Impacts of Intercycle Demand Fluctuations on Delay Lee D. Han1; Jan-Mou Li2; and Tom Urbanik3 Abstract: This paper demonstrates that in addition to intracycle demand fluctuation, which is already a consideration in many delay models, intercycle demand variance also impacts average delay at signalized intersections. Webster-type delay models treat demand fluctuation over the whole analysis period, often 15 min or longer, as if it were just within a single cycle. Such an approach is fine if used judiciously, one might presume. However, results from Monte Carlo simulations with the incremental queue accumulation 共IQA兲 method indicate that Webster-type delay models will underestimate the average delay under heavy traffic conditions. As unutilized capacity at a signalized intersection cannot be saved or carried over to be used by succeeding cycles when demand surges due to normal fluctuation, better understanding of the patterns of intercycle demand variance is important. Simulation results demonstrate that different patterns of intercycle demand variance can result in different levels of average delay. A low-to-high demand pattern will cause a higher average delay than a high-to-low pattern would, even though the overall demand level is exactly the same. It is therefore clear that neglecting intercycle demand variance may lead to significant inaccuracy and, hence, suboptimal signal timing decisions. DOI: 10.1061/共ASCE兲0733-947X共2009兲135:5共288兲 CE Database subject headings: Intersections; Traffic signals; Monte Carlo method; Delay time; Traffic capacity. Introduction Queueing theory has been the primary basis of delay analysis at signalized intersections. According to Newell 共1965兲, the simplest models of traffic flow through intersections were considered by Clayton 共1941兲, and perhaps by other researchers even earlier. In these early queueing models, vehicles were assumed to arrive at regularly spaced time intervals with a mean-time headway of 1 / q, where q = average flow rate over a certain time period. The vehicles form a queue during the red phase, R, at a traffic light and then during the subsequent green phase, G, depart at regularly spaced intervals with a time headway of 1 / s, where s = saturation flow rate, until either the end of the green time or when the queue has fully dissipated. The assumption that traffic arrivals and departures are uniformly distributed is an important part of Webster’s work 共Webster 1958兲, which attempts to attribute the average vehicular delay at signalized intersection to three main components, or terms, i.e., uniform delay, random delay, and empirical errors. A very similar formulation for delay estimation is later employed by the 1985 edition 共TRB 1985兲 and subsequent updates of the Highway Capacity Manual 共HCM兲 共TRB 1994, 1997, 2000兲. The first term in each of these delay formulas represents uni- form delay, which can be and is derived from simple queueingbased analysis. By assuming uniform arrivals within a signal cycle, or intracycle, and by ignoring the discrete nature of vehicles, traffic can be considered as a continuous flow arriving at a uniform rate of q. At some point in time the flow is dammed up for a period of R; it is then released at a rate of s until the buildup has dissipated. A tool in the form of queue accumulation diagram 共QAD兲 as depicted in Fig. 1, has been quite useful for such analyses. The first term of Webster’s delay model, with all the simplicity in its algebraic form, has stood the test of time. Because neither the world nor traffic at a signalized intersection is deterministic, researchers have endeavored to introduce stochastic terms into delay models in order to estimate delay more realistically. To this end, the second term of Webster’s model makes some allowance for the random nature of the arrivals. Webster further employed Monte Carlo simulations to devise a third term to fit a wide range of flow conditions. According to the description in Appendix 2 of Webster’s report 共Webster 1958兲, the randomness of the arrivals was assumed. Traffic is assumed to arrive at the intersection at random. In fact, the actual distribution obtained from observations on the road could be used but random traffic has the advantage that it can be generated artificially using tables of random numbers to derive the intervals between successive vehicles. 