TAG Unit 3.5.7: The Reliability Sub-Objective

July 2008

1. The Reliability Sub-objective
   1.1 Introduction

2. Measures of Reliability

3. Private Vehicle Travel
   3.1 Introduction
   3.2 Inter Urban Motorway and Dual Carriageway Variability
   3.3 Urban Road Variability
   3.4 Other Road Types

4. Public Transport
   4.1 Valuation
   4.2 Theory and Practical Constraints
   4.3 Consequences of Delay from a Rail Perspective
   4.4 Variation in Delay
   4.5 Lateness Factor
   4.6 Early Arrivals
   4.7 Reliability

5. References

6. Further Information

7. Document Provenance

8. Annex A - General Principles

9. Annex B - Highway Reliability in Urban Areas Approach

10. Annex C - Local Survey for the Calibration of Urban Variability Models

11. Annex D - Calculating Averages and Variance

12. Annex E - Highway Reliability - Incident Assessment Approach using INCA

13. Annex F - The Stress Based Approach to the Assessment of Reliability Impacts of Road Proposals

1. The Reliability Sub-objective

1.1 Introduction

1.1.1 In this sub-objective, the impact of a proposal on improving journey time reliability for transport users is assessed.

1.1.2 The term reliability will be used in this document to refer to travel time variability or journey time variability.

1.1.3 This Unit represents the current state of knowledge. This is a rapidly developing area where we are likely to learn from further research .The recommendations on reliability set out in this Unit are made within the context of what is achievable using the existing knowledge base.

1.1.4 Section 2 discusses the measure of reliability for different modes. The appraisal of travel time variability is described in section 3. Section 4 looks at the appraisal of public transport. A discussion of the theoretical basis for the valuation of reliability is given in Annex A. The approach used to appraise reliability in urban areas is described in Annex B. Annex C discusses the collection of local data to evaluate travel time variability. The calculation of mean and variance of travel time is explained in Annex D. Incident assessment using the updated INCA software is described in Annex E and finally, the stress based approach which is used in appraising reliability on single carriageways is described in Annex F.

2. Measures of Reliability

2.1.1 Reliability is defined as variation in journey times that drivers are unable to predict. Hence reliability is confined to random effects. It arises from either variability in recurrent congestion at the same period each day - Day to Day Variability (DTDV) or variability in non-recurrent congestion such as incidents. It excludes predictable variation relating to varying levels of demand by time of day, day of week, and seasonal effects which travellers are assumed to be aware of.

2.1.2 Hence any calculations of changes in reliability must first remove the effects that are attributable to predictable variation. Once we are left with only unpredictable variation, the appraisal of transport schemes or policies should aim to place a value on any changes to this journey time variability because of the extra costs it incurs on drivers and passengers.

2.1.3 Current theoretical approaches (see Annex A) suggest that slightly different definitions of 'reliability' are used for public transport and private vehicle travel.

2.1.4 For most public transport journeys, the existence of timetabled arrival times means that it is usual to consider reliability in terms of lateness, defined as the difference between travellers' actual and timetabled arrival times. Adopting this definition means that arrival before the timetabled arrival time is usually ignored. This is based on the assumption that the operation of public transport generally acts to avoid early arrival. Two measures of lateness must be considered: average lateness; and the variability of lateness, measured by the standard deviation of lateness.

2.1.5 For journeys by private road vehicles (including road goods vehicles), it is reasonable to expect travellers to be aware of the average journey time, including variations caused by factors such as different traffic conditions at different times of the day. Thus reliability should be measured in terms of the unpredictable variability in travel times about these averages, measured by the standard deviation of travel time.

2.1.6 To estimate the monetised benefit of changes in the variability of lateness (for public transport) or of journey time (for private road vehicles), money values are needed. The concept of the reliability ratio enables changes in variability of lateness or of journey time to be expressed in monetary terms. The reliability ratio is defined as:

Reliability Ratio = Value of SD of travel time / Value of travel time
or
Reliability Ratio = Value of SD of lateness / Value of lateness

2.1.7 In addition, for public transport, the calculation of the monetised benefit of changes in average lateness also requires suitable money values. The value of average lateness may also be expressed in relation to the value of travel time:

Value of lateness = factor * value of travel time

Note that it is possible to estimate the benefits of changes in average lateness without also estimating the benefits of changes in the variability of lateness.

