Connectivity Reliability and Approachability Indicators in Public Transport

1. Introduction

In the following, we will talk more specifically about buses, but the equations and reasoning apply to all types of public transport. For users, the reliability of public transport and the level of transport supply are essential. Fleet managers evaluate various reliability factors and the transport supply. A number of studies (Section 2, State of the art) have analyzed many dimensions of reliability; generally, only the shortest path linking an origin to a destination is considered, although all the paths connecting origins to destinations contribute to reliability. Our contribution (summarized in Section 3) consists in characterizing the contribution to reliability of “k” paths for each Origin-Destination (OD) pair; these paths are found, at every departure time, thanks to a “k” shortest paths algorithm [1]; this algorithm is implemented in the space-time, taking into account timetables and a limit on the maximum tolerable waiting time (Appendix A). We provide fleet managers with a set of indicators (Section 4), useful for identifying the most disadvantaged Origin-Destination pairs, and on the basis of which service improvement projects can be compared. In Section 5, equations provide the contribution of the various paths to reliability and unreliability (defined here as one minus reliability). An application (Section 6) to the bus network of a city in Brittany shows how to understand and use these indicators; the impact of node closure is shown in Appendix C.

2. State of the Art

Wakabayashi and Iida defined reliability as ‘‘the probability of a device performing its purpose adequately for the period intended under the operating conditions encountered’’ [2] – applied to transport networks, it is, according to Nicholson, “the probability that a trip can be successfully finished within a specified time interval”[3]. Reliability is generally used to describe the stability, certainty, and predictability of travel conditions [4].

Understanding the causes and consequences of unreliability, assessing reliability, possibly model it, proposing its enhancement, are useful for developing networks and allocating capacity, for the appraisal of transportation projects, for passenger route choice, as well as for diagnosing critical locations in transportation infrastructure.

In a public transportation network, the main causes of unreliability are, according to Mahdavi et al. [5], failure on physical infrastructure or on fleet service and uncertainty on travel demand and travel time.

2.1. Failure in the Connection Between Origin and Destinations

Bell and Iida defined connectivity reliability as the probability of having a connection between a pair of nodes when one or more links are removed [6]. Similar definitions are proposed in [7,8,9]. Kim gave flow reliability, and topological reliability indicators, based on the supply at the OD level and the bus route level [10].

Vulnerability, the opposite of reliability, is addressed in [11,12,13,14,15,16]; Cats and Jenelius established a vulnerability curve showing the relationship between network performance and the degradation of route or link capacities [17]; for a rail network, Liu [18,19] determined the economic value of reserve capacity and determined its impact for enhancing reliability and robustness: the disturbances cause train delays as well as cancellations. Taylor and D’Este developed a method for diagnosing critical points in transportation infrastructure [20].

However, only one path per OD pair is considered, whereas all paths would have to be considered, for determining whether or not the failure of a link or of a public transport vehicle leads to a disconnection of the OD pair.

2.2. Uncertainty on Travel Time or Travel Demand

A trip may not be completed on time due to an increase in travel time. Indicators measuring the variability of travel time distribution define travel time reliability, which intervenes in the overall reliability at (at least) three levels:

• As a component in project appraisal [21]. For an economic appraisal, it is necessary, according to Taylor, to estimate the value of travel time reliability [22]; Benezech and Coulombel estimated the value of the service interval and the value of service reliability [23].
• To suggest ways of enhancing overall reliability. Muñoz, after analyzing the sources of irregularity, proposed to regularize headways, thereby improving reliability [24]. Artan and Sahin developed a stochastic model, to be used in the timetable design phase, to prevent the impact of cumulative delays on reliability [25].
• To offer passengers the most reliable routes. Waiting time reliability has an impact on the passenger route choice- this was evaluated by Shelat [26]; Khani developed a stochastic algorithm capable of proposing the most reliable route (in terms of time) [27] - in addition, proposed backup itineraries are proposed by Redmond [28].

Excessive passenger demand may also prevent a trip from being completed in time. To reduce this negative effect, Liu optimized the reserve capacity [19]. A number of works, initially carried out for road networks, could be adapted to public transportation networks: Clark and Walting modeled network travel time reliability based on daily demand variability [29]; Chen [30], and Yang [31] defined capacity reliability as the probability that a network can meet a certain traffic demand at a required level of service, while taking into account drivers’ route choice behavior.

All this argues in favor of introducing a stochastic term into the shortest-path algorithm - we have not yet introduced it in the work presented here.

2.3. Reliability and Accessibility

According to Kim and Song, accessibility refers to the ease with which passengers can make their journeys between the network’s origin and destination stations [32].

Travel time reliability and accessibility are closely related concepts [20]; Bimpou and Ferguson integrated day-to-day travel time reliability into the measurement of accessibility [33].

Connectivity reliability and accessibility are also closely related. Kim and Song combined accessibility and topological reliability in a single indicator [34]. Taylor proposed an approachability index, equal to the ratio between the Euclidean distance between an Origin and a Destination and the actual length of the path connecting Origin and Destination [35]. This indicator, valid for a single path, is here generalized for an OD connected by several paths.

3. Research Contribution

The various paths that exist between an origin and a destination contribute to the transport supply and reliability - we clarify and characterize these contributions here; this should help fleet managers to improve service, and passengers to use the service.

