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Akcelik & Associates Pty Ltd

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REPRINT

Time-Dependent Expressions for Delay, Stop

Rate and Queue Length at Traffic Signals

R. AKÇELIK

REFERENCE:

AKÇELIK, R. (1980). Time-Dependent Expressions for Delay, Stop Rate and Queue

Length at Traffic Signals. Internal Report AIR 367–1. Australian Road Research

Board, Vermont South, Australia.

NOTE:

This report is related to the intersection analysis methodology used in the SIDRA

INTERSECTION software. Since the publication of this report, many related aspects of the

traffic model have been further developed in later versions of SIDRA INTERSECTION.

Though some aspects of this report may be outdated, this reprint is provided as a record of

important aspects of the SIDRA INTERSECTION software, and in order to promote software

assessment and further research. This report was originally published by the Australian

Road Research Board.

©

Akcelik and Associates Pty Ltd / www.sidrasolutions.com

PO Box 1075G, Greythorn Victoria 3104, Australia

Email: info@sidrasolutions.com

ARRB INTERNAL REPORT

AIR 367-1

October 1980

TIME – DEPENDENT EXPRESSIONS FOR DELAY, STOP RATE AND

QUEUE LENGTH AT TRAFFIC SIGNALS

by

R. AKÇELIK

Senior Research Scientist

367 Traffic Signal Control Techniques

This reprint has been prepared by retyping the report including some minor revisions and reformatting.

Rahmi Akçelik

14 September 2011

AUSTRALIAN ROAD RESEARCH BOARD

INTERNAL REPORT

500 BURWOOD HIGHWAY

VERMONT SOUTH, VICTORIA

AIR 367-1 Akçelik, R.

REPORT SUMMARY

THE PURPOSE OF THIS REPORT

Is to present new expressions for delay, number of stops and queue length at traffic signals operating in a fixed-

time mode so as to provide simple working models to predict traffic operating characteristics in both

undersaturated (below capacity) and temporarily oversaturated (above capacity) conditions.

THIS REPORT SHOULD INTEREST

traffic engineers and researchers concerned with the design and operation of traffic signals.

THE MAJOR CONCLUSION OF THE REPORT ARE

(a) the limitations of the existing formulae for delay, number of stops and queue length based on the steady-

state (equilibrium) flow assumptions for undersaturated signals and the deterministic formulae for

oversaturated signals can be overcome by using a co ordinate transformation technique; the use of the

overflow queue concept provides a good basis for this purpose;

(b) the new expressions predict finite values of delay, queue length and number of stops for flows near

capacity with a smooth transition from below-capacity to above-capacity conditions; this is the contrast

with the existing steady-stare expressions for undersaturated signals which predict infinite values for

flows just below capacity and the deterministic expressions for oversaturated signals which predict zero

(or very small) values for flows just above capacity; the difference is important since most signals

operate under near-capacity conditions during peak periods of traffic.

AS A CONSEQUENCE OF THE WORK REPORTED, THE FOLLOWING ACTION IS RECOMMENDED

(a) incorporate the new expressions (simplified forms) given in this paper in the relevant section of the

revised Signalised Intersection Capacity and Timing Guide (to replace ARR 79);

(b) extend to the method described to develop more comprehensive formulae to predict traffic operating

characteristics in consecutive time periods;

(c) study overflow queues at co-ordinated and vehicle actuated traffic signals; and

(d) study the optimum signal timings derived using the new expressions in comparison with the timings

derived using the traditional (Webster, Miller,etc.) expressions.

RELATED ARRB RESEARCH

P312 Effects of Traffic Patterns on Fuel Consumptions and Emission

P331 Analysis of SCATS

P347 Signalised Intersection Survey Method

FS1068 Australian Traffic Signalling

FS1094 Signalised Intersection Capacity and Timing Guide Revision

INFORMATION RETRIEVAL

AKÇELIK, R. (1980): TIME-DEPENDENT EXPRESSIONS FOR DELAY, STOP RATE AND QUEUE LENGTH

AT TRAFFIC SIGNALS. Australian Road Research Board. Internal Report AIR 367-1.

KEYWORDS : Traffic signal / traffic control / junction / design (overall design) / traffic flow/ capacity (road,

footway) / congestion (traffic) / delay / queue / stops* / fixed time (signals) / linked signals / mathematical model

ABSTRACT : This paper presents new approximate expressions for predicting delay, stop rate and queue

length at traffic signals operating in a fixed-time mode for flow conditions which last for a finite period of time.

The emphasis is on the use of the overflow queue concept which facilitates a better understanding of the

transition from undersaturated conditions to oversaturated conditions, and allows for the derivation of simpler

expressions which apply to both conditions. Firstly, deterministic expressions for delay, stop rate and queue

length are derived for oversaturated conditions. Steady-state expressions for undersaturated signals are

summarised. Then the limitations of steady-state and deterministic expressions for near-capacity conditions are

discussed. Transition functions which overcome these limitations are given. These functions have been derived

using the Whiting / Kimber-Hollis co-ordinate transformation technique. Each one of the new expressions for

delay, stop rate and queue length is presented in a simple form consisting of a uniform component and an

overflow component. The latter is an explicit function of the average overflow queue. Transition functions for the

average overflow queues at isolated as well as co-ordinated signals are presented.

*

Non-IRRD keywords

ACKNOWLEDGEMENTS: The author would like to thank Mr. D.I. Robertson of the Transport and Road

Research Laboratory, U.K., Prof. K. Ohno of Kyoto University, Japan, and Dr. A.J. Miller of the Commonwealth

Scientific and Industrial Research Organisation, Australia for their valuable comments on subjects related to

various aspects of this paper.