1 Associate Professor, Dept. of Civil and Environmental Engineering, Univ. of Tennessee, 112 Perkins Hall, Knoxville, TN 37996-2010. E-mail: lhan@utk.edu 2 Postdoctoral Research Fellow, Univ. of Idaho, 115 Engineering Physics Building, Moscow, ID 83844-0901. E-mail: jli@uidaho.edu 3 Professor and Goodrich Chair of Excellence, Dept. of Civil and Environmental Engineering, Univ. of Tennessee, 219B Perkins Hall, Knoxville, TN 37996-2010. E-mail: turbanik@utk.edu Note. Discussion open until October 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this paper was submitted for review and possible publication on March 9, 2007; approved on August 11, 2008. This paper is part of the Journal of Transportation Engineering, Vol. 135, No. 5, May 1, 2009. ©ASCE, ISSN 0733-947X/2009/5-288–296/$25.00. This implies, in a rather subtle and largely unnoticed manner, that the random nature of vehicular arrivals within a cycle 共intracycle兲 and that among cycles 共intercycle兲 can be considered identical and are, thus, represented with identical statistical distribution. In fact, no delay model, Webster’s or else, distinguished intercycle and intracycle randomness until Han and Li 共2007兲, simultaneously readdressing the cycle-length optimization problem with Monte Carlo simulations. One of the advantages of this implied assumption is one could simplify the analysis and treat the entire study period of, say an hour, as a single signal cycle with the same average demand of q throughout. The flip side, however, is the errors this assumption introduces when intercycle randomness 288 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org Cum ulativ e Num ber Queueing Diagram D(τ) A (τ) Slope = q Slope = s Q(0) Red Green Red Queue Length, Q(t) QAD Slope = q Slope = q-s Q(0) 0 R t0 R+G Tim e, t Fig. 1. Demand fluctuations seen at different timescales exists. As unused capacity at a signalized intersection cannot be carried over from one cycle to succeeding ones, if intercycle demand fluctuation exists, the delay model has to be formulated to address the factor of randomness beyond the boundary of a single signal cycle. Many studies have analyzed the impact of fluctuating demand on average delay, but none has distinguished the randomness of demand within and among cycles. Akcelik and Rouphail 共1993兲 applied symmetrical triangular and parabolic functions to represent demand over the total flow period. Heidemann 共1994兲 as- JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 / 289 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org 1600 1400 1200 Flow Rate (v ph) 1000 800 600 400 Flow Rat e 200 Minut e Flow Rat e 15-m in Flow Rat e 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Dem and Flow Rate Tim e (m in) 0 15 30 45 60 Tim e (m in) Fig. 2. Queueing diagram and QAD with an initial queue at t = 0 sumed the number of vehicles arriving during a time interval to follow the Poisson arrival process and the arrivals for different but equal-length time intervals to be identically and independently distributed. However, he did not approach the subject from a signal-cycle perspective, and he did not consider nonidentical distribution cases from one interval to the next. Han 共1996兲 proposed a similar approach to handle time-varying demands where the overall analysis period 共usually 1 h兲 is divided into a sequence of subperiods 共usually 5 – 15 min兲 with traffic demands constant throughout all subperiods. Although many studies on delay at signalized intersections have considered demand fluctuation within a cycle 共intracycle兲, they have often implicitly treated demand over multiple cycles 共intercycle兲 to be the same and, consequently, have reduced the analysis for a longer period, e.g., 15 min, to a single cycle. This paper distinguishes between intra- and intercycle demand fluctuations 共see Fig. 1兲 and recognizes the potentially significant impact of delay underestimation when intercycle demand fluctuation is unaccounted for, as in all previous models. The remainder of this paper presents the approach used to analyze the intercycle demand fluctuations; the Monte Carlo simulations performed, with detailed descriptions of various scenarios; the results, observations, and discussions of the analyses; and the conclusions. Analytical Contemplation Queueing analysis is employed to assess the impact of intercycle demand fluctuations on delay, in comparison with the case of intracycle demand fluctuations already studied