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Queuing Theory

Queuing Theory. Lectures 14 (Sections 14.1 and 14,2, Textbook ) . Fundamental of Queuing Theory. Arrivals Uniform or random Departures Uniform or random Service rate Departure channels Discipline FIFO and LIFO are most popular FIFO is more prevalent in traffic engineering.

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Queuing Theory

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  1. Queuing Theory Lectures 14 (Sections 14.1 and 14,2, Textbook)

  2. Fundamental of Queuing Theory • Arrivals • Uniform or random • Departures • Uniform or random • Service rate • Departure channels • Discipline • FIFO and LIFO are most popular • FIFO is more prevalent in traffic engineering

  3. Fundamental of Queuing Theory (Cont’d) Number of service channels Arrival rate nature • Popular notations: • D/D/1, M/D/1, M/M/1, M/M/N • D = deterministic distribution • M = exponential distribution Departure rate nature

  4. Queuing model coverage • D/D/1 • M/M/1 • M/M/2 • M/D/1

  5. D/D/1 Queue Uniform arrival and departure Departure Rate Vehicles Arrival Rate Delay of nth arriving vehicle Maximum queue length Total vehicle delay Queue length at time t1 Maximum delay Time t1

  6. Exercise 1. At a signalized intersection, assume vehicles arrive at a uniform rate of 540 vph on a single approach lane, and the saturation flow rate (uniform) is 1800 vph for the approach. The cycle length is 120 sec with 80 sec of effective red and 40 sec of effective green. What is the total delay and average delay per vehicle? Solution: Assume at time t the queue is cleared sec The total number of accumulated vehicles at t=114 sec is veh sec The total delay is sec/veh The average delay is

  7. Total vehicle delay D/D/1 Queue Two different uniform arrival rates and one departure rate Vehicles Departure Rate Delay of nth arriving vehicle Arrival Rate Maximum queue length Maximum delay Queue length at time t1 Time t1

  8. M/M/1 Queue • Poisson (random) arrival. l arrival per unit of time. • Exponential service times. m served per unit of time • First-in, first-out queue discipline • Unlimited length queue • Service utilization factor • Steady-state when l < m

  9. Poisson arrival and Exponential service time • For a length of time t, the probability of n arrivals of time in t: • Exponential service time (the time between services) Poisson Arrival Assumptions or Rules: The number of arrival in the system is very large. Impact of a single arrival on the performance of the system is very small. All arrivals are independent, Number of arrivals Time t Probability of zero arrivals during t Time interval t

  10. M/M/1 Queue

  11. Example 14.1

  12. Solution of Example 14.1

  13. M/M/2 Queue

  14. Example 14.2

  15. Solution of Example 14.2

  16. Solution of Example 14.2 (Cont’d)

  17. M/D/1 Queue Random arrival and uniform departure Vehicles Departure Rate Arrival Rate Delay of nth arriving vehicle Maximum queue length Total vehicle delay Queue length at time t1 Maximum delay Time t1

  18. M/D/1 Queue • Average number of arrival during t E(At) = lt • Probability of 0 in system • Probability if 1 in system • Probability of n in system (n>2) • Average number of customer in system • Average number of customer in waiting line • Expected time each customer spends in the system • Expected time each customer spends in the queue

  19. Exercise 2. Vehicles arrive at a signalized intersection in a Poisson distribution with average arrival rate of 540 vehicles per hour per lane. The saturation flow rate during the effective green is a uniform rate of 1800 vehicles per hour per lane. The cycle length is 100 seconds with effective red of 50 sec and effective green is 50 sec. Assume the effective red starts at t=0 a. What is the expected number of vehicles in queue when t = 60 sec? b. What is the expected waiting time for a vehicle arrive the signalized intersection at t=60 sec? Solution a. b. Waiting time Queue = 9 – 5 = 4 veh

  20. Exercise 3. Vehicles arrive at a toll plaza in a Poisson distribution with average arrival rate of 1152 vehicles per hour. The service at the toll plaza is at a uniform rate of 1280 vehicles per hour. What is the average number of vehicles in the toll plaza? What is the average queue length What is the average time each driver spends in the toll plaza? Solution: • The average number of vehicles • The average queue length • Average time each driver spends in the toll plaza veh

  21. Parking Lectures 15 (Section 9.3, Textbook) (6-23-11)

  22. Parking • Background • Type of Parking • Types of Parking Studies • Parking Measurements and Analysis • Design, Operations, and Other Considerations

  23. Background • Parking affects mode choice. • Parking affects many institutions, business centers, transit systems as well as traffic circulation in CBD. • Parking has certain direct economic impact. • Parking program offers many benefits • Reduce the cost of development • Encourage shared parking • Improve urban design • Support historic preservation • Encourage developers to reduce parking demand (encourage transit)

  24. Type of Parking • Public • Curb-side parking • Off-street parking • Lots • Decks • Exclusive parking structure • Private • Home or apartment building garages • Stalls and driveways • Affiliate-specific parking (i.e., permit required)

  25. Type of Parking Studies • Financial feasibility • Functional design (Our focus) • Structural design • Demand study (Our focus) • Comprehensive • Limited • Site-specific

  26. Assessment of site parking demand

  27. Example 9.5

  28. Example 9.5 (Cont’d)

  29. Solution of Example 9.5

  30. Solution of Example 9.5 (Cont’d) (1) Two separated and independent parking lots: Office Weighted average = 28.04% Cinema (2) One combined parking lot: Much higher utilization rate

  31. Parking Measurement and Analysis

  32. Example License Plate Survey Data and Encoding Average duration min All min Permit min Visitor

  33. Design, Operation, and Other Considerations

  34. Design, Operation, and Other Considerations

  35. Design, Operation, and Other Considerations

  36. Design, Operation, and Other Considerations

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