Akçelik, R. AIR 367-1

CONTENTS

Page

1. INTRODUCTION ............................................................................................................... 1

2. DETERMINISTIC EXPRESSIONS FOR OVERSATURATED SIGNALS ......................... 2

3. STEADY-STATE EXPRESSIONS FOR UNDERSATURATED SIGNALS ....................... 6

4. TRANSITION FUNCTIONS ............................................................................................... 6

4.1 Average Overflow Queue ........................................................................................... 8

4.2 Total Delay ............................................................................................................... 10

4.3 Stop Rate .................................................................................................................. 12

4.4 Average Queue Length at the Start of the Green Period ......................................... 12

5. SIMPLIFIED EXPRESSIONS ......................................................................................... 13

6. CO-ORDINATED SIGNALS ............................................................................................ 17

7. CONCLUSION ................................................................................................................. 17

REFERENCES ................................................................................................................ 18

APPENDIX -NOTE ON VARIOUS DELAY EXPRESSIONS ........................................... 19

AIR 367-1 1

1. INTRODUCTION

The normal analysis of congestion is as a steady-state phenomenon potentially in long-run equilibrium.

... For an intersection there might be a flow/delay relation which will relate an average equilibrium delay

to a flow which will be expected to continue for some time without change. (This type of relation is) often

shown to rise to infinite time as the rate of flow approaches some finite value. Clearly so long as the

steady state assumption is retained, that the same rate of flow might reasonably be supposed to

continue forever, this is just what would happen, but as flow approaches “capacity” this postulate

becomes increasingly implausible. In such a situation the standing queue increases steadily in length

and the time required to reach the steady state becomes longer, until on the extreme, there will be an

infinite queue and it will take an infinite time to reach the steady state if flow is actually equal to or greater

than “capacity”. In fact (congestion) is always a peaked phenomenon lasting only for a limited length of

time. Failure to take account of the temporary nature of (congestion) can lead to analytical problems.

(Neuburger 1971)

Various expressions for predicting the queue length, delay and stops at isolated, undersaturated

signals were discussed in a paper by the author (Akçelik 1980). It was shown that, as in the case of

delay and queue length, an expression to predict the stop rate (average number of stops per vehicle)

could be derived using the overflow queue concept. Miller’s (1968) approximate expression for the

average overflow queue, i.e. the average number of vehicles left in the queue at the end of the green

period, at isolated undersaturated signals, was recommended for use in the stop rate formula because

of its explicit and direct formulation. It was also shown that an equivalent average overflow queue

value could be calculated using the queue length prediction method given by Webster (1958). The

importance of the average overflow queue parameter in various expressions for delay at

undersaturated signals, and other discussions on the overflow queue concept can be found in Haight

(1959), Miller (1964), Allsop (1972), McNeil and Weiss (1974), Ohno (1978) and Sosin (1980).

It appears that there has been an over-emphasis on some detailed aspects of the theoretical

formulations of delay at undersaturated signals without due consideration to their practical relevance.

As aptly stated by Allsop (1972):

‘The appreciation of the relevance of these theoretical probabilistic arrival models was valuable in

enabling probability theory to be applied to the analysis of the traffic signal queue. The mathematical

fascination of such analyses has, however, caused them to be pursued to a level of intricacy far greater

than is likely to find practical application. The behaviour of real traffic does not correspond closely

enough to any of these very specific mathematical models to warrant, for practical purposes, the very

refined solutions that have been obtained.

A major limitation of the expressions given in the papers referred to above is that they are valid for

steady-state (equilibrium) conditions only. This means that the demand flow is, on average, less that

the capacity available, i.e. undersaturated conditions apply. Following Miller’s (1968) suggestion about

the accuracy of his average overflow queue formula, the formulae given in Akçelik (1980) were

considered to give good approximation for degrees of saturation (flow / capacity ratios) up to 0.95. The

steady-state expressions are independent of the length of time period they are applied to. However, in

oversaturated conditions, which last for a finite period of time, it becomes important that the analyses

are based on the use of time-dependent flow and timing parameters.

There is now a growing amount of literature on temporarily oversaturated signals (e.g. Rahman 1973,

Gazis 1974, Catling 1977, Yagar 1977, Branston 1978, Pignataro, et al. 1978, Mayne 1979,

Michalopoulos 1979, Roberston 1979, Kimber and Hollis 1979, Luk and Sims 1979, Michalopoulos

and Meyer 1980). Formulae given in some of these publications for predicting delays and queue

lengths appear to be mathematically involved, especially those for consecutive time periods (e.g. see

Mayne 1979). A simpler formulation for delay at isolated signals is adopted by Robertson (1979). This

is based on a co-ordinate transformation (or, transition curve) technique which was conceived by

Whiting for use in TRANSYT program and developed in detail for a variety of queuing situations by

Kimber and Hollis (1979).

2 Akçelik, R.

The purpose of this paper is to present new expressions for delay, stop rate and queue length which

have been derived using the co-ordinate transformation technique. The emphasis in the new

expressions is on the use of the overflow queue concept. This facilitates a better understanding of the

transition from undersaturated (steady-state) conditions to (temporarily) oversaturated conditions, and

allows for the derivation of simpler expressions which apply to both conditions.

The expressions given in this paper are intended for practical use in a way similar to the use of the

traditional formulae by Webster (1958) and Miller (1968) which were limited to undersaturated cases.

For this reason, the formulae given in this paper are based on the consideration of a single time period

with zero initial queue.

A major contribution of this paper is considered to be presenting expressions to predict the stop rate at

traffic signals. The importance of stops as a measure of performance has been discussed elsewhere

(Akçelik 1980, 1981a).

The expressions for delay, stop rate and queue length are derived below for oversaturated conditions

using a deterministic approach first. Steady-state expressions for undersaturated signals are

summarised. Then the limitations of steady-state and deterministic expressions for near capacity

conditions are considered and the co-ordinate transformation technique is utilised to obtain transition

functions which are of use in both below and above capacity conditions. Simplified forms of these

functions are presented for practical use. These expressions have been used in the new Australian

Guide for Traffic signal capacity and Timing Analysis (Akçelik 1981b).

