C onsider a signal-controlled road network for which a given tim ing plan is im plem ented in each of the consecutive time slices form ing the w hole time span. The following notation can be stated first.
3.3.1. General
L et T be the time span com prising a series of time slices denoted by
T ' = [ t C ; we have T = VJ T ' = u [f ' , .
i i
Let G( N , L , R , S ) be a directed road network, where A is a set of N j nodes
each of which represents one of N j fixed-tim e signal controlled junctions, that is,
A = { 1, 2, ... , Ayj , L is a set of A ^ links, that is, L = { 1, 2, ... , A ^|
^ is a set of origin nodes and 5 is a set of destination nodes.
For each signal controlled junction m in , let L be the corresponding set of
the links leading to that Junction. 3.3.2. Signal timings
The following notation is used and is supposed to be fixed within any one given tim e
slice. For each signal controlled junction m \n N \ let be respectively
the num bers of signal groups and stages at junction m , and let
f = I 1, 2, ..., N I be the set o f num bers of signal groups at junction m ,
S = I 1 , 2 ...Ny,,, } be the set of num bers of stages at junction m ,
, i e , j e • â--^ = 1 o r 0 } be the stage-signal group
m
incidence matrix at junction m , where a - = 1 if signal group j has green in the
stage I at junction m , and a - = 0 otherw ise; and
. « e L,„ . ,/■ e = 1 o r 0 J be the link-signal group
incidence matrix at junction m , where 6 . = 1 if the traffic stream a is controlled
by the signal group j at junction m , and 6 - = 0 otherw ise; and
g be the specified minimum green for signal group j at junction m , \f j e P ^ ,
be the specified minimum green for stage i at junction m , \/ i e S ^ ,
be a common multiplier applied to the average arrival flow for each link a in
û) be the time at which the stage i begins at junction m expressed as a proportion
of common cycle time relative to an arbitrary time origin to the network as a
whole, V / e ,
Cji^n be the clearance time between the end of green for signal group j and the start
of green for incompatible signal group / at junction m, for j ^ I , \f j , I e P ^ , and
y , / ) be the successor function which is defined by H eydecker (1992), a
collection of numbers 0 and 1 for each pair of incom patible signal groups at junction
m\ where _ / , / ) = 0 , if the start of green for the signal group j precedes
that of /, and / , / ) = 1 otherwise, \/ j , I e P .
For the whole road network G , let
Ç be the reciprocal of the com mon cycle time c ,
0 = [ ; V 7 e , V m G ] be the vector of starts of green, w here
elem ent is start of next green for signal group j at junction m as a proportion of
com m on cycle time relative to an arbitrary tim e origin to the network as a whole, and
(j) = [ ; V ; G , V m 6 j be the vector of durations of green, w here
elem ent is the duration of green as a proportion of com m on cycle time for signal
group 7 at junction m.
For each link a \n L , let
e be the extra effective green for all links which equals green end lag ^2 m inus
0^^ be start of next effective green for link a expressed as a proportion of com m on
cycle time relative to an arbitrary time origin to the netw ork as a w hole, the
relationship between 0^ and 0y^ is 6^= V a g L ,
jm
be the duration of effective green for link a, which is expressed as a proportion
of common cycle time, the relationship betw een and is
Y . Kjm + e C . V a 6 L , and
jm
be the maximum degree of saturation for link a.
3.3.3. Flows
For each link a \n L , the traffic conditions in each tim eslice T \ t e T'' are
described by the following indicators. Let
/^(f) be the arrival rate of traffic at the dow nstream end of link a at time t,
/^(f) be the cum ulative amount of traffic arriving at the dow nstream end of link a
over the time interval [ t \ t ] ,
^ - [ ^ ^ ^ ^ ] be the vector of the average flow on link a,
be the saturation flow on link a,
be capacity for link a, which can be expressed by definition as p^= , and
3.3.4. Indicators of traffic conditions
Four indicators of traffic conditions are used, each of which is a function of
t , t e T ' . For each link a in L , let
be the rate of delay on link a over the interval [ t \ t ] ,
be the average delay to a vehicle arriving on link a during the interval [t % t] ,
L^(t) be the num ber of queuing vehicles on link a af tim e t, and
S^{t) be the number of stops per unit tim e on link a over the interval [ t \ t ] .
3.3.5. User equilibrium traffic assignment For the road network G {N , L , R , S ) , let
W = { w = { r , s ) \ \ / r e R , \ / s e S } be the set o f origin-destination pairs.
P ^ be the set of paths between the origin-destination pair w in W ,
D = Djj, ; V w G W be the vector o f travel dem ands, where elem ent is
the travel demand between origin-destination pair w,
/ = J /^ ; V/7 G , V w e W J be the vector o f path flows, where elem ent
is traffic flow on path p ,
cO = be the vector of undelayed link travel times, where elem ent
is the undelayed travel time for link a, and
c{ q , \\f ) = [ c^{ ] be the vector of link travel times, where elem ent
signal setting variables, \j/ = ( Ç , 9 , (j) ) , which is the sum of the undelayed
0
travel tim e under prevailing traffic condition, i.e. c and the average delay to a
vehicle at the end of the link leading to the dow nstream signal-controlled junction over
the tim eslice [ t ‘ , t ] , i.e. d^{t) , therefore by definition for each link a in L , if
CaiÇa ’ V) is averaged over the interval [ r f ] then
Let Ô = J 0^^ ; V a e L , V / ? G P ^ , V w e W j b e the link/path incidence
matrix where 8 = 1 if link a is in path p , and 8 = 0 otherwise,
A = J ; V /? G , V w G W j be the origin-destination/path incidence
matrix where A = 1 if path p connects origin-destination pair w , and A^^= 0
otherw ise, and
C = [ ; V p G , V w G W ] be the vector of path travel times, where
elem ent is the travel time on path p , and the relationship between C and
Cciiqa , Y) is Cp= Y) ’ V) and in vector form C ^ = 8 where
a e L
the superscript T is the matrix transpose operator.