1. 1 CEE 8207 Summer 2013 L#6 Queue

2. 2

3. Queueing System Provide a mean to estimate important measures of Highway Performance  Travel time  Speed Affects Roadway Design  Required left-turn bay length Queueing System Components  CEE 8207 Summer 2013 L#6 Queue 3

4. Arrival Process/Pattern Average Arrival Rate Statistical distribution of interarrival times or headways Traffic  usually Random arrival  Poisson Distribution CEE 8207 Summer 2013 L#6 Queue 4

5. Service Process/Pattern & # of Servers Service Process/Pattern  Average Service Rate vs. Service rate is.  Distribution of Service Time # of Servers  # of channels to perform service  i.e – Grocery Cashier CEE 8207 Summer 2013 L#6 Queue 5

6. Server Capacity & Queue Discipline Server Capacity  How much can. i.e. Queue discipline  Means by which next customer/entity is selected FIFO – First In, First Out  i.e: LIFO – Last In, First Out  i.e: Priority CEE 8207 Summer 2013 L#6 Queue 6

7. Queue Process (cont’) Kendal’s Notation  To describe Queue Process in easy way  i / j / n i: arrival process j: service process n: # of servers  For arrival process and service process If Random → M If Deterministic → D  In Traffic Arrival Rate =. Service Rate =. CEE 8207 Summer 2013 L#6 Queue 7

8. 8

9. Example of Queue Process Signalized Intersection  Arrival Process/Pattern Vehicle, Random Arrival  Service Process/Pattern Green Signal Timing (i.e; 1 veh/2sec when green)  Service Channel, # of Servers # of lanes for corresponding movement  System Capacity.  Queue Discipline. CEE 8207 Summer 2013 L#6 Queue 9

10. 10 CEE 8207 Summer 2013 L#6 Queue Queue Diagram

11. An incident occurs on a freeway Freeway Capacity before the incident : 4,000 veh/hour Constant Flow : 2,900 veh/hour of morning commute Due to the incident, 8:00 – Freeway closed 8:12 – Partial open 2,000 veh/hour 8:31 – Freeway fully open Assume D/D/1 Queue, FIFO STEP I – Determine Arrival Pattern Queue Example STEP II – Determine Service Pattern STEP III – Cumulative Concept Find Time of Queue Dissipation Longest Queue Length Total Delay Average Delay per Vehicle Longest Wait of any vehicle STEP IV – Queue Diagram Can find all the answers CEE 8207 Summer 2013 L#6 Queue 11

12. Step I – Arrival Pattern We need to determine which one is the arrival process in this problem set Constant Morning Commute Flow 2,900 veh/hour = 48.33 veh/min CEE 8207 Summer 2013 L#6 Queue 12

13. Step II – Service Pattern Freeway capacity is Service Rate  4,000 veh/hour Due to the incident  8:00 – 8:12 Freeway Closed → 0 veh/hour  8:13 – 8:30 Partial Open with 2,000 veh/hour = 33.33 veh/min  8:31 – Normal Operation with 4,000 veh/hour = 66.67 veh/min CEE 8207 Summer 2013 L#6 Queue 13

14. Step III – Cumulative Concept Need to express Step I and II in Equation Arrival Pattern  No change  48.33t t : time in min Service Pattern  Some changes due to the incident  0 t ≤ 12  33.33(t- 12) 12 < t ≤ 31  633.33 + 66.67(t-31) t > 31 Remember that there is only ONE server – freeway lane CEE 8207 Summer 2013 L#6 Queue 14

15. Step IV – Queue Diagram CEE 8207 Summer 2013 L#6 Queue 15

16. Step IV (2) Queue Dissipation Point Arrival # = Service # 633.33+66.67(t-31) = 48.33t t = 78.16 min 78.16 3777.5 CEE 8207 Summer 2013 L#6 Queue 16

17. Step IV (3) Max Delay and Queue Length Max diff btw arrival and service lines Data on Points (t, # of vehicles) Point A = (31, 633.33) Point B = (31, 1498.33) Point C = (13.1, 633.33) t = 31 min, the 633.33 rd vehicle Longest Queue Longest Delay A C B Longest Queue = 1498.33 – 633.33 = 865 (vehicles) Longest Delay = 31 – 13.1 = 17.9 (minutes) CEE 8207 Summer 2013 L#6 Queue 17

18. Step IV (4) Total Delay, Average Delay Area btw arrival and service lines A & C : Triangle B: (Trapezoid) – (Triangle) Triangle Area = 0.5×(length)×(height) Trapezoid Area = 0.5×(height) ×(Length1 + Length2) B Total Delay = 37,604.2 veh- min C A Average Delay = Total Delay / total # of Vehicles = 37,604.2 / 3777.5 = 9.95 min/veh CEE 8207 Summer 2013 L#6 Queue 18