ASSESSING EXPECTED ACCURACY OF PROBE VEHICLE TRAVEL TIME REPORTS

Transcription

1 ASSESSING EXPECTED ACCURACY OF PROBE VEHICLE TRAVEL TIME REPORTS By Bruce Hellinga, 1 P.E., and Liping Fu 2 (Reviewed by the Urban Tranportation Diviion) ABSTRACT: The ue of probe vehicle to provide etimate of link travel time ha been uggeted a a mean of obtaining travel time within ignalized network for ue in advanced travel information ytem. Pat reearch in the literature ha provided contradictory concluion regarding the expected accuracy of thee probe-baed etimate, and conequently ha etimated different level of market penetration of probe vehicle required to utain accurate data within an advanced traveler information ytem. Thi paper examine the effect of ampling bia on the accuracy of the probe etimate. An analytical expreion i derived on the bai of queuing theory to prove that bia in arrival time ditribution and/or in the proportion of probe aociated with each link departure turning movement will lead to a ytematic bia in the ample etimate of the mean delay. Subequently, the potential for and impact of ampling bia on a ignalized link i examined by imulating an arterial corridor. The analytical derivation and the imulation analyi how that the reliability of probe-baed average link travel time i highly affected by ampling bia. Furthermore, thi analyi how that the contradictory concluion of previou reearch are directly related to the preence or abence of ample bia. INTRODUCTION The ucceful wide cale deployment of advanced traveler information ytem (ATIS) and advanced traffic management ytem depend on the ability to obtain and ubequently dieminate information that accurately reflect network traffic condition. Many different technique for aeing traffic condition have been propoed. However, one method in particular, namely, the ue of vehicle that are capable of tranmitting link travel time to the traffic management center, ha received coniderable attention. The ue of probe vehicle enable a ample of the travel time experienced by all vehicle travering the link to be obtained. Thi paper examine the ue of probe vehicle on ignalized link and addree the critical quetion of How accurately do the probe vehicle travel time (ample) reflect the travel time of all the vehicle (population) that travered the link? A number of reearcher have previouly invetigated the expected reliability of probe travel time report. Van Aerde et al. (1993) developed an analytical expreion for the reliability of probe travel time for ignalized link and verified thee expreion uing imulated data. Thee expreion, which aume that probe report repreent an independent random ample from the traffic tream, indicate that a the number of probe report in a period increae, the ample mean approache the population mean. The ame aumption wa ued by other reearcher in determining the required level of market penetration or number of probe vehicle (Boyce et al. 1991a,b; Turner and Holder 1995; Srinivaan and Jovani 1996). Sen et al. (1997a,b) examined field data collected from probe vehicle a part of the ADVANCE project. On the bai of a tatitical analyi of probe link travel time, they found that probe report are not independent, and therefore, regardle of the ample ize, the ample mean doe not approach 1 At. Prof., Dept. of Civ. Engrg., Univ. of Waterloo, Waterloo, ON, Canada N2L 3G1. bhellinga@uwaterloo.ca 2 At. Prof., Dept. of Civ. Engrg., Univ. of Waterloo, Waterloo, ON, Canada N2L 3G1. lfu@uwaterloo.ca Note. Dicuion open until May 1, To extend the cloing date one month, a written requet mut be filed with the ASCE Manager of Journal. The manucript for thi paper wa ubmitted for review and poible publication on December 23, Thi paper i part of the Journal of Tranportation Engineering, Vol. 125, No. 6, November/ December, ASCE, ISSN X/99/ /$8.00 $.50 per page. Paper No the population mean. They concluded that only a mall number of probe report within a 5-min interval yield a tandard error that i not ubtantially improved by making the number of probe much larger. The concluion reached by Van Aerde et al. (1993) and Sen et al. (1997) appear to be contradictory. Thi paper will how that both reult are indeed correct, but each i appropriate only for pecific traffic and ampling condition, and neither reult can be held a a generalization for all traffic network condition. The remainder of thi paper conit of three component. In the next ection it i hown from fundamental queuing theory that bia in the