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Simulating travel reliability


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Simulating travel reliability
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Travel time (Traffic engineering)--Mathematical models
Traffic flow--Mathematical models   ( lcsh )
Commuting--Econometric models   ( lcsh )
letter   ( marcgt )

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Simulating travel reliability
Washington, D.C.
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Travel time (Traffic engineering)--Mathematical models
Traffic flow--Mathematical models
Commuting--Econometric models
Noland, Robert B
1 773
t Center for Urban Transportation Research Publications [USF].


Simulating Travel Reliability . by Robert Noland Kenneth Small Xuehao Chu


. . .. Paper No. 9 6 0 4 5 0 PRE PRINT Duplication of this preprint for publication or sale is strictly prohibited without prior written permission of the Transportation Research Board Title: Simulating Travel Reliability Author(s): Robert B. Noland Kenneth A. Small Pia Maria Koskenoja Xuehao Chu . Transp ortation Research Board 7S'h Annual Meeting January 7-11, 1996 Washington, DC


SIMULATING TRAVEL RELIABILITY R obert B. Noland U S Env i ronmental Protect ion Agency E n ergy and Transportation Sectors Division Office ofPolicy Development 401 M St., SW, mail code 2126 Washingt on, DC 20460 Kenneth A. Small Pia Maria Koskenoja Institute ofTransportation Studies University of California, Irvine Irvine, CA 92717 XuehaoChu Center for Urban Transportation Researc h University of South Florida Tarnpa, FL 336 20 revised: January 5, 1996 This work has been funded by a. grant from California Partners f o r Advanced Transit and Highways (PATH) To be prese nted at th e Annual Meeting of the Transportation R e searc h Board, Washington DC. Jan 7-11. 1 996. A previous v ers ion was presented at the Annual Meeting of the Regional S c ienc e Assoc iation International, Niagara Falls, Ontario, Canada, November 1 7 20, 1994 and a t .J..n International Sympos ium on the Latest Developments in T rans port E c onomics and T heir Po lic y Implicatio ns, Seoul South Korea. Ju l y 28 1995 ..


ABSTRACT We present a simulation model designed to detenni ne the impact on congestion of policies for dealing with non-recurrent congestion (i.e., travel time uncertainty). The model combines a supply side model of congestion delay with a discrete choice econometric demand side model that predicts scheduling choices for mornin g trips. The supply model exogenously specifies given inputs related to uncertain travei times (the probability, severity, and duration of non-recunent events). From these a distribution of travel times is then generated, from which a mean, a standard deviation, and a probability of arriving late are calculated. The demand side model uses these outputs from the supply model as independent variables and choices are forecast using sample enumeration and a synthetic sample of work start times and free flow travel times. The process is iterated until a stable congestion pattern is achieved. We report on the components of expected cost and the average travel delay for selected sim ulations.


1 INTRODUCTION One of the great rediscoveries of transportat ion analysis is the im p ortance of reliability in travel times. Based on instinct and direct statements of trave l e rs (Pras hker 1979), trave l demand analysts have long suspected that reliability should be one of the important components of travel demand models and that unreliability is one of th e primary c osts of road congestion Yet only very recen t ly have suc h models succeeded in finding measurable effects (Small, 1992, pp 35-36). Stated-prefere n ce techni q ues have open e d the way to some solid empirica l estimates of travel time reli ab ili ty. Most of these measure how much peop l e are deterred by a higher standard deviati on of travel times, rela tive to a h igher mean trave l time (Ba tes, 1990; Black and To wriss 1993; Abdel-Aty et al. 1 994) Thes e measurements, however, do not necessarily distinguish among different reasons for aversion to unreliability. There are at lea st two such r easons. The first, which we call expected scheduling cost, is th e desire t o lower the likelihood of arriving at the destination at an inc o nveni ent t ime; many a uthors have focused specifica lly on the prob a bility that o n e w ill be l ate f or an activity wit h a defini t e starting time The second, which we ca ll p lanni ng cost, is the pure n uisance of not being abl e to plan one's activities p r ecise ly beca use of unce rtainty about wh e n a trip will b e completed. These two reasons have quite d i fferen t impl i cations for ho w people respond t o various policies. F o r exarnpie, measures that change the degree of re li abili ty, suc h as quick-response teams to clear up acci dents, might simply make p e ople happier by reducing planning costs, or they might induce complex changes in th e timin g o f people's trips by chan g in g the scheduling calculus. The same is true of m e asures, such as a d vanced trave le r inf ormati on systems. that improve inf ormat i o n about travel conditio ns. The p o ssib ility o f people changing th e timing of their trips leads to a funher question of just bow a highway corridor or nerwor k will re-equilibrate when re li ability or informatio n changes A so phisticated literature has developed to analyze e q uilibria whe n people choose their


trip schedules endogenously, but it sel dom addresses reliability. Furtheunore most of this literature asswnes greatly simplified models on both the supp l y and demand sides; usually the supply side is one or a few bottlenecks and the demand side is deterministic cost minimization with one or a few types of travelers. Exceptions on the supply side are Ben -Aki va et al. (1986) and Vythoulkas (1990), who consider networks of some complex i ty Exceptions on the demand side are De Palma eta!. (!983) and Chu (1993). Chu applies a discrete choice model of travel choice that is rich in specification and that allows for unlimited heterogeneity of preferences. In this paper we develop and e m pirically estimate a demand model whose components include both scheduling costs and planning cos t s. We then develop an equilibration model in which travel schedules congestion at each point in time, and the degree of reliability in travel times are all derived endogenously, given exogenous parameters on the probability, severity, and duration of an incident that reduces capacity This equilibration model is an extension of Chu' s, and therefore retains the advantage of a rich demand side specification The analysis of the stated preference questions led to a behavioral model of schedule choice in the face of uncertainty. Implicit in this mode l are estimates of the costs of various characteristics of the travel schedule: mean travel time, average "schedule delay early" (defined as time spent at work prior to the preferred work start time) average "schedule delay late" (defined as number of minutes that an arrival is later than that preferred work s tart time), probability of being lat e, and standard deviation of travel time. The re s ults s how t h at, consis t e n t with prior research people are modera tely averse to arriving early a t work and more averse to arriving late, with a substantial d i screte pena l ty t o b eing lat e at all in addition to a per-minute cost of lateness. The results show that once these scheduling costs are taken into account, there is little additional residual cost to uncertainty per se. In order to assess the practical importance of these findings, we performed simulations in which both recWTent and non-recWTent congestion was generated by a se t o f hypothetical commuters making their scheduling choices according to the model esti m ated as just described In these simulations, cong est ion results from insufficient capacity and uncertainty results from 2


