Chapter 2 Transportation Supply Models

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1 Chapter 2 Transportation Supply Models 2.1 Introduction This chapter deals with the mathematical models simulating transportation supply systems. In broad terms a transportation supply model can be defined as a model, or rather a system of models, simulating the performances and flows resulting from user demand and the technical and organizational aspects of the physical transportation supply. Transportation supply models combine traffic flow theory and network flow theory models. The former are used to analyze and simulate the performances of the main supply elements, the latter to represent the topological and functional structure of the system. Therefore, in Sect. 2.2 we present some of the basic results of traffic flow theory. Section 2.3 covers the constituent elements of a transportation network supply model: such elements form an abstract model of transportation supply (transportation network) which combines network flow theory with the functions that express dependence between transportation flows and costs on the network. This is followed by some general indications on the applications of network models in Sect Specific models for transportation systems with continuous services (such as road systems) are described in Sect ; models for discrete or scheduled services (such as bus, train, or airplane) are described in Sect Throughout this chapter, as stated in Chap. 1, it is assumed that the transportation system is intraperiod (within-day) stationary (unless otherwise stated); extensions of supply models to intraperiod dynamic systems are dealt with in Chap Fundamentals of Traffic Flow Theory 1 Models derived from traffic flow theory simulate the effects of interactions between vehicles using the same transportation facility (or the same service) at the same time. For simplicity s sake, the models presented refer to vehicle flow, although most of them can be applied to other types of users, such as trains, planes, and pedestrians. In the sections below we describe stationary uninterrupted flow models (nonstationary models are introduced in Chap. 7), followed by models of interrupted flow, derived from queuing theory. 1 Giulio Erberto Cantarella is co-author of this section. E. Cascetta, Transportation Systems Analysis, Springer Optimization and Its Applications 29, DOI / _2, Springer Science+Business Media, LLC

2 30 2 Transportation Supply Models Uninterrupted Flows Multiple vehicles using the same facility may interact with each other and the effect of their interaction will increase with the number of vehicles. This phenomenon, called congestion, occurs in most transportation systems, generally worsening the overall performances of the facility, such as the mean speed or travel time. Indeed, it may happen that a vehicle is forced to move at less than its desired speed if it encounters a slower vehicle. The higher the number of vehicles on the infrastructure, the more likely this condition is to happen. This circumstance may also occur in transportation systems with scheduled services: the higher the number of vehicles on the infrastructure, the more likely out-of-schedule vehicles are to cause a delay to other vehicles. In general, stochastic models may be used to characterize in a probabilistic sense an interaction event that causes a delay. For congested systems with continuous services it is very often sufficient to adopt the aggregate deterministic models described below; they may be applied in areas far away from interruptions such as intersections and toll booths Fundamental Variables Several variables can be observed in a traffic stream, that is, a sequence of cars moving along a road segment referred to as a link, a. In principle, although all variables should be related to link a, to simplify the notation the subscript a may be implied. The fundamental variables are as follows (see Fig. 2.1). τ The time at which the traffic is observed L a The length of road segment corresponding to link a s A point along a link, or rather, its abscissa increasing (from a given origin, usually located at the beginning of the link) along the traffic direction (s [0,L a ]) i An index denoting an observed vehicle v i (s, τ) The speed of vehicle i at time τ while traversing point (abscissa) s For traffic observed at point s during time interval [τ,τ + Δτ], several variables can be defined (see Fig. 2.1) asfollows. h i (s) The headway between vehicles i and i 1 crossing point s m(s τ,τ + Δτ) The number of vehicles traversing point s during time interval [τ,τ + Δτ] h(s) = i=1,...,m h i(s)/m(s τ,τ + Δτ) The mean headway, among all vehicles crossing point s during time interval [τ,τ + Δτ] v τ (s) = i=1,...,m v i(s)/m(s τ,τ + Δτ) The time mean speed, among all vehicles crossing point s during time interval [τ,τ + Δτ] Similarly, for traffic observed at timeτ between points s and s +Δs, the following variables can be defined.

3 2.2 Fundamentals of Traffic Flow Theory 31 Fig. 2.1 Vehicle trajectories and traffic variables sp i (τ) The spacing between vehicles i and I 1 at time τ n(τ s,s + Δs) The number of vehicles at time τ between points s and s + Δs sp(τ) = i=1,...,n sp i(τ)/n(τ s,s + Δs) The mean spacing, among all vehicles between points s and s + Δs at time τ v s (τ) = i=1,...,n v i/n(τ s,s + Δs) The space mean speed, among all vehicles between points s and s + Δs at time τ During time interval [τ,τ + Δτ] between points s and s + Δs, a general flow conservation equation can be written: where Δn(s,s + Δs,τ,τ + Δτ) + Δm(s,s + Δs,τ,τ + Δτ) = Δz(s, s + Δs,τ,τ + Δτ) (2.2.1) Δn(s,s + Δs,τ,τ + Δτ) = n(τ + Δτ s,s + Δs) n(τ s,s + Δs) is the variation in the number of vehicles between points s and s + Δs during Δτ Δm(s,s + Δs,τ,τ + Δτ) = m(s + Δs τ,τ + Δτ) m(s τ,τ + Δτ) is the variation in the number of vehicles during time interval [τ,τ + Δτ] over space Δs Δz(s, s + Δs,τ,τ + Δτ) is the number of entering minus exiting vehicles (if any) during time interval [τ,τ + Δτ], due to entry/exit points (e.g., on/off ramps), between points s and s + Δs In the example of Fig. 2.1 there are no vehicles entering/exiting in the segment Δs; then Δz = 0(Δn is equal to 1 and Δm is equal to 1). With the observed quantities two relevant variables, flow and density, can be introduced: f(s τ,τ + Δτ) = m(s τ,τ + Δτ)/Δτ is the flow of vehicles crossing point s during time interval [τ,τ + Δτ], measured in vehicles per unit of time k(τ s,s + Δs) = n(τ s,s + Δs)/Δs is the density between points s and s + Δs at time τ, measured in vehicles per unit of length

