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Evaluating the assumption of independent turning probabilities
Transportation Research Part B 40 (2006) 903–916 www.elsevier.com/locate/trb
Evaluating the assumption of independent turning probabilities Hillel Bar-Gera a
a,*
, Pitu B. Mirchandani b, Fan Wu
b
Department of Industrial Engineering and Management, Ben-Gurion University, P.O. Box 653, Be’er-Sheva 84105, Israel b Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721-0020, United States Received 23 August 2005; received in revised form 2 December 2005; accepted 10 January 2006
Abstract Several urban traffic models make the convenient assumption that turning probabilities are independent, meaning that the probability of turning right (or left or going straight through) at the downstream intersection is the same for all travelers on that roadway, regardless of their origin or destination. In reality most travelers make turns according to planned routes from origins to destinations. The research reported here identifies and quantifies the deviations that result from this assumption of independent turning probabilities. An analysis of this type requires a set of reasonably realistic ‘‘original’’ route flows, which were obtained by a static user-equilibrium traffic assignment and an entropy maximization condition for most likely route flows. These flows are compared with those route flows resulting from the Assumption of Independent Turning Probabilities (ITP). A small subnetwork of 3 km by 5 km in Tucson, Arizona, was chosen as a case study. An overall ‘‘typical ratio’’ of 2.2 between original route flows and ITP route flows was obtained. Aggregating route flows to origin–destination flows led to an overall ‘‘typical ratio’’ of 1.7. Such deviations are particularly high for routes that go back-and-forth, reaching a ratio of more than 3 in certain time periods. Substantial deviations for origins and destinations that are on the same border of the subnetwork are also observed in the analyses. In addition, under the ITP assumption, morning rush hour traffic peaking is the same in all directions, while in the original flows some directions do not exhibit a peak in the morning rush hour period. Overall, the conclusion of the paper is that the assumption of independent turning probabilities leads to substantial deviations both at the route level and at the origin–destination level, even for such a small network of the case study. These deviations are particularly detrimental when a network is being modeled and studied for route-based measures of effectiveness such as the number and types of routes passing a point – for monitoring specified vehicles and/or managing detouring strategies. 2006 Elsevier Ltd. All rights reserved. Keywords: Route choice; User-equilibrium; Entropy maximization; Turning proportions; Markovian traffic flow models
1. Introduction The assumption of Independent Turning Probabilities (ITP) is that the probability of making a specific turning movement (straight, right or left) at the downstream intersection is the same for all travelers, *
Corresponding author. Tel.: +972 8 6461398; fax: +972 8 6472958. E-mail address: [email protected] (H. Bar-Gera).
0191-2615/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2006.01.001
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regardless of their origin or destination. The ITP assumption leads to a Markovian traffic flow model (e.g., Berman et al., 1995; Yang and Zhou, 1998). The ITP assumption is also used quite often in urban traffic studies (e.g., Lan and Davis, 1999; Thomas and Upchurch, 2000; Hansen et al., 2000; Sacks et al., 2002; Jones et al., 2004; Sisiopiku et al., 2004; Chang and Sun, 2004). Describing traffic propagation by specifying turning proportions, assuming ITP, is an available option in many urban traffic software packages, including microscopic simulation models (e.g. CORSIM 5.1, 2000; VISSIM 3.7, 2003; AIMSUN 4.1, 2002) as well as macroscopic models (e.g., TRANSYT-7F Wallace et al., 1998). While some software packages offer other alternatives including specification of an origin–destination matrix or even route flows, there are software packages (e.g., CORSIM 5.1) where the user traffic equilibrium principle (Wardrop, 1952) is followed to determine route flows from OD flows, which are subsequently translated to link flows and turning proportions and then, in the actual simulation of traffic, vehicles are moved from link to link based on these turning probabilities under the ITP assumption (Park and Qi, 2004, p. 7). In reality most travelers make turns according to their planned routes; which they choose in order to arrive efficiently from an origin to a destination, given their perception of the transportation network and its performance. The importance of routes and route choice is well recognized in the vast literature on the subject, starting with the seminal work of Beckmann et al. (1956) on the mathematical formulation of the static user-equilibrium model. The focus on routes continues in more recent studies of dynamic traffic assignment, both in analytical models (e.g., Ran and Boyce, 1996) as well as in mesoscopic simulation models (e.g., Hu and Mahmassani, 1997; Ben-Akiva et al., 1997). Appropriate consideration of route flows is particularly important when it has direct impact on model results, for example, when choosing facility locations to maximize the number of travelers that will pass at least one facility along their route. Relevant facilities in this type of problems can be traffic detectors, truck inspection stations, billboards, gasoline stations, automatic teller machines, and more. Utilizing the ITP assumption (i.e. the Markov model) in these problems as proposed in various studies (e.g., Berman et al., 1995; Yang and Zhou, 1998), is quite inappropriate as will be demonstrated later in this paper. The focus on turning proportions in urban traffic studies and models may be due in part to the simplification of model formulation, derivations, computations, and input specification. A more critical reason is data availability, considering the lack of effective direct methods for collecting data about route flows. In contrast, turning flows in urban intersections are counted traditionally by human observers, sometimes with the assistance of video cameras. In addition, turning proportions can be estimated statistically from automatic traffic detectors (e.g. Bell, 1991; Lan and Davis, 1999; Nobe, 2002). The need for estimating OD flows in networks from link flows has been recognized by various authors (e.g., Bell, 1991; Zhang et al., 2003; Muthuswamy et al., 2005) who also discuss the associated difficulties. Different OD and route flow patterns can lead to the same total link flows and corresponding turning proportions. However, route flow patterns have important effects on traffic behavior in terms of platoons, signal coordination, weaving patterns, and more. As will be demonstrated below, the ITP assumption creates deviations in route flows, which are likely to become more noticeable as microscopic simulations are applied to larger networks with thousands nodes and links (e.g., Sisiopiku et al., 2004) and in applications which require better representation of route flows. In order to decide whether the simplifying ITP assumption is appropriate or not in a specific study with its particular circumstances, it is not enough to know that in theory deviations may exist, but rather it is necessary to understand the type and magnitude of the deviations that can be expected. Therefore, the purpose of our study is to evaluate the deviations that result from the ITP assumption. An ideal situation for examining the deviations that result from the ITP assumption is when real route flows over a certain network are known. In that case, we can compute the turning proportions associated with these real route flows. We can also compute the total flows from every origin according to the real route flows. Under the ITP assumption, the flow on every route is simply the total flow from the origin times the product of the turning proportions of all the turns along the route. The resulting ITP route flows can then be compared with the original real route flows in various ways. In the absence of data on real route flows, a similar analysis can be conducted using any set of reasonably realistic route flows. At the current state of the field of traffic models, a suitable source for route flow information seems to be route choice models that are based on real network data and on analyses of travel behavior surveys.
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One of the more popular route choice models is the static User-Equilibrium (UE) model (Wardrop, 1952; Beckmann et al., 1956). This model assumes that each traveler chooses a route of shortest travel time (or generalized cost), given the congested conditions that result from the choices of all the travelers in the system. Under appropriate conditions (separable monotone link travel costs), a solution of the UE model determines total link flows uniquely. The UE route flows, however, are not uniquely determined. Among all possible UE route flow solutions, the entropy maximizing (MEUE) solution is considered the most likely one (Rossi et al., 1989). Recently, a new primal algorithm has been presented (Bar-Gera, in press) that can find the MEUE solution effectively even for large networks. Route choice models are typically applied over relatively large metropolitan regions. Our case study is based on a network for the metropolitan region of Tucson, in conjunction with origin–destination flows for eight time periods during the day (6:00–7:00, 7:00–8:00, 8:00–9:00, 9:00–10:00, 10:00–14:00, 14:00– 16:00, 16:00–18:00, and 18:00–21:00). Each time period is treated separately, leading to eight sets of route flows over the region. The availability of MEUE route flows from these models, allows determining the flows on route segments in a smaller area within the region. In our case the smaller area of interest is 3 by 5 km in the vicinity of the University of Arizona at Tucson, where the network is instrumented to obtain second by second detectors data. The effects of the ITP assumption are studied for this smaller area. 2. Definitions and terminology The traffic network consists of a set of nodes N and a set of links A. A subset of the nodes are zone centroids, Z N. We assume w.l.o.g that there is exactly one link from every zone and exactly one link to every zone; these are referred to as origin links and destination links respectively. We now need to formally define routes, route segments, route flow, etc. A (simple) route segment is a sequence of (distinct) nodes [v1, . . . , vk] such that [vl, vl+1] 2 A "1 6 l 6 k 1, and vl 62 Z "l : 2 6 l 6 k 1 as route segments are not allowed to pass through zone centroids. In particular, any route segment with two nodes, [i, j], is simply the link from node i to node j. (We assume that there is only one link, if any, between every pair of nodes, and that there are no links from a node to itself.) Similarly, every route segment with three nodes, m = [i, j, k] is the turning movement on the route from (arrival) link ma = [i, j] through node (intersection) j to (departure) link md = [j, k]. The set of all turns in the network is denoted by M A · A. A route segment denoted as [v] indicates a single-node route segment at node v. The first node of route segment s is considered its tail and denoted by st, and the last node is considered the route’s head denoted by sh. By definition, a [at, ah] for every link a 2 A and at, ah 2 N. The set of all route segments from node i to node j is denoted by Rij. If route segment s1 = [i = v1, . . . , vn = j] 2 Rij is followed by route segment s2 = [j = u1, . . . , um = k] 2 Rjk then the combination (concatenation) of the two segments is denoted by (s1 + s2) = [i = v1, . . . , vn1, vn = j = u1, u2, . . . , um = k]. The statement s1 s2 means that route segment s1 is part of route segment s2. A route is a segment that starts from a zone (its origin) and ends at a zone (its destination).SThe S set of all routes that connect origin p to destination q is denoted by Rpq, and the set of all routes by R = p2Z q2ZRpq. Correspondingly, the flow in units of vehicles per hour (vph) along route r 2 Rpq from origin P p to destination q is denoted by hr, and h denotes the vector of all route flows. The segment flow gs ðhÞ ¼ r2R:sr hr , is the aggregation of all route flows that share route segment s. Observe that if route segment s is in fact a complete route, s 2 Rpq, p, q 2 Z, then since route segments cannot pass through zones the only route that includes s is s itself, and hence gs(h) = hs. The total flow through turning movement m is gm(h). The total flow through link a will be also denoted by fa(h) = ga(h), and the vector of total link flows is denoted by f. As in most route choice models, the link generalized costs ta(fa) will be assumed as separable, strictly positive and monotonically increasing functions of total link flows. The Origin–Destination flow (OD flow) from each origin p 2 Z to every destination q 2 Z will P be denoted by d pq ðhÞ ¼ r2Rpq hr where d denotes the array of OD flows. These definitions are applicable whether the network represents an entire region, or whether the network represents a smaller ‘‘window’’ area. When it will be necessary to distinguish between the two networks, NM, AM, ZM will refer to the main regional network, and NS, AS, ZS will refer to the smaller ‘‘window’’ sub-model network.
