Effect of time of day and day of the week on congestion duration and breakdown: a case study at a bottleneck in salem, NH

Effect of time of day and day of the week on congestion duration and breakdown A case study at a bottleneck in Salem, NH Q3 ww sciencedirect com 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2[.]

JTTE110_proof ■ 11 January 2017 ■ 1/10 j o u r n a l o f t r a f fi c a n d t r a n s p o r t a t i o n e n g i n e e r i n g ( e n g l i s h e d i t i o n ) ; x ( x ) : e1 Available online at www.sciencedirect.com 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 ScienceDirect journal homepage: www.elsevier.com/locate/jtte Original research paper Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH Q3 Eric M Laflamme a,*, Paul J Ossenbruggen b a b Department of Mathematics, Plymouth State University, Plymouth, NH 03264, USA Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA highlights
Features of recurrent congestion and recovery are analyzed via regression models
Probability of recurrent congestion increases between AM- and PM-rush periods
Effect of time-of-day on congestion duration depends on the day-of-the-week article info abstract Article history: This work uses regression models to analyze two characteristics of recurrent congestion: Available online xxx breakdown, the transition from freely flowing conditions to a congested state, and duration, the time between the onset and clearance of recurrent congestion First, we apply a binary logistic regression model where a continuous measurement for traffic flow and a dichoto- Keywords: mous categorical variable for time-of-day (AM- or PM-rush hours) is used to predict the Stochastic models probability of breakdown Second, we apply an ordinary least squares regression model Ordinary least squares regression where categorical variables for time-of-day (AM- or PM-rush hours) and day-of-the-week Binary logistic regression (MondayeThursday or Friday) are used to predict recurrent congestion duration Models are Congestion duration fitted to data collected from a bottleneck on I-93 in Salem, NH, over a period of months Breakdown Results from the breakdown model, predict probabilities of recurrent congestion, are consistent with observed traffic and illustrate an upshift in breakdown probabilities between the AM- and PM-rush periods Results from the regression model for congestion duration reveal the presences of significant interaction between time-of-day and day-of-the-week Thus, the effect of time-of-day on congestion duration depends on the day-of-the-week This work provides a simplification of recurrent congestion and recovery, very noisy processes Simplification, conveying complex relationships with simple statistical summaries-facts, is a practical and powerful tool for traffic administrators to use in the decision-making process © 2017 Periodical Offices of Chang'an University Publishing services by Elsevier B.V on behalf of Owner This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/) * Corresponding author Tel.: þ1 603 724 5336 E-mail addresses: emlaflamme@plymouth.edu (E.M Laflamme), pjo@unh.edu (P.J Ossenbruggen) Peer review under responsibility of Periodical Offices of Chang'an University http://dx.doi.org/10.1016/j.jtte.2016.08.004 2095-7564/© 2017 Periodical Offices of Chang'an University Publishing services by Elsevier B.V on behalf of Owner This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 JTTE110_proof ■ 11 January 2017 ■ 2/10 2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 Introduction Traffic congestion costs American drivers billions of dollars in wasted fuel and loss of productivity Among all types of congestion, recurrent congestion, congestion occurring every day, accounts for roughly half of the congestion experienced by Americans (U.S Department of Transportation, 2016) Furthermore, these recurring congestion events on highways account for 40% of the total delay, more than the delay from construction and traffic incidents (non-recurring events) combined (Paniati, 2003) Clearly, recurrent congestion is a problem and methods must be developed to alleviate the situation But, before expensive construction or remedial measures are employed, it is critical that administrators identify trends/characteristics of recurring congestion on their roadways As stated by Rao and Rao (2012), identifying characteristics of congestion events will serve as a guide to administrators to help them choose appropriate measures to mitigate such congestion Along these same lines, Hao et al (2007) stated that in order to control or alleviate congestion effectively, researchers must investigate its key features So, what are these characteristics of recurrent congestion, and the key features of these events pursued by researchers? Key features may refer to the underlying causes of congestion For example, both high flow during peak-hours (Downs, 2004) and physical structures such as bottlenecks (Ban et al., 2007) have been shown to be responsible for recurrent congestion Other features include congestion dynamics, the evolvement of the congestion event such as queue propagation (Newell, 1993) and shock-wave analysis (Bertini and Cassidy, 2002; Kerner, 2004) Other common features of congestion are the stochastic nature of breakdown, the transition from freely flowing traffic to a congested state, and the factors that trigger these events (Elefteriadou et al., 1995) Other researches are