Optimal Signal Timing for an Oversaturated Intersection

Optimal signal timing for an oversaturated intersection

Tang-Hsien Chang

_{, Jen-Ting Lin}

Department of Transportation Science, Tamkang University, P.O. Box 7-876, Taipei 10617, Taiwan, ROC Received 5 September 1998; accepted 16 June 1999

Abstract

Trac congestion occurs frequently at downtown intersections during rush hours, at road construction zones as well as at accident sites. Under such circumstances, trac ¯ow exceeds intersection capacity causing queuing of automobiles that cannot be eliminated in one signal cycle. In this paper, we present a timing decision methodology which considers the whole oversaturation period. Discrete dynamic optimi-zation models are developed and an algorithm to solve them is presented. The optimal cycle length and the optimal assigned green time for each approach are determined for the case of two-phase control. The application of the performance index model to certain multi-phase signals in common use is also intro-duced. Evaluation results indicate that the proposed discrete type performance index model is a more appropriate design for congested trac signal timing control. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Multilevel design strategies are a novel trend in trac signal control (Gartner et al., 1995). Among the promising design features are included the ability to avoid and relieve congestion. Many basic signal theories have been studied in recent decades, including those developed by Webster (1958), May (1965) and Allsop (1972) and that described in the Highway Capacity Manual (1985). However, relatively few of those models have addressed congestion relief strat-egies. Neither control systems nor commonly used software such as SOAP (1985) and TRANSYT (1987) can adequately handle oversaturated trac. The performance of these conventional signal systems deteriorates under heavy trac conditions (Tarno€ and Parsonson, 1981; Cronje, 1983; Elahi et al., 1991). While addressing the limitations of conventional signal control systems, Cronje (1983) developed a model for optimizing ®xed-time signalized intersections that can be applied to

*_{Corresponding author. Tel.: +886-2-2363-1004; fax: +886-2-2622-1135.}

E-mail address: [email protected] (T.-H. Chang).

0191-2615/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S0191-2615(99)00034-X

Nomenclature

ai the number of lanes on approach or movement i

B the control gain

c cycle length

d the average delay per vehicle in a cycle

D total delay in a cycle

D…k† total delay in cycle state k

Di…k ‡ 1† delay per lane of approach or movement i in cycle state k

F number of stops

g e€ective green time

gmax; gmin upper limit and lower limit of e€ective green time

g…k† green time in cycle state k

gi…k† e€ective green time for approach or movement i in cycle state k

gi max; gi min upper limit and lower limit of e€ective green time for approach or movement i

G…k† green-time adjusted factor due to stop penalty in k

Gi…k† green-time adjusted factor due to stop penalty in k for approach or movement i

H Hamiltonian formula

k the pointer of a cycle state in a sequence of cycles during the saturation period

K stop penalty factor

li queue length of approach or movement i

l…k† queue length at the beginning of cycle state k

li…k ‡ 1† queue length of approach or movement i at the end of cycle state k, equal to the

queue length of approach or movement i at the beginning of cycle state k ‡ 1

N terminative state

PI performance index

PI…k† total performance index in state k

PIi…k† performance index of approach or movement i in state k

q input ¯ow rate

q…k† input ¯ow rate in cycle state k

qi…k† input ¯ow rate of approach or movement i in cycle state k

s saturated ¯ow rate as de®ned by Webster (1958)

si saturated ¯ow rate of approach or movement i

umax; umin upper and lower limit of control variables

u…k† control variables in state k

Wi…k† exogenous variable of approach or movement i in state k

W…k† exogenous variables in state k

Yi…k† exogenous variable of approach or movement i in state k

Z…k† exogenous variables in state k

k…k† Lagrange multiplier in state k

k g/c, ratio of e€ective green time

undersaturated and oversaturated conditions. That investigation also compared microscopic and macroscopic models to determine delay and the number of stops, indicating that the macroscopic approach is suciently accurate for practical purposes. In a related work, Elahi et al. (1991) developed a knowledge-based system SCII. For near- and over-saturated conditions, SCII adopts the deterministic model proposed by Newell (1982), in which the e€ects of random variations are neglected since arriving and queuing vehicles provide a steady source of inputs. Elahi et al. (1991) also indicated that the TEXAS model and NETSIM have no optimization capability. Although capable of providing optimal design, SOAP84 heavily depends on Webster's (1958) approach. When searching for timing optimization, the above models only plan for the next single cycle after the executing one, not concurrently for the entire congestion period.

The timing design of isolated signals is a prerequisite for trac control. This paper presents a novel strategy for the timing decision of isolated signals during congestion or oversaturation. A macroscopic and deterministic model is developed. The underlying notion of the delay formula is derived from the interactive relationship between the delay in a signal cycle and the following cycle. This relationship lacks a conventional signal timing formula. Dispersing a whole queue in one cycle in oversaturated conditions is problematic owing to the maximum cycle length con-straint. The remaining automobiles in queues cause delay in each cycle stage, i.e. the delay in a cycle a€ects the delay in the subsequent cycle. Conventional timing strategies consider only the optimization of a single cycle, commonly referred to as static systems, and are obviously unsat-isfactory. Optimal control timing should be designed to regulate the trac for minimum delay during the entire oversaturated period. A dynamic theory of optimization has to be developed to resolve such a cycle-chaining problem.

Regarding the one-by-one structure of cycle-chaining states, the conventional delay formula can be modi®ed into a state-dependent form called `state space equations'. An optimal control methodology can then be applied to determine optimal timing from the constructed state space equation of an intersection. Herein, we focus mainly on minimizing total intersection delay during the entire oversaturated period, not per cycle only. The proposed model is formulated as a discrete type operation. Gazis (1964), Gazis and Potts (1965), Green (1968), Burhardt (1971), Kaltenbach and Koivo (1974), Dans and Gazis (1976) and Michalopoulos and Stephanopolos (1977, 1978) constructed similar models for oversaturation control. But their models are all continuous types and do not address the problem of optimizing cycle length. Gazis (1964) proposed that, during an oversaturated period, the queues in all approaches should be allowed to disperse completely and simultaneously, thereby minimizing the total delay (Green, 1968). This method focuses on en-suring that the green time does not have any loss in any cycle during the oversaturated period. This type of control is terminated when completely dispersing the queues of all approaches. Michalopoulos and Stephanopolos (1977, 1978) proposed an ecient two-stage timing method, termed `bang-bang control', for the controlled signal. Their method attempts to ®nd an optimal switch-over point during the oversaturated period to interchange the timing of the approaches. For example, during the ®rst stage, the procedure is as follows: set maximal green time to the approach having a maximal arrival rate and minimal green time to the minimal arrival rate ap-proach. At the optimal switch-over point, switch the maximal green time to the minimal arrival approach and the minimal green time to the maximal arrival approach.

