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Queuing

Queuing. CEE 320 Anne Goodchild. Example 2. You are now in line to get into the Arena. There are 3 operating turnstiles with one ticket-taker each. On average it takes 3 seconds for a ticket-taker to process your ticket and allow entry. The average arrival rate is 40 persons/minute.

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Queuing

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  1. Queuing CEE 320Anne Goodchild

  2. Example 2 You are now in line to get into the Arena. There are 3 operating turnstiles with one ticket-taker each. On average it takes 3 seconds for a ticket-taker to process your ticket and allow entry. The average arrival rate is 40 persons/minute. Find the average length of queue, average waiting time in queue assuming M/M/N queuing.

  3. Example 2 • N = 3 • Departure rate: μ = 3 seconds/person or 20 persons/minute • Arrival rate: λ = 40 persons/minute • ρ = 40/20 = 2.0 • ρ/N = 2.0/3 = 0.667 < 1 so we can use the other equations • P0 = 1/(20/0! + 21/1! + 22/2! + 23/3!(1-2/3)) = 0.1111 • Q-bar = (0.1111)(24)/(3!*3)*(1/(1 – 2/3)2) = 0.88 people • W-bar = 0.072 – 1/20 = 0.022 minutes = 1.32 seconds

  4. Example 3 You are now inside the Arena. They are passing out Harry the Husky doggy bags as a free giveaway. There is only one person passing these out and a line has formed behind her. It takes her exactly 6 seconds to hand out a doggy bag and the arrival rate averages 9 people/minute. Find the average length of queue, average waiting time in queue, and average time spent in the system assuming M/D/1 queuing.

  5. Example 3 • N = 1 • Departure rate: μ = 6 seconds/person or 10 persons/minute • Arrival rate: λ = 9 persons/minute • ρ = 9/10 = 0.9 • Q-bar = (0.9)2/(2(1 – 0.9)) = 4.05 people • W-bar = 0.9/(2(10)(1 – 0.9)) = 0.45 minutes = 27 seconds • T-bar = (2 – 0.9)/((2(10)(1 – 0.9) = 0.55 minutes = 33 seconds

  6. Course logistics • Projects returned Wednesday • HW5 due today • Kelly to lecture Wednesday • No class Friday • Kelly OH tomorrow • Prof. Goodchild no OH Thursday • HW6 due November 3

  7. Performance Management CEE 320Anne Goodchild

  8. I-5 Average Daily Traffic from the WSDOT 2001 Annual Traffic Report

  9. Freeway Defined • A divided highway with full control of access and two or more lanes for the exclusive use of traffic in each direction.

  10. Definitions – Flow Characteristics • Undersaturated • Traffic flow that is unaffected by upstream or downstream conditions. • Queue discharge • Traffic flow that has just passed through a bottleneck • Oversaturated • Traffic flow that is influencedby the effects of a downstream bottleneck.

  11. Speed vs. Flow SfFree Flow Speed Uncongested Flow Sm Speed (mph) Optimal flow, capacity, vm Congested Flow Flow (veh/hr)

  12. Metrics for measurement • Average speed • Variance • How to provide a measurement for a region?

  13. Definitions – Free-Flow Speed • Free-Flow Speed (FFS) • The mean speed of passenger cars that can be accommodated under low to moderate flow rates on a uniform freeway segment under prevailing roadway and traffic conditions. • Factors affecting free-flow speed • Lane width • Lateral clearance • Number of lanes • Interchange density • Geometric design

  14. Definitions • Passenger car equivalents • Trucks and RVs behave differently • Baseline is a freeway with all passenger cars • Traffic is expressed in passenger cars per lane per hour (pc/ln/hr or pcplph) • Driver population • Non-commuters suck more at driving • They may affect capacity • Capacity • Corresponds to LOS E and v/c = 1.0

  15. Definitions – Level of Service (LOS) • Chief measure of “quality of service” • Describes operational conditions within a traffic stream. • Does not include safety • Different measures for different facilities • Six measures (A through F) • Used for performance analysis, investment decisions • Freeway LOS • Based on traffic density • Assumptions for freeway segment • No interaction with adjacent facilities (streets, other freeways) • Free-flow conditions exist on either side of the facility being analyzed • Outside the influence or ramps and weaving areas

  16. Definitions • Freeway Capacity • The maximum sustained 15-min flow rate, expressed in passenger cars per hour per lane, that can be accommodated by a uniform freeway segment under prevailing traffic and roadway conditions in one direction of flow.

  17. LOS A Free-flow operation LOS B Reasonably free flow Ability to maneuver is only slightly restricted Effects of minor incidents still easily absorbed Levels of Service From Highway Capacity Manual, 2000

  18. LOS C Speeds at or near FFS Freedom to maneuver is noticeably restricted Queues may form behind any significant blockage. LOS D Speeds decline slightly with increasing flows Density increases more quickly Freedom to maneuver is more noticeably limited Minor incidents create queuing Levels of Service From Highway Capacity Manual, 2000

  19. LOS E Operation near or at capacity No usable gaps in the traffic stream Operations extremely volatile Any disruption causes queuing LOS F Breakdown in flow Queues form behind breakdown points Demand > capacity Levels of Service From Highway Capacity Manual, 2000

  20. Performance comparison • Different for different types of roadways • Arterial • Highway • Local streets • Bridges • Rural roads • Mountain roads • …..

  21. TTI Urban Mobility Report • http://mobility.tamu.edu/ums/congestion_data/national_congestion_tables.stm • http://ops.fhwa.dot.gov/congestion_report_04/appendix_C.htm

  22. Travel time index • the ratio of average peak travel time to a free-flow travel time (in this report, the travel time at 60 mph for freeways). For example, a value of 1.20 means that average peak travel times are 20 percent longer than free-flow travel times.

  23. Percent of congested travel • the ratio of congested travel to total travel. The analysis uses vehicle-miles of travel (VMT); person-miles of travel could also be used if person flows are of interest and widely available. The percent of congested travel is a relative measure of the amount of travel affected by congestion.

  24. Delay • the additional travel time that is incurred when actual travel times are greater than free-flow travel times. The delay is expressed in several different ways, including total delay in vehicle-hours, total delay per 1,000 VMT, and share of delay by time period, day of week, or speed range.

  25. Buffer index • the extra time (buffer) most travelers add to their average travel time when planning trips. For example, a buffer index of 40 percent means that a traveler should budget an additional 8-minute buffer for a 20-minute average peak travel time to ensure on-time arrival most of the time (95 percent in this report).

  26. Planning time index • Statistically defined as the 95th percentile travel time index, this measure also represents the extra time most travelers add to a free-flow travel time when planning trips. For example, a planning time index of 1.60 means that travelers plan for an additional 60 percent travel time above the free-flow travel time to ensure on-time arrival most of the time (95 percent in this report).

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