1. Queuing Analysis
Two analytical techniques can be employed to study queuing processes:
Shock wave analysis
Demand-capacity process is deterministic
Suited to evaluating the space occupied by the queuing processes and their interactions
Queuing analysis
Deterministic (input-output) or stochastic
Vertical queues are assumed

2. Objectives
1. To estimate queuing characteristics when demand exceeds capacity at a bottleneck.
For example,
a. Average number of vehicles in a queue at a given time
b. Average delay per vehicle
c. Total hours of delay
d. Duration of queuing
2. To determine idle time (of system, employees, etc.)

3. Queuing Process Queuing system Arrivals Departures Queue Server
Stream of customers
who demand service
Line of customers
waiting to be served

4. Queuing Application Examples
Bank teller problem
Transportation applications
Toll booth
Check-in counter at airport
Blockage on highway
Signal at intersection

5. Queuing Analysis in Traffic Applications
Arrivals
Departures
Queue
Server
arrival rate determined
by traffic demand
departure rate determined
by facility capacity
How long the queue is going to be?
How much delay?
Duration of congestion? …

6. Queuing Characteristics/Classifications
Arrivals: l (rate, distribution)
Servicing: m (rate, distribution)
Queue discipline: (e.g.FIFO)
Number of channels (m)
Arrival rate=demand=input
Service rate=capacity=departure=output
Served in random order

7. Queue discipline First come (in) – first served (out) (FIFO)
First in, last out (FILO)
Random service (SIRO: served in random order)
Priority
Inpatient customer
Infinite (∞)or finite lines

8. Queuing Models Deterministic Stochastic
Arrival and servicing are uniform (same interval, no randomness)
arrival and service rates may vary over time
Can be used to temporary blockage, temporary overloading, and periodical interruption such as signal
Stochastic
constant long-term arrival and service rates
Arrival and service conform to certain random distribution
short-term random fluctuation causes queue (even with arrival rate <service rate)

9. Deterministic Queuing
arrival l
Slope: flow rate 1
Cumulative vehicles
Point of queue
accumulation m2
Point of queue
dissipation
Waiting time(delay) 1
Queue length
departure m1 1
Time

10. Deterministic Queuing Example
Vehicles arrive at the entrance to a park with a single gate where a ranger distributes a free brochure. Park opens at 8:00 a.m. and arrivals are 480 veh/hr. After 20 min, arrival rate declines to 120 veh/hr for remainder of day. Time required to distribute a brochure is 15 seconds.
Describe operational characteristics.

11. Arrival and Departure Rates
l(1) = 480 veh/hr / 60 min/hr
= 8 veh/min
for t < 20
l(2) = 120 veh/hr / 60 min/hr
= 2 veh/min
for t > 20
m= 60 sec/min / 15 sec/veh
= 4 veh/min for all t

12. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time

13. Total vehicle arrivals @ t = 2010 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time (min)
Total vehicle t = 20
8t = 8 (20) = 160 veh
(20, 160)
Arrival curve 1 8 1

14. Total vehicle arrivals @ t > 2010 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
Total vehicle t > 20
(t - 20)
Arrival curve 2 2 1
(20, 160)
Arrival curve 1

15. Vehicle departures @ all t10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
Arrival curve 2
(20, 160) 4
Arrival curve 1 1
Departure curve
Vehicle all t 4t

16. Vehicle departures @ all t10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
Arrival curve 2
Point of queue
dissipation
(20, 160)
Arrival curve 1
Departure curve
Vehicle all t 4t

17. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
Arrival curve 2
Point of queue
dissipation
(20, 160)
Arrival curve 1
Departure curve
arrivals = departures
(t-20) = 4t
t = 60 min

18. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
(60, 240)
Arrival curve 2
Point of queue
dissipation
(20, 160)
Arrival curve 1
Departure curve
Total vehicles
4t = 4(60) = 240 veh
=8(20)+2(40)

19. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
(60, 240)
Arrival curve 2
Point of queue
dissipation
(20, 160)
Longest vehicle delay
(FIFO)
Arrival curve 1
Departure curve
Longest vehicle queue