• Relation between the reliability logarithm of a path and its length. The Shannon information of an event, measuring its amount of uncertainty, is the absolute value of the logarithm of its probability [36]. Applied to the event “no failure on the path”, it is the absolute value of the logarithm of the path reliability. It appears to be the product of the path length and the failure rate (which is the opposite of the logarithm of the reliability of a unit-length element). Provided that only one path per OD is considered, this extends to the geometric mean of path reliability for all connected ODs, which is therefore related to the average path length.
• Concept of “OD length”, based on the OD reliability logarithm. When an OD pair is connected by several paths in space or time, the previous relation holds when replacing the path length by an “OD length”, defined so that the product of this length and the failure rate is equal to the absolute value of the OD reliability logarithm. The OD length is less than the shortest path length- the reduction in length quantifies the contribution of alternative paths.
• Independence and equivalence degrees. For each path linking a given OD, the “independence coefficient” quantifies the genuinely new part of the path in relation to previous paths. The coefficient equals 1 when the path is entirely distinct from previous paths and is lower when there are overlaps; the “equivalence degree” of a path, relative to the first (shortest) path, combines its independence degree and its lengthening relative to the shortest path. The sum of the equivalence degrees over the different paths directly appears in the equation giving the OD unreliability.
• Approachability indicator, characterizing the transport supply, useful for service improvement decisions and comparisons between projects.

4. Power Indicators of Reliability and Transport Supply

A path φ(ω) effectively connects an OD ω when all its elements operate. Let us consider the case where all elements have the same operating probability p. The elements of a path being disjoint, the events corresponding to the fact that an element operates, are independent. The reliability of path φ $\left(\omega \right)$ , denoted $R\left(\phi \left(\omega \right)\right)$ is then the product of the probabilities of these events:

$$R\left(\phi \left(\omega \right)\right)=p^{N\left(\phi \right(\omega \left)\right)}$$

(1)

where N(φ(ω)) is the number of elements of the path φ(ω).

Let us assume that the number of failures of an element is distributed according to a Poisson distribution with a parameter λ (the failure rate); the probability of the event ” zero failure” is e-λ, i.e., p=e-λ , or λ =-Log(p).

The reciprocal function of Equation (1) relates the number of elements to reliability:

The absolute value of the logarithm of the probability of an event, which measures its uncertainty, is known as its Shannon’s information [36]. According to Equation (2), path φ(ω) has a Shannon information equal to the product of its number of elements and the failure rate.

Let us consider the set (a_i) of elements in the network. When elements a_i have different operating probabilities p(a_i) ,all strictly less than 1, let us consider the most reliable element, called a₀, and define p such that p=p(a₀). We can say that the reliability of any element a_i is the same as that of r_i independent elements (with r_i≥1) similar to a₀, provided that:

Thus the element a_i “counts” for r_i elements. Equations (1) and (2) remain valid if replacing N(φ(ω)) by the following new count of elements:

$$\mathrm{New} \mathrm{count}={\sum }_{\mathrm{i}=1}^{\mathrm{N}\left(\mathsf{\phi }\left(\mathsf{\omega }\right)\right)}{\mathrm{r}}_{\mathrm{i}},$$

(4)

In the following, for sake of simplicity, we will continue to refer to N(φ(ω)) instead of the new count.

Let φ₁(ω) be, among all paths linking ω, the shortest one (i.e., with the smallest number of elements); it is also the most reliable.

Let N_COD be the number of connected ODs of the network. Applying Equation (2) to the shortest path φ₁(ω) of each connected OD ω, then taking the arithmetic average, we obtain as the first member of the new equation, $\overline{\mathbf{N}}$ the arithmetic average of the numbers of elements N(φ₁(ω)), and in the 2nd member the arithmetic average of the logarithms of R(φ₁(ω)), that is to say the logarithm of the geometric average (denoted ${\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)$) of the R(φ₁(ω)):

$$\sum _{\omega =1}^{N_{COD}}\frac{1}{N_{COD}}.N[{\phi }_1(\omega )]=\overline{N}={\displaystyle \frac{Log{\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right))}{Log\left(p\right)}}=-{\displaystyle \frac{Log{\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right))}{\lambda }}$$

(5)

$\overline{N}$ , in the first member, does not depend on p. Note that, in addition to $\overline{N}$ , we need to count the unconnected ODs.

When an estimation of the traffic demand per OD is available, we replace in Equation (5), the coefficient 1/N_COD by the proportion of the demand of OD ω; the result $\overline{N}$ becomes the arithmetic mean (over travellers) of the numbers of elements of the shortest paths.

Different indicators are built according to whether we consider a path as a set of buses, stops, links or kilometers. Let us note p_bus, p_stop, p_link, p_length the probabilities that a bus, a stop, a link or “a kilometer” are operating These probabilities replace the probability “p” used in Equations (1)–(5).

Five indicators are proposed for any path φ:

• Bus Availability Reliability:

$${\mathrm{R}}_{\mathrm{bus}}\left(\mathsf{\phi }\right(\mathsf{\omega }\left)\right)={\mathrm{p}}_{\mathrm{b}\mathrm{u}\mathrm{s}}^{N\left[\mathsf{\phi }\right(\omega \left)\right]}$$

(6)

Here, N(φ(ω)) is the number of buses successively taken. When there is a prior to join the bus first bus stop, or a final walk to join the destination we increase the number of buses N(φ(ω)) by one or two units.

2.

Bus Stop Reliability

$${\mathrm{R}}_{\mathrm{stop}}\left(\mathsf{\phi }\right(\mathsf{\omega }\left)\right)={\mathrm{p}}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p}}^{N\left[\mathsf{\phi }\right(\omega \left)\right]}$$

(7)

Here, N(φ(ω)) is the number of bus stops encountered. There is no need to modify N(φ(ω)) when there is an initial or a final walk.

3.

