional reliability for a certain urban public transport service, the divergence between both two points of view of the operator and the user must be considered. This harmonization is difficult. Moreover, it is noted that the repeated congestion in time and space, for which the user has sufficient prior information, should not be interpreted as non-reliability of the physical network or services.

2.2. Ways to Improve Functional Reliability

Actions aimed at improving the functional reliability of urban public transportation are intended to increase the attractiveness of public transport and usually depend on tactical and operative decisions. This attractivity increase is mainly based on reducing service variability. That is obtained by operation according to the planned, publicly known schedule, as faithfully as possible. These actions correspond to a systemic treatment of the performance of physical resources that must be mobilized to satisfy the estimated travel demands addressed to the services’ network. Also at the systemic level, the improvement of the reliability of urban public transportation aims at the actions of continuous monitoring of the actual circulation and, when needed, the interventions required in case of disruptive events on the planned circulation.

Functional reliability also reflects the results of dynamic actions of the competent authority aimed at increasing the performance of the system components [28]. Some of these actions on the components whose positive impacts are found in the functional reliability of the urban public transport network are as follows:

• Improving the physical performance of existing infrastructures or network developments to obtain additional capacities or improve the quality of existing ones. However, capacity increases are generally expensive and time-consuming, and their political acceptance is often uncertain and difficult.
• Better incident management by infrastructure managers and better organization of corrective and preventive maintenance operations through proactive monitoring of the status of network components accompanied by adequate communication with users.
• Differentiated charging of infrastructure use for different areas of the network to increase the overall reliability of the network (difficult to achieve) or for categories of users and/or intervals of the day, week, or year (practiced with good results in reducing the variability of the demand on the network components)
• Informing users in advance and in real-time to mitigate the negative effects of the lack of reliability.

All these actions and others like them (which, however, exceed the objectives of the study) through the positive effects on the system’s components improve the performance of the physical network. As such, premises are created for the service network, which also has a more advanced physical network as its support, to become more attractive.

This study aims to find the solutions for the functional reliability improvement of an urban tram line with an exclusive way. As already mentioned, we aim to increase the probability of respecting the planned schedule and respecting the planned headways between vehicles. The motivation for the choice resides in the fact that the degree of comfort of the travel is also reflected in the vehicle headways (in terms of loading degree of tram capacity). Moreover, the headways are important for the user in his/her daily activity planning.

2.3. Framework for Assessing the Functional Reliability

The choice of the urban public transport line was inspired by a tram line with an exclusive way in the capital of Romania.

It is the most efficient line of the tram network in the city of Bucharest. In terms of track construction and transport capacity, it has been widely publicized since its commissioning (more than two decades ago). It is served exclusively by modern, high-capacity trams. The passenger flows taken over are the highest recorded on the city tram network. Yet, during the exploitation period, there were episodes where the loss of functionality produced large dissatisfaction among the users. These, widely publicized, did not go unnoticed. However, solutions for improving the actions in cases of similar disruptive events have not been put in place.

To provide general conclusions from this study, we simplify the presentation by omitting some particularities, and we consider that tram line as a hypothetical urban public tram line.

Let us consider that the planned periodic schedule is for a certain operating period of a working day, in which the maximum recorded flows on the studied line are from B end-stop to A end-stop of the tram line (Figure 1).

The number N of trams with capacity q, necessary to carry the maximum number of passengers, $V_{max},$ during the time $\Theta ,$ ($\Theta \ge T$) is [34].

$$N={⌈{\displaystyle \frac{V_{max}}{\Theta }}.{\displaystyle \frac{T}{q\gamma }}⌉}^z,$$

(3)

where T is a full cycle of one tram operation of the line A–B–A), $\gamma$ sub-unitarian loading degree of tram capacity, $q$. $⌈z⌉$—upper integer value of z.

The first term in Equation (3) is the number of carried passengers per hour of the requested duration, $\Theta$, and the second one is the inverse ratio stating the carried passengers in a total full cycle, T.