random "incidents" which reduce cap a city in a specified manner. These simul ations show that slightly less than half of in cr ease in travel cos t s caused by inci dents is due to increased travel time; the rest is due to sc hed uling costs, prim ari l y increased probabili ty of arriving late The latter occurs despite a small tende ncy for people to adju s t to increas ing uncertai nty by leaving for work earlier, which does occur and imposes an additional cost. Followin g a system ati c review of the literature, we describe i n turn our theo retical model of t ra vele r choice, a stated -prefere nce survey we cond ucted t o obt ain empiri ca l meas urements of its parameters, the full simu l a tion m odel, and simula t ion resu l ts 2. REVIEW OF R ELATED LITERATURE 2.1 Scheduling Choice Many autho r s have postula ted traveler cos t functions that include costs of arriving either earlier or later than a preferred arrival time such as a work start time. Only a few have been estimated empirically. Smal l (19 8 2) estimates how commuters who h ave an official work start time choose their us u al travel schedules from among twe l ve possible five-minute i ntervals. The discret e choice spe c ific atio n ass ume s a fixed pen a lty (d isu tility) for arriving later than 2.5 minu t es prior to the work start time I t also a s sumes additional per-minute p enalties for arriving at w o r k either early (schedule delay early, SDE) or late (schedule delay late, SDL). Small fin ds the se pen alties to vary systematically w ith p ersonal and occupat i onal characte ri stics; on ave rag e the per-minu t e disutility of SDL i s greate r than th a t of SDE, and the fixed late ne s s penal t y is e quivalent to about 5 m inutes o f travel time (Smai l 198 2 mode l I ). C o sslett (1977) Hendri ckson and Plank (1984) and Mannering and Hamed ( 1 990) prov ide addi t io nal emp irica l measurements, the latter usin g more aggregated choice sets. Mannering and Hamed's study is unique in focus in g on the trip from work to home rather than the opposite 3


2.2 Reliability A number of a u thors have focused on the fact that unreliable tra v el times create the possibility of being lateforwork (Gaver, 1968; Abkowitz, 1981; Polak, 1987; Bates, 1990). Typically they specify scheduling costs similarly to Small (1982) as described above although usually including only a scheduling -dela y or a lateness p en alty. They then derive conditions for how travelers adjust their scheduling choices to differing amounts of reliability For example, Gaver (1968) postula tes utility to b e a linear function of trav el time, schedule delay ear ly, and schedule delay late; he then derives the scheduling choice th at maximizes the expected value of this utility given a particular distribution of uncertain trave l time Polak (1987) also takes this approach, bu t adds a concave transformation to utility in order to represe n t ri sk aversion. Our theoretical model (described below and disc ussed in detail in Noland and Small, 1995) is an extension ofthe models of Gaver (1968) and Po lak(! 987); our key innovation is the addition of a discrete lateness penalty and accounting for the time varying patt ern of co ngestion over a the course of the morning commute. We also solve the model for the expeeted costs of a morning comm ute trip To this expected cost function we hypothesize the existence of an additional term which we call the planning cost a function of the standard deviat ion of travel times Empirica l m easurements have typi cally not d istinguished between expeeted scheduling costs and planning costs. Black and Towriss (1993) assume that (expected) trave l c ost is simply a linear function of mean travel time and its standard dev iation, and estimate the parame ters A sim ilar approach is taken by Abdel-Aty et aL (1994). Bot h sets of authors estimate th i s cost function from stated-preference data. Using anew stated preference study by Kosk enoja (1994) and Small et al. (1995)1, we estimate a model that separates thes e components. We do this by estimating the underlying cost function prior to taking its expectation over the d i stribution of uncertain travel t imes. That is. the I Additional modelling details irom s

survey questions specify departure time from home as we ll as a distribution of travel times. We are therefore able t o estimate separate coefficients on sc hedul e delay earl y (SDE), probability of being late (PL), and standard deviation of travel times (S). The first two of these coefficients ace solely related to scheduling co sts, whereas the third represe nts plannirlg cost. 2.3 Equilibrium Modeling with Endogenous Scheduling There is an e xte ns ive literature on modeling equilibria or dynamic adjustment paths using a simple deterministic demand structure. On the supply side, most suc h papers use a bottleneck model that is basically that of Vickrey ( 1969) except usually simplified by making everyone's desired arrival times identical. An example is Arnott et al. (1990). These models are reviewed by Small (1992). One author, Henderson ( 1977; 1981), uses instead a supply model which applies a conventional static speed-flow curve to each cohort of travelers. Chu (1995) demonstrates that it . is essential to define that cohort by their arrival time at their destination rather than by their departure times from home; otherwise, anomalous possibilities occur and equilibria do not exist. .. Chu also shows that the Vickrey bottleneck model appears as a limiting case of the Henderson model in which the speed-flow curve becomes kinked (i.e the elasticity of travel time with respect to vehicle-capacity ratio becomes infinite). Chu ( 1 993) inves tiga tes equilibrium behavior in a model w ith a Henderson -type supply . side, and in which th e simple detenninistic demand-side specifications just discussed are replaced by a discrete -c hoice model o f scheduling very similar to that of Small (1982) Here. we extend Chu's model to incorporate reliability. This is done by making two chan ges. First, we substitute the demand mode l estimated from Koske noja 's data for that used by Chu. Second we make capacity dependent on the probability that a non-recurrent incident will occur, as well as levels for the severity and duration of the incident when it does occur. In this way trave l times become stochastic ; equi lib ration the n occurs with people responding to the entire distribution of trave l times. 5


' 3. THEORETICAL DEMAND MODEL We begjn with a theory of scheduling costs under uncertain travel times. As in Noland and Small (1995), we build upon prior work by Gaver (1968), Polak (1987), and Bates (1990) by postulating a cost function for a commuter with a particular preferred arrival time at work, which empirically is taken t o be the official work start time. This scheduling cost function, C5 is that of Small (1982): Cs = df + /J(SDE) + y(SDL) + 6D c (1) where Tis travel time, SD E and SDL are schedule delay early and late respectively, ac<:ording to the definitio n in Small (1982), and D L is equal to I when SOL> 0 and 0 otherwise (X. i s the cost of travel time, f3 andy are the costs per minute of arrivittg early and late, respectively, and e is an additional discrete lateness penalty. We define three elements oftotal commute time, T. The first is the free. flow travel time, T r. which occilrs if the highway has no congestion The extra travel time due to congestion is defined as T x This i s minimwn congested travel time that the commute r knows will occur, i.e., recurrent congestion. The added time due to non-recurrent congest ion, due for example to incident related delays (Lindley, 1987; Schrank et. a!, 1993), is defined as Tro a random variable. We define a probabiliry distribution with a mean and standard deviation for this variable; fot simplicity we a ssume it i s independent of the amount of recurrent congestion and of the time of day of travel. The official work start time tw, and t h e home departure time, [h, are u sed to defme the maximum eariy arrival time, T 0 : (2) This is the "head stan t.ime, origjnally defined by Gaver ( 1968). This enables us to rewrite the cost function as follows : 6