Link Reliability

$${\mathrm{R}}_{\mathrm{link}}\left(\mathsf{\phi }\right(\mathsf{\omega }\left)\right)={\mathrm{p}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^{N\left[\mathsf{\phi }\right(\omega \left)\right]}$$

(8)

Here, N(φ(ω)) is the number of links of the path φ (equal to the number of stops minus one). N(φ(ω)) is increased by one or two units when there are prior or/and final walks.

4.

Travel Length Reliability

For some types of failure, the average failure rate is constant by length unit, which implies an exponential distribution of failures; Then the operating probability of link a_i, of length L_i,, is linked to the failure rate on this link, denoted λ_i, by the equation:

$$p\left(a_i\right)=e^{{-\mathsf{\lambda }}_i·L_i}$$

(9)

As above, for sake of simplicity, we consider that the failure rate by unit length is the same for all links i.e., λi= λ whatever “i”. The corresponding operating probability per length unit is p_length , such that

Let L(φ) be the length of path φ. The travel length reliability for path φ is:

$${\mathrm{R}}_{\mathrm{length}}\left(\mathsf{\phi }\right)={\left(p_{length}\right)}^{{\sum _{i=1}^{N\left(\mathsf{\phi }\right)}L}_i\left(a_i\right)}={\left(p_{length}\right)}^{\mathrm{L}\left(\mathsf{\phi }\right)}$$

(11)

5.

Reliability Standard Indicator of transport supply.

The reliability of an OD pair obviously decreases as the Euclidean distance between origin and destination increases. To focus on the quality of transport supply, and compare ODs independently of their Euclidean distance, we propose an indicator equal, for a path φ(ω), to the ratio between the actual length of the path and the Euclidean distance of ω.- exactly the inverse of the approachability index proposed by [35] Michael Taylor (2010). The Euclidean distance is denoted D_min(ω); let us define L_standard(a_i, ω) the standard length of element a_i for OD ω, equal to the ratio L_standard(a_i)/D_min(ω); let us define, the sinuosity of path φ(ω) (denoted L_standard (φ(ω))) by the ratio L(φ(ω))/D_min(ω). The sinuosity of the path would be minimal and equal to 1 for a perfectly straight path. An indicator, called “standard indicator” is defined by replacing, in Equation (11), the length by the standard length and the probability p by a value p_standard, to be specified later:

$${\mathrm{R}}_{\mathrm{standard}}\left(\mathsf{\phi }\right)=p_{standard}^{L\left(\phi \right)/D_{min}\left(\omega \right)}$$

(12)

Let $\overline{\mathrm{S}\mathrm{I}\mathrm{N}}$ be the average, over all connected ODs, of sinuosity L(φ₁(ω))/D_min(ω) of the first paths (the less sinuous, also, the shortest) connecting the ODs. For improving the equity of transport supply, the bus fleet manager should improve the service on ODs (or nodes) where the standard indicator is low.

Selecting the value of p_standard

We propose two adjustments for p_standard (described in Appendix B) for two different purposes:

A low standard indicator may be due to topological constraints on the road network (sinuosity, one-way roads), poor bus line design (long, winding bus lines), or poor bus linee supply (insufficient number of lines and line-to-line connections, insufficient service frequency). Comparing the value of this indicator with that of the bus reliability indicator (which is only sensitive to the third cause) can help discriminate the cause. To facilitate comparison between the two indicators, p_standard is adjusted so as to obtain, on average over the shortest paths, the same value for both indicators.

To enable comparisons between two service improvement projects: increasing frequency and creating a more direct route.

5. Contribution to Reliability and Unreliability of a Path

Paths are found thanks to an algorithm, (see Appendix A) giving, for a given departure time, up to k paths by OD pair, taking into account the timetables and a constraint on the maximum admissible waiting time.

5.1. Summary

• The reliability of an OD is computed from the reliability of the various paths linking it, thanks to the inclusion/exclusion principle (Equation (13)).
• The absolute logarithm of OD reliability, divided by the failure rate, defines the “OD number of elements”, smaller than, or equal to the number of elements of the shortest path linking the OD (Equation (14)).
• The absolute logarithm of the geometric mean of OD reliability (over all connected OD pairs), divided by the failure rate, is the arithmetic mean of the “OD numbers of elements” (Equation (15)); it is smaller than or equal to the mean of the numbers of elements of the shortest paths, the reduction in number of elements giving the contribution of alternative paths to reliability.
• Contribution of alternative paths is also equivalent to a decrease in the failure rate.
• OD unreliability, defined here as “One minus reliability”, is expressed, thanks to the chain rule in probability, as the product of the contributions of the paths linking the OD (Equations (19) and (26)); this reveals two characteristics of each path: its degrees of independence (Equation (21)), and of equivalence (Equation (23)). The sum of the equivalence degrees, over all paths linking an OD, quantifies the contribution of alternative paths to unreliability (Equation (27)); these sums per OD are used in (Equation (28)), which gives the unreliability logarithm for all connected ODs.

5.2. Equations on Reliability

Let M be the number of elements of the network, and a_i be an element. Let x_i be such that x_i =1, with the probability p, when element a_i operates - otherwise x_i =0, with the probability 1- p. A state of the network is described by a vector {x₁… x_i …x_M}. The number of such states is 2^M; the set of such states is denoted S. The event “path φ_j is operating” (respectively “is off”) corresponds to a subset S_j of S (respectively to its complement $\overline{{\mathit{S}}_{\mathit{j}}}$). The probability of S_j is the path reliability R(φ_j(ω)).

