222 European Journal of Operational Research 71 (1993) 222-234 North-Holland

Individual controlled Optimal design

junctions:

Giuseppe Bruno and Gennaro Improta Dipartimento di Informatica e Sistemistica, Universitd di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy

Received September 1991; revised October 1992

Abstract: The phase-based approach is a recent evolution in the field of undersaturated isolated junction control optimization. Using the incompatibilities among the streams, it proceedes calculating the control variable values without the external individuation of a stage matrix. One of these methods, based on a binary linear mathematical model, performs the simultaneous optimization of cycle time, green splits and green scheduling. The model, however, requires a preliminary decision about the composition of the streams and their assignment to the lanes. The flow ratios of the streams, indeed, have to be supposed constant during the cycle. In this paper an idea to remove this last hypothesis is described. It is based on a simple model regarding the behaviour of the users belonging to streams coming from the same access. A numerical example using a test junction is shown; the obtained results are compared to the ones deriving from a standard phase-based method.

Keywords: Traffic control; Signal optimization; Integer Programming

1. Basic definitions

A junction is defined &olated if there are no ' interactions' with it and the other surrounding junctions. In general this means that the mutual distances are sufficient to eliminate the 'p la toon effect ' generated by the traffic signals on the vehicle flows.

Interacting junctions can be treated as isolated only if the control objective can be assumed to be independent of the characteristics of the arrivals. This is true in the case of capacity factor maximization or cycle time minimization.

A junction is formed by a set of approaches and a crossing area. An approach is a part of a road leading to the junction such as the traffic in it has right of way simultaneously and a vehicle joining the back of the queue can expect to pass the signal at roughly the same time whichever lane it chooses.

The traffic at a junction is usually divided into streams. A stream is formed by all the users who cross the junction from the same approach. A stream can be formed by a set of manoeuvres which share the same approach.

When considering the control system, users are divided into groups, which are sets of streams that receive identical signals from the controller. The streams in a group may therefore use different approaches. In many practical cases each group is formed by one stream.

We assume, in general, that the vehicle departures occur at a constant time interval. An effective green-time g during which the vehicles can leave the stop line, and an effective red-time r during which no departure occurs are hence considered.

Correspondence to: G. Bruno, Dipartimento di Informatica e Sistemistica, Universitfi di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy.

0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

G. Improta / Individual controlled junctions : Optimal design 2 2 3

Let s(t) be the flow leaving the stop line during the green and amber periods. A saturation flow S is defined, for each stream, as the average flow that can cross the junction in the unitary time when a queue is present at the approach.

Let G, R, A be respectively the displayed green, red and amber for a stream. We define (Webster, 1958; Webster and Cobbe, 1966):

g=(1/s)fC+As(t) dt, r = G + R + A - g . o

We also define lost time:

l = G + A - g .

The cycle time, c, is the least time during which a complete succession of signals occurs. The cycle time is, obviously, equal to the sum of the effective green and the effective red for each group.

Let q(t) be the flow reaching the stop line. We define undersaturated approach as an approach for which

f~ fo c + A q(t) dt <_ s(t) dt.

An undersaturated junction is a junction for which all the approaches are undersaturated. The cycle can be subdivided into stages, elementary periods during which particular sets of groups

receive green simultaneously. The order in which n groups receive right of way in m stages can be specified by a Boolean matrix

T(m • n), called stage matrix, such that tij = 1 if group j has right of way in stage i; tij = 0 , otherwise. Two groups that can safely cross the junction at the same time are called compatible; otherwise they

are called incompatible or conflict. The clearance time for two incompatible groups is the minimum time between the end of the amber of

one and the beginning of the green time of the other. The incompatibilities among n groups can be expressed by a Boolean matrix A(n .n), called

incompatibility matrix or conflict matrix [a = 1 if groups i,j are incompatible; aii= 0 otherwise]. An incompatibility graph can be defined from matrix A. It is an undirected graph with n nodes (one node for each group) and one edge ij for each pair of incompatible groups i and j.

The edges corresponding to a set of groups that are mutually incompatible determine on the incompatibility graph a clique, i.e. a complete subgraph. For this reason a set of groups that are mutually incompatible is called a clique. The clique matrix, D = (dik), associated to an incompatibility graph is a Boolean matrix in which djk = 1 if stream k belongs to clique j and d/k = otherwise.

In a similar way a matrix and a graph of compatibility, as well as cliques of compatibility, can be defined (compatibigty matrix, compatibility graph, compatibility cliques).

