Total Transportation System Solutions (T2S2)

INTRODUCTION

One of the fundamental paradigms of traffic engineering is that trip arrivals can be predicted using a Poisson distribution.  This paradigm is so ubiquitous that it is used in practically every traffic flow theory text.  Can trip arrivals be modeled by Poisson in every instance? For a multilane facility, does Poisson work not only overall but for individual lanes?  The purpose of this research is to test the hypothesis that trips arrive, in all cases, in a Poisson distribution.  To test this hypothesis, fast food drive through lanes were used because of their operational simplicity, as they have, for all practical purposes, no outside influences affecting driver behavior or the traffic stream, and because they exist in both single lane and multilane form

LITERATURE REVIEW

An extensive literature review was performed, and there were few references to trip arrival distributions.  Given the wide acceptance of the Poisson distribution for trip arrivals in the profession, this is not surprising.  There were three papers in the 1980s and 1990s that advocated different distribution models.  The first paper was published in 1983 by Mahalel & Hakkert, who advocated the use of a Markovian process to model arrival patterns on a one-way multi-lane facility such as one direction of a freeway. (1)  Subsequently, Akcelik, in research published by the Australian Road Research Board and presented at the 1994 Annual Meeting of the Transportation Research Board, discusses three different arrival headway distributions:  M1 (negative exponential), M2 (shifted negative exponential), and M3 (bunched exponential), with M1 and M2 being random arrival models, and concludes that the M3 gives a better representation of real-life arrival patterns. (2)  That same year, Alfa of the University of Manitoba published a paper using Markovian Arrival Processes for modeling traffic queues at an actuated signal.  (3)  Since 1994, however, there have been no further references to other distribution types.  In 2000, the Belgian team of Vandaele et. al. reaffirmed the supremacy of Poisson distribution by using it in their queueing models. (4)  

METHODOLOGY

This study consists of several components; to test variability of arrivals by time of day; to test variability of arrivals by number of channels; to test variability of arrivals by individual channels of a multi-channel system.  Every thirty seconds, the amount of vehicles queued at the order window was recorded, and readings were taken for one hour.  The changes to the queue were calculated, and recorded as the number of arrivals and departures.  Thirty seconds was the incremental time chosen because, based on observations, the typical time a vehicle spends ordering at a drive through exceeds 30 seconds, and thus the probability of a vehicle arriving and departing without being recording as being queued is small.

To test variability of time, a single channel drive through was used because it is the simplest and easiest to collect data at.  Because the data collector did not know what to expect, using a simple site allowed the data collector the opportunity to make changes to the data collection technique without compromising the dataset.  The first site chosen (Site 1) was a Taco Bell inFort Gratiot,Michigan.  (Figure 1) Located on a five lane regional arterial, it also had a secondary entrance to another arterial which added randomness to its vehicle stream.  In addition, the order kiosk is at the far rear of the property, while the pick-up window is at the front, far enough away that queueing at the pick-up window has a negligible effect on arrivals or departures from the order kiosk.  Data was collected for an hour in the afternoon and an hour in the evening on the same day to determine if arrival patterns vary by time of day.  The same procedure was repeated the following day at a nearby McDonald’s inPort Huron(Site 2) with dual side-by-side order kiosks, with data being collected at the same times as it was at Taco Bell. (Figure 2) Queue data was collected for each individual kiosk, and that data was combined into for each of the two hours, and those hours were compared to each other to determine if the results obtained at the single channel drive through were also valid for a dual channel one.  The composite results for the McDonald’s were then compared to those for the Taco Bell to determine if the overall patterns observed for the single channel were the same as for the dual channel.  Data was then collected at a second nearby dual-channel McDonald’s inFortGratiot(Site 3) at the same time of day and on the same day of the week to compare the variability by site.  (Figure 3) A final data set was taken at a third dual channel McDonald’s inPort HuronTownship(Site 4) on the same day of week but at a different time to test the variability of different sites by different times.  (Figure 4)

FIGURE 1 Study site 1, single channel Taco Bell, Fort Gratiot, MI.

 FIGURE 2 Study site 2, dual-channel McDonald’s, Port Huron, MI.