When k paths {φ_j (ω), j=1,k} link an OD ω, the reliability R(ω) is the probability that at least one path is operating: it is equal to the probability of the union of subsets S_j. Such a probability (without double counting) is given by an equation due to de Moivre known as the inclusion-exclusion principle [37]:

$$\begin{gathered} R\left(\omega \right)=Proba\left(\bigcup S_j\right) \\ =\sum _{j=1}^{k}proba\left(S_j\right)-\sum _{j,j^{\mathrm{\text{'}}}}proba\left(S_j\bigcap S_{j^{\mathrm{\text{'}}}}\right) \\ +\sum _{j,j^{\mathrm{\text{'}}},j\mathrm{”}}proba\left(S_j\bigcap S_{j^{\mathrm{\text{'}}}}\bigcap S_{j^{\mathrm{\text{'}}\mathrm{\text{'}}}}\right)..+{(-1)}^{k-1}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\left(\bigcap _{j=1}^k{S}_j\right) \end{gathered}$$

(13)

Let us apply, like in Equation (2), the reciprocal function of reliability, i.e., the function Log(.)/Log(p) on R(ω); the result, denoted N’(ω,p), defines the “OD number of elements” for OD ω, , which generalizes at the OD level the number of element of a path:

From Equations (2) and (14); it comes that the reliability provided by the k paths is equal to that which would result from a single path with a number of elements N’(ω,p). As R(ω) greater than the reliability of the shortest path R(φ₁), and as the function (-Log(.) is decreasing), then N’(ω,p) is less than N(φ₁);

With the length reliability indicator, N’(ω,p_length) is called “OD number of elements” of ω . According to Equation (14), the Shannon information of ω is the product of N’(ω,p) and failure rate (which is the information of an element).

Let R^geometric(p) be the geometric mean (over all connected ODs) of the R(ω). R^geometric(p) is necessary greater than ${\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)$ .

In line with Equation (5), applying Equation (14) for all connected ODs, then taking the arithmetic mean, we obtain:

$$\stackrel{̿}{\mathrm{N}}(p)={\displaystyle \frac{\mathrm{L}\mathrm{o}\mathrm{g}\left({\mathrm{R}}^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)\right)}{\mathrm{L}\mathrm{o}\mathrm{g}\left(p\right)}}=-{\displaystyle \frac{\mathrm{L}\mathrm{o}\mathrm{g}\left({\mathrm{R}}^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)\right)}{\mathsf{\lambda }}})$$

(15)

where $\stackrel{̿}{\mathrm{N}}\left(p\right)$ is the arithmetic mean of the N’(ω,p) and ${\mathrm{R}}^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)$ is the geometric mean of the R(ω).

$\stackrel{̿}{\mathrm{N}}\left(p\right)$ is smaller than $\overline{\mathrm{N}}$ because ${\mathrm{R}}^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)$ is greater than ${\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(p\right)$ ^.

The contribution of alternative paths is equivalent to reducing the mean of the numbers of elements from $\overline{\mathrm{N}}$ to $\stackrel{̿}{\mathrm{N}}\left(p\right)$ .

Equation (15) applies to all five indicators, replacing, p by p_bus, p_stop, p_link, p_length, p_standard. Note that for the standard indicator, $\stackrel{̿}{\mathrm{N}}\left(p_{standard}\right)$ is the “average sinuosity” of the network; it indicates the level of the transport supply on the network and is used to compare different networks or different improvement levels on the same network.

In the case where only two paths connect ω, the equation giving N’(ω, p) is simple; we demonstrated (but the demonstration is not included here) that N’(ω, p ) is a decreasing function of p.

In the case where all paths share a common part, the contribution of this part to N’(ω, p) remains equal to its number of elements - there is no reduction on these common elements. In the case of two paths, we demonstrated (the demonstration is not included here) that, when p tends to 1, N’(ω, p), decreases to the number of elements of the common part - thus to zero when paths are disjoint.

These properties spreads over to the arithmetic mean $\stackrel{̿}{\mathrm{N}}\left(p\right)$ . Plotting on the same figure $\overline{\mathrm{N}}$ and $\stackrel{̿}{\mathrm{N}}\left(p\right)$) embodies the contribution of alternative paths. The following figure displays the evolution of $\overline{\mathrm{N}}$ and $\stackrel{̿}{\mathrm{N}}\left(p\right)$ according to p in the case of the travel length reliability (defined in Equation (11)); the “OD number of elements” is then the “ OD length”; it was computed for the bus network of Rennes in Britanny ( see Section 6).

Figure 1. Average of “OD lengths” on the Rennes bus network according to the operating probability.

Figure 1. Average of “OD lengths” on the Rennes bus network according to the operating probability.

The arithmetic average of the shortest paths lengths ($\overline{\mathbf{N}}$ =14.5 kilometers) is displayed by the green horizontal straight line. $\stackrel{̿}{\mathrm{N}}\left(p\right)$ , the average of the “lengths” of the OD pairs, is displayed by the blue dashed curve (up to two paths) or the red curve (up to five paths). In the case “up to two paths, when p ten. ds to 1, $\stackrel{̿}{\mathrm{N}}\left(p\right)$) tends to 11.2 kilometers which is the average lengths of overlaps - note that for ODs connected by a single path, the entire length of the path is counted as overlap.

When p tends to zero, $\stackrel{̿}{\mathrm{N}}\left(p\right)$ tends to $\overline{\mathrm{N}}$ (no reduction).

Remark.

It’s also interesting to analyze reliability at the level of an individual node (or set of nodes).For a given node, let us define “node reliability” as the average (either geometric or arithmetic) of the R(ω) for all ODs whose origin or destination is that node. Equations (5) and (15) generalize easily to the node level.

