Fundamentals of Queueing Therory with Applications in Traffic Flow Description Where is Nofretete? Several 100m queue length of visitors for the Berlin Museum, October Source: Berliner Zeitung/Markus Wächter,

You are in a waiting queue! What does queueing theory say about this? arrival process disappointed customers give up service facility leaving the system A/ B / C / Y / Z Arrival distribution (exponentiell, deterministi, general… ) service distribution (exponentiell, … ) number of service counters capacity limit discipline served custumers

Queue notation UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) A / B / X / Y / Z Warteschlangentheorie, Vorlesung, Einführung

Queue discipline Most common:first in, first out - fifo Public clerks:last in, first out – lafo Very Important Person:service with priority statistical service:service with statistical selection

Waiting time in queue t q = mean (inverse) number of waitingtime arrival rate waiting elements in queue 1 λ q n qn q Little´s Formula = result: Only the number of waiting people and the arrival rate is important. Signs with fixed waiting times along a queue are possible like along the queue for the statue of liberty

Summary basic relations UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Warteschlangentheorie, Vorlesung, Einführung

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Explanation of the Pollaczek- Khintchine Formula UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) For the mean number of elements in an M/G/1- system the Pollaczek-Khintchine Formula holds Warteschlangentheorie, Vorlesung, Systeme mit allg. Abfertigung utilization rate

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Application of the Pollaczek- Khintchine fomula to a M/D/1 system σ = 0 gives ρ = utilization rate transformation into mean time spend in the system with Little´s formula waiting time at service counter

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) road as service station in a queueing system Verkehrsleittechnik Strecke als Regelkreis Elements of a queueing system road section as waiting space queue arrival service (station) characteristics: arrival/departure rate stationary/unstationary capacity queue length

total time spend in a M/D/1 system after Pollaczek - Khintchine mean time in service counter service distribution traffic intensity division by mean length several service counters in series gives or

balance for generation and recombination process UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) n+1 n-3 n-2 n-1 n μ n+1 λ n-1 μnμn λnλn Warteschlangentheorie, Vorlesung, Vernetzte Systeme Stapelweise Eingabe

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Stationary generation and recombination processes UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Initial system Stationary solution as recursion Warteschlangentheorie, Vorlesung, Markovprozesse 2

non stationary queuing theory continuum approximation Balance equation Taylor expansion restriction to second order (Fokker Planck equation) Drift term Diffusion term

Fokker Planck equation and stochastic equivalent Langevin equation Stochastic equivalent equation of motion (Langevin equation) with fluctuating force ensemble average

Quelle: amp2005.blog.lemonde.fr/files/langevin_by_picasso.jpg und Paul Langevin * January † December french physicist - studied at the Ecole Supériere de Physique et de Chimie Industrielles de la Ville de Paris - career at this school, director at last - since 1909 professor for physics at the Collège de France - student of Pierre (†1906) and Marie Curie (†1934). He was a friend of the family and he had 1910 an affaire with Marie Curie. - in the 30‘s and 40´s years he belonged to a bohemian in Paris with Picasso. - applied firstly in 1916 the Piezo electricity of quartz crystals by constructing the first ultra sonic object detector (Sonar) Paul Langevin painted by Pablo Picasso, 1938

free flow traffic cluster number of vehicles within the cluster: n minimum cluster size: n crit P (n+1) P(n) P(n-1) adhesion rate: inverse time gap q discharge rate: Continuum approximation Traffic breakdown description balance: gives with

Probability and temporal drop of finding n anywhere below n crit = Probability flow over n crit First passage time

Summary of traffic flow breakdown description potential n ß < 0 ß = 0 ß > 0 n crit 0 ß < 0 stable ß = 0 bistable ß > 0 unstable

Measurement sites

Speed Traffic Flow v1v1 v2v2 q1q1 q2q2 Δt = 5 min 1)speed drop: Δv > 15 km/h 2)speed after drop: v 2 < 75 km/h 3)minimum traffic flow: q 1 > 1000 veh/h result: breakdown y/n at q 1 Definition of Traffic Breakdown

Demonstration of two 5 - h - periods on two cross sections of the A9 München - Holledau UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) mit SBA ohne SBA Kühne, Verkehrsablauf an SBA, Uni Innsbruck

Text UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Weiterer Text mit SBA ohne SBA q [Fz/min] Vpkw [km/h] Comparison of two q –v Diagrams from 5 minutes intervals [A9 München – Holledau, Zeitraum , d.h Messwerte] Kühne, Verkehrsablauf an SBA, Uni Innsbruck

Transformation of the Fokker-Planck equation by separation into a Schrödinger equation with the boundary conditions General solution of the Fokker Planck Equation with respect to first passage time calculation reflecting boundary at n = 0 absorbing boundary at n = n crit

ground state excited states The eigenvalues can be calculated from the remaining boundary conditions The eigenfunctions, which automatically fulfill the absorbing boundary conditions, are Eigenfunctions ground state excited states

Eigen values

First passage time distribution Starting with the completeness relation for the eigenfunctions the first passage time distribution density is given by Calculating the increment and comparing with the normalization of the eigenfunctions allows the continuum approximation which leads to

-0,500,5 First passage time probability density First passage time cumulative distribution 0n crit n stable 0 n bistable n crit 0 unstable n crit n Cumulative first passage time

traffic flow class [veh/h] probability of traffic breakdown with traffic control without traffic control

cumulative first passage time distribution

Cumulative breakdown probability distribution

Warteschlangentheorie, Vorlesung, Einführung Waiting time at traffic signals

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Time gaps when discharging at traffic signals after (1974) Δt [s] = 2,10 / n+1,47 after (1987) Δt [s] = 2,03 / n+1,60 Time difference [s] Vehicle -Position

= inflow vehicle/s = max. discharge vehicle/s (= 0.5 vehicle/s) Number of lined up passenger cars time

Warteschlangentheorie, Vorlesung, Einführung Total waiting time during red Waiting time at a traffic signal as queueing problem with random inflow and deterministic discharge

Total waiting time during green from queueing theory Webster-Formula for total waiting time at a traffic signal Waiting time at a traffic signal as queueing problem with random inflow and deterministic discharge

Comparison between Webster- Formula and Simulation Results