2.1.8 Evidence on values for these measures is of variable quality. However, some broad conclusions presented in the PDFH^[1] can be drawn, as follows: the value of average lateness for public transport is broadly the same as the value of time spent waiting for public transport, that is, 2.5 the value of in-vehicle time; the value of the reliability ratio ranges from 0.6 to 1.5 for public and about 0.8 for private passenger travel.

2.1.9 In order to use these results it is necessary to be able to specify how a transport intervention will affect public transport lateness and/or private vehicle reliability.

2.1.10 For multi modal studies, highway and public transport reliability should be appraised separately, employing the methods currently available for each mode.

2.1.11 What is therefore required on the highway side to assess reliability is:

• to model the impact of incidents on average journey time, and
• to model the level of variability (remaining with the approach of representing this by the standard deviation of travel time) associated with both incidents and DTDV.

How this can be done in practice is discussed in Section 3.

2.1.12 For public transport and heavy rail in particular, it is certainly the case that standard modelling will make use of scheduled journey times (though the treatment of waiting time might be done differently). The implication therefore is that in respect of reliability both the mean extra time and the standard deviation need to be modelled.

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Footnote 1: PDFH is a technical document, summarizing research on the various factors affecting forecasts of demand for passenger rail services, published by the Passenger Demand Forecasting Council. It is not a public document and is only available on subscription from the Association of Train Operating Companies.

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3. Private Vehicle Travel

3.1 Introduction

3.1.1 A key element used in a highway scheme appraisal is the expected average journey time on the day of the week and at the time of day in question. This can be complemented by an assessment of reliability, which may reflect:

• the consequences for subsequent activities should unexpected variability arise;
• the likelihood of encountering an incident which reduces capacity and
• other implicit effects which cause unreliability and variability in the average journey times.

3.1.2 Reliability can be measured by the standard deviation of journey time at a given time of day, or by the coefficient of variation (CV) which is defined as the ratio of the standard deviation of journey time to the average journey time. Either measure can be used when appraising proposals to improve the reliability of private vehicle travel.

3.1.3 Typically, road journeys that are repeated on a number of occasions are likely to take slightly less time than the average with a small number of trips encountering significant delays. The latter have a disproportionate impact on the standard deviation. The following boxed example (also used in Annex C) demonstrates this point through the use of a simplified example.

3.1.4 The Reliability Ratio expresses the value of variability of journey times, measured using the SD of journey time at a particular time of day, in comparison to the value of journey time. There is a limited amount of evidence on the values to be applied to the standard deviation of travel time - the 'value of reliability'. This comes from a variety of sources, but has usefully been pulled together through a workshop arranged by the Netherlands Ministry of Transport (The value of Reliability in Transport, 2005). Using the value of time in Unit 3.5.6, the value of the standard deviation of journey times can be calculated using the recommend reliability ratio values below:

3.1.5 The way in which the change in the level of JTV is forecast will, in the light of current knowledge, vary according to the context. Different methodologies have been developed for inter urban motorway and dual carriageway roads, urban roads, and other roads, as discussed below. In appraising travel time reliability on highway schemes, it is important to distinguish whether the scheme being appraised is an Urban Road (defined usually as having a speed limit of 30 or 40 miles per hour) or Inter Urban Road (which usually have a speed limit of 50 plus miles per hour). On Inter Urban Roads (taken to include all Highways Agency roads) it is also important to further distinguish between Motorway roads; Dual carriageway roads and single carriageway roads.

3.2 Inter Urban Motorway and Dual Carriageway Variability

3.2.1 Research (Arup, 2004) has shown that as long as demand is below capacity, incidents will be the main source of JTV, and DTDV is much less important except in urban areas where the two effects cannot be readily separated. The additional delays caused by congestion unrelated to incidents and any associated variability can be assumed to be allowed for in the journey time forecasts. However, in the case of delays due to incidents a separate element for average delays will usually need to be added to the variability element.

3.2.2 The research led to the development of a computer program INCA and associated advice note, which provides details of how to estimate the monetised benefits of measures affecting journey time variability covering incidents on motorways and dual carriageways. The model derivation assumes a dual carriageway layout and the parameters are based on data for motorways only. It is therefore not suitable for single carriageways, though the model may be used for dual carriageways as well as motorways. The resulting estimates of benefits cannot be taken to be as robust as those for time savings or accident reductions, for example. However, it is thought that a monetary benefit estimate will be of more value to decision makers than the qualitative score that can be presented in the Appraisal Summary Table. INCA reflects how delays caused by incidents vary according to the severity and length of the incident, the number of lanes blocked and the volume of traffic at the time. Changing the number of lanes available to traffic changes both the probability of encountering an incident (or its aftermath) and the delays caused by incidents.