5.3. Interpretation of the Increase in Reliability due to Alternative Paths

For a single OD ω, the contribution of the alternative paths is equivalent to an increase of the operating probability p up to a greater probability p’ –or equivalently to a reduction of the failure rate-Log(p) to –Log(p’). The equivalent operating probability p’ with which the reliability of the only shortest path φ₁ would become R(ω) is such (after rewriting of Equation (2)) that:

N(φ₁)= Log_’(R(ω))/Log(p’)

(16)

$$\mathrm{i}.\mathrm{e}: p^{\mathrm{\text{'}}}={\left(\mathrm{R}\left(\mathsf{\omega }\right)\right)}^{1/\mathrm{N}\left({\mathsf{\phi }}_1\right)})$$

(17)

For the whole set of connected ODs, in line with Equation (5), the value p’ which convenes is such that:

$$\overline{N}.\mathrm{Log}(p’)=\mathrm{Log} ({\mathrm{R}}^{\mathrm{geometric}} (p)) \mathrm{or} p^{\text{'}}={\left({\mathrm{R}}^{\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}(p)\right)}^{1/\overline{\mathrm{N}}})$$

(18)

5.4. Equations on Unreliability

In this subsection, unreliability, i.e., the quantity “1-reliability”, is decomposed into a product of factors, each of which is related to the unreliability of a path. This decomposition shows the impact of any path.

The event “path j is off” corresponds to the set $\overline{S_j}$ . An OD is disconnected when the 1st, 2nd, ... kth paths are simultaneously off; this corresponds to the set $\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_k}$ . The probability of this set is referred as “unreliability”, denoted V(ω), and is equal to 1-R(ω).

Using the chain rule of probability that combines conditional probabilities [38], the unreliability of OD is factored as:

$$\begin{gathered} \mathrm{V}\left(\omega \right)=Proba\left(\overline{S_1}\cap \overline{S_2\cap }\dots \cap \overline{S_k}\right)=proba\left(\overline{S_1}\right).{\prod }_{\mathrm{j}=2}^{\mathrm{k}}{\displaystyle \frac{Proba\left(\overline{S_1}\cap \overline{S_2\cap }\dots \cap \overline{S_j}\right)}{Proba\left(\overline{S_1}\cap \overline{S_2\cap }\dots \cap \overline{S_{j-1}}\right)}} \\ =proba\left(\overline{S_1}\right).{\prod }_{j=2}^{k}proba\overline{{(S}_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right) \end{gathered}$$

(19)

The jth factor of the product, $proba(\overline{S_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)$ , is equal to $proba\left(\overline{S_j}\right)$ =V(φ_j (ω)) only when the path φ_j(ω) is disjoint from all previous paths, since in this case the condition” the previous paths do not operate”, is independent of whether φ_j (ω) operates or not. Otherwise, $proba(\overline{S_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)>proba(\overline{S_j})$ . The full demonstration is not given here. Intuitively, the condition $\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}$ means that, whatever z (1≤z≤j-1), path z doesn’t operate; now, the more overlaps there are between paths z and j, the more likely it is that the element of z that doesn’t operate belongs to path j, so there is an increase in the probability that path j doesn’t operate.

Let us write this jth factor as:

$$proba(\overline{S_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)={\left(\mathrm{V}\left({\phi }_j\left(\omega \right)\right)\right)}^{\beta \left(p,{\phi }_j\left(\omega \right)\right)}$$

(20)

$$\mathrm{thus}: \beta (p,{\phi }_j(\omega ))={\displaystyle \frac{Log\left(proba(\overline{S_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)\right)}{Log\left(proba\left(\overline{S_j}\right)\right)}}$$

(21)

β(p,φj(ω)) is equal to 1 when path φ_j(ω) is disjoint from all previous paths (independence), less than 1 otherwise. - Recall that $\mathrm{V}\left({\phi }_j\left(\omega \right)\right)=proba(\overline{S_j}$)

In order to compare the contribution φ_j(ω) with that of φ₁(ω) let us write:

$$proba(\overline{S_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)={\left(\mathrm{V}\left({\phi }_1\right(\omega \left)\right)\right)}^{\gamma \left(p,{\phi }_j\left(\omega \right)\right)}$$

(22)

$$\gamma \left(p,{\phi }_j\left(\omega \right)\right)={\displaystyle \frac{Log\left(proba(\overline{S_j}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)\right)}{Log\left(proba\left(\overline{S_1}\right)\right)}}$$

(23)

The denominator of the equation of γ is related to $\overline{{\mathit{S}}_1}$ (instead of $\overline{{\mathit{S}}_{\mathit{j}}}$ for β). As the most reliable (thus the least vulnerable) path has been put first, thus:

As the conditional probability $proba(.\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)$ is itself a probability distribution, the complement (with respect to value 1) of this jth factor is $proba({\mathrm{S}}_{\mathrm{j}}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)$ . This last term corresponds to the increase in reliability for ω when moving from j-1 to j paths - it combines terms already calculated when applying the inclusion/exclusion principle (Equation (13))

Let us note α(p, ${\phi }_j$ (ω)) the following ratio:

$$\mathsf{\alpha }(p,{\phi }_j(\mathsf{\omega }\left)\right)=\left(proba({\mathrm{S}}_{\mathrm{j}}\left|\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)/proba(S_j\right)$$

(24)

We can also write:

$$\mathsf{\alpha }(p,{\phi }_j(\mathsf{\omega }\left)\right)={\displaystyle \frac{proba(\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\cap {\mathrm{S}}_{\mathrm{j}})}{proba\left(\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right).\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\left({\mathrm{S}}_{\mathrm{j}}\right)}}={\displaystyle \frac{proba(\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}|{\mathrm{S}}_{\mathrm{j}})}{proba\left(\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}\right)}}$$

(25)

In information theory, the logarithm in basis 2 of α(p, φ _j(ω)) is known as the “pointwise mutual information” between the events: $\overline{S_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}}$ and S_j [39].