Continuous type models are limited in that the switch-over point does not necessarily occur at the end of a cycle, neither does the termination of the oversaturated period occur only at the end

of the ®nal cycle. On the other hand, the switch-over points determined by a discrete model occur exactly at the termination of a cycle. Discrete operation provides a smooth, regular, and ordered transfer of control. Calculating delay is more reliable. In addition, the penalty level incurred by vehicle stops is easily incorporated. It is more suitable for calculating the optimal cycle length and setting the optimal green time for each approach.

Details of the two-phase model are described below. Certain multi-phase signals in common use are introduced later.

2. Two-phase timing plan for oversaturation control 2.1. Subjective function with state space representation

The following describes the two proposed models: one is a basic discrete minimal delay model, and the other a performance index model. The former is to manifest the complexity of the con-tinuous delay model developed by Michalopoulos and Stephanopolos (1977, 1978), and dem-onstrate that pure delay models are ine€ective in searching optimal cycle length. The latter is suggested to be more appropriate in studying oversaturation control.

2.1.1. Discrete minimal delay model

Fig. 1 illustrates the situation of queue during oversaturation. The graph there represents a queue l…k ‡ 1† left when the green time terminates at a certain cycle state k. (`Cycle state' denotes the cycle in a sequence of cycles during a saturation period.) The delay in the graph can geo-metrically be calculated as

D ˆ1

2 2
l…k†c



‡ q…k†c2_{ÿ sg}2_k†_{: …1†}

This equation meets the May's delay formula (May, 1965) d ˆ c…1 ÿ k†2

2…1 ÿ kx† …2†

since k ˆ g=c; x ˆ qc=sg, and D ˆ d
qc, the area of the graph in Fig. 1. The continuous delay model developed by Michalopoulos and Stephanopolos (1977, 1978) is also consistent with this approach.

In the case of a cross intersection with a two-phase signal control as Fig. 2 illustrates, during oversaturation, the queue and dispersion situation is as indicated in Fig. 3. Without loss of

generality, it is assumed herein that the cumulative demand on all the approaches is a linear asymptotic function of time and that the cumulative output curves do not intersect the cumulative input curves for any of the approaches. This fact implies that no queue becomes negative or zero before the end of the oversaturated period. If a queue becomes negative while the signal is green, the designed green time becomes invalid due to the waste of control time.

According to Fig. 3, the relation of the queue lengths between state k and k ‡ 1 can be rep-resented by the following equations:

l1…k ‡ 1† ˆ l1…k† ‡ q1…k ÿ 1†g2…k ÿ 1† ‡ ‰q1…k† ÿ s1Š
‰c ÿ g2…k†Š; …3a†

l2…k ‡ 1† ˆ l2…k† ‡ q2…k ÿ 1†g2…k ÿ 1† ‡ ‰q2…k† ÿ s2Š
‰c ÿ g2…k†Š; …3b†

Fig. 3. Queue and delay of a four-leg intersection with two-phase control. Fig. 2. Four-leg intersection with two-phase signal control.

l3…k ‡ 1† ˆ l3…k† ‡ q3…k†c ÿ s3g2…k†; …3c†

l4…k ‡ 1† ˆ l4…k† ‡ q4…k†c ÿ s4g2…k†: …3d†

Also, from Fig. 3, the delay of approach 1 can be stated as D1…k ‡ 1† ˆ1₂ 2l1…k†c  ‡ 2q1…k ÿ 1†g2…k ÿ 1†c ‡ q1…k†c2ÿ s1c2‡ s1g22…k†  : …4† Thus, D1…k ‡ 2† ˆ1₂2l1…k‡ 1†c ‡ 2q1…k†g2…k†c ‡ q1…k ‡ 1†c2ÿ s1c2‡ s1g22…k ‡ 1†  : …5†

Incorporate (3a) into (5); to obtain

D1…k ‡ 2† ˆ D1…k ‡ 1† ‡1₂s1g22…k ‡ 1† ‡ W1…k ‡ 1† …6†

in which,

W1…k ‡ 1† ˆ1₂q1…k†c2ÿ s1g22…k† ÿ 2s1c2‡ 2s1g2…k†c ‡ q1…k ‡ 1†c2: …7†

Similarly, for approach 2, 3, 4 which give

D2…k ‡ 2† ˆ D2…k ‡ 1† ‡1₂s2g22…k ‡ 1† ‡ W2…k ‡ 1†; …8† D3…k ‡ 2† ˆ D3…k ‡ 1† ÿ1₂s3g22…k ‡ 1† ‡ W3…k ‡ 1†; …9† D4…k ‡ 2† ˆ D4…k ‡ 1† ÿ1₂s4g22…k ‡ 1† ‡ W4…k ‡ 1†; …10† where W2…k ‡ 1† ˆ1₂q2…k†c2ÿ s2g22…k† ÿ 2s2c2‡ 2s2g2…k†c ‡ q2…k ‡ 1†c2; …11† W3…k ‡ 1† ˆ1₂q3…k†c2‡ s3g22…k† ÿ 2s3g2…k†c ‡ q3…k ‡ 1†c2; …12† W4…k ‡ 1† ˆ1₂q4…k†c2‡ s4g2₂…k† ÿ 2s4g2…k†c ‡ q4…k ‡ 1†c2: …13†

Suppose that approach 1 has a1 lanes, approach 2 has a2 lanes, approach 3 has a3 lanes, and

approach 4 has a4 lanes, the total delay, summed from (6), (8), (9) and (10), should be

D…k ‡ 2† ˆ D…k ‡ 1† ‡1

2…a1s1‡ a2s2ÿ a3s3ÿ a4s4†g22…k ‡ 1† ‡ a1W1…k ‡ 1†

‡ a2W2…k ‡ 1† ‡ a3W3…k ‡ 1† ‡ a4W4…k ‡ 1†:

…14† The above equation can be equivalently restated as a state space expression

D…k ‡ 1† ˆ D…k† ‡ Bu…k† ‡ W…k†; …15† where D…k† is the state variable; B the control gain; u…k† the control variable and W…k† is the exogenous variable completely known before triggering the state