The next step is to calculate the expectation of this cost function using a specified distribution function. Many authors, including Richardson & Taylor (I 978) have fit log-normal curves to travel-time variance data; Giuliano (1989) has verified that non-recurrent congestion follows a log-normal distribution. Unfortunately, the log-normal distribution is analytically intractable Instead we calculate it h ere for a uniform distribution (like Polak, 1987) and an exponential distribution ( l ike Gaver, 1968) Our empirical model uses neither of these distributions but instead gen erates T from a s u pply mode l. with random capacity reductions (as specified in the discussion o f our simulati o n methodology below) The uniform distribution for T r is defined by the probability density function f{T ,)= 1/ T m fo r 0 ,;; T r ,;; T m and f{T,)=O otherwise. Its mean is Y,T m and its standard deviatio n is T ml.Jfi The exp e cted cost is 1 T1., EC5 = T JqT,) dT, (4) m 0 Assuming that 0 ,;; Te,;; T m the chosen departure time can lead to either early or late arrival depending on the realization of the random variab l e T ; expected cost is the n T I T, l T ECs = a(T r + T, + 2m) + T Jp(T.T )dT,+-T f[y(T, -T,) +B]dT mo mr ( 5 ) [ T m ] I I [m-' ()') = a Tr+Tx+ 2 +T(6'(T .. -T ,)] + 2T "' + r 1, T m (6) = aE(T) + 6'!\ + j3E(SDE ) + yE(SDL}, (7) w here PL = CTm-T el f Tm is the p robability of arriving l a te. W e see that for any given choice T e of t rave l sch edule. exp e cted scheduli n g c o s t i s s i m ply the swn of ex!)ected costs of trave l time sc hedule d elay eariy, schedu l e delay l ate, and lateness penalty Note that sin ce r andomness in T, makes e ithe r early or late arrival possible. all fo u r terms are positive. 1


When T0 < 0 or T0> Tm, Noland and Small ( 1 995) show that equation (7) still holds although certain terms are then equal to zero because E(SDE}=O in t he case T e < 0 and E (SDL) e = 0 in the caseT0> Tm The expo ne n tial distribution forT, is defi ned by the probab ili ty den s i ty function I (-T./\ Te, one can rewrite (II) as: EC5 = aE(T) + PE(SDE) + ,(SDL) + BPt ( 12) where PL = e -rh is the probability of arriving late. Equation (12) is identical t o (7). In addition to producing mis matched schedules, travel time uncertainty may also impose an inconven ience d ue to the inability t o plan one 's activities exactly. W e cal l this "planning cost and a ssume it is a function of the stan dard d e viationS o f uncertain trave l timeT witl1 coe fficient c;. Tot a l expected c os t is therefore (13) = aE(T) + PE(SDE) + .r(SDL).;. 61\ + of(S). (14) This is the basic model that we estimate in secti o n 6. Our expectation i s that <

In Noland and Small (1995}, we permit head-start T e to be chosen to minimize expect ed scheduling cost, EC5 However, here we assume i nstead that head-start is chosen from a random utility model in which disutility i s p roportio nal to EC. We fmd that this leads to similar qualitative behavior such as a shift toward earlier schedules in response to increased standard deviation of travel time. 4 STATED PREFERENCE SURVEY AND DATA COLLECTION In order to empirically estima te the trade-offs among reliability, mean travel time, and scheduling decisions a stated-preference survey was administered to a sample of more than 700 commuters in the Los Angeles region who bad a keady ta k en part in a recent pane l study (see Koskenoja (1994) and Small eta!. (1995) for mo r e details on the survey methodology). Th is strategy enabled us to take advantage of i nfonnation already compiled about employer, work start time, and travel cond it ions. It also provided an 80 percen t response rate ultimately resulting in 543 usable questionnaires. Nine stated preference choices were asked of each respondent; from this we obtained 4340 usable observatio ns on binary choice. Ea ch SP choice is between two alternative commutes to work, each with a spe c ifie d distribut i on of travel times and a specified departure time from h ome. Departu re time was presented in minu tes prior to the "usual arrival time," w h ich was ascertai ned from a prev ious question about t h e commuter's actual situation and which here t akes the rol e of work start t ime i n the theory developed above. The travel mode is not spe c ified. A sam ple question is show n i n Figure l. T he empirical demand model reponed in the nex t section is estimated from th e answers to the SP q uestions. The question fomlat i s a c omp romise between the need to desc ri be a travel time distribution that would be realistic to the and the need to keep the question simple enough to be understood. Based 0n the experience of Black and Tov.Tiss ( 1993 ), who srudied 3 since tbe estimation procedure is based on utility maximizati o n the coefficientS would aJI be negalive when specified in a utility func tion 9


different question formats with this tradeoff in mind, the travel time distribution in the curren t study is described as a five point discrete distribution, where each possible travel t ime has an equal probability. The possible travel times were determined by choosing a log-normal distribution with a given mean and standard deviation, then picking the 1st, 3rd, 5th, 7th, and 9th . decile points, each rounded to the nearest minute. The standard deviation was chosen to be larger for those commuters whose current actual travel time was longer. To represent a travel t ime dis tri bution as a discrete distribution is clearly a simplification. Two aspects could be problematic. First, it restricts the dornain of the probability distribut i on, creating an artificial certainty as to the maximmn possible delay tha t coul d occur Second, one cannot adequately capture the skewness of the underlying distribution by only five points. To counteract any hidden skewness effects, all the sets of trav e l t imes we p resented to respondents are derived from distributions with the same skewness, which means we cannot study the effects of third or higher moments of the travel time dis tribu tion. In order to reduce some ofthe problems that have been i dent ified in stated preference questions (Bonsall, 1985; Bradley and Kroes, 1990), the questions were designed to be realistic and relevant to the respondents. For example, the distributions presented were customized so as not t o deviate too far from the r esponden t 's current mean travel time. In order to avoid "affinnatio n bias," the questions were designed as abstract alternatives w ith no obvious way to pro mote any particular poli t ica l philosophy through t he answer. Finally, following a pilo t study i n which questions about tolls elicited responses that were clearly political statements, the price attribute was dropped from th e design; this means that we can measure rat ios of cost coefficients but not the actual costs. As for the design of the i ndependent variables, Hensher and Barnard ( 1990) demonstrate that an orthogonal design will contain some combinations of attribute levels for which one alte rn ative completely dominates another making that choice not very informative and possibly boring the respondent. As an alternative to orthogonal statistical design we selected from a 10