Usually the set of groups having the right to cross the junction during a stage forms a maximal compatibility clique. Any ordered set of cliques respecting various imposed constraints is a possible stage sequence or, merely, sequence.

Two dependent variables are of great importance in defining the performances of a single junction: the capacity factor f , i.e., for an assigned signal setting, the maximum common multiplier for all the rates of flow arriving at junction, for which undersaturation still holds; the delay d, i.e. the difference between the travel time of a vehicle crossing the signalized junction and the time it would have taken if no other traffic were present and the stream in which it travels had constant right of way.

A third variable used in traffic control calculations is the critical cycle time, i.e. the least cycle time which gives a useful operating condition for the junction.

2. Mathematical programming methods for individual junction control

According to the general statements previously introduced, mathematical programming methodologies for single junction control assume as possible objectives of optimization: for heavily congested junctions,

224 G. Improta / Individual controlled junctions: Optimal design

capacity factor maximization (linear); if the capacity factor is satisfactory, the total rate of delay minimization (nonlinear but convex, by introducing as variable the reciprocal of the cycle time, or linear if using piecewise linearization of delay functions); the critical cycle time calculation (linear).

Mathematical programming based methodologies for single function control can be grouped into two classes. In the first class, the flows and saturation flows of the streams being known, a stage matrix is preliminarily fixed and the corresponding optimal green times for the stages calculated. This is called a stage-based approach (Allsop, 1971, 1972; Yagar, 1974; Improta, 1978). In the second class, the flows and saturation flows are still assumed known and the optimal timing is evaluated from the knowledge of the crossing compatibilities of the different streams (see Improta, 1991).

It is possible to divide the methods belonging to the second class (see Allsop, 1983) into two subclasses. The first subclass is based on the analysis of possible sequences of compatibility cliques and can be regarded as a natural evolution of the stage-based approach. Any clique is a possible stage, any sequence of cliques respecting some imposed constraints is a possible stage succession. The optimization problem consists of calculating the optimal clique sequence, the green time for each clique and the cycle time (Stoffers, 1968; Zuzarte Tully, 1977; Zuzarte Tully and Murchland, 1978; Cantarella and Improta, 1981).

An endpoint and the green time for each stream are used in the second subclass to determine the optimal cycle time, the green timing and the green scheduling. This is called a phase-based approach or stream-based approach. In this alternative approach the durations of the signal indications for the groups are considered directly rather than through the intermediary of stages.

Two seminal phase-based methods belonging to this approach have been proposed in recent years. In the first method (Gallivan and Heydecker, 1983; Heydecker and Dudgeon, 1987; Moiler, 1987) the green timing and scheduling is represented, in the complex plane, by two variables for each group, one representing the start and the other the duration of the green interval. This method is less constrained if compared to stage-based methods but the determination of the optimal solution requires the complete enumeration of all the possible orders of incompatible groups.

The other method is based on a mathematical programming model by which control variables (cycle time, green timing and scheduling) are calculated simultaneously. This model is formulated representing on a time axis the control variables, measured in time units (Improta and Cantarella, 1982, 1984) or defined as a proportion of the cycle time (Cantarella and Improta, 1988). All the constraints are linear and the nonoverlapping conditions among pairs of incompatible groups are modelled using binary variables.

An alternative formulation of this method using only real variables requires nonlinear constraints (Cantarella and Improta, 1983).

3. A classification of existing formulations and some possible developments for single junction control optimization

It can be useful to classify the single junction control problems using the control variables set. Five problems can be arranged in order of increasing difficulty: A. Calculation of the green times of the stages. B. Calculation of the green times and the cycle time. C. Control system design of the junction:

C1. stage-based method; C2. phase-based method.

D. Control of the junction with nonconstant flow ratios. E. Controlled junction design.

Table 1 shows, for each problem, the set of the variables to be calculated, the set of the independent variables and the possible objectives (performance indices).

G. Improta / Individual controlled junctions : Optimal design

Table 1 Classification of single junction control problems

225

Problem Independent variables Dependent variables Objective variables

A q s A Tc g B q s A T gc fmax C q s A g c T Cmi n D q A g c T s dmi n E q g c T s A

In the first two problems (A, B), the stage matrix is fixed either empirically or using some existing mathematical method (Zuzarte Tully, 1977).

The stage-based methods (C1) determine the control variables through a sequential technique while the phase-based methods (C2) use models that let one simultaneously calculate the green times and the phases.