3.2.3 For motorways and dual carriageways, alternative routes avoiding particular sections usually have limited capacity making it difficult for large numbers of drivers to divert if they encounter delays due to an incident. In the absence of significant "transient excess demand" (temporary periods of demand exceeding capacity), incidents are the main source of unpredictable variability and the methods set out in the INCA advice should be used. However, it is important to note that the research underlying this advice currently incorporate what are intended to be conservative assumptions, which will be refined in due course. Benefits should be calculated by INCA only for motorways and dual carriageways and should always be identified separately from other economic benefits in the Appraisal Summary Table (see the appraisal process Unit 2.5). The total benefits figure in the INCA final summary report should be reported in the assessment column of the Appraisal Summary Table under the reliability sub-objective.

3.3 Urban Road Variability

3.3.1 Models predicting journey time variability from all sources have been developed for urban areas. In such areas alternative routes are more readily available than on motorways and there are many ways for drivers to divert away from incidents which reduce capacity on a particular route. This affects the relative importance of incident and day to day variability (DTDV) effects.

3.3.2 Initially models were developed for the London Congestion Charging study in 1993. These were modified using additional data collected in Leeds (2003) with these improvements reported in Arup (2004). In 2007, Hyder Consulting in collaboration with Ian Black and John Fearon were commissioned by the DfT to further develop the travel time variability relationships for a wider sample of urban routes. Theses routes are spread over the 10 largest urban areas in England as identified in DfT's Public Service Agreement (PSA). The analysis in the research was based on ITIS/CJAM (Congestion and Journey-time Acquisition and Monitoring System). See Annex F for more details on the analysis. The form of model developed forecasts the Standard Deviation of Journey Time from Journey Time (t) and Distance (d) for each origin to destination flow. Under the further assumptions that distances and free-flow speeds do not change as a result of the scheme, the change in journey time variability (represented by Δσ_ij) is given by:

Δσ_ij = 0.0018 (t _ij2 ^2.02 - t_ij1 ^2.02) d_ij ^-1.41

where

• t_ij1 and t_ij2 are the journey times before and after the change for the journey from i to j (seconds)
• Δσ_ij is the change in standard deviation of journey time for the journey from i to j (seconds)
• d_ij is the journey distance from i to j (km)

The reliability benefit applying the rule of a half is therefore calculated using:

Note that the value of reliability (VOR) is obtained by multiplying the value of time by the reliability ratio and T_ij1 and T_ij2 are number of trips before and after the change.

3.4 Other Road Types

3.4.1 For journeys predominantly on single carriageways outside urban areas, it is not currently possible to estimate monetised reliability benefits. Instead, the assessment of changes in reliability should be based on changes in 'stress', the ratio of the annual average daily traffic (AADT) flow to the Congestion Reference Flow (a definition of capacity). Reliability of road journey times is believed (on the basis of work carried out for DfT's ITEA Division) to decline as flows approach capacity. Thus, 'stress', is, with some limitations, considered to be a reasonable proxy for reliability. Detailed advice on stress, including the definition of Congestion Reference Flow, is provided in DMRB Vol 5.1.3. The method to be used is described in detail in Annex F below.

3.4.2 A worksheet is provided so that values for improved reliability can now be calculated and results presented in a consistent manner.

3.4.3 Experience to date has shown that schemes which reduce congestion, by reducing the journey time spent queuing, also reduce the variability of journey times. The effect is largest on motorways operating nearest capacity. The effect is less in urban areas, because more alternative routes are available in urban areas.

4. Public Transport

4.1 Valuation

4.1.1 It is likely that some passengers, particularly infrequent travellers, will be unaware of predictable variation. Indeed evidence from the Passenger Demand Forecasting Handbook (PDFH) for rail suggests that only 25% of passengers are aware of advertised delay^[2]. For those passengers unaware of this predictable delay the increase in journey time can be treated as unpredictable variation.

4.1.2 This is equivalent to a lateness factor of 2.5 for all advertised delay. This is because the predictable and unpredictable lateness factors (1 and 3 respectively), when weighted according to the 25/75 split, produce an average lateness factor of 2.5. In the absence of similar evidence for private travel the following recommendation applies only to rail travel.

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Footnote 2: This has been generalized from research carried out by SDG in 1995 on delays caused by advertised engineering works.