complete factorial design the largest subse t of non-dominated alternat ives i n order to form the choice sets. The three attrib utes specified in th e SP design were ass igned three l evels (hi gh, medium and l ow ) leading to a 33 matrix of attrib ute combinations. Out ofth\s matrix of27 attribute combiJ:!lltions the largest subset of non-dominated attribute combinations was chosen. TI1is subset consists of seven attribute combinations, which are presented in Table l. Randomly drawn pairs of the three attr ibute levels were assigned to each individual to create nine repea ted measure SP questions Respondents were not presented th e same pair twice. The respondents were also div i ded into 5 gro ups based on their usual commuting time from home to work with each group having a separate set of attribute com binat ions de signed for it. Departure time levels were calculated as a lin ear combination of the mean travel time and tbc standard deviation to determine three departure time levels The lowest of the three levels was the departur e time being equal to the ex pected travel time. The medium level was the expected trave l time plus one s t andard deviatio n while the highest level was the expe c ted travel time plus two stand ard deviati ons. As mentioned above, the 5 travel times were computed by choosing a log-normal distribution. The variance of the corresponding normal distribution was assumed to be constant at 0.3. The 5 points were chosen as the 1st, 3rd, 5th 7th, and 9th deciles of the chosen log normal distribut i on. The actual values used were, however, approxim at e due ro rounding to zero deci mal places Durin g our est imations (see Sec ti on 6), we recalcu la te the standard deviatio n based on the actua l round ed values presented to the respondent. Small eta!. (1995) contains a compl ete se r of the alternatives used in th e s urvey. 5. EMPIRICAL DEMAND MODEL 5.1 Analysis M ethodo l ogy The estimatio n procedure used to analyze our dat a se t is a binary logit model. This is a q ualitat i ve choice model applicable to s iruations where the dependent variable is discrete, i .e the 11


set of choice alternatives is limited and exhaustive and the alternatives are mutually exclusi ve. The model calculates a probability that the decision maker will choose a particular alternative from the set of alternatives, given the observed data. Coefficients from our demand model, specified in equation 14, are estimated. Stated preference des i gns are well suited to discrete choice binary logit models; the design provides the survey respondent with an actual choice of two hypothetical s ituati ons. One drawback in aualyzing SP surveys is that we use a repeated measures apprpach to obtain a large number of usable observa t ions. This can resu l t in an upward bias in out -statistics. Unfortunately, there are no easy procedures for correcting this bias. I nstead we note tbat the true -statistic (i.e. true standard error divided by estimated coefficient) lies between the reported statistic and an "adjusted !-statistic" which treats the repeated observations as though their error were perfectly correlated. The adjusted !-sta t istic is foW'ld, as suggested by Louviere and Woodworth ( 1983), by dividing the reported !-statistic by the square root of the number of repeated measures ( .J9 in our data set). To correspond with our analytical model (equation 14) we need to determine the expected . schedu l e delay (both e arly and la te) and the lateness probability from these q ues tio ns. Schedule delay, early and late, as used in our model were based on the definition in Small (1982). We modified that framework slightly in order to match the format of the questio n shown in F igure 1 in whic h we used the words "usua l arrival time" instead of "officia l work start time" as the basis for representing people's most preferred arrival time; we did this t o avoid having to make elaborate descrip t ions of how t o cou n t time in the e lev a tor, walking through the office, and so fort h. The stated-preference q ues tion format specifies "departure [ T a) minutes before your usual arrival time", where a specific number is inserted forT a ; that num ber is therefore. take n to be a measure of tw 111 in our notation above (The notation T a i ndica tes minutes ahead of desired arrival time.) The definitio n of early and l ate schedule delay can be formally defined as: 12


{ T.-'f;,if > 0, SDL= 0 otherwise; {1;-T., if >0, SDE. = O othei'Wise. (18) (19) T a and the five values ofT i are stated directly in the question. The expectations ofT, SDE and SDL are derived by s umming, over the five possible values and d iv iding T1 SDE, or SDL, b y five. For example, using the sample question in Figure 1 for choice A we have three possi bili t ies of arriving early since the dep artur e time i s 15 minutes before usual arriva l time, (i e Ta is 15 minutes) and travel times (T,) are 12, 13, 14, 16, and 20. I n three cases one can arrive early by 3, 2, and I minu te. Therefore to calcu l ate E(SDE) we sum the early arrivals (3 + 2 + I 0 + 0 ) and divide by 5 to get a value ofE(SDE) equal to 6/5 = 1.2 minu tes. I n choice B, E(SDE) is 1.8 minutes E(SDL) is calculat e d in a similar m a nne r and wou l d be 1. 2 m inu tes in choice A and 2 mi n ute s in choice B. The lateness p robability is detem1ined discretely by counting the number of poss i ble travel times that will result i n a l ate arrival and dividi n g by 5 Using the samp l e q uesti o n in F igure I, cho ice A h as 2 possib il ities o f arriving late (16 and 20 m inute tJ:ave J times) which result in a 40% late ness probability (PL = 0 .40 ) Choice B also h as a 40% l atenes s probab ility (PL = .. 0.40). The design of the SP questi ons prov ided onl y three d iscre t e lev els of l ateness probability: 0%, 20 %, and 40%. TI1e standard devia tion o f the travel time is defined in the usual way as the sum of five terms [Ti-E(T)]2 divide d by 5. 5 .2 Estimation Results O ur first step in analyzing th e re sults is to estimate a simple mode l th at contains o n l y the trade-off bet ween mean trave l time and the standard deviation of tr avel time The res ul t i s s hown in Table 2 column 1 Both attributes are highly significant in explaining c hoice and both estimated coeffi c ients have the expected nega tive sign (i.e. the larg er the trave l time and/or the 1 3


standard deviation, the less desirable the alternative). For comparison, column 2 shows the results of Black and Towriss (1993). They also include money in their estimations. A useful comparison is the ratio of the coefficient of standard deviation to that ofE(T). We find the ratio to be 1.32 while Black and Towriss have a ratio of0.55. Thus our estimation indicate s that each minute of standard deviation is about 30% more costly than each minute of mean travel time. One of the key differences between our study and Black and Towriss is that they did not specify the head start time as we do; it is natural that they found people less averse to trave l time variat ion because in the survey people can anticipate its effects by adjusting their schedules.4 We can also take that into account, but to do so we need to estimate the full model of equation (14). Table 3, column 1, shows the resu lt of estimating th e full model of equation (14) with planning cost assumed proportional to the standard deviation of travel time. Planning cost has a positive and significant coefficient, which is contrary to the theory. The coefficient for the mean travel time is also less than that for E(SDE) which implies that people prefer to be in traffic than to arrive early. As an altemative w e specify planning cost as proportional to the coefficient of variation (standard deviat ion div ided by mean travel time) and find that while this is statistically insignificant, it does have the appropriate sign. Tllis is shown in Table 3, column 2, and henceforth will be referred to as our "Basic Model". All coefficients in the basic model have t h e expected negative signs. They also have the expected relative magnitudes: E(SDE) is less onerous than E(T) which is less than E(SDL). However, i t is surp rising that the relative magnitude ofE(SDL) was not much greater than E(T). In fact, the adj u sted T-Stat implies that it may not even be significant. The adjusted T -Stats for the lateness probability and the coefficient of variation are also not significant. It is possible that E(SDL), PL, and the cOefficient of variation are too highly correiated LO distinguish their effects separa tel y. Column 3 !Table 3) shows that when PL is removed the 4Biack and Towriss (l993) did include a money cost in their survey. As mentioned previously we found a large politically motivated bias in our pre-test and so excluded a cost variable from our fmal survey design. 1 4


