CONVERTING STATIC TO DYNAMIC ASSIGNMENT MODELS ...

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Sean Carmichael. Hague Consulting Group. Frank Hofman,. Hank Taale ... Increasing computer power and continued developments in software have.resulted in ...
CONVERTING STATIC TO DYNAMIC ASSIGNMENT MODELS: PRELIMINARY FINDINGS Tom van Vuren, Sean Carmichael Hague Consulting Group Frank Hofman, Hank Taale Netherlands Ministry of Transport I.

INTRODUCTION

Congestion keeps increasing, and will most likely keep doing so for the foreseeable future. Together with a change in the public acceptance of the impacts of traffic congestion on many aspects o f life this has led to more prominent congestion-related transport policy objectives. The time dimension has entered the thinking framework, not only through the reaiisation that time-of-travel choice has been a neglected aspect of individuals' overall transport decision-making, but also through a desire to make better u s e of existing transport facilities over the whole day. Appropriate modelling teclmiques are required, and there is a growing aspiration to use dynamic rather than static network models for forecasting and assessment. (In addition to these within-day dynanfics, there is a growing interest in day-to-day dynamics, reflecting daily variations in demand mad supply conditions, learning processes, and longer term transport dynamics, acknowledging the different time scales of changes in e.g. route, mode, destination and location choice; these are outside the scope of this paper). Dynamic assignment packages have been around for two decades, and have been used sporadically for practical studies. Increasing computer power and continued developments in software have.resulted in easier access to these packages, whilst the wish to account for within-day dynamics has increased interest further. Parallel to this practical rising profile for dynamic network modelling, academic .research has continued to improve our understanding o f the defmitiun of dynamic equilibrium. In practical reality there are many existing static network models in operation, which represent a high level of investment (in terms of data and staff expertise). It would be desirable to use this investment "by converting static models to equivalent dynamic representations, aiming to maintain the main inputs and to stay close to the level of calibration achieved with the original, static model (at least at the time-aggregate level). Hence, conversion procedures are required. In addition, extra data inputs are required for the operation of dynamic network models. In particular, the fixed, hourly departure rate per OD pair, which suffices for static assignment models, must be broken down into a departure profile, so that the departure time process is properly reflected and the buildup and dissipation o f queues can be modelled in more detail.

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This paper reports on a projects that required conversion of static to dynamic representations, encompassing the QBLOK static assignment model (Bakker et al, 1994) and two dynamic models, CONTRAM (Taylor, 1990) and 3DAS (De Romph et al, 1992). The aim o f the paper is twofold: to discuss the conceptual and practical issues involved in the transfer from an existing static assignment models to equivalent dynamic representations, mad to serve as a resource paper for those embarking on such an exercise. This paper does not allow sufficient space for a detailed discussion of each of the assignment models involved. The references should be consulted for more comprehensive model descriptions. 2.

CONCEPTUAL CONSIDERATIONS

2.1

Levels of dynamics

The term dynamic assignment modelling is in use for a large number of assignment packages that, on closer inspection, have subtly different ways of accounting for the dynamics of traffic in calculating routes and flow patterns, and the resulting level of equilibrium between flows and travel times: • the dynamics of travel conditions during the peak on route choice may be represented, without aiming for an equilibrated traffic state For example, INTEGRATION (Van Aerde, 1995), DYNEMO (Schwerdtfeger, 1984) or METANET (Papageorgiou & Korsialos, 1998) estimate routes on the basis of current and future traffic conditions, without accounting for feedback oi changing conditions on choice. The resulting flow pattern is not in any kind of equilibrinm. • the dynamics of travel conditions may be taken into account, but their effects on route choice at equilibrium are only represented in a time-period aggregated way. For example, through time profiling of demand at origins and junctions en-route, TRIPS allows the modelling of changing queues and delays during the peak period, but their effect on route choice is only represented through the improved calculations o f average travei conditions during the whole period. The resulting flow pattern is a static equilibrium for the whole peak. • the dynamics o f travel conditions are represented and taken into account in route choice, which aims to achieve a dynamic equilibrium in which no-one can improve his/her travel time by changing route. Examples of such models are 3DAS and CONTRAM. This is the most complete way of dealing with dynamics in the assignment model. These different levels of incorporating dynamics in the assignment model are of importance for their suitability for application in different contexts. Models that calculate an equilibrium (static or dynamic) are mainly suitable for longer term applications in which either the drivers have enough learning opportunity to obtain sufficient information about all alternatives to settle down to at least a semblance of equilibrinm, unless there is control over the information provided to drivers (e.g. through advanced electronic route guidance systems). Non-equilibrium models are most applicable in situations where dynamics are important, but the concept of equilibrium is unreasonable. This is often so when "normal" travel conditions are disrupted.