This means that the average tram’s headway, ${\overline{I}}_{\gamma }={\displaystyle \frac{T}{N}}$ is

$${\overline{I}}_{\gamma }={\displaystyle \frac{\Theta q\gamma }{V_{max}}}$$

(4)

A couple of notices are needed concerning the results of applying Equation (4), as follows:

(a)

If ${\overline{I}}_{\gamma }$ is smaller than the time between any two successive stops of the tram line, with tram capacity $q$, then a tram with a larger capacity is needed, or a larger loading degree of using capacity is accepted (that means the planned comfort is worsening),

(b)

If ${\overline{I}}_{\gamma }$ does not comply with the below condition:

$${\overline{I}}_{\gamma }>max\left({\displaystyle \frac{{\tau }_1}{n_A}},{\displaystyle \frac{{\tau }_2}{n_{\mathrm{B}}}}\right)$$

(5)

(where ${\tau }_1$, ${\tau }_2$ are the technological time of trams in the two end-stops of the tram line, A and respective B, and $n_{\mathrm{A}}$, respective $n_{\mathrm{B}}$, are places for trams’ waiting), this means that additional waiting time occurs that increases the total time of the round cycle, T (with direct impacts on the size of the number N of trams fleet, needed to take over the flow $V_{max}$). The average rate of arrivals (which is the average trams’ headway) at the end of the line ${\overline{I}}_{\gamma }$ should be higher than the time to release a waiting space at the end-stop of the line, ${\displaystyle \frac{{\tau }_1}{n_A}}$ or ${\displaystyle \frac{{\tau }_2}{n_{\mathrm{B}}}}$.

If specifications (a) and (b) were not considered in the schedule planning, then the actions to comply with them can be interpreted as improving actions of the functional reliability of circulation on the tram line.

As mentioned, the functional reliability of the service on the A-B line is defined by reference to the average circulation interval $\overline{I}$, which also includes the comfort condition, according to the relation, that is:

$${\overline{I}}_{\gamma }\le \overline{I}\le {\overline{I}}_{{\gamma }_c},$$

(6)

which means by the reliability function

$$P\left(S\right)=P({\overline{I}}_{\gamma }\le \overline{I}\le {\overline{I}}_{{\gamma }_c}),$$

(7)

where ${\overline{I}}_{{\gamma }_c}$ is the maximum threshold value of the headway corresponding to a loading degree ${\gamma }_c$ that does not severely impact the comfort ($\gamma <{\gamma }_c<1).$

The trams’ headway in current operation, I, is of the nature of a random variable. If we admit that I correspond to a normal distribution, having an average value $\overline{I}$ and the mean square deviation, ${\sigma }_I$, then, the condition that the planned interval ${\overline{I}}_{\gamma }$ does not exceed the maximum value ${\overline{I}}_{{\gamma }_c}$ (Figure 2) is:

$${\overline{I}}_{\gamma }+3{\sigma }_I\le {\overline{I}}_{{\gamma }_c}.$$

(8)

This means that the positive variations of the headways between the trams (the negative ones are annihilated as soon as they appear by respecting the planned time of departures from each of the line’s stops) must be characterized by the values of the standard deviation, ${\sigma }_I$:

$${\sigma }_I\le {\displaystyle \frac{1}{3}}{\displaystyle \frac{\Theta q}{V_{max}}}\left({\gamma }_c-\gamma \right),$$

(9)

The standard deviation, in the case of a normal distribution, ${\sigma }_I$, and its interdependency to the loading degree in Equation (9) are obtained from combining the Equations (8) and (4). However, the statistical investigations on the actual distribution of the headway between trams need additional research.

All the above operational considerations are the basic frame of functional reliability assessment.

3. Results and Discussion

3.1. Anticipated Occasional Overloads

3.1.1. Solution Modeling

When the size of passenger flows exceeds the value of the transport capacity $\mathcal{C}$ (Figure 1) provided by the planned supply, it means that the functional reliability of the service with the number of trams N (resulting from Equation (3)) can no longer be ensured.

The marketing service of the public transport operator anticipates the lines in the network and also the periods with such demand that exceed the capacity considered in designing the public planned periodic schedule. These tasks generated, for example, by sports, cultural, and commercial events of large dimensions, are known in advance, anticipated occasionally, and are the basis of the redesign of the circulation schedule. During this paper, these demands are called overloads. The redesigned schedule must be made known/public by appropriate means of information.

Figure 3 shows the estimated passenger flows to end-stop A of the line where, for example, a sports event with many spectators is expected.

Passenger flows during this period $\Omega$ are constantly increasing from B to A. Major flows occur in stops $m_1$ și $m_2$ along the route. These could be stops where passengers from two subway lines access the tram line to reach A.