coefficients oftbe other two increase, and the ordinary T Stat (t h oug h not the adjusted T-Stat) of the coefficient of variation becomes significant. When the coefficient of variation is removed from the mode l (Table 3, column 4) the other coefficients do not noticeably change. The coefficient on late ness probability does increase slightly, indicating that it is picking up some of the explanatory power of the coefficient of variation. This result seems to suggest that our hypothesized "planning cost" is not important a variable in tbe commut ers' choice as the other variab l es. Alternatively, we may not have specified an appropriate functional form for planning cost. I n any case, it appears that muc h o f the uncertainty inhe ren t i n unreliab l e com m uting trips is better explained by the schedule delay and lateness probability variables. The relative importance of the schedule d e lay variables v.<\th respect to travel time was fust detected by Small (1982). We present his model for comparison in column 5. This model contained no uncertain t y in travel time, and therefore no planning cost (coefficient of variation) The bottom of T able 3 shows the ratios of the coeffic ients for E (SDE) and E (SDL) relative to travel time for each model. As can be seen, other than for the model in column 1 and column 3, we get very similar results to the mod e l that Small estimated The model in column I has a very hig h ratio but this probab l y a mis-spec i fication f rom including the standard devia t ion in the model with scheduling costs. T h e model in colum n 3 bas a large ratio for expected schedule delay l ate, because the lateness probability is not inc l uded. These results are generally encouraging and seem to indicate that the respondents to the questionnaire interpreted the trade offs in the SP questions appropriately. These mode l s s how that all the components of the scheduling cost, C5 in equation ( 1 4) are important determinants of the trave l c h oices indivi d uals make We believe these are the underlying factors behind the aversion to travel time uncerta inty fo und by other researchers. such as Black and Towriss (1993}. Whether planning cost is a signific ant factor when scheduling variables are taken into account remains unproven due to the sta t ist ic al insignificanc e of the 15


coefficient of variation in our bas ic model. It may be that it is a Jesser factor whose importance is too small for us to measure. The model with the highest p' (and the largest likelihood function) and that matches our theoretical formulation is the basic mode l in column 2 of Table 3. For this reason we will utilize this model in the simulations presented next. 6. SIMULATIONS WITH ENDOGENOUS CONGESTION We now combine the demanchna l ysis of the basic model ofT able 3, cohum12 with a s u pply side model of a congested highway corridor to s i mulate the effect of non-recurren t events on actual congestion patterns Th is procedure will allow us to examine scheduling shifts due to either a reduction in incident probabil ities or an expansion of capacity. We also examine the expected costs to commuters of these policy options. Firs t we discuss the basic simulation procedure and methodology We then briefly present the travel conditions generated by the sinllllations. This is f o llowed by an analysis of the pattern of scheduling shifts and the re lative components of tota l t ravel costs. 6.1 Simulation Methodology A simu l ation was performed to determ ine how uncertain capac ity w ill affect the equilibrium pattern of congestion, total commuter costs, and the per person average trave l delay. The simulatio n model used is essentially that of Chu ( 1993), but mod ified to account for random events that reduce the capacity of the hig hwa y facility We also substitute our demand model for his Capacity reducing events r esult in non-recurrent congestion and may be due to accidents, minor incidents such as b r eakdowns, or adverse weather conditions. T he probability of a capacity red ucing incident occurring can be considered an exogenous policy variable. For example, spec i fic measures to reduce the probability of an inci dent may include a state vehic l e i n spection program or increased enforcement of traffic r egulations. T h e simu l ations analyzed in thi s report focus on changes in this variable and the l evel o f capacity. 1 6


The simulation is an iterative process that balances the demand model with a supply side model of congestion. The demand model is applied to a synthetic sample of 5000 individua ls. We assume that "work start" times are eq_ual to the time that individual's depart the highway facility (i.e ., it is a "highway egress" time). Each individual would actually face some addi t ional tra vel time between their egress and their arrival at work This time would vary across the popu la ti on and hence we assume that our "work sta rt" (or "highway egress") time is normally distributed with a mean = 8:00am an d standard deviat ion= 60 minutes. Th is is opposed to h aving discrete "work start" times at fixed intervals, such a s 8:00am, 8:30am, Free flow times were generated from a dis tri bution with mean = 20 minutes and standard deviation = 5 mi n utes (these are used in our calculation of the coefficient of variation which has total travel time i n the denominator). Although our respondents faced only two cho ic es in th e SP questions, we assume th at w he n faced wi th a large r choice set they would apply a multinomiallogit choice rule to that larger set using the same estima ted utility function In our simulation there are eleven choices, ranging from E(SD E ) = 20 minutes t o E(SDL) = 20 min utes. The intermediate values for both E(SDE) and E(SDL) are 15, 10, 5, and 3 minutes, as well as th e expectation of arriving on-time (0 minutes). Alternatively, we can express t his as a choice of E(SD) ranging from -20 t o +20. F or each member of th e synthetic sample, the demand m ode l det ermines t he probabilities of each of th e eleve n possible va lue s of expect ed schedule delay, E(SD) The probab ility that a given individual will travel d ur i ng a sp e cified time interval is then calculated by summing ea ch schedule dela y interval with that individual's uni q u e "wo rk start" t ime. This i s done for each individual in the synthetic sample and gives us for each individual th e p robability of trave lling during a given time slot For example, if one individual has a "work s tart' t ime of 8:30 am (540 minutes) the proba bility thatE(SD) is -20 m inutes will be equivalent t o t he p robab ility that thar individual tra vels i n the t ime interval between 8:10 and 8:20am. Sample en um erat ion ( as described i n B en-Akiva and Lerman, 1985), which cons ist s of summing the choice probabilities 17


for each individual in the synthetic sample allows us to determine the estima ted traffi c volume for each 10 minute time slot. Our supply model appli es the fo llowi ng simp le speed-flow re lat i onship t o eac h time s l ot: (20) where T is the trave l tim e in minutes, V i s the number of vehicles leaving t he highway per hour, Cis the capacity of the fac ility, 6 i s the e l asti c i ty parameter, lis the l e ngth of the facili ty (assumed t o be equal t o 5 miles) and TO a n d T l are con s tants T he s upply mod e l of equation (20) has a long history in tr an s portat i on engineering and economics datin g b a c k at least to t he U.S. Bureau of Public Roads (1964). It was inco r porated into the U rban Transportation P l anning Process compu ter software used widely in the U .S. (Bran ston, 1976, p. 230) and h as also been used in many economic models of congestion including Vickrey (1963), Mohring (1979), and Kraus ( 1981 ), with values of ran ging fro m 2.5 to 5. Small ( 1 992, pp. 70-73) finds t h a t equat ion (2) fits quite well the data from a dynamic simulat ion of city streets in To ronto (wi t h s = 4.08) and t h e data from an aggregate analysis o f Boston express ro ads (with 6 = 3 27). Since t h e precise function is less important fo r our purposes than its g ene r al ability t o measure rapidly increasin g congest ion, we fo reg o an extens i ve empirica l estima t ion and simply used t he parameters of U S Burea u of P u bli c Road s (1964), n amely: s = 4 and T I JT O = 0. 15. We also set TO=J O minutes/ mile co r epresent a Iree-flow speed o f 6 0 miles pe r h our. It i s assumed that the vol ume used in (20) is calculated a t the po int w h e r e the flow lea ves the highway, as defined by Chu (1995 ) The capacity is assumed equal t o 1200 vehicles/hour except for random red uct ions due to i nciden ts. It is these random capacity reductions that make T stochastic We assume that the probabil ity o f an incident is the same fo r every iO minute increment o f clock time. We ais o assume thar each incident is indepe n d e nt oi other inci dentS. excep t that tOr we assw ne t ha t oni y one incident can within a given time imenr a i. \Ve also 13