by earlier researchers. Following Newell’s fluid model, let A共␶兲 be the cumulative number of arrivals at time ␶ and let D共␶兲 be the cumulative number of departures at time ␶. Then for the single cycle depicted in Fig. 2, A共␶兲, D共␶兲, and Q共␶兲 can all be derived for any given ␶ within that single cycle. Under previous assumptions, the total delay of all vehicles in the queue during the cycle, R + G, is the area under the QAD curve in Fig. 2. This is what the first term in Webster’s model was based on. When the randomness was added to the arrival, i.e., assuming q follows a certain kind of stochastic distribution, a 290 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org Vehicles Cum Accumulated ulativ e Num ber of Vehicles 14 250 12 200 10 150 8 6 100 4 50 2 0 0 0 10 20 30 40 50 60 70 0 100 200 300 400 500 600 700 800 900 Time Tim e, in s econds Tim e,Time in s econds (a ) An a lys is Pe riod = 60 s e con d s (b) An a lys is Pe riod = 900 s e con d s 1000 Fig. 3. Demand variations within a 1-min cycle and for 15 min individual minute 共and perhaps cycle兲 will have a different average demand for that minute 共or cycle兲 as a result of the stochastic fluctuation of demand over time. The delay for 15 one-minute cycles, each with an identical demand rate of 1,000 vph, will differ 共greatly in fact兲 from the delay for 15 one-minute cycles with nonidentical demand rates of 861, 935, 1,049, . . . , 1,203 vph, even though the average demand 共over 15 min兲 for both cases are the same at 1,000 vph. Another way to look at this problem is this: Let A1共␶兲 represent the arrival curve in the first period; q1 follows a certain Cum ulativ e Num ber of Vehicles Webster-type of delay model could be derived. As Webster-type delay models were derived from a single cycle, the assumption of randomness was really for the entire analysis period. If the analysis period were 60 s, the arrival curve would look like the one in Fig. 3共a兲; if the period were 15 min, the arrival curve 共based on the assumption that the entire period had a single stochastic distribution and a fixed mean兲 would resemble the one in Fig. 3共b兲. Unfortunately, empirical data have shown that demand does not remain nearly stationary over a long period of 15 min. In fact, Fig. 1 is closer to what may actually occur. With an average flow rate of 1,000 vehicles per hour 共vph兲 over the 15-min period, each Case 1 End of A nalysis Period Case 2 0 Tim e, t Fig. 4. Two cases of different demand patterns JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 / 291 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org Cum ulativ e Num ber of Vehicles μL gh Hi to w Lo μA μH w Lo to End of A nalysis Period μH gh Hi μL 0 T Tim e, t Fig. 5. Low-to-high and high-to-low demand patterns A rriv al A (τ) Departure D(τ) End of A nalysis Period from a single cycle to the whole analysis period. In Case 2, the arrival rate is shown to have changed over time, even though the average arrival rates for both cases are identical over the whole analysis period. Considering average delay, the two cases will be different and may very well have different levels of service 共LOS兲. The average delay in Case 2 will be larger than that in Case 1. In fact, Case 2 may experience some cycle failures toward the latter part of the analysis period. The reason intercycle variance, as opposed to the intracycle Cum ulativ e Num ber of Vehicles Accumulated Vehicles distribution, say N共␮1 , ␴21兲. Let A2共␶兲 represent the arrival curve in the succeeding period, with q2 following a slightly different distribution, e.g., N共␮2 , ␴22兲. When the single-cycle approach is employed to analyze the whole 共two-cycle兲 period, there can be a third arrival function, A3共␶兲. Even if q3 also follows normal distribution, e.g., N共␮3 , ␴23兲, it cannot be the summation of q1 and q2. That is, ␮3 will not equal ␮1 + ␮2, and ␴23 will not equal ␴21 + ␴22. Fig. 4 further illustrates this situation. Case 1 shows a common approach that basically extends the same average demand rate Queue Q(τ) Time Tim e, t Q(T) T Fig. 6. Case of oversaturation 292 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org Cum ulativ e Num ber of Vehicles Case 3 End of A nalysis Period Case 1 Case 2 0 T Tim e, t Fig. 7. Three different demand patterns with same average demand Webster’s Delay Model variance that has been studied quite thoroughly, should be emphasized is this: The underutilized