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Bus Priority Using pre-signals
Transpn Res.-A, Vol. 32, No. 8, pp. 563±583, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0965-8564/98 $Ðsee front matter
Pergamon PII: S0965-8564(98)00008-1
BUS PRIORITY USING PRE-SIGNALS JIANPING WU* and NICK HOUNSELL
Department of Civil and Environmental Engineering, University of Southampton,Southampton,U.K., SO17 1BJ (Received 23 July 1996; in revised form 14 January 1998) AbstractÐThe need to provide ecient public transport services in urban areas has led to the implementation of bus priority measures in many congested cities. Much interest has recently centred on priority at signal controlled junctions, including the concept of pre-signals, where trac signals are installed at or near the end of a with-¯ow bus lane to provide buses with priority access to the downstream junction. Although a number of pre-signals have now been installed in the U.K., particularly in London, there has been very little published research into their design, operation and optimisation. This paper addresses these points through the development of analytical procedures which allow pre-implementation evaluation of speci®c categories of pre-signals. The paper initially sets out three categories of pre-signal, which have dierent operating characteristics, dierent requirements for signalling and dierent impacts on capacity and delay. Key issues concerning signalling arrangements for these categories are then discussed, together with a summary of the analytical approach adopted and the assumptions required. Equations are developed to allow appropriate signal timings to be calculated for pre-signalised intersections. Further equations are then developed to enable delays to priority and nonpriority trac, with and without pre-signals, to be estimated with delay being taken here as the key performance criterion. The paper concludes with three application examples illustrating how the equations are applied and the impacts of pre-signals in dierent situations.The analyses con®rm the potential bene®ts of pre-signals, where these signals apply to non-priority trac only. Where buses are also subject to a pre-signal, it is shown that disbene®ts to buses can often occur, unless bus detectors are used to gain priority signalling. # 1998 Elsevier Science Ltd. All rights reserved Keywords: pre-signal, queue relocation, bus priority, public transport, trac management 1. INTRODUCTION
In busy urban areas where trac congestion is severe and bus frequencies are high, priorities are often provided to buses to reduce bus passengers' journey time and to improve regularity. Many bus priority methods have been studied since the 1960s (Robertson and Vincent, 1975; King, 1983; Hounsell and McDonald, 1988; Roberts and Buchanan, 1993; Oakes et al, 1994; Tee et al, 1994; Fox et al, 1995; Oakes and Metzger, 1995). The traditional bus lane has, since the 1970s, become widely accepted as a means of both allowing buses to overtake existing queues and reserving road space, and therefore capacity, for buses to combat trac growth and consequent delay over time. As a bus priority strategy, the concept of a pre-signal was ®rst documented in the U.K. in the Department of Transport in the report Keeping Buses Moving in December 1991 (Oakes et al, 1994) although the techniques has been used for many years in some European cities. Since 1993 a number of schemes have been designed in detail and implemented (Oakes et al, 1994), particularly in London. Fig. 1 shows an example of a pre-signal operating at Shepherds Bush in London. The implementation of pre-signals in U.K. cities is becoming signi®cant. Taking London only, 14 pre-signals are now in place since the ®rst scheme was introduced in Shepherd's Bush in 1993. A further 20±25 pre-signals are planned for the coming years to contribute to London's Bus Priority Network. Many pre-signals have had site-speci®c aspects and operate with additional trac detection for real time control. However, there remains a strong need for a methodology for the design and evaluation of pre-signals at the pre-implementation stage, to ensure that optimum deployment is achieved. The research described in this paper focuses on this point. A pre-signal aims to give buses priority access into a bus advance area of the main junction stop line so as to avoid the trac queue and reduce bus delay at the signal controlled junction. There are two particular cases where pre-signals can achieve major bene®ts for buses. The ®rst is where buses must move from kerbside bus stops to the centre of the road to turn right at the main *Author for correspondence. Fax: 01703 593152; e-mail: [email protected]
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junction. Without pre-signals they must force their way through congested trac queues. The presignal can provide an empty reservoir in which they can position themselves at will. The second case is using a pre-signal as a device for local queue relocation. The approach to a congested, trac signal controlled junction may not always be a practical location for a bus lane, particularly if the road width is inadequate. However, it may sometimes be possible to provide a bus lane further upstream. In such a circumstance a pre-signal could be installed at the end of the bus lane to control the non-priority trac and keep the reservoir area between the pre-signal and main signal clear to allow buses clear access to the main signal (Roberts and Buchanan, 1993). This may also be bene®cial for more conventional with-¯ow bus lane approaching a junction which incorporates a `setback' for general trac. Three dierent categories of pre-signal design have been suggested and/or implemented in the U.K. (Roberts and Buchanan, 1993; Oakes et al, 1994; Tee et al., 1994) each with its own operating characteristics (these are denoted categories A, B and C in this paper). In category A (Fig. 2) the pre-signal controls only the non-priority trac with buses uncontrolled (or subject to `give way control'). Figure 2 is a typical layout of pre-signal category A. In contrast, in pre-signal category B, buses are also controlled by a pre-signal at the end of the bus lane. When the pre-signal in the bus lane turns red, non-priority vehicles receive right of way, and vice versa. A typical layout of a pre-signal category B is shown in Fig. 3. An intermediate layout, not directly considered here, is where the bus has to give way to non-priority trac at the end of the bus lane.
Fig. 1. A pre-signal controlled junction in london (courtesy of London Transport Buses)
Fig. 2. Typical layout of pre-signal category A.
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Fig. 3. Typical layout of pre-signal category B.