Let α(p, φ ₁(ω))= β(p, φ ₁(ω))= γ(p, φ ₁(ω))=1 (whatever p is). The new factorization of V(ω) is:

$$\begin{gathered} V\left(\omega \right)=\prod _{j=1}^k\left(1-\alpha (p,{\phi }_j(\omega \left)\right).R\left({\phi }_j\right(\omega )\right)=\prod _{j=1}^k{(1-R({\phi }_{\mathrm{j}}\left(\omega \right))}^{\beta (p,{\phi }_j(\omega \left)\right)} \\ =\prod _{j1}^k{(1-R({\phi }_1\left(\omega \right))}^{\gamma (p,{\phi }_j(\omega \left)\right)} \end{gathered}$$

(26)

α(p, φ _j(ω)), β(p, φ _j(ω)) and γ(p, φ _j(ω)) are called “independence coefficient”, “independence degree” and “equivalent degree”.

The inequality β(p,φ _j(ω))≤ α(p,φ _j(ω)) is obtained by studying Equation (26); β is an increasing function of α.

When path φ_j is disjoint from the previous paths; the event S_j is independent of $\overline{{(S}_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}})$ then, the condition “|S_j” of Equation (25) has no impact on proba $\overline{{(S}_1}\cap \overline{S_2}\cap \dots \cap \overline{S_{j-1}})$ , then α(p, φ _j(ω))=1.

Otherwise, the path φ_j(ω) has a common part with a previous path. α(p, φ _j(ω)) and β(p, φ _j(ω)) are smaller than 1; we demonstrated in the case j=2, (but the demonstration is not included here), that they decrease when the overlap between φ _j and the previous path increases. When path φ _j includes a previous path φ _j’, φ _j does not contribute to unreliability (nore reliability); indeed, in this case ${{\mathrm{S}}_{\mathrm{j}}\subset \mathrm{S}}_{\mathrm{j}\mathrm{\text{'}}}$ thus $\overline{S_{j\text{'}}}\subset \overline{S_j}$ then:

$Proba\left(\overline{S_1}\cap \overline{S_2\cap }\dots \cap \overline{S_j}\right)=Proba\left(\overline{S_1}\cap \overline{S_2\cap }\dots \cap \overline{S_{j-1}}\right)$ implying that the jth factor of Equation (19) or Equation (26) equals 1; then α(p, φ _j(ω)) = β(p, φ_j(ω))=0.

In the case j=2 we demonstrated (the demonstration is not included here), that, when p increases from 0 to 1, α and β decrease, from 1 to the proportion, in path φ₁, of the number of elements which do not belong to path φ₂.

By applying the logarithm function to both members of Equation (26), and replacing 1-R(φ₁(ω)) by V(φ ₁(ω)) and 1-R(φ _j(ω)) by V(φ _j(ω)), it comes:

$$\begin{gathered} \mathrm{L}\mathrm{o}\mathrm{g}\left(\mathrm{V}\left(\mathsf{\omega }\right)\right)=\sum _{j=1}^k\beta \left(p, {\phi }_j\left(\omega \right)\right).\mathrm{L}\mathrm{o}\mathrm{g}\left(\mathrm{V}\left({\mathsf{\phi }}_{\mathrm{j}}\left(\mathsf{\omega }\right)\right)\right) \\ =\left[\sum _{j=1}^k\gamma (p, {\phi }_j(\omega \left)\right)\right].\mathrm{L}\mathrm{o}\mathrm{g}\left(\mathrm{V}\left({\mathsf{\phi }}_1\left(\mathsf{\omega }\right)\right)\right) \end{gathered}$$

(27)

The numbering of the paths 2…k has an impact on the factorization in Equations (24) and (25) and thus on the coefficients α, β, γ, which are therefore not unique; however, when the purpose is to evaluate the gain due an additional path, this new path is in all cases the last one, so the coefficients α, β, γ, for this last path are not disputable. Fortunately, the sum (from 1 to k) of the equivalence degrees γ does not depend on the numbering of paths 2...k: indeed, according to Equation (27), this sum is the ratio Log(V(ω))/Log(V(φ₁(ω))). This sum is necessarily less than (or equal to) k. It is as if there were, for ω, only Ʃ_j=1..k γ(p, φ _j(ω)) paths independent and equivalent to φ₁.

The geometric mean of the unreliability is obtained by multiplying the products of factors of Equation (26) whatever ω (ω between 1 and N_COD), then taking the N_COD root; its logarithm is obtained by the following equation:

$$Log{V}^{geometric}=\sum _{\omega =1}^{N_{COD}}{\displaystyle \frac{1}{N_{COD}}}.\left[\sum _{j=1}^k\gamma \left(p,{\phi }_j\left(\omega \right)\right)\right].Log(V({\phi }_1(\omega ))$$

(28)

The average (over connected ODs) of the sums (from 1 to k) of the equivalence degrees is the average of the ratios Log(V(ω))/Log(V(φ₁(ω))); This average, always between 1 and k, characterizes the impact of alternative paths: it is close to 1 when there are few alternative paths, or when these paths largely overlap, or are much longer than the shortest path.

When the traffic demand per OD is known, unreliability is calculated per user; Equation (28) gives the arithmetic mean of the logarithms of unreliability per user, provided that the coefficient 1/N_COD is replaced by the proportion of the OD demand ω.

6. Results: Application on a City in Brittany

Greater Rennes area in Brittany, France includes 43 communes, served by 148 bus lines; there are 707 bus stops, 680 of which were taken into account here; each bus stop is considered here as as both an origin and a destination, giving 680*679 ODs; the network is represented here.