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2.2

Solution methods

The conceptual difference between equilibrium and non-equilibrium dynamic assignment models is reflected in the solution methods used: • equilibrium methods rely on iterative solution methods, t6 reflect the mutual effect of traffic flow on travel times and vice versa; • non-equilibrium methods avoid such iterations. Within the iterative procedures two implementations appear to have evolved: • the consecutive loading of (packets of) vehicles to routes calculated through the network, taking account of conditions when traffic arrives each point, dealing with all origin and destination pairs in turn, and recalculating traffic conditions after the loading of each new packet. This method was pioneered in CONTRAM, and has since been copied into, e.g. DYNDART and HWYSIM, both'developed in The Netherlands; • the calculation and loading o f routes for all OD pairs through a 3-dimensional timespace network, and the combination of iterations on the basis of estimated flow rates or densities per link. This is a more intuitive extension o f the favoured combinatorial solution methods for (2-dimensional) static equilibrium assignment, such as the Method of Successive Averages or the Frank-Wolfe algorithm. This is applied in 3DAS. Non-equilibrium methods tend to rely on time-propagation, with different levels of traffic detail in the aggregation of individual vehicles to flows and in the represent~ttion o f traffic conditions on links and at junctions. As we are concerned here with converting static to dynamic equilibrium models, these latter methods are not discussed further. However, note that various of the applications for which a transfer from static to dynamic models may be desirable (e.g. incident management and some types of information systems), might be better addressed through non-equilibrium models.

2.3

Speed-flow-density relationships

Network modellers are familiar with flow-delay relationships (or their converse, speedflow relationships). In practice, the latter are often based on simple piecewise linear relationships, with possible problems in the following respects: • the uncongested part of the curve is often flat, so that at lower levels of demand the assignment may degenerate into an all-or-nothing assignment; • the discontinuities in the curve are hard to defend from a conceptual point of view (although the practical implications may be modest); • there is no natural way of dealing with speed when estimated flows exceed capacity (whilst such conditions become more and more relevant in forecasting). Flow delay relationships are generally represented by mathematical functionsf(x), with x depending on the flow on a link and its capacity. A large number of these have been proposed and are in use in static network modelling, and Spiess (1990) gives a good insight in some of the conditions these functions need to satisfy so that the assignment and its solution process behave well:

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.fix) is" strictly increasing, necessary for the assignment to converge to a unique solution.



./'(a9 exists a~Td is strictly increasb*g, to ensure the desirable convexity o f the congestion function.

• ./'(a? > O. which guarantees uniqueness of the link volumes at equilibrium. The functional tbrms o f the speed-flow and flow-delay relationships are driven by computational requirements, and less so by any behavioural theory and observations in reality. In the time-ag~egate representation o f a static assignment this is acceptable, and perhaps even unavoidable. Dynamic assiglmaent, however, aims to reflect better the actual network conditions and the relationship between network conditions and route choice. Often, this means a return to the fundamental diagram, illustrated in Figure I. Innnediately two observations can be made: • the relationship between flow and speed is not continuously decreasing; rather, fllere are two regimes, an uncongested regime in which speeds fall with increasing.flow, and a congested regime in which both speeds and flow fall, ultimately to a static queue at which both flow and speed are 0; • the relationship between speed and density, on the other hand, is strictly decreasing.