D…k† ˆ a1D1…k† ‡ a2D2…k† ‡ a3D3…k† ‡ a4D4…k†; …16†

B ˆ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4†; …17†

u…k† ˆ g2

2…k†; …18†

W…k† ˆ a1W1…k† ‡ a2W2…k† ‡ a3W3…k† ‡ a4W4…k†: …19†

2.1.2. Performance index model

During oversaturation, the number of stops and arrivals will increase. If the stop factor is considered in the utilization of penalty, the model becomes more reasonable (SOAP, 1985; TRANSYT-7F, 1987, 1991). In general, the performance index of signal control is expressed as

PI ˆ D ‡ KF : …20†

Applying basic Eq. (20) with (6) and following the derivative procedure of the discrete minimal delay model, we have

PI…k ‡ 1† ˆ PI…k† ‡ Bu…k† ‡ Z…k†; …21†

where PI…k† ˆ a1PI1…k† ‡ a2PI2…k† ‡ a3PI3…k† ‡ a4PI4…k†; …22† B ˆ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4†; …23† u…k† ˆ …g2…k† ‡ G…k††2; …24† Z…k† ˆ ÿ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4†G2…k† ‡ a1W1…k† ‡ a2W2…k† ‡ a3W3…k† ‡ a4W4…k† ‡ K…a1Y1…k† ‡ a2Y2…k† ‡ a3Y3…k† ‡ a4Y4…k†† …25†

satisfying the relationships:

PI1…k ‡ 1† ˆ PI1…k† ‡1₂s1g22…k† ‡ K…s1ÿ q1…k††g2…k† ‡ W1…k† ‡ KY1…k†; …26†

PI2…k ‡ 1† ˆ PI2…k† ‡1₂s2g2₂…k† ‡ K…s2ÿ q2…k††g2…k† ‡ W2…k† ‡ KY2…k†; …27†

PI3…k ‡ 1† ˆ PI3…k† ÿ1₂s3g22…k† ÿ Ks3g2…k† ‡ W3…k† ‡ KY3…k†; …28†

Y1…k† ˆ ÿq1…k ÿ 1†g1…k ÿ 1† ‡ 2q1…k†c ÿ s1c; …30† Y2…k† ˆ ÿq2…k ÿ 1†g1…k ÿ 1† ‡ 2q2…k†c ÿ s2c; …31† Y3…k† ˆ ÿq3…k ÿ 1†c ‡ 2q3…k†c; …32† Y4…k† ˆ ÿq4…k ÿ 1†c ‡ q4…k†c; …33† G…k† ˆK…a1s1‡ a2_as2ÿ a3s3ÿ a4s4ÿ a1q1…k† ÿ a2q2…k†† 1s1‡ a2s2ÿ a3s3ÿ a4s4† : …34†

2.2. Objective function and solution approach

First, the objective function is set only to minimize the total delay of the entire oversaturated period. The function is assigned to be a quadratic form as below (Anderson and Moore, 1990; Kuo, 1991)

MIN J ˆ1₂…D…N††2‡1₂XN

kˆ2

…D…k††2 …35†

in which, N is the terminative state of the oversaturated period. Minimizing (35) subjected to (15), based on the optimal control theory, involves equivalently to minimizing the Hamiltonian for-mula (Kuo, 1991; Luenberger, 1979). The Hamiltonian forfor-mula is de®ned as

H ˆ1

2…D…k††2‡ k…k ‡ 1†‰D…k† ‡ Bu…k† ‡ W…k†Š; …36†

where the adjoint variables, k…k†; k ˆ 0; 1; 2; . . . are functions of time. Now, the task is to ®nd a satisfactory value of the control variable such that (36) is minimal in the subjection of (15). According to the optimal control theory, if an extreme in H exists, it must satisfy the following conditions: 1: o……1=2†D2…N†† oD…N† ˆ k…N† ) D…N† ˆ k…N†; …37† 2: _oD…k†oH ˆ k…k† ) D…k† ‡ k…k ‡ 1† ˆ k…k†; …38† 3: oH ok…k ‡ 1†ˆ D…k ‡ 1† ) D…k ‡ 1† ˆ D…k† ‡ Bu…k† ‡ W…k†; …39† 4: _ou…k†oH ˆ 0: …40†

Eq. (36) obviously indicates that only the single control variable u…k† can minimize H. In signal control, the control variable is related to the green time which should be taken as the value

between predetermined upper and lower limits, i.e. gmin6 g 6 gmax. As de®ned in (18), g2…k† is the

only control variable here

u…k† ˆ fg2

2…k† gj 2 min6 g2…k† 6 g2 maxg: …41†

Eq. (36) clearly reveals that the relation between H and the control variable u…k† is linear. This leads to oH=ou…k† ˆ k…k ‡ 1†B 6ˆ 0. Thus, to minimize H, the control at each time point should be

taken with u…k† ˆ g2

2 min, (i.e. g2…k† ˆ g2 min, g1…k† ˆ c ÿ g2 min) when k…k ‡ 1†B > 0; u…k† ˆ g2 max2 ,

(i.e. g2…k† ˆ g2 max, g1…k† ˆ c ÿ g2 max) when k…k ‡ 1†B < 0.

This proposition is clearly a bang-bang control with a two-stage operation, from g2 max

switching to g2 min or from g2 minswitching to g2 max. The time at which the switching is required is

termed the `switch-over' point. For example in Figs. 2 and 3, at the ®rst stage, the maximal green

time g2 maxis set to approaches 3 and 4 which are associated with higher ¯ow rates; then the green

time for the other pair of approaches (with lower ¯ow rates) is determined if the cycle length is given. When the switch-over point evaluated by the described above requirement is reached, the

second stage begins, at which point the minimal green time g2 minis switched to substitute g2 maxfor

the control of approaches 3 and 4. The control is terminated at liˆ 0 …i ˆ 1; 2; 3 or 4†:

If the objective function is to minimize the performance index previously described, the func-tion can also be constructed as (35)±(40), but previous D…k† should be replaced by PI…k† and W…k† be replaced by Z…k†, respectively. This results in the control variable

u…k† ˆ …g2…k† ‡ G…k††2 …42†

and, which should be operated with the following two situations:

(i) When k…k ‡ 1†B > 0, u…k† ˆ MIN…g2…k† ‡ G…k††2. Since only green time g2 can be

con-trolled, MIN…g2…k† ‡ G…k††2 implies that g2…k† ˆ ÿG…k† if ÿG…k† is located in the interval

…g2 min; g2 max†; g2…k† ˆ g2 maxif ÿG…k† P g2 max; and g2…k† ˆ g2 min if ÿG…k† 6 g2 min.