4 32 2 Transportation Supply Models Flow and density are related to mean headway and mean spacing through the following relations. f(s τ,τ + Δτ) = 1/h(s) k(τ s,s + Δs) = 1/sp(τ) Note that if observations are perfectly synchronized with vehicles, the near-equality in the previous two equations becomes a proper equality. Moreover, if the general flow conservation equation (2.2.1) is divided by Δτ,the following equation is obtained. where Δn/Δτ + Δf = Δe (2.2.2) Δf (s, s + Δs,τ,τ + Δτ) = Δm(s,s + Δs,τ,τ + Δτ)/Δτ is the variation of the flow over space Δe(s, s + Δs,τ,τ + Δτ) = Δz(s, s + Δs,τ,τ + Δτ)/Δτ is the (net) entering/ exiting flow Finally, dividing by Δs, we obtain a further formulation of (2.2.1) (useful for comparisons with nonstationary models based on the fluid-dynamic analogy described in Chap. 7) that expresses the role of variation in density: where Δk/Δτ + Δf /Δs = Δe/Δs (2.2.3) Δk(s, s + Δs,τ,τ + Δτ) = Δn(s,s + Δs,τ,τ + Δτ)/Δs is the variation of the density over time Model Formulation In this subsection we describe several deterministic models developed under the assumption of stationarity, formally introduced below. Extensions to nonstationarity conditions are reported in Chap. 7 (some information on stochastic models is reported in the bibliographical note). In formulating such models it is assumed that a traffic stream (a discrete sequence of vehicles) is represented as a continuous (onedimensional) fluid. Traffic flow is called stationary during a time interval [τ,τ + Δτ] between points s and s + Δs if flow is (on average) independent of point s, and density is independent of time τ (other definitions are possible): f(s τ,τ + Δτ) = f k(τ s,s + Δs) = k Note that this condition is chiefly theoretical and in practice can be observed only approximately for mean values in space or time. It is nevertheless useful in that it

5 2.2 Fundamentals of Traffic Flow Theory 33 Fig. 2.2 Vehicle trajectories and traffic variables for stationary (deterministic) flows allows effective analysis of the phenomenon. In this case, the time mean speed is independent of location and the space mean speed is independent of time: v τ (s) = v τ v s (τ) = v s In the case of stationarity, both terms in the left side of the conservation equation (2.2.3) are identically null, anyhow other flow conservation conditions may be formulated. Hence, let n = k Δs be the number, time-independent due to the assumption of stationarity, of vehicles on the stretch of road between cross-sections s and s + Δs, and let v s be the space mean speed of these vehicles. The vehicle that at time τ is at the start of the stretch of road, cross-section s, will reach the end, cross-section s + Δs, on average at time τ + Δτ, with Δτ = Δs/v s. Due to the assumption of stationarity, the number of vehicles crossing each cross-section during time Δτ is equal to f Δτ. Thus the number of vehicles contained at time τ on section [s,s +Δs] is equal to the number of vehicles traversing cross-section s +Δs during the time interval [τ,τ + Δτ ] (see Fig. 2.2); that is, kδs = fδτ = fδs/v s. Hence, under stationary conditions, flow, density, and space mean speed must satisfy the stationary flow conservation equation: f = kv (2.2.4) where v = v s is the space mean speed, simply called speed for further analysis of stationary conditions. 2 2 It is worth noting that the time mean speed is not less than the space mean speed, as can be shown because the two speeds are related by the equation v τ = v s + σ 2 / v s,whereσ 2 is the variance of speed among vehicles. In Fig. 2.2 σ 2 = 0, hence v τ = v s.

6 34 2 Transportation Supply Models Fig. 2.3 Relationship between speed and flow In stationary conditions, empirical relationships can be observed between each pair of variables: flow, density, and speed. In general, observations are rather scattered (see Fig. 2.3 for an example of a speed flow empirical relationship) and various models may be adopted to describe such empirical relationships. These models are generally given the name fundamental diagram (of traffic flow) (see Fig. 2.4) and are specified by the following relations. v = V(k) (2.2.5) f = f(k) (2.2.6) f = f(v) (2.2.7) Although only a model representation of empirical observations, this diagram permits some useful considerations to be made. It shows that flow may be zero under two conditions: when density is zero (no vehicles on the road) or when speed is zero (vehicles are not moving). The latter corresponds in reality to a stop-and-go condition. In the first case the speed assumes the theoretical maximum value, free-flow speed v 0, whereas in the second the density assumes the theoretical maximum value jam density, k jam. Therefore, a traffic stream may be modeled through a partially compressible fluid, that is, a fluid that can be compressed up to a maximum value. The peak of the speed flow (and density flow) curve occurs at the theoretical maximum flow, capacity Q of the facility; the corresponding speed v c and density k c are referred to as the critical speed and the critical density. Thus any value of flow (except the capacity) may occur under two different conditions: low speed and high density and high speed and low density. The first condition represents an unstable state for the traffic stream, where any increase in density will cause a decrease in speed and thus in flow. This action produces another increase in density and so on until traffic becomes jammed. Conversely, the second condition is a stable state because any increase in density will cause a decrease in speed and an increase in

7 2.2 Fundamentals of Traffic Flow Theory 35 Fig. 2.4 Fundamental diagram of traffic flow flow. At capacity (or at critical speed or density) the stream is nonstable, this being a boundary condition between the other two. These results show that flow cannot be used as the unique parameter describing the state of a traffic stream; speed and density, instead, can univocally identify the prevailing traffic condition. For this reason the relation v = V(k) is preferred to study traffic stream characteristics. Mathematical formulations have been widely proposed for the fundamental diagram, based on single regime or multiregime functions. An example of a single regime function is Greenshields linear model: V(k)= v 0 (1 k/k jam ) or Underwood s exponential model (useful for low densities): V(k)= v 0 e k/k c. An example of a multiregime function is Greenberg s model: V(k)= a 1 ln(a 2 /k) V(k)= a 1 ln(a 2 /k min ) for k>k min for k k min where a 1,a 2 and k min k jam are constants to be calibrated. Starting from the speed density relationship, the flow density relationship, f = f(k), may be easily derived by using the flow conservation equation under station-

8 36 2 Transportation Supply Models ary conditions, or fundamental conservation equation (2.2.4): Greenshields linear model yields: In this case the capacity is given by f(k)= V(k)k f(k)= v 0 (k k 2 /k jam ) Q = v 0 k jam /4 Moreover the flow speed relationship can be obtained by introducing the inverse speed density relationship: k = V 1 (v), thus f(v)= V ( k = V 1 (v) ) V 1 (v) = v V 1 (v) For example, Greenshields linear model yields: V 1 (v) = k jam (1 v/v 0 ) thus f(v)= k jam (v v 2 /v 0 ) In general, the flow speed relationship may be inverted by only considering two different relationships, one in a stable regime, v [v c,v o ], and the other in an unstable regime, v [0,v c ]. Greenshield s linear model leads to: v stable (f ) = v 0 2 v unstable (f ) = v 0 2 ( f/(v 0 k jam ) ) = v 0 ( ) f/q 2 ( ) 1 1 f/q In the particular case that one can assume the flow regime is always stable, with reference to relation v = v stable (f ) the corresponding relationship between travel time t and flow may be defined (some examples of this type of empirical relationship may be found in Sect. 2.4): t = t(f)= L/v stable (f ) (2.2.8) Queuing Models The average delay experienced by vehicles that queue to cross a flow interruption point (intersections, toll barriers, merging sections, etc.) is affected by the number of vehicles waiting. This phenomenon may be analyzed with models derived from queuing theory, developed to simulate any waiting or user queue formation at a server (administrative counter, bank counter, etc.). The subject is treated below with reference to generic users, at the same time highlighting the similarities with uninterrupted flow.