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the the travel time distribution for that time inierval including the mean travel time and the standard deviation of travel time.6 This distribution was then fed back into the demand mode l (see Figure 2 for a flowchart of the simulation process). This allows us to calculate a new distribution of expected schedule delays for each individual. The demand model also uses the ratio of standard deviation to the mean travel time, the latter also includes the free flow travel time for each individual. From this the demand model allocates each individual stochastically to a clock time interval and we can enumerate over the entire synthetic sample. This process continues until the number of individuals in each time interval remains essentially constant (or changes by a very small amount) from one iteration to the next. After convergence is achieve we evaluate the congestion profile, the average travel delay, and the total cost. 6.2 Travel Conditions Generated by Simulations The travel conditions generated by the sim u lations are a function of our assumptions about incident probability levels, the severity of those incidents, the probability of a given le vel of severity occurring, and the incident duration. The travel conditions give the values that are used in the demand model and that represent the eq uilib rium le vel of the system. Here we review these values and b riefly discuss their realism and the rationale behind h.ow they are generated. Average travel times vary with the exogenous distribution of "work start" times specifi e d previously. This results in a peak travel time between ll and 15 minutes, depending on the probability of an incident occurring and a 5 minute off-peak travel time Off-peak times do not vary because the capacity reduction (from incidents) does not result in any congestion during the offpeak periods. The trave l delay which occurs only because of non-recurrent congestion, ranges from peaks of about 2.5 minutes up to 5.5 minutes in our simulations. 0We initially used a random montecarl o process to generate incident probabilities and durarion leveJs. However. we found that given the constraints of processing time we could n o t eiiminate random fluctuations which created l

The standard deviation of travel time and the coefficient of variation also vary over the peak. The maximum standard dev i ation and coefficient of variation occur at the most congested time. This is because any reduction in capacity at this time will have a much greater impact on travel times than a capacity red uc tion when traffic volumes are less. Incidents during off-peak hours \vill n ot have any impact on travel times since the r e is ample capacity, even after an incident causes a reduction in capacity. The increase in standard deviation (and the coefficient of variation) over the peak occurs despite mode ling a co nstant incident probability for eac h clock interval. The coefficient of variation ranges up to about 0.14 which matches empilical measurements ranging from 0.08 to 0.2 as reported by Bates ( 1990) The probability of arriving at work la te for any given choice of schedule shows an exp ected pattern. As the p roba bili ty of a capac ity reduct ion increas es la tene ss probability increases. The simulations generate a maximum lateness probability of slightly over 40% in the on-time case (with a 25% incident probability); th is i s a good match to the range of lat e ness probabilities in our stated preference ques tions which h ad three leve l s of 0%, 20%, and 40%. Lateness probability is greater than 0% only when there is traffic congestion. 6-3 Travel Delay and Scheduling Shifts from Incident Reduction and Capacity Expansion Reduct ions in non-rec urrent delay (expressed as incident probab ilit ies) can decrease average travel times. Increases in capacity can have a similar effect. Both may also have an . impact on schedulin g choices w hich may reduce the benefits of reductions in peak trave l time by allowing more commuters to travel during peak hours There may be reductions in scheduling costs associa t ed with any shift t o the peak. Figure 3 shows how average t ravel delay is red uced as the i ncid\mt probab il ity decreases. Both total delay and delay due on l y to non-recurrent cong e stio n decrease with decreasing incident probability. The relat ionship is essentially linear and directly rela te d to the probability 21


of an incident occurring. Obviously, policies that reduce the probability of an incident blocking capacity will result in a decrease in average travel times. Figure 4 shows th e effect on averag e travel delay for a doubling o f highway capacity from 1200 vehicles per hour to 2400 vehicles per hour. This is obviously very effective at reducing travel delay. Figure 3 showed that for a capacity of 1200 vehicles per hour, eliminating incident probabilities results in a reduction in average delay to about 2 5 minutes per vehicle. This is comparable to increasing the capacity, as shown in Figure 4, to about 1400' vehicles per hour for an incident probability level of20%. While we don't know what the costs of reducing the incident would be, we do know that freeway capacity expansions are genera lly very costly. Therefore, if reducing travel delay is the only objective, this show s the relative trade-offi; of two possible alternative strategies for reducing delay. Scheduling cosiS involved in commuting decisions may be as important as traveltime costs. Figure 5 shows that reducing the probability of an incident results in significant shifts in schedules: many commuters who previou sly plann ed lO arrive early or late now choose to instead arrive at their desired work start time. About 400 (out of 5000) more commuter s choose to arrive with schedule delay of zero when incident probabilities are zero, compared to an incident probability level of25%. For comparison, a simulation using the values calculated by Small (1982) which does not account for variance in travel times resulted in a larger shift to earlier travel periods than the current model. This can probably be attributed to the relatively larger coeffi cient for schedule delay late in S m all's model (see Table 3, column 5). Other than this differenc e the scheduling shifts of the current model show the same bas i c pattern. Such shifts do not occur as a result of increasing capacity. Figure 6 shows the difference in schedule delay choices between capacity level 1200 and 2400 fur each incident probability level. The greatest shift occurs with an incident probability of25% with a very small increase of abou t 50 (out of 5000) commuters choosing to arriv e with no schedule delay (compared to about 400 in the above case with incident probability reductions) 22


Despite the s ch eduling benefits of incident reductions, the overall congestion profile does not really change. There is a s ligh t increase i n peak travel w hen there are no i nc idents, but i t is essentially negligible and will have only a minor impac t on in creasing average travel d e la ys; therefore, the sched uling co st reduct ions do n ot seem to be off-set by significantly more congestion at the p eak. Components of Total Travel Costs The expected travel co sts can be calcul ated usin g the demand model (Table 3 column !) and the equ ilibrium tr avel conditi ons ge ne rated by the siotulations These are calculated for d iff erent i ncide n t p rob ability levels and differen t capacity leve l s. Table 4 shows the ave rage to ta l cost per trip. This i n creases as the incident probabi lity increases. The percent co n trib u t ion from e ac h component i s a l s o shown. These are components related to travel time, schedule de l ay early and late l ateness probability, and the coefficient of variation. Schedule delay early cos ts make u p th e largest segment of the to t al costs, b ut this prop ortion decreases w ith inc reasing incident probability When incident p r o bab ili ty is high, travel time costs a cco unt for the largest proportion of total costs because of the high level of n on recurrent congestion. The "planni ng cos t as indicated by the coefficient of variatio n is relativ e l y .minor, bu t doe s increase with increasing i nciden t probab ility Th e cos t s associated wi th t he probability of arriv ing late also in crease. TI1e major reduc tion in total costs witl t decreasing prob ability of an incident can be attributed to decreases i n costs of schedule delay early and l atene ss probability Ta ble 5 shows a similar br eakdown for s i m u l at i o ns increasing leve l s of c apa c i ty. T otal costs decr ease by abou t the same amou nt when capac ity i s dou bled from 1200 t o 2400 as in the case w h en incide nt probabilities are reduced from 25% to 0%. The source of the dec r ease in costs is how ever different. When capacity i s increa sed the main reduc ti on comes from redu c ti ons i n the trave l t ime costs associate d with both r ecurren t and non-recurr ent c o ngestion. 23