Figure I: Fundamental diagram for QBLOK, 3DAS and CONTRAM Speed/Flow

'~

- -" r ~ ' n ~ 5 ! ~

SpeedlDens~,

./

\

Row

Flow/DensRy

q=p'v where: q is flow p is density v is speed [~nzlly

Hence, in many dynamic applications the speed-density rather than the speed-flow relationship is the basis for calculation of travel conditions and the attractiveness of muting alternatives. Transfer of static to dynamic model forms requires therefore the estimation of speed density curves from conventional flow-delay or speed-flow curves (if as in this case a compatible base year representation is sought). If the flow delay curve and speed-density curves are of simple mathematical forms, an algebraic

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transfomaation may be possible z. However, in many practical applications this is not tbasible, and then a more pragmatic approach is required, as follows: • identification of key points in the curves • ensurance of coincidence of travel conditions at these points • avoidance of structural over- or underestimation of travel conditions between these points. 3.

P R A C T I C A L CONSIDERATIONS

3.1

The definition of capacity

In Section 2.2 the conversion of flow-delay to speed-density relationships was discussed. One practical aspect in such a transfer remains: the definition of capacity. In static assignment models capacity is a somewhat arbitrary point on the flow-delay or speed-flow curve at which conditions have deteriorated, although speeds remain certainly higher than 0 (and delays less than infinity). For example, COBA9 speed flow curves for Motorways show a minimum speed of 45 km/h at capacity and the wellknown BPR curve estimates travel times at capacity to be twice that at free flow. Hence, traffic flows in excess of capacity can occur, which experience 'reasonable traffic conditions. From a computational point this is necessary to enable the solution algorithm to find an equilibrium pattern, from a conceptual point this is defensible as during parts of an hourly assignment period flows in excess of capacity can indeed be observed. In effect, the estimation of capacity is not unique, but depends on the length of time interval considered. As discussed by Sachse (1993) and supported by data analyses, capacities for short time intervals (say, 1 minute) can be 50% higher than those estimated for a longer, hourly interval. This has an impact on the conversion of static capacity values to dynamic representations. hi the fundamental diagram capacity cannot be exceeded: at that point maximum flow throughput is achieved beyond which the speed-flow curve bends back on itsel£ This is also the point where critical density occurs, and the uncongested and congested traffic regimes meet. The meaning of capacity differs in the static and dynamic representations. Setting the dynamic value equal to the capacity from the static assignment would imply that all flow levels up to capacity would encounter uncongested conditions. So what to do? There are competing pressures on the conversion of a static capacity value to an input to a dynamic representation based on speed-density relationships2: •

as the time intervals considered in dynamic assignment are generally smaller than in static assignment (typically 15 min as opposed to 1 hour), the maximum flow rate should be higher than the static capacity;

Clearly, the detailing &such an approach dependson the curves involved. Likeflow-delaycurves, many of the speed-densitycurves in practical use have been chosen for their desirablemathematical properties (sometimesbacked up by (coincidental)behaviouralconsiderations). Del CastiUoand Benitez (1995) provide an interestingoverview, and discussthe properties of relevance. -' We are awarethat in an ideal world the static capacitywould be constructedfrom the fundarnentaI diagram, rather than the other way around.

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to represent the increasing congestion at flow levels approaching static capacity, in the equivalent speed-density curve at least some of this should ideally take place in the congested regime, and hence the maximum flow rate should be less than capacity (say anywhere between a practical capacity of 85% and 100%). In the end, this thorny issue has been approached in our study through sensitivity testing across the range of 85-99% of static capacity. Based on a comparison of absolute values of total travel time and queuing time plus their relative values for the static and dynamic models, dynamic capacities in our application would need to be less than 90% of their static values to obtain comparable results. However, this will be affected by the method used for estimation of static capacities.