Figure 2. Map of the Greater Rennes area bus network (extracted from https://www.star.fr).

Figure 2. Map of the Greater Rennes area bus network (extracted from https://www.star.fr).

The algorithm searched for a maximum of k=5 paths per OD; The criterion to be minimized was the number of connections between buses; the maximum waiting time allowed was one hour; for a user departure time of 8:00, 90.6% of ODs were connected. This might seem low, but it is not unusual for some nodes not to be connected at certain times of the day. An average of 2.48 paths were found per connected OD - here, paths that are identical in space, simply shifted in time, have been eliminated (see Appendix A).

6.1. Reliability due to the Shortest Path and to Alternative Paths for Four Indicators

When several paths connect the same OD, the path placed first (the shortest) depends on the indicator: for the link, bus, length or standard indicators, it is respectively the path with the fewest links, buses taken in succession, the shortest path in length (which is also the least sinuous).

p_length was arbitrarily set to 96%, which corresponds to a failure rate λ per kilometer equal to 0.383 (i.e., Log(0.96)= -0.383=- λ ).With the length reliability indicator, the arithmetic mean of the reliability of the shortest paths was 0.581; to facilitate comparisons between indicators (see Appendix B, Equation (29)), P_link , P_bus and P_standard were chosen to obtain the same average reliability; this occurred for P_link=97.1% , P_bus =83.1% and P_standard =70.8%. This does not imply that, after inclusion of alternative paths, average reliabilities remain equal, because the gain in reliability due to alternative paths depends on the indicator chosen.

Table 1. Reliability and Standard Indicators for four indicators (shortest paths and all paths) (no initial/final walks).

	Link plink=97.1%	Bus pbus=80.3%	Length plength=96%	Standard pstandard=70.8%
${\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{A}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}$	0.581	0.581	0.581	0.581
Rarithmetic	0.680	0.673	0.674	0.680
$\overline{\mathrm{N}}$ (*)	19.40	2.53	14.54	1.65
Average of Average (**)	23.24	2.95	15.55 (kms)	1.71
$\stackrel{̿}{\mathrm{N}}\left(p\right)$ (***)	14.09	1.89	10.91	1.22
β	1.532	1.432	1.466	1.467
γ	1. 423	1.367	1.424	1.417

(*) average, over all connected ODs, of the numbers of elements of the shortest paths, (**) average, over all connected ODs, of the average numbers of path elements per OD, (***) average, over all connected ODs, of the “OD numbers of elements.”.

Table 1. Reliability and Standard Indicators for four indicators (shortest paths and all paths) (no initial/final walks).

	Link plink=97.1%	Bus pbus=80.3%	Length plength=96%	Standard pstandard=70.8%
${\mathrm{R}}_{\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}^{\mathrm{A}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}$	0.581	0.581	0.581	0.581
Rarithmetic	0.680	0.673	0.674	0.680
$\overline{\mathrm{N}}$ (*)	19.40	2.53	14.54	1.65
Average of Average (**)	23.24	2.95	15.55 (kms)	1.71
$\stackrel{̿}{\mathrm{N}}\left(p\right)$ (***)	14.09	1.89	10.91	1.22
β	1.532	1.432	1.466	1.467
γ	1. 423	1.367	1.424	1.417

(*) average, over all connected ODs, of the numbers of elements of the shortest paths, (**) average, over all connected ODs, of the average numbers of path elements per OD, (***) average, over all connected ODs, of the “OD numbers of elements.”.

The average length of the shortest paths is 14.54 kilometers; the average sinuosity for the less sinuous (shortest) paths per OD is 1.65 (see heading “ $\overline{\mathrm{N}}$ ” above).

The average, over all connected ODs, of the average numbers of path elements for the OD, is higher than the average on the shortest paths alone (heading $\overline{\mathrm{N}}$) - alternative paths having a higher number of elements than the shortest paths.

The improvement provided by the alternative paths is equivalent to a reduction in length from $\overline{\mathrm{N}}$ =14.54 kilometers (average length of the shortest paths) to 10.91 kilometers ($\stackrel{̿}{\mathrm{N}}\left(p\right)$ the average of “OD length”) and a reduction in sinuosity from $\overline{\mathrm{N}}$ =1.65 (shortest paths) to $\stackrel{̿}{\mathrm{N}}\left(p\right)$ ⁼1.22.

The overlaps between the 2.48 paths per connected OD, lead to an average sum of independence degrees between 1.43 and 1.53 (according to the indicator), as if there were, between 1.43 and 1.53 independent (i.e., disjoint) paths per connected OD. The average sum of equivalence degrees is between 1.37 and 1.42, as if there were, on average per OD, between 1.37 and 1.42 paths independent and equivalent to the first (shortest) one.

6.2. Gain When Initial and/or Final Walks Are Allowed

Allowing initial or final walks (here up to 400 meters), increases the percentage of connected ODs (from 90.6% to 90.6% to 96.9%)%; and increases the number of paths per connected OD (from 2.48 to 3.92).

The average, over all connected ODs, of the average numbers of path elements per OD is necessarily higher than the average of the “OD numbers of elements” -see Table 2.

As expected, all the figures in Table 2 show an improvement (and quantify it) on those in Table 1.

6.3. Reliability Along Bus Line #9

The network is structured around nodes where many bus lines connect. Users entering the network at these nodes are favoured in terms of the bus reliability indicator. Conversely, the closure of such a node has a major negative impact on both the number of connected ODSs and the reliability of those still connected (this point is developed in Appendix C). A study of the evolution of reliability along a bus line (the 8.3 kilometer- long line # 9), shows how to understand and use the various indicators.

