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1.2251.225J (ESD 205) Transportation Flow SystemsJ (ESD 205) Transportation Flow Systems
Lecture 1Lecture 1Cumulative Plots & TimeCumulative Plots & Time--Space DiagramsSpace Diagrams
Prof.Prof. IsmailIsmail Chabini and Prof.Chabini and Prof. Amedeo OdoniAmedeo Odoni
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Cumulative PlotsCumulative PlotsObserver: count the total number of vehicles,N(t), that passed in
front of him/her during time interval [0,t].
Lane 2
Lane 1
Link entry Link exit
Observer
1
2
3
4
5
time
N(t)
0
N(t)
)(~
q(t)
tN
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Observations onObservations on N(t)N(t)
N(t) is a step function. (not smooth)~~ (
( tN ) is a smooth approximation ofN(t) tNd ) exists.dt~
((
tq ) = tNd ) : instantaneous flow at time t.dt
~N(T) N(0) N(T) N(0)Average flow =
T T
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ArrivalArrival--Departure Cumulative PlotsDeparture Cumulative Plots
Single lane
Observer A:A(t) Observer D:D(t)
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D(t)
A(t)
t0 t1A-1(n)
n1
D-1(n)time
N(t)
00
t
?
n?
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Accumulated Items:Accumulated Items: Q(t) = A(t)Q(t) = A(t)--D(t)D(t)??
Q(t): number of items (cars, planes) accumulated between the twoobservers.
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 )
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Waiting Under FIFO OrderWaiting Under FIFO Order
Vehicles depart in the same order as they entered a link (i.e. segmentof road) (First-In-First-Out) FIFO
Item n is observed at the entrance of a link at timeA-1(n).Item n is observed at the exit of a link at timeD-1(n).Waiting time of the item n: w(n) = D-1(n) - A-1(n)
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Q(t)Q(t),, w(n)w(n), and Elemental Waiting, and Elemental Waiting
D(t)
A(t)
t0
elemental waitings
t+dt
N(t)
time0 A-1(n) D-1(n)
n w(n)n+dn
t1
n1
t
Q(t)
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Total Waiting TimeTotal Waiting Time
Elemental waiting during [t, t+dt]: Q(t)dt
( )
t1
( (
Total waiting during [t0, t1]: Area =
t1
dttQ = t0 ( tA ) tD ))dtt0
Elemental waiting during [n, n+dn]: w(n)dnTotal waiting during [0, n1]: Area = n1 dnnw( )0Average wait suffered by vehicle arriving between t0 and t1
Area =Area t1 t0 =Q 1W =( ( ( (tA 1) tA ) t1 t0 tA 1) tA ) 0 0
Q = W (Queuing formula)
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TimeTime--Space Diagram: Analysis at a Fixed PositionSpace Diagram: Analysis at a Fixed Position
position
h1 h2 h4h3
0
L
x
timet0
T
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Flows and HeadwaysFlows and Headways
m(x): number of vehicles that passed in front of an observer atpositionx during time interval [0,T]. (ex. m(x)=5)
( (Flow rate: xq ) = xm )T
Headway hj(x): time separation of consecutive vehiclesxm )(
hj (x)Average headway: h (x) = j=1
(xm ) 1What is the relationship between q(x) and h (x) ?
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Flow Rate vs. Average HeadwayFlow Rate vs. Average Headway
m(x)
IfTis large, T hj (x)j=1
m(x)
hj (x)1 = T j=1 =h (x)Then,xq ) xm ) xm )( ( (
1(xq ) This is intuitively correct.
h (x)
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)
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TimeTime--Space Diagram: Analysis at Fixed TimeSpace Diagram: Analysis at Fixed Time
position
t
s1
s2
L
0 timet0
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Density and SpacingDensity and Spacing
n(t):number of vehicles in a stretch of lengthL at time t.
(
(
Density tk )= tn )
L
si(t):spacing between vehicle i and vehicle i+1.tn )(
L si (t)i=1 tn )(si (t)
1( = L i=1 = ts )
( ( (tk ) tn ) tn )1
( tk )ts )
(Is this intuitive?)(
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Lecture 1 SummaryLecture 1 Summary
Cumulative plots:A(t), D(t), Q(t), w(n) Q = W
Time-Space Diagram: Analysis at a fixed position1
xq ) (h (x)
Time-Space Diagram: Analysis at a fixed time1
tk ) (ts )(
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