# Queuing

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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/
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

P0  Ph  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

Queue Analysis – Numerical

         1.0


• M/D/1
2
– Average length of queue             Q
21   

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?

Pn  
0.20 veh min  t            n
e  0.20 veh m int
n!
Fall 2008
CEE 320

From HCM 2000
Example Calculations

Exactly 2:   P2  
0.20  15 2 e 0.20 15    0.224  22 .4%
2!
Less than 2:   Pn  2  P0  P1  0.1992

P(0)=e-.2*15=0.0498, P(1)=0.1494

More than 2:   Pn  2  1  P0  P1  P2  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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