| The multiple of income flow | Average Delay | Standard Deviation |
| 1.0 times | 42.76 seconds / vehicle·hour | 0.95 seconds / vehicle·hour |
| 1.2 times | 47.22 seconds / vehicle·hour | 1.56seconds / vehicle·hour |
| 1.4 times | 51.99 seconds / vehicle·hour | 2.54 seconds / vehicle·hour |
| 1.6 times | 61.54 seconds / vehicle·hour | 3.81 seconds / vehicle·hour |
| 1.8 times | 81.28 seconds / vehicle·hour | 8.30 seconds / vehicle·hour |
+
+# Emergency Case
+
+Our model can simulate an emergency. In Figure 14, one of the cars breaks down and block an entire lane. However, the traffic circle still works, but the average delay time has increased by $10\mathrm{~s}$ .
+
+The self-adaptivity of our model lets us adjust the light timestamps and reduce the traffic jam in an emergency case. However, because of limited time, we cannot describe the adaptivity here.
+
+
+Figure 14. Breakdown of a vehicle slows the traffic, but traffic still circulates.
+
+# Conclusion
+
+To estimate the overall performance of a traffic circle with a specific vehicle flow, we develop two simulation models. The first uses a Markov process to consider the entire flow, the second devotes its attention to the individual behavior of each vehicle.
+
+We choose five objectives to evaluate the control method and convert them to a combined expense. We apply this standard to a real-life traffic circle with typical traffic control device setups.
+
+We offer an optimization model to select traffic devices and determine the green-light period when traffic lights are used. In addition, we introduce orientation signs as a thoroughly new measure to bring efficiency. The flexibility of these solutions is proved when confronted with accidents.
+
+# References
+
+Federal Highway Administration, U.S. Department of Transportation. 2008. The importance of transit. Chapter 14 in 2004 Status of the Nation's Highways, Bridges, and Transit: Conditions and Performance. http://www.fhwa.dot.gov/policy/2004cpr/chap14.htm.
+Fitzpatrick, Kay. 2000. Accident Mitigation Guide for Congested Rural Two-lane Highways. Transportation Research Board, National Research Council.
+Garber, N.J., and L.A. Hoel. 2002. Traffic and Highway Engineering. 3rd ed. Pacific Grove, CA: Brooks/Cole.
+Hangzhou Public Security Bureau Traffic Police Detachment. 2006. March on the city's briefing on road traffic accidents. www.hzpolice.gov.cn/ tabid/67/InfoID/9343/Default.aspx.
+Hubacher, Markus, and Roland Allenbach. 2002. Safety-related aspects of traffic lights. http://www.bfu.ch/PDFLib/676_68.pdf, http://www.bfu.ch/English/Forschung/Forschungsergebnisse/pdfResults/r48e.pdf.
+Maher, Mike. 2008. The optimization of signal settings on a signalized roundabout using the cross-entropy method. Computer-Aided Civil and Infrastructure Engineering. 23: 76-85.
+Picken, Andrew. 2008. Businesses want Sheriffhall flyover. *Edinburgh Evening News* (26 June 2008). http://edinburghnews.scotsman.com/edinburgh/Businesses-want-Sheriffhall--flyover.4225078. jp.
+SetupWeasel. 1999. Traffic circles. http://www.bbc.co.uk/dna/h2g2/A199451.
+Traffic Light Wizard. n.d. http://trafficlightwizard.com.
+Transport for London Street Management—London Road Safety Unit 5. Do traffic signals at roundabouts save lives? IHT Transportation Professional (April 2005). http://www.tfl.gov.uk/ assets/downloads/SignalsatRoundabouts-TransportationProfessiona-Article.pdf .
+
+TuffRhino.com. n.d. www.tuffrhino.com/Yield_Sign_p/sts-r1-2.htm.
+
+Wang, Shen Lin. 2005. People look forward to the traffic lights at an early date, "induction." (Chinese). People's Government of Kaifeng.
+
+http://www.kaifeng.gov.cn/html/ 4028817b1d926237011d9332cfe300bb/3214.html.
+
+Wikipedia. 2009. List of variations in traffic light signalling and operation.
+
+http://en.wikipedia.org/wiki/unusual Uses_ofTrafficLights.
+
+Ye, Ran. 2001. California power-saving traffic lights out of new tactics to use energy-saving foam. (Chinese). news.eastday.com (15 August 2001).
+
+http://www.shjubao.cn/epublish/gb/paper148/20010815/class014800004/hwz462812.htm.