Even if the phase-based methods represent an evolution in the single junction control optimization, they require that:

- the manoeuvres must be grouped in streams and, then, the matrix of incompatibilities must be fixed;

- each stream must be assigned to some lanes and, moreover, the flow ratios must be supposed constant during the whole cycle time.

These limitative hypothesis are overcome in the last two problems (D, E). The first one takes into account the possibility of variation of flow ratios during the cycle, in dependence on different assign- ments of streams to lanes. In the second one the matrix of incompatibilities (or, alternatively, the matrix of compatibilities) becomes a decision variable. The grouping of manoeuvres in streams determines an increase of the saturation flow and of the incompatibilities of the stream. The first effect generates a decrease of the delay while the second one can produce an increase of the delay.

In dependence on the different possibilities of forming the streams, different matrices of incompatibil- ities and, then, different optimal values of control variables and performance indices can be obtained. The optimal control junction design consists of the choice of the optimal lay-out of the junction which can lead to the best values of the used performance index.

At present, in the literature, there are no proposed methods able to solve the last two problems, (D and E).

4. A phase-based model: SICCO (Single Intersection Capacity factor and Cycle time Optimization)

We refer to the model used by Improta and Cantarella (1982, 1984), Cantarella and Improta (1988) and, for simplicity of notation, we suppose that each group is formed by just one stream.

We assume the following input variables: n : The number of the streams. c : The cycle time ( z = l / c ) . A : The incompatibility matrix. li~ : The clearance time between the streams i,j. w/j : A binary variable associated to a pair of i , j incompatible streams, such that w~j = 0 if the green of

i precedes the green of j. Further, for each stream k:

qk = The arrival rate. S k = The saturation flow. Yk = qk /Sk = the flow ratio.

2 2 6 G. Improta / Individual controUed junctions: Optimal design

l k = The lost time; it is a part of the green-amber interval unused because of the starting and stopping transients.

We assume the following output variable: f : The function capacity factor.

Further, for each stream k, as proportion of the cycle time: uk: The starting time of green. vk: The ending time of amber. gk: The effective green. rk: The effective red, so that

gk + rk = l , Vk -- u~ = gk + lk " z.

The mathematical model for the maximization of the capacity factor is then:

f Max!

subject to

l i j + u i - u J + ( f Y J + Z l J ) < - w i j } V(i, j ) : = 1 ,

lji + Uy -- U i + ( fYi + zli) <- (1 - wij ) aij i 0 < t J k < l

wit = 0 /1

The first set of constraints includes the junction under-saturation condition and the nonoverlapping condition among the green times of conflict streams.

The model can be solved using the SICCO procedure (Cantarella and Improta, 1988) whose steps are described in the following.

As all the streams belonging to the clique j are mutually incompatible, the sum of their green times must be less than or equal to the duration of the cycle:

Cj = Eajk (cgk + t~) _< C Vj. (1) k

According to the under-saturation conditions (g~ > Yk, Vk), assigning g~ = f - Y k , we obtain from (1):

f<_ ( c - ~kdjklk) / (C • ~kdjkyk) Vj. (2)

Assuming at least one of the relations (1) to be verified as equality, it is possible to determine a relationship between the maximum capacity factor and the minimum value of the cycle; a method for determining the green times and the scheduling can be deduced consequently. For simplicity, we assume that there are no constraints on minimum green times and maximum red times.

Let:

Yj = EdjkYk, (3) k

L~ = ~ d j k l k. (4) k

The maximum capacity factor for a fixed cycle c is

fmax = f ( c ) = m!n ((1 -- L J c ) / Y j ) , (5) Y

while, for a fixed capacity factor f , the minimum cycle is

C m i n = c ( f ) = max ( L J ( 1 - fl~j)). (6) )

G. Improta / Individual controlled junctions : Optimal design 227

The clique j which determines the value f for an assigned c (or viceversa the value c for an assigned f ) is the critical clique.

After determining f and c from one of (5) or (6), the green time gk assigned to each stream k is then

gk =fYk" (7)

The green times of streams not belonging to the critical cliques can obviously be further increased observing the constraints of compatibilities. As a consequence of this property, more than one scheduling with the same maximum capacity factor is possible.

The increase of green times for 'noncritical' streams can be performed in order to minimize the total delay with fixed values of capacity factor and cycle.

The optimal scheduling of the green times, once they have been computed from (7), is carried out (if there are no odd holes in the incompatibility graph) by orienting the incompatibility graph with a branch and bound technique as described in Cantarella and Improta (1985, 1988). Moyer (1991) has recently proposed an alternative procedure in order to performe this step.