FIGURE 3 Study site 3, dual-channel McDonald’s, Fort Gratiot, MI.

FIGURE 4 Study site 4, dual-channel McDonald’s, Port Huron Twp, MI.

ANALYSIS

A total of six hours of data were collected at the four study sites, two at the single channel site and four at the three dual-channel sites.  In order to remove any variation in lane choice at a particular location from compromising the ability to compare from site to site for the dual-channel sites, data was collected and recorded for each individual lane, and the separate datasets were also combined to create composite datasets.  Thus, the fours hours of data collected yielded twelve datasets- four for the left lane, four for the right lane, and four for both lanes combined.  With the addition of the two datasets collected at the single channel site, a total of fourteen datasets were compiled.  A summary of the total arrivals and departures for each of the fourteen datasets is presented in Table 1.  As can be seen from the table, the number of arrivals and departures for any given data set may not be the same.  Where arrivals are greater than departures, a queue was present at the end of the dataset hour, resulting in arriving vehicles being trapped in line at the end of the study period.  Conversely, where departures for a study hour exceed arrivals, vehicles were already queued in line when the study period began, having arrived at some unknown point in time prior to the study period.  Thus, the hourly flow during a particular study period is the lesser of either the arrivals or the departures for that study period.

TABLE 1  Arrivals and Departures

Because of the locations of the various drive-thru kiosks studied with respect to the pick-up window, i.e. far enough away to be minimally affected by queueing at the window, it was decided to test a Poisson distribution for the departures from the kiosk, as well, as departures from the kiosk are, for all practical purposes, arrivals at the pick-up window.  Such a test will allow us to determine if the Poisson distribution is valid for not only arrivals from off-site onto the site, but from one point of the site to another.

A Poisson distribution was calculated for both the arrivals and departures for all fourteen datasets.  These distributions were then compared to the actual distributions using the Wilcoxin Rank Sum test, a non-parametric goodness of fit test, for both a 90% and a 95% level of confidence (LOC).  The test statistic, z_a, is 1.282 for a 90% LOC and 1.645 for a 95% LOC.  Using this test, the null hypothesis that the distribution is a Poisson distribution, is rejected when the calculated z is greater than z_a .  The results of this calculation for all fourteen arrival and departure datasets are shown in Table 2.

As can be seen from Table 2, the null hypothesis, that the distributions are Poisson, cannot be rejected for any of the datasets, either arrival or departure, at a 95% LOC.  For one dataset, the total arrivals for Sample 1 at Site 3, the null hypothesis can be rejected at a 90% LOC.  It should be noted that the Z statistic for the departures for this dataset is the worst of all fourteen datasets.  This particular dataset is the only dataset where the Z statistic for arrivals exceeds 0.700, and is only one of two where the Z statistic for the arrivals is larger than the Z statistic for the departures, the other being the dataset for the left lane of Sample 1 at Site 2, where the Z value for the arrivals is 0.578 and that for the departures is 0.525.  Other then the two aforementioned datasets, the Z statistic for the departures is greater than that for arrivals for every dataset, and while there is only one dataset that had a Z statistic for arrivals greater than 0.700, eight datasets had a Z statistic for departures greater than 0.700, and four had a Z statistic for departures greater than 1.000.

TABLE 2  Results of Wilcoxin Rank Sum test

NOTE:  Reject H_O, this is a Poisson distribution, if z>z_a. z_a=1.645 for a 95% LOC; z_a=1.282 for a 90% LOC.

A further analysis of Table 2 reveals a trend of a lack of trends.  There does not appear to be any correlation between the dataset size and the goodness-of-fit.  For example, Sample 1 from Site 2 shows an increase in the Z statistic (i.e. worsening goodness-of-fit) as the dataset size increases, for both arrivals and departures, while the other sample from the same site, Sample 2, does not show any correlation at all, despite being an almost identical sample.  There does not appear to be any correlation between time of day and the value of the Z statistic.  Both samples from Sites 1 and 2 were taken at corresponding times of day, e.g. Sample 1 from Site 1 was taken at the same time of day as Sample 1 from Site 2, and Sample 2 from Site 1 was taken at the same time of day as Sample 2 from Site 2.  At Site 1, the Z statistic for both arrivals and departures is worse for Sample 2; the same trend did not occur at Site 2.  Sample 1 from Site 3 was taken at the same time as Sample 2 from both Sites 1 and Sites 2, and there is no consistency in the values of the Z statistic across the three samples.  There is also no consistency with respect to a particular day of week, as the four samples taken on the same day of the week, the samples from Sites 2, 3, and 4, do not have any unique patterns that differentiate them from Site 1.  Since there does not appear to be day of the week characteristics, it is logical to assume that there are no patterns unique to dual-lane drive-throughs, as the same three sites from which samples were taken on the same day of the week are also the three dual channel sites.

