1. Queuing
CEE 320 Anne Goodchild

2. Outline Fundamentals Poisson Distribution Notation Applications
Analysis
Graphical
Numerical
Example

3. 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

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

5. Not Activated
Arrivals
Departures
server

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

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

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

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

11. 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?

12. 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

13. 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

14. Example Graph

15. Example Graph

16. Example: Arrival Intervals

17. 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

18. 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

19. Queue times depend on variability

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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) =
P(1) =
P(0) + P(1) = =
More than 2
P(n>2) = 1 – ( ) =
More than 2:

26. 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.

27. 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

28. 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.

29. 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 = < 1 so we can use the other equations
P0 = 1/(20/0! + 21/1! + 22/2! + 23/3!(1-2/3)) =
Q-bar = (0.1111)(24)/(3!*3)*(1/(1 – 2/3)2) = 0.88 people
T-bar = ( )/40 = minutes = 4.32 seconds
W-bar = – 1/20 = minutes = 1.32 seconds

30. 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.

31. 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

32. 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 National Research Council, Washington, D.C.