probe ample lead to a ample mean that doe not aymptotically approach the population mean, regardle of the ample ize. Following thi, imulation data are ued to illutrate the impact of ample bia on a ignalized arterial. Finally, concluion are made regarding the importance of thee finding for the deign of probe-baed ATIS and advanced traffic management ytem. ANALYTICAL ESTIMATION OF EXPECTED DELAY OF PROBE VEHICLES Thi ection provide a theoretical etimate of the mean travel time experienced by a probe vehicle travering a ignalized arterial link. The objective i to how that the mean travel time of the probe vehicle (the ubpopulation from which ample are taken for etimation) may be different from the mean travel time of all the vehicle (population). The travel time that a vehicle experience when travering a ignalized link conit of two component, namely, the running time and the delay caued by the ignal control. In the following theoretical derivation, we will aume that the mean running time of the probe vehicle and the general vehicle are the ame and we will focu on the difference in mean delay between probe and all vehicle. Aumption and Notation The delay that a probe vehicle experience when it travel through a ignalized approach depend on a number of factor, including the arrival flow rate and ditribution, ignal timing, and the time when the vehicle arrive at the approach. In a real application environment, many of thee factor are random variable. A a reult, the travel time reported by probe 524 / JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999

2 vehicle would likely be ubject to large variation. For the purpoe of illutrating the effect of ample bia, we will conider an idealized interection approach coniting of a ingle through lane controlled by a ignal with known timing. The approach ha unlimited pace for queuing and ha a contant aturation flow rate. Furthermore, it i aumed that vehicle arrival at the approach are uniformly ditributed and conit only of paenger car unit (pcu). The other notation are decribed a follow: Signal Timing Parameter c y = cycle time () g = effective green interval () r = effective red interval () = g/c y Arrival Flow q g = average arrival flow rate during effective green interval (pcu/) q r = average arrival flow rate during effective red interval (pcu/) q = average arrival flow rate during cycle time (pcu/). Defined a qgg qrr q = (1) Capacity = aturation flow rate (pcu/) c a = capacity (pcu/), determined by x = degree of aturation during the cycle time, defined a q/c a Probe Vehicle Flow P g = proportion of probe vehicle among all vehicle arriving during effective green interval. The probe arrival rate during effective green interval i therefore q g P g (probe pcu/) P r = proportion of probe vehicle among all vehicle arriving during effective red interval. The probe arrival rate during the red interval i therefore q r P r (probe pcu/) q p = average probe arrival flow rate during cycle time (pcu/), defined a c y the arrival time of a general vehicle ampled from all arriving vehicle (population). For a vehicle arriving at the approach at a given time t, it delay (noted a d for general vehicle and d p for probe vehicle) can be determined baed on determinitic queuing theory (a hown in Fig. 2). q r r 1 t if 0 < t r d = d = qr r tc t p (4) if r < t t c t r c 0 if t c < t cy where t c = time when the queue i cleared and can be determined by q r t c = r 1 (5) q g In the cae that a vehicle i randomly elected from the arriving flow, it delay would alo be a random variable with it ditribution depending on the ditribution of it arrival time [(3)] (Fig. 3). Denote D p a the delay of a randomly elected probe vehicle, and D a the delay of a randomly elected general vehicle. The following ection dicue the derivation of the ditribution of D p. Note that the ditribution of D can be eaily obtained from the ditribution of D p by etting P r = P g = 1.0. Firt, the ampled vehicle may experience no delay and the probability of thi outcome i equal to the probability that the vehicle arrive during the time interval [t c, c y ], which can be determined from (3) Pgqg Pgqg qrr P(D p =0)=P(t c < T < c y)=(cy t c) = ge qpcy qpcy qg (6) Pgqgge Prqrr q = (2) p = ratio of the proportion of probe vehicle arrival during the effective green interval to the proportion of probe vehicle arrival during the effective red interval, defined a = P g /P r c y Ditribution of Delay Fig. 1 illutrate the arrival rate for all vehicle