1. Queueing analysis Basics Methodologies Models Queueing process
Classification of queueing analysis
Methodologies
Deterministic queueing analysis
Stochastic queueing analysis
Models
Delays at signalized intersections

2. Glossary Queueing process 排队过程 Deterministic queueing analysis 确定型排队分析
Stochastic queueing analysis 随机型排队分析
Arrival 到达
Service 服务
Queue discipline 排队规则
Cumulative vehicles-time diagram 累计车辆数-时间曲线
Traffic intensity 交通强度
Steady state 稳态
Time-dependent 与时间有关的，随时间变化的

3. Basics Queueing process Arrival: arrival distribution and mean value
Service: service distribution and mean value
Queue discipline: FIFO, FILO, SIRO, etc.
Queueing process at signalized intersections
Arrival: arrivals of vehicles
Service: departures of vehicles
Queue discipline: FIFO
Classification of queueing analysis
Deterministic queueing analysis
Stochastic queueing analysis

4. Deterministic queueing analysis
Both the arrival and service distributions are deterministic
Arrivals of vehicles: constant or varying rate
Departures of vehicles: constant or varying rate
Applications: signalized intersections, incidents

5. s Flow rate l Time Cumulative vehicles l s Time Red: Arrival: l
Service rate: 0
Green:
Service rate: s (Queue present)
l (Otherwise) s
Flow rate l
Time
Performance measures:
Number of vehicles queued
Percent of vehicles queued
Maximum queue length
Average queue length (queue)
Average queue length
Maximum individual delay
Average delay (queue)
Average delay
Total delay
Arrivals
Cumulative vehicles l s
Departures
Time

6. Stochastic queueing analysis
Either the arrival and/or service distributions are probabilistic
Arrivals of vehicles: constant, Poisson, etc.
Departures of vehicles: constant, Poisson, etc.
Traffic intensity: r = l/m
The necessity condition: r < 1.0
l = average arrival rate
m = average departure rate
Code identification scheme: M/D/1(∞,FIFO)
Single-channel and multichannel problems

7. Delays at signalized intersections
Saturation states at signalized intersections
Undersaturation: demand is less than capacity in a specified analysis period (usually 15min), but there exists one or more cycles that the demand is more than capacity.
Saturation: demand = capacity
Oversaturation: demand is more than capacity in the analysis time interval and most of cycles.
First two = steady state, which can be solved by queueing theory
Components of delay models: uniform, incremental (random and oversaturation), and initial queue delays

8. Queue development process during one signal cycle
(Adapted from McNeil 1968).

9. Uniform delay (d1)
Derived from deterministic queueing analysis
See textbook2, p.126
Incremental delay (d2)
Complicated because of initial queues in some cycles
Initial queue delay (d3)
Initial queue in the analysis period

10. Steady-state delay models
Webster’s model, etc.
Time-dependent delay models
Akcelik’s models, HCM models