1.225J (ESD 205) Transportation Flow Systems
Lecture 1
1
Cumulative Plots & Time-Space Diagrams
Diagrams
Prof. Ismail Chabini and Prof. Amedeo Odoni
Cumulative Plots
‰ Observer: count the total number of vehicles, N(t), that passed in
front of him/her during time interval [0,t].
Link entry
Link exit
Lane 1
Lane 2
Observer
N(t)
~
N (t )
5
4
3
2
1
0
1.225, 10/29/02
N(t)
q(t)
t1
t2 t
t3
t4
t5
time
Lecture 1, Page 2
Observations on N(t)
‰ N(t) is a step function. (not smooth)
~
N
‰ (t ) is a smooth approximation of N(t)
~
dN (t )
exists.
dt
~
dN (
t )
‰ q (t ) =
: instantaneous flow at time t.
dt
~
N (T ) − N (0) N (T ) − N (0)
≅
‰ Average flow =
T
T
1.225, 10/29/02
Lecture 1, Page 3
Arrival-Departure Cumulative Plots
Single lane
Observer A: A(t)
Observer D: D(t)
A(t)
N(t)
D(t)
n1
?
n
0
?
0 t0
1.225, 10/29/02
A-1(n)
t
D-1(n) t1
time
Lecture 1, Page 4
Accumulated Items: Q(t) = A(t) - D(t) ?
‰ Q(t): number of items (cars, planes) accumulated between the two
observers.
‰ Q(t) = Q(0) + [A(t) - A(0)] - [D(t) - D(0)]
= (Q(0) + D(0) - A(0)) + A(t) - D(t)
= A(t) - D(t)
( if Q(0)+D(0)-A(0) = 0 )
1.225, 10/29/02
Lecture 1, Page 5
Waiting Under FIFO Order
‰ Vehicles depart in the same order as they entered a link (i.e. segment
of road) ≡ (First-In-First-Out) FIFO
‰ Item n is observed at the entrance of a link at time A-1(n).
‰ Item n is observed at the exit of a link at time D-1(n).
‰ Waiting time of the item n: w(n) = D-1(n) - A-1(n)
1.225, 10/29/02
Lecture 1, Page 6
Q(t), w(n), and Elemental Waiting
N(t)
A(t)
D(t)
n1
Q(t)
n+dn
w(n)
elemental waitings
n
0 t0
1.225, 10/29/02
A-1(n)
t
D-1(n) t1
t+dt
time
Lecture 1, Page 7
Total Waiting Time
‰ Elemental waiting during [t, t+dt]: Q(t)dt
t
t
t0
t0
‰ Total waiting during [t0, t1]: Area = ∫ 1 Q(t )dt = ∫ 1 ( A(t ) − D (t ))dt
‰ Elemental waiting during [n, n+dn]: w(n)dn
n1
‰ Total waiting during [0, n1]: Area = ∫ w(n)dn
0
‰ Average wait suffered by vehicle arriving between t0 and t1
t1 − t0
Area
Area
1
W =
=
×
=Q×
A(t1 ) − A(t0 ) t1 − t0 A(t1 ) − A(t0 )
λ
Q = λ ×W
1.225, 10/29/02
(Queuing formula)
Lecture 1, Page 8
Time-Space Diagram: Analysis at a Fixed Position
position
L
h1
x
h2
h3
h4
time
0
t0
1.225, 10/29/02
T
Lecture 1, Page 9
Flows and Headways
‰ m(x): number of vehicles that passed in front of an observer at
position x during time interval [0,T]. (ex. m(x)=5)
m( x )
‰ Flow rate: q ( x) =
T
‰ Headway hj(x): time separation of consecutive vehicles
m( x)
∑ h j (x)
‰ Average headway:
h (x) =
j =1
m(x) − 1
‰ What is the relationship between q(x) and h (x) ?
1.225, 10/29/02
Lecture 1, Page 10
Flow Rate vs. Average Headway
‰ If T is large, T ≈
m( x)
∑ h (x)
j =1
j
m( x)
1
T
=
≈
‰ Then,
q ( x ) m( x )
1
q (x) ≈
h (x)
∑ h (x)
j =1
j
m(
x)
= h (x)
This is intuitively correct.
‰ q(x) is also called volume in traffic flow systems circles (i.e. 1.225)
‰ q(x) is also called frequency in scheduled systems circles (i.e.
1.224)
1.225, 10/29/02
Lecture 1, Page 11
Time-Space Diagram: Analysis at Fixed Time
position
L
s1
s2
0
1.225, 10/29/02
t0
t
time
Lecture 1, Page 12
Density and Spacing
‰ n(t):number of vehicles in a stretch of length L at time t.
‰ Density k (t ) =
n(t )
L
‰ si(t):spacing between vehicle i and vehicle i+1.
n (t )
∑ s (t)
‰ L ≈
i
i=1
n (t )
∑ s (t)
i
1
L
i=1
=
≈
= s (t )
‰
k (t ) n(t )
n(t )
‰ k (t )≈
1
s (t )
1.225, 10/29/02
(Is this intuitive?)
Lecture 1, Page 13
Lecture 1 Summary
‰ Cumulative plots: A(t), D(t), Q(t), w(n)
Q = λ ×W
‰ Time-Space Diagram: Analysis at a fixed position
q( x) ≈
1
h (x)
‰ Time-Space Diagram: Analysis at a fixed time
k (t ) ≈
1.225, 10/29/02
1
s (t )
Lecture 1, Page 14