The number of lanes for general trac in the bus advance area is usually greater than the number of lanes available upstream of the pre-signal. (An equal number of lanes is shown in Figs 2 and 3 for illustration). This variation has an important eect on pre-signal operation and bene®ts, as will be shown later in the paper. With both categories A and B, the signal timings at the presignal and the main junction are co-ordinated for maximum eciency (for category B, the main junction signals could be co-ordinated either with the bus or with the non-priority trac pre-signal, according to priority eciency). A pre-signal category C can also be de®ned which has a similar layout as category B, except that detectors are installed in the bus lane. When a bus approaches the pre-signal, a red signal is shown to the non-priority trac stream to enable the bus to access to the main signal without impedance. The bus lane pre-signal then turns to red, allowing non-priority trac to ®ll the bus advance area behind the bus after it has selected the appropriate lane at the main signal. The amount of bus delay in pre-signal category C depends on whether there is co-ordination for buses between the pre-signal and the main signal. For example, if buses can call the green in both the pre-signal and the main signal, the bus has full priority, and may experience no delay at all at the junction. However, if buses can only call the green at the pre-signal, and may be delayed by a red light at the main signal, the delay to buses will depend on the timing of the main signal. Delays to non-priority vehicles are more complicated to analyse for Category C, and, a microscopic simulation model may be required to study such a VA or semi-VA (only bus's arrival is detected) controlled junction. In this paper, only the pre-signal categories A and B with ®xed time controls are considered in detail. The following sections describe an analytical approach developed for determining optimum pre-signal timings and vehicle delays in pre-signal controlled approaches. The terms and symbols used in the paper are summarised in Appendix A at the back of the paper. 2. PROBLEM REVIEW AND ASSUMPTIONS
There are two potential problems which need to be guarded against when designing pre-signals. The ®rst is the potential waste of junction capacity by not releasing trac from the pre-signals suciently early to make use of green time at the main junction. The second is that the relocated trac queue may block back to the upstream junction (Roberts and Buchanan, 1993). Both the problems can usually be addressed by careful signal setting at both the pre-signal and the main signal, and the correct estimation of the re-located trac queue at the pre-signal. To ensure no potential capacity waste at the junction, the following two assumptions were made while setting signal times for pre-signal and main signal. Assumption 1 The non-priority vehicles arriving at the pre-signal stop line during the red time of the main signal, rm , will be fully discharged into the main signal stop line to ®ll up the bus advance area before the main signal green time starts. This has the form
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rm d rm ÿ rp Sl Np
1
where rp and rm are the red time at the pre-signal and main signal respectively (see Fig. 4), d is the rate of demand of non-priority vehicles in the approach (vehicles/s). Sl is the lane based saturation ¯ow (vehicles/s/lane) and Np the number of lanes at the pre-signal for non-priority vehicles. This assumption implies co-ordination of the pre-signal and the main signal for non-priority trac and ensures no loss of junction capacity due to the pre-signal. Assumption 2 Junction capacity equals demand. In other words, there will be no residual queues at the main signal stop line at the end of green time, gm . This is given by the following equation. gm Sl NmC d dB
2
where gm is the green time at the main signal and C the cycle time. Nm is the number of lanes at the main signal stop line and dB the rate of bus ¯ow (buses/s). d and Sl are as de®ned earlier (also see Glossary in Appendix). This situation describes a degree of saturation of one. Where is less than one, a bus lane (or pre-signal) could not usually be justi®ed; when is greater than one for a period of time, there would be signi®cant queuing alongside the bus lane which itself should provide greater bene®t to buses. However, the operation (and bene®t) of the pre-signal would remain as analysed in the paper. With these assumptions there is also no consideration of stochastic processes (e.g. in vehicle arrivals and departures). This is justi®ed for this case of pre-implementation evaluation because of the ¯ow regime of interest ( equals one) and because of the proximity of the main junction and the pre-signal, (which would produce minimal platoon departure). However, for real life pre-signal operation stochastic eects may be sucient to warrant vehicle detection to ensure that maximum bene®ts are obtained from the pre-signal (this has been necessary in some London intersections). 3. PRE-SIGNAL AND MAIN SIGNAL TIMING
In this section, equations are developed to calculate signal times for pre-signal categories A and B, given known signal cycle, C, of the main junction. Pre-signal and main signal timing for pre-signalised intersections are controlled by the two assumptions de®ned in the last section. With assumption 2 or eqn (2), we have gm C
d dB Sl Nm
3
From Fig. 4, we have rm C ÿ g m
Fig. 4. The relationship of rp ; gp ; rm ; gm and C.
4
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Substituting eqn (3) into eqn (4) yields rm C1 ÿ
d dB Sl Nm
5
For pre-signal timing, from assumption 1 or eqn (1), we have rp 1 ÿ
d rm Sl Np
6
From Fig. 4, we have gp C ÿ rp
7
Substituting (6) into (7) yields g p gm
d rm Sl Np
8
Further, in pre-signal category B, the bus lane is also controlled by a pre-signal which has the following relationship with the non-priority trac pre-signal. gpB rp
9
rpB gp
10
and
where gpB and rpB are the green and red time for the bus lane pre-signal in pre-signal category B. Substituting eqns (6) and (8) into eqns (9) and (10) respectively, we have gpB 1 ÿ
d rm Sl Np
11
d rm Sl Nm
12
and rpB gm
4. THE BENEFITS AND DISBENEFITS TO BUSES IN PRE-SIGNALLED JUNCTIONS
The purpose of a pre-signal is to give buses priority at signal controlled intersections. In this section, we will discuss the bene®ts and disbene®ts to buses in both pre-signal categories A and B. The bene®ts to buses or bus delay saving is de®ned as the dierence in bus delay with and without pre-signals. 4.1. Bus delay without pre-signal Assuming buses and non-priority vehicles arrive randomly, they have the same average delay without a pre-signal. Figure 5 shows the cumulative arrival and departure at a normal (without pre-signal) signal controlled intersection. The total trac delay per cycle without pre-signal, D(normal), equals the area of triangle ADF, which is given 1 D normal C2 d dB ÿ gm C d dB 2
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Fig. 5. The cumulative arrival/departure in normal signalised intersections.
or 1 D normal C d dB rm 2 The average delay per vehicle without pre-signal, d(normal), is given 1
d normal 2
C d dB rm C d dB
or d normal
rm 2
13
eqn (13) is the same as the ®rst term of Webster's expression (Webster and Ellson, 1965) when 1. 4.2. Bus delay and delay saving with pre-signal category A In pre-signal category A, non-priority vehicles are held by the pre-signal for a time period of rp in each cycle to give priority to buses moving into the bus advance area without any impedance. As shown in Fig. 6, line AH is the cumulative bus arrival, AG the non-priority vehicle arrival and AD the total arrival without pre-signal. With pre-signal category A, the total arrival follows the line of ABCD instead of AD at the main signal. Buses arriving during time rp have total delay ABEF (see Fig. 6) at the main signal. This part of bus delay, D CA1, is D CA1 ABEF
14
Buses arriving during gp have no priority access to the bus advance area. This part of bus delay, D CA2, is given by: D CA2 BCDE
gp dB C d gp dB
4.2.1. The total bus delay with pre-signal category A. A, D CA, can be shown as:
15
The total bus delay in pre-signal category
D CA D CA1 D CA2
16
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Fig. 6. The cumulative arrival/departure at the main signal with pre-signal category A.