2. DETERMINISTIC EXPRESSIONS FOR OVERSATURATED

SIGNALS

Methods to derived deterministic expressions for delay and queue length at oversaturated signals, i.e.

when the average arrival flow rate exceeds capacity, are given in earlier publications by May (1965),

May and Keller (1976), Neuburger (1971) and Gazis (1974). The basic model presented below

assumes a constant arrival (demand) flow rate, q, which persists during a time interval of length, t. It

also assumes, in the deterministic sense, that the queue length at the start of this time interval is zero.

The capacity, Q = s g / c (s = saturation flow, g / c = effective green time / cycle time ratio), which is

constant as determined by the fixed-time operation of signals, is exceeded by the arrival flow rate, q,

by an amount equal to z

Q = q - Q (where z = x - 1, x = q / Q is the degree of saturation, i.e. flow /

capacity ratio). Thus, the oversaturation queue is assumed to increase linearly from zero to a

maximum value of (z Q t) at the end of the time interval, t. Provided the arrival flow rate in the

subsequent interval is less than the capacity (q

2

< Q), the oversaturated queue will be cleared at time

t

2

= (z / z

2

) t after the end of interval t, where z

2

= 1 - x

2

> 0 and x

2

= q

2

/ Q < 1.0.

In the derivations below, q

2

= 0, and hence t

2

= z t = (x - 1) t, is used so as to exclude the delays and

stops to vehicles which arrive during the subsequent period.

The basic model is illustrated in Figure 1. The cumulative numbers of arrivals and departures are

shown in the top part of Figure 1 for the following example:

cycle time, c = 120 s, effective green time, g = 30 s,

arrival flow, q = 360 veh/h, arrivals per cycle, q c = 12 veh, saturation flow,

s = 1200 veh/h, departures per cycle, s g = 10 veh, capacity,

Q =300 veh/h, degree of saturation, x = 1.2, z = x - 1 = 0.2, and

excess flow, z Q = q - Q = 60 veh/h (q c - sg = 2 veh per cycle)

A time interval of t = 10 min (5 cycles) is considered. The queue at the end of interval t is z Q t =

10 veh, and the time to clear this queue is t

2

= z t = 0.2 x 10 = 2 min (1 cycle).

The following can be seen in the top part of Figure 1:

(a) The area between the cumulative arrival curve (linear with slope q) and departure curve

(sawtooth with zero slope during the red period and slope s during the green period) represents

the total delay. The horizontal distance between the two curves represents the delay to an

individual vehicle (e.g. AA' = 116 s for vehicle 23).

(b) The vertical distance between the two curves represents the queue length (number of vehicles in

queue) at a given time (e.g. BB'). The overflow queue is the queue at the end of a signal cycle

(e.g. CC’ = 8 veh at the end of the fourth cycle).

AIR 367-1 3

Figure 1 - Deterministic model of oversaturated signals (an example)

4 Akçelik, R.

The idealised time-distance trajectories of individual vehicles are shown in the bottom part of Figure 1.

The assumption of the uniform arrivals (constant headways, 1 / q) is used as in the case of deriving

the uniform component of delay, stops and queue length at undersaturated signals (Akçelik 1980).

The relationship between the overflow queue and multiple stops, i.e. more than one stop per vehicle,

can be observed from Figure 1. For example, an overflow queue of 2 vehicles at the end of the first

cycle corresponds to vehicles 11 and 12 which make two stops before clearing the intersection.

Similarly, an overflow queue of 4 vehicles at the end of the second cycle corresponds to vehicles 21 to

24 which are stopped twice, and so on (initial stops by vehicles 13 and 25, 26 in the first two cycles

are neglected as explained below).

General expressions for the overflow queue, delay and stops in an oversaturated cycle are given

below using the parameters shown in Figure 2.

Overflow queue at the end of the ith cycle:

n

i

= n

i-1

+ q c + s g (1)

Total delay during the ith cycle:

D

i

= n

i-1

c + 0.5 (q c

2

+ s g

2

) (2)

Total number of stops during the ith cycle:

H

i

= n

i-1

+ q c (3)

Theoretically, the number of stops per cycle is (n

i-1

+ q r) / (1 - y), where y = q / s (flow ratio).

Equation (3) gives a smaller number of stops because it neglects the stops made by vehicles which

arrive at the back of a moving queue from the previous cycle, i.e. only the stops by (q c) vehicles

arriving in a cycle are accounted for. For example, the initial stops by vehicle 13 in the second cycle,

vehicles 25 and 26 in the third cycle, vehicles 37 to 39 in the fourth cycle, etc., are neglected (shown

by dotted lines in the bottom part of Figure 1). This assumption was made so as to compensate for

drivers who may adjust their speed to avoid a stop when faced with a red signal and moving queue

ahead.

Figure 2 - An oversaturated signal cycle

Q r

AIR 367-1 5

The following deterministic expressions (subscript d is used to indicate this) for the average overflow

queue, total delay and stop rate as experienced by (q t) vehicles during the period t + t

2

= x t

(in x t / c cycles) are derived from Equations (1) to (3).

Average overflow queue is given by:

N

d

= 0.5 (q - Q) t = 0.5 z Q t (4)

where

N

d

= average overflow queue in vehicles total number of vehicles queued in all lanes),

q = arrival flow which persists during the time interval t (q in vehicles per hour, t in hours), and

Q = capacity (in vehicles per hour).

Total delay is given by:

D

d

= 0.5 q r + N

d

x (5)

where

D

d

= total delay (in vehicle-hours per hour, or simply ‘vehicles’), and

q r = total number of vehicles which arrive during the red period (q = flow in vehicles per second, r =

effective red time in seconds)

The average delay per vehicle can be calculated from d = D / q where q is in vehicles per second.