9 2.2 Fundamentals of Traffic Flow Theory 37 Fig. 2.5 Fundamental variables for queuing systems Fundamental Variables The main variables that describe queuing phenomena are: τ The time at which the system is observed τ i Thearrivaltimeofuseri h i = τ i τ i 1 The headway between successive users i and i 1 joining the queue at times τ i and τ i 1 m IN (τ, τ + Δτ) Number of users joining the queue during [τ,τ + Δτ] m OUT (τ, τ + Δτ) Number of users leaving the queue during [τ,τ + Δτ] h(τ, τ + Δτ) = i=1,...,m h i/m IN (τ, τ + Δτ) Mean headway between all vehicles joining the queue in the time interval [τ,τ + Δτ] n(τ) Number of users waiting to exit (queue length) at time τ With reference to observable quantities, flow variables may be introduced. u(τ, τ + Δτ) = m IN (τ, τ + Δτ)/Δτ arrival (entering) flow during [τ,τ + Δτ] w(τ,τ + Δτ) = m OUT (τ, τ + Δτ)/Δτ exiting flow during [τ,τ + Δτ] Note that the main difference with the basic variables of running links is that space (s, Δs) is no longer explicitly referred to because it is irrelevant. Some of the above variables are shown in Fig With reference to the service activity, let: t s,i Be service time of user i t s (τ, τ + Δτ) Average service time among all users joining the queue in time interval [τ,τ + Δτ] tw i Total waiting time (pure waiting plus service time) of user i tw(τ, τ + Δτ) Average total waiting time among all users joining the queue in time interval [τ,τ + Δτ]

10 38 2 Transportation Supply Models Fig. 2.6 Fluid approximation of deterministic queuing systems Q(τ, τ + Δτ) = 1/t s (τ, τ + Δτ) the (transversal 3 ) capacity or maximum exit flow, that is, the maximum number of users that may be served in the time unit, assumed constant during [τ,τ + Δτ] for simplicity s sake (otherwise Δτ can be redefined) The capacity constraint on exiting flow is expressed by w Q. A general conservation equation, similar to (2.2.1) and (2.2.2) introduced for uninterrupted flow, holds in this case: n(τ) + m IN (τ, τ + Δτ) = m OUT (τ, τ + Δτ) + n(τ + Δτ). (2.2.9) Moreover, dividing by Δτ we obtain: Δn/Δτ + [ w(τ,τ + Δτ) u(τ, τ + Δτ) ] = 0. (2.2.10) In the following subsection we describe several deterministic models developed under the assumption that the headway between two consecutive vehicles and the service time are represented by deterministic variables. This is followed by a subsection on stochastic models developed using random variables. In formulating such models, as in the case of uninterrupted flow models, we assume arrival at the queue is represented as a continuous (one-dimensional) fluid. 3 In some cases it is also necessary to introduce longitudinal capacity, that is, the maximum number of users that may form the queue.

11 2.2 Fundamentals of Traffic Flow Theory 39 Fig. 2.7 Cumulative arrival and departure curves Deterministic Models Deterministic models are based on the assumptions that arrival and departure times are deterministic variables. According to the fluid approximation introduced above, the conservation equation (2.2.10) forδτ 0 becomes (see Fig. 2.6): dn(τ) = u(τ) w(τ) dt Deterministic queuing systems can also be analyzed through the cumulative number of users that have arrived at the server by time τ, and the cumulative number of users that have departed from the server (leaving the queue) at time τ, as expressed by two functions termed arrival curve A(τ), and departure curve D(τ) A(τ), respectively; see Fig Queue length n(τ) at any time τ is given by: n(τ) = A(τ) D(τ) (2.2.11) provided that the queue at time 0 is given by n(0) = A(0) 0 with D(0) = 0. The arrival and departure functions are linked to entering and exiting users by the following relationships. m IN (τ, τ + Δτ) = A(τ + Δτ) A(τ) (2.2.12) m OUT (τ, τ + Δτ) = D(τ + Δτ) D(τ) (2.2.13) The flow conservation equation (2.2.9) can also be obtained by subtracting member by member the relationships (2.2.12) and (2.2.13) and taking into account (2.2.11). The limit for Δτ 0of(2.2.12) and (2.2.13) leads to (see Fig. 2.7): u(τ) = da(τ) dτ

12 40 2 Transportation Supply Models Fig. 2.8 Undersaturated queuing system w(τ) = dd(τ) dτ If during time interval [τ 0,τ 0 + Δτ] the entering flow is constant over time, u(τ) =ū, then the queuing system is named (flow-)stationary and the arrival function A(τ) is linear with slope given by ū: A(τ) = A(τ 0 ) +ū (τ τ 0 ) τ [τ 0,τ 0 + Δτ] The exit flow may be equal to the entering flow ū, or to the capacity Q as described below. 4 (a) Undersaturation When the arrival flow is less than capacity (ū<q)the system is undersaturated. In this case, if there is a queue at time τ 0, its length decreases with time and vanishes after a time Δτ 0 defined as (see Fig. 2.8) Δτ 0 = n(τ 0 )/(Q ū) (2.2.14) Before time τ 0 + Δτ 0, the queue length is linearly decreasing with τ and the exiting flow w is equal to capacity Q: n(τ) = n(τ 0 ) (Q ū)(τ τ 0 ) w = Q (2.2.15) D(τ) = D(τ 0 ) + Q(τ τ 0 ) After time τ 0 + Δτ 0 the queue length is zero and the exiting flow w is equal to the arrival flow ū: n(τ 0 + Δτ 0 ) = 0 4 In stationary queuing models used on transportation networks, the inflow ū can be substituted with the flow f a of the link representing the queuing system.