The areas of ABEF and BCDE (see Fig. 6) can be estimated by the following equations. Detailed derivation of the equations can be found in Appendix B. rp dB gm rp dB 2rm ÿ rp 2 C d dB
17
1 frm gp C 2d 2rm gp C rp C gm ÿ gp d dB rm g2p 2dB g 2C d dB
18
ABEF
BCDE
The total bus delay with pre-signal category A, D CA, therefore has the form D CA
1 d 2dB 2rm C3 ÿ rp rm C2 2C d dB C d gp dB 2d dB rm C3 ÿ r2p gm C 3db rm gp C2
19
with consideration of eqns (14±18). 4.2.2. The average bus delay with pre-signal category A. category A, d CA, has the form d CA
D CA C dB
The average bus delay in pre-signal 20
or d CA
1 f d dB 2rm C2 ÿ rp rm C 2d rm C2 ÿ r2p gm 2dB rm gp Cg 2C d dB C d gp dB 21
with substitution of eqn (19). Without the pre-signal, parameter rp equals 0 and gp equals C. Equation (21) then becomes d CA which has the same form as eqn (13).
rm 2
22
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Jianping Wu and N. Hounsell
4.2.3. Bus delay saving with pre-signal category A. As has been de®ned earlier, the bus delay saving in pre-signal category A, d CA, is the dierence between the average bus delay without pre-signal and that with pre-signal category A. The general form of it is d CA d normal ÿ d CA
23
Substituting eqns (13) and (21) into eqn (23) yields d CA
gm r2p 2d 2C d dB C d gp dB
24
eqn (24) shows that the bus delay saving increases as the parameters gm and rp increase. 4.3. Bus delay and delay saving with pre-signal category B In pre-signal category B, when buses receive the green signal, non-priority vehicles are held on red, and vice versa. Bus delay with pre-signal category B consists of (i) delay at the bus lane presignal, and (ii) delay at the main signal. Figure 7 shows the cumulative arrival and departure of buses at the bus lane pre-signal. Line AC in eqn (7) indicates the queue of buses which arrived during time rpB gp of last signal cycle, CE the cumulative bus arrivals and AJB the departures at the bus lane pre-signal. The total bus delay per cycle for the bus lane pre-signal, D CB1, is therefore the sum of the areas ACJ and BED, i.e. D CB1 ACJ BED
25
In Fig. 8, line AJB is the cumulative bus arrival from the bus lane pre-signal to the main signal and BFD the non-priority vehicle arrival from the pre-signal to the main signal. Because all the buses will take the front positions of the trac queue at the main signal and discharge ®rst during the green time gm , the total bus delay at the main signal with pre-signal category B; D CB2, is the area of AJBEK in Fig. 8, i.e. D CB2 AJBEK
4.3.1. The total bus delay with pre-signal category B. category B; D CB, has the general form
26
The total bus delay per cycle with pre-signal
Fig. 7. The cumulative arrivals/departures of buses at the bus lane pre-signal with pre-signal category B.
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Fig. 8. The cumulative arrival/departure at the main signal with pre-signal category B.
D CB D CB1 D CB2 or D CB ACJ BED AJBEK This becomes D CB
C dB 2rm gp ÿ rp 2
27
by consideration of the areas of ACJ, BED and AJBEK. Detailed derivation of eqn (27) can be found in Appendix B. 4.3.2. The average bus delay with pre-signal category B. category B, d CB, has the form d CB
The average bus delay with pre-signal
D CB C dB
Substitution of eqn (27) yields 1 d CB 2rm gp ÿ rp 2
28
4.3.3. The Bus Delay Savings with Pre-signal Category B. The bus delay saving with pre-signal category B, d(CB), is de®ned as the dierence between the average delay per vehicle without the pre-signal, d(normal), and the average bus delay with the pre-signal category B, d(CB). This can be expressed as d CB d normal ÿ d CB
29
1 d CB rm gp ÿ rp 2
29
or
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Jianping Wu and N. Hounsell
5. THE BENEFIT AND DISBENEFITS TO NON-PRIORITY VEHICLES IN PRE-SIGNALISED JUNCTIONS
This section develops equations to estimate the impact of pre-signals on non-priority trac. 5.1. Non-priority vehicle delay in pre-signal For non-priority vehicles, the total delay consists of two parts, the delay at the pre-signal and that at the main signal. There is no dierence on delay of non-priority vehicles at pre-signal between categories A and B. As shown in Fig. 9, line AC is the arrival of non-priority vehicles and area ABD the total non-priority vehicles delay at the pre-signal, Dn(P), which has the form 1 Dn P rm rp d 2
30
5.2. The delay and delay savings of non-priority vehicles with pre-signal category A 5.2.1. Non-priority vehicle delay in main signal with pre-signal category A. As shown in Fig. 6, the area BCDE is the total delay at the main signal for all non-priority vehicles and buses those arriving during time gp . Therefore, the total non-priority vehicle delay at the main signal with presignal category A, Dn CA2, has the form Dn CA2 BCDE
C d C d gp dB
With consideration of eqn (18), it becomes Dn CA2
d frm gp C 2d 2rm gp C rp C gm ÿ gp d dB rm g2p 2dB g 31 2 d dB C d gp dB
5.2.2. Total non-priority vehicle delay with pre-signal category A. The total non-priority vehicle delay with pre-signal category A, Dn CA , is the sum of delays at the pre-signal and main signal. Dn CA Dn P Dn CA2 With consideration of eqn (30) and eqn (31), this becomes Dn CA
1 frm C 3 3d r2p C2 2C rp rm gp C 2d dB rm gp C2 d 2dB g 2C d dB C d gp dB 32
Fig. 9. The cumulative arrival/departure of non-priority vehicles at the pre-signal.
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5.2.3. Average non-priority vehicle delay with pre-signal category A. The average non-priority vehicle delay with pre-signal category A, dn CA, has the general form dn CA
Dn CA C d
Substitution of eqn (32) yields dn CA
1 frm C2 2d r2p C 2C rp rm gp d dB rm gp C 2dB g 2C d dB C d gp dB
33
5.2.4. Delay savings of non-priority vehicles with pre-signal category A. The delay savings of nonpriority vehicles with pre-signal category A, d CA, is de®ned as the dierence between the delay without pre-signal and that with pre-signal category A. The general form is dn CA d normal ÿ dn CA Consideration of eqns (13) and (33) yields dn CA ÿ
r2p gm d dB 2C d dB C d gp dB
34
As we can see, eqn (34) gives a negative value. This means that the average delay for non-priority vehicle has a small increase with pre-signal category A. The reason for this can be explained by comparing Figs 10 and 11. Assuming all buses and non-priority vehicles arrive randomly, an average trac queue at the main signal will have the pattern as shown in Fig. 10. With pre-signal category A, buses arriving during time rp are given priority to access the bus advance area. Therefore, these buses will take the front positions of the queue at the main signal. The trac queue at the main signal will have the pattern as shown in Fig. 11. More non-priority vehicles will locate to the rear part of the trac queue compared to the without pre-signal case. There is a reduction in average delay to buses (as shown in eqn (24)) and, meanwhile, an increase of average delay to non-priority vehicles with pre-signal category A (as shown in eqn (34)). However, the total vehicle (including buses and non-priority vehicles) delay with pre-signal category A in the approach remains the same. This hypothesis can be veri®ed as below. Without the pre-signal, the total vehicle (including buses and non-priority vehicles) delay, TD, is 1 TD rm C d dB 2
35
With pre-signal category A, the total delay, TD CA, will be the sum of total bus delay and total non-priority vehicle delay. This can be shown that
Fig. 10. Vehicle locations without pre-signal.
Fig. 11. Vehicle locations with pre-signal category A.