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3. Obtaining original sets of route flows We obtained original sets of route flows using a static User-Equilibrium (UE) model for the Tucson Metropolitan region shown in Fig. 1, containing 879 zones, 3426 nodes, and 9901 links. The model was solved for eight different time periods along the day: 6:00–7:00, 7:00–8:00, 8:00–9:00, 9:00–10:00, 10:00–14:00, 14:00– 16:00, 16:00–18:00, and 18:00–21:00. The UE traffic assignment problem is formulated mathematically as X Z fa ðhÞ ½UE min T ðfðhÞÞ ¼ ta ðxÞ dx s:t:
X
0
a2A
hr ¼ d^pq
ð1Þ
8p; q 2 Z;
r2Rpq
h P 0; where ^ d are given total OD flows; ta(fa) is the link travel time function which is typically of the form 4 t0a ð1 þ 0:15 ðfa =k a Þ Þ, where t0a is the free flow travel time on link a; and ka is the ‘‘capacity’’ of link a. Solution methods for the UE assignment problem can be divided into three main categories: link-based, origin-based, and route-based. In link-based methods only the aggregated total link flows f are stored during the iterative solution process to minimize T(f(h)). In route-based methods, the set of used routes and their flows h are stored explicitly. In an origin-based solution (Bar-Gera, 2002) routes are implicitly described by a set of restricting a-cyclic subnetworks {Ap}p2Z, one for each origin; as well P as origin-based approach proportions apa 2 [0, 1] such that "a 62 Ap, apa = 0 and "p 2 Z; v 2 N; v 5 p, a2Ap :ah ¼v apa ¼ 1. This implicit set of routes from origin p is the set of all routes within Ap, that is Rpq[Ap] = {r 2 Rpq : a r ) a 2 Ap}. The implicit route flow interpretation is Y hr ¼ d^pq apa 8p; q 2 Z; r 2 Rpq . ð2Þ ar
Origin-based assignment offers high levels of convergence for large-scale networks with reasonable computation times and memory requirements (Bar-Gera, 2002). The results reported here are based on origin-based assignment solutions for the eight periods. Among all possible optimal UE route flow solutions, the entropy maximizing route flow vector is considered to be the most likely one. The Maximum Entropy User-Equilibrium (MEUE) problem is to find this route flow vector given a vector of UE total link flows f*. The problem is formulated mathematically as
150 145
km
140 135 130 125 290
300
310
320
km
Fig. 1. Road network for the Tucson metropolitan regional model. The rectangle shows the ‘‘window’’ sub-model area.
H. Bar-Gera et al. / Transportation Research Part B 40 (2006) 903–916
½MEUE
max s:t:
EðhÞ ¼ X
XX X p2Z
907
hr logðhr =d^pq Þ
q2Z r2Rpq
hr ¼ d^pq
8p; q 2 Z; ð3Þ
r2Rpq
X
hr ¼
fa
8a 2 A;
r2R: ra
h P 0: As shown in (Bar-Gera, in press), the main implication of entropy maximizing may be illustrated by the situation shown in Fig. 2: with two OD pairs (1–9 and 2–10) and two alternative routes for each OD pair (r1 ; r01 from 1 to 9 and r2 ; r02 from 2 to 10). Shifting the same amount of flow d from r1 to r01 and from r2 to r02 does not change the total link flows. A preliminary condition for entropy maximization is the consideration of a consistent set of routes, R0, such that in every situation similar to Fig. 2 with r1, r2 2 R0, the alternative routes are also included, r01 ; r02 2 R0 . Route flows in this situation are considered to be consistent, or proportional, if hr01 =hr1 ¼ hr02 =hr2 , which, in this case, is the primal optimality condition for entropy maximization. This condition simply means that the proportion of flow using segment [4, 5, 7] relative to segment [4, 6, 7] is the same for both OD pairs. So if, for example, the OD flow from 1 to 9 is 100 vph and from 2 to 10 is 400 vph; and if the flow through segment [4, 5, 7] is 200 vph while the flow through segment [4, 6, 7] is 300 vph, then in an entropy maximizing solution r1 = 40; r01 ¼ 60; r2 = 160; and r02 ¼ 240. It is quite clear that in real life this condition is not always satisfied precisely, but as a reasonable behavioral assumption it could be the most likely among all UE flows. The fundamental understanding that results from the consideration of consistency as presented above provides the basis for: (i) practical methods for identifying a consistent set of routes (which are likely to be the exact set of UE routes) and (ii) efficient primal methods for entropy maximization (Bar-Gera, in press). After identifying regional route flows for a certain time period, we extracted the relevant data for the submodel ‘‘window’’ area. This 3 by 5 km area is shown by the rectangle in the center of Fig. 1, and in further detail in Fig. 3. It includes 144 nodes, 474 links, 21 internal zones and 36 external zones. The nodes in the sub-model are a subset of the nodes from the main model NS NM. The links of the submodel are also derived directly from the main regional model, AS = AM \ (NS · NS). Zones in the sub-model can be either internal zones from the original model, Z SI ¼ Z M \ N S , represented in the figure by circles; or external border nodes, Z SE , represented in the figure by squares, which were all regular nodes in the regional model. Z S ¼ Z SI [ Z SE .
Fig. 2. Consistent consideration of alternative route segments.
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km
139
138
137
136
303
304
305
306 km
307
308
309
Fig. 3. The sub-model ‘‘window’’ area. Circles represent internal zones. Squares represent external border zones.
Every sub-model route, s 2 Rpq, p, q 2 ZS, can as a regional route segment, with its flow induced P be viewed M M by the regional model. That is hSs ¼ gM ðh Þ ¼ h . These sub-model route flows are the basis for the M s r2R :sr r analyses described in the subsequent sections. General properties of the models obtained by the process described above are given in Table 1, demonstrating the wide range of congestion conditions spanned by the eight time periods. Convergence properties of these solutions are given in Table 2, showing that all solutions are very well converged. Convergence in terms of satisfying the UE conditions is measured by the maximum excess cost over all used routes in the solution, where the excess cost is defined as the difference between the route cost and the minimum alternative cost for the same OD pair. Convergence in terms of entropy maximization is measured by the duality gap as a percentage of the total entropy.
Table 1 Key attributes of the route flow solutions Time period
Main model (regional)
Sub-model (window)
Total OD flow
Number of routes
No. of links fa > ka
Total OD flow
Number of routes
6–7 7–8 8–9 9–10 10–14 14–16 16–18 18–21
232,223 721,004 235,598 190,337 221,070 284,187 317,914 177,790
949,896 237,486,584 862,594 664,902 849,805 1,573,906 2,921,370 686,044
186 2783 99 17 18 111 219 4
38,245 105,716 36,057 27,791 34,044 43,711 48,251 25,304
2534 2818 2274 2132 2306 2776 3395 2074
Table 2 Convergence properties of route flow solutions Time period
Maximum excess cost
Total entropy
Duality gap (EM)
Duality gap (%)
6–7 7–8 8–9 9–10 10–14 14–16 16–18 18–21
2.68E09 4.59E11 9.77E14 3.50E12 5.49E12 3.07E12 1.21E13 7.11E14
33,408 358,865 23,433 10,447 15,571 37,317 61,788 7757
4.4E10 1132 182 162 153 287 8 2.3E09
0.00 0.32 0.78 1.55 0.98 0.77 0.01 0.00
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4. The assumption of independent turning probabilities The analyses proposed in this section are applicable to any network. The results shown are for the submodel network of Fig. 3. The starting point of the discussion in this section is a given vector of ‘‘original’’ route flows h, that utilizes a finite set of routes R0, so that hr = 0 for all r 62 R0. According to the definitions in Section 2, gm(h) is the total flow through turning movement m, and gma ðhÞ the total flow on the arrival link of m; therefore, the turning proportion of m is y m ðhÞ ¼ gm ðhÞ=gma ðhÞ. Denote the set of turning proportions for all turns in the network by y. Let Op(h) = g[p](h) be the total flow from origin p. Denote the set of all origin totals by O. The assumption of independent turning probabilities (ITP) suggests that given a vector of origin totals O and a set of turning proportions y the resultant flow on route r should be Y hITP ym. ð4Þ r ðO; yÞ ¼ Ort mr
To illustrate the terminology consider the stylized example network shown in Fig. 4, together with the information in Table 3 showing the eight routes that comprise R0 and their associated original route flows. From these data we can determine that g[5,2] = h[C,4,5,2,3,B] + h[D,6,5,2,1,A] = 8 + 12 = 20, g[5,2,1] = h[D,6,5,2,1,A] = 12 and g[5,2,3] = h[C,4,5,2,3,B] = 8; therefore, the turning proportions are y[5,2,1] = 0.6 and y[5,2,3] = 0.4. The origin totals are: OA = 15; OB = 10; OC = 28; and OD = 13. hITP ½A;1;4;5;6;D ¼ OA y ½A;1;4 y ½1;4;5 y ½4;5;6 y ½5;6;D ¼ 15
10 10 10 þ 20 10 þ 20 ¼ 4:93. 10 þ 5 10 þ 6 10 þ 8 þ 20 10 þ 20
ð5Þ
This is clearly different from the original route flow h[A,1,4,5,6,D] = 10. Indeed, in general hITP(O(h), y(h)) 5 h. Using Eq. (4) we can easily find the ITP route flows on every route in R0, and compare the results with the original route flows. Fig. 5 shows the results of this comparison for the peak time period 7:00–8:00. The horizontal axis shows the original route flow, and the vertical axis shows the ITP route flow. Notice that the scale on both axes is logarithmic, thus deviations from the one-to-one line are much greater then they may appear in the figure. While there is a modest correlation between the original flows and the ITP flows, the spread is very substantial. In addition, in most cases the original flow is greater than the ITP flow because under the ITP assumption 21% of the total flow uses routes that are not in R0. For other time periods we observe fairly similar patterns, with a similar spread, and 20–25% of the flow on routes that are not in the set R0 for the specific period. In addition to the graphical analysis, we desired an aggregate measure of deviation, for which we chose the entropy-based ‘‘Typical Ratio’’, ETR. This is defined as follows: P i y i j lnðxi =y i Þj P ETRðx; yÞ ¼ exp . ð6Þ iyi
Fig. 4. Stylized example network.