(ii) When k…k ‡ 1†B < 0, u…k† ˆ MAX…g2…k† ‡ G…k††2. u…k† may be veri®ed as a concave curve

by taking twice di€erential of u…k† with g2…k†. This implies that maximal u…k†, g2…k† should be at

its boundary, g2…k† ˆ g2 minor g2…k† ˆ g2 max. If ÿG…k† is located in the interval …g2 min; g2 max†, the

choice is dependent upon MAXf…g2 min‡ G…k††2; …g2 max‡ G…k††2g, i.e. g2…k† ˆ g2 min when

…g2 min‡ G…k††2> …g2 max‡ G…k††2g and g2…k† ˆ g2 maxwhen …g2 max‡ G…k††2g > …g2 min‡ G…k††2. In

addition, if ÿG…k† P g2 max, g2…k† ˆ g2 min is selected. If ÿG…k† 6 g2 min, g2…k† ˆ g2 max. De®nitely,

g1…k† ˆ c ÿ g2…k†.

Obviously, the control by the performance index model di€ers somewhat from that by the

discrete minimal delay model. In condition (i), when ÿG…k† drops into the interval …g2 min; g2 max†,

there exists an otherwise condition from `bang-bang control', i.e. the optimal green time may not be at the assigned boundary. The fact that such an outcome is seldom but still possible during oversaturation (the outcome is much relying on the factor K) accounts for why a `bang-bang like control' is denoted as such a model's control to discriminate the real `bang-bang control'. 3. Algorithm for solving the timing models

Based on the solution approach and the conditions described in Section 2.2, the algorithm for solving the discrete minimal delay timing model is arranged in the following steps:

Step 1. Let k ˆ 2; and initiate k…2† with a positive value.

Step 2. When k…k†B > 0, g2…k ÿ 1† ˆ g2 min; when k…k†B < 0, g2…k ÿ 1† ˆ g2 max.

Step 3. Calculate D…k†, li…k† (i ˆ 1, 2, 3, 4). Check the queue length of each approach, if

spil-lover, adjust k…2† and back to step 1.

Step 4. If li…k† < 0 (i ˆ 1, 2, 3, 4), go to step 7.

Step 5. If l1…k† < 0, l2…k† < 0, l3…k† > 0 and l4…k† > 0, next step; otherwise, employing Eq. (38)

calculate k…k ‡ 1† and reset k ˆ k ‡ 1; then go to step 2.

Step 6. Let g2…k† ˆ g2…k ÿ 1†. Calculate l3…k ‡ 1† and l4…k ‡ 1†. If l3…k ‡ 1† < 0 and

l4…k ‡ 1† < 0, next step; otherwise, adjust k…2† then go to step 1.

Step 7. Calculate total delay: sum of D…j† from j ˆ 2 to k ÿ 1. Also, show g2…j† …j ˆ 1; . . . ; k ÿ 1†; then stop.

Step 1 aims to give priority of dispersion with a maximal green time to the approaches with the maximum ¯ow rate. In step 6, if the calculation falls into `otherwise', this indicates that there is some green time loss, because the dispersal curve intersects the arrival curve before achieving the termination. Thus, k…2† should be adjusted. Based on Eq. (39), move D…k† to the right, and substitute k…k† by its elder generations till k ˆ 2; then implies

k…k† ˆ k…2† ÿ ‰D…2† ‡ D…3† ‡


‡ D…k ÿ 1†Š: …43†

Therefore, in order to make k…k† < 0, k…2† should be reduced in the next iteration.

As for the algorithm for solving the performance index model, except for step 2 needing to be replaced by (i) and (ii) described in Section 2.2, the procedure is similar to that for solving the delay timing model.

4. Evaluation with a case

Based on the complicated description above, a simpli®ed case is now presented to demonstrate how the model is employed.

4.1. Complexity of the continuous delay model

As mentioned earlier, Michalopoulos and Stephanopolos (1977, 1978) proposed a continuous signal timing model for oversaturated control. The model attempts to minimize the total delay of the entire oversaturated period. Its cycle length is ®xed at 150 s …c ˆ 150†. Also mentioned earlier was its de®ciency, i.e. the possible mis-timing of its switch-over point before a cycle is complete. To explain this phenomenon, the case in their paper is applied herein. Assume an intersection of two one-way streets with a two-phase signal control. One street, denoted as approach 1, has two lanes; the other, denoted as approach 2, has a single lane. No left-turn trac is considered.

Approach 1 is with s1 ˆ 1400 pcu/h, g1 maxˆ 0:65c, g1 minˆ 0:4c, and approach 2, s2 ˆ 1000 pcu/h,

g2 maxˆ 0:6c, g2 minˆ 0:35c. Table 1 lists the input volumes (extracted from Michalopoulos and

Stephanopolos, 1978). While corresponding to the previously described algorithm for the discrete minimal delay timing model, Table 2 summarizes those results.

According to that table, at the eighth cycle, the control strategy is switched from g2 minto g2 max.

Notably, the oversaturation control is terminated at the 16th cycle. The total delay summed from approaches 1 and 2 is 737,914 veh-s, which is equivalent to 283.29 s/veh in average delay.

According to Michalopoulos and Stephanopolos (1978), the switch-over point of their con-tinuous delay model is at 994 s, and termination at 2558 s. However, in Table 2, the switch-over Table 2

Queue length and control strategy by the discrete minimal delay model

Cycle sequence …k† Queue length on

approach 1 l…k ‡ 1† Queue length onapproach 2 l…k ‡ 1† Green time onapproach 2 g2…k†

1 26 28 52.5 2 48 57 52.5 3 59 73 52.5 4 63 89 52.5 5 60 97 52.5 6 54 105 52.5 7 43 107 52.5 8 38 100 90.0 9 38 90 90.0 10 35 80 90.0 11 32 68 90.0 12 27 56 90.0 13 22 43 90.0 14 15 30 90.0 15 9 16 90.0 16 1 3 90.0 17 )7 11 90.0

Total delay from 1st

cycle to 16th 402,624 (s) 335,290 (s)

Table 1

Five-minute cumulative volumes

Approach 1 (veh/lane) Approach 2 (veh/lane) Time (s)

121 86 300 205 147 600 268 192 900 318 227 1200 359 257 1500 396 283 1800 430 307 2100 462 330 2400 492 352 2700 523 373 3000 552 394 3300 582 415 3600 611 436 3900 640 457 4200

point of the discrete delay model is at 1050 s and termination at 2400 s. The switch-over point and the termination of the continuous model are not located at the ends of their corresponding cycles. This means these two cycle lengths are probably not a constant at 150 s. Consequently, the op-eration would be problematic for a general signal controller. In addition, the oversaturation control time in the continuous model is longer than in the discrete model, indicating that the discrete model is better than the continuous model.