and for probe vehicle only. Conider the cae that a probe vehicle i randomly ampled from all probe vehicle arriving at the approach during the cycle time. The arrival time of the ampled probe vehicle, noted a T, would be a tepwie uniformly ditributed random variable, and it ditribution can be decribed by (3) FIG. 1. Illutration of Arrival Rate for Probe Vehicle and All Vehicle during Red and Green Interval Prqr qpcy if 0 < t r f (t) = (3) T Pgqg q p c y if r < t c y It hould be noted that if both P r and P g are replaced by 1.0 in (3), the reult i the probability denity function (PDF) of FIG. 2. Determinitic Queuing Diagram JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999 / 525

3 FIG. 3. Time Uniform Queue Delay a Function of Vehicle Arrival Second, a hown in (4), the vehicle may experience a delay that decreae linearly from q r r/ to zero when the arrival time increae from r to t c. The probability that the vehicle would experience a delay greater than zero and le than or equal to q r r/ i therefore equal to the probability that the vehicle arrive during the time interval [r, t c ] [(7)] qrr Pgqg P 0<Dp = P(r < T t c)=(tc r) qpcy q r P q = ( q ) q c r g g g p y (7) Becaue the arrival time i uniformly ditributed over the arrival time interval [r, t c ], delay hould alo be uniformly ditributed with it PDF a hown in (8) P 0<D < p qrr P q q r g g r Dp p p qpcy( q g) ca f (d )= =, 0<d qrr 0 Similarly, the probability that the vehicle would experience a delay greater than q r r/ and le than or equal to r can be determined baed on (4) qrr Prqr Prqrr P < D p < r = P(0 < T < r) =(r 0) = qpcy qpcy (9) The PDF of the delay within thi regime i P qrr < d p < r P q q r r r r Dp p p qpcy( q r) f (d )= =, <d r qrr r (8) (10) In ummary, the delay of ampled probe vehicle i a mixed dicrete and continuou random variable with it PMF repreented by (6), (7), and (9), and the PDF repreented by (8) and (10). Mean Delay of Probe Vehicle With the given ditribution function, the mean delay of probe vehicle E[D p ] can be obtained through the following mathematical expectation: (q r /)r r E[D ]=0P(D =0) f (x)xdx f (x)xdx (11) p p Dp Dp 0 (q r /)r Baed on (8) and (10), (11) can be rewritten a r P q q P q q E[D p]= 1 (12) 2q c q q r r r g g r 2 2 p y r g Conider a more idealized ituation where no platoon progreion exit and the arrival flow rate during the effective green interval i the ame a the arrival rate during the effective red interval (i.e., q g = q r = q). Then (12) can be implified to (13) E[D ]= p r 1 x ( 1) q 2c 1 y 1 ( 1) x ( 1) = E[D], (x 1.0) 1 ( 1) (13) From (13), it can be oberved that if the probe arrival ratio i equal to 1 (i.e., a randomly elected probe can be conidered a a general vehicle), the reulting equation i the wellknown expreion for uniform delay for all vehicle (Hurdle 1984; Teply et al. 1995). Thi implie that the expected delay of the probe i equal to the expected delay of all vehicle (population). However, when the proportion of probe arrival during the green interval P g i not equal to the proportion of probe vehicle arrival during the red interval P r, then 1 and E[D p ]ino longer equal to the expected mean of the population E[D]. Phyically, thi mean that if a diproportionate number of probe report are received from probe that arrive during the green or red interval, then the ample i no longer random, but i biaed. Conequently, the average delay computed from the probe report i alo biaed and will aymptotically approach the probe mean E[D p ], but not the population mean E[D], even if the number of probe report i very large. The theoretical analyi in thi ection ha examined an idealized ignalized interection in which the delay aociated with only a ingle link departure movement ha been conidered. However, the reult are equally applicable to interection with more than one outbound movement (e.g., left-turn, through, and right-turn movement), if each outbound movement i conidered individually. The next ection examine the potential ource of bia that are likely to be encountered in field condition, and addree the iue of multiple outbound movement. SOURCES OF SAMPLE BIAS An important practical iue then i to determine under what condition the probe ample can be conidered a biaed ample. The previou ection ha hown, baed on a theoretical derivation, that for an idealized interection arrival time bia in the ample will lead to a ytematic bia in the etimate of the population mean. If we conider that the objective i to etimate