Queuing CEE 320 Anne Goodchild

Outline Fundamentals Poisson Distribution Notation Applications Analysis Graphical Numerical Example

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

Activated Upstream of bottleneck/server Downstream Arrivals Departures Server/bottleneck Direction of flow

Not Activated Arrivals Departures server

Flow Analysis Bottleneck active Service rate is capacity Downstream flow is determined by bottleneck service rate Arrival rate > departure rate Queue present

Flow Analysis Bottle neck not active Arrival rate < departure rate No queue present Service rate = arrival rate Downstream flow equals upstream flow

http://trafficlab.ce.gatech.edu/freewayapp/RoadApplet.html

Fundamentals of Queuing Theory Arrivals Arrival rate (veh/sec) Uniform Poisson Time between arrivals (sec) Constant Negative exponential Service Service rate Service times

Queue Discipline First In First Out (FIFO) Last In First Out (LIFO) prevalent in traffic engineering Last In First Out (LIFO)

Queue Analysis – Graphical D/D/1 Queue Departure Rate Delay of nth arriving vehicle Arrival Rate Total vehicle delay Maximum queue Vehicles Maximum delay Won’t really ask you to do this – it’s basically an exercise in geometry Queue at time, t1 t1 Time Where is capacity?

Poisson Distribution Good for modeling random events Count distribution Uses discrete values Different than a continuous distribution 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

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

Example Graph

Example Graph

Example: Arrival Intervals

Queue Notation Popular notations: D/D/1, M/D/1, M/M/1, M/M/N Number of service channels Popular notations: D/D/1, M/D/1, M/M/1, M/M/N D = deterministic M = some distribution Arrival rate nature Departure rate nature Exponential distribution of times between vehicle arrivals = Poisson arrivals

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

Queue times depend on variability

Queue Analysis – Numerical Steady state assumption Queue Analysis – Numerical M/D/1 Average length of queue Average time waiting in queue Average time spent in system λ = arrival rate μ = departure rate =traffic intensity

Queue Analysis – Numerical M/M/1 Average length of queue Average time waiting in queue Average time spent in system λ = arrival rate μ = departure rate =traffic intensity

Queue Analysis – Numerical M/M/N Average length of queue Average time waiting in queue Average time spent in system λ = arrival rate μ = departure rate =traffic intensity

M/M/N – More Stuff Probability of having no vehicles Probability of having n vehicles Probability of being in a queue 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: Exactly 2 vehicles arrive in a 15 minute interval? Less than 2 vehicles arrive in a 15 minute interval? More than 2 vehicles arrive in a 15 minute interval? From HCM 2000

Example Calculations Exactly 2: Less than 2: P(0)=e-.2*15=0.0498, P(1)=0.1494 Less than 2 P(0) = e-(0.20)(15) = 0.0498 P(1) = 0.1494 P(0) + P(1) = 0.0498 + 0.1494 = 0.1992 More than 2 P(n>2) = 1 – (0.1992 + 0.224) = 0.5768 More than 2:

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.

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

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.

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

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.

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

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.