devoted to measures/metrics that quantify their extent, duration, and intensity of congestion (Shaw, 2003) Such measures include level-of-service, congestion duration (the time between onset of breakdown and clearance of congestion event, or, alternatively, the time that the travel rate indicates congested travel on a segment), travel time index, etc In all cases, no matter the characteristic, researchers aim to better understand recurrent congestion and identify the underlying dynamics of the process With these works in mind, authors focus on two particular characteristics of recurrent congestion: breakdown and duration Authors will only focus on recurrent events, and, going forward, “congestion” will always means “recurrent congestion” As mentioned above, breakdown refers to the transition from a freely flowing traffic state to a congested state This is the standard definition and commonly used phrase of breakdown given by Kerner (2009) Of course, identifying breakdown relies on how we define “freely flowing” and “congested” The precise criteria for identifying freeflow and congestion, and thus how we define breakdown, will be discussed later in Section 2.4 Congestion duration refers to the time between the onset of breakdown and clearance of a congested event This too relies on how we define a congested traffic state This definition and the precise criteria used to identify congestion duration will be discussed later in Section 2.5 Why did authors choose these particular congestion characteristics? First, congestion duration is chosen because, as a “time-based” measure of congestion, it is in keeping with the common perception of the congestion problem (Rao and Rao, 2012), yet the explicit investigation of congestion duration's statistical characteristics has attracted only limited attention in the literature Vlahogianni et al (2011) state that the dynamics of congestion duration may contain useful information about intraday traffic operations and should be further explored Second, breakdown was chosen because, despite receiving ample treatment in the literature (Elefteriadou et al., 1995; Persaud et al., 1998), it remains a controversial topic That is, while the stochastic nature of traffic breakdown has been verified (Elefteriadou et al., 1995), the mechanism or trigger of breakdown is still mysterious It is our goal to gain some insight into these two aspects of recurrent congestion In this work, authors will use two separate regression analyses where breakdown and congestion duration are used as respective response variables We will then introduce explanatory variables derived from traffic stream data to identify underlying factors associated with both breakdown and duration Ideally, from our models, identifying significant predictors of breakdown and congestion duration will lead to a better understanding of recurrent congestion The remainder of this paper is organized as follows Section presents authors' materials and methods including statistical (regression) model forms, the raw traffic stream data from authors' collection site, preprocessing procedures, data aggregation, and extraction of the requisite variables for model fitting Section presents authors' results from model-fitting and interpretation of these results Lastly, section is the conclusion of authors' study Materials and methods 2.1 Statistical models: binary logistic regression model for breakdown probability A traditional approach to identifying probability of breakdown is the use of a generalized linear model (GLM) GLMs have the basic form for random response variable Y gmi ị ẳ Xi b (1) where the mean value mi is given by E(Yi), g($) is a smooth monotonic “link” function, Xi is the ith row of model matrix X, and b is a vector of unknown model parameters GLMs assume that the Yi are independent and Yi ~ some exponential family distribution To model breakdown probability, the authors let Y represent the traffic state where Y ¼ denotes congested traffic and Y ¼ denotes freely flowing traffic Then, mi is interpreted as the probability, p, of Yi taking on the value of one, Pr(Y ¼ 1), and the authors use the canonical link function with the
form of Eq (1) p  (2) gpị ẳ ln 1p Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 3/10 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Such models are known as binary logistic regression models For further details regarding logistic models and GLMs, the reader is directed to Hosmer et al (1989) Several analyses have been performed in which probability of breakdown is identified as a function of flow measurements Athol and Bullen (1973) suggested that the expected time to breakdown is a declining function of flow Lorenz and Elefteriadou (2001) illustrated the probabilistic nature of breakdown and showed empirical evidence that probability of congestion increases with increasing flow rate Other studies have similarly shown that congestion breakdown occurs with some probability at various flow rates, that breakdown is, in fact, a stochastic process (Dong and Mahmassani, 2009a,b; Elefteriadou et al., 1995; Evans et al., 2001) Persaud et al (1998) explored the relationship between flow and the probability of breakdown empirically, by visual assessment Ossenbruggen (2016), used stochastic differential equations to predict breakdown probability based on average flows for discretized time-of-day periods Following these analyses, traffic flow (volume) will be used