13 2.2 Fundamentals of Traffic Flow Theory 41 Fig. 2.9 Oversaturated queuing system w =ū (2.2.16) D(τ) = A(τ) = A(τ 0 ) +ū(τ τ 0 ) (b) Oversaturation When the arrival flow rate is larger than capacity, ū Q, the system is oversaturated. In this case queue length linearly increases with time τ and the exiting flow is equal to the capacity Q (see Fig. 2.9): n(τ 0 ) = n(τ 0 ) + (ū Q)(τ τ 0 ) w = Q (2.2.17) D(τ) = D(τ 0 ) + Q(τ τ 0 ) (c) General Condition By comparing (2.2.15) through (2.2.17) it is possible to formulate this general equation for calculating the queue length at generic time instant τ : n(τ) = max { 0, ( n(τ 0 ) + (ū Q)(τ τ 0 ) )} (2.2.18) With the above results, any general case can be analyzed by modeling a sequence of periods during which arrival flow and capacity are constant. An important case is that of the queuing system at traffic lights which may be considered a sequence of undersaturated (green) and oversaturated (red) periods with zero capacity (see p. 73: Application of Queuing Models). The delay can be defined as the time needed for a user to leave the system (passing the server), accounting for the time spent queuing (pure waiting). Thus the delay is the sum of two terms: where tw = t s + tw q

14 42 2 Transportation Supply Models Fig Deterministic delay function at a server tw is the total delay t s = 1/Q is the average service time (time spent at the server) tw q is the queuing delay (time spent in the queue) In undersaturated conditions (ū<q)if the queue length at the beginning of period is zero (it remains equal to zero), the queuing delay is equal to zero, tw q (u) = 0, and the total delay is equal to the average service time: tw(ū) = t s In oversaturated conditions (ū Q), the queue length, and respective delay, would tend to infinity in the theoretical case of a stationary phenomenon lasting for an infinite time. In practice, however, oversaturated conditions last only for a finite period T. If the queue length is equal to zero at the beginning of the period, it will reach a value (ū Q) T at the end of the period. Thus, the average queue over the whole period T is: (ū Q)T n = 2 In this case the average queuing delay is x/q, and average total delay is (see Fig. 2.10): tw(ū) = t s + (ū Q)T 2Q (2.2.19)

15 2.2 Fundamentals of Traffic Flow Theory Stochastic Models Stochastic models arise when the variables of the problem (e.g., user arrivals, service times of the server, etc.) cannot be assumed deterministic, due to the observed fluctuations, as is often the case, especially in transportation systems. If the system is undersaturated, it can be analyzed through (stochastic) queuing theory which includes the particular case of the deterministic models illustrated above. Some of the results of this theory are briefly reported below, without any claim to being exhaustive. It is particularly necessary to specify the stochastic process describing the sequence of user arrivals (arrival pattern), the stochastic process describing the sequence of service times (service pattern) and the queue discipline. Arrival and service processes are usually assumed to be stationary renewal processes, in other words with stable characteristics in time that are independent of the past: that is, headways between successive arrivals and successive service times are independently distributed random variables with time-constant parameters. Let N be a random variable describing the queue length, and n the realization of N. The characteristics of a queuing phenomenon can be redefined in the following concise notation, where a a/b/c(d,e) denotes the type of arrival pattern, that is, the variable which describes time intervals between two successive arrivals: D = Deterministic variable M = Negative exponential random variable E = Erlang random variable G = General distribution random variable b denotes the type of service pattern, such as a c is the number of service channels: {1, 2,...} d is the queue storage limit: {,n max } or longitudinal capacity e denotes the queuing discipline: FIFO = First In First Out (i.e., service in order of arrival) LIFO = Last In First Out (i.e., the last user is the first served) SIRO = Service In Random Order HIFO = High In First Out (i.e., the user with the maximum value of an indicator is the first served) Fields d and e, if defined respectively by (no constraint on maximum queue length) and by FIFO, are generally omitted. In the following we report the main results for the M/M/1 (, FIFO) and the M/G/1 (, FIFO) queuing systems, which are commonly used for simulating transportation facilities, such as signalized intersections.

16 44 2 Transportation Supply Models Some definitions or notation differ from those traditionally adopted in dealing with queuing theory (the relative symbols are in brackets) so as to be consistent with those adopted above. The parameters defining the phenomenon are as follows. u, (λ) The arrival rate or the expected value of the arrival flow Q = 1/t s,(μ) The service rate (or capacity) of the system, the inverse of the expected service time u/q, (ρ) The traffic intensity ratio or utilization factor n A value of the random variable N, number of users present in the system, consisting of the number of users queuing plus the user present at the server, if any (the significance of the symbol n is thus slightly different) tw A value of the random variable TW, the time spent in the system or overall delay, consisting of queuing time plus service time (a) M/M/1 (, FIFO) Systems In undersaturated conditions (u/q < 1): u Q E[N]= 1 Q u VAR[N]= = u Q u u Q (1 u Q )2 (2.2.20) According to Little s formula, the expected number of users in the system E[N] is the product of the average time in the system (expected value of delay) E[TW] multiplied by arrival rate u: from which: E[N]=uE[TW] (2.2.21) E[TW]= 1 (2.2.22) Q u The expected time spent in the queue E[tw q ] (or queuing delay) is given by the difference between the expected delay E[tw] and the average service time t s = 1/Q: E[TW q ]= 1 Q u 1 Q = u Q(Q u). (2.2.23) According to Little s second formula, the expected value of the number of users in the queue E[N q ] is the product of the expected queuing delay E[TW q ] multiplied by the arrival rate u: and then: E[N q ]=ue[tw q ] (2.2.24) E[N q ]= u 2 Q(Q u) (2.2.25)

19 2.3 Congested Network Models 47 Fig Example of a graph and link path incidence matrix where w i is the homogenization coefficient of the users of class i with respect to their influence on link performances. For example, for road flows, automobiles are usually the reference vehicle type (w i = 1) and the other vehicle flows are trans-

21 2.3 Congested Network Models 49 Fig Transportation network with link and path flows transportation link cost, is a variable synthesizing (the average value of) the different performance variables borne and perceived by the users in travel-related choice and, more particularly, in path choices (see Sect ). Thus, the transportation link

24 52 2 Transportation Supply Models Fig Transportation network with link and path costs The expression (2.3.4) can also be formulated in vector format by introducing the vector of additive path costs g ADD (see Fig. 2.13): g ADD = Δ T c (2.3.5)

28 56 2 Transportation Supply Models Fig Schematic representation of supply models study, and the flow propagation model defines the relationship among path and link flows. The link performance model expresses for each element (link) the relationships among performances, physical and functional characteristics, and flow of users. The impact model simulates the main external impacts of the supply system. Finally, the path performance model defines the relationship between the performances of single elements (links) and those of a whole trip (path) between any origin destination pair. 2.4 Applications of Transportation Supply Models Network models and related algorithms are powerful tools for modeling transportation systems. A network model is a simplified mathematical description of the phys-

29 2.4 Applications of Transportation Supply Models 57 Fig Functional phases for the construction of an urban bimodal network model ical phenomena relevant to the analysis, design, and evaluation of a given system. Thus transportation network models depend on the purpose for which they are used. Building a network model usually requires a sequence of operations whose general criteria are described in the following. A schematic representation of the main activities in the case of a bimodal supply system (road and transit urban systems) is depicted in Fig In the most general case, a supply network model is built through the following phases. (a) Delimitation of the study area (b) Zoning (c) Selection of relevant supply elements (basic network) (d) Graph construction (e) Identification of performance and cost functions (f) Identification of impact functions Phases (a), (b), and (c) relate to the relevant supply system definition. They are described, respectively, in Sect of Chap. 1 and are not repeated here. The rest of this section introduces some general considerations related to phases (d), (e), and

32 60 2 Transportation Supply Models Fig Example of a graph representing part of an urban road system

34 62 2 Transportation Supply Models Fig Graphs for a road intersection if the link represents only the waiting time for a maneuver, tr a and mc a are zero, and the same consideration is true for monetary costs and waiting times on most pedestrian links. If an individual link represents both the trip along a road section and queuing at the intersection, its cost function will include both travel time tr a and queuing time tw a.