capacity cannot be “saved” for or carried over to succeeding cycles, where the capacity would be needed when demand surges randomly. In order to obtain realistic estimates of average delay, one may average out the varying demand within a cycle by some statistical methods, but one cannot and should not do the same for varying demand in the case of intercycle fluctuations. That is, delay models which were derived from the queueing analysis within a single cycle, e.g., Webster and Webster-like models, should not be extended to an analysis period beyond a single cycle unless intercycle demand fluctuation is minimal to nonexistent. Failing this, the Webster type model, when misused, will underestimate average delay and potentially lead to incorrect LOS projection. The impacts of intercycle demand variance can also result from the patterns of the variance. Fig. 5 shows two different demand patterns; each consists of two different arrival rates, i.e., ␮H and ␮L, within the analysis period, although the overall arrival rate for both cases is the same, ␮A. The average arrival rates are from low to high for one case and high to low for the other. If T equals the cycle length, then the average delay can be approximated by using ␮A as the demand for both cases. But if T is a longer duration of, say, 15 min or even 1 h, which spans over many cycles, the average delay may be quite different. where d = average delay per vehicle 共s兲; ␾ = cycle length 共s兲; ␭ = proportion of the cycle, which is effectively green of the phase under consideration 共i.e., ge / ␾兲; ge = effective green time 共s兲; q = traffic demand; s = saturation flow rate; and X = lane group demand/capacity, or v / c, ratio or degree of saturation; this is the ratio of the actual flow to the maximum flow that can pass through the intersection and is given by X = q / ␭s. Verification with Monte Carlo Simulations Table 1. Mean-Time Headway 共in Seconds兲 under Different Demand Levels To verify the concerns posed in the previous section, several scenarios were designed to examine the impact of intercycle variation on average delay via Monte Carlo simulation. The results from the simulation are compared to those from Webster and from the HCM 2000 共TRB 2000兲 delay models, as detailed in the following. The incremental queue accumulation 共IQA兲 method was employed to calculate the delay within the system. Webster’s model, which is based on a single-cycle analysis, is expressed as d= 冉冊 ␾共1 − ␭兲2 ␾ X2 + − 0.65 2 2共1 − ␭X兲 2q共1 − X兲 q 1/3 X2+5␭ 共1兲 HCM 2000 Delay Model When an initial queue is nonexistent, the HCM 2000 model for average control delay per vehicle for a given lane group can be simplified as Case1 Demand 共vph兲 300 600 900 12 6 4 Case 2 Case 3 First section Second section First section Second section 15 9 6 10 4.5 3 10 4.5 3 15 9 6 JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 / 293 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org Table 2. Average Delay Estimated by Different Models Demand 共vph兲 T 共min兲 Case 1 共uniform兲 2 300 6.00 600 10.20 900 14.93 15 300 6.00 600 10.20 900 14.93 30 300 6.00 600 10.20 900 14.93 60 300 6.00 600 10.20 900 14.93 a Value of 15.73 s was calculated under a demand of 899 vph. 冉 冊 0.5␾ 1 − d= ge ␾ Webstera HCM 2000 Case 2 共low-high兲 Case 3 共high-low兲 9.00 11.25 ⬎15.73 9.00 11.25 ⬎15.73 9.00 11.25 ⬎15.73 9.00 11.25 ⬎15.73 9.98 14.67 25.95 10.00 15.15 45.00 10.00 15.20 57.43 10.00 15.22 75.00 6.30 12.11 17.07 6.16 12.73 43.80 6.30 12.97 86.07 6.30 12.97 161.07 6.30 12.60 19.93 6.44 13.06 68.90 6.30 12.97 124.33 6.30 12.97 236.83 2 ge 1 − min共1,X兲 ␾ 冋 冑 PF + 900T X − 1 + 共X − 1兲2 + 8kIX cT 册 共2兲 where d = control delay per vehicle 共s兲; P = proportion of vehicles arriving on green; T = duration of analysis period 共h兲; k = incremental delay factor; I = upstream filtering/metering adjustment factor; c = lane group capacity 共vph兲; and PF= uniform delay progression adjustment factor, which accounts for effects of signal progression. Some Assumptions Normal, Pearson Type III, and negative exponential timeheadway distributions for high, intermediate, and low demand conditions, respectively 共May 1990兲, were used for the Monte Carlo simulation runs. Other assumptions include the following: 1. The site is an isolated signalized intersection of two one-way one-lane roads; 2. Arrivals in the two approaches are assumed to be similar so that delay in only one approach