In the SICCO method, each stream, characterized by an arrival rate and a saturation flow, is the basic element in control variable calculations; the flow ratio is assumed constant.

Let us consider two adjacent streams and suppose that their scheduled green times partially overlap. If the flow ratios of the streams are not equal, the model used by SICCO does not represent the behaviour of the users well.

In fact, it is reasonable that, during the common green period, if there is no physical separation, the vehicles of the two adjacent streams indifferently use the available lanes, decreasing in this way the length of the queues and balancing, therefore, according to Wardrop's first principle (Wardrop, 1952), the own flow ratios.

If we do not take into account this possibility, two kinds of mistakes can arise: the first mistake is that the calculated capacity factor value can turn out to be less than the real one; the second mistake is that the obtained scheduling might not be the optimal one.

5. A simple model to describe the behaviour of the users

We introduce the possibility that streams coming from the same access can share the assigned lanes during common green periods.

We suppose that a scheduling has been carried out for a given junction and that the adjacent streams s,t have green times partially overlapping. We can distinguish, in this scheduling, a period gC in which both streams have green (common green) and two periods g~ and g~ in which respectively either s or t has green.

To represent the phenomenon of 'shared lanes', we can refer to a model in which the streams s and t (arrival rates: q, and qt) are artificially divided into two streams [s i, s c] and It i, t c] where the first ones [s i, t i] use the green periods gis and g~ while the second ones [s c, t c] use the common green gC.

Of course, it is necessary that

_ i q~=q~+qCs and qt -q t +q[.

In other words, the model uses a new group (arrival rate: qC + qC) with a saturation flow S s + S t and two streams (arrival rates: q~, q]), respectively characterized by saturation flows S s and S t.

Figure 1 and Table 2 summarize the principal elements of the introduced model. In this representation, the streams s ~, t ~ and sOU t c are mutually incompatible: the capacity factor

corresponding to the new scheme can be calculated using the procedure SICCO replacing the streams s and t with the streams s c U t c, s i, t i, whose flow ratios are, respectively:

( q C + q ~ ) / ( S s + S t ) ' i " q , / S s , q ] / S , .

228 G. lmprota / Individual controlled junctions." Optimal design

g s

s t r e a m s

r s

stream t r t gt

I C g g

I . I I

1 gt

s t r e a m

a r r i v a l r a t e

s a t u r a t i o n f l o w

i S

i q

S

S 9

C C s U t

C C

q s U q t

S U S s t

Figure 1. The introduced elements in the new scheme

i t

I

qt

S t

Table 2 Correspondences between real situation and the proposed model for two adjacent streams

Stream Arr. rate Arr. rate in gC Arr. rate in gi Satur. flow

s qs qC q~ Ss t ql q~ q~ St

Model representation

New stream Arr. rate Sat. flow Incompatibilies

s i q~ s s s i t i s c to t c t i q~ st s i , , New group Arr. rate Sat. flow t i * *

sCUtC qsC + qtC Ss + S t sCUt c , ,

As a consequence of the in t roduc t ion of the artificial streams, a conflict clique modif icat ion occurs. Each clique conta in ing s or t or both becames a clique in which the s t ream s is subst i tu ted by the streams s i and s ~ U t c while the s t ream t is subst i tu ted by the s treams t and s c u t ~.

Therefore , the Yj for the cliques conta in ing the s t ream s or t change, assuming new values Y' . Let p be the s t ream s ~ U t c. The difference 8Yj = Y / - Yj, for a clique con ta in ing s, is given by

8 Y j = Y i ' - Yj = ( d j l y 1 + " ' " + d j p y p + dysiYsi " " + d j . y , , ) - ( d j l y 1 + ' ' - + d j s y s + " " + d j n Y n )

q~ qC + qC qs q Cs + q Ct q Cs qCSs + qtCSs - q ~ S s - q c S t q t S s - q c S t - + . . . . ( 8 )

ss ss+s, s, s ,+s, &(&+s,) ss(s,+s,)

For any clique k conta in ing t expression (8) must be obviously modif ied replacing s with t and viceversa. Consequent ly we have

q ~ S , c - - qt Ss

8 Y ~ = Y ~ - Y k S , ( S s + S t ) . (9)

As a consequence of (8) and (9) we can in t roduce the following theorem.

i k e .