as the primary predictor of breakdown In addition to flow, and because breakdown probability is likely influenced by time-of-day, a time variable will be introduced into authors' regression model In the spirit of simplification, our time-of-day measure will be a categorical variable distinguishing between two time sectors: AM-rush (7 a.m.e2 p.m.) and PM-rush (2 p.m.e7 p.m.) These periods were not chosen arbitrarily, but based on several factors First, and most simply, the vast majority, more than 90%, of congestion events occur between a.m and p.m Second, within these 12 h, the AM-rush and PM-rush periods were distinguished/identified based on a visual comparison of average flows throughout the day (Fig 1) This figure illustrates how average flows within each period are similar in terms of magnitude and variability, yet dissimilar to one another That is, average flows from p.m to p.m (PMrush) are higher and have more variability as compared to flows from a.m to p.m (AM-rush) Third, identification of these periods was further based on a previous analysis where a piecewise linear function was fitted to daily average flows In this investigation, there was a distinct and sharp increase in average flow at p.m., lending credence to the choice of p.m as a break point between AM- and PM-rush periods Furthermore, this analysis revealed a nearly flat trend in average flow from around a.m up to p.m., while flows after p.m were more volatile and higher until about p.m These estimated trends reinforced the AM- and PMrush periods identified visually In Ossenbruggen (2016), this piecewise linear modeling approach and these exact time periods were used to calibrate a stochastic differential equation model used to predict traffic breakdown In fact, these same AM- and PM-rush time periods were identified as significant predictors of breakdown Fourth, in a study of both recurrent and non-recurrent congestion events, Hallenbeck et al (2003) identified p.m as a natural break between midday and afternoon peak periods While this work does not perfectly agree with our choice of p.m as break point, it is very similar and we feel our choice is justified based on trends in average flows Also, this analysis supports our decision to simplify time-of-day in terms of a categorical variable with just a few variables Fifth, in an analysis of recurring congestion, Mazzenga and Demetsky (2009) identified AM- and PM-peak periods between a.m Q1 and a.m and between p.m and p.m., respectively Again, this does not agree with our periods exactly, but supports the use of simplified time-of-day categories The authors are confident that the time periods capture the recurring events and distinguish the morning and afternoon breakdowns As a final note, the authors acknowledge that time-of-day likely serves as a proxy for more complicated, “lurking,” secondary traffic characteristics (aggressive driving, weaving, etc.) that cannot be extracted from the traffic stream data In Fig 1, vertical lines represent breaks between AM-rush and PM-rush periods Fig illustrates the homogeneity of traffic flows within each time period In addition to a flow main effect and a time sector dichotomous variable, the interaction between the time and the continuous flow variable will be included in our model specification This will allow for a possible shift (change in intercept) and a change in slope (actually, a change in the “log odds”) between the two time sectors Fig e Average flow and variability by time-of-day Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 4/10 4 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 So, for our breakdown model, the probability of breakdown Pr(Y ¼ 1), denoted p, may be expressed as follow pẳ expb0 ỵ b1 q ỵ b2 d þ b3 qdÞ þ expðb0 þ b1 q þ b2 d ỵ b3 qdị (3) where q is the flow, d is the dichotomous variable distinguishing AM- or PM-rush hours, qd is the interaction variable The procedure for identifying our binary response variable, freeflow or congested state, and the associated predictor variables will be discussed later in Section 2.4 In terms of statistical methodology, Persaud et al (2001) is especially relevant as the authors use logistic regression models based on 3-lane, 1-min flow averages to predict congestion events Also, Ossenbruggen (2016) uses a logistic form for breakdown probability as a component of a stochastic differential equation to explain the congestion process To our knowledge, no statistical analyses have been performed using our methodology and exact combination of predictors 2.2 Statistical models: ordinary least squares regression for congestion duration To analyze congestion duration, we use an ordinary least squares (OLS) regression platform Specifically, our response, W is congestion duration in minutes, a continuous variable measured from the onset of breakdown to the clearance of the congestion event The authors chose two explanatory variables in model: time-of-day and day-of-the-week Since most drivers have seen some association between time-of-day and the presence/severity of the congestion, considering time-ofday as a potential predictor of congestion duration makes intuitive sense Furthermore, since flow follows a distinct pattern daily, time-of-day may be considered a simple proxy for traffic flow In Tebaldi et al (2002), day-of-the-week was used as a primary predictor of traffic flow in their hierarchical regression models In the literature, however, day-of-the-week