needs to be simulated; 3. A pretimed, two-phase signal control is running with cycle length 60 s with an effective green time of 30 s, and an effective red time of 30 s; and 4. At the onset of effective green time, queued vehicles discharge at a saturation flow rate, s, of 1,800 vph, or 0.5 vehicle/ s. As traffic is assumed to arrive at the intersection randomly, further assumptions for the HCM 2000 delay model include these: 1. Arrival type is 3 共random兲; 2. Each approach sustains a 4 s / cycle lost time; 3. Uniform delay progression adjustment factor PF= 1; 4. Incremental delay factor k = 0.5 for pretimed controller settings; and 5. No upstream filtering/metering exists, so the adjustment factor I = 1. Incremental Queue Accumulation „IQA… Method The IQA method originally proposed by Strong and Rouphail 共2006兲 was used to implement the HCM model with more flex- ibility. It extends the usability of the HCM to better reflect conditions commonly found in the field without the plethora of limiting assumptions that are required by the current HCM 2000 method. This method suggests that equal-sized time slices be used, adding/subtracting the number of arrivals/departures during each time slice to the queue at the start of the time slice and resulting in the queue at the end of the time slice. Even though the concept of the IQA method is intuitive, some characteristics of this method are introduced here due to its novelty. The method 1. Uses equal-sized time slices during the analysis period; 2. Examines the queue accumulation every time slice; 3. Calculates the uniform delay component; 4. Is consistent with the model in HCM 2000 and Webster’s model; and 5. Is fully capable of handling variable arrival rates in different parts of the cycle. The IQA method is a more generalized approach to calculating the queue accumulation area using multiple trapezoids, and it simplifies the calculation of trapezoids, which represent the periods of time during the cycle when the inflow and outflow rates are not changing. Because the boundaries of each time slice fall squarely on points where signal status and traffic flow rate change, IQA is considered suitable for this research and was used for this purpose. To calculate average delay in oversaturated or successive cycle failure conditions, one has to estimate and project the delay for queued vehicles that could not depart by the end of the analysis period. At the end of the analysis period, or time T, as shown in Fig. 6, it is evident that a nontrivial number of queued vehicles, Q共T兲, have to depart after T. The total delay for each of these queued vehicles was estimated based on their projected departure times. Simulation of Hypothetical Cases Traditional Webster-type delay models do not consider intercycle demand changes, even though many of them do consider intracycle demand fluctuation. This approach is fine if the analysis period is limited to a single cycle and is not extended to a longer period, or if demand holds relatively steady throughout the analysis period, unlike those in Fig. 1. To test how Webster and HCM 2000 delay models may be “off” when intercycle demand fluctuation is a factor, three very simple and, obviously, hypothetical 294 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2009 Downloaded 06 May 2010 to 160.36.192.127. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org demand patterns were designed for this purpose 共see Fig. 7兲. All of the three patterns share the exact average demand over the analysis period, with Case 1 representing the traditional straightline approach showing no intercycle variance throughout the analysis period, whereas Cases 2 and 3 each have exactly one change in demand level during the analysis period. In Case 2, the mean of arrival rate for the first section is lower than that in the second section. In contrast, the mean of arrival rate for the first section of Case 3 is higher than that in the second section. To further simplify the analysis and simulation, the two sections in both Cases 2 and 3 were assumed to be of the same length of time. Simulated vehicle arrivals in each case were generated according to the mean-time headway as listed in Table 1. Three demand levels of 300, 600, and 900 vph were used to represent light, intermediate, and heavy traffic conditions, respectively. For the simplest case of a longer-than-one-cycle analysis period, a