35 2.4 Applications of Transportation Supply Models 63 Fig Explicit representation of parking supply In the most general case, the monetary cost term mc a includes the cost items that are perceived by the user. Because users do not usually perceive other consumption (motor oil, tires, etc.), in applications monetary costs are usually identified as the

39 2.4 Applications of Transportation Supply Models 67 Fig Hyperbolic travel time cost function S a is the percentage of length of a occupied by parking Pv a is a dummy variable of 1 if the pavement of link a is asphalt, 0 otherwise f a is the equivalent flow on link a in equiv. vehicles/hour The travel time on link a may thus be calculated by multiplying the time obtainable from (2.4.5) by a corrective factor c(l a ), which makes allowance for the effect of transient motions at the ends of the link (in the case of stopping at intersections): tr a = L a c(l a ) = L a 1 (2.4.6) v a v a 1 exp( E 02 L a ) where L a is the road section length in km. A further example of link travel time function is the hyperbolic expression given by Davidson, which also holds for interrupted flow (delays at intersections are thus included): { tra = (L a /v 0a )(1 + γf a /(Q a f a )) for f a δq a (2.4.7) tr a = tangent approximation for f a >δq a with δ<1 and Q a = link capacity. Also see Fig In this last case the tangent approximation is necessary because tr a tends to for f a going to Q a. This condition is unrealistic because the oversaturated period has a finite duration. Waiting Links (a) Toll-Barrier Links In the case of links representing queuing systems, it is assumed that average waiting time is the only significant time performance variable. In

40 68 2 Transportation Supply Models simple cases (e.g., a link corresponds to all toll lanes), the average undersaturation waiting time can be obtained by using a stochastic queuing model: tw u a (f a) = T s + ( Ts 2 + σ s 2 ) f a 2 1 (2.4.8) 1 f a /Q a where T s is the average service time for each toll lane σs 2 is the variance of the service time at the pay-point Q a = N a /T s is the link (toll-barrier) capacity equal to the product of the number of lanes (N a ) by the capacity of each lane (1/T s ) Expression (2.4.8) is derived from the assumption of a queuing system M/G/1 (, FIFO) with Poisson arrivals and general service time (see Sect ). The values of T s and σs 2 depend on various factors such as the tolling structure (fixed, variable) and the payment method (manual, automatic, etc.). Note that the average waiting time obtained through (2.4.8) is larger than the average service time T s even though the arriving flow is lower than the system s capacity. This effect derives from the presence of random fluctuations in the headways between user arrivals and service times. Hence the delay expressed by (2.4.8) is known as stochastic delay. Moreover, the average delay computed with (2.4.8) tends to infinity as the flow f a tends to capacity (i.e., if f a /Q a tends to one). This would be the case if the arrivals flow f a remained equal to capacity for an infinite time, which does not occur in reality. In order to avoid unrealistic waiting times and for reasons of theoretical and computational convenience, two different methods can be adopted. The first, and less precise, method assumes that (2.4.8) holds for flow values up to a fraction α of the capacity, for example, f a 0.95Q a. For higher values, the curve is extended following its linear approximation, that is, in a straight line passing through the point of coordinates αq a, tw(αq a ) with angular coefficient equal to the derivative of (2.4.8) computed at this point: with tw a (f a ) = tw a (αq a ) + K(f a αq a ) (2.4.9) K = T 2 s + σ 2 s 2 1 (1 α) 2 Figure 2.22 shows the relationships (2.4.8) and (2.4.9) for some values of the parameters. A more rigorous method is based on calculating oversaturation delay using a deterministic queuing model with an arrival rate equal to f a, deterministic service times equal to T s and an oversaturation period equal to the reference period duration T (see Sect ). The deterministic average (oversaturation) delay tw d a is then equal to: ( ) tw d a = T fa T s + 1 Q a 2 which, for a given capacity, is a linear function of the arrivals flow f a. (2.4.10)

41 2.4 Applications of Transportation Supply Models 69 Fig Waiting time functions (2.4.8)and(2.4.9) at toll-barrier links Note that in this case the assumption of intraperiod stationarity is challenged because even if the arrivals flow rate f a and capacity 1/T s are constant over the whole reference period T, the waiting time is different for users arriving in different instants of the reference period. In static models it is assumed that users perceive the average waiting time. Intraperiod dynamic models, discussed in Chap. 7, remove this assumption. The average delay tw a can be calculated by combining the stochastic undersaturation average delay tw u a expressed by (2.4.8) with the deterministic average oversaturation delay tw d a, expressed by (2.4.10). The combined delay function is such that the deterministic delay function is its oblique asymptote (see Fig. 2.23). The following equation results. tw a (f a ) = T s + ( T 2 s + σ 2) f a [( fa + Q T { fa 1 4 Q a ) 2 + 4(f ] a/q a ) 1/2 } Q a T (2.4.11) (b) Signal-Controlled Intersection Links Queuing and delay phenomena at signalized intersections can be obtained from the queuing theory results reported in Sect In fact, signalized intersections are a particular case of servers for which capacity is periodically equal to zero (when the signal is red). During such times the system is necessarily oversaturated. The simplest case is that of a signal-controlled intersection not interacting with adjacent ones (isolated intersection), without lanes reserved for right or left turns.

42 70 2 Transportation Supply Models Fig Under- and oversaturation waiting time functions for toll barrier links Fig Discharge flow from signal-controlled intersection in relation to cycle phases Below we first introduce the assumptions and variables for each access as well as the most widely used calculation method. We then present the various models for calculating delays at intersections.