the population mean link travel time, where the population conit of all vehicle travering the link regardle of the movement ued to exit the link, then for a typical ignalized link, the ample of probe report can become biaed in two way: 1. Firt, the ditribution of link entry time for probe may differ from the population. The time at which a vehicle enter the uptream end of a link depend largely on the turning movement required to acce the link and the traffic control impacting that movement. For example, conider the network illutrated in Fig. 4 in which the interection bounding the uptream end of the link i ignalized and conit of four approache. Vehicle acceing the link for which the delay i being meaured do o via one of three poible movement, namely, a 526 / JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999

4 movement experience a different average delay. If the proportion of probe vehicle varie with each exit movement, then the ample i biaed a a reult of over- or underampling an exit movement that experience a delay that i greater than or le than the population average. Conequently, for a link bounded by a four-leg interection at the uptream and downtream end, there exit nine ditinct ubpopulation (one aociated with each combination of entry and exit movement), each with it own et of travel time characteritic. If the proportion of probe varie acro thee nine ubpopulation, then a bia will reult. EFFECT OF SAMPLE BIAS WITHIN SAMPLE NETWORK FIG. 4. Arrival Time Bia a Function of Link Entry Movement left turn from the cro treet; a right turn from the cro treet; or a through movement from the main treet. Each of thee movement i controlled by the uptream traffic ignal and by gap acceptance behavior for oppoed movement (i.e., left turn on green, and right turn on red). A illutrated in Fig. 4, the ditribution of link entry time i different for each of the three movement. Conequently, if the proportion of probe vehicle varie with each movement, then it alo follow that the probe link entry time ditribution will be different from that aociated with the population. Becaue delay at the downtream interection i a function of arrival time, a bia in arrival time will alo reult in a bia in delay. 2. The econd caue of bia i aociated with the movement ued to exit the link (i.e., left turn; through, or right turn) at the downtream interection. Typically, each To illutrate the potential magnitude of the ample bia, a imple linear network wa modeled uing the INTEGRATION traffic imulation model (Van Aerde et al. 1996). The network, illutrated in Fig. 5, conit of a ingle arterial roadway that i interected by two cro treet. Each interection i controlled by a fixed-time ignal (c y = 120, g = 75 for the arterial, g = 37 for the cro treet, and offet = 0 ). The network i modeled for 1 h with contant (time invariant) demand. Vehicle are generated at all origin zone with negative exponentially ditributed headway. The O-D traffic demand between each of the ix zone are provided in Table 1. The application of thee demand to the network reult in the interection approache experiencing V/C ratio ranging from approximately 0.3 to Conequently, all of the interection approache operate in an underaturated mode. The time to travere each link egment, the unique vehicle identification number, the time when the vehicle departed the link (i.e., time of probe report), and the vehicle origin and detination were recorded for each vehicle. Thi log repreented the travel time experienced by the entire vehicle population. Three probe ampling cenario were defined a follow: 1. Unbiaed: All O-D pair were ampled at the ame level of market penetration. FIG. 5. Arterial Tet Network Configuration JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999 / 527

5 Origin zone (1) Total TABLE 1. 1 (2) ,475 a Vehicle per hour. O-D Traffic Demand for Tet Network a 2 (3) Detination Zone 3 (4) (5) ,300 5 (6) (7) Total (8) 1, , , Biaed 1 6: Only vehicle traveling between Origin 1 and Detination 6 were ampled. 