was not commonly used to distinguish traffic stream characteristics In fact, most analyses used data collected from weekday traffic interchangeably Germane to this analysis, Falcocchio and Levinson (2015) summarized trends in recurrent congestion duration by both time-of-day and day-of-the-week The authors found endof-week traffic more severe, a perspective supported by the own driving experience at the collection site and from an initial data investigation Under the OLS regression format, a congestion duration length W then has the following form W ¼ b0 þ b1 x þ b2 d þ b3 xd þ e The study of non-recurrent congestion duration (incident duration) has received ample treatment in the literature Analyses by Garib et al (1997), Giuliano (1989), and Sullivan (1997) used lognormal distributions to describe freeway incident duration Conditional models for incident duration have been pursued by Jones et al (1991), for example Nam and Mannering (2000) used hazard-based models to find the likelihood that an incident will end in the next short time period given its continuing duration Similarly, Stathopoulos and Karlaftis (2002) used a probabilistic log-logistic functional form to describe incident durations A number of works have used OLS regression models to investigate the association between incident duration and certain traffic stream variables (Garib et al., 1997; Gomez, 2005) To our knowledge, however, no statistical analyses have been performed using our methodology to analyze recurrent congestion duration 2.3 Data To fit the regression models, the authors use real-world traffic stream data collected by the New Hampshire (NH) Department of Transportation at a collection site in Salem, NH, along the northbound lane of I-93 just north of an off-ramp, exit 1, and just south of an on-ramp A bottleneck occurs here as, immediately north of this location (downstream), I-93 is physically constricted from three to two lanes (Fig 2) In addition to this physical bottleneck, traffic volumes at this site exceed 100,000 vehicles per day (VPD) which far surpass the 60,000e70,000 VPD that the roadway was designed to accommodate (U.S Department of Transportation, 2016) Because of this heavy, daily flow at the bottleneck, recurrent congestion occurs here As stated by Brilon et al (2005), such sites are ideal for the collection of congestion data Data collection occurred between April and November 30, 2010 During this time, side-fire radar devices intermittently measured traffic at irregular but frequent time periods about apart Data observations (raw data) consist of the following measurements: vehicle counts, average speed, occupancy, and speed (spot speed) of individual vehicles observed over the interval Since incidents of recurrent congestion are the sole focus, data obtained during weekends and holidays are omitted from the analysis Also, because of scheduled maintenance as well as unscheduled “gaps” where the radar devices stopped collecting data (some lasted for several days), many other days were omitted In the end, 128 complete days of quality data were retained Without any (4) where x is the categorical variable distinguishing day-of-theweek, xd is the interaction variable, and e  Nð0; se Þ and independent The procedure for identifying congestion duration and the associated predictor variables will be discussed later in Section 2.5 Such a model (continuous response with two categorical predictors) could also be analyzed using an analysis of variance (ANOVA) format In fact, with such dichotomous predictors, the two model forms and their corresponding hypothesis tests produce equivalent results Fig e Illustration of collection site structure along northbound lanes of I-93 Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 5/10 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 records of non-recurrent events during the collection period, the authors assume that all congestion events observed on these days are recurrent Next, because radar data were collected over very short, irregular time intervals, these measurements were aggregated into uniform intervals of 15 15-min intervals are recommended to ensure “stable” flow rates that are especially suitable for macroscopic/speed-flow analyses (Smith and Ulmer, 2003; TRB, 2010) Specifically, harmonic averages were calculated from flow and spot speed observations within each 15-min interval to produce an aggregated flow rate (q) and aggregated speed (u) in units of vehicles per minute (vpm) and miles per hour (mph), respectively (Daganzo, 1997) Thus, for each day, aggregation yields ut and qt (speed and flow, respectively) where t represents time-of-day with t ¼ 1, 2, …, 96 Fig illustrates the speed and flow aggregates produced for one week in April Also, Fig illustrates the stochastic nature of breakdown, how high flows typically, but not necessarily, result in breakdown (and high, sustained flows are more likely to result in breakdown) Lastly, the authors note that the figure includes the flow profiles during a weekend (the third and fourth mounds) where, as is typically the case, no breakdowns occur This supports the decision to remove weekend days from the analysis of recurrent congestion In Fig 3, black dots indicate speeds less than 50 mph, the critical threshold speed Notice that breakdown corresponds to sustained, high traffic flows on the first, fifth, and seventh days Also, notice that no breakdown occurs on the