**1. Introduction**

Reliability is a major criterion for assessing selected elements of technical infrastructure such as transmission [1], information technology (IT) [2] or energy [3] infrastructure. The reliability of road infrastructure is also the subject of many studies, because of the role the parameter plays in traffic performance [4,5] and the safety of road users [6–8]. In the case of road safety, speed tests and testing of speed's impact on road safety measures are very important. In the tests [9] floating car data were used to achieve the goal. Ensuring the reliability of road infrastructure at a level acceptable to road users is a key aspect of planning and design decisions [10,11].

There are not many reliability analyses of two-way highways in scientific literature. Instead, researchers focus mainly on dual carriageways, i.e., motorways [12–14], expressways [15–18] or other dual carriageways [19,20], inter alia, analyzing the impact of Intelligent Transport System (ITS) solutions [21–23]. In simulation analyses and field research [24], the impact of selected parameters on the level of service (LOS) under heterogeneous traffic conditions for a two-lane highway was identified. The work [25] also analyzed (LOS) on the basis of estimation of passenger car unit values. The research [26] also pointed out

**Citation:** Ostrowski, K.; Budzynski, M. Measures of Functional Reliability of Two-Lane Highways. *Energies* **2021**, *14*, 4577. https://doi.org/10.3390/ en14154577

Academic Editor: Stefania Santini

Received: 31 May 2021 Accepted: 16 July 2021 Published: 28 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the variability of traffic flow on individual road lanes. Some studies concern themselves with sections of motorways and expressways in urban areas [27–29]. Obtaining reliable data is an extremely important aspect of reliability studies. The work [30] presents various techniques for examining road traffic parameters. It compares the pneumatic tube detector method, video capturing method, moving observer method and the classic manual method. The studies [31] indicate the effectiveness of combining the moving observer method and digital image processing. The work [32] presents the effects of using stationary devices along the road to collect road traffic data. The research [33] provides an example of an effective use of video traffic monitoring. Modern techniques allow the use of Bluetooth technology to collect data on traffic parameters [34–36] and Lidar technology to collect data on road and its surroundings parameters [37–39].

Road traffic parameters depend on many factors, including the driver's psychophysiological characteristics, road and meteorological conditions [40]. A very important aspect influencing traffic conditions is constituted by road geometry, including the parameters of horizontal curves [41]. One factor related to driver behaviour is the distance between vehicles [42].

An example of research conducted on two-lane highways is provided by reliability analyses carried out on Poland's road network [43]. These studies were undertaken on higher standard roads managed by the General Directorate for National Roads and Motorways, with speed limits of 70–90 km/h, and a typical lane width of 3.5 m (with or without a hard shoulder). In Poland, these roads account for over 86% of all national roads, including motorways, expressways and accelerated main roads. According to the standards specified in the American method [44], these are first-class roads on which drivers expect travel speeds close to the speed limit. In Germany, a similar approach applies to roads with a similar function marked as EKLII and EKLIII [45].

The project [43] and work [14] also present studies on dual-carriageways, i.e., motorways, expressways and roads of lower technical class, on which speed limits in Poland are 140 km/h, 120 km/h and 100 km/h respectively. In Poland, motorways are roads of the highest technical standard, where traffic can be joined only through interchanges. In the case of expressways and other dual carriageways of the lower technical class, traffic can be joined through interchanges or through intersections (usually signalised).

The analyzed two-lane highways mainly support traffic functions typical of roads of higher technical classes, although they have a limited capacity (max. 3200 veh/h according to [44], approx. 2600 veh/h according to [45]) compared to high-speed roads (highways, expressways). Reliability, measured in terms of travel speed or time, is highly variable on the analyzed roads and depends on the time of day, day of the week or month. It also varies in the longer term (analysis by year). Therefore, it is necessary to identify the most important factors that influence their reliability levels and to indicate the best reference level for analyses conducted on two-lane highways.

The main aim of the analyses presented in the paper is to answer the research questions:


The answers will provide the foundation for an effective transformation of the existing road network, enabling the attainment of a standard of travel that will be acceptable to road users.

The paper is divided into several parts which include a review of literature providing a description of the reliability measures used, selected results of empirical studies conducted on Poland's two-lane highways, and reliability studies conducted on a selected road section where use is made of GPS data. At the end of the paper, conclusions are drawn and directions for further work are stated.

#### **2. Materials and Methods**

#### *2.1. Reliability Measures*

Travel time reliability depends on a benchmark and therefore has no fixed value. Its value is influenced by a number of factors of various origins [46,47], including traffic factors (traffic intensity, types of vehicles), geometry, road's location and type of surroundings, the knowledge of which is necessary in order to identify the reliability process, interactions between the variables and the correct interpretation of the results.