+
+
+Yichen Huang, Zeyuan Allen Zhu, Tianyi Mao, and team advisor Jun Ye.
+
+# Pseudo-Finite Jackson Networks and Simulation: A Roundabout Approach to Traffic Control
+
+Anna Lieb
+
+Anil Damle
+
+Geoffrey Peterson
+
+Dept. of Applied Mathematics
+
+University of Colorado
+
+Boulder CO
+
+Advisor: Anne Dougherty
+
+# Summary
+
+Roundabouts, a foreign concept a generation ago, are an increasingly common sight in the U.S. In principle, they reduce accidents and delays. A natural question is, "What is the best method to control traffic flow within a roundabout?" Using mathematics, we distill the essential features of a roundabout into a system that can be analyzed, manipulated, and optimized for a wide variety of situations. As the metric of effective flow, we choose time spent in the system.
+
+We use Jackson networks to create an analytic model. A roundabout can be thought of as a network of queues, where the entry queues receive external arrivals that move into the roundabout queue before exiting the system. We assume that arrival rates are constant and that there is an equilibrium state. If certain conditions are met, a closed-form stationary distribution can be found. The parameters values can be obtained empirically: how often cars arrive at an entrance (external arrival rate), how quickly they enter the roundabout (internal arrival rate), and how quickly they exit (departure rate). We control traffic by thinning the internal arrival process with a "signal" parameter that represents the fraction of time that a signal light is green.
+
+A pitfall of this formulation is that restricting the capacity of the round-
+
+about queue to a finite limit destroys the useful analytic properties. So we utilize a "pseudo-finite" capacity formulation, where we allow the roundabout queue to receive a theoretically infinite number of cars, but we optimize over the signal parameter to create a steady state in which a minimal number of waiting cars is overwhelmingly likely. Using lower bound calculations, we prove that a yield sign produces the optimal behavior for all admissible parameter values. The analytic solution, however, sacrifices important aspects of a real roundabout, such as time-dependent flow.
+
+To test the theoretical conclusions, we develop a computer simulation that incorporates more parameters: roundabout radius; car length, spacing, and speed; period of traffic signals; and time-dependent inflow rates. We model individual vehicles stochastically as they move through the system, resulting in more-realistic output. In addition to comparing yield and traffic-signal control, we also examine varied input rates, nonstandard roundabout configurations, and the relationships among traffic-flow volume, radius size, and average total time. However, our simulation is limited to a single-lane roundabout. This model is also compromised by the very stochasticity that enhances its realism. Since it is nondeterministic, randomness may mask the true behavior. Another drawback is that the computational cost of minimization is enormous. However, we verify that a yield sign is almost always the best form of flow control.
+
+# Introduction
+
+A report from the Wisconsin Dept. of Transportation notes that "to many, the idea of replacing four-way signaling with a roundabout seems like replacing hot dogs with crepes at the ballpark" [McLawhorn 2002]. For many Americans, the roundabout (traffic circle, rotary) is a foreign idea, even though the first one was built in New York in 1903. Roundabouts fell out of favor in the U.S.; but since midcentury, as studies showed how much safer and more efficient they can be, there has been a resurgence in their construction [National Cooperative Highway Research Program 1998]. Half of the states in the U.S. now have roundabouts, more than 1,000 installations. One study indicated that, on average, fatal crashes decreased $90\%$ after traditional traffic lights were replaced by roundabouts [Arizona Department of Transportation n.d.].
+
+A crucial aspect of efficiency and safety is entry. Until the 1920s, "yield-to-right" rules gave right of way to incoming cars, which tended to cause "locking" and delays at high traffic volumes. British studies indicated that adopting "priority-to-the-circle" rules allows more cars to move through the circle more quickly and diminishes accident rates. The deflection of entering traffic serves to prevent excessive speed within the roundabout and to reduce further the incidence of accidents [National Cooperative Highway Research Program 1998]. So in modern roundabouts, incoming traffic
+
+yields to traffic in the circle (and changes direction to some extent). With that rule, entry may be governed in various ways. The simplest and most common is a yield sign at each entry point. The U.S. Dept. of Transportation advises that roundabouts "should never be planned for metering or signalization" [Robinson et al. 2000].
+
+We develop a mathematical model for flow in roundabouts. We introduce assumptions used in determining the key parameter inputs and developing a metric for "effectiveness." We subsequently formulate and solve a simple analytic model of networked queues in an equilibrium state. After discussing limitations of the analytic model, we adapt it into a computer simulation that allows for detailed analysis and can be used to optimize the flow-control method.