third and fourth days, which is a weekend 2.4 model Extraction of variables for binary logistic regression From the speed and flow aggregates, the variables required by the regression models can be easily extracted First, to identify a response variable Y to distinguish between congested and freely flowing traffic (to identify a traffic breakdown), a fixed speed threshold u* is typically chosen to identify the transition between them (Banks, 2006; Brilon et al., 2005; Geistefeldt and Brilon, 2009; Habbib-Mattar et al., 2009; Lorenz and Elefteriadou, 2001; Yeon et al., 2009) No standard approach exists for identifying u*, but based on bimodal speed aggregates, u* ¼ 50 mph was chose This transitional threshold is similar to those of Brilon et al (2005), Geistefeldt and Brilon (2009), and Lorenz and Elefteriadou (2001), Yeon et al (2009), who used fixed values of 47 mph (75 kph), 50 mph, 43 mph (70 kph), and 56 mph, respectively So, for some time t, ut > u* and utỵ1 > u* indicates breakdown at time t and thus Yt ¼ Among others, Lorenz and Elefteriadou (2001) used a similar approach to identify breakdown from traffic stream data If, on the other hand, ut > u* and utỵ1 > u*, no breakdown occurs at time t and thus Yt ¼ This simple rule was applied to create a response Yt for each time t in the data Because the GLM form assumes the independent response values, we must select an independent set of Y values for model fitting First, since congested observations are typically separated by long time intervals and periods of freeflow, the authors may assume that the congested responses are, by their rare nature, independent of one another Next, the authors take a random sample of uncongested responses This random sampling “removes” the serial correlation from these data and destroys any time structure that exists in the aggregated values The authors then combine the two sets of responses, the congested and sampled uncongested values, to form an independent set of Y values suitable for our model form Next, for each response Yt considered, the corresponding flow aggregate is observed at time t So, when breakdown occurs at time interval t, qt represents the flow observed immediately prior to breakdown (commonly called a “breakdown flow”) Kondyli et al (2013), who pursued a variety of breakdown identification methods (speed-based, occupancybased, and volume-occupancy/correlation-based), identified these “breakdown flows” in a similar situation When breakdown does not occur at time t, qt represents a traffic flow under freely flowing conditions Lastly, for each response Yt, a corresponding categorical variable for time-of-day is simply identified based on time markers retained from the aggregation process Despite starting with a time-ordered, correlated set of aggregated values, we have extracted a set of data that contains no temporal structure By considering an independent set of response values (see above) and their corresponding flow and time-of-day values, we have extracted an independent set of data that is suitable for a regression analysis 2.5 Extraction of variables for duration model For our duration modeling, the authors must identify a response variable W that represents the time between onset of Fig e Speed and flow aggregates for one week (April 1, 2010eApril 7, 2010) Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 6/10 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 breakdown and clearance of breakdown, the time required by the system to clear a congestion event As previously discussed, time of breakdown is first identified via transition from sustained speeds above to below the threshold u* Then, clearance is similarly identified via transition from sustained speeds below to above the threshold ut < u* and utỵ1 > u* indicate a clearance at time t Then, for each congestion event, a duration W is simply calculated by the time between onset and clearance This definition agrees with Rao and Rao (2012) who stated that the duration of congestion can be determined by measuring reduced travel speeds over a period of time Other more complicated techniques exist for duration estimation by Elefteriadou et al (2011a,b) who used a wavelet transform method to identify the start and end time of congestion event occurrence, but this simpler method was applied For all congestion events, a histogram of congestion durations is given below (Fig 4) This figure reveals a very rightskewed distribution with some very long congestion events The median congestion duration is 120 Lastly, based on time and date markers retained from the raw data, time-of-day and day-of-the-week categorical variables were identified for each congestion event These categorical variables were matched with each duration amount to create a dataset to be used for our duration regression model Because there were one or two congestion events per day, which were separated by a period of freeflow, each duration measurement was assumed to be independent In an initial data investigation, no temporal dependence between durations was observed Thus, this data is appropriate for our regression model Results and discussion Based on the variables extracted from the I-93 data, the logistic and OLS regression models for breakdown and duration, respectively, model fitting via maximum likelihood estimation was performed using R statistical software Fig e Frequency distribution (histogram) for congestion durations 3.1 Binary logistic