3.2

Departure time profiling

Departure profiles are an essential additional input to dynamic assignment models, compared with static models based on hourly flow rates. Their derivation in practice from observations is complex and costly: • observed profiles from ATCs only provide information on resultant traffic levels onstreet, not on departure profiles; • observation through household or roadside interviews is expensive. The Netherlands National Travel Survey is an excellent source of information on departure profiles, also allowing an investigation of beneficial segmentation. The Survey is based on travel diaries completed by 80-150,000 persons each per year. This was the source used to determine departure profiles for input to the dynamic assignment models. Two bases for segmentation have been assessed: • segmentation on the basis of trip purpose; • segmentation on the basis of trip length. The former is a more conventional way of splitting up a demand matrix, and if route choice is to be affected by policy (e.g. business travellers will be less influenced by tolling levels) this segmentation may be preferable in a static context. However, in a dynamic assignment there are reasons for distinguishing trip distance bands instead: • if arrival time considerations play, long distance trips may be expected to have a distinctly earlier peak than short trips; • this behaviour leads to a concentration of traffic at bottlenecks during the peak, a characteristic which would be diffused if this mechanism was not properly catered for.

Figures 2 and 3 show the departure profiles by trip purpose and distance band respectively. These are very revealing (although not surprising): • a distinct profiIe can be recognised for the three purposes distinguished. Commuting peaks earliest and trails off towards the end of the peak period. Business trips exhibit a later peak and remain relatively stable thereafter, whilst other trips show a slow increase during the peak, with the highest value at the end of the period.

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the five distance bands also show distinct profiles. As expected, the longest trips depart earliest (the 50+ km band peaking some 90 rain earlier than the 0-5 km band), and peaking also becomes more pronounced for the shorter trips. The impacts of both methods of segmentation have been assessed through sensitivity tests, described in section 4.

Figure 2: 4-hour AM demand profile by purpose (source: OVG) 0.3 0.25 .~

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0.2 0.15

=_. 0.1" 0.05 0

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Figure 3: 4-hour AM demand profile by distance band (source: OVG) 0.25

--~0-5km

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. ---=-- 5-10 km 0.20

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3.3

The basis for assessment

A final consideration in the application of dynamic assignment methods is the determination of the basis for assessment. In static assignments conditions before and (to a lesser extent) after the assessed period are of limited importance: all traffic departing in a period is assumed to reach its destination in that same period (otherwise the underlying equilibrium assumptions would be violated). However, in dynamic assignments travellers departing in the peak can be followed through their whole trip, so that the post-peak period, and conditions there, are of relevance. The pre-peak period also matters, as traffic departing towards the end of this period will occur in the peak and affect travel conditions then. This loss of independence between time periods affects the estimation of aggregate travel conditions and raises the issue of the most appropriate basis for assessment. In assessing aggregate travel conditions in a dynamic context the following distinctions are of importance 3: • total or average travel conditions (e.g. total travel time, or distance) encountered by traffic on the network in lhe peak period; • total or average travel conditions (e.g. total travel time, or distance) encountered by traffic departing in the peak period. In dynamic assignments the latter is conventionally calculated by enabling queued traffic to dissipate in "empty" periods subsequent to the peak. Aggregation of travel times and distances can then take place by simply adding over links and time periods. The error through omitting impacts by traffic already on the network at the beginning of the modelled period could be minimised by allowing a long initial period of virtually free flow conditions, but this is rarely done in practice. The error due to the simplification of (empty) network conditions encountered by traffic still on the network m the post-peak is generally assumed to be small, as the proportion of such traffic is small, compared with overall demand. This, of course, may change in future, highly congested scenarios. For a better estimation of true travel conditions encountered by traffic departing in the peak, the pre-peak and post-peak period should be modelled explicitly, but then arty aggregation of travel conditions would need to take place at an OD and departure time specific level, as traffic departing in the pre- and post peak would invalidate arty summation over links and time periods. Alternatively, the former (average travel conditions in the peak period itself) could also be estimated. Then, traffic departing in the pre-peak would need to be loaded as this might occur on parts of the network in the peak and affect driving conditions. Aggregation could still take place at the link, rather than at the OD level, summing over the peak period only4. A final consideration is that the calculation of travel conditions encountered by traffic on the network in the peak is not directly suitable for economic appraisal, as the demand ~ Apart from a possiblecorrection for residual queuesthese values are equal for static assignments. 4 Again, post-peakconditionscould be modelledexplicitly,but because summationonly takes place over the peak itself, conditionare only of relevance for routing, and not for level of service.

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in the peak would vary with congestion. Again, this could be overcome by OD-matrixbased, rather than link-based assessments of aggregate travel conditions. 4.