Although there are no general trends applicable to all the datasets, there is a trend that is applicable to the dual-channel drive throughs with respect to the performance of individual lanes.  For every sample collected at a dual-channel location, the left lane had a worse Z statistic than the right lane, for both arrivals and departures.  There are two plausible explanations for this phenomenon.  First, people prefer the left lane over the right and are more willing to wait in queues in the left lane, a phenomenon observed by the data collector.  An unscientific poll on Facebook in which a photograph of an empty dual drive through was shown and individuals were asked to state the lane they would choose; more chose the left than the right. (Figure 5)  Reasons given for the preference of a particular lane include travel distance, familiarity, type of vehicle, approach direction, parking lot configuration, and political affiliation.  Interestingly, some of those choosing the left lane stated that they would choose the left even if there were cars in the left and none in the right.  The datasets from the three dual-channel sites support this assumption, as the left lane was preferred for all datasets and the right lane was unoccupied more of the time than the right lane; in other words, people preferred the left lane even though it is occupied more.  (Table 3)  The second explanation is one of the reasons given for choosing the left lane in the unscientific poll- the left lane is shorter than the right lane.  Because it is shorter, it is more susceptible to queueing from the pick-up window than the right lane, and this queueing negatively impacts arrival and departure patterns, causing a worse fit with a Poisson distribution.

FIGURE 5  Photograph used in unscientific Facebook poll.

TABLE 3  Lane preference and occupancy

CONCLUSIONS

The test hypothesis was that arrival distributions are Poisson in all situations.  This hypothesis was tested for arrivals at a drive-through kiosk of a fast food restaurant, and was also as for arrivals at the pick-up window.  The distribution of the arrivals at the pick-up window was determined by determining the distribution of the departures at the kiosk, as a departing vehicle from the kiosk becomes an arriving vehicle at the pick-up window, as it is a closed system between the kiosk and the window.  The Poisson distribution was tested not only for a single lane drive through, but a dual-lane one as well, and was not only tested at the dual lane drive-throughs for total arrivals and departures, but also for arrivals and departures for each lane.  Based on this limited study, the null hypothesis cannot be rejected at a 95% LOC for any dataset, although it can be rejected at a 90% LOC for one dataset.  Furthermore, the data revealed that there is no correlation between the size of the dataset and how well Poisson fits the distribution.  There are no apparent unique characteristics of trip arrival patterns with respect to time of day, day of the week, or number of lanes.

There are several trends that have been observed that should be noted.  First, Poisson does not fit the distribution of arrivals at the pick-up window as well as it does the distribution of arrivals at the kiosk.  One possible explanation of this is that the activity at the pick-up window is more complicated than at the kiosk, which requires a larger time period at the window which leads to backups, queueing, and a breakdown of the random distribution.  A good way to verify this hypothesis would be to study the arrival patterns on toll roads with manned and unmanned booths, with the manned booth being analogous to the pick-up window and the unmanned booth being analogous to the kiosk, and see if the trend exists there as well.  Second, in every case, Poisson did not fit the distribution of the left lane as well as it did the right lane.  The implication of this is that, on a multilane facility, every lane may have a unique distribution.  Mahalel & Hakkert suggested this phenomenon in their study.  Due to the limited scope of this study, it cannot be deduced if this is a phenomenon unique to dual-channel drive through lanes, for the reasons stated before, or if it occurs in other situations where there is more than one lane to choose from, as argued by Mahalel & Hakkert.  Finally, in every case, the left lane is the preferred lane, even though is occupied more than the right.  There has been little research on lane choice, and it would be useful to determine if this is a phenomenon unique to restaurant drive through or if it occurs on the roadway network.