+
+# Assumptions
+
+Exponential arrivals/departures: Arrivals and departures follow a Poisson process, with exponentially-distributed interarrival times.
+
+Local variable selection: External forces such as weather, special events, or acts of God may alter the system, but we do not address these factors.
+
+Unbounded output: Although blockages would affect the roundabout, we assume that cars can always leave the roundabout at their exits.
+
+Yield and stop sign equivalence: In terms of efficiency, a stop sign performs only as well as (or worse) than a yield sign, so a yield sign is preferable. Stop signs may, however, may be appropriate for safety, for example, with high pedestrian traffic.
+
+# Effectiveness
+
+The most effective roundabout design minimizes delay.
+
+# Analytic Formulation
+
+Our analytic model consists of a network of $\mathrm{M} / \mathrm{M} / s$ queues (queues with Markovian arrivals, Markovian departures, and $s$ servers), known as a Jackson network. (Figure 1). We give the model's parameters in Table 1. We assume that there is a steady state and no explicit time dependence. Effectiveness is quantified by the probability of few cars waiting, which can be calculated from the stationary distribution of the Jackson network. For the most effective roundabout, the most likely stationary states will be those with the fewest total cars, implying that delay is minimized.
+
+
+Figure 1. Visual schematic of queuing network.
+
+Table 1. Summary of parameters to the analytic model.
+
+| Variable | Description |
| P(t) | Population of U.S. in year t |
| pl(t) | Landlines per person in U.S. |
| h(t) | Handsets per landline |
| πa(t) | Percentage of landline owners with corded phones |
| ea(t) | Yearly Energy Consumption (YEC) (kWh) by corded phones |
| πb(t) | Percentage of landline owners with cordless phones |
| eb(t) | YEC by cordless phones |
| πc(t) | Percentage of landline owners with combination cordless phone/answering machines |
| ec(t) | YEC by combination cordless phone/answering machines |
| πd(t) | Percentage of landline owners with separate answering machines |
| ed(t) | YEC by separate answering machines |
+
+Due to a lack of relevant data, we make several assumptions:
+
+- All yearly energy consumption functions are constant over time. Because corded phones draw their energy solely from phone lines, there is little room for variation in their power draws, so this at least seems reasonable. However, answering machines, cordless phones, and combinations of the two do not have this restriction, and it seems likely that they are becoming more energy efficient with time. However, no data were available to support this hypothesis, so we fixed YEC based on available sources.
+- The adoption of cordless vs. corded phones and answering machines no doubt follows its own sigmoidal curve, but again no data are available. So the variables $h, \pi_{a}, \pi_{b}, \pi_{c}, \pi_{d}$ are all modeled as first-order linear approximations.
+
+Regardless, results produced by the model agree well with available data for energy consumption.
+
+# Energy Cost of Cellphones
+
+The energy cost for cellphones can be modeled as
+
+$$
+E _ {C} (t) = P p _ {c} (E _ {c 1} + E _ {c 2}),
+$$
+
+where
+
+$$
+E _ {c 1} (t) = f _ {C} \left(C _ {\text {c h a r g e}} t _ {\text {c h a r g e}} + C _ {\text {s t a n d b y}} t _ {\text {s t a n d b y}}\right)
+$$
+
+and
+
+$$
+E _ {c 2} (t) = R _ {\mathrm {c e l l}} R (t).
+$$
+
+Table 2 describes each relevant variable.
+
+Table 2. Variables and their meanings.
+
+| Variable | Description |
| P(t) | Population of U.S. in year t |
| pc(t) | Number of cellphones per person |
| Ec1 | YEC by cellphones and chargers |
| Ec2 | YEC by cellphone recyclers |
| fC | Frequency of cellphone charging |
| Charge | Charger wattage during charging |
| tcharge | Daily charger time spent charging |
| Cstandby | Charger wattage during standby |
| tstandby | Daily charger time spent in standby |
| Rcell | Energy needed to recycle one cellphone battery |
| R(t) | Percentage of cellphones recycled in year t |
+
+The immediate contributions to cellphone energy consumption are charging the phone and leaving the charger plugged in with no phone attached. It is difficult to find data on cellphone charging frequency. Rosen et al. [2001] argue that people charge their phone 50 times each year at their residence (noting that many people charge the phone in their car); but this figure seems very low. Newer phones with a multitude of features require more-frequent charging. Since charging the cellphone has developed into a habit for most people, we assume that people charge the phone every night and keep the charger attached to an outlet all the time.