General factors influencing road reliability include:


Considering the above division, it is possible to introduce a classification [48] that assigns the indicated factors to three different groups (Figure 1) on account of:


**Figure 1.** Interactions between sources of failure.

The diagram in Figure 1 shows interactions between the main sources of failure and the relationship between demand (traffic) and supply (infrastructure). The authors of paper [71] point out that the indicated interactions are the main determinants of road functionality. Both demand and supply vary over time, as both traffic and road capacity are influenced by various deterministic and random factors. Weather conditions, especially adverse ones, such as prolonged rainfall, snowfall, etc., have a significant impact on drivers' behaviour. The research [72–74] shows that road and intersection capacity decreases in such conditions by as much as 20%. The type of road surroundings resulting from the road's location often translates into the type of trips [75], which also determines drivers' behaviour. In cities and agglomerations where short trips, mostly related to commuting to work, shops, schools, and other facilities, predominate, the behaviour and expectations of vehicle drivers regarding traffic conditions and network reliability are completely different from such expectations outside cities, or during long-distance travel [O5] spread over a longer period of time. In the worst case scenario, all of these variables may affect travel time reliability. This situation occurs in the common sections of all three circles.

Over the past few decades, many studies have been conducted in the USA on existing roads to describe the reliability of travel times. In the research into and evaluation of reliability, generally available models were used, their modifications were created or completely new solutions were developed. Table 1 [76] shows an example of the application of reliability measures in practice, i.e., it lists selected US transport agencies and identifies indicators used by them to describe the functional reliability of roads.

Current methods of analysis [61] can be divided into:


**Table 1.** Reliability metrics used by selected US transport agencies.


Below, the authors present selected methods of analysis, including their advantages and disadvantages.

#### 2.1.1. Statistical Methods

One of the oldest approaches to the description of travel time reliability used by Abishai Polus in 1979 [77] was based on a simple measure—standard deviation *δ*, showing the variability of the metric's value in relation to its mean value (Equation (1)). The author was one of the first to indicate the travel time variable as the best measure with which to describe a road's functional reliability.

$$\delta = \sqrt{\frac{1}{n-1} \cdot \sum\_{i=1}^{n} \left(t\_i - \overline{t}\right)^2} \tag{1}$$

where:

*n*—number of travels, *t*i—*i*-travel time, *t*—average travel time.

The author defined a road's functional reliability using a measure of variability, i.e., travel time variance [78]. The higher the variance, the less reliable the road (Equation (2)).

$$R = \frac{1}{\left[Var(x)\right]^{\frac{1}{2}}} = \frac{1}{\left[E(x^2) - (E(x))^2\right]^{\frac{1}{2}}} \tag{2}$$

where: *R*—road's functional reliability, *x*—reliability measure (in this case–travel time), *Var*(*x*)—variance of reliability measure, *E*(*x*)—expected value of reliability measure.

The simplicity of the approach accounts for its advantage, while its low usefulness is a disadvantage because in most cases the empirical distributions describing the variability of travel time are not symmetrical and show considerable skewness. However, this did not prevent the development of these methods, and in subsequent editions, recommendations for reliability were developed based on the ranges of skewness as presented in publications [12,13,18]. The value of the standard deviation was used to build subsequent measures, such as the time window (Equation (3)) and the coefficient of variability (Equation (4)).

$$Time\,\,window = \mathbb{F} \pm \delta\tag{3}$$

$$\text{Coefficient of variability} = \frac{\delta}{\overline{t}} \cdot 100\% \tag{4}$$

where: *t*— arithmetic mean of all travel times, *δ*—standard deviation of travel time.

The time window can have two values—one lower and another greater than the arithmetic mean by the value ±*δ*. The road user receives information about possible travel time discrepancies. In order to increase the scope of analyses, the *δ* may be multiplied. The travel time variation coefficient can be used to compare travel time variability between days, weeks, or road sections.

A measure that allows the comparison of traffic conditions between peak and off-peak times is the index of variation (Equation (5)).

$$\text{Index of qualitative variation} = \frac{V\_{1,+95} - V\_{1,-95}}{V\_{0,+95} - V\_{0,-95}}\tag{5}$$

where: *V*1,+95, *V*1,−95—upper and lower values between which there are 95% of travel times during the peak traffic period, *V*0,+95, *V*0,−95—upper and lower values between which 95% of travel times lie outside the peak hours.

The index of qualitative variation may apply to roads close to urban agglomerations, where increased traffic may occur because of urban traffic peaks. Due to the larger discrepancy between the peak and off-peak traffic, the value of the index is often greater than 1.0.