Table 3 Routes and route flows for stylized example Origin A
Origin B
Origin C
Origin D
Route r
Flow hr
Route r
Flow hr
Route r
Flow hr
Route r
Flow hr
[A, 1, 4, 5, 6, D] [A, 1, 2, 3, 6, D]
10 5
[B, 3, 2, 1, 4, C] [B, 3, 6, 5, 4, C]
6 4
[C, 4, 5, 2, 3, B] [C, 4, 5, 6, D]
8 20
[D, 6, 5, 2, 1, A] [D, 6, 5, 4, C]
12 1
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H. Bar-Gera et al. / Transportation Research Part B 40 (2006) 903–916 3
10
2
10
ITP route flow
1
10
0
10
–1
10
–2
10
–3
10 –3 10
–2
10
–1
0
1
10 10 10 Original route flow
2
10
3
10
Fig. 5. Comparison of the original route flows and the route flows obtained under the assumption of Independent Turning Probabilities (ITP) for the morning peak period 7:00–8:00.
This measure provides a ‘‘typical ratio’’ in the sense that, for example, ETR(1, 4) = ETR(2, 0.5) = ETR((1, 4, 1, 1, 4, 1), (4, 1, 4, 4, 1, 4)) = 4. The advantages of ETR are: (i) it ignores all zero components of y since limx!0(x Æ ln(x)) = 0; (ii) it is invariant to artificial divisions of the data into components, say by splitting an origin into two; and (iii) it is not too sensitive to outliers, in comparison, for example, to the root mean squared difference. Therefore, ETR(hITP(O(h), y(h)), h) provides a reasonable aggregate measure for the overall deviation between the original route flows and the ITP route flows. Note that like other measures of deviation, ETR does not distinguish between ‘‘bias’’ in the sense of a constant ratio as in the case of ETR((1, 1), (4, 4)) = 4, and distribution differences as in the case of ETR((1, 4), (4, 1)) = 4. The results for all time periods were analyzed, and the ETR ratios were very similar, with a typical ratio of 2.2. This ratio is a combination of contributions from the distribution mismatch between MEUE and ITP route flows as well as the overall constant ratio of 1.25–1.33 that results from the 20–25% ITP flow on non-UE routes as discussed above. In a comparison of flows on individual routes, a shift of flow from route r1 to route r2 is interpreted as a deviation of the same magnitude, regardless of whether the routes r1 and r2 are similar or not. To better understand the significance of the differences in route flows, the following subsections consider the effect of the ITP assumption on aggregated flows including (i) total link flows, (ii) OD flows, (iii) grouped OD flows, and (iv) back-and-forth flows, as defined below. 4.1. Total link flows In this subsection we show that the ITP assumption does not change total link flows. In fact, the following theorem shows that total flows on all links in the network are uniquely determined by any combination of origin total flows O and feasible turning proportions y. By feasible turning proportions we mean that: (1) the sum of P turning proportions is one from any link that is not a destination link, m:ma ¼a y m ¼ 1 8a 2 A; ah 62 Z, (since routes and route segments cannot pass through zones, there are no turns from destination links); and (2) any origin link and any turn with strictly positive (non-zero) proportion are part of a route r 2 Rpq from origin p 2 Z to destination q 2 Z, such that all the turning proportions along the route are strictly positive (non-zero), ym > 0 "m r. These conditions of feasibility are maintained by any turning proportions that are generated from route flows. To be consistent with a given vector of origin total flows O and a vector of turning proportions y, the vector of total link flows must satisfy the following set of equations: 8 at 2 Z; < Oat ; P ð7Þ fa ¼ f y ; otherwise: ma m : m:md ¼a
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Notice that without the second feasibility condition there could be a cycle in the network, say [1, 2, 3, 4, 1], with turning proportions all equal to one along the cycle, y[1,2,3] = y[2,3,4] = y[3,4,1] = y[4,1,2] = 1. In this case Eq. (7) can be satisfied with any arbitrary flow along the cycle [1, 2, 3, 4, 1], and therefore the solution is not unique. Theorem 1. If the vector of turning proportions y is feasible, then there is a unique solution for (7). Proof. It is always possible to find a solution for (7) using ITP route flows that can be generated from any vector of total origin flows and any feasible vector of turning proportions. To prove that the solution is unique, first let A0 ðyÞ ¼ fa 2 A : at 2 Zg [ fa 2 A : 9v 2 N y ½v;at ;ah > 0g be the set of all origin links together with all departure links of turns with non-zero proportion. Clearly in any solution for (7) the flow on any other link must be zero, fa = 0 "a 62 A 0 (y). So it is sufficient to consider in (7) only the links in A 0 (y), and show that the equations for these links are linearly independent. For brevity, let A 0 = A 0 (y). b where we define the right hand side vector O b ¼ fO b ¼ O, b a g 0 and the Eq. (7) can be rewritten as f ðI TÞ a2A b ¼ f Tb aa0 g 0 0 as follows: matrix T a;a 2A Oat ; at 2 Z; ba ¼ O ð8Þ 0; otherwise, ( y ½at ;ah ;a0 ; ah ¼ a0t ; h ð9Þ Tb aa0 ¼ 0; otherwise, Tb aa0 is the probability for vehicles to go from link a = [at, ah] to a0 ¼ ½a0t ; a0h . Therefore, if a vector f describes the numbers (or continuous amounts) of vehicles that are presently located on all the links in the network, b describes the situation once all vehicles are propagated one link forward according to the ITP then f T b In that respect, T b can be viewed as assumption. Vehicles that are on destination links in f, vanish in f T. a semi-transition probability matrix. Therefore, the sum of all vehicles in the network in f, that is kfk1 where P b . kxk1 ¼ i jxi j is the L1 norm, cannot be less than their sum after the transition, kf Tk 1 According to the second feasibility requirement on turning proportions, for every a 2 A 0 we can choose a continuing route from a to a destination with non-zero proportions on all of its turns. Let ka denote the number of nodes in the chosen continuing route for link a. When vehicles on link a are propagated ka links forward according to the ITP assumption, a strictly positive fraction of them will reach a destination and vanish, so that only la < 1 of the flow remains on the network. For example, in Fig. 4 k[C,4] = 3 because of the route [C, 4, 5, 6, D], and since y[C,4,5] = 1; y[4,5,6] = 0.78; and y[5,6,D] = 1 then l[C,4] = 0.22. Therefore, if b k1 k 6 l kfk . k 1 ¼ maxa2A0 k a and l ¼ maxa2A0 la < 1 then kf T 1 1 0 Furthermore, let some k2 > log(jA 0 j)/(log(l)) then k 2 logðlÞ < logðjA lk2 < 1=jA0 j. Then P jÞ2 ) 0:5 0 k 1 k 2 k 1 k 2 k2 b b kf T k2 6 kf T k1 6 l kfk1 < kfk1 =jA j 6 kfk1 6 kfk2 , where kxk2 ¼ ½ i xi is the L2 norm and kxk1 = maxijxij is the L1 norm. b that is f T b ¼ a f for a (possibly complex) scalar a. Then, Suppose that f is an eigenvector of T, b k1 k2 k ¼ kakk1 k2 kfk , which means that kak < 1. But f ðI TÞ b ¼ ð1 aÞ f, which means that f kfk2 > kf T 2 2 b with eigenvalue 1 a 5 0. Since this is true for all eignevectors, all eigenvalues is also an eigenvector of I T b are not zero, which means that I T b is a linearly independent matrix. h of I T Conclusion: If y(h1) = y(h2) and O(h1) = O(h2) then f(h1) = f(h2). In particular, f(hITP(O(h), y(h))) = f(h), meaning that ITP total link flows are always identical to the original total link flows. In particular, the total flow to any specific destination, which is the flow on the single link to that destination, is not modified by the ITP assumption. 4.2. Total OD flows The next question is whether P the ITP assumption changes total OD flows. It is straight forward to find the original OD flows, d pq ðhÞ ¼ r2Rpq hr , since R0 is finite. However, the ITP assumption suggests that flow may use cyclic routes, for example using the data in Fig. 4 and Table 3 hITP ½A;1;2;3;6;5;2;3;B ¼ OA y ½A;1;2 y ½1;2;3 y ½2;3;6 y ½3;6;5 y ½6;5;2 y ½5;2;3 y ½2;3;B ¼ 15 0:333 1 0:385 0:444 0:706 0:4 0:615 ¼ 0:149. This example also
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demonstrates that the ITP assumption can change total OD flows, since in the original route flows there is zero flow from origin A to destination B. The number of cyclic routes is infinite since for every route with one cycle there is also a route where the cycle is traversed twice, thrice, and so on. Indeed, typically, most of the flow can be captured by constraining the routes maximum length, yet even enumerating all the routes within a given length limit is likely to be computationally impractical. Nevertheless, the ITP assumption leads to a Markov process model to compute OD flows, where the probability of going to next state (next link) given current state (current link) is independent of past states (links). In this model, stages represent the number of links passed so far along the route. Thus at stage one, P 1a represents the probability that link a will be the first link along the route of a random vehicle in the population, which is 8 < POat ; at 2 Z; Op p2Z P 1a ¼ ð10Þ : 0; otherwise. At stage k, P ka is the probability that link a will be the kth link along the route of a random vehicle in the population. Unlike the definition in (9), to create a Markov process we need a proper transition probability matrix. Therefore, we consider destination links as absorbing states, meaning that once a vehicle arrives at a destination, it stays there. A transition probability matrix T = {Taa 0 }, whose dimensions equal the number of links in the network, is defined for vehicles going from link a = [at, ah] to a0 ¼ ½a0t ; a0h in a similar way to (9) as follows: 8 ah ¼ a0t ; y 0 ; > < ½at ;ah ;ah ð11Þ T aa0 ¼ 1; a ¼ a0 ; ah 2 Z; > : 0; otherwise. For each power k, Tk is the transition probability after k stages, and as k grows to infinity Tka;a0 converges to zero for all en-route links, that is, non-destination links a 0 with a0h 62 Z. Define Oat ; a ¼ a0 ; at 2 Z; 0 Daa0 ¼ ð12Þ 0; otherwise, Dk ¼ D0 Tk . Note that Dkaa0 ¼ 0 for every non-origin link a with at 62 Z. In the limit: ( d at a0h ðhITP Þ; at ; a0h 2 Z; 1 Daa0 ¼ 0; otherwise:
ð13Þ
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k we can evaluate the total amount of ITP flow on routes with more than k links by P ForPa finite k at 2Z a0h 62Z Daa0 . Fig. 6 shows how the volume of en-route flow decreases as k increases. Each line represents one of the eight time periods. In all cases the value reaches practically zero for k P 256. To be on the conservative side, results reported here are for k = 213. We assume that for such large k the approximation Dk D1 is sufficiently good and allows to determine d(hITP) according to Eq. (14). Fig. 7 shows a comparison of the original OD flows d(h) on the horizontal axis with the approximated ITP OD flows d(hITP) on the vertical axis for the peak period of 7:00–8:00. As in Fig. 5, both axes are on logarithmic scale, hence deviations from the one-to-one line may appear smaller than they really are. In this case the total OD flow is the same, and as a result the spread is more or less symmetric around the one-to-one line. It is clear that while there is some correlation between the original OD flows and the ITP OD flows, the deviations are very substantial. The aggregate measure of deviation, ETR(d(hITP), d(h)), indicates a typical ratio of 1.7 between ITP OD flows and original OD flows. This is slightly smaller than at the route level, but still quite substantial. We observed fairly similar patterns with similar spreads and nearly the same ETR deviation value for the seven other time periods. Deviations of the type described above could be, theoretically, a result of switches between neighboring origins, or between neighboring destinations, and thus have a relatively limited impact on analyses of interest. To
