1. Queuing and Transportation
Transportation Logistics
Prof. Goodchild
Spring 2009

2. Two ways to address queues
Make an analytical model of customers needing service, and use that model to predict queue lengths and waiting times
Steady state assumption
Simulation

3. Definitions
Customers — independent entities that arrive at random times to a Server and wait for some kind of service, then leave.
Server — can only service one customer at a time; length of time to provide service depends on type of service;
Arrival time: time customer arrives at the back of the queue
Departure time: time customer leaves server
Inter-arrival time: time between successive arrivals of customers
Service time: time for server to serve one customer (amount of time you are delayed if no one else present)
Queue — customers that have arrived at server but are waiting for their service to start are in the queue.
Queue Length at time t — number of customers in the queue at time t.

4. Total Time in System
Service time: the amount of time you would be delayed if no other customers required service
Waiting time: the amount of time you have to wait because others also want service
The price you pay for others
Total Time in System = Service time + Waiting time

5. Queue Discipline FIFO LIFO Random Priority Traffic intersection
Elevator
Airplane
Random
Fluids
Priority

6. Transportation Applications
Traffic congestion
Being serviced at:
Border
Toll plaza
Bus stop
Goods waiting at a distribution center
Marine terminal ….

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

8. Not Activated
Arrivals
Departures
server

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

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

11. Queue Analysis – Graphical
Departure
Rate
Delay of nth arriving vehicle
Arrival
Rate
Total vehicle delay
Maximum queue
Cumulative Number of Items
Maximum delay
Won’t really ask you to do this – it’s basically an exercise in geometry
Queue at time, t1 t1
Time

12. Queue Notation Popular notations: D/D/1, M/D/1, M/M/1, M/M/N
Number of
servers
Popular notations:
D/D/1, M/D/1, M/M/1, M/M/N
D = deterministic
M = other distribution
Arrival rate
Departure rate
Exponential distribution of times between vehicle arrivals = Poisson arrivals

13. Poisson Distribution Good for modeling random events
Standard deviation equals the mean
Count distribution
Uses discrete values
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

14. Example Graph

15. Example Graph

16. If we assume Poisson arrival process
Inter-arrival times are exponentially distributed

17. Example: Arrival Intervals

18. Little’s Formula (1961)
T = time spent by a customer in the queueing system
 = arrival rate
N = number of customers in the system
The long-term average number of customers in a stable system N, is equal to the long-term average arrival rate, λ, multiplied by the long-term average time a customer spends in the system, T
Steady state assumption

19. Steady State Analysis 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

20. Queue Analysis 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

21. Queue Analysis D/D/1 Average length of queue
Average time waiting in queue
Average time spent in the system

22. Queue Analysis 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 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. Queue times depend on variability
items
time

25. Can’t store extra capacity
No reservoir for storing capacity
If capacity goes unused, it is wasted

26. Queue times depend on variability
Delay will be very different depending on the arrival PATTERN, not just number of arrivals

27. limitations
There are many cases when we want to consider changes to the arrival rate
This is difficult to do when you are limited to steady state assumptions
Limited number of distributions that provide a closed form expression

28. Simulation
In general we are interested in the variability in arrival rates or service times
If these are constantly varying a steady state assumption is fine
The alternative is to use a discrete event simulation framework and keep track of individual customers
Microsimulation of an intersection
Queue simulation

29. Queue simulation Simulation based approach Track vehicles
Step through time
Can change arrival rates, service times, with knowledge of previous system state
Border wizard

30. Examples Marine terminal Rail infrastructure International border
Airport terminal

31. Port gate and terminal stacks

32. Observed data

33. Theoretical wait times

34. Rail line as server
Bottleneck activation

35. Airport of the Future Separates queue into two different processes
Check in
Bag check
Allows travelers to enter mid-stream

36. Change in terminal processingK B
Baggage flow behind the counter
Queue approaching the counter P

37. Previous system

38. Airport of the future

39. Service times got worse!
Does not include wait time, only measured from arrival at check-in desk

40. Total times do improve with AF
Encourages travelers to check-in online
Reduces perceived wait time
Start process sooner
Can’t see a big queue
Reduces employee requirements
Improves space utilization

41. Border as server

42. Observations from one day

43. Regression equations by DOW

44. Regression equations by Season

45. Number of very long delays

46. Proportion of very long delays

47. Transportation Realities
For many systems other factors influence delay
Queing can be used to model wait times
Appropriate tool can be identified on a case by case basis
Be sure you understand the theoretical and practical framework