1. 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 Summer L#6 Queue

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

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

4. Server Capacity & Queue Discipline
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
Priority
CEE Summer L#6 Queue

5. 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 Summer L#6 Queue

6. CEE Summer L#6 Queue

7. 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 Summer L#6 Queue

8. Queue Diagram
CEE Summer L#6 Queue

9. Queue Example Find 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
STEP II – Determine Service Pattern
STEP III – Cumulative Concept
STEP IV – Queue Diagram
Can find all the answers
Find
Time of Queue Dissipation
Longest Queue Length
Total Delay
Average Delay per Vehicle
Longest Wait of any vehicle
CEE Summer L#6 Queue

10. Step I – Arrival Pattern
We need to determine which one is the in this problem set
Constant
2,900 veh/hour =
CEE Summer L#6 Queue

11. Step II – Service Pattern
: Service Rate .
Due to the incident
8:00 – 8:12
8:13 – 8:30
8:31 after
CEE Summer L#6 Queue

12. Step III – Cumulative Concept
Need to express Step I and II in Equation
Arrival Pattern
Service Pattern
Remember that there is –
CEE Summer L#6 Queue

13. Step IV – Queue Diagram
CEE Summer L#6 Queue

14. Step IV (2) Queue Dissipation Point 3777.5 78.16
CEE Summer L#6 Queue

15. Step IV (3) Max Delay and Queue Length
Max diff btw arrival and service lines B
Longest Queue C A
Longest Delay
Longest Queue =
Longest Delay =
CEE Summer L#6 Queue

16. Step IV (4) Total Delay, Average DelayC
Area btw arrival and service lines B A
Total Delay =
A & C : Triangle
B: (Trapezoid) – (Triangle)
Triangle Area = 0.5×(length)×(height)
Trapezoid Area = 0.5×(height) ×(Length1 + Length2)
Average Delay
= Total Delay / total # of Vehicles =
CEE Summer L#6 Queue