regression model for probability of breakdown The binary logistic regression model was fit to the following data: flow, the continuous explanatory variable; time sector designation, the dichotomous categorical explanatory variable; and congestion state, the binary response Based on individual p-value analyses (using a 5% significance level) and Chi-squared tests for change in deviance, both the flow and the dichotomous time variables were found to be highly significant explanatory variables in the prediction of breakdown The interaction term, however, was not found to be significant and thus omitted from the model Thus, in terms of the probability curve for the two time sectors (AM- and PM-rush), (1) there is a significant shift, and (2) there is no significant change in trend Residual deviances indicate no significant evidence of lack-of-fit of the model Based on our fitted models, an “S” shaped curve repreb was senting probability of breakdown for observed flows p produced for both time periods (Fig 5) 95% confidence bands were included to illustrate uncertainty associated with probabilities at certain flow values 95% confidence bands (light and dark gray bands) are included to illustrate variability associated with logistic curves First, the authors analyze the time-of-day variable The fitted coefficient associated with this dichotomous variable was found to be highly significant (z ¼ 6.750, p z 0) The odds ratio, given by d OR ẳ exp b b ị, is the relative increase in odds of congestion when going from AM-rush (d ¼ 0) to PM-rush (d ¼ 1) The fitted parameter estimate of b b ¼ 2.17 corresponds to an odds ratio of 8.76, or about an 8-fold increase in the odds of breakdown when going from AM-to PM-rush When considering the 95% confidence interval for b b , (1.54, 2.80), this odds ratio is between 4.66 and 16.44 in 95% confident, clearly a significant (non-zero) increase in odds between time periods From the plot of the logistic curves for AM- and PM-rush periods (Fig 5), this change in odds is observed in the clear shift Fig e Fitted logistic curves for AM- and PM-rush time periods Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 7/10 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 between the two curves While some overlap between confidence bands occurs at high flows, this is simply because the uncertainty associated with the AM-rush period (dark bands indicate 95% confidence for AM-rush period) increases at high flows where few observations exist It can be concluded that, for any flow, the probability of breakdown during the PM-rush is higher than that during the AM-rush (that PM-rush traffic behavior is generally more likely to be congested) Observed traffic behavior supports this finding as congestion is much more common during the PMrush period than during the AM-rush In fact, of the congestion events considered, about 80% are observed during the PM-rush period Certainly, flow is the most important driver of congestion, but our result suggests that time-of-day may play a role The authors suspect that driver behavior during these periods may contribute to this discrepancy, that PM-rush commuters likely drive more aggressively In any event, the results warrant further investigation and analysis Lastly, steps to minimize congestion duration for the AM-rush, say, may not be effective for the PM-rush This result illustrates the limitation of a single, fixed capacity value as prescribed by the Highway Capacity Manual (HCM) According to the HCM guidelines, the freeway capacity for a single lane of traffic is 2000 vehicles per hour (vph) Since our traffic observations were taken across three traffic lanes, the HCM capacity for these three lanes is 6000 vph or 100 vehicles per minute (vpm) At that flow, for flows of 100 vpm, the probability of breakdown is somewhat low (around 20%) during the AM-rush period, while probability of breakdown during the PM-rush period for the same flow is about 70% Since the interaction term was not significant and omitted from our model (b3 was not found to be significantly different from zero), it is assumed that both the AM- and PM-rush periods share a common trend That is, the change in odds of breakdown for a one unit increase in flow is the same for both AM- and PM-rush periods This shared change in odds is the OR ẳ exp b b ị From our odds ratio associated with b1 , given by d fitted model results, the coefficient estimate of b b ¼ 0.03 corresponds to an odds ratio of 1.031 Thus, for both periods, the probability of breakdown increases 3.1% for every one additional vehicle per minute From the plot above, both periods have probabilities that progress at the same rate relative to flow (Fig 5) From an administrator's point of view, there may be a practical application of these results Most simply, results from the breakdown model suggest that measures to remedy recurrent congestion be focused on PM-rush periods, when congestion is more likely For example, hard shoulder running may be implemented during these congestion-prone hours to mitigate expected congestion This practice is a proven technique for congestion mitigation in Northern Virginia (Mazzenga and Demetsky, 2009), and the extension of its use is recommended (Bauer et al., 2004) Furthermore, this practice has been shown to effectively ease congestion without increasing accident rates (ITS International, 2013) Or, administrators may opt to introduce variable speed limits (VSLs) during these congestion-prone hours Such VSLs are used extensively in Europe with great success (Mazzenga and Demetsky, 2009) As a final example, administrators may implement ramp metering during the PM-rush It has been shown that ramp metering is effective at limiting the number of vehicles entering a freeway and can help to prevent recurrent congestion (Texas A&M Transportation Institute, 2016) 3.2 Ordinary least squares regression model for congestion duration The least squares regression model was fit to the following data: time-of-day, a dichotomous explanatory variable distinguishing AM- and PM-rush time sectors; day-of-the-week, a categorical explanatory variable; and congestion duration, the continuous response Initially, the categorical variable representing day-of-the-week contained a level (category) for each weekday, Monday through Friday However, after several model-fitting exercises, it was determined that most days were not significantly different from one another In fact, based on an analysis of variance, data collected from Monday, Tuesday, Wednesday, and Thursday are nearly identical However, it was found that MondayeThursday data was significantly different from Friday data, so a dichotomous variable was used in lieu of a categorical variable for day-ofweek Also, initial models suggested a lack of normality among the residuals To remedy this, a transformation of the response was pursued Because duration times are highly right-skewed, and because variance seems to increase with duration, a log-transformation was applied (Fig 4) Therefore, the slightly modified model is as follow (5) logWị ẳ b0 ỵ b1 d1 ỵ b2 d2 ỵ b3 d1 d2 ỵ e where d1 is a dichotomous variable distinguishing AM- and PM-rush periods, d2 is a dichotomous variable distinguishing Fridays from all other days (d2 ¼ if Friday, d2 ¼ if Monday, Tuesday, Wednesday, or Thursday), d1d2 is the interaction term Similarly, Garib et al (1997) used linear regression models to predict the log of duration of traffic incidents observed from California freeways (non-recurrent events) While the model fits the data surprisingly well (R2 suggests the model form explains about 67% of the total variation observed in log(Duration) measurements), a “better” model was not pursued While attempting to find a model that maximizes the explained variation is a useful exercise, our aim is to identify significant explanatory variables and their implications on congestion duration Of primary importance from our fitted result is the statistical significance of the interaction term (t ¼ 3.277, pvalue ¼ 0.00189) Both main effects, time-of-day and day-ofthe-week, are also found to be statistically significant (t ¼ 4.323, p-value ¼ 7.15e-05; t ¼ 4.455, p-value ¼ 4.61e-05, respectively), although main effects are typically ignored in the presence of interaction That is, if two terms interact, changes in both explanatory variables will have an effect on the response outcome In our case, the effect on the mean outcome, log(Duration), from a change in time-of-day depends on the level of day-of-the-week This effect is most easily seen in the plots below (Fig 6) Interaction lines calculated from the fitted regression model are included The two plots above represent the same phenomenon from two different perspectives: grouping the data by day-of-theweek (Fig 6(a)) and time-of-day (Fig 6(b)) In either case, a dis- Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 8/10 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 Fig e Boxplot for congestion duration (a) MondayeThursday and Friday traffic by time-of-day (b) AM- and PM-rush periods for day-of-the-week ordinal interaction is marked by the clearly non-parallel interaction lines Fig 6(b) is probably the most interpretable as the moving along the x-axis represents transition from AM-to PM-rush periods Here, when going from AM-to PMrush periods (x-axis grouping), the change in log(Duration) depends on day-of-the-week For MondayeThursday, there is an increase in log(Duration), while there is a decrease in log(Duration) for Fridays Analysis of the fitted parameters allows us to precisely quantify this effect After converting back to the original scale of measurement (duration in minutes instead of log(Duration)), calculations reveal that for MondayeThursday traffic, transitioning from AM-rush period to PM-rush period results in more than a 2-fold increase (273%) in duration On the other hand, for Friday traffic, transitioning from AM-to PM-rush periods results in a 59% decrease in duration Similarly, from Fig 6(a), there is a distinct change in log(Duration) when going from Monday to Thursday to Fri categories, and the magnitude of the change depends on the time-of-day For the AM-rush, this change (from Monday to Thursday to Fri.) is a steep increase in log(Duration) For the PM-rush period, this change is increasing as well, but more modestly Results from our duration modeling are a bit surprising The authors can speculate that Friday “evening” commutes may be occurring earlier in the day Since our analysis did not identify a significant number of congestion events in the early afternoon, though, this idea was abandoned Perhaps the evening commute is spread across a wide time span and results in fewer long congestion events This conjecture would agree with the results presented in this analysis In any event, this result may warrant a more day-specific analysis of congestion events