The line operator must configure its offer to take over the estimated flows, restricted to the conditions of the highest commercial efficiency. The significant difference between the flows of the B–$m_1$ section and those of the $m_1$–A section guides the operator to a solution that ensures both the continuity of the movement from B to A and the best possible commercial efficiency. This last goal of best commercial efficiency is not achievable through a uniform offer from B to A, that is, by increasing the number of circulated trams on the entire B–A–B line. The solution remains to introduce an additional transport offer in the section $m_1$–A. This solution, added to the supply on the B–A line, must be sized to take the entire flow V from B to A.

Implementation of the additional operational line, on the direction $m_1$ to A for a short turn (which we call “line II”), apart from the operational line from B to A (called “line I”), is conditioned by two simultaneous technical interventions [29,30]:

(a)

introduction of the bi-directional trams into circulation (with driving cabs at both ends of the tram and access doors for passengers on each side), and

(b)

installing a crossover line between the two tracks in the zone of the tram stop $m_1$ (red line, in Figure 4).

In Figure 4, we present only a single type of crossover line between the two tracks, for simplification reasons, but other configurations are possible [35]. In any case, the generality of the analysis is not jeopardized. In distributing the tasks between the two lines that will operate on the section $m_1$–A, the PTO is concerned with the continuity of the transport from B to A (provided by line I) and with minimizing the operational effort required to take overflow V, too.

The adopted solution aims to distribute the tasks between the two lines according to their potential productivity. This means that (assuming the same type of tram on both lines) the following relationship exists:

$${\displaystyle \frac{V_I}{V_{II}}}={\displaystyle \frac{t}{T}}$$

(10)

where, $V_I$, $V_{II}$ represent the flows to be taken over, respectively, by line I and line II, T, t—the total operational cycle of trams for line I, respectively, for line II (Figure 5).

Because $V_I+V_{II}=V$ is the size of the flow to be picked up, the tasks of each line are:

$$V_I=V{\displaystyle \frac{t}{T+t}},\mathrm{respective}{V}_{II}=V{\displaystyle \frac{T}{T+t}},$$

(11)

that means the number of trams operating the line I, is:

$$N_I=⌈{\displaystyle \frac{V_{I}T}{\Omega q\gamma }}⌉,$$

(12)

respectively, the line II, is:

$$N_{II}=⌈{\displaystyle \frac{V_{II}t}{\Omega q\gamma }}⌉,$$

(13)

Regarding the $N_I$ value, a clarification must be made. For the continuity of the movement from B to A, it must be that $N_I\ge N_I^{\prime }$, $N_I^{\prime }$ representing the number of trams required to take over the $V_I^{\prime }$ flow from $m_1$ (Figure 3).

The hypothesis $N_I^{\prime }\ge N_I$, means that $N_I^{\prime }$ trams running the line I take over a flow $V_I^{\prime }$. Line II would have the task of taking over a flow $V_{II}^{\prime }=V-V_I^{\prime }$ ($V_{II}^{\prime }>V_{II}$).

The average trams’ headway for the line I is ${\overline{I}}_{\gamma }={\displaystyle \frac{T}{N_I}}$ and for the line II is ${\overline{i}}_{\gamma }={\displaystyle \frac{t}{N_{II}}}$ (this headway is for specified conditions, when the number of trams $N_I^{\prime }$, respectively $N_{II}^{\prime }=N-N_I^{\prime }$, are introduced in circulation), which means that the headway between trams on section $m_1$ to A is

$${\overline{\mathfrak{I}}}_{\gamma }={\displaystyle \frac{T}{N_I+\frac{N_{II}T}{t}}}$$

(14)

We define the functional reliability of the service differently:

• for the section B to $m_1$ for the line I,

  $$P\left(S\right)=P({\overline{I}}_{\gamma }\le \overline{I}\le {\overline{I}}_{{\gamma }_c}),$$

  (15)
• for the section $m_1$to A, for the lines I and II,

  $$P\left(S\right)=P(I_u\le \overline{\mathfrak{I}}\le {\overline{\mathfrak{I}}}_{{\gamma }_c}),$$

  (16)

  where $I_u$ is the minimum headway between the successive occupation of a stop by the following trams.

At the end of the event, trams of lines I and II must take major flows in the reverse direction, from A to B in a certain time for outbound flow, ${\Omega }_d<\Omega$.