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Jianping Wu and N. Hounsell
TD CA D CA Dn CA
36
Substituting eqns (19) and (32) into eqn (36)) yields 1 TD CA rm C d dB 2
37
The total vehicle delay in eqns (35) and (37) are the same. 5.3. The delay and delay savings of non-priority vehicles with pre-signal category B 5.3.1. Non-priority vehicle delay at the main signal with pre-signal category B. The area BFDE in Fig. 8 indicates the total delay of non-priority vehicles at the main signal with pre-signal category B, Dn CB2, which has the form: Dn CB2
d rm gp d rm gp gm C dB 2 d dB
38
Detailed derivation of eqn (38) can be found in Appendix B. 5.3.2. Total non-priority vehicle delay with pre-signal category B. The total non-priority vehicle delay with pre-signal category B, Dn CB, is the sum of delays at the pre-signal and the main signal. The general form of this is Dn CB Dn P Dn CB2 or Dn CB
C d rm d C dB 2 d dB
39
with consideration of eqns (30) and (38). 5.3.3. Average non-priority vehicle delay with pre-signal category B. vehicle delay with pre-signal category B, dn CB, is given by dn CB
The average non-priority
Dn CB C d
Substitution of eqn (39) yields dn CB
rm d C dB 2 d dB
40
5.3.4. Delay savings for non-priority vehicles with pre-signal category B. The delay savings for the non-priority vehicles with pre-signal category B, dn CB, is de®ned as the dierence in vehicle delay without pre-signal and that with pre-signal category B. This has a general form dn CB d normal ÿ dn CB This becomes dn CB ÿ With consideration of eqn (13) and eqn (40).
gm dB 2 d dB
41
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As we can see, eqn (41) gives a negative value. This means that there is an increase in average delay for non-priority vehicles with pre-signal category B. The reason for this is similar to that with pre-signal category A. In the case of pre-signal category B, all the buses queued in the bus lane pre-signal are discharged into the bus advance area during time gpB rp when the nonpriority vehicles are held by the pre-signal red light. Therefore, all the buses will take the front positions of the queue at the main signal. The trac queue will have a pattern as shown in Fig. 12. Although there are no non-priority vehicles which will be held to the next cycle at the main signal stop line, the change of the position of non-priority vehicles in the trac queue leads to a slight increase in their average vehicle delay. Unlike pre-signal category A, the total vehicle delay (including buses and non-priority vehicles) with pre-signal category B also increased because the buses arriving at the bus lane pre-signal during time rpB gp are not able to discharge into the bus advance area and leave the main signal stop line until green time gpB rp in the following signal cycle. This hypothesis is veri®ed as below. Without the pre-signal, the total delay has the form as shown in eqn (35). With pre-signal category B, the total delay, TD CB, is the sum of total bus delay and total non-priority vehicle delay. TD CB D CB Dn CB Substitution of eqns (27) and (39) yields gm C 2dB 1 TD CB rm C d dB gp C dB ÿ 2 d dB 2
42
The extra total vehicle delay with pre-signal category B per cycle, TD CB, is the dierence of the total vehicle delay without pre-signal and that with pre-signal category B. This is TD CB gp C dB ÿ
gm C 2dB 2 d dB
43
by comparing the eqns (42) and (35). 6. THE BUS ADVANCE AREA AND THE RELOCATED TRAFFIC QUEUE LENGTH
The size of bus advance area depends on the layout of the approach and can be estimated by the following equation. N mL BAA d dB rm H or L BAA
d dB rm H Nm
44
where L BAA is the length of the bus advance area and H the average vehicle queued headway (a default value of 5.5 m may be used when no data available) from the approach. Nm , d , dB , and rm are as de®ned earlier (also see Glossary in Appendix). The relocated trac queue of non-priority trac due to the pre-signal has its maximum length at the end of red time, rm , and may be estimated by the following equation.
Fig. 12. Vehicle locations with pre-signal category B.
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Jianping Wu and N. Hounsell
Q relocated
rm d H Np
45
where Q relocated is the estimated maximum queue length (m) backwards from the non-priority trac pre-signal stop line. rm , d , Np and H are de®ned as earlier (also see Glossary in Appendix). Equation (45) is useful to estimate the potential impact of the locally relocated trac queue to the upstream trac. 7. APPLICATION EXAMPLES
Three examples are presented here to demonstrate how to use the equations developed in this paper for signal time setting; estimations of delay and delay savings and the lengths of the relocated trac queue and of the bus advance area required when designing pre-signalised junctions. 7.1. The example approaches The example approach 1, as shown in Fig. 13, has the following data: Np 1; Nm 2; C 100 s; Sl 2000=3600; d 1000=3600, dB 60=3600. Assuming a bus has a pcu=2.0, the converted dB 120=3600 (pcu/s). The example approach 2, as shown in Fig. 14, has the following data: Np 2; Nm 2; C 100 s; Sl 2000=3600; d 1000=3600, dB 60=3600. Assuming a bus has a pcu=2.0, the converted dB 120=3600 (pcu/s). The example approach 3, as shown in Fig. 15, has the following data: Np 2; Nm 3; C 100 s; Sl 2000=3600; d 2000=3600, dB 60=3600. Assuming a bus has a pcu=2.0, the converted dB 120=3600 (pcu/s).
Fig. 13. Example approach 1.
Fig. 14. Example approach 2.
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Fig. 15. Example approach 3.
7.2. Signal timing To set the signal times at the pre-signal and main signal, the following equations are used: . . . .
Green time in main signal: Equation (3) Red time in main signal: Equation (5) Green time in pre-signal: Equation (8) Red time in pre-signal: Equation (6)
Further, the signal times in bus lane with pre-signal category B are . Green time in bus lane pre-signal: Equation (11) . Red time in bus lane pre-signal: Equation (12) Table 1 summarises the signal times of the above three examples. 7.3. Bus delay and delay saving To estimate vehicle delay and delay savings with both pre-signal categories A and B, the following equations are used. . . . . . . . .