two-cycle analysis was selected in which the first section mentioned previously would be the first cycle, and the second section is the second cycle. In addition, analysis periods of 15 min, which is typical for HCM 2000, 30 min, and 60 min were also used for comparison purposes. Results and Discussions Table 2 tabulates the results from Monte Carlo simulation, for Cases 1, 2, and 3, and from Webster and HCM 2000 models under the prescribed hypothetical conditions. The first impression is that neither the results from Case 1 nor those from the Webster model changed at all as the analysis period T increased from 2 to 60 min. This verifies what was presented previously, that like Case 1, the Webster model does not consider any intercycle demand fluctuations. The results from Cases 2 and 3 do show higher levels of average delay than those from Case 1 as a result of a single intercycle demand change. The increases in delay, however, were not significant in light 共an increase of merely 5%兲 and intermediate 共an increase between 19 and 27%兲 flow conditions. The results did not worsen as T increased. The reason is that the fluctuation of demand from one cycle to the next, under light and intermediate traffic, never reached the same serious tandem cycle-failure situation as those in Fig. 6. Therefore, the average delay never quite got out of control. Results from Webster and HCM 2000 models, in general, are higher than those from the three cases under light and intermediate traffic. Under heavy traffic conditions, HCM 2000 projects higher delay than Case 1, Webster, and even Cases 2 and 3 for T = 0.0333 h. It is unclear why HCM 2000 yields significant higher delay than the other models, though. Under heavy traffic, i.e., 900 vph, as T increases, results from Cases 2 and 3 reflect serious cycle failures and, hence, increasingly undesirable delay levels, which eventually reached an increase of 979% for Case 2 and one of 1487% for Case 3 in comparison with Case 1, when T = 1 h. The fact that the resultant delay from HCM 2000 under heavy traffic increases as T goes from 2 to 60 min indicates some attempt to account for intercycle demand fluctuation. The values of the estimated delay, which are significantly lower than those from Cases 2 and 3 when T is large, may indicate that the simple inclusion of T in the model’s second term in a linear fashion is insufficient; or, perhaps, the explanation is as simple as the result of oversimplification in the design of the two cases. More complicated and realistic cases will have to be designed to test this thoroughly. Between Cases 2 and 3, it is clear that Case 2, which squandered away unused capacity during the first half of the analysis period, resulting in a 47% higher level of delay than that of Case 3, which had its cycle failures in the first half of T, but had extra capacity in the second half available to accommodate the queued traffic. Conclusions This paper recognizes and demonstrates, with a simple example, that intercycle demand fluctuation has a significant impact on the delay at a signalized intersection. Webster-type delay models treat the variance over the whole analysis period as if it were within a single cycle. Such an approach is fine if used judiciously. Simulation results indicate, however, that this type of delay model will underestimate the average delay under heavy traffic conditions. As unutilized capacity at a signalized intersection cannot be saved or carried over to be used by succeeding cycles when demand surges due to fluctuation, the pattern of intercycle demand variance is important. Simulation results demonstrate that different patterns of intercycle demand variance can result in different levels of average delay. A low-to-high demand pattern will cause a higher average delay than a high-to-low pattern would, even though the overall demand level is exactly the same. This paper points out the importance of intercycle demand variance on delay analysis, especially under heavy traffic conditions. Neglecting intercycle demand variance may lead to significant inaccuracy and, hence, suboptimal signal timing decisions. Further research is needed to investigate the patterns of intercycle demand variance in the real world and to revise existing delay models to handle intercycle demand fluctuations. References Akcelik, R., and Rouphail, N. 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