Since little time before the start of the event, the passenger flow decreases significantly, which means that the $N_{II}$, the number of trams on line II is withdrawn from circulation, as they appear in A. At the same time, the reduction of general passenger flow after the evening peak (a little time before the end of the mass event) requires the adaptation of the offer of line I. That is, the reduction by $\mathsf{\Delta }{N}_I$ of the number of trams in circulation on line I. The $N_{II}+\mathsf{\Delta }{N}_I$ trams withdrawn from circulation are either directed to the proximity depot or stationed on the loop, in A. In both situations, the trams are to enter circulation at the appropriate time.

At the beginning of the period of outbound dispatch, ${\Omega }_d$, the number of trams $N_{II}+\mathsf{\Delta }{N}_I$ are available to receive the flow of spectators. They will be dispatched at the minimum interval $\tau$. To these numbers, there are added the trams of line I (that are arriving at the average interval $\overline{I_I^{\prime }}={\displaystyle \frac{T^{\prime }}{(N_I-\mathsf{\Delta }{N}_I)}},$ with $T^{\prime }<T$).

It follows that, regardless of the time ${\Omega }_d$ in which the operator planned to ensure the offer from A to B, there is an additional time, ${\mathsf{\Delta }\Omega }_d$:

$${\mathsf{\Delta }\Omega }_d=\left(N_{II}+\mathsf{\Delta }{N}_I\right)\tau +{\mathsf{\Delta }N}_I^{\prime }\tau ,$$

(17)

in which $N_{II}$ trams will be dispatched on line II, and $\mathsf{\Delta }{N}_I+\mathsf{\Delta }{N}_I^{\prime }$ trams, which will be dispatched on line I, at the minimum headway, $\tau$.

The $\mathsf{\Delta }{N}_I^{\prime }$ is the additional number of trams on the line I that arrive in A during the time $\left(N_{II}+\mathsf{\Delta }{N}_I\right)\tau$, that is

$$\mathsf{\Delta }{N}^{\prime }={\displaystyle \frac{\left(N_{II}+\mathsf{\Delta }{N}_I\right)\tau }{\overline{I_I^{\prime }}}},$$

(18)

Later, during the interval ${(\Omega }_d-\mathsf{\Delta }{\Omega }_d),$ the frequency of tram dispatches from A depends on the relations between ${\Omega }_d$ and the values of the total cycle of trams on line II, $t^{\prime }$(which is shorter, $t^{\prime }<t$), respectively, of line I, $T^{\prime }$ ($T^{\prime }<T$). The diminished values of $t^{\prime }$ and $T^{\prime }$ are those corresponding to the relax period, which follows after the evening peak period.

Boarding of spectators in A terminus stop can be completed from a single platform, alternatively, to B, for the line I, respectively to $m_1$, for the line II (Figure 6), or from two different platforms (Figure 7).

In both cases, the passenger queues for lines I and II, respectively, must be distinct and strictly follow the First-In-First-Out rule.

In both cases, the intervals between successive dispatching are small (as confirmed in the numerical experiment, presented below). As such, defining functional reliability by reporting at such intervals the supply of tramways would be irrelevant to the users. Therefore, functional reliability must be defined concerning the major interest of the potential user of either of the two lines. Namely, the user to be able to travel back from the recently concluded mass event, also by tram. For this purpose, the information about the waiting time from the moment of his/her attachment in the queue of each of the two lines until access is allowed to the respective boarding platform is essential.

If the boarding process along the entire chain is rigorously directed so that the order of arrival in the queue is preserved without exception, then the information about the waiting time can be calculated and displayed at the entrance to each of the two formed queues. The schedule of the tram dispatches and the counting of the user number in each queue are the necessary elements for the periodic information update to users (at the moment of his/her decision to attach or not to the waiting queue).

If the number of users is V and the interval between dispatches, $\tau$ and ${\tau }_u^{\prime }$, respectively, then the estimated waiting time shown for the newcomer in each queue is

$${\omega }_{ea}=⌈{\displaystyle \frac{V}{q_t\gamma }}⌉\tau +\tau ,$$

(19)

for alternative shipments from the same platform, respectively

$${\omega }_{ea}=⌈{\displaystyle \frac{V}{q_t\gamma }}⌉{\tau }_u^{\prime },$$

(20)

for simultaneous boarding at two platforms dedicated to line I, respectively, line II (with the notice that there are different values for line I, respectively, line II corresponding to the values of V and $V_I$ or $V_{II}$ respectively).