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see if this is the case for the ITP assumption, or whether deviations are more spread, the next subsection examines further aggregation of OD flows into zone groups. 4.3. Grouped OD flows In the ‘‘window’’ model there are two different types of zones: type I: actual zones from the regional model, located in the interior of the ‘‘window’’; and type II: border zones added to the ‘‘window’’ network in order to represent a point of entry to the window, or a point of exit from the window. We consider all interior zones as one group, and divide border zones into four groups: north border, east border, south border and west border. Fig. 8 consists of 25 subplots, each showing the flows from one zone group to another zone group for the eight time periods of the day, both for the original flows and the ITP flows. Even at this extremely aggregate level, deviations are quite substantial. The most evident differences are in flows from the north border to itself and from the south border to itself. In the original data these flows are practically zero, except for the morning peak (7:00-8:00) when congestion leads travelers to use detouring routes that enter and exit our ‘‘window’’ from the same border. The ITP
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assumption implies that such routes are used much more frequently, by as many as 1000 vehicles during the morning peak, which also contradicts the intuition about real traffic behavior. In addition, the ITP assumption overestimates flows from internal zones to themselves, as ITP assumes that travelers wander around the network until they encounter a destination, possibly relatively close to their origin. Further, the ITP assumption leads to significantly different results that do not follow the overall traffic patterns. In our case study, the original flows exhibit morning peaks except flows from the south border to the east border and vice versa. The ITP assumption does not exhibit these exceptions in the traffic patterns, but exhibit a similar peaking behavior on all directions of traffic. 4.4. Back-and-forth flows Our final test on the ITP assumption concerns the amount of traffic that is assigned to routes that go backand-forth, that is, routes that include an eastbound link and a westbound link. (Origin links and destination links are not considered for this purpose, only internal links.) Back-and-forth routes can be a sensible result of detours that travelers may make for various reasons, such as to avoid congestion. However, the ITP assumption implies that many travelers may wander around, turning from one link to the next link without any particular destination and, as a result, they are much more likely to follow back-and-forth routes. Determining the total amount of flow on back-and-forth routes within the original route flows is straightforward, since the set R0 is finite and pre-specified. Computing the equivalent value under the ITP assumption is a little more complicated. We employ again a Markov process model, similar to the one described above for identifying OD flows, only with four times as many states. Each state is defined by the current link, as before, as well as a history flag that indicates whether the route traversed thus far included links in the east or west or both directions. The history flag receives one of four possible values: (1) no previous east links and no previous west links (nE&nW); (2) one or more previous east links but no previous west links (E&nW); (3) one or more previous west links but no previous east links (nE&W); and (4) one or more previous east links as well as one or more
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previous west links (E&W). In the transition to link a 0 , the new value of the flag depends upon the value of the flag in the previous stage and the direction of the current link as shown in Table 4. Back-and-forth flows under the ITP assumption are those that end in absorbing states with flag ‘‘E&W’’. Fig. 9 shows the total amount of flow on back-and-forth routes (in the east–west direction) for all time periods both according to the original route flows and according to the ITP assumption. The figure shows that the amount of such flows under the ITP assumption is always substantially higher. In the morning peak period (7:00–8:00) it is ‘‘only’’ 1.5 times higher, while in the afternoon peak (16:00–18:00), as well as in most other time periods, it is about three times higher. These results provide another indication that the ITP assumption is problematic and not realistic. 5. Conclusions We examined the assumption of independent turning probabilities in a case study of a small traffic network in Tucson, Arizona, using eight sets of route flows over this network for different time periods of the day. We compared these ‘‘original’’ route flows with those obtained under the assumption of independent turning probabilities and found substantial differences. Differences remain substantial when the flows are aggregated by origin–destination pairs, and even when further aggregated into five zone groups. Differences are particularly substantial for flows that enter and exit the area from the same border, and when there are directional patterns in the peaking behavior of traffic. As expected, total link flows were identical. The conclusion of our findings is that even with a relatively small area, 3 · 5 km, the assumption of independent turning probabilities may lead to substantial deviations from true traffic behavior, especially with respect to route flows. So, for studying detouring behavior, and analyzing or making decisions on route-based behavior, the ITP assumption is not appropriate. On the other hand, for analysis with limited
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scope of studying link flows or link capacities, the ITP assumption may suffice. However, for majority of cases for analyzing or simulating route-based flows and behaviors, such as ‘‘capturing’’ travelers’ visibility of bill boards, analysis of traffic patterns based on the ITP assumption may lead to erroneous results. Acknowledgments The authors wish to thank David Boyce for reviewing several previous versions of this paper. The authors also like to acknowledge the US–Israel Binational Science Foundation grant #2002145 and the Federal Highway Administration Contract DTFH61-05-P-00104 for their partial support for this research. Suggestions by the reviewers of the paper are also appreciated. References AIMSUN 4.1, 2002, User Manual, Transport Simulation Systems. Bar-Gera, H., 2002. Origin-based algorithm for the traffic assignment problem. Transportation Science 36 (4), 398–417. Bar-Gera, H., in press. Primal method for determining the most likely route flows in large road networks. Transportation Science. Beckmann, M., McGuire, C.B., Winston, C.B., 1956. Studies in the Economics of Transportation. Yale University Press, New Haven, CT. Bell, M.G.H., 1991. The real time estimation of origin–destination flows in the presence of platoon dispersion. Transportation Research Part B 25 (2/3), 115–125. Ben-Akiva, M., Bierlaire, M., Bottom, J., Koutsopoulos, H., Mishalani, R., Papageorgiou, M., Pouliezos, A., 1997. Development of a route guidance generation system for real-time application. In: Transportation Systems. Berman, O., Krass, D., Xu, C.W., 1995. Locating discretionary service facilities based on probabilistic customer flows. Transportation Science 29 (3), 276–290. Chang, T.H., Sun, G.Y., 2004. Modeling and optimization of an oversaturated signalized network. Transportation Research Part B 38 (8), 687–707. CORSIM 5.1, 2000, User Manual, FHWA, Washington, DC. Hansen, B.G., Martin, P.T., Perrin, H.J., 2000. SCOOT real-time adaptive control in a CORSIM simulation environment. Transportation Research Record 1727, 27–31. Hu, T.Y., Mahmassani, H.S., 1997. Day-to-day evolution of network flows under real-time information and reactive signal control. Transportation Research Part C 5 (1), 51–69. Jones, S.L., Sullivan, A., Anderson, M., Malave, D., Cheekoti, N., 2004. Traffic simulation software comparison study. University Transportation Center for Alabama Report 02217. Lan, C.J., Davis, G.A., 1999. Real-time estimation of turning movement proportions from partial counts on urban networks. Transportation Research Part C 7 (5), 305–327. Muthuswamy, S., Levinson, D., Michalopoulos, P., Davis, G., 2005. Improving the estimation of travel demand for traffic simulation. University of Minnesota, Department of Civil Engineering, Report number CTS 04-11. Nobe, S., 2002. On-line estimation of traffic split parameters based on lane counts. Ph.D. dissertation, Systems and Industrial Engineering, The University of Arizona. Park, B., Qi, H., 2004. Development and evaluation of a calibration and validation procedure for microscopic simulation models. Virginia Transportation Research Council, VTRC 05-CR1. Ran, B., Boyce, D., 1996. Modeling Dynamic Transportation Networks, second ed. Springer-Verlag, Berlin. Rossi, T.F., McNeil, S., Hendrickson, C., 1989. Entropy model for consistent impact fee assessment. Journal of Urban Planning and Development/ASCE 115 (2), 51–63. Sacks, J., Rouphail, N.M., Park, B., Thakuriah, P., 2002. Statistically-based validation of computer simulation models in traffic operations and management. Journal of Transportation and Statistics 5 (1), 1–24. Sisiopiku, V.P., Jones, S.L., Sullivan, A.J., Patharkar, S.S., Tang, X., 2004. Regional traffic simulation for emergency preparedness. University Transportation Center for Alabama (UTCA) Report 03226. Thomas, G.B., Upchurch, J.E., 2000. Simulation of detector locations on an arterial street management system. In: Proceedings of the Mid-Continent Transportation Symposium, pp. 202–206. VISSIM 3.7, 2003, User Manual, PTV Planung Transport Verkehr AG. Wallace, C.E., Courage, K.G., Hadi, M.A., Gan, A.C., 1998. TRANSYT-7F User’s Guide. Transportation Research Center, University of Florida, Gainesville. Wardrop, J.G., 1952. Some theoretical aspects of road traffic research. In: Proceedings of the Institution of Civil Engineers, Part II, vol. 1, pp. 325–378. Yang, H., Zhou, J., 1998. Optimal traffic counting locations for origin–destination matrix estimation. Transportation Research Part B 32 (2), 109–126. Zhang, L., McHale, G., Zhang, Y., 2003. Modeling and validating CORSIM freeway origin–destination volumes. Transportation Research Record 1856, 135–142.