43 2.4 Applications of Transportation Supply Models 71 It is common to divide the cycle length into two time intervals (Fig illustrates the quantities associated with a traffic-light cycle). The effective green time equals the green plus yellow time minus the lost time, during which departures occur at a constant service rate, given by the inverse of saturation flow. The effective red time is the difference between cycle length and the effective green time, during which no departures occur. Below, to simplify the notation, we omit the index of link a. Moreover, to facilitate application of the results in Sect , the symbol ū instead of f is used for the arrivals flow. Let: T c be the cycle length for the whole intersection G be the effective green time for an approach R = T c G be the effective red time for the approach μ = G/T c be the effective green/cycle ratio for the approach The number of vehicles arriving at the approach during time interval T c is given by the following equation. m IN (τ, τ + T c ) =ū T c The maximum number of users that may leave the approach, during time interval T c, is given by: S G = μ S T c where S is the saturation flow of the intersection approach, that is, the maximum number of equivalent vehicles which in the time unit could cross the intersection if the traffic lights were always green (μ = 1). Alternatively, the saturation flow may be defined as the maximum discharge rate that may be sustained by a queue during the green amber time. Hence the actual capacity of the approach is given by: Q = S G = μ S T c Thus, the approach can be defined undersaturated if: that is: ū T c <μ S T c ū<μ S (2.4.12) On the other hand the approach is defined oversaturated if: ū μ S (2.4.13) The saturation flow rate of an intersection can in principle be obtained through specific traffic surveys; in practice, however, empirical models based on average results are often used. The Highway Capacity Manual (HCM) describes one of the

44 72 2 Transportation Supply Models Fig Typical lane groups for the HCM method for calculating saturation flow most popular methods. To apply this method, it is necessary to determine appropriate lane groups. A lane group is defined as one or more lanes of an intersection approach serving one or more traffic movements with which a single value of saturation flow, capacity, and delay can be associated. Both the geometry of the intersection and the distribution of traffic movements are taken into account to segment the intersection into lane groups. In general, the smallest number of lane groups that adequately describes the operation of the intersection is used. Figure 2.25 shows some common lane group schemes suggested by the HCM. The saturation flow rate of an intersection is computed from an ideal saturation flow rate, usually 1900 equivalent passenger cars per hour of green time per lane (pcphgpl), adjusted for a variety of prevailing conditions that are not ideal. The method can be summarized by the following expression, S = S 0 N F w F HV F g F p F bb F a F RT F LT

46 74 2 Transportation Supply Models ADJUSTMENT FACTOR FOR AVERAGE LANE WIDTH F w Averagelanewidth,W (FT) F w ADJUSTMENT FACTOR FOR HEAVY VEHICLES F HV Percentageofheavy vehicles(%) F HW Percentage of heavy vehicles (%) F HW ADJUSTMENT FACTOR FOR APPROACH GRADE F g Grade (%) F g ADJUSTMENT FACTOR FOR PARKING F p F p No. of parking maneuvers per hour No. of lanes in lane group No parking or more ADJUSTMENT FACTOR FOR BUS BLOCKAGE F bb F bb No. of buses stopping per hour No. of lanes in lane group or more ADJUSTMENT FACTOR FOR AREA TYPE F a Type of area F a CBD (Center Business District) All other areas Fig Adjustment factors in the HCM method for saturation flow I T c + R τ I T c + R + G (2.4.15) The time period (within the green) in which the queue is exhausted is (see (2.4.15)): Δτ 0 = ū(1 μ)t c (S u) The queue in undersaturation conditions therefore shows a periodic time trend, with zero values at the end of effective green time (i.e., at the beginning of the red interval) and maximum values at the end of the effective red interval (see Fig. 2.27). However, if the system is in oversaturation conditions (ū μ S),thetotal queue length is obtained by summing the queue length in undersaturation to the queue length in oversaturation (see Fig. 2.28). The queue length in undersaturation, n u (τ), is obtained once again by (2.4.14) and (2.4.15), for an arrivals rate equal to capacity

47 2.4 Applications of Transportation Supply Models 75 Fig Deterministic queuing model for signalized intersections, undersaturated conditions Fig Deterministic queuing model for signalized intersections, oversaturated conditions (ū = μ S): n R u (τ) = μ S(τ I T c) I T c τ I T c + R (2.4.16) n G u (τ) = μ S(1 μ)t c S(1 μ)(τ I T c R) I T c + R τ I T c + R + G (2.4.17) The oversaturated queue length can be computed with the queue obtained from (2.2.18) with Q = μ S, τ 0 = 0 and n(τ 0 ) = 0(seeFig.2.28): n 0 (τ) = (ū μ S)τ (2.4.18) The expressions of queue length allow us to determine the deterministic delay at intersections, as described below. For undersaturated conditions ū<μs, the average individual delay tw US can easily be obtained from the evolution over time of the queue length, as described

48 76 2 Transportation Supply Models Fig Deterministic delay function at a signalized intersection by (2.4.14) and (2.4.15): tw US = T c[1 μ] 2 2[1 ū/s] (2.4.19) In oversaturated conditions, ū>μs, for the deterministic case, the queue length, and respective delay, would tend theoretically to infinity. In practice, however, oversaturation lasts only for a finite period of time T, and the average delay tw OS can be calculated from the evolution over time of queue length as described by (2.4.16) through (2.4.18): tw OS = T c[1 μ] + T [ ] (ū/μs) 1 (2.4.20) 2 2 Note that the first term is the value of (2.4.19) forū = μ S. The delay for the arrival flows can be computed through (2.4.19) forū<μ S, and through (2.4.20) for ū μ S, as depicted in Fig Note that the diagram depicted in Fig shows an increase in average delay also for flows below the capacity. This is due to the increase in the undersaturated delay expressed by (2.4.19). Stochastic delay models are based on the results of queuing theory. More precisely, a signalized intersection is considered to be a M/G/1 (, FIFO) system. Therefore, the average delay is (see Sect ): tw st q (u) = (ū/μs)2 2ū(1 ū/μs) (2.4.21)

49 2.4 Applications of Transportation Supply Models 77 Fig The Webster delay model Overall Delay Models The total (mean individual) delay equals the sum of the deterministic and the stochastic terms (introduced in the previous section), and sometimes, of terms calibrated through experimental observations. One of the best known expressions is Webster s three-term formula, proposed for an isolated intersection under the assumption of random (Poisson) arrivals and undersaturation conditions (f a /Q a < 1) (see Fig. 2.30): tw a (f a ) = T c(1 μ) 2 2(1 f a /S a ) + (f a /Q a ) 2 2f a (1 f a /Q a ) 0.65 ( Q a /fa 2 ) 1/3(fa /Q a ) 2+μ (2.4.22a) where T c is the cycle length μ is the effective green to cycle length ratio for the lane group represented by link a Q a is the capacity of the lane group represented by link a The first term expresses the deterministic delay (see (2.4.19)), the second is the stochastic delay due to the randomness of the arrivals (see (2.4.21)), and the third term is an adjustment term obtained by simulation results. This term amounts to about 10% of the sum of the other two, hence its established use in practical applications of Webster s two-term formula: [ Tc (1 μ) 2 tw a (f a ) = 0.9 2(1 f a /S a ) + (f a /Q a ) 2 ] (2.4.22b) 2f a (1 f a /Q a )