Thus, the functional reliability of the withdrawal offer is

$$P\left(S\right)=P({\omega }_{ec}\le {\omega }_e),$$

(21)

where ${\omega }_{ec}$ is the actual waiting time for passengers, ${\omega }_e$—expected waiting time (${\omega }_{ea}$ or ${\omega }_{es}$, related to each line).

3.1.2. Numerical Experiment

Physical and operational data:

• total length of the tram route, L_A–B = 9 km.
• length of the section with mass flow, $l_{m_1-\mathrm{A}}$ = 5.4 km.
• total cycle for the round trip of tram on route A–B, T = 90 min.
• total cycle for the round trip of tram on route $m_1-\mathrm{A}$, t = 60 min.
• total estimated demand in A, V = 12,000 passengers.
• passenger flow in the interchange station $m_1$, $V^{\prime }=$ 3000 passengers.
• time of tram line offer for inbound flows in A, $\Omega$ = 150 min.
• tram capacity, q = 350 persons/tram (including seating).
• average number of passengers per tram, $q.\gamma =$ 300 passengers/tram.

Results:

• Case a. Bi-directional trams and crossover line in $m_1$.

From Equation (11), $V_I=$ 4800 passengers > $V^{\prime }=$ 3000 passengers, respectively $V_{II}$ = 7200 passengers.

From Equations (12) and (13) by incrementing to the higher integer value, $N_I=$ 10 trams and $N_{II}$ = 10 trams, with the average headways of ${\overline{I}}_{\gamma }$ = 9 min for the line I, and ${\overline{\iota }}_{\gamma }=$ 6 min, for the line II. Thus, by overlapping the circulation of the trams on the section $m_1-\mathrm{A}$, the average headway, results ${\overline{\Im }}_{\gamma }$ = 3.6 $\cong$ 4 min, computed by Equation (14).

• Case b. No action is taken.

Without the contribution of line II (bi-directional trams are not used, nor is the crossover line in $m_1$ zone is implemented), it means that the entire load, 12,000 passengers to be transported in 150 min, would fall to line B–A–B (line I), which would require the movement of $N_l$ = 24 trams (according to the Equation (12) with $V_I=V$). That is, 4 additional properly staffed trams and an additional mileage resulting from the Equation (22), i.e.,

$${\displaystyle \frac{\Omega }{T}}\left(N_l-N_I\right)2L_{A-B}={\displaystyle \frac{\Omega }{t}}{N}_{II}2l_{m_1-B},$$

(22)

of about 300 − 270 = 30 km.

Thus, from the point of view of productive technical efficiency, the solution proposed to satisfy the occasional expected overload is translated by putting into circulation a smaller number of trams (20 instead of 24) and by reducing the total distance of the trams needed in circulation (270 km instead of 300 km).

In terms of commercial efficiency, the proposed solution is useful, too. Thus, on the section ${\mathrm{B}-m}_1$, a ratio between the offered capacity and the maximum required one is registered as ${\displaystyle \raisebox{1ex}{$V_I$}\!\left/ \!\raisebox{-1ex}{$V^{\prime }$}\right.}=$ 1.6 (compared to ${\displaystyle \raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$V^{\prime }$}\right.}=$ 4, in case line II would not operate).

Due to the trams’ headway (${\displaystyle \raisebox{1ex}{$\mathrm{T}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{N}$}\right.}$), is less than 4 min, the exact timetable is unnecessary [36].

Consequently, the functional reliability of circulation is defined by the probability of ensuring compliance with the headway limits (Equation (16)). Practically, the functional reliability of the dispatches corresponds to the quasi-continuous presentation of the trams at the line stops.

• Case c. Case of outgoing flows from A

For the outgoing flow of spectators, when the general transport demand has decreased, the total cycle T and t for the trams of the two lines will have lower values than those used in the calculation of the passenger inflow. By reducing the stopping times at the line stops and at the ends of the lines, we admit that they become $T^{\prime }$ = 75 min and $t^{\prime }$ = 45 min. These values correspond to a commercial speed of 14.4 km/h on the line I of length 18 km, and on the line II of 10.8 km, respectively.

We assume that the same active fleet is used to take over the outflow loads as at the inflow ($N_I$ = 10 trams and $N_{II}$ = 10 trams).

During the event, line II no longer operates. The $N_{II}$ = 10 trams of the line are waiting to re-enter in circulation with the start of the spectator evacuation task. As well as the $\mathsf{\Delta }{N}_I$ = 4 trams are withdrawn from circulation gradually, at the end of the $\Omega$ period. At the end-stop A, the number of $N_{II}+\mathsf{\Delta }{N}_I$ = 14 trams are waiting to pick up passengers.

