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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

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Queuing

CEE 320Anne 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

Downstream

Upstream of bottleneck/server

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
Fundamentals of Queuing Theory
• Arrivals
• Arrival rate (veh/sec)
• Uniform
• Poisson
• Time between arrivals (sec)
• Constant
• Negative exponential
• Service
• Service rate
• Service times
• Constant
• Negative exponential
Queue Discipline
• First In First Out (FIFO)
• prevalent in traffic engineering
• Last In First Out (LIFO)

Total vehicle delay

Queue Analysis – Graphical

D/D/1 Queue

Departure

Rate

Delay of nth arriving vehicle

Arrival

Rate

Maximum queue

Vehicles

Maximum delay

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
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
Queue Notation

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

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 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

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.