If our results are confirmed, and MondayeThursday traffic experiences longer congestion events, perhaps administrators should implement congestion mitigation techniques during these days or possibly extend these mitigation techniques later into the evening, beyond traditional “rush hours” As mentioned above, these methods to remedy such congestion may be hard shoulder running, VSLs, or ramp metering Lastly, the regression/model fitting diagnostics suggest that the regression assumptions are all met (that our model specification is appropriate for this data) Plots of residuals versus fitted values show constant variance and normality plots show the residuals to be normally distributed Also, there was no indication of correlated errors, further evidence that the duration response values are completely uncorrelated and suitable for the OLS format Conclusions Recurrent congestion is a complicated process that is likely triggered by a variety of interconnected conditions That said, understanding the frequency and duration of these events is critical to highway administrators and decision-makers The goal of this study is not to create a real-time forecasting tool, but rather to identify model structures that reveal the importance, driving mechanisms of the congestion process Such structures will contain information that is easily interpreted and understood In our binary logistic regression model to predict probability of breakdown by flow, results show a distinct shift by time-of-day For any flow, the PM-rush period has higher probability of breakdown than the AM-rush period Overall, based on the parameter estimate associated with the time-ofday, there is an estimated 8-fold increase in the odds of breakdown when going from AM-to PM-rush So, while flow is the primary predictor of congestion, our result indicates that time-of-day is likely a factor as well From an administrator's point of view, this may suggest that measures to remedy recurrent congestion should be focused on PM-rush periods As previously mentioned, these remedial measures may include hard shoulder running, variable speed limits (VSLs), or ramp metering By considering time-of-day as a simple proxy for secondary traffic characteristics, the result from our logistic regression analysis may suggest that driver behavior affects congestion and roadway capacity That is, one could speculate that there are more aggressive drivers during the PM-rush hour, and these aggressive drivers are prone to excessive weaving, etc With more detailed data, future work could investigate these Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JTTE110_proof ■ 11 January 2017 ■ 9/10 J Traffic Transp Eng (Engl Ed.) 2017; x (x): 1e10 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 specific, underlying factors rather than time-of-day categories In our duration model, an ordinary least squares regression model is used to predict congestion duration using time-ofday and day-of-the-week categorical variables After an initial investigation, it suffices to use dichotomous variables for time-of-day, AM- or PM-rush, and day-of-the-week, MondayeThursday or Friday The primary result from this model is the statistically significant interaction term This suggests that when going from AM-to PM-rush periods, the change in congestion duration depends on day-of-the-week In fact, this change is increasing for MondayeThursday and decreasing for Fridays This surprising result may suggest the need for more day-specific analysis of congestion events Ultimately, administrators may choose to focus congestion mitigation techniques (hard shoulder running, VSLs, ramp metering, etc.) during MondayeThursday, evening commute hours This work represents an empirical study on breakdown and delay at a known bottleneck in Salem, NH Our respective model specifications successfully identify significant predictors that shed light upon the congestion process A limitation of this analysis, however, is that our conclusions are restricted to the one location from which data was collected That is, our results are not transferable to other locations, even those very near our location or in other directions Data from additional bottlenecks will allow for transferable results, a general model that could explain the congestion process across diverse locations That says, the results here are justified and illuminate some congestion phenomena that are likely not site-specific Uncited reference Q2 Texas A and M Transportation Institute, 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and a PhD in Mathematics from the University of New Hampshire His areas of research are transportation and extreme value theory related to climate change projections Please cite this article in press as: Laflamme, E.M., Ossenbruggen, P.J., Effect of time-of-day and day-of-the-week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017), http://dx.doi.org/10.1016/j.jtte.2016.08.004 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 ... P.J., Effect of time -of- day and day -of- the- week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition)... Effect of time -of- day and day -of- the- week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation Engineering (English Edition) (2017),... Laflamme, E.M., Ossenbruggen, P.J., Effect of time -of- day and day -of- the- week on congestion duration and breakdown: A case study at a bottleneck in Salem, NH, Journal of Traffic and Transportation