Let us also admit that from the moment the passengers show up at the boarding stop in an estimated period ${\Omega }_d$ = 60 min, the use of the line is of interest to the users. A longer duration would direct them either to other public transport offers or to walking (for their destinations, with times less or like ${\Omega }_d$).

We assume that the 14 trams can be dispatched successively at an average interval of ${\tau }_u$ = 3 min, higher than the boarding time of $q.\gamma$ = 300 passengers/tram (the trams are equipped with double doors and a lowered floor). It follows that this rate of dispatching of the 14 trams can be maintained for a duration of 14 × 3 = 42 min. According to Equation (18), the additional number of trams, $∆N^{\prime },$ arrives in B from line I at the interval $\overline{I_I^{\prime }}$ = 75/6 min. However, the additional trams $∆N^{\prime }=$3 trams can be dispatched in the same ${\tau }_u=$3 min. It means that, for a duration of ${∆\Omega }_d=$ 51 min (according to Equation (17)) the number of 17 trams is dispatched at the interval of 3 min. Since ${∆\Omega }_d>t^{\prime }$, it means that in end-stop B there are at least 3 trams to continue the 3-min pace of dispatches, in the remaining period ${\Omega }_d-\mathsf{\Delta }{\Omega }_d$ = 9 min.

In conclusion, during the proposed outgoing flow period, ${\Omega }_d$, the offer of the operator of the two lines makes it possible to take over a maximum of 20 × 300 = 6000 passengers (when the load level is $\gamma <{\gamma }_c$). The performance of the service is assured by correct management of dispatches; that is, by ensuring the planned load degree of the transport capacity of each tram ($q.\gamma$).

It should be noted that even after the expiration of the ${\Omega }_d$ period, using the same active fleet, the same pace of dispatches can be maintained (especially since $\overline{I_I^{\prime }}$ is gradually reduced to 7.5 min, by putting into circulation the number $∆N^{\prime }$ of trams).

Since the trams of lines I and II are identical, to avoid the frustrations of the differential pace of advancement of passengers waiting to board the two lines (corresponding to lines I and II), it is possible to maintain the same alternation of dispatches (every 6 min) to $m_1$, respectively, B. That is the same number of dispatches to the two destinations. The total volume of passengers is the same.

The invoked equity of the pace of advancement of the two boarding lines can also be achieved by allocating distinct platforms to each line (Figure 7). Boarding the tram of line I is performed simultaneously with boarding the tram of line II.

The same number of trams are dispatched at the interval ${\tau }_u^{\prime }$ = 4 min, which means that a headway on the common route is of 2 min and results in 7 trams being dispatched to B and the same number of 7 trams being dispatched to $m_1$, for 28 min (according to note a), of the Equation (4).

During this time, 2 other trams arrive from A to B, which means that in total 16 trams are dispatched in 32 min from B. Then (after these 16 trams dispatching) the following trams from A arrive in B at minutes 37.5 and 50, and from $m_1$ at minutes 45, 49, 53, 57 (before 60, which it assumed to be the end of the ${\Omega }_d$ period). Thus, there are additional 6 trams, for which the schedule and destination could be set according to the length of the queues in B platforms.

These 22 trams can carry the maximum possible volume of dispatching of 6600 passengers (with the degree of loading $\gamma <{\gamma }_c$).

Figure 8 illustrates the dynamics of the outgoing flow’s requests from the arrival of the first applicants in the boarding area of A stop, until the moment ${\Omega }_{sos}$.

The average passenger arriving intensity, $\lambda =V_d/{\Omega }_{sos}$ being larger than the takeover offer, $\mu$, $\lambda >\mu ={\displaystyle \raisebox{1ex}{$C_d$}\!\left/ \!\raisebox{-1ex}{${\Omega }_d$}\right.}$, means that the number of users waiting for boarding increases according to the relation $\left(\lambda -\mu \right)t$, with the maximum value for $t={\Omega }_{sos}$. The clearance duration of this queue is $\left(\lambda -\mu \right){\Omega }_{sos}/\mu$.