4.2. Control of the performance index model

Also, this study applies the case in Section 4.1 to discover the control (bang-bang like control) of the performance index model. The control is the same as the previous strategy for the discrete minimal delay model shown in Table 2. The total delay is extended to 1,046,886 veh-s (equivalent to 363.51 s/veh in average) because the performance index model includes a stop-penalty item. 4.3. Comparison of equal time-sharing control and bang-bang like control

At oversaturation, in the bang-bang control, the basic strategy involves assigning the sequence of the maximal green time and minimal green time to a relevant approach. Thus, the trac of the relevant approach is controlled in two stages: ®rst with maximal green time and then with minimal green time, or vice versa. Regarding the bang-bang like control, results di€er somewhat from the bang-bang control, as described in Section 2.2. For nearly all conventional timing designs, the timing split is distributed with the Webster's frame, which maintains the green time ratio with respect to the following formula (Webster, 1958):

ga=gbˆ …qa=sa†=…qb=sb† …44†

in which ga; qa and sa are the green time volume, and saturation ¯ow rate of approach a, and

gb; qb and sb the green time, volume, and saturation ¯ow rate of approach b. Obviously, at

oversaturation, ga : gbshould be 1:1. With conventional control, the green time for each approach

should be 0.5c during oversaturation and equally distributed. Thus, the signal timing becomes equal time-sharing for oversaturation control. The case in Section 4.1 is applied to compare the two strategies. Also investigated herein are the results based on the performance index model and c ˆ 155 s involving both the two strategies of equal time-sharing control and bang-bang like control. This study also determines the total delay arising from the conventional equal time-sharing control to be 449.07 s/veh; meanwhile, for bang-bang like control, it is only 362.78 s/veh. This observation con®rms Michalopoulos and Stephanopolos' hypothesis on applying bang-bang control (as well as the bang-bang like control proposed herein) to the oversaturation operation to be feasible. Clearly, conventional controls for congested intersections are invalid.

In terms of robustness, bang-bang like control is superior to conventional equal time-sharing control. Providing a 5% detection or prediction error in trac volume is acceptable, similar to the above case, we ®nd no di€erence in the switch-over point when applying bang-bang like control; but, it changes when applying equal time-sharing control. This indicates that bang-bang like control is more stable than equal time-sharing control.

4.4. Optimal cycle length

The above discussion con®rms that the cycle length remains constant. In fact, the total inter-section delay is a function not only of green time g…k†, but also of cycle length c. For other op-erational reasons, cycle lengths generally have upper and lower limits. Under normal conditions, the imposed cycle length ranges from 60 to 180 s.

The previous case still applies to the analysis of the optimal cycle length. The left column in Table 3 lists the switch-over points, termination, and average delays with respect to the minimal delay model in each given cycle, varying from 60 to 180 s. This table reveals that the average delay decreases as the cycle length descends. This phenomenon resembles the model's ®nding for an undersaturated situation, which can be veri®ed from the May's formula presented in Eq. (2). This observation con®rms that the delay in c ˆ 120 is twice the delay in c ˆ 60. However, the de-scendent slope of the model's ®nding applied for an undersaturated situation is steeper than in an oversaturated one. Obviously, the optimal cycle length in the minimal delay model should be as small as possible. This ®nding implies that, at oversaturation, the cycle length is set in c ˆ 60. In fact, a short cycle causes more stops (including full stops and partial stops, see TRANSYT-7F User Guide, 1991), resulting in increased operating cost, exhaust energy and pollution. Therefore, pertinent indicates that long cycle lengths are common in practice. Nevertheless, extremely long cycle lengths may also waste time and become unfair. To attain more reasonable control, except Table 3

Switch-over point, termination and average delay in each given cycle Cycle length

(s) Minimal delay model_Switch-over Performance index model

point (s) Termination(s) Averagedelay (s/veh) Switch-overpoint (s) Termination(s) Averagedelay (s/veh)

180 1080 2340 292.81 1080 2340 365.40 175 1050 2275 295.71 1050 2275 370.29 170 1020 2380 294.70 1020 2380 370.34 165 990 2310 299.39 990 2310 377.60 160 1120 2240 284.45 ± ± ± 155 1085 2325 284.04 1085 2325 362.78 150 1050 2400 283.29 1050 2400 363.51 145 1015 2320 288.53 1015 2320 371.96 140 980 2380 288.30 980 2380 373.65 130 1040 2340 280.14 1040 2340 368.47 125 1000 2375 279.72 1000 2375 370.40 115 1035 2300 274.44 1035 2300 369.50 110 990 2310 277.92 990 2310 377.40 105 1050 2310 268.33 1050 2310 386.18 100 1000 2400 268.88 1000 2400 373.02 90 990 2340 268.56 990 2340 381.66 80 960 2320 269.40 960 2320 394.00 75 975 2325 265.43 975 2325 395.20 70 980 2380 259.58 980 2380 394.28 65 975 2340 260.76 975 2340 404.65 60 960 2340 260.42 960 2340 414.15

for the delay term, considering appropriate stops penalty into the timing determination to reduce relative stops is acceptable. This perspective suggests that the performance index model is more appropriate than the minimal delay model for congestion control. Searching for the optimal cycle length in the discrete performance index model is much more complicated than by a trial and error method. The right column of Table 3 summarizes the searching results while assuming that K ˆ 30. On this occasion, the optimal cycle length is 155 s. It is not optimal for the cycle length to become small. This ®nding also re¯ects the fact that Michalopoulos and Stephanopolos' (1977, 1978) continuous delay model has diculty in searching for an optimal cycle length for over-saturation control since those investigators failed to consider stops penalty into their model.

Previous discussion of bang-bang like control did not consider the variation of the lengths of maximal and minimal green time. Maximal and minimal green time are normally regulated by laws in many countries. The regulation can be considered only as constraints. From the mathe-matical process depicted in previous sections or a theoretical perspective, maximal and minimal green times are actually a boundary condition. An attempt is currently underway to change the boundary values before determining what one cycle length for the optimal control is. Though, the assigned boundary must be within legality. Table 4 lists the details of analysis. In this case, we can

conclude that the ®nal optimal control should be set in C ˆ 160 s, g2 minˆ 0:25c and

g2 maxˆ 0:65c. The average delay is 320.43 s. Table 4 also indicates that not all provided cycle

length can adhere to the warrant of simultaneous dispersion. In other words, not all provided cycle lengths can be applied to oversaturation control.