3. Biaed 2 4: Only vehicle traveling between Origin 2 and Detination 4 were ampled. For each ampling cenario, imulation were conducted for 22 level of market penetration (i.e., in 0.02 increment and in 0.1 increment). For each tet, probe report were aggregated into 5-min period (12 period in total). The mean egment travel time computed on the bai of the probe report, and the number of probe report within each period were recorded for each tet. The reult are examined for vehicle traveling eatbound on link Segment 1, 2, and 3 only (Fig. 5). Segment 1 repreent a link that i not affected by traffic ignal at either it uptream or downtream, and for which only a ingle entry movement and exit movement i poible. Therefore, a biaed ample (baed on over- or underampling a ubpopulation) i not poible. Furthermore, becaue the link i not controlled by a ignal, the link travel time are not expected to exhibit a great amount of variation. Segment 2 repreent a link for which bia can only arie a a reult of over- or underampling the downtream exit movement. Segment 3, having a ignalized interection at both the uptream and downtream boundarie, i uceptible to bia a a reult of over- or underampling any of the nine poible combination of entry and exit movement. Fig. 6 illutrate the mean travel time for Segment 1 etimated from probe report received during each 5-min period, a a function of the number of probe report. The 95% confidence limit (C.L.) of the etimated mean egment travel time are alo illutrated. Thee C.L. are computed from (14) uing the entire vehicle population under the aumption that probe report repreent a randomly elected ample from a ingle infinite population. For Segment 1, there i no opportunity for bia by over- or underampling a pecific entry or exit movement, and o thi aumption i valid var(x i ) C.L. of x = x z (14) n where x = mean egment travel time computed from the entire population of vehicle travering the egment during the 1-h imulation; var(x i ) = variance of the individual vehicle egment travel time about x ; n = number of probe report received during the time interval; and z = normal tandard deviate aociated with the confidence limit (i.e., z = 1.96 for a confidence limit of 95%). Three obervation can be made from Fig. 6. Firt, the mean travel time are all within a very mall range of between 23 and 28. Thi i expected becaue no ignal impact thi link. Second, the 5-min average travel time are ditributed between the confidence limit with no apparent bia. Third, the confidence limit of the ample mean how an initial rapid reduction in the error of the etimate a the number of probe report increae. Furthermore, the error tend to zero a the number of report approache infinity. Fig. 7 illutrate the mean 5-min travel time obtained from probe report for biaed and unbiaed ample. From thee reult it i evident that having a biaed ample reult in mean travel time etimate that do not repreent the travel time experience of the population. In particular, when ampling from only thoe vehicle traveling from Origin 1 to Detination 6, the reulting ample mean travel time are much larger than the population mean travel time. Thi i expected a thi ample of vehicle i required to make a left turn at the downtream end of Segment 2, and conequently, experience a much greater delay than the population of vehicle travering Segment 2. Fig. 8 illutrate the effect of bia in the arrival time ditribution between probe vehicle and general vehicle travering Segment 3. Mean travel time from two ample are preented. The unbiaed ample reflect the experience of the true population. The biaed ample only reflect vehicle that are traveling between Origin 2 and Detination 4. From the reult in Fig. 8 it i evident that the biaed ample travel time conitently underetimate the population travel time. While the biae preented in thi example network can be conidered to be extreme, they do erve to illutrate the po- FIG. 6. Mean 5-min Travel Time and 95% Confidence Limit for Unbiaed Sample (Segment 1) 528 / JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999

6 FIG. 7. Mean 5-min Travel Time from Unbiaed and Biaed Sample a Function of Number of Probe Report (Segment 2) FIG. 8. Mean 5-min Travel Time from Unbiaed and Biaed Sample a Function of Number of Probe Report (Segment 3) tential extent of the problem. Furthermore, the reult from Sen et al. (1997b), dicued earlier in thi paper, are baed on field data in which all probe vehicle travered a et of link by uing the ame link entry and link exit movement. Thu, the ample ued in their analyi ha a level of bia that i imilar to that aociated with the example provided in thi paper. The finding decribed within thi paper demontrate that the concluion of Sen et al. that a mall number of probe report within a 5-minute interval yield a tandard error that i not ubtantially improved by making the number of probe much larger, hould not be generalized. CONCLUSIONS AND RECOMMENDATIONS Previou reearch