Scheduling costs and lateness probability remain essentially the same. The "planning cost" is again negligib le, as is its relative decrease with increasing capacity. These cost calculations are averages over all the clock intervals of the simulations. This averages peak and off-peak travelers together. Those choosing to travel at peak periods, due perhaps to job or other constraints, will face higher total costs, than those traveling at off-peak hours. The relative contribution of the various components will also differ. Table 6 shows the cost components for an o ff-peak period (between the clock interval6:35 am to 6:45 arn). Travel time costs due to non-recurrent capacity reducti ons are negligible. T h ere is also very little variation in total costs as the inci dent probability increa ses. Most of the increase is due to the costs of the probability of arriving late increasing. During the peak period, travel time costs are a significant fraction of the total costs, which increase significantly as the incident probability increases (see Table 7). Lateness probability costs also show an increase while scheduling costs do not change much and their total percent contribution decreases. 7. CONCLUSIONS Our model p rov ides a way to simulate complex interactions arnong scheduling choices reliability of travel t i mes, and congestion. It allows f or g reat h eterogeneity of travelers, considerable complexity in the determinants of scheduling choices, and a variety of underlying processes for generating ran domness i n travel t imes. At the sam e t i me, the actual choices made and the resulting travel conditions are fully endogenous. As expected, people shift their schedu le s earlier in response to increases in travel time dispersion. The main motivation for doing this is to offset the increased probability of being late. This shift is not very larg e, and it does not seem to appreciably change the extent of peaking, so the amount of congestion on those days when travel time is low (i.e., when non-recurring congestion is absent) is barely affected. 24


Scheduling ac co unts for an important parr of the costs o f congestion and o f unreliability. As the probability o f a capacity-reducing incident is increased in our model, commuters' total travel costs increases. Nearly half the increase (44 percent) is due to the extra travel time due to in cidents, and almost as much (3 7 percent) is due to the extra probability of late arrival at work; the remaining 15 percent is due to other sc h eduling considerations such as spending more time at work before work begins. This implies that greater flexibility at the workplace would substantially reduce the costs of unreliability to commuters. Once the costs of non-optimal sched u les are taken into acco u nt, uncertainty in travel time has only a very small additional cost. As just no t ed, costs rise as incident probability is . . increased, and slig htly under h a l f of this is due to increased average travei time. Of t h e rest, 92 . percent is explained by costs of early or lat e arrival at work; the "planning cost or res i dual pure cost of uncertai nty accounts for o nly 9 percent. 1l1ere f ore we are able to explain most of people's aversion to uncertainty in t enns of their being unab le to avoid the costs of early or la te arrival. This is an important finding because earlier studies measuring people's aversion to uncertainty have not distinguished among the causes. .. . . . : -.:";.-..:.,: :.. Our results suggest that one of the more effectixe::oo.yises of goverrunent p 0 licy to redu ce . th e costs as s ociated with unreliability is t o e ncoura ge more flexi ble :work schedules. Lat ( atri '(a l . . and adherence t o strict schedules seems t o b e the greatest source of both stress and the costs of unreliability. Many o ccupa ti onal categories and pro f essions may req u ire empioyees to have coordinated schedules; this is obviously dependent on the nature of the specific business or professional activity. It is therefore difficult for po li cy inter ventions to mandate the remova l of strict work schedules. Probably the best th a t can be done is to encourage flexible work schedules. We have shown how reducing the probability of incidents and non-recurrent conge s t ion can affect the c o sts of sched ule de lav Policies aimed a t reducing in cident orobabilitv mav be .. . .. more effective at reducing costs to society t han policies increasing capacity. There is an assumption that capacity increases can also reduce the probability of incidents and their severity


(i.e., tl1e number of lanes blocked); h owever, our review of the literature d i d not show any clear indi cation that this is the case. There is a clear need for research to analyze the incid ence severity and duratio n of non-recurrent events in b oth congested and unconges ted conditions and the effect on travel time variance. Methodologies for determining the cost of specific policies and how they reduce travel time variance are needed to perform cost/benefit analysi s of alte rna t ive m ethods for decreasing non-recurrent conges tion. For example, what would be the effects on travel time variance, travel costs, and traveler benefits of a expansion re l ative t o a freeway service patrol that detects and removes incidents rap id ly? We found that our hypothesized "planning cost" did not seem to account for a large fractio n of the total costs of unreliable travel. However, i f advanced traveler information systems become widely available, it coul d effect these costs; that is, people will need to plan to use them Future research could determine whether the benefits of these systems will exeeed both the monetary c o sts and p l anning costs of using t hem. 26


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TABLEJ DESIGN OF A'ITRIBUI'E LEVELS FOR SP QUESTIONS mean travel time Standard deviation of departure time travel time high medium low medium high low high low mediwn medium m ediu m medium low high medium medium low high low medium high 30


TABLE2 SIMPLE MODEL COMPARED WITH BLACK AND TOWRISS MODEL Simple Black and Model Towriss m ode l (cars only) (1993) {I) (2) E(travel time) -0.0996 -0 0635 T-Stat (-17.517) (-8 90) Adj. (-5.839) -standard deviation 0.1263 0.0352 T -Sta t (-12.669) (-3.17) Ad j T-Stat (-4.223) -Money 0.0082 T-Stat -(-6.34) N -r40 -2826.5 -l p 0.0598 l No te: Tho measure of fitness was computed as p 1-(L(B)-K)IL(O), where K equals the number of estimated parameters, L(B) is the log likelinood value evaluated at tile estimated parame ters, and L(O) = 3008 3 is the log-likelihood value evaluated setting all coefficients equal to zero. Sample size is equalro N above .. 31