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4.2 Theory and Practical Constraints

4.2.1 For projects where passenger performance improvements are to be fully assessed, a detailed reliability assessment should provide evidence on how improvements in reliability are to be achieved and the expected consequential effects within the principles set out below.

4.2.2 Due to the existence of timetables for public transport the analyst should focus on delay, rather than journey time in appraising reliability. This practical constraint comes from the rail industry performance data and not from something inherent in all public transport. For rail we use the term performance to capture the impact of both punctuality and reliability. Punctuality concerns whether trains arrive at their scheduled time. When a train fails to run, or does not stop at all its scheduled destinations, it is said to contribute to unreliability (the rate of cancelled trains.

4.2.3 In contrast to the highway case, where the traveller has the possibility of continuous adjustment of his departure time, most public transport is characterised by the existence of a timetable, with only discrete possibilities for departure. As can be expected, this leads to further disutility associated with the service interval. The disutility resulting from discrepancies between Preferred Arrival Time (PAT) and scheduled arrival time would exist even with 100% reliability, therefore the costs associated with discrete departure times can effectively be ignored when valuing the effects of a change in reliability. When public transport timetables do not correspond well to the timing of an individual user's activities (i.e. their PAT), some travellers may actually find the variation beneficial.

4.2.4 Ideally valuation of reliability would be based on the difference between the passenger's PAT and their actual arrival time. However we have very little knowledge about passengers' PATs. For this reason we adopt the scheduled arrival time of a service as a proxy for the passenger's PAT. It is the difference between scheduled arrival time and actual arrival time which is used to measure a passenger's delay.

4.3 Consequences of Delay from a Rail Perspective

4.3.1 Disutility from unpredictable variation has two elements. The first and most significant element is the consequences of arriving late (missing meetings, unproductive use of time, etc). This is measured using mean delay and valued by applying the appropriate lateness factor. Mean delay cannot be interpreted as a component of journey time because of its uncertain nature. Mean delay should contain only unpredictable delay. Any predictable delay should be removed from the calculation and treated as additional journey time.

4.3.2 Research has assessed the perceived weighting of lateness relative to in-vehicle time in order to place a value on the consequences of arriving late. Estimates of the value of this lateness factor vary between 1 and 5, according to a number of variables such as the type of service and the journey purpose.

4.4 Variation in Delay

4.4.1 The second element of disutility arises from the additional cognitive burden of unpredictable variation and can be described as the intrinsic irritation factor arising from variation in arrival times. This is most accurately measured using the standard deviation of journey time and valued according to the appropriate reliability ratio.

4.4.2 The reliability ratio allows us to represent the variability of delay, measured by the standard deviation of delay, in terms of an equivalent change in mean delay. The recommended reliability ratio values are shown below:

4.4.3 If the reliability ratio has a value of, for example 0.5, then a 1 minute reduction in the standard deviation of delay is equivalent to a 0.5 minute reduction in mean delay.

4.4.4 Given that it is rare that we ever have a complete knowledge of the delay distribution with which to calculate the standard deviation of journey time an alternative method can be used. Research by Bates et al^[3] has suggested that it is the "pure" lateness effect which tends to dominate the calculations, because the effect of variability is less important given that rail passengers have already made some "compromises" in selecting arrival or departure time of their preferred scheduled train. Indeed, as noted earlier, some travellers may find that variability brings them closer to their preferred arrival time than an "on-time" arrival would. Consequently a 20% uplift of the lateness factor is an acceptable proxy for the additional disutility incurred as a result of variability in delay.

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Footnote 3: 'The Valuation of Reliability for Personal Travel' by Bates et al, Transportation Research Part E 37 (2001).

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4.5 Lateness Factor

4.5.1 Therefore a central lateness factor of three, which includes the uplift of 20% for a change in variability, should be used in the general case. Where sufficient evidence can be provided to justify the application of a different lateness factor a value higher or lower than 3 should be adopted. In the general case one minute of average lateness is valued by passengers as being equivalent to three minutes of scheduled journey time. This conversion to scheduled journey time allows us to place a monetary value on reliability using the appropriate value of time.

4.5.2 Where no delay data is available for an intermediate station the analyst should use delay data from the final destination. In this case it may be appropriate to use a different lateness factor. But a robust rationale should be provided for any departure from the recommended central factor of 3.

4.6 Early Arrivals

4.6.1 The theory of Bates et al states that early arrival contributes to variability. They recommend that early arrival is given the same weight as late arrival but the opposite sign. Early arrivals are recorded but not included in the Public Performance Measure (PPM). It is therefore recommended that early arrivals are treated as on time and, as a result, excluded from calculations of mean delay and variance of delay. It is recognised that this is not the ideal theoretical approach, or the method outlined by Bates et al, but that it represents a pragmatic approach.