Evaluating the assumption of independent turning probabilities Hillel Bar-Gera a
a,*
, Pitu B. Mirchandani b, Fan Wu
b
Department of Industrial Engineering and Management, Ben-Gurion University, P.O. Box 653, Be’er-Sheva 84105, Israel b Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721-0020, United States Received 23 August 2005; received in revised form 2 December 2005; accepted 10 January 2006
Abstract Several urban traffic models make the convenient assumption that turning probabilities are independent, meaning that the probability of turning right (or left or going straight through) at the downstream intersection is the same for all travelers on that roadway, regardless of their origin or destination. In reality most travelers make turns according to planned routes from origins to destinations. The research reported here identifies and quantifies the deviations that result from this assumption of independent turning probabilities. An analysis of this type requires a set of reasonably realistic ‘‘original’’ route flows, which were obtained by a static user-equilibrium traffic assignment and an entropy maximization condition for most likely route flows. These flows are compared with those route flows resulting from the Assumption of Independent Turning Probabilities (ITP). A small subnetwork of 3 km by 5 km in Tucson, Arizona, was chosen as a case study. An overall ‘‘typical ratio’’ of 2.2 between original route flows and ITP route flows was obtained. Aggregating route flows to origin–destination flows led to an overall ‘‘typical ratio’’ of 1.7. Such deviations are particularly high for routes that go back-and-forth, reaching a ratio of more than 3 in certain time periods. Substantial deviations for origins and destinations that are on the same border of the subnetwork are also observed in the analyses. In addition, under the ITP assumption, morning rush hour traffic peaking is the same in all directions, while in the original flows some directions do not exhibit a peak in the morning rush hour period. Overall, the conclusion of the paper is that the assumption of independent turning probabilities leads to substantial deviations both at the route level and at the origin–destination level, even for such a small network of the case study. These deviations are particularly detrimental when a network is being modeled and studied for route-based measures of effectiveness such as the number and types of routes passing a point – for monitoring specified vehicles and/or managing detouring strategies. 2006 Elsevier Ltd. All rights reserved. Keywords: Route choice; User-equilibrium; Entropy maximization; Turning proportions; Markovian traffic flow models
1. Introduction The assumption of Independent Turning Probabilities (ITP) is that the probability of making a specific turning movement (straight, right or left) at the downstream intersection is the same for all travelers, *
Corresponding author. Tel.: +972 8 6461398; fax: +972 8 6472958. E-mail address: [email protected] (H. Bar-Gera).
0191-2615/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2006.01.001
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regardless of their origin or destination. The ITP assumption leads to a Markovian traffic flow model (e.g., Berman et al., 1995; Yang and Zhou, 1998). The ITP assumption is also used quite often in urban traffic studies (e.g., Lan and Davis, 1999; Thomas and Upchurch, 2000; Hansen et al., 2000; Sacks et al., 2002; Jones et al., 2004; Sisiopiku et al., 2004; Chang and Sun, 2004). Describing traffic propagation by specifying turning proportions, assuming ITP, is an available option in many urban traffic software packages, including microscopic simulation models (e.g. CORSIM 5.1, 2000; VISSIM 3.7, 2003; AIMSUN 4.1, 2002) as well as macroscopic models (e.g., TRANSYT-7F Wallace et al., 1998). While some software packages offer other alternatives including specification of an origin–destination matrix or even route flows, there are software packages (e.g., CORSIM 5.1) where the user traffic equilibrium principle (Wardrop, 1952) is followed to determine route flows from OD flows, which are subsequently translated to link flows and turning proportions and then, in the actual simulation of traffic, vehicles are moved from link to link based on these turning probabilities under the ITP assumption (Park and Qi, 2004, p. 7). In reality most travelers make turns according to their planned routes; which they choose in order to arrive efficiently from an origin to a destination, given their perception of the transportation network and its performance. The importance of routes and route choice is well recognized in the vast literature on the subject, starting with the seminal work of Beckmann et al. (1956) on the mathematical formulation of the static user-equilibrium model. The focus on routes continues in more recent studies of dynamic traffic assignment, both in analytical models (e.g., Ran and Boyce, 1996) as well as in mesoscopic simulation models (e.g., Hu and Mahmassani, 1997; Ben-Akiva et al., 1997). Appropriate consideration of route flows is particularly important when it has direct impact on model results, for example, when choosing facility locations to maximize the number of travelers that will pass at least one facility along their route. Relevant facilities in this type of problems can be traffic detectors, truck inspection stations, billboards, gasoline stations, automatic teller machines, and more. Utilizing the ITP assumption (i.e. the Markov model) in these problems as proposed in various studies (e.g., Berman et al., 1995; Yang and Zhou, 1998), is quite inappropriate as will be demonstrated later in this paper. The focus on turning proportions in urban traffic studies and models may be due in part to the simplification of model formulation, derivations, computations, and input specification. A more critical reason is data availability, considering the lack of effective direct methods for collecting data about route flows. In contrast, turning flows in urban intersections are counted traditionally by human observers, sometimes with the assistance of video cameras. In addition, turning proportions can be estimated statistically from automatic traffic detectors (e.g. Bell, 1991; Lan and Davis, 1999; Nobe, 2002). The need for estimating OD flows in networks from link flows has been recognized by various authors (e.g., Bell, 1991; Zhang et al., 2003; Muthuswamy et al., 2005) who also discuss the associated difficulties. Different OD and route flow patterns can lead to the same total link flows and corresponding turning proportions. However, route flow patterns have important effects on traffic behavior in terms of platoons, signal coordination, weaving patterns, and more. As will be demonstrated below, the ITP assumption creates deviations in route flows, which are likely to become more noticeable as microscopic simulations are applied to larger networks with thousands nodes and links (e.g., Sisiopiku et al., 2004) and in applications which require better representation of route flows. In order to decide whether the simplifying ITP assumption is appropriate or not in a specific study with its particular circumstances, it is not enough to know that in theory deviations may exist, but rather it is necessary to understand the type and magnitude of the deviations that can be expected. Therefore, the purpose of our study is to evaluate the deviations that result from the ITP assumption. An ideal situation for examining the deviations that result from the ITP assumption is when real route flows over a certain network are known. In that case, we can compute the turning proportions associated with these real route flows. We can also compute the total flows from every origin according to the real route flows. Under the ITP assumption, the flow on every route is simply the total flow from the origin times the product of the turning proportions of all the turns along the route. The resulting ITP route flows can then be compared with the original real route flows in various ways. In the absence of data on real route flows, a similar analysis can be conducted using any set of reasonably realistic route flows. At the current state of the field of traffic models, a suitable source for route flow information seems to be route choice models that are based on real network data and on analyses of travel behavior surveys.
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One of the more popular route choice models is the static User-Equilibrium (UE) model (Wardrop, 1952; Beckmann et al., 1956). This model assumes that each traveler chooses a route of shortest travel time (or generalized cost), given the congested conditions that result from the choices of all the travelers in the system. Under appropriate conditions (separable monotone link travel costs), a solution of the UE model determines total link flows uniquely. The UE route flows, however, are not uniquely determined. Among all possible UE route flow solutions, the entropy maximizing (MEUE) solution is considered the most likely one (Rossi et al., 1989). Recently, a new primal algorithm has been presented (Bar-Gera, in press) that can find the MEUE solution effectively even for large networks. Route choice models are typically applied over relatively large metropolitan regions. Our case study is based on a network for the metropolitan region of Tucson, in conjunction with origin–destination flows for eight time periods during the day (6:00–7:00, 7:00–8:00, 8:00–9:00, 9:00–10:00, 10:00–14:00, 14:00– 16:00, 16:00–18:00, and 18:00–21:00). Each time period is treated separately, leading to eight sets of route flows over the region. The availability of MEUE route flows from these models, allows determining the flows on route segments in a smaller area within the region. In our case the smaller area of interest is 3 by 5 km in the vicinity of the University of Arizona at Tucson, where the network is instrumented to obtain second by second detectors data. The effects of the ITP assumption are studied for this smaller area. 2. Definitions and terminology The traffic network consists of a set of nodes N and a set of links A. A subset of the nodes are zone centroids, Z N. We assume w.l.o.g that there is exactly one link from every zone and exactly one link to every zone; these are referred to as origin links and destination links respectively. We now need to formally define routes, route segments, route flow, etc. A (simple) route segment is a sequence of (distinct) nodes [v1, . . . , vk] such that [vl, vl+1] 2 A "1 6 l 6 k 1, and vl 62 Z "l : 2 6 l 6 k 1 as route segments are not allowed to pass through zone centroids. In particular, any route segment with two nodes, [i, j], is simply the link from node i to node j. (We assume that there is only one link, if any, between every pair of nodes, and that there are no links from a node to itself.) Similarly, every route segment with three nodes, m = [i, j, k] is the turning movement on the route from (arrival) link ma = [i, j] through node (intersection) j to (departure) link md = [j, k]. The set of all turns in the network is denoted by M A · A. A route segment denoted as [v] indicates a single-node route segment at node v. The first node of route segment s is considered its tail and denoted by st, and the last node is considered the route’s head denoted by sh. By definition, a [at, ah] for every link a 2 A and at, ah 2 N. The set of all route segments from node i to node j is denoted by Rij. If route segment s1 = [i = v1, . . . , vn = j] 2 Rij is followed by route segment s2 = [j = u1, . . . , um = k] 2 Rjk then the combination (concatenation) of the two segments is denoted by (s1 + s2) = [i = v1, . . . , vn1, vn = j = u1, u2, . . . , um = k]. The statement s1 s2 means that route segment s1 is part of route segment s2. A route is a segment that starts from a zone (its origin) and ends at a zone (its destination).SThe S set of all routes that connect origin p to destination q is denoted by Rpq, and the set of all routes by R = p2Z q2ZRpq. Correspondingly, the flow in units of vehicles per hour (vph) along route r 2 Rpq from origin P p to destination q is denoted by hr, and h denotes the vector of all route flows. The segment flow gs ðhÞ ¼ r2R:sr hr , is the aggregation of all route flows that share route segment s. Observe that if route segment s is in fact a complete route, s 2 Rpq, p, q 2 Z, then since route segments cannot pass through zones the only route that includes s is s itself, and hence gs(h) = hs. The total flow through turning movement m is gm(h). The total flow through link a will be also denoted by fa(h) = ga(h), and the vector of total link flows is denoted by f. As in most route choice models, the link generalized costs ta(fa) will be assumed as separable, strictly positive and monotonically increasing functions of total link flows. The Origin–Destination flow (OD flow) from each origin p 2 Z to every destination q 2 Z will P be denoted by d pq ðhÞ ¼ r2Rpq hr where d denotes the array of OD flows. These definitions are applicable whether the network represents an entire region, or whether the network represents a smaller ‘‘window’’ area. When it will be necessary to distinguish between the two networks, NM, AM, ZM will refer to the main regional network, and NS, AS, ZS will refer to the smaller ‘‘window’’ sub-model network.