Figure 3. Line #9 (in blue) in Rennes, from Cleunay to Saint-Laurent (extracted from https://www.star.fr).

Figure 3. Line #9 (in blue) in Rennes, from Cleunay to Saint-Laurent (extracted from https://www.star.fr).

The node reliability is the arithmetic average of the reliability of ODs that have this node as origin or destination. The reliability of successive bus stops along bus line # 9 is shown in Figure 4 for four reliability indicators.

When a node (a bus stop) is served by very many lines, the shortest paths from this node to very many destinations consist in a single bus. These shortest paths have a single element for bus reliability, resulting in a high bus reliability indicator (shortest paths only). This is particularly the case at “République”, with 19 bus lines. The other bus stops on line #9 generally have a bus reliability (shortest paths only) of just over 0.6, (at p_bus =80%), slightly higher than the average bus reliability (shortest paths only) for the whole network (0.581); this is due to the fact that the density of bus lines is higher in the city (where line #9 is located) than in the rest of Greater Rennes.

The increase in bus reliability between the curves (Bus 80% shortest path) and (Bus 80% all paths) is due to alternative paths. The existence of alternative paths is favored when several lines serve a bus stop, implying several ways to reach it. For such nodes, the bus reliability indicator (all paths) is high. This is the case at “Cité Judiciaire” and “Hôtel Dieu”, served by 5 lines as shown in Figure 3.

The standard indicator is close to the bus reliability indicator, firstly because it has been calibrated to be equal to it, on average over the network, and secondly because, for an OD, a good (respectively bad) offer generally allows more direct (respectively less direct) paths, which results in a high (respectively low) standard indicator, as it is the case for the bus reliability indicator. We note that, at République, the standard indicator is not as high as bus reliability, perhaps because the paths are not as direct, due to the presence of the river “La Vilaine” which imposes topological constraints on the road network

Length reliability is greater than 0.736, higher than that of the Rennes network (0.674). Indeed, as line#9 is more central than average, the paths to or from a node on this line are shorter than average. This is particularly true for nodes that are central to line#9.

7. Limitations and Future Work

This work has a number of limitations, which can be overcome in future research.

• Regarding the shortest path algorithm. The introduction of a stochastic term in the travel time would make the connections between buses more realistic. On the other hand, certain paths should be eliminated when they include irrelevant detours, whose sole purpose is to circumvent the maximum waiting time constraint, by replacing part of the waiting time with unnecessary travel time on these detours. This could be avoided by calculating the shortest paths in space (without time constraints) - these paths have no detours - and then proposing the elimination of paths in space-time that are much longer than the shortest paths
• Regarding the network reliability. The geometric average of ODs reliability over-weights ODs with very low reliability; conversely, the geometric average of ODs unreliability over-weights ODs with very high reliability. this argues in favour of replacing geometric averages with arithmetic averages.
• Regarding the ODs with the poorest service. Our approach doesn’t tell us why certain ODs aren’t connected at a given time. An additional tool is needed to propose either an increase in frequencies, the creation of a new line or the extension of an existing line.

8. Conclusions

In this paper, we investigated the connectivity reliability in a public transportation network, by examining, on the basis of real lines and schedules, the {space-time} paths that connect origins and destinations. A shortest path algorithm provides, for each Origin-Destination pair, up to ‘k’ paths in the space-time domain. The reliability of a path is a power function of the reliability of its elements. The length of a path appears equal to the opposite of the logarithm of the path reliability, divided by the failure rate (which is the opposite of the logarithm of the reliability of a path element). The “OD number of elements” or the “OD length”, a new concept, is defined as the opposite of the OD reliability logarithm, divided by the failure rate. When several paths connect an OD, the OD length is less than the length of the shortest path –the reduction in length characterizing the contribution of alternative paths. This extends to the average length over all connected ODs, provided the geometric average is taken into account.

The “standard indicator” is based on the ratio {actual number of kilometers/Euclidean distance} of a path, i.e., its sinuosity. For an OD, the more alternative paths there are in time or space, the higher the standard indicator, the less “OD sinuosity”- This new concept makes it possible to compare the equity of transport supply on different ODs, whatever their Euclidean length, or even to compare different networks. Service improvement projects can be compared on the basis of their performance against this indicator. Fleet managers could define an expected quality of service per OD by a threshold on “OD sinuosity”, and select the project that leads to the largest number of ODs below this threshold (or to the largest number of users concerned, when the demand per OD is known).

Depending on the type of failure, reliability depends on the availability of links, buses, bus stops, or of the successive kilometers of the path. Then, the shortest path is either the shortest path in terms of number of links, buses, stops or length.

In Greater Rennes area (Brittany) the shortest paths connecting the ODs have an average length of 14.54 kilometers. The increased reliability obtained with the alternative paths is that which would have been obtained with a single path reduced in average, to a length of 10.91 kilometers (calculated with a reliability probability of 0.96 per kilometer). While the actual average sinuosity of the shortest paths is equal to 1.65, it is reduced to 1.22 with the alternative paths.

Unreliability is defined here as one minus reliability. The geometric mean of OD unreliability is decomposed into a product of factors, each factor being the unreliability contributed by a path in an OD. This is valid even when paths overlap, thanks to a new concept, the “degree of independence” (between 0 and 1 for each path), which measures the overlap between paths. The more overlaps there are, the lower the degree of independence. The “degree of equivalence”, combines the degree of independence with the possibly lower reliability of the alternative path (compared with the shortest path). In the application, there were, in average, 2.48 paths (including overlaps) per connected OD, but the sums of the degrees of equivalence, was only, in average 1.42.