Queuing

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Queuing
CEE 320Anne Goodchild
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Outline

• Fundamentals
• Poisson Distribution
• Notation
• Applications
• Analysis
• Graphical
• Numerical
• Example

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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

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Activated
Downstream
Upstream of bottleneck/server
Arrivals
Departures
Server/bottleneck
Direction of flow
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Not Activated
Arrivals
Departures
server
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Flow Analysis

• Bottleneck active
• Service rate is capacity
• Downstream flow is determined by bottleneck
  service rate
• Arrival rate gt departure rate
• Queue present

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Flow Analysis

• Bottle neck not active
• Arrival rate lt departure rate
• No queue present
• Service rate arrival rate
• Downstream flow equals upstream flow

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• http//trafficlab.ce.gatech.edu/freewayapp/RoadApp
  let.html

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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

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Queue Discipline

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

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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?
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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
? average arrival rate
t duration of time over which vehicles are counted
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Poisson Ideas

• Probability of exactly 4 vehicles arriving
• P(n4)
• Probability of less than 4 vehicles arriving
• P(nlt4) P(0) P(1) P(2) P(3)
• Probability of 4 or more vehicles arriving
• P(n4) 1 P(nlt4) 1 - P(0) P(1) P(2)
  P(3)
• Amount of time between arrival of successive
  vehicles

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Example Graph
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Example Graph
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Example Arrival Intervals
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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
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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

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Queue times depend on variability
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Queue Analysis Numerical
Steady state assumption

• M/D/1
• Average length of queue
• Average time waiting in queue
• Average time spent in system

? arrival rate µ departure rate ?traffic
intensity
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Queue Analysis Numerical

• M/M/1
• Average length of queue
• Average time waiting in queue
• Average time spent in system

? arrival rate µ departure rate ?traffic
intensity
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Queue Analysis Numerical

• M/M/N
• Average length of queue
• Average time waiting in queue
• Average time spent in system

? arrival rate µ departure rate ?traffic
intensity
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M/M/N More Stuff

• Probability of having no vehicles
• Probability of having n vehicles
• Probability of being in a queue

? arrival rate µ departure rate ?traffic
intensity
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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?

From HCM 2000
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Example Calculations
Exactly 2
Less than 2
P(0)e-.2150.0498, P(1)0.1494
More than 2
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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.
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Example 1

• Departure rate µ 18 seconds/person or 3.33
  persons/minute
• Arrival rate ? 3 persons/minute
• ? 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

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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.
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Example 2

• N 3
• Departure rate µ 3 seconds/person or 20
  persons/minute
• Arrival rate ? 40 persons/minute
• ? 40/20 2.0
• ?/N 2.0/3 0.667 lt 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

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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.
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Example 3

• N 1
• Departure rate µ 6 seconds/person or 10
  persons/minute
• Arrival rate ? 9 persons/minute
• ? 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

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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.