4.7 Reliability

4.7.1 A measure of rail performance must also examine the rate of cancelled services or reliability. To make allowance for the total lateness caused by cancelled trains we usually multiply the service interval by 1.5. This cancellation factor is in line with the notion that in this case the delay impacts on waiting rather than in-vehicle time. Waiting time incurs higher disutility than in-vehicle time because of the additional discomfort involved. The resulting lateness should then be multiplied by the lateness factor of 3 to capture the full costs of poor performance.

5. References

Arup, Bates, J., Fearon, J. and Black, I. (2004) Frameworks for Modelling the Variability of Journey Times on the Highway Network. Department for Transport, UK.

Bates, J., Polak, J., Jones, P and A. Cook (2001) 'The Valuation of Reliability for Personal Travel', Transportation Research Part E 37.

DMRB Vol 5. HMSO, August 1997.

Hamer, R., De Jong, G., Kroes E and P, Warffemius (2005), The Value of Reliability in Transport.

Hyder Consulting, Fearon, J. and Black, I. (2007) Forecasting Travel Time Variability in Urban Areas. Department for Transport, UK.

Passenger Demand Hand Forecasting Handbook (PDFH).

6. Further Information

The following documents provide information that follows on directly from the key topics covered in this TAG Unit.

7. Document Provenance

This Transport Analysis Guidance (TAG) Unit is new guidance.

Technical queries and comments on this TAG Unit should be referred to:

Integrated Transport Economics and Appraisal (ITEA) Division
Department for Transport
Zone 3/06 Great Minster House
76 Marsham Street
London
SW1P 4DR

E-mail: itea@dft.gsi.gov.uk
Tel: 020 7944 6176
Fax: 020 7944 2198

8. Annex A - General Principles

8.1.1 Travel time variability (TTV) or Journey time variability (JTV) is defined as variation in journey times that travellers are unable to predict. Since the essence of any measure of variability (such as variance) relates the variations to the expected value, alternative definitions of the expected value will clearly have an impact. A failure to clarify this point in the past has led to much confusion of measurement. In general, it is sensible to remove as far as possible any non-random effects. The term travel time variability and journey time variability will be used interchangeably throughout this guidance as they both mean the same thing.

8.1.2 Travel time variability is an increasingly important issue. Travellers are sensitive to the consequences, such as prolonged waiting times, missed connections and arrival at the destination either before or after the desired or expected arrival time. This leads to an analysis in which the traveller is conceived of as choosing between travel alternatives each of which is characterised by a distribution of consequences, defined in terms of conventional generalised cost components (cost, travel time, etc.), together with the impact on timing constraints. The decision rule used to resolve choice in this situation has generally been some version of "maximum expected utility" (MEU) theory, in which travellers are assumed to choose the travel alternative that maximises the expected value of an appropriate defined utility function.

8.1.3 Within the transport field, the impact of travel time variability is primarily on departure time. The framework in general has been related to highway mode but can be expanded to take in the additional complexity of scheduled public transport services. The theory assumes that travellers choose the course of action which, bearing in mind the probabilities of different outcomes, has the highest value of expected utility.

8.1.4 The major source of the disutility associated with travel time variability is scheduling cost. Analysis is based on the model due to Small (1982) which specifies the following utility/generalised cost function.

U = β₁C + β₂SDE + β₃SDL + β₄D_L               (1)

Where

C is the travel time

SDE Schedule delay early is the amount of time one arrives early at the destination

SDL Schedule delay late is the amount of time one arrives late at the destination

DL = 1 for late arrival, 0 otherwise

8.1.5 SDE and SDL are defined with respect to a preferred arrival time (PAT), normally defined as the start time of an activity (e.g., work start time).

8.1.6 Noland and Small (1995) further developed the scheduling cost model to take travel time variability into account. This led to the following model, independent of the distribution of travel times:

U = β₁E(C) + β₂E(SDE) + β₃E(SDL) + β₄P_L               (2)

Where

E[X] is the expected value (mean) of X

PL is the probability of arriving late

8.1.7 If there is travel time variability then, with the reasonable assumption that β₂ < β₃, there is a need to allow a certain amount of slack time when choosing departure time in order to maximise expected utility by reducing the risk of late arrival and more importantly the probability of being late.

