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

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

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

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

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

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

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

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

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

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