Stop rate is given by:

h

d

= 1 + N

d

/ (s g) (6)

where

h

d

= stop rate (average number of stops per vehicle), and

s g = total number of departures (capacity) per cycle (s = saturation flow in vehicles per second,

g = effective green time in seconds, s g in vehicles).

The total number of stops can be calculated from H = h q, where H is in the same units as q.

Also of interest is the average queue length at the start of the green period. This is given by:

N

dr

= Q r + N

d

(7)

where Q, r, N

d

are as in the above expressions. The maximum (stationary) queue length occurs in the

last cycle during the time interval t (DD' in Figure 1), and is given by N

c

= 2 N

d

+ (s - q) g.

It is seen that Equations (5), (6) and (7) consist of two components:

(a) A uniform component which corresponds to the triangular area OAB in Figure 2 (which is seen to

be a fixed amount in each cycle in Figure 1). This is the at-capacity value of the statistic

concerned, and can be obtained from the uniform component of the expression for under-

saturated signals (Equations 9 to 11 given below) by putting x = 1 (u = y, or q = Q).

(b) A deterministic overflow component which corresponds to the area OBCD in Figure 2, and is

expressed as a function of the average overflow queue given by Equation (4). It can be shown

that the overflow components of average delay (d

o

= N

o

x / q) and stop rate (h

o

= N

o

/ (s g)) are

related as (h

o

= d

o

/ c) where c is the cycle time.

It should be noted that z > 0 and x > 1 in Equations (4) to (7). It is also important to note that N

d

, and

hence the overflow component of each expression, is a linear function of the time interval, t.

The following values are calculated from above expressions for the example shown in Figure 1

(using t = 1 / 6 h):

average overflow queue, N

d

= 5.0 veh,

total delay, D

d

= 10.5 veh-h/h,

average delay, d

d

= 105.0 sec,

stop rate, h

d

= 1.5 stops/veh

total number of stops, H

d

= 540 stops per hour,

average queue at the start of green, N

dr

= 12.5 veh, and

maximum queue, N

c

= 17.0 veh.

6 Akçelik, R.

3. STEADY-STATE EXPRESSIONS FOR UNDERSATURATED

SIGNALS

It was emphasised in an earlier publication (Akçelik 1980) that the expressions for delay, stop rate and

queue length at undersaturated signals (i.e. when the average arrival flow rate is below capacity) are

interrelated, and that each could be considered to have a uniform component and a random

component. The random component was expressed as a function of average overflow queue.

At undersaturated signals the overflow queues are due to random variations in arrival flow rates which

result in some cycles being oversaturated. However, because the arrival flow rate is, on average,

below capacity, these queues will be discharged in subsequent cycles. The average overflow queue

parameter represents the steady-state assumption which is the basis of the delay, stop rate and

queue length expressions given in Akçelik (1980). These are summarised below (subscript s is used

to denote steady-state).

Miller’s (1968) formula for predicting the average overflow queue at undersaturated fixed-time signals

is:

N

s

= 0.5 {exp [-1.33 sqrt (s g) (1 - x) / x]} / (1 - x) (8)

It is interesting to note that the numerical values of the average overflow queue given by

Wormleighton (1965) are very close to those given by Equation (8).

The Miller (1968) formula for delay is based on an explicit use of the average overflow queue

parameter, N

s

, given by Equation (8)

D

s

= 0.5 q c (1 - u)

2

/ (1 - y) + [(1 - u) / (1 - y)] N

s

(9)

where

u = green time ratio (effective green time /cycle time, g/c), and

y = flow ratio (flow/saturated flow, q/s).

Although a more complicated expression is possible as proposed by Ohno (1978), the accuracy of

Equation (9) is considered to be sufficient for all practical purposes (see the Appendix ).

The steady-state expression for stop rate is:

h

s

= f [(1 - u) / (1 - y) + N

s

/ (q c)] (10)

where f is a correction factor to allow for partial stops (slowdowns). A general purpose value of f = 0.9

was recommended in Akçelik (1980).

The average queue length at the start of green period is given by:

N

sr

= q r + N

s

(11)

where N

s

is given by Equation (8), and (q r) is the number of vehicles in queue at the start of green

period if uniform arrivals were assumed (q = average arrival flow rate in vehicles per second,

r = effective red time in seconds).

4. TRANSITION FUNCTIONS

Let us now consider how the two sets of expressions given above, namely, Equations (4) to (7) for

oversaturated conditions, and Equations (8) to (11) for undersaturated conditions, can be related

together. The limitations of these two sets of expressions in predicting traffic operating characteristics

for near-capacity conditions is discussed below in terms of the average overflow queue expressions.

The same considerations apply to the delay, stop rate and queue length expressions.

As seen in Figure 3, which shows that average overflow queues predicted by Equations (4) and (8),

for the numerical example given by Robertson (1979), a zero overflow queue value is predicted by the

deterministic expression for a flow rate equal to the capacity whereas the steady-state ( probabilistic )

expression predicts infinite overflow queue values for flows just below capacity. This indicates clearly

that neither set of equations given in the preceding sections can be expected to give reasonable

results for flows near capacity.

AIR 367-1 7

Figure 3 - Prediction of the average overflow queue by steady-state (x < 1) and

deterministic (x > 1) expressions

Figure 4 - Uniform, random and overflow components of a general performance function

8 Akçelik, R.

The problem is solved using the co-ordinate transformation technique described in detail by Kimber

and Hollis (1979). The technique was originally conceived by P.D. Whiting for deriving a ‘random

delay’ expression for the TRANSYT computer program (Robertson and Gower 1977, Robertson 1979,

Robertson, et al 1980). This technique transforms the steady-state function to a transition function

which has the line representing the deterministic function as its asymptote. The result is not only to

obtain realistic finite values of the performance function concerned for flows around capacity, but also

to add a random component to the deterministic oversaturation function.