Average bus delay with pre-signal category A:eqn (21) Bus delay saving with pre-signal category A: eqn (24) Average bus delay with pre-signal category B: eqn (28) Bus delay saving with pre-signal category B: eqn (29) Average non-priority vehicle delay with pre-signal category A: eqn (33) Non-priority vehicle delay saving with pre-signal category A: eqn (34) Average non-priority vehicle delay with pre-signal category B: eqn (40) Non-priority vehicle delay saving with pre-signal category B: eqn (41)
Vehicle delay and delay savings with pre-signal categories A and B are calculated for all the three example approaches and listed in Tables 2±5. It is clear from these examples that the pre-signal category B is not bene®cial using ®xed time signalling. Active bus detection and signal response is likely to be needed to provide the required bene®ts. Table 1. Signal timings for the three examples Example approaches Example 1 Example 2 Example 3
rm (Seconds)
gm (Seconds)
rp (Seconds)
gp (Seconds)
72 72 65
28 28 35
36 54 32.5
64 46 67.5
For the pre-signal category B,rpB gp ,andgpB rp
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Jianping Wu and N. Hounsell Table 2. Bus delay and delay saving for pre-signal category A
Examples
One Two Three
Without pre-signal (sec/bus)
With pre-signal (sec/bus)
Bus delay saving (sec/bus)
Bus delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
34.50 32.55 30.80
1.50 3.45 1.70
4.2% 9.6% 5.2%
Table 3. Bus delay and delay saving for pre-signal category B Examples
One Two Three
Without pre-signal (sec/bus)
With pre-signal (sec/bus)
Bus delay saving (sec/bus)
Bus delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
86.00 68.00 82.50
ÿ50.00 ÿ32.00 ÿ50.00
ÿ139% ÿ89% ÿ154%
Table 4. Non-priority vehicle delay and delay saving for pre-signal category A Examples
One Two Three
Without pre-signal (sec/veh)
With pre-signal (sec/ veh)
Non-priority vehicle delay saving (sec/ veh)
Non-priority vehicle delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
36.18 36.41 32.60
ÿ0.18 ÿ0.41 ÿ0.10
ÿ0.50% ÿ1.14% ÿ0.30%
Table 5. Non-priority vehicle delay and delay saving for pre-signal category B Examples
One Two Three
Without pre-signal (sec/ veh)
With pre-signal (sec/ veh)
Non-priority vehicle delay saving (sec/ veh)
Non-priority vehicle delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
37.50 37.50 33.50
ÿ1.50 ÿ1.50 ÿ1.00
ÿ4.16% ÿ4.16% ÿ3.08%
Table 6. The relocated queue length of the three examples Examples
One Two Three
The maximum relocated queue length with pre-signals (Meters) 110 55 99
7.4. The relocated queue length at pre-signal The length of the relocated queue is critical sometimes for a short link with very close signal controlled junctions. Equation (45) can be used to estimate the maximum queue length of the relocated trac queue. Table 6 lists the results of the three examples with the assumption that H (average vehicle length) equals 5.5 m. There is no dierence in the relocated trac queue length of non-priority vehicles between pre-signal categories A and B.
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Table 7. The length of the bus advance area for the three examples Examples
Length of the bus advance area with pre-signals (Meters)
One Two Three
61.5 61.5 70.2
7.5. The size of bus advance area The size of bus advance area is an important element to design a pre-signal controlled junction. Equation (44) can be used to estimate the size of bus advance area. Table 7 lists the results of the three examples with the assumption that H (average vehicle queued headway) equals 5.5 m. There is no dierence in the size of bus advance area between presignal categories A and B. 8. CONCLUSIONS
This paper has illustrated how an analytical approach can be used for the pre-implementation evaluation of pre-signals. The approach covers issues of design, capacity, signal settings, delay and delay saving estimation for priority and non-priority trac, calculation of the relocated trac queue at pre-signal, and the design of the bus advance area. Examples have shown how delay savings to buses can be achieved with the category A pre-signal (where buses are unsignalised at the pre-signal) without signi®cant disbene®t to non-priority trac. Delay savings to buses are highest where there is a long red period at the non-priority trac pre-signal, which is possible when the proportion of green time at the main signal is low. However, category B pre-signals (where buses are signalised at the pre-signal) are shown to generally cause disbene®t to buses unless special bus detectors are installed to give signal priority to buses. In pre-signal category A, the bus delay saving is caused by the relocation of vehicles in the trac queue at the main signal, as shown by Fig. 11. There is a delay saving to buses and a small delay increase to non-priority vehicles. However, the total delay of the whole approach remains unchanged compared to without pre-signal. In pre-signal category B, although all buses can move to the front of the trac queue at the main signal, those buses arriving during time rpB gp are held at the bus lane pre-signal until the time gpB of the following cycle, extra bus delay therefore results. This also results in a overall delay increase for the approach as shown in eqn (43). The analytical approach has been based on two assumptions which are justi®ed for pre-implementation evaluation when an estimate of bene®ts is required. However, the approach excludes stochastic processes which can be potentially important for real time control where trac demand is at or near capacity. Pre-signal implementation may therefore require close attention to be paid to vehicle detection if the best performance is to be achieved. Overall, the approach developed in this paper for pre-signal timing, estimating delays, lengths of the relocated trac queue and bus advance area is considered to provide a good basis for evaluating pre-signal designs and could readily be extended to other signalling schemes of this nature.
REFERENCES Fox K., Montgomery F. and Shepherd S. (1995) Integrated ATT strategies for urban arteries; DRIVE II project PRIMAVERA: 2, Bus priority in SCOOT and SPOT using TIRIS. Trac Engineering and Control 36, 356±361. Hounsell N. and McDonald M. (1988) Evaluation of bus lanes. Contract Report 87. Transport and Road Research Laboratory, Department of Transport, Crowthorne, U.K. King G. N. (1983) Bus priority in LondonÐtechniques for the 1980's. Proceedings of Seminar K: Trac operation and management, PTRC 11th Annual Meeting. London. Oakes J. and Metzger D. (1995) Park view pre-signals in Uxbridge road, England. Trac Engineering and Control 36, 62±67. Oakes J., ThellMann A. M. and Kelly I. T. (1994) Innovative bus priority measures. Proceedings of Seminar J, Trac Management and Road Safety, 22nd PTRC European Transport Summer Annual Meeting, University of WARWICK, U.K., vol.381, pp.301±312. Roberts M. and Buchanan C. (1993) Bus priorityÐthe solution and west London demonstration project.Proceedings of Seminar C, Highways and Planning, 21st PTRC European Transport Summer Annual Meeting, vol.365, pp.49±57.
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Robertson D. I. and Vincent R. A. (1975) Bus priority in a network of ®xed-time signals. Laboratory Report, Transport and Road Research Laboratory, Department of Transport, Crowthorne, U.K. Tee A., Cuthbertson T. and Carson G. (1994) Public transport initiatives in Surrey. Trac Engineering and Control 35, 70± 73. Webster F. V. and Ellson P. B. (1965) Trac signals for high-speed roads.Road Research Technical Paper No. 74.Road Research Laboratory, HMSO, London.
APPENDIX A
A.1. Glossary and symbols
The terms used in this paper have the following de®nitions: Pre-signals: Bus advance area:
Trac signals at or near the end of bus lane to provide buses with priority access to the downstream junction. Signals apply to non-priority trac and may apply to priority trac count on strategy. The area between the end of a with-¯ow bus lane and the main junction stop line into which buses gain preferential access through the use of pre-signals.
The symbols used in this paper are listed as below alphabetically with the Greek symbols appearing at the end: gp gm gpB Nm Np rm rp rpB Sl d dB
Eective green time at non-priority trac lane pre-signal Eective green time at main signal Eective green time at bus lane pre-signal Number of trac lanes in the bus advance area Number of trac lanes for non-priority trac at pre-signal stop line Eective red time at main signal Eective red time at non-priority trac lane pre-signal Eective red time at bus lane pre-signal Lane based saturation ¯ow (vehicles/s/lane) Rate of trac demand (vehicles/s) Rate of bus ¯ow (buses/s) Degree of saturation
APPENDIX B
B.1. The derivation of the main equations in the paper B.1.1. The area of ABEF in Figs 6. Figure B1 is the same as Fig. 6 with some added lines for equation derivation purposes.