+
+Rosen et al. [2001] observe that the average time to charge a cellphone is 2 hrs, which seems low in comparison to other data, which suggest 3-4 hrs to charge to $80\%$ and an additional 8 hrs to charge to $100\%$ . However, a phone charged every night is unlikely to have a nearly-empty battery. We assume that the overnight charging does not affect the 2-hr charging time. That $50\%$ of cellphone batteries are lithium-ion batteries, which do not allow for overcharging, justifies this assumption [Fishbein 2002; Rosen et al. 2001]. Once a lithium-ion battery is charged, the power drawn differs negligibly from that when no phone is connected to the charger [Rosen et al. 2001]. Therefore, we feel justified in adopting Rosen et al.'s statistic.
+
+To model the energy cost of recycling used cellphone batteries, we consider the batteries to be recycled by the Rechargeable Battery Recycling Corporation, justified by its significant market share and the fact that it recycles batteries in the U.S. [Office Depot 2004].
+
+# Energy Optimization
+
+Given the above functions for energy costs for cellular and landline telephone usage, we can optimize energy consumption. A Pseudo U.S. with the approximate size of the U.S. would likely have a similar distribution of household size.
+
+Let $H_{m}$ be the number of households with $m$ members and $l_{m}$ the fraction of households with $m$ members that have landline service. If we assume that the communication needs of every family are satisfied by either having a landline or by each member possessing a cellphone, the numbers of required cellphones $T_{c}$ and landline phones $T_{l}$ can be calculated as
+
+$$
+T _ {l} = \sum_ {m = 1} ^ {7} l _ {m} H _ {m}, \qquad T _ {c} = \sum_ {m = 1} ^ {7} m (1 - l _ {m}) H _ {m}.
+$$
+
+We believe that in the absence of a landline, members of a household will not share cellphones.
+
+The total telephony energy demand of the proposed plan for Pseudo U.S. is
+
+$$
+E (t) = E _ {l} (t) + E _ {c} (t).
+$$
+
+Using only landlines would minimize the number of telephone units required; however, landline phones and their companion technologies are much less energy-efficient than cellphones. Using only cellphones would maximize the number of telephone units required; and though the energy cost per unit is reduced, the overall increase in units may have deleterious consequences. Therefore, we optimize the variables $l_{m}$ to yield the best communications strategy from an energy perspective.
+
+We could modify the above summations to consider roles played by cellphones that are not achievable by a landline. For example, suppose that a single landline cannot serve a large family. If $n$ is the number of people a single landline can serve in a household, we may assume that a family of $m$ with one landline will need to purchase $(m - n)$ cellphones. Then we have
+
+$$
+T _ {c} = \sum_ {m = 1} ^ {7} m (1 - l _ {m}) H _ {m} + \sum_ {m = n + 1} ^ {7} l _ {m} H _ {m} (m - n),
+$$
+
+where the second term gives the fraction of families too large to be served by a single landline. Implicit in this formula is an assumption that no family obtains a second landline. This is reasonable, since the average number of landlines per household in the U.S. is only 1.118 [Eisner 2008].
+
+Likewise, we could further complicate the cost function by asserting that not every family member requires a cellphone if a landline is absent.
+
+However, we find that such a modification does not enrich the conclusions of our optimization.
+
+# Results
+
+# Energy Consumption
+
+Using the above information, we create an energy consumption function:
+
+$$
+E (t) = E _ {c} (t) + E _ {l} (t).
+$$
+
+To make this specific, we must estimate parameter values for $A, \dots, G$ in (1) and (2). Using an optimization algorithm described in the methods section below, we arrive at the conclusions in Table 3.
+
+Table 3. Values of parameters, as fitted from data in Eisner [2008].
+
+| Variable | Value |
| P(t) | Population growth as predicted by the Census Bureau |
| pl(t) | As defined in (1) |
| h(t) | 1.89E-3t + 1.076, t ≤ 40; -1.20E-3+ 1.152, t > 40 |
| πa(t) | 1 - πb(t) - πc(t) |
| ea(t) | 20 kWh |
| πb(t) | max(0,1.45E-2t - 1.45E-1), t ≤ 40; .44,t > 40 |
| eb(t) | 28 kWh |
| πc(t) | max(0,1.07E-2t - 1.07E-1), t ≤ 40; .32,t > 40 |
| ec(t) | 36 kWh |
| πd(t) | max(0,2.31-2t - 2.31E-1), t ≤ 40; .69,t > 40 |
| ed(t) | 36 kWh |
+
+Table 5.