Figure 4 shows a general performance function, P, which represents any one of the delay (total or

average, D, d) , stops (rate or total number, h, H), or the average queue length (N) functions. This can

be expressed as:

P = P

u

+ P

o

(12)

where

P

u

= uniform component, and

P

o

= overflow component.

The uniform component, P

u

, corresponds to the first terms of Equations (5) to (7) for x > 1, and

Equations (9) to (11) for x < 1. As shown in Figure 4, the uniform component is constant for x > 1, and

is equal to the value at capacity, P

u

(x = 1 ).

The overflow component can be expressed as a function of the average overflow queue, P

o

= f(N

o

), as

in the second terms of Equations (5) to (7) and Equations (9) to (11). The value of N

o

can be

predicted by a transition function as explained below. As shown in Figure 4, for flows above capacity

(x > 1), the overflow component, P

o

, is the sum of a deterministic component, P

d

, and a random

component, P

r

. For flows below capacity (x < 1), the overflow component is equivalent to the random

component, P

o

= P

r

since P

d

=0. It is also seen in Figure 4 that the overflow component is zero below

a certain degree of saturation (x < x

o

) since N

o

≅ 0 as predicted by Equation (8).

4.1 AVERAGE OVERFLOW QUEUE

In order to facilitate the derivation of a transition function for the average overflow queue, the following

expression has been developed by the author for undersaturated signals, which is a simple

approximation to Miller’s steady-state expression (Equation 8):

N

s

= 1.5 (x - x

o

) / (1 - x) for x > x

o

(13)

= 0 x ≤ x

o

where x

o

is a degree of saturation below which the average overflow queue is approximately zero, and

is given by:

x

o

= 0.67 + s g / 600 (13a)

where (s g) is in vehicles.

Two other steady-state expressions for the average overflow queue at undersaturated signals are of

interest. McNeil’s (1968) upper-bound expression (for Poisson arrivals):

N

s

= 0.5 / (1 - x) (14)

and Miller’s (1963) approximate expression:

N

s

= (x - 0.5) / (1 - x) for x > 0.5 (15)

= 0 x ≤ 0.5

This is also considered to be an upper-bound expression, it is simply 0.5 less than the value given by

Equation (14) for x > 0.5.

AIR 367-1 9

Figure 5 - Average overflow queues predicted by various steady-state (x < 1) expressions

The overflow queues for undersaturated conditions predicted by Equations (8), (13), (14), and (15) are

shown in Figure 5. It is important to note that:

(a) Equation (14) predicts non-zero overflow queues for all degrees of saturation, whereas the other

three expressions predict zero overflow queues for degrees of saturation below a certain value,

and

(b) Equations (14) and (15) are not dependent on the (s g) parameter which is an indicator of the

relative importance of various movements at an intersection (high s g values for major

movements and low sg values for minor movements).

It appears that these two effects combined may have some impact on signal timing optimisation when

there are marked differences between major and minor movement characteristics.

The transition function for average overflow queue which is obtained using the co-ordinate

transformation technique to relate Equation (13) to Equation (4) is:

N

o

= 0.25 Q t [z + sqrt (z

2

+ 12 (x - x

o

) / (Q t))] for x > x

o

(16)

= 0 x ≤ x

o

where

Q t = throughput, i.e. the maximum number of vehicles which can be discharged during the time

interval, t (Q is the capacity in vehicles per hour, t in hours),

z = x - 1 (x = q / Q, degree of saturation, q is the average arrival flow rate which persists during

the time interval, t),

x

o

= 0.67 + s g / 600 as in Equation (13).

10 Akçelik, R.

It should be noted, for the correct use of Equation (16), that the parameter z has a negative value for

flows below capacity (x < 1).

It should also be noted that the derivation of Equation (16), as well as other transition functions given

in this section, involve some minor simplifying assumptions.

The transition function for average overflow queue based on Equation (14) is:

N

o

= 0.25 Q t [z + sqrt (z

2

+ 4 x / (Q t))] (17)

This expression is of interest because of its relevance to the delay function used in the TRANSYT

computer program (Robertson and Gower 1977). As in Equation (14), this function predicts non-zero

overflow queues for all degrees of saturation, and can be considered to be an upper-bound

expression.

The difference between the average overflow queue functions obtained from Equations (16) and (17)

can be seen in Figure 6 which illustrates examples for s g = 10 and s g = 80 when Q t = 30 s g

(e.g. for t = 1 h, c = 120 s).

4.2 TOTAL DELAY

To derive a transition function for total delay, let us replace random delay term of Equation (9) by the

following approximate term:

D

r

= ((1 - u) / (1 - y)) N

s

≅ N

s

x (18)

With this approximation, if N

o

is predicted by the upper-bound expression, Equation (14), the random

delay is:

D

r

= 0.5 x

2

/ (1 - x) (19)

This is the second term of Webster’s (1958) delay formula.

Since Equation (18) is the same as second term of Equation (5), the transition function is also of the

same form. Therefore, total delay can be expressed as:

D = D

u

+ D

o

(20)

where the uniform delay component is:

D

u

= 0.5 q c (1 - u)

2

/ (1 - y) for x < 1 (20a)

= 0.5 q r x ≥ 1

and the overflow component is:

D

o

= N

o

x (20b)

where N

o

is given by Equation (16).

If N

o

given by Equation (17) is used, D

o

corresponds to the transition function given by Robertson

(1979) which was obtained directly from Equation (19). In this case, Equation (20) can be considered

to be an upper-bound expression for total delay.

It should be noted that the approximation introduced in Equation (18) is relevant to degrees of

saturation, x, in the range between about 0.65 and 0.95 because N

o

is zero below about 0.65 (or x

o

in

Equation (16)) and the steady-state expressions are valid for x up to 0.95. The effect of this

approximation is considered to be negligible for most practical purposes (see the Appendix).