For $V_d=C_d=$ 6000 passengers and ${\Omega }_{sos}=$ 45 min, ${\Omega }_d=$ 60 min (the period for which the offered capacity was previously calculated) results in $\lambda \cong$133 passengers/minute and $\mu$ = 100 passengers/minute, which leads to a duration of line clearance of 15 min (an average boarding wait time for the last passengers attached to the line, that is tolerable, in the authors’ opinion).

Waiting times are even shorter for users attached later in the queue. To prove this statement, consider, for example, the user who arrived at time t = 15 min., when the queue length is about 500 passengers. Since the boarding rate is $\mu$ = 100 passengers/minute, the waiting time of this user in the queue is 5 min. But according to Equation (20), the estimated waiting time for boarding is ${\omega }_{es}=⌈{\displaystyle \frac{500}{300}}⌉$·4 = 8 min. This time was used in the definition of the functional reliability (including for the last passenger of the 500, attached to the queue), and therefore the statement is demonstrated.

3.2. Random Disruptions in Planned Trams’ Circulation

Events of the nature of risks or uncertainties that affect the operation of the components of the public urban transportation system may also negatively affect the functional reliability of the announced service. The interruption of the electricity supply, in the case of a tram line, for example, paralyzes the movement of all trams in both directions of traffic (except in the rare case where they would be equipped with batteries or supercapacitors). Other specific malfunctions of the running track (rail break or track blocking) or of a tram unable to run are likely to block traffic, initially locally. So that then, depending on the duration, the blockage spreads over shorter or longer distances with potential traffic disruptions on the entire line.

Especially when the effects produced cannot be counteracted by the parking reserves of the trams at the ends of the line. In fact, in the case of unidirectional trams, this is the only possibility to mitigate or compensate for deviations from the schedule in the case of disruptive events.

In the case of bi-directional trams together with the crossover links between the tracks corresponding to the two directions of traffic (the diagonal in ${\mathrm{m}}_1$, for example), the effects of the functionality degradation of the scheduled service can be significantly diminished.

Figure 9 and Figure 10 illustrate the evaluation frame of negative effects on the planned schedule if, in the same area of the line $Z_l$ a disruptive event occurs that interrupts circulation (for the same duration $\mathsf{\Delta }$) in the direction from $Z_l$ to B. Figure 9 corresponds to the breakdown of a tram that can no longer run and for which it is required a duration $\mathsf{\Delta }$ of the intervention for restarting circulation.

Similarly, Figure 10 corresponds to a rail failure or an obstacle in the track area that blocks the movement in the direction from $Z_l$ to B.

Both figures present solutions for the adaptation of tram circulation in the situation of that random event. The rescheduling solutions are specific. Just as the effects of planned circulation disruption are specific. It is easy to notice that the negative effects are larger when the tram breaks down (a more likely situation) than those caused by the loss of track functionality.

However, not only does the nature and duration of the disruptive event bring particularity in the size of the disturbances produced in the scheduled circulation, but also the moment and the position related to the two end-stops of the line when the event occurs on the line yield particularity. Of course, the positioning in relation to the $m_1$ zone is a matter of record. That is why Figure 11 (in which the area where it was possible to pass from one path to another was eliminated) highlights the different effects of the disruptive event in zone $Z_l$, respectively, zone $Z_k$.

The solutions to reduce the disruptive effects of random negative events can include the extreme solution of stopping both traffic directions in a certain area. Moreover, the picture is not complete.

Concerning this diversity of disruptive events with specific negative consequences and with specific solutions to limit deviations from the scheduled circulation, we appreciate that references to the functional reliability of the service during the disruptive event and after its clearance and the resuming of the scheduled circulation remain the responsibility of the TPO. It is the one who must promptly inform the users about the best estimation of the disruption, the maximum headway of trams during the event, and the estimated moment for the return to planned scheduled circulation, depending on each specific situation and using the most appropriate means.

However, our current analysis has limitations. Future research will address the type of disruptions, locations of disruptions, their frequency, durations, and number of trams affected, as recorded in PTO statistics (but also in other sources, such as the municipal traffic surveillance system).

4. Conclusions

• Increasing the attractiveness of urban public transportation as a solution for limiting the negative effects of individual motorized mobility in the urban environment requires an increasingly better quality of the offer presented to users. The compliance norms for all trained resources, which are in an upward dynamic due to technological progress, are defined for both the technical performances and those of convenience, comfort, safety, and security of the urban public transportation system. However, the transformation of these potential performances into the actual ones, i.e., those revealed to the users, depends on the operating technology adopted by the network manager and the chosen solution for adapting the offer to the estimated demand. From here, it derives the attributes of the service that can be found in the quality of presentation made publicly.
• According to a fundamental principle of good practice, the public service provider must ensure that the offer presented to potential users never exceeds the performance of the service that can be provided with certainty.

