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Queuing

CEE 320Anne Goodchild

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Outline

1. Fundamentals

2. Poisson Distribution

3. Notation

4. Applications

5. Analysisa. Graphical

b. Numerical

6. 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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Server/bottleneck

Arrivals Departures

Activated

Upstream of bottleneck/server Downstream

Direction of flow

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server

Arrivals Departures

Not Activated

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Flow Analysis

• Bottleneck active– Service rate is capacity– Downstream flow is determined by bottleneck

service rate– Arrival rate > departure rate– Queue present

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Flow Analysis

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

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

ArrivalRate

DepartureRate

Time

Ve

hicl

es

t1

Queue at time, t1

Maximum delay

Maximum queue

D/D/1 Queue

Delay of nth arriving vehicle

Total vehicle delay

Where is capacity?

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Poisson Distribution

• Good for modeling random events• Count distribution

– Uses discrete values– Different than a continuous distribution

!n

etnP

tn

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

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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

36000

!00 qtt

t

eeet

thPP

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Example Graph

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Arrivals in 15 minutes

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ilit

y o

f O

ccu

ran

ce

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0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Arrivals in 15 minutes

Pro

bab

ilit

y o

f O

ccu

ran

ce

Mean = 0.2 vehicles/minute

Mean = 0.5 vehicles/minute

Example Graph

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Example: Arrival Intervals

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 2 4 6 8 10 12 14 16 18 20

Time Between Arrivals (minutes)

Pro

bab

ilit

y o

f E

xced

ance

Mean = 0.2 vehicles/minute

Mean = 0.5 vehicles/minute

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Queue Notation

• Popular notations:– D/D/1, M/D/1, M/M/1, M/M/N– D = deterministic– M = some distribution

NYX //

Arrival rate nature

Departure rate nature

Number ofservice channels

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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

• M/D/1– Average length of queue

– Average time waiting in queue

– Average time spent in system

0.1

12

2

Q

12

1w

1

2

2

1t

λ = arrival rate μ = departure rate =traffic intensity

Steady state assumption

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Queue Analysis – Numerical

• M/M/1– Average length of queue

– Average time waiting in queue

– Average time spent in system

0.1

1

2

Q

1

w

1t

λ = 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

2

10

1

1

! NNN

PQ

N

1

Q

w

Q

t

0.1N

λ = 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

1

0

0

1!!

1N

n

N

c

n

c

c

NNn

P

Nnfor !

0 n

PP

n

n

Nnfor !

0 NN

PP

Nn

n

n

NNN

PP

N

Nn

1!

10

0.1N

λ = arrival rate μ = departure rate =traffic intensity

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Poisson Distribution Example

From HCM 2000

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?

!

minveh20.0 minveh20.0

n

etnP

tn

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Example Calculations

%4.22224.0

!2

1520.02

1520.02

e

PExactly 2:

Less than 2:

More than 2:

1992.0102 PPnP

5768.021012 PPPnP

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

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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: 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

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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: 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

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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: 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

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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.