E M P I R I C A L RESULTS

Here w e report on the conversion of the static assignment component in the Dutch National Model System to a dynamic representation. The ultimate aim of the exercise is to investigate whether policy decisions would be different if a dynamic assignment model is employed in this disaggregate demand model system. As in this application (as yet) no feedback to the demand component has taken place, the results presented here cannot yet give any indication of such effects. 4.1

Practical requirements

The application is extensive (and the largest CONTRAM and 3DAS applications ever reported): * 10,000 nodes • 25,000 links • 400 zones • .4 user classes • I million trips in the 2-hour AM peak period (excluding intrazonals) The computing requirements of the applications are considerable, but well within limits of relatively standard PC configurations: • 266MHz Penfium processor • 128MbRAM ' • 800 MB of disk space for the most resource-hungry run l~un times of dynamic assignment models are notoriously longer than static models, because of the complications of the time element in computation, particularly in achieving acceptable levels of convergence. Indicative run times for the three models tested are as in Table 1 (configuration as above): Table 1: Convergence and run time s u m m a r y for model applications model

no of convergence iterations indicator

QBLOK

20

RMSE (flow) = 9

stability travel time*** 99.9%

3DAS

30

Gap = 2%*

100.3%

CONTRAM

10

RMSE (flow)

=

16"*

99.9%

* insufficient convergence, 60-90 iterations desirable ** total RMSE divided by number of time slices *** quotient of total travel times in two final iterations

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in run time

practical implication

1 hour

several daring the day 14 hours ovemight 38 hours over weekend

the

As described above, the whole conversion process is aimed at a dynan~ic representation that uses the investment made in the static model and remains as close as possible to the flow and travel conditions estimated by the static model in the validated base year. The base year representations o f the three models have been compared on a number of outputs, at different levels of spatial and temporal aggregation, distinguishing road types m~d levels of service. Below we present only aggregate values. Any such comparison depends on a number of key assumptions made and these are discussed first. 4.2

Comparison of peak period definitions

Table 2: Impact of the basis for assessment on results

sum of link flows total travel distance total travel cost of which percentage queued , average speed

trips on network in trips departing peak period peak period 100 131 100 132 100 128 24 23 61 64

in

In Section 3 we discussed how in a dynamic model different bases for assessment could be employed. Here we compare an assessment based on trips on the network in the peak and trips departing in the peak. The general picture in Table 2 is as follows: • the latter assessment sees more trips per link (sum of link flows) as whole trips are assessed, rather than the proportion in the peak; • as a. consequence more distance and travel time are clocked up in the assessed period; however, there is little impact on general travel conditions, expressed through the percentage queuing and average speed. Hence, in this application the basis for assessment has little impact on the comparison of alternatives. However, the issue is of sufficient interest to bear in mind with any dynamic assignment model application, e.g. in terms of suitability for operational vs. economic analyses. 4.3

The impact of alternative departure profiles

The departure profile is a unique extra input to a dynamic assignment model. In Table 3 below the impact of a distance-based rather than a purpose based profile is illustrated. All network statistics increase by 25-50%, due to the larger proportion of traffic taking part in the peak period itself and the concentration of queues at bottlenecks close to the destination. Travel conditions deteriorate markedly, the speed dropping from 61 to 54 km/h and the proportion of queued traffic rising to 32%, even though the overall OD pattern and total demandremains fixed.

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Table 3: Impact of departure time profile on results purpose-based 100 100 100 24 61

sum of link flows total travel distance total travel cost of which percentage queued average speed 4.5

distance-based 124 128 146 32 54

Final comparison

Based on the optimum set of parameters from the sensitivity tests we show in Table 4 below the final comparison between static and dynamic model representations. The following conclusions may be drawn: • through a careful conversion of speed-flow-density curves a good transfer of a static model to a dynamic model representation may be achieved, which reproduces travel conditions in the validated static model well, in terms of fit between assigned flows, average speed and percentage queued time; • in addition to the conversion of speed-flow or flow-delay curves further fine-tuning of model-specific parameters is essential; • even though travel conditions may compare well (e.g. assigned flows and average speed), total travel distance and time may still be quite different between models. This is mainly due to the different basis for analysis: in QBLOK all traffic reaches its destination in the assessed period; whereas in the dynamic models only the part travelled during the peak period is assessed. Table 4: Final comparison of results

QBLOK total distance travelled total travel time ofwhich percentage queued average speed R-squared (link flow

5.

