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Queuing Fall 2008 CEE 320 CEE 320 Anne Goodchild Outline 1. Fundamentals 2. Poisson Distribution 3. Notation 4. Applications 5. Analysis a. Graphical b. Numerical 6. Example Fall 2008 CEE 320 Fundamentals of Queuing Theory • Microscopic traffic flow – Different analysis than theory of traffic flow – Intervals between vehicles is important – Rate of arrivals is important • Arrivals • Departures • Service rate Fall 2008 CEE 320 Activated Upstream of bottleneck/server Downstream Arrivals Departures Server/bottleneck Fall 2008 CEE 320 Direction of flow Not Activated Arrivals Departures server Fall 2008 CEE 320 Flow Analysis • Bottleneck active – Service rate is capacity – Downstream flow is determined by bottleneck service rate – Arrival rate > departure rate – Queue present Fall 2008 CEE 320 Flow Analysis • Bottle neck not active – Arrival rate < departure rate – No queue present – Service rate = arrival rate – Downstream flow equals upstream flow Fall 2008 CEE 320 • http://trafficlab.ce.gatech.edu/freewayapp/ RoadApplet.html Fall 2008 CEE 320 Fundamentals of Queuing Theory • Arrivals – Arrival rate (veh/sec) • Uniform • Poisson – Time between arrivals (sec) • Constant • Negative exponential • Service – Service rate – Service times • Constant Fall 2008 CEE 320 • Negative exponential Queue Discipline • First In First Out (FIFO) – prevalent in traffic engineering • Last In First Out (LIFO) Fall 2008 CEE 320 Queue Analysis – Graphical D/D/1 Queue Departure Rate Delay of nth arriving vehicle Arrival Rate Maximum queue Vehicles Maximum delay Total vehicle delay Queue at time, t1 t1 Time Fall 2008 CEE 320 Where is capacity? Poisson Distribution • Good for modeling random events • Count distribution – Uses discrete values – Different than a continuous distribution P n t n e t n! P(n) = probability of exactly n vehicles arriving over time t n = number of vehicles arriving over time t λ = average arrival rate t = duration of time over which vehicles are counted Fall 2008 CEE 320 Poisson Ideas • Probability of exactly 4 vehicles arriving – P(n=4) • Probability of less than 4 vehicles arriving – P(n<4) = P(0) + P(1) + P(2) + P(3) • Probability of 4 or more vehicles arriving – P(n≥4) = 1 – P(n<4) = 1 - P(0) + P(1) + P(2) + P(3) • Amount of time between arrival of successive vehicles P0 Ph t t 0 e t e t e qt 3600 0! Fall 2008 CEE 320 Example Graph 0.25 0.20 Probability of Occurance 0.15 0.10 0.05 0.00 Fall 2008 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 CEE 320 Arrivals in 15 minutes Example Graph 0.25 Mean = 0.2 vehicles/minute 0.20 Probability of Occurance Mean = 0.5 vehicles/minute 0.15 0.10 0.05 0.00 Fall 2008 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 CEE 320 Arrivals in 15 minutes Example: Arrival Intervals 1.0 0.9 Mean = 0.2 vehicles/minute 0.8 Mean = 0.5 vehicles/minute Probability of Excedance 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 2 4 6 8 10 12 14 16 18 20 Fall 2008 CEE 320 Time Between Arrivals (minutes) Queue Notation Number of Arrival rate nature service channels X /Y / N Departure rate nature • Popular notations: – D/D/1, M/D/1, M/M/1, M/M/N – D = deterministic – M = some distribution Fall 2008 CEE 320 Queuing Theory Applications • D/D/1 – Deterministic arrival rate and service times – Not typically observed in real applications but reasonable for approximations • M/D/1 – General arrival rate, but service times deterministic – Relevant for many applications • M/M/1 or M/M/N – General case for 1 or many servers Fall 2008 CEE 320 Queue times depend on variability Fall 2008 CEE 320 Steady state assumption Queue Analysis – Numerical 1.0 • M/D/1 2 – Average length of queue Q 21 1 – Average time waiting in queue w 1 2 1 2 – Average time spent in system t 1 2 λ = arrival rate μ = departure rate =traffic intensity Fall 2008 CEE 320 Queue Analysis – Numerical 1.0 • M/M/1 2 – Average length of queue Q 1 1 – Average time waiting in queue w 1 – Average time spent in system t λ = arrival rate μ = departure rate =traffic intensity Fall 2008 CEE 320 Queue Analysis – Numerical N 1.0 • M/M/N P0 N 1 1 – Average length of queue Q N! N 1 N 2 Q 1 – Average time waiting in queue w Q – Average time spent in system t λ = arrival rate μ = departure rate =traffic intensity Fall 2008 CEE 320 M/M/N – More Stuff N 1.0 – Probability of having no vehicles 1 P0 N 1 nc N 0 n ! N!1 N nc c – Probability of having n vehicles P0 n n P0 Pn for n N Pn n N for n N n! N N! – Probability of being in a queue P0 N 1 Pn N N! N 1 N Fall 2008 CEE 320 λ = arrival rate μ = departure rate =traffic intensity Poisson Distribution Example Vehicle arrivals at the Olympic National Park main gate are assumed Poisson distributed with an average arrival rate of 1 vehicle every 5 minutes. What is the probability of the following: 1. Exactly 2 vehicles arrive in a 15 minute interval? 2. Less than 2 vehicles arrive in a 15 minute interval? 3. More than 2 vehicles arrive in a 15 minute interval? Pn 0.20 veh min t n e 0.20 veh m int n! Fall 2008 CEE 320 From HCM 2000 Example Calculations Exactly 2: P2 0.20 15 2 e 0.20 15 0.224 22 .4% 2! Less than 2: Pn 2 P0 P1 0.1992 P(0)=e-.2*15=0.0498, P(1)=0.1494 More than 2: Pn 2 1 P0 P1 P2 0.5768 Fall 2008 CEE 320 Example 1 You are entering Bank of America Arena at Hec Edmunson Pavilion to watch a basketball game. There is only one ticket line to purchase tickets. Each ticket purchase takes an average of 18 seconds. The average arrival rate is 3 persons/minute. Find the average length of queue and average waiting time in queue assuming M/M/1 queuing. Fall 2008 CEE 320 Example 1 • Departure rate: μ = 18 seconds/person or 3.33 persons/minute • Arrival rate: λ = 3 persons/minute • ρ = 3/3.33 = 0.90 • Q-bar = 0.902/(1-0.90) = 8.1 people • W-bar = 3/3.33(3.33-3) = 2.73 minutes • T-bar = 1/(3.33 – 3) = 3.03 minutes Fall 2008 CEE 320 Example 2 You are now in line to get into the Arena. There are 3 operating turnstiles with one ticket-taker each. On average it takes 3 seconds for a ticket-taker to process your ticket and allow entry. The average arrival rate is 40 persons/minute. Find the average length of queue, average waiting time in queue assuming M/M/N queuing. Fall 2008 CEE 320 Example 2 • N=3 • Departure rate: μ = 3 seconds/person or 20 persons/minute • Arrival rate: λ = 40 persons/minute • ρ = 40/20 = 2.0 • ρ/N = 2.0/3 = 0.667 < 1 so we can use the other equations • P0 = 1/(20/0! + 21/1! + 22/2! + 23/3!(1-2/3)) = 0.1111 • Q-bar = (0.1111)(24)/(3!*3)*(1/(1 – 2/3)2) = 0.88 people • T-bar = (2 + 0.88)/40 = 0.072 minutes = 4.32 seconds • W-bar = 0.072 – 1/20 = 0.022 minutes = 1.32 seconds Fall 2008 CEE 320 Example 3 You are now inside the Arena. They are passing out Harry the Husky doggy bags as a free giveaway. There is only one person passing these out and a line has formed behind her. It takes her exactly 6 seconds to hand out a doggy bag and the arrival rate averages 9 people/minute. Find the average length of queue, average waiting time in queue, and average time spent in the system assuming M/D/1 queuing. Fall 2008 CEE 320 Example 3 • N=1 • Departure rate: μ = 6 seconds/person or 10 persons/minute • Arrival rate: λ = 9 persons/minute • ρ = 9/10 = 0.9 • Q-bar = (0.9)2/(2(1 – 0.9)) = 4.05 people • W-bar = 0.9/(2(10)(1 – 0.9)) = 0.45 minutes = 27 seconds • T-bar = (2 – 0.9)/((2(10)(1 – 0.9) = 0.55 minutes = 33 seconds Fall 2008 CEE 320 Primary References • Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2003). Principles of Highway Engineering and Traffic Analysis, Third Edition (Draft). Chapter 5 • Transportation Research Board. (2000). Highway Capacity Manual 2000. National Research Council, Washington, D.C. Fall 2008 CEE 320

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