Fig. B1. The cumulative arrival/departure at the main signal with pre-signal category A (same as Fig. 6).
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ABEF ABKF KEF
B1
1 1 ABKF rm ÿ rp rm rp dB 2rm ÿ rp rp dB 2 2
B2
1 KEF KE rp dB 2
B3
From Fig. B1, we have
Because, KE Sl Nm rp dB or rp dB Sl Nm
B4
rp dB 2 2Sl Nm
B5
Sl Nm
C d dB gm
B6
KEF
gm rp dB 2 2C d dB
B7
KE eqn (B3) becomes KEF From eqn (2) in the main body of the paper, we have
Therefore,
The area of ABEF therefore has the form ABEF
rp dB gm rp dB 2rm ÿ rp 2 c d dB
B8
B.1.2. The area of BCDE in Fig. 6. As shown in Fig. B1, the area of BCDE has the general form BCDE BCK LDF ÿ LDC ÿ KEF
B9
1 BCK rm ÿ rp rm d dB ÿ rp dB 2
B10
LDF
gm C d dB 2
B11
LDC
gm gm d dB 2
B12
The area of BCDE therefore has the form 1 gm rp dB 2 BCDE frm ÿ rp rm d dB ÿ rp dB gm C d dB ÿ g2m d dB ÿ g 2 C d dB
B13
by considering eqn (B10),(B11),(B12) and (B7). Or, in dierent format, BCDE
1 frm gp C 2d 2rm gp C rp C gm ÿ gp d dB rm g2p 2dB g 2C d dB
B14
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Jianping Wu and N. Hounsell
Fig. B2. The cumulative arrival/departure of buses at the bus lane pre-signal with pre-signal category B (same as Fig. 7).
Fig. B3. The cumulative arrival/departure at the main signal with pre-signal category B (same as Fig. 8).
B.1.3. The derivation of equation (27). As shown in Figs B2 and B3, the buses delay at the bus lane pre-signal equals the sum of the areas of ACJ and BED. The buses' delay at the main signal is the area of AJBEK, which can further break down to the areas of AJBL and LBEK. Therefore, the total bus delay with pre-signal category B is D CB ACJ BED AJBL LBEK
B15
As area ACJ plus area AJBL equals to area ACBL (see Fig. B3), eqn (B15) becomes D CB ACBL BED LBEK
B16
1 1 ACBL AC BLrp gp dB C dB rp 2 2
B17
1 1 1 BED gp ED gp gp dB g2p dB 2 2 2
B18
From Fig. B3, we have
Bus priority using pre-signals 1 1 1 LBEK BE LKC dB 2LK NEc dB 2 rm ÿ rp NEC dB 2 2 2
583 B19
Because NE Sl Nm C dB or NE
C dB Sl Nm
B20
eqn (B19) becomes 1 C dB LBEK 2 rm ÿ rp C dB 2 Sl Nm
B21
1 gm dB LBEK 2 rm ÿ rp C dB 2 d dB
B22
This further becomes
by considering eqn (B6). Therefore, eqn (16) has the form D CB
dB gm C dB C gp rp g2P 2C rm ÿ rp 2 d dB
B23
C dB 2rm gg ÿ rp 2
B24
or, in dierent format D CB
B.1.4. The derivation of equation (38). As shown in Fig. B3, the total non-priority vehicle delay at the main signal Dn CB2 has the form Dn CB2 BFDE
B25
Dn CB2 BFDE BFN NMDE ÿ FMD
B26
1 1 BFN rm ÿ rp FN rm ÿ rp rm d 2 2
B27
1 1 gm dB NMDE MD NEMN gm C d 2 2 d dB
B28
From Fig. B3, we have
by referring eqn (B20) and eqn (B6), and 1 1 FMD gm gm d g2m d 2 2
B29
d gm Bd f rm ÿ rp rm gm C ÿ g2m g 2 d dB
B30
d rm gp d rm gp gm C dB 2 d dB
B31
Therefore, Dn CB2 has the form Dn CB2 or, in a simpler format Dn CB2
Pergamon PII: S0965-8564(98)00008-1
BUS PRIORITY USING PRE-SIGNALS JIANPING WU* and NICK HOUNSELL
Department of Civil and Environmental Engineering, University of Southampton,Southampton,U.K., SO17 1BJ (Received 23 July 1996; in revised form 14 January 1998) AbstractÐThe need to provide ecient public transport services in urban areas has led to the implementation of bus priority measures in many congested cities. Much interest has recently centred on priority at signal controlled junctions, including the concept of pre-signals, where trac signals are installed at or near the end of a with-¯ow bus lane to provide buses with priority access to the downstream junction. Although a number of pre-signals have now been installed in the U.K., particularly in London, there has been very little published research into their design, operation and optimisation. This paper addresses these points through the development of analytical procedures which allow pre-implementation evaluation of speci®c categories of pre-signals. The paper initially sets out three categories of pre-signal, which have dierent operating characteristics, dierent requirements for signalling and dierent impacts on capacity and delay. Key issues concerning signalling arrangements for these categories are then discussed, together with a summary of the analytical approach adopted and the assumptions required. Equations are developed to allow appropriate signal timings to be calculated for pre-signalised intersections. Further equations are then developed to enable delays to priority and nonpriority trac, with and without pre-signals, to be estimated with delay being taken here as the key performance criterion. The paper concludes with three application examples illustrating how the equations are applied and the impacts of pre-signals in dierent situations.The analyses con®rm the potential bene®ts of pre-signals, where these signals apply to non-priority trac only. Where buses are also subject to a pre-signal, it is shown that disbene®ts to buses can often occur, unless bus detectors are used to gain priority signalling. # 1998 Elsevier Science Ltd. All rights reserved Keywords: pre-signal, queue relocation, bus priority, public transport, trac management 1. INTRODUCTION
In busy urban areas where trac congestion is severe and bus frequencies are high, priorities are often provided to buses to reduce bus passengers' journey time and to improve regularity. Many bus priority methods have been studied since the 1960s (Robertson and Vincent, 1975; King, 1983; Hounsell and McDonald, 1988; Roberts and Buchanan, 1993; Oakes et al, 1994; Tee et al, 1994; Fox et al, 1995; Oakes and Metzger, 1995). The traditional bus lane has, since the 1970s, become widely accepted as a means of both allowing buses to overtake existing queues and reserving road space, and therefore capacity, for buses to combat trac growth and consequent delay over time. As a bus priority strategy, the concept of a pre-signal was ®rst documented in the U.K. in the Department of Transport in the report Keeping Buses Moving in December 1991 (Oakes et al, 1994) although the techniques has been used for many years in some European cities. Since 1993 a number of schemes have been designed in detail and implemented (Oakes et al, 1994), particularly in London. Fig. 1 shows an example of a pre-signal operating at Shepherds Bush in London. The implementation of pre-signals in U.K. cities is becoming signi®cant. Taking London only, 14 pre-signals are now in place since the ®rst scheme was introduced in Shepherd's Bush in 1993. A further 20±25 pre-signals are planned for the coming years to contribute to London's Bus Priority Network. Many pre-signals have had site-speci®c aspects and operate with additional trac detection for real time control. However, there remains a strong need for a methodology for the design and evaluation of pre-signals at the pre-implementation stage, to ensure that optimum deployment is achieved. The research described in this paper focuses on this point. A pre-signal aims to give buses priority access into a bus advance area of the main junction stop line so as to avoid the trac queue and reduce bus delay at the signal controlled junction. There are two particular cases where pre-signals can achieve major bene®ts for buses. The ®rst is where buses must move from kerbside bus stops to the centre of the road to turn right at the main *Author for correspondence. Fax: 01703 593152; e-mail: [email protected]
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Jianping Wu and N. Hounsell