100 100 21 62

3DAS 88 86 19 63 0.950

CONTRAM 80 81. 24 61 0.979

CONCLUSIONS

In this paper we have addressed conceptual and practical issues playing when converting static to dynamic assignment models. Throughout such a conversion the following objective has played a major role: • to maintain the investment made in the static model, in terms of underlying data and level of validation with respect to observations. Network topology is relatively easily transferred. The main issue is the conversion of static speed-flow or flow-delay relationships into dynamic speed-density curves. Although sometimes a mathematical procedure is possible, in practice this generally requires a more pragmatic approach:

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• • •

identification of key points in the curves ensurance of coincidence of travel conditions at these points avoidance of structural over- or underestimation of travel conditions between these points.

A second important step in the conversion of the model is in the transfer of important parameters that are unique to the model or subject to interpretation. We have carried out sensitivity tests to obtain best values, but further conceptual and theoretical work in this area would be beneficial: • maximum (jam) density; • critical density; • capacity (or maximum flow). Two elements are additional in dynanlic as opposed to static assignment: • departure time profile; • basis for assessment. Our tests have shown that the departure profile has considerable impact on the travel conditions in a dynamic assignment. In particular, trip distance as distinguishing variable affects the occurrence and level of congestion, as it alters the temporal distribution of arrivals at bottlenecks. It can easily be obtained through skimming the network or a crow-fly analysis and should be considered in any OD matrix segmentation for dynmnic assignment. Trips departing in the peak period are also those on the network in static models, but this is not necessarily the case in dynamic applications. Hence, the basis for assessment is important: •. traffic on the network in the 15eakperiod; • traffic departing in the peak period. This also raises the issue whether the pre- and post peak conditions may be ignored, as is usually done in dynamic applications, and i f a summation over links, rather than ODpairs is the most suitable way of analysing. This is an issue that deserves further consideration. Tile bottom line is, however, that a robust conversion of a static to a dynamic assigmnent model has proved possible, through a careful conversion of speed-flowdensity curves, plus further fine-tuning of model-specific parameters. 6.

REFERENCES

Bakker, DM, PH Mijjer, AJ Daly, PC Vrolijk & F Hofman (1994) "Prediction and evaluation of the effects of traffic management measures on congestion and vehicle queues" PTRC Summer Annual Meeting, Proceedings of Seminar H, Warwick, England, 1994, pp 13-25. De Romph E, HJM van Grol and R Hamerslag (1992) "3DAS - 3-Dimensional ASsignment - A dynamic assignment model for short-term predictions", 39th North American Meeting of the Regional Science Association International, Chicago.

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Del Castillo JM & FG Benitez (1995) "On the functional form of the speed-density relationship --I: General theory", Transportation Research, Vo129B, No 5, pp 373-389. Papageorgiou M & A Korsialos (1998) "Short term traffic forecasting with METANET", paper presented at D'ACCORD Short Term Forecasting Workshop, Delft, February 1998. Sachse T (1993) "The influence of the time interval on the determination of capacity", PTRC Summer Annual Meeting, Proceedings of Seminar D, Manchester, 1993, pp 139150. Schwerdtfeger Th (1984) "DYNEMO: a model for the simulation of traffic flow in motorway networks", in: Volmuller J and R Hamerslag (eds) "Proceedings of Ninth International Symposium on Transportation and Traffic Theory", VNU Science Press, Utrecht, pp 65-87. Spiess, H (1990) "Conical volume-delay functions", Transportation Science, Vol 24, No 2, 1990. Taylor N B (1990) "CONTRAM 5: an enhanced trafftc assignment model", TRL Report RR 249, Transport Research Laboratory, Crowthorne. Van Aerde M (1995) "INTEGRATION: a model for simulating IVHS integrated traffic networks", Users guide for model version 1.5c, Kingston, Ontario, Canada.

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