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3. Obtaining original sets of route flows We obtained original sets of route flows using a static User-Equilibrium (UE) model for the Tucson Metropolitan region shown in Fig. 1, containing 879 zones, 3426 nodes, and 9901 links. The model was solved for eight different time periods along the day: 6:00–7:00, 7:00–8:00, 8:00–9:00, 9:00–10:00, 10:00–14:00, 14:00– 16:00, 16:00–18:00, and 18:00–21:00. The UE traffic assignment problem is formulated mathematically as X Z fa ðhÞ ½UE min T ðfðhÞÞ ¼ ta ðxÞ dx s:t:
X
0
a2A
hr ¼ d^pq
ð1Þ
8p; q 2 Z;
r2Rpq
h P 0; where ^ d are given total OD flows; ta(fa) is the link travel time function which is typically of the form 4 t0a ð1 þ 0:15 ðfa =k a Þ Þ, where t0a is the free flow travel time on link a; and ka is the ‘‘capacity’’ of link a. Solution methods for the UE assignment problem can be divided into three main categories: link-based, origin-based, and route-based. In link-based methods only the aggregated total link flows f are stored during the iterative solution process to minimize T(f(h)). In route-based methods, the set of used routes and their flows h are stored explicitly. In an origin-based solution (Bar-Gera, 2002) routes are implicitly described by a set of restricting a-cyclic subnetworks {Ap}p2Z, one for each origin; as well P as origin-based approach proportions apa 2 [0, 1] such that "a 62 Ap, apa = 0 and "p 2 Z; v 2 N; v 5 p, a2Ap :ah ¼v apa ¼ 1. This implicit set of routes from origin p is the set of all routes within Ap, that is Rpq[Ap] = {r 2 Rpq : a r ) a 2 Ap}. The implicit route flow interpretation is Y hr ¼ d^pq apa 8p; q 2 Z; r 2 Rpq . ð2Þ ar
Origin-based assignment offers high levels of convergence for large-scale networks with reasonable computation times and memory requirements (Bar-Gera, 2002). The results reported here are based on origin-based assignment solutions for the eight periods. Among all possible optimal UE route flow solutions, the entropy maximizing route flow vector is considered to be the most likely one. The Maximum Entropy User-Equilibrium (MEUE) problem is to find this route flow vector given a vector of UE total link flows f*. The problem is formulated mathematically as
150 145
km
140 135 130 125 290
300
310
320
km
Fig. 1. Road network for the Tucson metropolitan regional model. The rectangle shows the ‘‘window’’ sub-model area.
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½MEUE
max s:t:
EðhÞ ¼ X
XX X p2Z
907
hr logðhr =d^pq Þ
q2Z r2Rpq
hr ¼ d^pq
8p; q 2 Z; ð3Þ
r2Rpq
X
hr ¼
fa
8a 2 A;
r2R: ra
h P 0: As shown in (Bar-Gera, in press), the main implication of entropy maximizing may be illustrated by the situation shown in Fig. 2: with two OD pairs (1–9 and 2–10) and two alternative routes for each OD pair (r1 ; r01 from 1 to 9 and r2 ; r02 from 2 to 10). Shifting the same amount of flow d from r1 to r01 and from r2 to r02 does not change the total link flows. A preliminary condition for entropy maximization is the consideration of a consistent set of routes, R0, such that in every situation similar to Fig. 2 with r1, r2 2 R0, the alternative routes are also included, r01 ; r02 2 R0 . Route flows in this situation are considered to be consistent, or proportional, if hr01 =hr1 ¼ hr02 =hr2 , which, in this case, is the primal optimality condition for entropy maximization. This condition simply means that the proportion of flow using segment [4, 5, 7] relative to segment [4, 6, 7] is the same for both OD pairs. So if, for example, the OD flow from 1 to 9 is 100 vph and from 2 to 10 is 400 vph; and if the flow through segment [4, 5, 7] is 200 vph while the flow through segment [4, 6, 7] is 300 vph, then in an entropy maximizing solution r1 = 40; r01 ¼ 60; r2 = 160; and r02 ¼ 240. It is quite clear that in real life this condition is not always satisfied precisely, but as a reasonable behavioral assumption it could be the most likely among all UE flows. The fundamental understanding that results from the consideration of consistency as presented above provides the basis for: (i) practical methods for identifying a consistent set of routes (which are likely to be the exact set of UE routes) and (ii) efficient primal methods for entropy maximization (Bar-Gera, in press). After identifying regional route flows for a certain time period, we extracted the relevant data for the submodel ‘‘window’’ area. This 3 by 5 km area is shown by the rectangle in the center of Fig. 1, and in further detail in Fig. 3. It includes 144 nodes, 474 links, 21 internal zones and 36 external zones. The nodes in the sub-model are a subset of the nodes from the main model NS NM. The links of the submodel are also derived directly from the main regional model, AS = AM \ (NS · NS). Zones in the sub-model can be either internal zones from the original model, Z SI ¼ Z M \ N S , represented in the figure by circles; or external border nodes, Z SE , represented in the figure by squares, which were all regular nodes in the regional model. Z S ¼ Z SI [ Z SE .
Fig. 2. Consistent consideration of alternative route segments.
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km
139
138
137
136
303
304
305
306 km
307
308
309
Fig. 3. The sub-model ‘‘window’’ area. Circles represent internal zones. Squares represent external border zones.
Every sub-model route, s 2 Rpq, p, q 2 ZS, can as a regional route segment, with its flow induced P be viewed M M by the regional model. That is hSs ¼ gM ðh Þ ¼ h . These sub-model route flows are the basis for the M s r2R :sr r analyses described in the subsequent sections. General properties of the models obtained by the process described above are given in Table 1, demonstrating the wide range of congestion conditions spanned by the eight time periods. Convergence properties of these solutions are given in Table 2, showing that all solutions are very well converged. Convergence in terms of satisfying the UE conditions is measured by the maximum excess cost over all used routes in the solution, where the excess cost is defined as the difference between the route cost and the minimum alternative cost for the same OD pair. Convergence in terms of entropy maximization is measured by the duality gap as a percentage of the total entropy.
Table 1 Key attributes of the route flow solutions Time period
Main model (regional)
Sub-model (window)
Total OD flow
Number of routes
No. of links fa > ka
Total OD flow
Number of routes
6–7 7–8 8–9 9–10 10–14 14–16 16–18 18–21
232,223 721,004 235,598 190,337 221,070 284,187 317,914 177,790
949,896 237,486,584 862,594 664,902 849,805 1,573,906 2,921,370 686,044
186 2783 99 17 18 111 219 4
38,245 105,716 36,057 27,791 34,044 43,711 48,251 25,304
2534 2818 2274 2132 2306 2776 3395 2074
Table 2 Convergence properties of route flow solutions Time period
Maximum excess cost
Total entropy
Duality gap (EM)
Duality gap (%)
6–7 7–8 8–9 9–10 10–14 14–16 16–18 18–21
2.68E09 4.59E11 9.77E14 3.50E12 5.49E12 3.07E12 1.21E13 7.11E14
33,408 358,865 23,433 10,447 15,571 37,317 61,788 7757
4.4E10 1132 182 162 153 287 8 2.3E09
0.00 0.32 0.78 1.55 0.98 0.77 0.01 0.00
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4. The assumption of independent turning probabilities The analyses proposed in this section are applicable to any network. The results shown are for the submodel network of Fig. 3. The starting point of the discussion in this section is a given vector of ‘‘original’’ route flows h, that utilizes a finite set of routes R0, so that hr = 0 for all r 62 R0. According to the definitions in Section 2, gm(h) is the total flow through turning movement m, and gma ðhÞ the total flow on the arrival link of m; therefore, the turning proportion of m is y m ðhÞ ¼ gm ðhÞ=gma ðhÞ. Denote the set of turning proportions for all turns in the network by y. Let Op(h) = g[p](h) be the total flow from origin p. Denote the set of all origin totals by O. The assumption of independent turning probabilities (ITP) suggests that given a vector of origin totals O and a set of turning proportions y the resultant flow on route r should be Y hITP ym. ð4Þ r ðO; yÞ ¼ Ort mr
To illustrate the terminology consider the stylized example network shown in Fig. 4, together with the information in Table 3 showing the eight routes that comprise R0 and their associated original route flows. From these data we can determine that g[5,2] = h[C,4,5,2,3,B] + h[D,6,5,2,1,A] = 8 + 12 = 20, g[5,2,1] = h[D,6,5,2,1,A] = 12 and g[5,2,3] = h[C,4,5,2,3,B] = 8; therefore, the turning proportions are y[5,2,1] = 0.6 and y[5,2,3] = 0.4. The origin totals are: OA = 15; OB = 10; OC = 28; and OD = 13. hITP ½A;1;4;5;6;D ¼ OA y ½A;1;4 y ½1;4;5 y ½4;5;6 y ½5;6;D ¼ 15
10 10 10 þ 20 10 þ 20 ¼ 4:93. 10 þ 5 10 þ 6 10 þ 8 þ 20 10 þ 20
ð5Þ
This is clearly different from the original route flow h[A,1,4,5,6,D] = 10. Indeed, in general hITP(O(h), y(h)) 5 h. Using Eq. (4) we can easily find the ITP route flows on every route in R0, and compare the results with the original route flows. Fig. 5 shows the results of this comparison for the peak time period 7:00–8:00. The horizontal axis shows the original route flow, and the vertical axis shows the ITP route flow. Notice that the scale on both axes is logarithmic, thus deviations from the one-to-one line are much greater then they may appear in the figure. While there is a modest correlation between the original flows and the ITP flows, the spread is very substantial. In addition, in most cases the original flow is greater than the ITP flow because under the ITP assumption 21% of the total flow uses routes that are not in R0. For other time periods we observe fairly similar patterns, with a similar spread, and 20–25% of the flow on routes that are not in the set R0 for the specific period. In addition to the graphical analysis, we desired an aggregate measure of deviation, for which we chose the entropy-based ‘‘Typical Ratio’’, ETR. This is defined as follows: P i y i j lnðxi =y i Þj P ETRðx; yÞ ¼ exp . ð6Þ iyi
Fig. 4. Stylized example network.
Table 3 Routes and route flows for stylized example Origin A
Origin B
Origin C
Origin D
Route r
Flow hr
Route r
Flow hr
Route r
Flow hr
Route r
Flow hr
[A, 1, 4, 5, 6, D] [A, 1, 2, 3, 6, D]
10 5
[B, 3, 2, 1, 4, C] [B, 3, 6, 5, 4, C]
6 4
[C, 4, 5, 2, 3, B] [C, 4, 5, 6, D]
8 20
[D, 6, 5, 2, 1, A] [D, 6, 5, 4, C]
12 1
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10
2
10
ITP route flow
1
10
0
10
–1
10
–2
10
–3
10 –3 10
–2
10
–1
0
1
10 10 10 Original route flow
2
10
3
10
Fig. 5. Comparison of the original route flows and the route flows obtained under the assumption of Independent Turning Probabilities (ITP) for the morning peak period 7:00–8:00.