T AB LE3 RESULTS OF MODEL ES TIMATIO NS with Basi c w ithout without Small stan dard Mod e l lateness coefficient (19 82), dev ia t io n probab ility of variati o n mode l!* (I) (2) (3) (4) ( 5) E(travel time) -0.0556 -0.1051 -0.128 5 0.0976 -0.106 T-Stat (-4.6 56) ( 10.148) ( 15.451) (-!'1.052) ( 2.79) Adj T-Stat ( 1.552) (-3.383) (-5. 150 ) ( 3.684) E (SDE) -0.1311 -0 .0 93 1 -0 0966 -0.0 9 4 5 -.065 T-S tat (11.38 6) ( -10.6 06) ( 1 1.004 ) ( 1 0.854) (-9 2 9 ) Adj. T S tat ( -3.79 5) (-3 .535) (-3 .66 8 ) (-3.618) E(SDL) -0 3036 -0.1299 -0 2 807 -0.1280 -0 .254 T-Sta t (-5. 085) (-2.69 4 ) (-10.594) (-2.656) ( 8.47) Adj. T -Stat ( -1.695 ) (-0. 898 ) (-3.531) (-0.885) -lateness probability -2.56 4 -1.3466 -1.529 -0. 58 T -Stat ( 6 426) ( 3.704) -(-4.495) (-2.76) Adj. T-Sta t (-2.1 42) ( -1.235) -(-1.498) -coef. of v ariation 0.346 3 0 .6674 -T -S tat (-1.403) (-2.908) --Adj. T -Stat . ( -0.4 67) (-0.969) --stan dard d e viation 0.1510 ---T-Stat ( 5 .098) -Adj. T -Stat (1. 6 99) -27 4 7.3 -2759.6 -27 6 6.5 -2760.6 2 p 0.085 1 0.0810 0.07 9 0 0.0810 !:;Qefficie n t Ratios E(S DE) I E(T) 2.358 0. 88 6 0.752 0.968 0.613 E(SD L)/E(T) 5.460 1.236 2.184 1.311 2.396 model aiso contains coeffieienfS tO adjust for rounding errors in reponed measurements. The variabfe definitions are somewhat diffemtt also. The travel time, SDE and SOL variables are actUal reported values as opposed to expected values: the Jatc:ness probability was a dummy variable for those choices involving actuat1y oTTiving at work late. whos e expectat io n w o u l d be the lateness probability in the c onte xt of the present paper 32


TABLE4 Components of Total Cost by Incident Probability (capacity= 1200) Incident Probability Average Travel E(SDE) E(SDL) Coef. of Lateness Cost T ime Variation probability ($/trip) 0 $1.5 1 27.92% 43.88% 10.9 1 % 0.00% 17 .29 % 0.1 $1.89 30 83% 36.86% 9.51% l.l3% 2 1. 67% 0.15 $2.07 32.00% 34.54% 9.08% 1.41% 22.97% 0.2 $2.24 33.04% 32.68% 8 75% 1.57% 23. 96% 0.25 $2.39 33.99% 31.17% 8.5 0% 1.65% 24.69% Difference between $0.88 44.33% 9.52% 4.39% 4.46% 37.30% highest & lowest incident probabilities TABLES Components of Total Cost by Capacity (Incident Probability= 0.2) Capacity Average Trave l E (SD B ) E (SD L ) Coe f. of Late n ess (vehic l es/hr) Cost Time Variation probability ($/trip} 1200 $2.24 33.04% 32 68% 8.75% 1.57% 23. 96% 1 500 $1.76 18.08% 4 1.67% 1 0.75% 1.02% 28.49% 1800 $1.56 1 0 10% 47.20% 1 1.90% 0.6 1 % 30. 18% 2100 $1.46 5.90% 50 37% 12.55% 0.37% 30.81% 2 4 00 $1.4 1 3.60% 5 2 .03% 12.91% 0 23% 3 1.23% Differe.nce betwee n ($0 .8 3) 83.68% 0.6 1 % 1.6 1 % 3.88% 11.44% highest & lowest capacity


TABLE6 Components of Total Cost During Off-peak Intervals, by Incident Probability (capacity = 11.00) Incident Probab il ity Average Travel E(SDE) E(SDL) Coef. of Lateness Cost Time Variation probab ility ($/trip) 0 $1.11 2.86% 60.09% 14.19% 0.00% 22. 86% 0.1 $1.27 3.65% 55.77% 1 2.85% 0.17% 27.56% 0.\5 $1.34 4 03% 54.44% 12.39% 0.23% 28.9 1 % 0.2 $1.40 4.42% 53.48% 12.01% 0 27% 29.82% 0 .25 $1.46 4.81% 52 79% 11.70% 0 31% 30.38% Difference between $0.35 1 1.05% 2 9.42% 3 73% 1.30% 5 4.49 % highest & l owest incident probabilities TABLE 7 Components of Total Cost During Peak I ntervals, by Incident Probability (capacity= 1200) Inciden t Probability Ave rage Travel E{SDE) E (SDL) Coef. of Lateness Cos t Time Variation probab ili ty ($/trip) 0 $2.02 144. 73% 32.82% 8 78% 0 00% 13.67% 0.1 $2.62 4 6 .86% 26.40% 7.60% !. 6 1 % 7 54% 0. 1 5 $2.89 47.72% 24.34% 7.25% 1.96% 18.74% 0.2 $3.16 48.47% 22.72% 6.99% '? 1 ,) Q 19.69% 0 .25 $3.40 49.13% 2 1.3 7% 6 80% 2.20% 20.50% Difference between $1.38 155.57% 4.61% 3.90% 5.42% 30.50% highest & lowest incident i probabilities I I I


FIGURE I SAIVIPLE STATED PREFERENCE QUESTION Time : minutes 12 13 14 16 20 Departure 15 minutes before your usual arrival time Please circle your choice: A 35 Tim e minutes 5 7 9 12 18 Departure 1 0 minutes before your usual arrival time ... .. B


F I G URE 2 FLOWCHART O F S IM U L AT ION PROCEDURE om o auze r.onao."esuu s P a rameters and End Simulation wltti Estimated C o nve rgen ce? Parameters Choices Syntheti c 'Work Sta rt uo T rave l T ime Standard Times Deviat i on, and Laten e s s Pro b abi l i t y Incide n t Ca l cu l ated 10,000 S e verity Leve l s a n d times? Inc i dent Durat i on l -..a o cu auon 0 1 Ca l cu lati on of T rave l T raffic Vol u mes Time for each C lock for each Cl o c k T ime Interva l T i me Int erva l 36


F I G U R E 3 Average Travel Delay, Capac ity= 1200 .... 4 r ' ...0 2 r-0 0 1 0.2 0 3 Incid ent (10 miu.Jte i n tervals) I ... del ay -o-non-r ecurTent deiay I : 37


FIGURE4 Average Travel Delay, Incident Probability = 0.2 3 1 0 00 1500 2000 2500 CapaciiY I Average dalay +non-recurrent delay I 38


FIGURE 5 Sche du le De lay Dist r ibution, b y i nciden t prob ability, capacity= 1200 0 (f) 1 0 00 f 0 .e u -z 500 -20 . -10 0 Schedule de la y 1 0 I ... 0 -+-0 1 ... 0.15 -a-0 2 -+{).25 I 39 2 0 3 0


FIGURES Change In Schedule Delay, capacity= 1 200 relative to capacity= 2400, by incident probability "" o 2 0 f 0 I -40 -80 -10 0 Schodulo delay 10 l o ... o.t ,.. o ts .,.o.2 -o-o.25JI 20 30