Finally, with respect to the one dataset where the null hypothesis can be rejected at a 90% LOC, no conclusions can be drawn as to why Poisson can be rejected for this dataset.  It is possible that this is a site specific anomaly, as this site is several miles away from the other three.  It is also possible that this is a time-specific anomaly, as this is the only sample that was taken at a time different than all the other samples.  Thus, it is not possible to determine if this is a site characteristic, a time characteristic, or neither.  In order to determine a possible explanation, it is necessary to either take another sample at the same site but at the same time of other sample, or take a sample at one of the other sites at the same time.

REFERENCES

1. Mahalel, D., and A.S. Hakkert.  Traffic Arrival Patterns on a Cross Section of a Multilane Highway. Transportation Research Part A, Volume 17, issue 4, July 1983, pp. 263-270.
2. Akcelik, R.  Estimation of Green Times and Cycle Time for Vehicle-Actuated Signals.  Presented at the 73^rdAnnual Meeting of the Transportation Research Board,Washington,D.C., 1994.
3. Alfa, A.S.  Modelling Traffic Queues at a Signalized Intersection with Vehicle-Actuated Control and Markovian Arrival Processes.  Computers Math Applic., Vol. 30, No. 8, pp. 105-119, 1995.
4. Vandaele, N., T. Van Woensel, and A. Verbruggen.  A Queueing Based Traffic Flow Model.  Transportation Research Part D 5 (2000) 121-135.

ABSTRACT  

One of the paradigms of traffic engineering is that arrival patterns can be represented by a Poisson distribution.  Is this paradigm valid in all cases?

In 1983 Mahalel and Hakkert postulated that, for a multilane highway, the distributions are lane dependent and that a Markovian process is a superior way to model arrival patterns.  In 1994 Alfa used Markovian arrival processes for modeling traffic queues at actuated signals.  That same year, Akcelik presented a paper at the Annual Meeting of the Transportation Research Board that advocated the use of a “bunched exponential” distribution model for arrival patterns.  Since then, there has been no further research to challenge the Poisson paradigm.

The test hypothesis for this study is that Poisson is a valid distribution model.  To test it, both single channel and multi-channel fast food drive through lanes were used.  Queue data were collected every thirty seconds for one hour at four sites, three dual-channel and one single channel; at the single channel and one of the multi-channel locations, data was collected for two different hours of the day, but the same hours at both sites, to allow for validation of the procedure.  Datasets at each of these two sites were compared to each other; first, site specific comparisons with respect to time, and then time specific comparisons with respect to site, in order to determine if there were any site specific or time specific data characteristics.  Data was then collected at two additional dual-channel sites, both on the same day as the day the data was collected at the first dual-channel site, with the data for the first new site collected at the same time as the first site, and the data for the second new site collected at a different time.  At the dual-channel sites, data was collected independently for each lane and then added together to model the site.

A total of fourteen datasets were obtained.  From these fourteen datasets, the following inferences were drawn: there are no site specific arrival patterns; there are no time specific arrival patterns; there are no day of the week specific arrival patterns; Poisson does not fit the arrival pattern at the pick-up window as well as at the kiosk; there are no patterns specific to the number of channels; Poisson does not fit the arrival pattern in the left lane of the dual channel sites as well as the right lane; drivers prefer the left lane even though the right is unoccupied more frequently; for all datasets, the Poisson distribution could not be rejected at a 95% LOC, although it could for one dataset for a 90% LOC, although it cannot be determined if this dataset is a random anomaly.

Due to the consistent difference in performance of Poisson between the pick-up window and the kiosk, further tests of this phenomenon at an analogous location, such as manned and unmanned toll booths, should be considered.  Furthermore, the consistent variation in the goodness-of-fit of Poisson between the left lane and the right lane is also worthy of more study.

ACKNOWLEDGEMENTS

The author would like to thank Victoria Schrader for her assistance in data collection and those who responded to the unscientific Facebook survey and provided insight as to lane choice.

CITE AS:

Schrader, M.H. (2013)  Is poisson always valid?  Total Transporation System Solutions.