5. Multi-phase timing plans

The above discussion con®rms that the discrete type performance index model is quite ap-propriate for trac signal timing design. The following introduces the performance index model as applied in commonly employed multi-phase signals.

Herein, two common types of multi-phases control for a four-leg intersection are presented. Fig. 4 displays a three-phase signal arrangement for a left-turn trac protection. Fig. 5 illustrates the queue and dispersion situation. Based on Fig. 5 and following the process presented in Section 2.1, it yields the form

PI…k ‡ 2† ˆ PI…k ‡ 1† ‡ f …g1; g3; k ‡ 1†: …45†

Table 5 provides the detail formulae. Also, Fig. 6 illustrates a four-phase signal with left-turn protection and Fig. 7, the queue and dispersion situation. From Fig. 7, we can infer that

PI…k ‡ 2† ˆ PI…k ‡ 1† ‡ f …g1; g3; g4; k ‡ 1†: …46†

Table 6 presents the detail formulae of the four-phase model. Solving the multi-phase models for optimal control timing resembles the description for the two-phase model in Section 3. Never-theless, the control variables in the multi-phase models are not single. Three-phase signals have two control variables and four-phase signals have three. Such a provision leads to a multiple-variables-single-object problem. The decision procedure is quite complicated. In the following, we only brie¯y provide the solving algorithm of the four-phase model. Eq. (46) can be rewritten as Eq. (21), but

Table 4

Relation between average delay and cycle length under di€erent green timea

Cycle length g_0.25c2 min _{0.30c 0.35c 0.4c} g2 max 0.55c 0.6c 0.65c 0.55c 0.6c 0.65c 0.55c 0.6c 0.65c 0.55c 0.6c 0.65c 180 375.70 360.73* ± 396.24 355.14 341.01* 393.38 365.40 355.37* 406.43 389.3* 382.87* 175 376.43* 365.84* ± 400.04* 360.39* 346.77* 397.32 370.29* 360.68* 394.14* 377.97* 371.24 170 375.76 ± ± 398.81 361.92 ± 395.63 370.34 ± 391.39 376.89 372.83 165 382.05 ± ± 372.50 368.77* ± 401.94* 377.60* 345.57 397.86 383.20 377.60* 160 381.04 341.98* 320.43 379.37* 375.80* 333.63* 400.51* 361.77* ± 408.20 382.41 384.23* 155 388.95 344.49 330.86* 379.32 378.53* 337.08 384.47 362.78 346.54 402.97 390.25* 370.73 150 388.45 346.42 ± 378.53 353.94* ± 383.16 363.51 356.44* 401.49 385.16* 371.20 145 395.88* 355.71* ± 386.34 357.46 ± 390.91 371.96 ± 394.27 382.95 379.33* 140 395.99 358.60 316.98* 386.23 359.71 329.38* 390.50 373.65 ± 391.45 383.36 366.23 130 409.14 ± 333.85* 399.71 377.00* 340.89* 403.51 368.47 347.52 405.43 384.07 382.17* 125 406.12* 344.65 ± 399.90 351.80 ± 403.41 370.40 ± 404.53 384.86 370.08

a_{Note: `*' expresses that at termination a queue still exists in a certain approach.`±' expresses that the termination fails in the given condition.}

Chang, J.-T. Lin /Transportation Research Part B 34 (2000) 471±491 485

B ˆ₂1 …a1s1‡ a2s2ÿ a…a33ss33ÿ a‡ a44ss44†ÿ a5s5ÿ a6s6† …a5s5‡ a6s6ÿ a7s7ÿ a8s8† 2 4 3 5 T ; …47† u…k† ˆ …g1…k† ‡ G1…k†† 2 …g3…k† ‡ g4…k† ‡ G34…k††2 …g4…k† ‡ G4…k††2 2 4 3 5: …48†

The objective function attempts to minimize the performance index and the Hamiltonian formula

are denoted as Section 2. Since Eq. (48) includes three implicit variables of green time g1, g3 and

g4, to ®nd minimal H from oH=ou…k† ˆ 0, a Hessian matrix X should be checked.

X ˆ o 2_H=…og 1†2 o2H=og1og3 o2H=og1og4 o2_H=og 1og3 o2H=…og3†2 o2H=og3og4 o2_H=og 1og4 o2H=og3og4 o2H=…og4†2 2 4 3 5 ˆ k
b0 1 k
b0 2 k
b0 2 0 k
b2 k
…b2‡ b3† 2 4 3 5; …49†

where bi denotes the three elements of vector B of Eq. (47).

Fig. 5. Queue and delay of a three-phase signal with left-turn protection. Fig. 4. Three-phase signal with left-turn protection.

(i) When X is positive de®nite, minimal u…k† is required. Since u…k† P 0, let u…k† ˆ 0 and then we have g1…k† ˆ ÿG1…k†; g3…k† ˆ G4…k† ÿ G34…k†; g4…k† ˆ ÿG4…k† and g2ˆ c ÿ g1ÿ g3ÿ g4ÿ G1…k†

…g1 min, g1 max†; …G4…k† ÿ G34…k††…g3 min; g3 max†; and ÿG4…k†…g4 min; g4 max†: If ÿG1…k† P g1 max,

g1…k† ˆ g1 max; if ÿG1…k† 6 g1 min; g1…k† ˆ g1 min; if …G4…k† ÿ G34…k†† P g3 max; g3…k† ˆ g3 max; . . . and

so on.

(ii) When X is negative de®nite, maximal u…k† is taken. Since u…k† is concave, this implies all gj…k†; j ˆ 1; 3; 4; should be at their boundaries, gj…k† ˆ gj min or gj…k† ˆ gj max. If ÿG1…k†…g1 min;

g1 max†; g1…k† is chosen from either g1 min or g1 maxsuch that MAXf…g1 min‡ G1…k††2; …g1 max_G1…k††2g.