50 78 2 Transportation Supply Models f f/q Akcelik Webster Fig Waiting time functions at a signalized intersection The delay given by (2.4.22) tends to infinity for an arrivals flow f a, which tends to capacity Q = μ S(seeFig.2.30). Thus Webster s formula cannot be used to simulate delays at oversaturated signalized intersections. To overcome this limit, it is possible to apply the two heuristic methods described for (2.4.8). The first method applies (2.4.22) for values of f a up to a percentage α of the capacity whereas for higher values a linear approximation of the function is used, thereby ensuring the continuity of the function and its first derivative: tw a (f a ) = tw a (αq a ) + d df tw a(f ).(f a αq a ) f a αq a (2.4.23) fa =αq a The second method computes the oversaturation delay combined with the stochastic delay, deforming the stochastic delay function so that it has an oblique asymptote defined by the deterministic delay. Based on these considerations, Akcelik s formula

51 2.4 Applications of Transportation Supply Models 79 was proposed: tw a (f a ) = 0.5T c(1 μ a ) 2 1 μ a X a X a 0.50 tw a (f a ) = 0.5T c(1 μ a ) 2 { T X a 1 1 μ a X a [ + (X a 1) 2 + 8(X ] a 0.5) 1/2 } μ a S a T tw a (f a ) = 0.5T c (1 μ a ) T { X a 1 [ + (X a 1) 2 + 8(X ] a 0.5) 1/2 } μ a S a T 0.50 X a 1 X a > 1 (2.4.24) where X a = f a /Q a is the flow/capacity ratio, the times tw a and T c are expressed in seconds, S a in pcph, and T is the duration of the oversaturation period in hours. Equation (2.4.24) is compared with the Webster formula in Fig for a value of T = 0.5 h. Note that application of the previous formulae for calculating saturation flows, capacities, and average waiting times (delays) in the case of multiple lane groups requires an exploded representation of the intersection with several links corresponding to the relevant lane groups and their maneuvers (see Fig. 2.18). For example, in the case of an exclusive right-turn lane a single link can represent such a movement and the associated delay. Sometimes, to simplify the representation, fewer links than lane groups are used; in this case the total capacity of all lane groups is associated with the single link and the resulting delay is associated with the whole flow. From a mathematical point of view the delay functions discussed so far are separable only if the traffic-signal regulation (assumed known) is such as to exclude interference between maneuvers represented by different links. For example, this is the case for the three-phase regulation scheme of a T-shaped intersection shown in Fig However, if the phases allow conflict points, for example, left turns from the opposite direction with through flows during the same phase, nonseparable cost functions may be necessary, which take account of the reciprocal reduction in saturation flow for maneuvers in conflict, such as for the two-phase scheme for the X intersection in Fig In general, if a single node represents the entire intersection, the effects of individual maneuvers and lane groups are impossible to distinguish and separable functions are adopted, with a single value of saturation flow, reduced to account for the interfering turns. In the case of control systems at signalized intersections, the control parameters (cycle length T c, ratio μ of green time to cycle length) depend on flows arriving at the access roads which converge at the intersection. In this case the delay functions are different and definitely nonseparable.

52 80 2 Transportation Supply Models Fig Examples of traffic light phases for 3- and 4-arm intersections Finally, in the case of networks of interacting intersections (i.e., so close as to affect one another), further regulation parameters must be introduced; hence, calculation of the delay cannot be performed with the formulae presented, but requires more detailed flow simulation models along the road sections joining a pair of adjacent intersections. (c) Priority Intersections To complete the survey of the delay functions, priority intersections (i.e., intersections regulated by give-way rules rather than traffic lights) need to be considered. Empirical functions are often used to express average delays; these functions are nonseparable in that right-of-way rules cause delays due to con-

53 2.4 Applications of Transportation Supply Models 81 Fig Flow conflicts for computing delays at a priority intersection flicts between flows. As an example, the delay corresponding to the maneuvers at a 4-arm intersection can be calculated by means of the following HCM function. tw a (f ) = exp ( ln(f conf ) A ( ln(f conf ) 6.92 )) (2.4.25) where tw a (f ) is the waiting time expressed in seconds f conf is the total conflicting flow, which varies according to the maneuver as showninfig.2.33 A = 1 if f conf > 1062 vehicles/h, 0 otherwise (d) Parking Links Monetary cost (fares) and search time are the most important performance attributes connected to links representing parking in a given area. In general, these attributes differ for links representing different parking types (facilities). The more sophisticated models of search time take into account the congestion effect through the ratio between the average occupancy of the parking facilities of type p, represented by link a, and the parking capacity Q l. The average search time can be calculated through a model assuming that available parking spaces of type p are uniformly distributed along a circuit, possibly

54 82 2 Transportation Supply Models mixed with parking spaces of different types (e.g., free and priced parking). If occupancy of a given parking type at the beginning and end of the reference period is inferior to capacity, the following expression can be obtained. where ts a (f a ) = L p 1 Qtot (Q p + 1) ln v s occ 2 (f a ) occ 1 Q p ( 1 + Qp occ 1 ) 1 + Q p occ 2 (f a ) (Q tot Q p ) Q p (2.4.26) ts a (f a ) is the search time in minutes f a is the flow on parking link a L p is the average length of a parking space v s is the average search speed for a free parking space occ 1 is the parking occupancy at the beginning of the reference period occ 2 is the parking occupancy at the end of the reference period, depending on flow assigned to the parking link and the turnover rate Q p is the parking capacity of type p corresponding to link a is the total capacity of all parking types mixed with type p in the zone Q tot If one or both occ are above capacity, similar but formally more complicated formulas can be obtained. These expressions are not reported here Supply Models for Scheduled Service Transportation Systems Discontinuous and nonsimultaneous transportation services can be accessed only at given points and are available only at given instants. Typical examples are scheduled services (buses, trains, airplanes, etc.), which can be used only between terminals (bus stops, stations, airports, etc.) and are available only at certain instants (departure times). Scheduled services can be represented by different supply models according to their characteristics and to the consequent assumptions on users behavior (see Sect ). The approach followed in this chapter is based upon the modeling of service lines, that is, a set of scheduled runs with equal characteristics. This approach is consistent with the assumption of intraperiod stationarity and with path choice behavior, typical of high frequency and irregular urban transit systems. If service frequency is low and/or it is assumed that the users choose specific runs, it is necessary to represent the service with a different graph known as a run graph or diachronic graph. This is usually the case with extraurban transportation services (airplanes, trains, etc.), which have low service frequencies and are largely punctual. In this case, however, the assumption of within-day stationarity does not hold. Indeed, the supply characteristics are often nonuniform within the reference period (arrival and departure times of single runs may be nonuniformly spaced). Furthermore, in order to simulate the traveler s behavior desired departure or arrival times should be introduced. For these reasons run-based supply models are described in Chap. 7 dealing with intraperiod dynamic systems.