AIR 367-1 11

Figure 6 - Average overflow queues predicted by two transition functions

12 Akçelik, R.

4.3 STOP RATE

The following transition function has been derived for the overflow component of stop rate by relating

the second terms of Equations ( 10 ) and ( 6 ):

h

o

= 0.25 (Q t / (s g)) [z + sqrt (z

2

+ 12 (x - x

o

) / (q t))] for x > x

o

(21)

= 0 x ≤ x

o

Therefore, the stop rate function can be expressed as:

h = f (h

u

+ h

o

) (22)

where the uniform component is:

h

u

= (1 - u) / (1 - y) for x < 1 (22a)

= 1.0 x ≥ 1

the overflow component h

o

is given by Equation (21), and f is the correction factor for partial stops as

in Equation (10).

The overflow component of stops rate based on the use of N

o

calculated from Equation (17), is the

same as Equation (21) except that 12 (x - x

o

) under the square root must be replaced by (4 x). If used

in Equation (22), this could be considered to give an upper-bound expression for stop rate.

4.4 AVERAGE QUEUE LENGTH AT THE START OF THE GREEN PERIOD

The average number of vehicles in the queue at the start of the green period (at the end of red period)

is given by:

N

r

= N

u

+ N

o

(23)

where the uniform component is

N

u

= q r for x < 1 (24)

= Q r x ≥ 1

and the overflow component, N

o

is simply the average overflow queue given by Equation (16), or

Equation (17).

Equation (23) gives the maximum stationary stop-line queue in an average cycle. The maximum

queue in any cycle can be calculated roughly as twice as the value of N.

The maximum back of the queue occurs sometime after the start of green as seen at the bottom part

of Figure 1. The average value of this queue can be estimated from

N

m

= (q r / (1 - y)) + N

o

(25)

See Equation (15) in Akçelik (1980).

N

m

is the physical end of the queue as perceived by the drivers. While the back of the queue increases

from N

r

at the start of green period (Equation 23) to N

m

at some time after the start of green period, the

front of the queue will be moving forward. At the time when N

m

is reached, all vehicles will be moving,

some beyond the stop line, i.e. will have already cleared the intersection.

AIR 367-1 13

5. SIMPLIFIED EXPRESSIONS

Equations (20) to (23) may appear to be a little cumbersome for use in practice. The condition of a

fixed value of the uniform component for x > 1 in these expressions can be relaxed at the expense of

obtaining slightly higher values of the performance function for oversaturated conditions. In the case of

the stop rate expression a further simplification is introduced which compensates for this effect. The

following expressions are recommended for practical use:

The total delay:

D = 0.5 q c (1 - u)

2

/ (1 - y) + N

o

x (26)

The stop rate:

h = f ((1 - u) / (1 - y) + N

o

/ (q c )) (27)

The average (stop-line) queue at the start of the green period:

N

r

= q r + N

o

(28)

It is seen that these expressions have the same form as the steady-state expressions for

undersaturated conditions (Equations 9 to 11), except for the second term Equation (26) due to the

approximation given by Equation (18). Of course, the difference is that the average overflow queue,

N

o

, which determines the overflow components of the above expressions, is to be calculated from the

transition function given by Equation (16), or alternatively from Equation (17) for an upper-bound

expression.

Equation (27) provides a convenient method of estimating stop rate at signals since separate

calculation of an overflow component, h

o

, from Equation (21) is avoided.

Figures 7 to 9 illustrate the average delay (d = D / q), stop rate, h, and average queue at the start of

the green period, N

r

, obtained from Equations (20) to (23), with N

o

calculated from Equation (16), as a

function of the degree of saturation, x. These diagrams are for the data used in an example by

Robertson (1979). The overflow components of each function (curve T) is shown separately at the top

part of each diagram together with the corresponding steady-state (curve S) and deterministic

oversaturation (line D) functions. The stops shown in Figure 8 are uncorrected for partial stops, i.e. not

multiplied by factor f.

The values of delay, stop rate and queue length obtained from simplified expressions (Equations 26

to 28) are shown in Figures 7 to 9 as dots in the bottom part of each diagram for the cases where

there is a difference from those given by respective Equations (20) to (23).

14 Akçelik, R.

Figure 7 - Average delay as a function of the degree of saturation

AIR 367-1 15

Figure 8 - Stop rate as a function of the degree of saturation

16 Akçelik, R.

Figure 9 - Average queue length at the start of green period as a function of the degree of saturation

r

AIR 367-1 17

6. CO-ORDINATED SIGNALS

It is interesting to note that a transition function was first used for co-ordinated signals rather than

isolated signals (Robertson and Gower 1977, Robertson et al. 1980). The evidence given in

Robertson (1969, App. 2) indicates that the effect of random variations in flow levels from cycle to

cycle is smaller at co-ordinated signals that at isolated signals, probably because of the structured

nature of arrival and departure patterns. In a way similar to the assumption made by Robertson

(1969) for deriving a random delay term for the TRANSYT program, it may be assumed that the

average overflow queue at undersaturated co-ordinated signals is half the average overflow queue at

undersaturated isolated signals with the same x and (s g) values. Although research is needed to

substantiate this, the assumption provides a reasonable working model in the meantime.

The transition function for average overflow queue at co-ordinated signals is then given by:

N

o

= 0.25 Q t [z + sqrt (z

2

+ 6 (x - x

o

) / (Q t))] for x > x

o

(29)

= 0 x ≤ x

o

This expression corresponds to Equation (16), the only difference is that the constant 12 under square

root is replaced by 6. The expression which corresponds to Equation (17) is obtained similarly by

replacing the constant 4 under the square root by 2:

N

o

= 0.25 Q t [z + sqrt (z

2

+ 2 x / (Q t))] (30)

The overflow components of delay, stop rate and queue length at co-ordinated signals have the same

form as those in Equations (26) to (28), i.e. (N

o

x), (N

o

/ qc) and (N

o

), respectively, with N

o

given by

Equation (29), or alternatively Equation (30).