If ÿG1…k† P g1 max, g1…k† ˆ g1 minis selected; if ÿG1…k† 6 g1 min, g1…k† ˆ g1 max. The choice for g4 is Table 5

A typical three-phase model

PI…k ‡ 2† ˆ PI…k ‡ 1† ‡ f …g1; g3; k ‡ 1† f …g1; g3; k ‡ 1† ˆ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4†…g1…k ‡ 1† ‡ G1…k ‡ 1††2‡1₂…a3s3‡ a4s4ÿ a5s5ÿ a6s6†…g3…k ‡ 1† ‡ G3…k ‡ 1††2‡ Z…k ‡ 1† G1…k ‡ 1† ˆ ÿc ÿ K ‡K…a_a1q1…k ‡ 1† ‡ a2q2…k ‡ 1†† 1s1‡ a2s2ÿ a3s3ÿ a4s4† G3…k ‡ 1† ˆ K ÿK…a_a3q3…k ‡ 1† ‡ a4q4…k ‡ 1†† 3s3‡ a4s4ÿ a5s5ÿ a6s6† Z…k ‡ 1† ˆ ÿ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4†…G21…k ‡ 1† ÿ c2† ÿ 1 2…a3s3‡ a4s4ÿ a5s5ÿ a6s6†G23…k ‡ 1† ‡X6 iˆ1 aiWi…k ‡ 1† ‡ K X6 iˆ1 aiYi…k ‡ 1† W1…k ‡ 1† ˆ1₂q1…k†c2ÿ s1g21…k† ÿ s1c2‡ q1…k ‡ 1†c2 W2…k ‡ 1† ˆ1₂ q2…k†c2  ÿ s2g21…k† ÿ s2c2‡ q2…k ‡ 1†c2  W3…k ‡ 1† ˆ1₂ s3…c h ÿ g1…k††2ÿ s3g23…k† ‡ q3…k†c2ÿ 2s3g2…k†c ‡ q3…k ‡ 1†c2 i W4…k ‡ 1† ˆ1₂ s4…c h ÿ g1…k††2ÿ s4g23…k† ‡ q4…k†c2ÿ 2s4g2…k†c ‡ q4…k ‡ 1†c2 i W5…k ‡ 1† ˆ1₂ q5…k†c2  ‡ s5g23…k† ÿ 2s5g3…k†c ‡ q5…k ‡ 1†c2  W6…k ‡ 1† ˆ1₂q6…k†c2‡ s6g23…k† ÿ 2s6g3…k†c ‡ q6…k ‡ 1†c2 Y1…k ‡ 1† ˆ q1…k ‡ 1†c ÿ q1…k†g1…k† Y2…k ‡ 1† ˆ q2…k ‡ 1†c ÿ q2…k†g1…k† Y3…k ‡ 1† ˆ ÿs3c ÿ q3…k†c ‡ 2q3…k ‡ 1†c ‡ q3…k†g3…k† Y4…k ‡ 1† ˆ ÿs4c ÿ q4…k†c ‡ 2q4…k ‡ 1†c ‡ q4…k†g3…k† Y5…k ‡ 1† ˆ ÿq5…k†c ‡ 2q5…k ‡ 1†c Y6…k ‡ 1† ˆ ÿq6…k†c ‡ 2q6…k ‡ 1†c

the same as choosing g1…k†. Finally, if …G4…k† ÿ G34†…g3 min; g3 max†; g3…k† is chosen from which

g3 min or g3 max such that MAXf…g3 min‡ g4‡ G34…k††2; …g3 max‡ g4‡ G34…k††2g. If G4…k† ÿ G34…k†

P g3 max, set g3…k† ˆ g3 min; if G4…k† ÿ G34…k† 6 g3 min, set g3…k† ˆ g3 max. Certainly, g2…k† ˆ c ÿ g1…k†

ÿg3…k† ÿ g4…k†. The interval …gmin; gmax† for left-turn phases is generally shorter than that for

through trac. The cycle set for four-phase signals is longer than for two-phase signals. Fig. 6. A four-phase signal with left-turn protection.

Otherwise, detX ˆ 0 implies that the control vector is out of control. At this occasion, Eq. (46)

should be changed to the alternative forms f …g2; g3; g4; k ‡ 1† or f …g1; g2; g3; k ‡ 1† or

f …g1; g2; g4; k ‡ 1†; then try to solve it again. When searching k…k† and tracing the optimal solution

within the entire oversaturated period, the proposed process described in Section 3 is referred. Table 6

A typical four-phase model

PI…k ‡ 2† ˆ PI…k ‡ 1† ‡ f …g1; g3; g4; k ‡ 1† f …g1; g3; g4; k ‡ 1† ˆ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4ÿ a5s5ÿ a6s6†…g1…k ‡ 1† ‡ G1…k ‡ 1††2‡1₂…a3s3‡ a4s4†…g3…k ‡ 1† ‡ g4…k ‡ 1† ‡ G34…k ‡ 1††2‡1₂…a5s5‡ a6s6ÿ a7s7ÿ a8s8†…g4…k ‡ 1† ‡ G4…k ‡ 1††2‡ Z…k ‡ 1† G1…k ‡ 1† ˆ ÿc ÿ K ‡_a K…a1q1…k ‡ 1† ‡ a2q2…k ‡ 1†† 1s1‡ a2s2ÿ a3s3ÿ a4s4ÿ a5s5ÿ a6s6† G34…k ‡ 1† ˆ K ÿK…a3q3…k ‡ 1† ‡ a_a 4q4…k ‡ 1†† 3s3‡ a4s4† G4…k ‡ 1† ˆ K ÿK…a_a5q5…k ‡ 1† ‡ a6q6…k ‡ 1†† 5s5‡ a6s6ÿ a7s7ÿ a8s8† Z…k ‡ 1† ˆ ÿ1₂…a1s1‡ a2s2ÿ a3s3ÿ a4s4ÿ a5s5ÿ a6s6†…G21…k ‡ 1† ÿ c2† ÿ1₂…a3s3‡ a4s4†G234…k ‡ 1† ÿ1₂…a5s5‡ a6s6ÿ a7s7ÿ a8s8†G24…k ‡ 1† ‡ X8 iˆ1 aiWi…k ‡ 1† ‡ K X8 iˆ1 aiYi…k ‡ 1† W1…k ‡ 1† ˆ 0:5 q1…k†c2  ÿ s1g21…k† ÿ s1c2‡ q1…k ‡ 1†c2  W2…k ‡ 1† ˆ 0:5 q 2…k†c2ÿ s2g21…k† ÿ s2c2‡ q2…k ‡ 1†c2 W3…k ‡ 1† ˆ 0:5 s3…c h ÿ g1…k††2ÿ s3…g3…k† ‡ g4…k††2‡ q3…k†c2ÿ 2s3g2…k†c ‡ q3…k ‡ 1†c2 i W4…k ‡ 1† ˆ 0:5 s4…c h ÿ g1…k††2ÿ s4…g3…k† ‡ g4…k††2‡ q4…k†c2ÿ 2s4g2…k†c ‡ q4…k ‡ 1†c2 i W5…k ‡ 1† ˆ 0:5 s5…c h ÿ g1…k††2ÿ s5g24…k† ‡ q5…k†c2ÿ 2s5g3…k†c ‡ q5…k ‡ 1†c2 i W6…k ‡ 1† ˆ 0:5 s6…c h ÿ g1…k††2ÿ s6g24…k† ‡ q6…k†c2ÿ 2s6g3…k†c ‡ q6…k ‡ 1†c2 i W7…k ‡ 1† ˆ 0:5 q7…k†c2  ‡ s7g24…k† ÿ 2s7g4…k†c ‡ q7…k ‡ 1†c2  W8…k ‡ 1† ˆ 0:5 q 8…k†c2‡ s8g24…k† ÿ 2s8g4…k†c ‡ q8…k ‡ 1†c2 Y1…k ‡ 1† ˆ q1…k ‡ 1†c ÿ q1…k†g1…k† Y2…k ‡ 1† ˆ q2…k ‡ 1†c ÿ q2…k†g1…k† Y3…k ‡ 1† ˆ ÿs3c ÿ q3…k†c ‡ 2q3…k ‡ 1†c ‡ q3…k†…g3…k† ‡ g4…k†† Y4…k ‡ 1† ˆ ÿs4c ÿ q4…k†c ‡ 2q4…k ‡ 1†c ‡ q4…k†…g3…k† ‡ g4…k†† Y5…k ‡ 1† ˆ ÿs5c ÿ q5…k†c ‡ 2q5…k ‡ 1†c ‡ q5…k†g4…k†† Y6…k ‡ 1† ˆ ÿs6c ÿ q6…k†c ‡ 2q6…k ‡ 1†c ‡ q6…k†g4…k†† Y7…k ‡ 1† ˆ ÿq7…k†c ‡ 2q7…k ‡ 1†c Y8…k ‡ 1† ˆ ÿq8…k†c ‡ 2q8…k ‡ 1†c