56 84 2 Transportation Supply Models Fig Line-based graph for urban transit systems where vector b a includes the relevant characteristics of the transit system represented by link a, and vector γ a comprises a set of parameters. The average speed is strongly dependent on the type of right-of-way. For exclusive right-of-way systems, such as trains, the average speed v a can be expressed as a function of the characteristics of the vehicles (weight, power, etc.), of the infrastructure (slope, radius of bends, etc.), of the circulation regulations on the physical section and the type of service represented. Relationships of this type can be deduced from mechanics for

57 2.4 Applications of Transportation Supply Models 85 which specialized texts should be referred to. For partial right-of-way systems, such as surface buses, the average speed depends on the level of protection (e.g., reserved bus lane) and the vehicle flows on the links corresponding to interfering movements. Performance functions of this type typically derive from descriptive models. The waiting time is the average time that users spend between their arrival at the stop/station and the arrival of the line (or lines) they board. Waiting time is usually expressed as a function of the line frequency ϕ ln, that is, the average number of runs of line ln in the reference period. When only one line is available the average waiting time Tw ln will depend on the regularity of vehicle arrivals and the pattern of users arrivals at the stop. It can be shown that, under the assumption that users arrive at the stop according to a Poisson process with a constant arrival rate 6 (consistent with the within-day stationarity assumption), the average waiting time is: Tw ln = θ ϕ ln (2.4.28) where θ is equal to 0.5 if the line is perfectly regular (i.e., the headways between successive vehicle arrivals are constant), and it is equal to 1 if the line is completely irregular (i.e., the headways between successive arrivals are distributed according to a negative exponential random variable); see Fig In the case of several attractive lines, that is, when the user waits at a diversion node m for the first vehicle among those belonging to a set of lines Ln m,the average waiting time can again be calculated with expression (2.4.28) byusingthe cumulated frequency Φ m of the set of attractive lines: Tw ln = θ with Φ m = Φ m ln Ln m ϕ ln (2.4.29) Expression (2.4.29) holds in principle when vehicle arrivals of all lines are completely irregular. In this case cumulated headways can still be modeled as a negative exponential random variable, with a parameter equal to the inverse of the sum of line frequencies. In practice, however, expression (2.4.29) is often used also for intermediate values of θ. These expressions of average waiting times are revisited in Sect on path choice models for transit systems. Access/egress times are also usually modeled through very simple performance functions analogous to expression (2.4.27): Ta ln = L ln v al (b ln,γ ln ) where v al represents the average speed of the access/egress mode. Also in the case of pedestrian systems, it is possible to introduce congestion phenomena and correlate 6 To be precise, it is assumed that users arrival is a Poisson process; that is, the intervals between two successive arrivals are distributed according to a negative exponential variable.

58 86 2 Transportation Supply Models Fig Arrivals and waiting times at a bus stop the generalized transportation cost with the pedestrian density on each section by using empirical expressions described in the literature. More detailed performance models introduce congestion effects with respect to user flows both on travel times and on comfort performance attributes. An example of the first type of function is that relating the dwelling time at a stop Td ln to the user flows boarding and alighting the vehicles of each line: where f al(a) f br(a) Q D is the user flow on the alighting link is the user flow on the boarding link is the door capacity of the vehicle γ 1,γ 2,γ 3 are parameters of the function ( ) fal(a) + f br(a) Td ln = γ 1 + γ 2 γ 3 (2.4.30) Q D

59 Reference Notes 87 Another example is the function relating the average waiting time to the flow of users staying on board and those waiting to board a single line. This function takes into account the refusal probability, that is, the probability that some users may not be able to get on the first arriving run of a given line because it is too crowded and have to wait longer for a subsequent one. In the case of a single attractive line l the waiting time function can be formally expressed as Tw ln = θ ( ) fb(.) + f w(.) (2.4.31) ϕ ln(.) where ϕ ln (.) is the actual available frequency of line ln, that is, the average number of runs of the line for which there are available places. It depends on the ratio between the demand for places sum of the user flow staying on board f b(.) and the user flow willing to board, f w(.) and the line capacity Q ln. This formula is valid only for f b(.) + f w(.) >Q ln. Note that both performance functions (2.4.30) and (2.4.31) are nonseparable, in that they depend on flows on links other than the one to which they refer. Discomfort functions relate the average riding discomfort on a given line section represented by link a, dc a, to the ratio between the flow on the link (average number of users on board) and the available line capacity Q a : Q ln dc a = γ 3 f a + γ 4 ( fa Q a ) γ5 (2.4.32) where, as usual, γ 3,γ 4, and γ 5 are positive parameters, usually with γ 5 larger than one expressing more-than-linear effect of crowding. Reference Notes The application of network theory to the modeling of transportation supply systems can be found in most texts dealing with mathematical models of transportation systems, such as Potts and Oliver (1972), Newell (1980), Sheffi (1985), Cascetta (1998), Ferrari (1996), and Ortuzar and Willumsen (2001). All of these, however, deal primarily or exclusively with road networks. The presentation of a general transportation supply model and its decomposition into submodels as described in Fig is original. Performance models and the traffic flow theory are dealt with in several books and scientific papers. The former include Pignataro (1973), the ITE manual (1982), May (1990), McShane and Roess (1990), the Highway Capacity Manual (2000), and the relevant entries in the encyclopaedia edited by Papageorgiou (1991). Among the latter, the pioneering work of Webster (1958), later expanded in Webster and Cobbe (1966) and those of Catling (1977), Kimber et al. (1977), Kimber and Hollis (1978), Robertson (1979), and Akcelik (1988) on waiting times at signalized intersections. In-depth examinations of some aspects of traffic flow theory can be found in Daganzo (1997).

60 88 2 Transportation Supply Models For a theoretical analysis of queuing theory, reference can be made to Newell (1971) and Kleinrock (1975). The work of Drake et al. (1967) reviews the main speed flow density relationships, and gives an example of their calibration. The linear model was proposed by Greenshields (1934). References to nonstationary traffic flow models are in part reported in the bibliographical note to Chap. 7. A review of the road network cost functions can be found in Branston (1976), Hurdle (1984), and Lupi (1996). The study of Cartenì and Punzo (2007) contains experimental speed flow relationships for urban roadways, reported in the text (2.4.5) and updates the work by Festa and Nuzzolo (1989). The cost function for parking links (2.4.26) was proposed by Bifulco (1993). Supply models for scheduled services have traditionally received less attention in the scientific community. The line representations of scheduled systems are described in Ferrari (1996) and in Nuzzolo and Russo (1997). Several authors, such as Seddon and Day (1974), Jolliffe and Hutchinson (1975), Montella and Cascetta (1978), and Cascetta and Montella (1979), have studied the relationships between waiting times and service regularity in urban transit systems. Congested performance models discussed in Sect have been proposed by Nuzzolo and Russo (1993), and other models for waiting time at congested bus stops are quoted in Bouzaiene-Ayari et al. (1998). Mechanics of motion is treated in detail in several classical books. For an updated bibliographical note see Cantarella (2001).

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