The uniform components (D

u

, h

u

, N

u

) can be calculated using a deterministic simulation model such as

the TRANSYT model. Simplified analytical expressions can also be used for this purpose, e.g. as in

MITROP computer program (Gartner, et al 1975). It is recommended that the overflow components of

stop rate and average queue length as described above are added to the stop rate and queue length

calculations of the TRANSYT program. The correction to stops would allow for multiple stops under

heavy traffic conditions, and would probably affect the optimum timing results.

7. CONCLUSION

The purpose of developing the time-dependent expressions given in this paper was to provide simple

working models to predict traffic operating characteristics at signals, rather than obtain theoretically

perfect but inevitably more complicated models. The expressions given in this paper are intended for

use in a fashion similar to the use of the traditional Webster (1958) and Miller (1968) formulae which

are limited to undersaturated cases.

It is possible to use the basic relationship given here to derive more comprehensive expressions to

predict conditions in consecutive time periods. Such expressions will appear to be more complicated

than those given in this paper, but can be implemented by means of a computer algorithm (e.g. see

Mayne 1979).

A study of the optimum signal timings obtained using the time-dependent expressions given in this

paper compared with those given by the Webster (1958) and Miller (1968) formulae is in progress.

18 Akçelik, R.

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Control 22 (2), pp. 54-65.

AKÇELIK, R. (1981b). Traffic Signals: Capacity and Timing Analysis. Australian Road Research

Board. Research Report ARR No. 123.

ALLSOP, R.E. (1972). Delay at a fixed time traffic signal – I: Theoretical analysis. Transp. Sci. 6 (3),

pp. 260-285.

BRANSTON, D. (1978). A comparison of observed and estimated queue lengths at oversaturated

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CATLING, I. (1977). A time-dependent approach to junction delays. Traffic Eng. Control 18 (11), pp.

520-523, 526.

GARTNER, N.H., LITTLE, J.D.C. and GABBY, H. (1975). Optimisation of traffic signal settings by

mixed-integer linear programming. Transp. Sci. 9 (4), pp. 321-363.

GAZIS, D.C. (1974). Oversaturated systems. In: GAZIS, D.C. (Ed.) Traffic Science, pp. 209-229

(Wiley: New York).

KIMBER, R.M. and HOLLIS, E.M. (1979). Traffic Queues and Delays at Road Junctions. Transp.

Road Res. Lab. (U.K.) TRRL Lab. Rep. LR 909.

LUK, J.Y.K. and SIMS, A.G. (1979). Oversaturated intersection control - a review. In: R. AKÇELIK

(Ed.) Signalised Intersection Capacity Workshop – Papers, Part 7. Australian Road Research Board,

Internal Report AIR 1094-1.

MAY, A.D. (1965). Traffic flow theory - the traffic engineer’s challenge. Proc. Inst. Traffic Eng. pp. 290-

303.

MAY, A.D. and KELLER, H.E.M. (1967). A deterministic queuing model. Transp. Res. 1 (2), pp. 117-

128.

MAYNE, A.J. (1979). Traffic signal settings for congested traffic. In: Research Directions in Computer

Control for Urban Traffic Systems, pp. 335-354 (American Soc. Civ. Eng: New York).

McNEIL, D.R. (1968). A solution to the fixed-cycle traffic light problem for compound Poisson arrivals.

J.Appl. Prob 5 (3), pp. 624-635.

McNEIL, D.R. and WEISS, G.H. (1974). Delay problems for isolated intersections. In: GAZIS, D.C.

(Ed.), Traffic Science, pp. 109-174. (J. Wiley: New York).

MICHALOPOULOS, P.G. (1979). Computerised signal control through improved modelling of traffic

dynamics. In: Research Directions in Computer Control of Urban Traffic Systems, pp. 324-334

(American Soc. Civ. Eng: New York).

MICHALOPOULOS, P.G. and MEYER, S.E. (1980). Settings for an isolated overload intersection.

Traffic Eng. Control 21 (12), pp. 580-586.

MILLER, A.J. (1963). Settings for fixed-cycle traffic signals. Oper. Res. Q. 14 (4), pp. 373-386.

MILLER, A.J. (1964). Settings for fixed-cycle traffic signals. Proc. 2nd ARRB Conf. 2 (1), pp. 342-365.

MILLER, A.J. (1968). Signalised Intersections - Capacity Guide. Australian Road Research Board

Bull. No. 4 (Reprinted as ARRB Research Report ARR No. 79, 1978).

NEUBURGER, H. (1971). The economics of heavily congested roads. Transp. Res. 5(4), pp 283-293.

OHNO, K. (1978). Computational algorithm for a fixed cycle traffic signal and new approximate

expressions for average delay. Transp. Sci. 12 (1), pp. 29-47.

PIGNATARO, L.J. et al. (1978). Traffic Control in Oversaturated Street Networks. NCHRP Rep. 194.

Transp. Res. Board, Washington D.C.

RAHMANN, W.R. (1973). Storage / output design of traffic signals. Aust. Rd. Res. 5 (1), pp. 38-43.

AIR 367-1 19

ROBERTSON, D.I. (1969). TRANSYT: A Network Study Tool. Transp. Road Res. Lab. (U.K.) TRRL

Lab. Rep. LR 253.

ROBERTSON, D.I. and GOWER, P. (1977). User Guide to TRANSYT Version 6. Transp. Road Res.

Lab. (U.K.) TRRL Supp. Rep. SR 255.

ROBERTSON, D.I. (1979). Traffic models and optimum strategies of control: a review. In: Inst.