20. What is Longest vehicle queue?10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
What is Longest vehicle queue?
Arrival curve 2
Departure curve
(20, 160)
Longest Vehicle Queue
Occurs at t = 20
Queue = Arrivals – Departures
= 8t – 4t
= 8(20) – 4(20)
= 160 – 80
=80 vehs
Arrival curve 1

21. What is Queue Length at t=40?10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
What is Queue Length at t=40?
Arrival curve 2
Arrivals
Departures
Arrival curve 1
Veh. In Queue = Arrivals – Departures
= [ (t-20)] – 4t
=[ (40-20)] - 4(40)
= 40 vehs
Departure curve

22. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
Arrival curve 2
Departure curve
(20, 160)
160 = 8 tEnter
tEnter = 20 min
160 = 4tExit
tExit = 40 min
Arrival curve 1
Longest vehicle delay
Delay = tExit – tEnter
Delay = 40 min – 20 min = 20 mins

23. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
What is delay of the 200th vehicle?
Delay = tExit – tEnter
Delay = 50 min – 40 min = 10 mins
Arrival curve 2
Departure curve
Arrival curve 1
200 = 160 – 2(tEnter - 20)
tEnter = 40 min
200 = 4tExit
tExit = 50 min

24. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
What is the TOTAL amount of Delay?
20 min
40 min
80 vehs

25. Total number of vehicles10 30 20 40 50 60
300
250
200
150
100
Total number of vehicles
Time
What is the average delay and queue length?
240
Total delay=2400veh mins
Little’s Formula

26. Deterministic Queuing Example: Signalized Intersection Uniform Delay Calculation
Uniform delay: delay due to red time under uniform arrivals

27. Signalized Intersection

28. Queuing Patterns

29. More Examples (with Equations)
LTs: Longest queue
Ts: time when LTs occurs
Tc: Total duration of congestion
D: total delay
W: average waiting time

30. Microscopic Analysis
Deterministic queuing analysis can be undertaken at the microscopic level
Requires arrival and departure times of individual vehicles
Computer simulation often used to calculate MOEs

31. Microscopic Analysis Example: Signalized intersection (Left turn)
Arrival to downstream is assumed to be 10secs after arrivals to upstream, which is the start of the left-turn lane and beyond the back of queue.

32. Stochastic Queuing Kendall’s notation Letter 1 / Letter 2 / Number
Letter 1: arrivals process
Letter 2: service process
Number of channels (servers)

33. Stochastic Queuing Traffic Density
ρ = λ/µ, where ρ < 1 (λ < µ)
ρ = traffic intensity
λ = mean arrival rate (vehicles per time interval)
µ = mean service rate per channel (vehicles per time interval)
λ and µ must have the same units

34. Kendall’s Notation Letter 1 / Letter 2 / Number
M: Random (negative exponential distribution)
D: Deterministic
E: Erlang distribution
G: Generalized arrival or departure
Number: the number of channels
Example: M/D/1 [or M/D/1(∞,FIFO)], M/M/1, M/M/n

35. M/D/1 Queuing Regime
Arrival times are exponentially distributed (arrivals on Poisson distribution)
Service rate deterministic (no random variation)
One service channel

36. M/D/1 Queuing Traffic Intensity: Average number in system:
(waiting and service)
Average Waiting Time:
Average time in system:

37. M/M/1 Queuing Arrivals and departures are exponentially distributed
One service channel

38. M/M/1 Queuing Regime Traffic Intensity: Average Queue Length:
Average Waiting Time:
Average time in system :

39. M/M/1 Queuing Regime

40. Little’s Law
Lq = lWq
Or E(m)=lE(w)

41. Multichannel M/M/n

42. M/M/1 Example Park scenario.
Arrivals: negative exponentially distributed, arrival rate = 180 vph (3 vehs/minute)
Park ranger takes an average of 15 sec to distribute brochures. The service time distribution is also negative exponential.

43. M/M/1 Example

44. Stochastic Queuing Example: Random Delay at Intersection
Classical Queuing System Applications:
Poison Arrival, General (Arbitrary, but continuous) service times (M,G,1)
Poisson Arrival, Erlang service times (M, Ea,1)
Poison Arrival, Uniform service times (M,D,1)