6. Conclusion

Trac congestion occurs frequently at downtown signalized intersections during rush hours, in commercial and industrial work zones as well as at trac incident sites. On such occasions, trac ¯ow exceeds intersection capacity, causing queuing of automobiles that cannot be eliminated in one signal cycle. According to our results, conventional signal control strategies are inadequate since the designed optimization timing is only considered for the next single cycle after the exe-cuting one, not concurrently for the entire congestion period. This paper presents a novel timing decision methodology which takes account of the whole oversaturated period. Two models are derived: one of basic discrete minimal delay model and the other a performance index model. Based on the results presented herein, we can conclude the following:

1. The performance index model is a more appropriate design for congested trac signal timing control and is superior to the pure minimal delay model;

2. Discrete type models are more applicable in practice than continuous models. Continuous models may activate a switch-over point not occurring at the end of a cycle, whereas the switch-over point in discrete models coincides exactly with that juncture;

3. While corresponding to the ®ndings of previous studies of signal timing for oversaturated in-tersections, bang-bang like control, by which signals are operated alternatively and sequential-ly, with minimal maximal green time, signi®cantly outperforms conventional equal time-sharing dispersion control; and

4. Not all provided cycle lengths are applicable to oversaturation control, since some may fail to meet the warrant of simultaneous dispersion indicated by Gazis (1964).

The discrete type performance index model proposed herein, which results in bang-bang like control, is quite appropriate for oversaturation control. The performance of this model is rather robust even when the input data appear to be slightly biased. The proposed model can also de-termine the optimal cycle length and the optimal assigned green time.

References

TRANSYT-7F, 1991. User Guide, Methodology for Optimizing Signal Timing, vol. 4. Transportation Research Center, University of Florida.

TRANSYT-7F, 1987. User's Manual, Release 5.0. Federal Highway Administration.

Allsop, R.E., 1972. Delay at a ®xed time trac signal I: theoretical analysis. Transportation Science 6 (3), 260±285. Anderson, B.D.O., Moore, J.B., 1990. Optimal Control-Linear Quadratic Methods. Prentice-Hall, Englewood Cli€s,

NJ.

Burhardt, K.K., 1971. Urban trac system optimization. Ph.D. Dissertation, University of Minnesota, Minneapolis, Minnesota.

Cronje, W.B., 1983. Optimization model for isolated signalized trac intersections. Transportation Research Record 905, 80±83.

Dans, G., Gazis, D.C., 1976. Optimal control of oversaturated store-and-forward transportation networks. Transportation Science 10 (1), 1±19.

Elahi, S.M., Radwan, A.E., Goul, K.M., 1991. Knowledge-based system for adaptive trac signal control. Transportation Research Record 1324, 115±122.

Gartner, N.H., Stamatiadis, C., Tarno€, P.J., 1995. Development of advanced trac signal control strategies for intelligent transportation systems: Multilevel design. Transportation Research Record 1494, 98±105.

Gazis, D.C., 1964. Optimal control of a system of oversaturated intersections. Operations Research 12, 815±831. Gazis, D.C., Potts, R.B., 1965. The oversaturated intersection. In: Proceedings of the Second International Symposium

on the Theory of Road Trac Flow. Organization for Economic Cooperation and Development, Paris, pp. 221±237. Green, D.H., 1968. Control of oversaturated intersections. Operational Research Quarterly 18 (2), 161±173.

1985. Highway capacity manual. Special Report 209. Transportation Research Board, National Research Council. Kaltenbach, M., Koivo, H.N., 1974. Modelling and control of urban trac ¯ow. In: Proceedings of the Joint

Automatic Control Conference. University of Texas, Houston, Texas, pp. 147±154. Kuo, B.C., 1991. Automatic Control Systems, 6th ed. Prentice-Hall, Englewood Cli€s, NJ.

Luenberger, D.G., 1979. Introduction to Dynamic Systems: Theory, Models, & Applications. Wiley, Chichester. May, A.D., Jr., 1965. trac ¯ow theory ± the trac engineer's challenge. In: Proceedings of the Institute Trac

Engineering. pp. 290±303.

Michalopoulos, P.G., Stephanopolos, G. Oversaturated signal system with queue length constraints-I. Transportation Research 11, 413±421.

Michalopoulos, P.G., Stephanopolos, G., May 1978. Optimal control of oversaturated intersections theoretical and practical considerations. In: Trac Engineering & Control. pp. 216±221.

Newell, G.F. 1982. Applications of Queuing Theory, 2nd ed. Chapman & Hall, London, England, pp. 287±300. SOAP 84, 1985. User's Manual. Federal Highway Administration.

Tarno€, P.J., Parsonson, P.S., 1981. Selecting trac signal control at individual intersections. NCHRP Report 233, TRB.

Webster, F.V., 1958. Trac signal settings. road research technical paper, No.39, Great Britain Road Research Laboratory, London.