Transp. Studies and U.S. Dept Transportation, Proceedings of International Symposium on Traffic

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Fuel Consumption. Transp. Road Res. Lab. (U.K.) TRRL Lab. Rep. LR 934.

SOSIN, J.A. (1980). Fully-saturated green periods at fixed-time time traffic signals. Traffic Eng. Control

21 (10), pp. 489-493, 499.

WEBSTER, F.V. (1958). Traffic Signal Settings. Road Res. Lab. Tech. Paper No. 39. HMSO,

London.

WORMLEIGHTON, R. (1965). Queues at a fixed time traffic signal with periodic random input.

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APPENDIX - A NOTE ON VARIOUS DELAY EXPRESSIONS

Ohno (1979) proposed the following steady-state expression for average delay at isolated fixed-time

signals as a good approximation to an ‘exact’ theoretical solution obtained using a special computer

algorithm for the’ Poisson arrival-constant saturation headway’ process.

d = 0.5 c (1 - u)

2

/ (1 - y) + ((1 - u) / (1 - y)) (N

o

/ q) + (A.1)

((1 - u) / (1 - y)) / (2 s) + ((1 - u) / (1 - y)

2

) / (2 s)

where

d = average delay per vehicle in seconds,

u = g / c = green time ratio, i.e. the proportion of the cycle that is effectively green

g = effective green time in seconds,

c = cycle time in seconds,

y = flow ratio (=q/s),

q = average arrival rate (flow) in vehicles per seconds,

s = saturation flow in vehicles per second,

N

o

= average overflow queue in vehicles obtained from Equation (8).

The Miller (1968) formula for average delay, which is obtained from Equation (9) as d = D / q, is

d = 0.5 c (1 - u)

2

/ (1 - y) + ((1 - u) / (1 - y)) (N

o

/ q) (A.2)

where x = (q c )/ (s g) is the degree of saturation.

It is seen that the difference between Equation (A.1) and Equation (A.2) is merely due to the third and

fourth terms; Equation (A.1) was called the modified Miller formula by Ohno.

The following average delay expression corresponds to the approximation given by Equation (18)

used to modify the second term of Equation (9):

d = 0.5 c (1 - u)

2

/ (1 - y) + N

o

x / q (A.3)

where x = degree of saturation ( = qc/sg)

Also of interest is the widely used Webster (1958) formula:

d = 0.5 c (1 - u)

2

/ (1 - y) + 0.5 x

2

/ (q (1 - x)) (A.4)

- 0.65 (c / q

2

)

1/3

x

(2+5u

)

20 Akçelik, R.

Average delays calculated from the above expressions are given in Tables A.1 and A.2 for the data

used in Ohno’s (1978) Figures 2 and 3. In Table A.1, c = 90 s, g = 45 s, s = 1.0 veh/s (3600 veh/h),

hence u = 0.50, sg = 45. The value of y is varied and the average delay, d, is calculated for each y

value. In Table A.2, c = 90 s, q = 0.4 veh/s (1400 veh/h), s = 1.0 veh/s, hence y = 0.40. The value of u

is varied and the average delay is calculated. In both tables, the degree of saturation, x is also given

as a better indicator of congestion levels. For Equations (A.1) and (A.2), N

o

is calculated from

Equation (8). For Equation (A.3), N

o

is calculated from Equation (13).

The following conclusions are drawn from a comparison of delay values from expressions given in

Tables A.1 and A.2:

(a) The absolute differences are very small (about one second only in most cases). Accepting the

results from Equation (A.1) as a basis for comparison, Equations (A.2) and (A.3) are found to

give slightly lower values, whereas the Webster formula (Equation A.4) is found to give slightly

higher values for high degree of saturation. However, the difference can hardly be said to be

practically significant.

(b) Ohno’s (1978) work indicates some large relative errors for expressions of the type Equations

(A.2) to (A.4). However, consideration of the relative differences alone may be misleading since

it is found that they increase as the absolute differences decrease. For example, Ohno’s (1978)

Figure 3 (for the data in Table A.2 of this paper) shows very large errors for low degrees of

saturation (high u values) in which case absolute delay values are of the order of 1 sec.

(c) Approximations introduced in this paper, namely Equation (13) for the average overflow queue

and Equation (18) for the random delay term are found to be satisfactory (compare the results

from Equations (A.2) and (A.3) given in Tables A.1 and A.2) indicating that the basis of time-

dependent expressions given in this paper are reliable.

Table A.1 - Average delay values (seconds) for c = 90 s, u = 0.50, s = 3600 veh/h

y = 0.10 0.20 0.30 0.40 0.45 0.47

x = 0.20 0.40 0.60 0.80 0.90 0.94

Eqn (A.1) *

13.1 14.8 17.0 20.4 25.5 32.1 Ohno (1978)

Eqn (A.2) *

12.5 14.1 16.1 19.3 24.2 30.7 Miller (1968)

Eqn (A.3) **

12.5 14.1 16.1 19.6 25.1 31.0 This paper

Eqn (A.4) 12.7 14.6 16.9 20.8 26.4 33.3 Webster (1958)

*

N

o

from Equation (8)

**

N

o

from Equation (13)

Table A.2 - Average delay values (seconds) for c = 90 s, u = 0.40, s = 3600 veh/h

u = 0.80 0.70 0.60 0.50 0.44 0.42

x = 0.50 0.57 0.67 0.80 0.90 0.95

Eqn (A.1) *

3.4 7.4 12.9 20.4 28.9 40.8 Ohno (1978)

Eqn (A.2) *

3.0 6.8 12.0 19.3 27.7 39.5 Miller (1968)

Eqn (A.3) **

3.0 6.8 12.0 19.6 28.7 39.8 This paper

Eqn (A.4) 3.5 7.5 13.0 20.8 29.8 42.4 Webster (1958)

*

N

o

from Equation (8)

**

N

o

from Equation (13)