+Values for variables and parameters in Table 2. Source: Rosen et al. [2001].
+
+| 1. Publication Title
+The UMAP Journal | 2. Publication Number | 3. Filing Date
+9/5/2009 |
| 0 | 1 | 9 | 7 | - | 3 | 6 | 2 | 2 |
| 4. Issue Frequency
+Quarterly | 5. Number of Issues Published Annually
+Four (4) | 6. Annual Subscription Price
+$125 |
| 7. Complete Mailing Address of Known Office of Publication (Not printer) (Street, city, county, state, and ZIP+4®)
+COMAP, Inc., 175 Middlesex Tpk., Suite 3B, Bedford, MA 01730 | Contact Person
+John Tomicek |
| Telephone (Include area code)
+781/862-7878 x130 |
+
+8. Complete Mailing Address of Headquarters or General Business Office of Publisher (Not printer)
+
+# SAME
+
+9. Full Names and Complete Mailing Addresses of Publisher, Editor, and Managing Editor (Do not leave blank)
+
+Publisher (Name and complete mailing address)
+
+Solomon Garfunkel, 175 Middlesex Tpk., Suite 3B, Bedford, MA 01730
+
+Editor (Name and complete mailing address)
+
+Paul Campbell, 700 College St., Beloit, WI 53511
+
+Managing Editor (Name and complete mailing address)
+
+Joyce Barnes, 175 Middlesex Tpk., Suite 3B, Bedford, MA 01730
+
+10. Owner (Do not leave blank. If the publication is owned by a corporation, give the name and address of the corporation immediately followed by the names and addresses of all stockholders owning or holding 1 percent or more of the total amount of stock. If not owned by a corporation, give the names and addresses of the individual owners. If owned by a partnership or other unincorporated firm, give its name and address as well as those of each individual owner. If the publication is published by a nonprofit organization, give its name and address.)
+
+| 13. Publication Title
+The UMAP Journal | 14. Issue Date for Circulation Data Below
+Sept 5, 2009 |
| 15. Extent and Nature of Circulation | Average No. Copies Each Issue During Preceding 12 Months | No. Copies of Single Issue Published Nearest to Filing Date |
| a. Total Number of Copies (Net press run) | 600 | 600 |
| b.PaCirculation (By Mail and Outside the Mail) | (1) | Mailed Outside-County Paid Subscriptions Stated on PS Form 3541(Include paid distribution above nominal rate, advertiser's proof copies, and exchange copies) | 507 | 474 |
| (2) | Mailed In-County Paid Subscriptions Stated on PS Form 3541 (Include paid distribution above nominal rate, advertiser's proof copies, and exchange copies) | 0 | 0 |
| (3) | Paid Distribution Outside the Mails Including Sales Through Dealers and Carriers, Street Vendors, Counter Sales, and Other Paid Distribution Outside USPS® | 50 | 50 |
| (4) | Paid Distribution by Other Classes of Mail Through the USPS (e.g. First-Class Mail®) | 0 | 0 |
| c. Total Paid Distribution (Sum of 15b (1), (2), (3), and (4)) | 557 | 524 |
| d. Free or Nominal Rate Distribution (By Mail a Outside the Mail) | (1) | Free or Nominal Rate Outside-County Copies included on PS Form 3541 | 0 | 0 |
| (2) | Free or Nominal Rate In-County Copies Included on PS Form 3541 | 20 | 38 |
| (3) | Free or Nominal Rate Copies Mailed at Other Classes Through the USPS (e.g. First-Class Mail) | 0 | 0 |
| (4) | Free or Nominal Rate Distribution Outside the Mail (Carriers or other means) | 0 | 0 |
| e. Total Free or Nominal Rate Distribution (Sum of 15d (1), (2), (3) and (4)) | 20 | 38 |
| f. Total Distribution (Sum of 15c and 15e) | 577 | 562 |
| g. Copies not Distributed (See Instructions to Publishers #4 (page #3)) | 23 | 38 |
| h. Total (Sum of 15f and g) | 600 | 600 |
| i. Percent Paid (15c divided by 15f times 100) | 97 | 93 |
+
+16. Publication of Statement of Ownership
+If the publication is a general publication, publication of this statement is required. Will be printed in the Third issue of this publication.
+
+□ Publication not required.
+
+