b

12

t

b 5

F i g u r e 2(a). L a y - o u t o f a t h r e e a r m j u n c t i o n

c

s - b - c

t - b

F i g u r e 2(b). I ncompa t ib i l i t y g r a p h a n d c l iques w h e n s a n d t d o n ' t s h a r e the i r l anes

c c s U t

G. lmprota / Individual controlled junct ions : Optimal design 229

c c 1 s U t s - b - c

c c l s U t t - b

F i g u r e 2(c). I ncompa t ib i l i t y g r a p h a n d c l iques w h e n s a n d t s h a r e the i r l anes

Theorem. For a given junction with control variables calculated using (7), if the streams s and t share their lanes, the capacity factor of the junction improves if:

(i) one of the streams (suppose s) belongs to the critical clique j, (lOa)

(ii) q~/Ss > qC/St, (lOb)

qtSs - q~S, (iii) Y ~ - Y~ > Sr( Ss + St) for all cliques k containing the stream t. (10c)

This is a direct consequence of (8). In fact, if (10a) and (10b) are satisfied, Yj of the critical clique decreases and, consequently, the capacity factor can increase. It is necessary, indeed, that the modifica- tions of the other cliques are such that the corresponding capacity factor value is not worse than the previous one. In particular, as each clique containing t has an increase of the corresponding Yk from (9), (10c) assures this condition.

In Figures 2a, 2b and 2c an example of the modifications of the cliques of incompatibility is provided for a three-arm junction.

230 G. Improta / Individual controlled junctions." Optimal design

6. Calculation of the arrival rates for the artificial streams

In this section we want to calculate the values of the arrival rates (q~, q~, q~, qC) associated to these artificial streams. In order to do that, it is necessary to distinguish two cases: (a) The green start of s precedes the green start of t. (b) The green start of t precedes the green start of s.

Considering case (a), for the calculation of q~ and qC we must distinguish three subcases (for q~ and q~ the calculations are similar taking into account that gC precedes g~): 1) The vehicles arrived at the junction during the red period (r~) and the green period (g~) are able to

leave the stop line during g~, i.e. qsrs < g i ( S s - q s ) . This means that, at the end of q~, there is no queue.

2) g~ is not so long that the queue vanishes, i.e. qsr~ i > gs(Ss - qs); however, the vehicles of the stream g~ arrived at the junction during r s, g~ and gC, are able to leave the stop line during gC, i.e.

qsrs < (g~ + g~)(Ss - qs). 3) The vehicles arrived during the cycle are not able to leave the stop line during the green time g~ + g~,

i.e. qsr~ > g~(S~ - qs). In dependence on these three different situations it is possible to determine the values q~, qC, q~, qCt

as schematically indicated in Table 3. The expressions in the case of the green start of t preceeding the green start of s (case (b)) can

obviously be obtained replacing t by s.

7. Evaluation of the green periods

Using the values of qi and qC obtained from the expressions indicated in Table 3, for all the adjacent streams respecting (10a)-(10c) it is possible to calculate the corresponding green times.

It is clear that the values of qi and qC, indeed, depend on the green times and the green scheduling which provides the values of g~, g~ and gC. On the other hand, the green times and the green scheduling depend on the values qi and qC and the incompatibilities of the corresponding streams.

Taking into account these considerations, it is possible to define an iterative procedure able to calculate green times and flows mutually consistent (see Figure 3).

Table 3 Calculation of q~, qs c, q~, q~ for situation (a)

Arr. rate < i Case 1: q,~r s _ g s ( S s - qs ) q t r t < g c ( S t - q t )

i q i e - qC . q i qs t q~ = qs q~ = "~tq,

c c gC rt + gC

c c qt qs qt q C = c q s q C = c

Case 2: qsrs > g~(Ss - qs ) q t r t > g c ( S t - q t )

qsrs < g s ( S s - q s ) q t r t <- g t ( S t - q t ) g i . gC

q i q i q~ = ° S S s q~ = ~ _ S t s t c

g i gC qCqC qC = qs - S Ss qC = q, _ ---~-S,

c

Case 3: qsr s > g s ( S s - qs ) q t r t > g t ( S t - q t ) Arrival rates cannot be calculated for saturation situation

G. Improta / lndit,idual controlled junctions : Optimal design 231

scheduling without

shared lanes

for each pair of streams satlsflng (10) :

I c calculation o£ q and q

/

g r e e n t l m e s

c a l c u l a t i o n s

Lgp-g I n o p - 1

y e s

g = g S s ~ p

F igu re 3, T h e i tera t ive p r o c e d u r e

The first step consists of the calculation of green times and green scheduling without considering the possibility of shared lanes: if the scheduling gives, for adjacent streams, overlapping green times, the procedure iteratively determines the values of qi and qC and, afterwards, new green times and a new scheduling and so on.