This measure provides a ‘‘typical ratio’’ in the sense that, for example, ETR(1, 4) = ETR(2, 0.5) = ETR((1, 4, 1, 1, 4, 1), (4, 1, 4, 4, 1, 4)) = 4. The advantages of ETR are: (i) it ignores all zero components of y since limx!0(x Æ ln(x)) = 0; (ii) it is invariant to artificial divisions of the data into components, say by splitting an origin into two; and (iii) it is not too sensitive to outliers, in comparison, for example, to the root mean squared difference. Therefore, ETR(hITP(O(h), y(h)), h) provides a reasonable aggregate measure for the overall deviation between the original route flows and the ITP route flows. Note that like other measures of deviation, ETR does not distinguish between ‘‘bias’’ in the sense of a constant ratio as in the case of ETR((1, 1), (4, 4)) = 4, and distribution differences as in the case of ETR((1, 4), (4, 1)) = 4. The results for all time periods were analyzed, and the ETR ratios were very similar, with a typical ratio of 2.2. This ratio is a combination of contributions from the distribution mismatch between MEUE and ITP route flows as well as the overall constant ratio of 1.25–1.33 that results from the 20–25% ITP flow on non-UE routes as discussed above. In a comparison of flows on individual routes, a shift of flow from route r1 to route r2 is interpreted as a deviation of the same magnitude, regardless of whether the routes r1 and r2 are similar or not. To better understand the significance of the differences in route flows, the following subsections consider the effect of the ITP assumption on aggregated flows including (i) total link flows, (ii) OD flows, (iii) grouped OD flows, and (iv) back-and-forth flows, as defined below. 4.1. Total link flows In this subsection we show that the ITP assumption does not change total link flows. In fact, the following theorem shows that total flows on all links in the network are uniquely determined by any combination of origin total flows O and feasible turning proportions y. By feasible turning proportions we mean that: (1) the sum of P turning proportions is one from any link that is not a destination link, m:ma ¼a y m ¼ 1 8a 2 A; ah 62 Z, (since routes and route segments cannot pass through zones, there are no turns from destination links); and (2) any origin link and any turn with strictly positive (non-zero) proportion are part of a route r 2 Rpq from origin p 2 Z to destination q 2 Z, such that all the turning proportions along the route are strictly positive (non-zero), ym > 0 "m r. These conditions of feasibility are maintained by any turning proportions that are generated from route flows. To be consistent with a given vector of origin total flows O and a vector of turning proportions y, the vector of total link flows must satisfy the following set of equations: 8 at 2 Z; < Oat ; P ð7Þ fa ¼ f y ; otherwise: ma m : m:md ¼a
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Notice that without the second feasibility condition there could be a cycle in the network, say [1, 2, 3, 4, 1], with turning proportions all equal to one along the cycle, y[1,2,3] = y[2,3,4] = y[3,4,1] = y[4,1,2] = 1. In this case Eq. (7) can be satisfied with any arbitrary flow along the cycle [1, 2, 3, 4, 1], and therefore the solution is not unique. Theorem 1. If the vector of turning proportions y is feasible, then there is a unique solution for (7). Proof. It is always possible to find a solution for (7) using ITP route flows that can be generated from any vector of total origin flows and any feasible vector of turning proportions. To prove that the solution is unique, first let A0 ðyÞ ¼ fa 2 A : at 2 Zg [ fa 2 A : 9v 2 N y ½v;at ;ah > 0g be the set of all origin links together with all departure links of turns with non-zero proportion. Clearly in any solution for (7) the flow on any other link must be zero, fa = 0 "a 62 A 0 (y). So it is sufficient to consider in (7) only the links in A 0 (y), and show that the equations for these links are linearly independent. For brevity, let A 0 = A 0 (y). b where we define the right hand side vector O b ¼ fO b ¼ O, b a g 0 and the Eq. (7) can be rewritten as f ðI TÞ a2A b ¼ f Tb aa0 g 0 0 as follows: matrix T a;a 2A Oat ; at 2 Z; ba ¼ O ð8Þ 0; otherwise, ( y ½at ;ah ;a0 ; ah ¼ a0t ; h ð9Þ Tb aa0 ¼ 0; otherwise, Tb aa0 is the probability for vehicles to go from link a = [at, ah] to a0 ¼ ½a0t ; a0h . Therefore, if a vector f describes the numbers (or continuous amounts) of vehicles that are presently located on all the links in the network, b describes the situation once all vehicles are propagated one link forward according to the ITP then f T b In that respect, T b can be viewed as assumption. Vehicles that are on destination links in f, vanish in f T. a semi-transition probability matrix. Therefore, the sum of all vehicles in the network in f, that is kfk1 where P b . kxk1 ¼ i jxi j is the L1 norm, cannot be less than their sum after the transition, kf Tk 1 According to the second feasibility requirement on turning proportions, for every a 2 A 0 we can choose a continuing route from a to a destination with non-zero proportions on all of its turns. Let ka denote the number of nodes in the chosen continuing route for link a. When vehicles on link a are propagated ka links forward according to the ITP assumption, a strictly positive fraction of them will reach a destination and vanish, so that only la < 1 of the flow remains on the network. For example, in Fig. 4 k[C,4] = 3 because of the route [C, 4, 5, 6, D], and since y[C,4,5] = 1; y[4,5,6] = 0.78; and y[5,6,D] = 1 then l[C,4] = 0.22. Therefore, if b k1 k 6 l kfk . k 1 ¼ maxa2A0 k a and l ¼ maxa2A0 la < 1 then kf T 1 1 0 Furthermore, let some k2 > log(jA 0 j)/(log(l)) then k 2 logðlÞ < logðjA lk2 < 1=jA0 j. Then P jÞ2 ) 0:5 0 k 1 k 2 k 1 k 2 k2 b b kf T k2 6 kf T k1 6 l kfk1 < kfk1 =jA j 6 kfk1 6 kfk2 , where kxk2 ¼ ½ i xi is the L2 norm and kxk1 = maxijxij is the L1 norm. b that is f T b ¼ a f for a (possibly complex) scalar a. Then, Suppose that f is an eigenvector of T, b k1 k2 k ¼ kakk1 k2 kfk , which means that kak < 1. But f ðI TÞ b ¼ ð1 aÞ f, which means that f kfk2 > kf T 2 2 b with eigenvalue 1 a 5 0. Since this is true for all eignevectors, all eigenvalues is also an eigenvector of I T b are not zero, which means that I T b is a linearly independent matrix. h of I T Conclusion: If y(h1) = y(h2) and O(h1) = O(h2) then f(h1) = f(h2). In particular, f(hITP(O(h), y(h))) = f(h), meaning that ITP total link flows are always identical to the original total link flows. In particular, the total flow to any specific destination, which is the flow on the single link to that destination, is not modified by the ITP assumption. 4.2. Total OD flows The next question is whether P the ITP assumption changes total OD flows. It is straight forward to find the original OD flows, d pq ðhÞ ¼ r2Rpq hr , since R0 is finite. However, the ITP assumption suggests that flow may use cyclic routes, for example using the data in Fig. 4 and Table 3 hITP ½A;1;2;3;6;5;2;3;B ¼ OA y ½A;1;2 y ½1;2;3 y ½2;3;6 y ½3;6;5 y ½6;5;2 y ½5;2;3 y ½2;3;B ¼ 15 0:333 1 0:385 0:444 0:706 0:4 0:615 ¼ 0:149. This example also
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demonstrates that the ITP assumption can change total OD flows, since in the original route flows there is zero flow from origin A to destination B. The number of cyclic routes is infinite since for every route with one cycle there is also a route where the cycle is traversed twice, thrice, and so on. Indeed, typically, most of the flow can be captured by constraining the routes maximum length, yet even enumerating all the routes within a given length limit is likely to be computationally impractical. Nevertheless, the ITP assumption leads to a Markov process model to compute OD flows, where the probability of going to next state (next link) given current state (current link) is independent of past states (links). In this model, stages represent the number of links passed so far along the route. Thus at stage one, P 1a represents the probability that link a will be the first link along the route of a random vehicle in the population, which is 8 < POat ; at 2 Z; Op p2Z P 1a ¼ ð10Þ : 0; otherwise. At stage k, P ka is the probability that link a will be the kth link along the route of a random vehicle in the population. Unlike the definition in (9), to create a Markov process we need a proper transition probability matrix. Therefore, we consider destination links as absorbing states, meaning that once a vehicle arrives at a destination, it stays there. A transition probability matrix T = {Taa 0 }, whose dimensions equal the number of links in the network, is defined for vehicles going from link a = [at, ah] to a0 ¼ ½a0t ; a0h in a similar way to (9) as follows: 8 ah ¼ a0t ; y 0 ; > < ½at ;ah ;ah ð11Þ T aa0 ¼ 1; a ¼ a0 ; ah 2 Z; > : 0; otherwise. For each power k, Tk is the transition probability after k stages, and as k grows to infinity Tka;a0 converges to zero for all en-route links, that is, non-destination links a 0 with a0h 62 Z. Define Oat ; a ¼ a0 ; at 2 Z; 0 Daa0 ¼ ð12Þ 0; otherwise, Dk ¼ D0 Tk . Note that Dkaa0 ¼ 0 for every non-origin link a with at 62 Z. In the limit: ( d at a0h ðhITP Þ; at ; a0h 2 Z; 1 Daa0 ¼ 0; otherwise:
ð13Þ
ð14Þ
k we can evaluate the total amount of ITP flow on routes with more than k links by P ForPa finite k at 2Z a0h 62Z Daa0 . Fig. 6 shows how the volume of en-route flow decreases as k increases. Each line represents one of the eight time periods. In all cases the value reaches practically zero for k P 256. To be on the conservative side, results reported here are for k = 213. We assume that for such large k the approximation Dk D1 is sufficiently good and allows to determine d(hITP) according to Eq. (14). Fig. 7 shows a comparison of the original OD flows d(h) on the horizontal axis with the approximated ITP OD flows d(hITP) on the vertical axis for the peak period of 7:00–8:00. As in Fig. 5, both axes are on logarithmic scale, hence deviations from the one-to-one line may appear smaller than they really are. In this case the total OD flow is the same, and as a result the spread is more or less symmetric around the one-to-one line. It is clear that while there is some correlation between the original OD flows and the ITP OD flows, the deviations are very substantial. The aggregate measure of deviation, ETR(d(hITP), d(h)), indicates a typical ratio of 1.7 between ITP OD flows and original OD flows. This is slightly smaller than at the route level, but still quite substantial. We observed fairly similar patterns with similar spreads and nearly the same ETR deviation value for the seven other time periods. Deviations of the type described above could be, theoretically, a result of switches between neighboring origins, or between neighboring destinations, and thus have a relatively limited impact on analyses of interest. To
Amount of flow on routes with more than k links
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913
10
10
0
10
–10
10
–20
10
–30
10
50
100
150
200
250
300
Limit on number of links in a route (k)
Fig. 6. Flow on routes with more than k links under the Assumption of Independent Turning Probabilities (ITP) reflecting the convergence of the Markov process to absorbing states. (Every line represents a model for a different time period.)