8.1.8 It has been shown empirically that if travellers are able to optimise their choice of departure time on a continuous basis, the sum of the terms [β₂ SDE + β₃ SDL] is closely related to the standard deviation of travel time. This provides some justification for the widespread use of standard deviation as the relevant component in the utility function to indicate the effect of travel time variability. Strictly speaking, however this relies on departure time being continuously variable (as with the car mode).

8.1.9 Most of public transport is characterised by the existence of a timetable, with only discrete possibilities for departure. As can be expected, this leads to further disutility associated with the service interval. The utility theory framework can be expanded to combine the continuous analysis and service interval analysis at some increase in complexity. For each advertised departure time, we can estimate the expected utility of travelling on that service. We then choose that departure time from the discrete set of services available that delivers the greatest expected utility.

8.1.10 While the underlying theory is compatible, the need for rail appraisal to take explicit account of the average delay relative to scheduled time tends to dominate the calculations, both because this delay appears to attract a greater level of disutility than would a corresponding increase in scheduled time, and because the effect of variability per se is less important in the light of the scheduling "compromises" which rail passengers have to make in any case. A further practical difference is the PDFH recommendation to ignore the effect of early arrivals.

8.1.11 For a fuller description of the of the theoretical background see Bates et al,.(2001). A discussion of the translation of theory into practical methodology for highway can be found in Arup (2004) and PDFH for public transport.

9. Annex B - Highway Reliability in Urban Areas Approach

9.1.1 In urban areas alternative routes are more readily available than on Motorways and there are many possibilities for avoiding incidents which reduce capacity on a particular route. This avoidance behaviour contributes to the day to day variability on the alternative routes and affects the balance between incident and day to day variability effects. Models predicting journey time variability from all sources are therefore the most relevant and prototype models using congestion indices were developed as part of the London Congestion Charging study in 1993.

An improved form of those models based on north London data was developed using additional survey data collected in Leeds (2003) as set out in Arup (2004). In 2007, Hyder Consulting in collaboration with Ian Black and John Fearon were commissioned by the DfT to further develop the travel time variability relationships for a wider sample of urban routes. Theses routes are spread over the 10 largest urban areas in England as identified in DfT's Public Service Agreement (PSA). The improved model is now available as set out below. Its derivation is set out in Hyder, 2007.

The recommended form of model forecasts the Coefficient of Variation (CV) from Distance (d) and Congestion Index (ci) terms for each origin to destination flow in the urban area. The Coefficient of Variation (CV) is the ratio of the standard deviation of travel time to the mean travel time.

CV = 0.16 ci ^1.02 d ^-0.39

9.1.2 The Congestion Index "ci" is defined as the ratio of mean travel time to free flow travel time, so that the model can be rearranged to forecast the Standard Deviation of Journey Time from Journey Time (t) and Distance (d). The areas on which the relationship was based comprised average free flow speeds of 37 to 47 kph (km/hr)^[4]. Using a constant average free flow speed of 44.5 kph and expressing this as 0.01236 km per second, the change in journey time variability (represented by Δσ_ij) is given, if distances do not change, by:

Δσ_ij = 0.0018 (t _ij2 ^2.02 - t_ij1 ^2.02) d_ij ^-1.41

where

• t_ij1 and t_ij2 are the journey times before and after the change for the journey from i to j (seconds)
• Δσ_ij is the change in standard deviation of journey time for the journey from i to j (seconds)
• d_ij is the journey distance from i to j (km)

The reliability benefit applying the rule of a half is therefore calculated using:

Note that the value of reliability (VOR) is obtained by multiplying the value of time by the reliability ratio and T_ij1 and T_ij2 are number of trips before and after the change.

9.1.3 The above model form implies lower standard deviations for higher speed journeys. The relationship may need to be recalibrated using local surveys where the variability benefits are crucial. Advice on the surveys needed to provide a local calibration of such models is provided in Annex C.

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Footnote 4: For consistent units in the equation the speed must be defined in terms of km per second.

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10. Annex C - Local Survey for the Calibration of Urban Variability Models

10.1.1 In 1993 a journey time survey in north London was conducted using the 'floating car' method, to produce a recommended relationship for forecasting Journey Time Variability (JTV) for the London Congestion Charging study. In 2002 Arup, Bates et al reviewed the research, re-analysed the data, and developed a new relationship incorporating data from an equivalent survey in Leeds. However, Hyder Consulting in 2007 re-evaluated the models using ITIS/CJAM data from the 10 largest urban areas in England as identified in DfT's Public Service Agreement Target. ITIS/CJAM collates and processes data provided by individual probe vehicles and a number of fleet management and tracking service providers. The models developed forecasts JTV along the surveyed route, as a function of variables which are already provided as inputs to the standard economic appraisal program TUBA.