The procedure stops when the values of the green times do not change from one step to the next: in this case, it means that the green times, calculated from a pair qi and qC, produce the same values qi and qC. The control of the convergency of the procedure is done calculating, for each stream, the difference between the assigned green times in two following steps p and p + 1 ( g p - g p + l ) . The procedure stops when the minimun among these differences is less than an arbitrarily fixed value e > 0.

This condition corresponds to the situation in which the sharing of the lanes does not affect the optimal regulation anymore.

8. Numerical example

The modified procedure has been proved on a simple test junction whose lay-out and incompatibility graph are shown in Figure 4. The flow data are reported in Table 4. The cycle time is assumed to be equal to 100 seconds.

The optimal scheduling performed by SICCO is shown in Figure 5 while in Table 5 the values of green times and the green scheduling are reported.

The critical clique is formed by the streams 2-3 whose value Y determines the capacity factor of the junction which is equal to 1.28. As stream 2 belongs to the critical clique, if it is considered that the vehicles belonging to stream 2 also use the lane assigned to stream 1 during the common green (71.4-85.7), then the clique 1U2-2-3 in the new scheme has Y-value lower than the previous one so that the capacity factor increases.

The scheduling carried out using the iterative procedure (see Figure 6 and Table 6) because of the reduction of the flow ratio of stream 2 assigns a shorter green period to the streams 2 and 4 and a longer

232 G. lmprota / Individual controlled junctions." Optimal design

2 1

If JL II

H 3

1 2 3

1

2 m

3 •

4 • •

4

Figure 4. Lay-out and conflict matrix for the test junction

Table 4 Flows and saturation flows of the streams

Stream Flow Satur. flow

1 200 3600 2 1200 1800 3 20O 1800 4 500 3600

2 5 SO 7 5 1 0 0 ( s e c )

s t r e a m : : :

0

I 71. ~ 8S. 7 100

Figure 5. Green times and scheduling for the junction of Figure 4 carried out by SICCO

G. Improta / Individual controlled junctions : Optimal design 233

Table 5 Scheduling obtained by SICCO

Stream Green start Green end Green time

1 71.4 100.0 28.6 2 0.0 85.7 85.7 3 85.7 100.0 14.3 4 0.0 71.4 41.2

s t r e a m

0 2S SO 7S 100 ( s e c )

I I I I 0 22 .8 7 1 . 8 1OO

Figure 6. Green times and scheduling for the junction of Figure 4 carried out by the modified procedure

Table 6 Scheduling performed by the new procedure with streams 1 and 2 sharing their lanes

Stream Green start Green end Green time

1 0.0 71.8 71.8 2 22.8 100.0 77.2 3 0.0 22.8 22.8 4 71.8 100.0 28.2

green period to stream 3. For the new scheduling the capacity factor value is 2.03 (+ 58%) and all the cliques are critical. In this case the stage succession is changed: in fact, gl preceeds gC in such a way that the vehicles belonging to stream 1 arrived at the junction during the red period cross the junction during gi. In this way the shared lanes during the green period can be used by the vehicles belonging to stream 2 which is a critical stream.

This condition also reduces the probability that vehicles belonging to stream 1 may block the departure line for the vehicles of stream 2.

9. Conclusions

Phase-based methods determine the control variables through a preliminary decision about the composition of the streams and their assignment to the lanes.

This approach does not take into account the possibility that vehicles belonging to some stream can use lanes assigned to adjacent streams when they have green simultaneously in order to minimize own waiting time.

In this paper it has been shown that, if we do not take into account this phenomenon, the calculations can determine a mistake in the values of the capacity factor and in the optimal scheduling.

234 G. lmprota / Individual controlled junctions: Optimal design

A first idea to remove this limitation has been proposed. It is based on a different representation of streams coming from the same access and on the application of an iterative procedure.

Further developments of the work concern the improvement of the adopted traffic model and the testing of the procedure on complex junctions in which more than two streams coming from the same access share lanes.

Moreover, the evaluation of possible mutual interactions among streams using shared lanes along the street leading to the junction needs further improvements. In particular the effects of this phenomenon on other performance indices (total delay) should be verified.

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