in the literature ha provided eemingly contradictory concluion regarding the accuracy of mean link travel time etimated from probe vehicle report. The apparent diagreement between thee reult give rie to confuion among practitioner and may lead to inappropriate ATIS deign deciion. Thi paper ha examined the iue of the accuracy of mean travel time a etimated from probe vehicle. More pecifically, thi paper ha hown that under condition when the probe vehicle repreent a biaed ample, the ample mean doe not approach the population mean. It ha alo been hown that for a typical link, which i bounded at both the uptream and downtream end by a ignalized interection, the population of vehicle can be divided into nine ubpopulation, each aociated with a unique turning movement combination for entering and exiting the link. If the proportion of probe vehicle varie between thee nine ubpopulation, the probe report repreent a biaed ample and error aociated with the ample mean may remain quite large, even when the ample ize i large. However, if the proportion of probe vehicle i nearly contant acro all nine ubpopulation, then the ample i unbiaed, and the probe report can be conidered to repreent independent ample from the population. In thi cae, tandard ampling theory hold, and the error of the ample mean decreae a the ample ize increae. Thu, the identification of ample bia enable the apparently inconitent reult of previou reearch reported in the literature to be more clearly interpreted. Specifically, the work of Sen et al. (1997b) can be een a applicable to cenario in which the probe report contitute a biaed ample of the population. Converely, the work of Van Aerde et al. (1993) i applicable when the probe report are an unbiaed ample of the population. It ha been hown that the degree to which the probe report repreent a biaed ample i critical in aeing the reliability of the ample mean a an etimate of the population mean. Therefore, it i recommended that further reearch focu on developing method that can be applied under field condition to quantify the degree of bia aociated with a ample of probe report. Furthermore, if thi bia can be quantified, it i recommended that method be developed by which the impact of thi bia can be reduced or eliminated to provide more accurate etimate of the population mean travel time. JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999 / 529

7 ACKNOWLEDGMENTS The writer gratefully acknowledge the financial upport provided by the Natural Science and Engineering Reearch Council of Canada. APPENDIX. REFERENCES Boyce, D., Hick, J., and Sen, A. K. (1991a). In-vehicle navigation ytem requirement for monitoring link travel time in a dynamic route guidance ytem. Proc., 70th Annu. Meeting of the Tranp. Re. Board. Boyce, D., Kiron, A., and Schofer, J. (1991b). Deign and implementation of ADVANCE: The Illinoi dynamic navigation and route guidance program. Proc., Vehicle Navigation and Information Sy. Conf., Vol. 1, Society of Automotive Engineer, Warrendale, Pa., Hurdle, V. F. (1984). Signalized interection delay modela primer for the uninitiated. Tranp. Re. Rec. 971, Tranportation Reearch Board, Wahington, D.C., Sen, A., Soot, S., Liga, J., and Tian, X. (1997a). Arterial link travel time etimation: Probe, detector and aignment-type model. Proc., 70th Annu. Meeting of the Tranp. Re. Board, Preprint No Sen, A., Thakuriah, P., Zhu, X., and Karr, A. (1997b). Frequency of probe report and variance of travel time etimate. J. Tranp. Engrg., ASCE, 123(4), Srinivaan, K. K., and Jovani, P. P. (1996). Overaturation delay etimate with conideration of peaking. Tranp. Re. Rec. 1537, Tranportation Reearch Board, Wahington, D.C., Teply, S., Allingham, D. I., Richardon, D. B., and Stephenon, B. W. (1995). Canadian capacity guide for ignalized interection. Intitute of Tranportation Engineer, Ditrict 7, Canada. Turner, M. S., and Holdener, D. J. (1995). Probe vehicle ample ize for real-time information: The Houton experience. Proc., Vehicle Navigation and Information Sy. (VNIS) Conf., IEEE, New York, 3 9. Van Aerde, M., Hellinga, B., Baker, M., and Rakha, H. (1996). INTE- GRATION: Overview of imulation feature. Proc., 75th Annu. Meeting of the Tranp. Re. Board, Tranportation Reearch Board, Wahington, D.C. Van Aerde, M., Hellinga, B., Yu, L., and Rakha, H. (1993). Vehicle probe a real-time ATMS ource of dynamic O-D and travel time data. Large urban ytemproc., ATMS Conf., / JOURNAL OF TRANSPORTATION ENGINEERING / NOVEMBER/DECEMBER 1999