3
10
2
10
ITP OD flow
1
10
0
10
–1
10
–2
10
–3
10 –3 10
–2
10
–1
10
0
1
10 10 Original OD flow
2
10
3
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Fig. 7. Comparison of the original OD flows and the OD flows obtained under the assumption of Independent Turning Probabilities (ITP) for the morning peak period 7:00–8:00.
see if this is the case for the ITP assumption, or whether deviations are more spread, the next subsection examines further aggregation of OD flows into zone groups. 4.3. Grouped OD flows In the ‘‘window’’ model there are two different types of zones: type I: actual zones from the regional model, located in the interior of the ‘‘window’’; and type II: border zones added to the ‘‘window’’ network in order to represent a point of entry to the window, or a point of exit from the window. We consider all interior zones as one group, and divide border zones into four groups: north border, east border, south border and west border. Fig. 8 consists of 25 subplots, each showing the flows from one zone group to another zone group for the eight time periods of the day, both for the original flows and the ITP flows. Even at this extremely aggregate level, deviations are quite substantial. The most evident differences are in flows from the north border to itself and from the south border to itself. In the original data these flows are practically zero, except for the morning peak (7:00-8:00) when congestion leads travelers to use detouring routes that enter and exit our ‘‘window’’ from the same border. The ITP
From From From From From South border North border West border East border internal zone
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Time of day Fig. 8. Comparison of aggregated original flows between zone groups and the comparable flows obtained under the assumption of Independent Turning Probabilities (ITP) for different time periods along the day. Solid lines – original flows; Dashed lines – ITP flows.
assumption implies that such routes are used much more frequently, by as many as 1000 vehicles during the morning peak, which also contradicts the intuition about real traffic behavior. In addition, the ITP assumption overestimates flows from internal zones to themselves, as ITP assumes that travelers wander around the network until they encounter a destination, possibly relatively close to their origin. Further, the ITP assumption leads to significantly different results that do not follow the overall traffic patterns. In our case study, the original flows exhibit morning peaks except flows from the south border to the east border and vice versa. The ITP assumption does not exhibit these exceptions in the traffic patterns, but exhibit a similar peaking behavior on all directions of traffic. 4.4. Back-and-forth flows Our final test on the ITP assumption concerns the amount of traffic that is assigned to routes that go backand-forth, that is, routes that include an eastbound link and a westbound link. (Origin links and destination links are not considered for this purpose, only internal links.) Back-and-forth routes can be a sensible result of detours that travelers may make for various reasons, such as to avoid congestion. However, the ITP assumption implies that many travelers may wander around, turning from one link to the next link without any particular destination and, as a result, they are much more likely to follow back-and-forth routes. Determining the total amount of flow on back-and-forth routes within the original route flows is straightforward, since the set R0 is finite and pre-specified. Computing the equivalent value under the ITP assumption is a little more complicated. We employ again a Markov process model, similar to the one described above for identifying OD flows, only with four times as many states. Each state is defined by the current link, as before, as well as a history flag that indicates whether the route traversed thus far included links in the east or west or both directions. The history flag receives one of four possible values: (1) no previous east links and no previous west links (nE&nW); (2) one or more previous east links but no previous west links (E&nW); (3) one or more previous west links but no previous east links (nE&W); and (4) one or more previous east links as well as one or more
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Table 4 History state flag as a function of the flag in the previous stage and the direction of the current link direction Previous flag
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Fig. 9. Total flow on back-and-forth routes using both links going east and links going west by time period according to the original route flows and those obtained under the assumption of Independent Turning Probabilities (ITP).
previous west links (E&W). In the transition to link a 0 , the new value of the flag depends upon the value of the flag in the previous stage and the direction of the current link as shown in Table 4. Back-and-forth flows under the ITP assumption are those that end in absorbing states with flag ‘‘E&W’’. Fig. 9 shows the total amount of flow on back-and-forth routes (in the east–west direction) for all time periods both according to the original route flows and according to the ITP assumption. The figure shows that the amount of such flows under the ITP assumption is always substantially higher. In the morning peak period (7:00–8:00) it is ‘‘only’’ 1.5 times higher, while in the afternoon peak (16:00–18:00), as well as in most other time periods, it is about three times higher. These results provide another indication that the ITP assumption is problematic and not realistic. 5. Conclusions We examined the assumption of independent turning probabilities in a case study of a small traffic network in Tucson, Arizona, using eight sets of route flows over this network for different time periods of the day. We compared these ‘‘original’’ route flows with those obtained under the assumption of independent turning probabilities and found substantial differences. Differences remain substantial when the flows are aggregated by origin–destination pairs, and even when further aggregated into five zone groups. Differences are particularly substantial for flows that enter and exit the area from the same border, and when there are directional patterns in the peaking behavior of traffic. As expected, total link flows were identical. The conclusion of our findings is that even with a relatively small area, 3 · 5 km, the assumption of independent turning probabilities may lead to substantial deviations from true traffic behavior, especially with respect to route flows. So, for studying detouring behavior, and analyzing or making decisions on route-based behavior, the ITP assumption is not appropriate. On the other hand, for analysis with limited
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scope of studying link flows or link capacities, the ITP assumption may suffice. However, for majority of cases for analyzing or simulating route-based flows and behaviors, such as ‘‘capturing’’ travelers’ visibility of bill boards, analysis of traffic patterns based on the ITP assumption may lead to erroneous results. Acknowledgments The authors wish to thank David Boyce for reviewing several previous versions of this paper. The authors also like to acknowledge the US–Israel Binational Science Foundation grant #2002145 and the Federal Highway Administration Contract DTFH61-05-P-00104 for their partial support for this research. Suggestions by the reviewers of the paper are also appreciated. References AIMSUN 4.1, 2002, User Manual, Transport Simulation Systems. Bar-Gera, H., 2002. Origin-based algorithm for the traffic assignment problem. Transportation Science 36 (4), 398–417. Bar-Gera, H., in press. Primal method for determining the most likely route flows in large road networks. Transportation Science. Beckmann, M., McGuire, C.B., Winston, C.B., 1956. Studies in the Economics of Transportation. Yale University Press, New Haven, CT. Bell, M.G.H., 1991. The real time estimation of origin–destination flows in the presence of platoon dispersion. Transportation Research Part B 25 (2/3), 115–125. Ben-Akiva, M., Bierlaire, M., Bottom, J., Koutsopoulos, H., Mishalani, R., Papageorgiou, M., Pouliezos, A., 1997. Development of a route guidance generation system for real-time application. In: Transportation Systems. Berman, O., Krass, D., Xu, C.W., 1995. Locating discretionary service facilities based on probabilistic customer flows. Transportation Science 29 (3), 276–290. Chang, T.H., Sun, G.Y., 2004. Modeling and optimization of an oversaturated signalized network. Transportation Research Part B 38 (8), 687–707. CORSIM 5.1, 2000, User Manual, FHWA, Washington, DC. Hansen, B.G., Martin, P.T., Perrin, H.J., 2000. SCOOT real-time adaptive control in a CORSIM simulation environment. Transportation Research Record 1727, 27–31. Hu, T.Y., Mahmassani, H.S., 1997. Day-to-day evolution of network flows under real-time information and reactive signal control. Transportation Research Part C 5 (1), 51–69. Jones, S.L., Sullivan, A., Anderson, M., Malave, D., Cheekoti, N., 2004. Traffic simulation software comparison study. University Transportation Center for Alabama Report 02217. Lan, C.J., Davis, G.A., 1999. Real-time estimation of turning movement proportions from partial counts on urban networks. Transportation Research Part C 7 (5), 305–327. Muthuswamy, S., Levinson, D., Michalopoulos, P., Davis, G., 2005. Improving the estimation of travel demand for traffic simulation. University of Minnesota, Department of Civil Engineering, Report number CTS 04-11. Nobe, S., 2002. On-line estimation of traffic split parameters based on lane counts. Ph.D. dissertation, Systems and Industrial Engineering, The University of Arizona. Park, B., Qi, H., 2004. Development and evaluation of a calibration and validation procedure for microscopic simulation models. Virginia Transportation Research Council, VTRC 05-CR1. Ran, B., Boyce, D., 1996. Modeling Dynamic Transportation Networks, second ed. Springer-Verlag, Berlin. Rossi, T.F., McNeil, S., Hendrickson, C., 1989. Entropy model for consistent impact fee assessment. Journal of Urban Planning and Development/ASCE 115 (2), 51–63. Sacks, J., Rouphail, N.M., Park, B., Thakuriah, P., 2002. Statistically-based validation of computer simulation models in traffic operations and management. Journal of Transportation and Statistics 5 (1), 1–24. Sisiopiku, V.P., Jones, S.L., Sullivan, A.J., Patharkar, S.S., Tang, X., 2004. Regional traffic simulation for emergency preparedness. University Transportation Center for Alabama (UTCA) Report 03226. Thomas, G.B., Upchurch, J.E., 2000. Simulation of detector locations on an arterial street management system. In: Proceedings of the Mid-Continent Transportation Symposium, pp. 202–206. VISSIM 3.7, 2003, User Manual, PTV Planung Transport Verkehr AG. Wallace, C.E., Courage, K.G., Hadi, M.A., Gan, A.C., 1998. TRANSYT-7F User’s Guide. Transportation Research Center, University of Florida, Gainesville. Wardrop, J.G., 1952. Some theoretical aspects of road traffic research. In: Proceedings of the Institution of Civil Engineers, Part II, vol. 1, pp. 325–378. Yang, H., Zhou, J., 1998. Optimal traffic counting locations for origin–destination matrix estimation. Transportation Research Part B 32 (2), 109–126. Zhang, L., McHale, G., Zhang, Y., 2003. Modeling and validating CORSIM freeway origin–destination volumes. Transportation Research Record 1856, 135–142.