The imperative of this principle requires that the manager of each public transport line make a careful analysis. In which, inevitably, they face multiple uncertainties. The spatial and temporal inconsistencies of the task to be performed are only partially possible to be estimated at the time of the service design, as well as the other multitude of unpredictable exogenous or endogenous events that can negatively affect the planned performance of the service. In the end, regardless of the thoroughness of the analysis, the conclusion is the same: full certainty cannot be obtained. Confronted with this reality, the TPO is obliged to present to its users not a certain value of the quality of the service but a range (in this range, the relevant indicators of the quality of the planned service provided will be set with the highest possible probability). Defining service performance in this probabilistic manner refers to the functional reliability. By addressing a range of indicators of good functioning, it means that service functional reliability is a different concept than the reliability of a product or a traffic network (which also has other peculiarities). Analogies are not helpful.

3.

Therefore, the functional reliability of public transportation is typically related to the operation of a system. An urban public transport line that brings together infrastructures, means of transport, human operators, and technologies adapted to the planned tasks meets the requirements of a system. Functional reliability is defined by the probability with which the success of the mission is ensured; that means that is systemic reliability. The mission of the service derives from the quality that the users of the service appreciate when they choose public transportation as an alternative to satisfy the need for mobility in urban space. The functional reliability of urban public transport is of the nature of a commitment of the service provider, which also includes the presentation quality and aims to reduce the user’s uncertainty related to the performance variability.

4.

For the planning of trips, especially the frequent and mandatory ones, functional reliability is distinguished by the importance given by the users (to respect the planned schedule assumed by the operator for each characteristic time). The planned schedule, to a good extent, also reflects the comfort conditions of the trip. The presented analyses on the functional reliability of a tram line consider compliance with the planned schedule for frequent flows, specific to a period of the day, as well as for other operation scenarios. Namely, the occasional overloads that involve inflows and outflows of mass flows in assumed and well-known periods, or the circulation disruptions, caused by unpredictable/random negative events.

For each studied case, the definition of functional reliability was accompanied by reasoned recommendations regarding possible improvements in the quality of the offer under the conditions of an accepted level of technical and commercial efficiency of the service.

5.

The introduction of reversible (bi-directional) trams and crossover connections between the tracks of the tram line, appropriately settled on the route, have been proven as effective solutions for both taking over occasional overloads as well as for maintaining a level of service in the event of a disruptive random event that temporarily interrupts the movement of trams in one of the two directions. The studied solutions confirm the possibilities of improving the functional reliability of the line. Simultaneously, with the improvement of technical productivity, there is an improvement of commercial efficiency, too. Both are of major interest for the transport operator. Thus, the technical improvements investigated together with the operational solutions meet the interests of both the users and the service provider.

6.

In the initial situation of the line, without any interventions (to track and trams), the functional reliability of the service is limited to the buffer time provided at the ends of the route. This is the common way to avoid the propagation to other directions of the disturbance from the planned schedule. Apart from the buffer time of a line service, there is also an additional margin of the loading degree of trams, if it is possible. After implementing the proposed solutions, the transport operator has several possibilities to improve the reliability of the urban public transport line service. Through proactive traffic monitoring, the operator needs to choose between effective decision-making alternatives. In the case of disruptive events of greater magnitude, there may be only the option of preserving functional reliability on certain sectors of the route. In such situations, the operator considers not only affected users from the stops of the line but also the possibilities of continuing the journey by using other lines of the public transport network. In future research, it would be necessary to establish the sectors of the line route for which the preservation of a certain functional reliability level is mandatory.

7.

Important for users’ perception of the functional reliability of a tram line’s service is previously advanced information on the periods when the offered service is disrupted (e.g., reconstruction, preventive or accidental maintenance works, overloads, etc.). Similarly to the daily car’s users (who do not interpret as unreliability the congestion that drastically reduces their average travel speed during certain, well-known periods), the users of the public transportation line should not interpret the deviations from the scheduled circulation (produced by disruptive events) as a lack of the functional reliability of the service. Of course, this is possible only if these random events are sufficiently rare, the users are promptly and fully informed of the event, and the rescheduling is tolerable and made publicly and respected.