junction. Without pre-signals they must force their way through congested trac queues. The presignal can provide an empty reservoir in which they can position themselves at will. The second case is using a pre-signal as a device for local queue relocation. The approach to a congested, trac signal controlled junction may not always be a practical location for a bus lane, particularly if the road width is inadequate. However, it may sometimes be possible to provide a bus lane further upstream. In such a circumstance a pre-signal could be installed at the end of the bus lane to control the non-priority trac and keep the reservoir area between the pre-signal and main signal clear to allow buses clear access to the main signal (Roberts and Buchanan, 1993). This may also be bene®cial for more conventional with-¯ow bus lane approaching a junction which incorporates a `setback' for general trac. Three dierent categories of pre-signal design have been suggested and/or implemented in the U.K. (Roberts and Buchanan, 1993; Oakes et al, 1994; Tee et al., 1994) each with its own operating characteristics (these are denoted categories A, B and C in this paper). In category A (Fig. 2) the pre-signal controls only the non-priority trac with buses uncontrolled (or subject to `give way control'). Figure 2 is a typical layout of pre-signal category A. In contrast, in pre-signal category B, buses are also controlled by a pre-signal at the end of the bus lane. When the pre-signal in the bus lane turns red, non-priority vehicles receive right of way, and vice versa. A typical layout of a pre-signal category B is shown in Fig. 3. An intermediate layout, not directly considered here, is where the bus has to give way to non-priority trac at the end of the bus lane.
Fig. 1. A pre-signal controlled junction in london (courtesy of London Transport Buses)
Fig. 2. Typical layout of pre-signal category A.
Bus priority using pre-signals
565
Fig. 3. Typical layout of pre-signal category B.
The number of lanes for general trac in the bus advance area is usually greater than the number of lanes available upstream of the pre-signal. (An equal number of lanes is shown in Figs 2 and 3 for illustration). This variation has an important eect on pre-signal operation and bene®ts, as will be shown later in the paper. With both categories A and B, the signal timings at the presignal and the main junction are co-ordinated for maximum eciency (for category B, the main junction signals could be co-ordinated either with the bus or with the non-priority trac pre-signal, according to priority eciency). A pre-signal category C can also be de®ned which has a similar layout as category B, except that detectors are installed in the bus lane. When a bus approaches the pre-signal, a red signal is shown to the non-priority trac stream to enable the bus to access to the main signal without impedance. The bus lane pre-signal then turns to red, allowing non-priority trac to ®ll the bus advance area behind the bus after it has selected the appropriate lane at the main signal. The amount of bus delay in pre-signal category C depends on whether there is co-ordination for buses between the pre-signal and the main signal. For example, if buses can call the green in both the pre-signal and the main signal, the bus has full priority, and may experience no delay at all at the junction. However, if buses can only call the green at the pre-signal, and may be delayed by a red light at the main signal, the delay to buses will depend on the timing of the main signal. Delays to non-priority vehicles are more complicated to analyse for Category C, and, a microscopic simulation model may be required to study such a VA or semi-VA (only bus's arrival is detected) controlled junction. In this paper, only the pre-signal categories A and B with ®xed time controls are considered in detail. The following sections describe an analytical approach developed for determining optimum pre-signal timings and vehicle delays in pre-signal controlled approaches. The terms and symbols used in the paper are summarised in Appendix A at the back of the paper. 2. PROBLEM REVIEW AND ASSUMPTIONS
There are two potential problems which need to be guarded against when designing pre-signals. The ®rst is the potential waste of junction capacity by not releasing trac from the pre-signals suciently early to make use of green time at the main junction. The second is that the relocated trac queue may block back to the upstream junction (Roberts and Buchanan, 1993). Both the problems can usually be addressed by careful signal setting at both the pre-signal and the main signal, and the correct estimation of the re-located trac queue at the pre-signal. To ensure no potential capacity waste at the junction, the following two assumptions were made while setting signal times for pre-signal and main signal. Assumption 1 The non-priority vehicles arriving at the pre-signal stop line during the red time of the main signal, rm , will be fully discharged into the main signal stop line to ®ll up the bus advance area before the main signal green time starts. This has the form
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Jianping Wu and N. Hounsell
rm d rm ÿ rp Sl Np
1
where rp and rm are the red time at the pre-signal and main signal respectively (see Fig. 4), d is the rate of demand of non-priority vehicles in the approach (vehicles/s). Sl is the lane based saturation ¯ow (vehicles/s/lane) and Np the number of lanes at the pre-signal for non-priority vehicles. This assumption implies co-ordination of the pre-signal and the main signal for non-priority trac and ensures no loss of junction capacity due to the pre-signal. Assumption 2 Junction capacity equals demand. In other words, there will be no residual queues at the main signal stop line at the end of green time, gm . This is given by the following equation. gm Sl NmC d dB
2
where gm is the green time at the main signal and C the cycle time. Nm is the number of lanes at the main signal stop line and dB the rate of bus ¯ow (buses/s). d and Sl are as de®ned earlier (also see Glossary in Appendix). This situation describes a degree of saturation of one. Where is less than one, a bus lane (or pre-signal) could not usually be justi®ed; when is greater than one for a period of time, there would be signi®cant queuing alongside the bus lane which itself should provide greater bene®t to buses. However, the operation (and bene®t) of the pre-signal would remain as analysed in the paper. With these assumptions there is also no consideration of stochastic processes (e.g. in vehicle arrivals and departures). This is justi®ed for this case of pre-implementation evaluation because of the ¯ow regime of interest ( equals one) and because of the proximity of the main junction and the pre-signal, (which would produce minimal platoon departure). However, for real life pre-signal operation stochastic eects may be sucient to warrant vehicle detection to ensure that maximum bene®ts are obtained from the pre-signal (this has been necessary in some London intersections). 3. PRE-SIGNAL AND MAIN SIGNAL TIMING
In this section, equations are developed to calculate signal times for pre-signal categories A and B, given known signal cycle, C, of the main junction. Pre-signal and main signal timing for pre-signalised intersections are controlled by the two assumptions de®ned in the last section. With assumption 2 or eqn (2), we have gm C
d dB Sl Nm
3
From Fig. 4, we have rm C ÿ g m
Fig. 4. The relationship of rp ; gp ; rm ; gm and C.