10.1.2 The Hyder et al model form can be used to estimate the effect of schemes and the order of magnitude of their variability benefits in urban areas. Although the model above can be used to estimate the effect of schemes and their reliability benefits in urban areas, a locally calibrated model or at least a local validation is preferable.

10.1.3 Where greater accuracy is required, and an appropriate circuit can be identified, data from established sources such as HATRIS and ITIS or a local survey similar to the Arups work and a locally calibrated model should be considered. The resulting data should be analysed to establish whether the relationship, which Arup, Bates et al developed from the Leeds and London data, is applicable or whether different parameters or in extreme cases different relationships should be used. Further guidance on this is available from DfT's ITEA Division.

11. Annex D - Calculating Averages and Variance

11.1 Private Vehicle Travel

11.1.1 Journey times vary due to a large number of factors including the time of day, the location of the origin and destination, the distance and the road or service types along the route. Such systematic variation has no relevance for JTV (except possibly where travellers making a "new" journey base their expectation of journey time on other journeys that they consider "similar").

11.1.2 JTV arises from unpredictable variation, and can occur on journeys by any mode. On the rail side, all variation arises from what are effectively operational anomalies. On the highway side, unpredictable variation arises from Day-to-day variability "DTDV", incidents and operational effects which cause anomalies for bus services.

11.1.3 The reliability of a journey to work by road, which normally takes 30 minutes but typically encounters delays of 20 minutes on one random weekday and 10 on another each week, can be derived by the following set of equations:

Where is the average journey time, x_n is the travel time on day n and n is the number of days used in the analysis.

Hence, Average journey time = 30*3/5 (3 normal days) + (30+20)/5 (long delay) + (30+10)/5 (shorter delay) = 36 minutes per trip.

The variance in the journey time is calculated by examining the average^[5] of the sum of the squares of the difference from the mean. This is as follows:

Hence Variance σ² is 1/5 ((50-36)² + (40 - 36)² + (30 - 36)² + (30 - 36)² + (30 - 36)²) = 64 minutes squared (of which 39 minutes squared (i.e. (50-36)²/5) comes from the longest delay).

The currently recommended measure of reliability, the standard deviation, equals 8 minutes per trip (the square root of the variance).

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Footnote 5: If the pattern under consideration is based on only a small number (n) of observed journey times when calculating variances the average of the squares of the difference from the mean should be multiplied by a factor n/(n - 1).

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11.2 Rail

11.2.1 While the basic results for a similar journey by rail are identical, the existence of a scheduled time (according to the rail timetable) means that we can also calculate average "lateness". Suppose the timetabled journey time is in fact 35 minutes. Then the "normal" journeys lasting 30 minutes will arrive early. The calculations will be different according to whether early arrival is treated as a) negative lateness or b) "on time" arrival. In the former case, we have

Average lateness = - 5 *3/5 (3 "normal" days: early arrivals) + (20 - 5)/5 (long delay) + (10 - 5)/5 (short delay) = 1 minute per trip.

As before, the variance turns out to be 64 minutes squared.

11.2.2 In the latter case, however, where we measure the lateness of early arrivals as zero, we have

Average lateness = 15/5 (long delay) + 5/5 (short delay) = 4 minutes per trip.

Variance of lateness = (15)² / 5 + (5)² /5 - mean lateness² = 45 + 5 - 16 = 34 minutes squared.

11.2.3 The method set out in paragraph 11.2.2 is recommended as it represents a pragmatic approach.

Worksheet D1.1 - Calculation of mean and variance of lateness (based on one week^[6])

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Footnote 6: While the illustration only shows one week, several weeks observations should be used of all journeys operated in the chosen period.
Footnote 7: If the pattern under consideration is based on only a small number (n) of observed journey times, when calculating variances the average of the squares of the difference from the mean should be multiplied by a factor n/(n - 1).

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12. Annex E - Highway Reliability - Incident Assessment Approach using INCA

12.1.1 The INCA software can be obtained by emailing INCA Support at inca@mottmac.com.

12.1.2 The INCA software automatically provides a summary of the INCA results. The table which is produced shows the total delays and journey time variability benefits which are also broken down by incident type.

