Optimization Approach
In the idealized situation where traffic arrives uniformly and deterministically to a
signalized intersection, one can model the traffic control system as a D/D/1 queuing
regime. The timing plan to minimize the total delay per cycle can be determined by
solving the following mathematical program:
(
)
(
)
2
,
min
2 1
/
0,
0,
i
n
i
i
C g
i
i
i
n
i
i
i
i
i
i
v
C
g
TD
v
s
g
L
C
g s
v C
i
g
i
−
=
−
+
=
−
≥
∀
≥
∀
∑
∑
(1)
where
TD
is the total delay per cycle obtained from a deterministic analysis of the arrival and
departure curves;
,
,
,
i
i
i
v s
g
are, respectively, the volume (in vehicles per second), saturation flow
(in vehicles per second), and green time (in seconds) for phase
i
;
C
is the cycle time (in
seconds);
L
is the total lost time of the timing plan (in seconds); and
n
is the total number of
phases of the timing plan. In situations where several approaches share the same phase, the
volume of the most critical approach, in terms of having the largest
/
i
i
v
s
ratio, is used. The
constraint
0
i
i
i
s g
v C
−
≥
is added to prevent the occurrence of overflow; otherwise the delay
will grow from cycle to cycle without a constant value.
In Appendix A, based on the first order conditions of (1), we show that the optimal strategy is to
provide each phase with a green time that it is just long enough to clear the traffic that arrives
within the cycle (i.e.,
/
i
i
i
g
v
C s
=
⋅
). The cycle time such determined is typically short. For an
n
phase operation, the optimal solution of (1) is:
(
)
1
/
/
/
1
/
n
i
i
i
i
i
i
n
i
i
i
i
i
n
i
i
i
L
C
v
s
g
v C
s
g
L
C
L v
s
g
v
s
=
−
=
⇒
+
=
=
−
∑
∑
∑
(2)
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According to this deterministic approach, plans with any excess green time deviate from the
optimal solution, thus resulting in an increase in total delay. In reality, vehicle arrivals have
variability. In cycles when the arrival rates are higher than the approach’s capacity (
/
i
i
g s
C
),
there is an overflow. The overflow must wait for the next cycle for discharge, thus incurring
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 Fall '10
 HKLO
 Optimization, Cycle Time, Trigraph, Lost Time, Webster delay formula

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