12.1.3 INCA carries out economic appraisal for a 60 year horizon using a method similar to that used in TUBA software with the number of modelled years limited to a maximum of six appraisal years, discounting the results to 2002 base year and then aggregating them for the final result worksheet. Data for the years between two modelled years is linearly interpolated between those two years. For years before the first modelled year it is extrapolated from the first two modelled years. Beyond the last modelled year, data is extrapolated using a horizontal line.

12.1.4 The final result worksheet produces 3 reports. Report 1 contains a summary of the scenario being modelled. Report 2 contains a summary of incident benefits (delay and JTV) by flow group and incident type. Report 3 contains the incident delays and travel time variability, identifying benefits by incident type.

13. Annex F - The Stress Based Approach to the Assessment of Reliability Impacts of Road Proposals

13.1.1 The stress based approach is only appropriate where the other approaches described above are not feasible. The change in stress is essentially a proxy for change in reliability. The approach does not provide a direct quantification of changes in reliability or reliability benefits. In addition, it is not a precise or comprehensive method and can only provide a very broad indication of the impact of a proposal on reliability.

13.1.2 This approach is based on the change in 'stress' (within the range 75% to 125%) as a result of the proposal, combined with the numbers of vehicles affected. Stress is the ratio of counted or measured annual average daily flow to the congestion reference flow. Where a proposal provides a new route, the approach takes account of improvements in reliability for those remaining on the old route as well as those transferring to the new. This approach is very similar to that taken in assessing time saving and vehicle operating cost benefits. Thus, proposals providing modest improvements for large volumes of traffic may be more highly rated than those providing large improvements for small volumes.

13.1.3 To take account of possible 'bottleneck' effects, where the effect of one link or junction operating close to capacity affects the reliability of an extended length of road, the method focuses on those key links/junctions, rather than the whole length of road.

13.1.4 Referring to the worksheet below, the following information needs to be provided, all for the year in which the proposal is implemented; for the key link on the existing road (the 'old route'):

• the percentage stress in the do minimum and do something scenarios - these may differ because the flow changes (if the proposal is a bypass, for example), because the Congestion Reference Flow changes (if the proposal is an on-line improvement, for example), or both (if the proposal is a bypass accompanied by traffic management on the old route, for example); and
• the do something annual average daily traffic flow;

and, where a new route is provided by the proposal, for the key link on the new route:

• the percentage stress in the do something scenario (clearly, there cannot be a new route in the do minimum scenario); and
• the do something annual average daily traffic flow.

The percentage stress in the do minimum and do something scenarios should be entered in the Quantitative column of the Appraisal Summary Table. Where the proposal provides a new route, the value for that route should be used.

13.1.5 The difference in stress should be calculated for the old and new routes (where appropriate). Note that the same do minimum value should be used for both calculations. If any stress value is less than 75% or greater than 125%, the calculation should be based on values of 75% or 125% as appropriate. The assessment for each route is the product of flow and difference in stress. These results are summed to provide the overall assessment.

13.1.6 Thus, it is not appropriate to present the numeric result of the calculations outlined above. Instead, the result should be used to assist in reaching an appropriate textual score, using the following guidelines:

• Values in excess of 3 million will usually be assessed as Large (Beneficial if the value is positive, Adverse if it is negative) - these will be high flow routes with moderate or large differences in stress, or moderate flow routes with large differences in stress;
• Values between 1 and 3 million will usually be assessed as Moderate - these will be high flow routes with small or moderate differences in stress, moderate flow routes with moderate differences in stress, or low flow routes with moderate or large differences in stress;
• Values between 200 thousand and 1 million will usually be assessed as Slight - these will be high and moderate flow routes with small differences in stress, and low flow routes with moderate differences in stress;
• Values less than 200 thousand will usually be assessed as Neutral.

13.1.7 Other considerations may justify a different assessment - they should be noted in the Qualitative assessment. For example, the performance of junctions is not included in the measure of stress.

13.1.8 This approach is not suitable for proposals affecting junctions alone. Nevertheless, such proposals on roads carrying large volumes of traffic may make a substantial contribution to reliability. In addition, the approach is not suitable for estimating changes in reliability during construction and maintenance. Where either of these considerations apply, a comment should be made in the Qualitative column, entering 'not applicable' in the Quantitative and Assessment columns.

Worksheet 1 - Economy: Reliability

Note: Where a new road route is provided, the Quantitative column should contain values a(i) and b(ii). Where no new road route is provided, use values a(i) and b(i).

Reference sources: _________________________________________________
Assessment scores: ________________________________________________
Qualitative comments: _______________________________________________