4
Bus priority using pre-signals
567
Substituting eqn (3) into eqn (4) yields rm C1 ÿ
d dB Sl Nm
5
For pre-signal timing, from assumption 1 or eqn (1), we have rp 1 ÿ
d rm Sl Np
6
From Fig. 4, we have gp C ÿ rp
7
Substituting (6) into (7) yields g p gm
d rm Sl Np
8
Further, in pre-signal category B, the bus lane is also controlled by a pre-signal which has the following relationship with the non-priority trac pre-signal. gpB rp
9
rpB gp
10
and
where gpB and rpB are the green and red time for the bus lane pre-signal in pre-signal category B. Substituting eqns (6) and (8) into eqns (9) and (10) respectively, we have gpB 1 ÿ
d rm Sl Np
11
d rm Sl Nm
12
and rpB gm
4. THE BENEFITS AND DISBENEFITS TO BUSES IN PRE-SIGNALLED JUNCTIONS
The purpose of a pre-signal is to give buses priority at signal controlled intersections. In this section, we will discuss the bene®ts and disbene®ts to buses in both pre-signal categories A and B. The bene®ts to buses or bus delay saving is de®ned as the dierence in bus delay with and without pre-signals. 4.1. Bus delay without pre-signal Assuming buses and non-priority vehicles arrive randomly, they have the same average delay without a pre-signal. Figure 5 shows the cumulative arrival and departure at a normal (without pre-signal) signal controlled intersection. The total trac delay per cycle without pre-signal, D(normal), equals the area of triangle ADF, which is given 1 D normal C2 d dB ÿ gm C d dB 2
568
Jianping Wu and N. Hounsell
Fig. 5. The cumulative arrival/departure in normal signalised intersections.
or 1 D normal C d dB rm 2 The average delay per vehicle without pre-signal, d(normal), is given 1
d normal 2
C d dB rm C d dB
or d normal
rm 2
13
eqn (13) is the same as the ®rst term of Webster's expression (Webster and Ellson, 1965) when 1. 4.2. Bus delay and delay saving with pre-signal category A In pre-signal category A, non-priority vehicles are held by the pre-signal for a time period of rp in each cycle to give priority to buses moving into the bus advance area without any impedance. As shown in Fig. 6, line AH is the cumulative bus arrival, AG the non-priority vehicle arrival and AD the total arrival without pre-signal. With pre-signal category A, the total arrival follows the line of ABCD instead of AD at the main signal. Buses arriving during time rp have total delay ABEF (see Fig. 6) at the main signal. This part of bus delay, D CA1, is D CA1 ABEF
14
Buses arriving during gp have no priority access to the bus advance area. This part of bus delay, D CA2, is given by: D CA2 BCDE
gp dB C d gp dB
4.2.1. The total bus delay with pre-signal category A. A, D CA, can be shown as:
15
The total bus delay in pre-signal category
D CA D CA1 D CA2
16
Bus priority using pre-signals
569
Fig. 6. The cumulative arrival/departure at the main signal with pre-signal category A.
The areas of ABEF and BCDE (see Fig. 6) can be estimated by the following equations. Detailed derivation of the equations can be found in Appendix B. rp dB gm rp dB 2rm ÿ rp 2 C d dB
17
1 frm gp C 2d 2rm gp C rp C gm ÿ gp d dB rm g2p 2dB g 2C d dB
18
ABEF
BCDE
The total bus delay with pre-signal category A, D CA, therefore has the form D CA
1 d 2dB 2rm C3 ÿ rp rm C2 2C d dB C d gp dB 2d dB rm C3 ÿ r2p gm C 3db rm gp C2
19
with consideration of eqns (14±18). 4.2.2. The average bus delay with pre-signal category A. category A, d CA, has the form d CA
D CA C dB
The average bus delay in pre-signal 20
or d CA
1 f d dB 2rm C2 ÿ rp rm C 2d rm C2 ÿ r2p gm 2dB rm gp Cg 2C d dB C d gp dB 21
with substitution of eqn (19). Without the pre-signal, parameter rp equals 0 and gp equals C. Equation (21) then becomes d CA which has the same form as eqn (13).
rm 2
22
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Jianping Wu and N. Hounsell
4.2.3. Bus delay saving with pre-signal category A. As has been de®ned earlier, the bus delay saving in pre-signal category A, d CA, is the dierence between the average bus delay without pre-signal and that with pre-signal category A. The general form of it is d CA d normal ÿ d CA
23
Substituting eqns (13) and (21) into eqn (23) yields d CA
gm r2p 2d 2C d dB C d gp dB
24
eqn (24) shows that the bus delay saving increases as the parameters gm and rp increase. 4.3. Bus delay and delay saving with pre-signal category B In pre-signal category B, when buses receive the green signal, non-priority vehicles are held on red, and vice versa. Bus delay with pre-signal category B consists of (i) delay at the bus lane presignal, and (ii) delay at the main signal. Figure 7 shows the cumulative arrival and departure of buses at the bus lane pre-signal. Line AC in eqn (7) indicates the queue of buses which arrived during time rpB gp of last signal cycle, CE the cumulative bus arrivals and AJB the departures at the bus lane pre-signal. The total bus delay per cycle for the bus lane pre-signal, D CB1, is therefore the sum of the areas ACJ and BED, i.e. D CB1 ACJ BED
25
In Fig. 8, line AJB is the cumulative bus arrival from the bus lane pre-signal to the main signal and BFD the non-priority vehicle arrival from the pre-signal to the main signal. Because all the buses will take the front positions of the trac queue at the main signal and discharge ®rst during the green time gm , the total bus delay at the main signal with pre-signal category B; D CB2, is the area of AJBEK in Fig. 8, i.e. D CB2 AJBEK
4.3.1. The total bus delay with pre-signal category B. category B; D CB, has the general form
26
The total bus delay per cycle with pre-signal
Fig. 7. The cumulative arrivals/departures of buses at the bus lane pre-signal with pre-signal category B.
Bus priority using pre-signals
571
Fig. 8. The cumulative arrival/departure at the main signal with pre-signal category B.
D CB D CB1 D CB2 or D CB ACJ BED AJBEK This becomes D CB
C dB 2rm gp ÿ rp 2
27
by consideration of the areas of ACJ, BED and AJBEK. Detailed derivation of eqn (27) can be found in Appendix B. 4.3.2. The average bus delay with pre-signal category B. category B, d CB, has the form d CB
The average bus delay with pre-signal
D CB C dB
Substitution of eqn (27) yields 1 d CB 2rm gp ÿ rp 2
28
4.3.3. The Bus Delay Savings with Pre-signal Category B. The bus delay saving with pre-signal category B, d(CB), is de®ned as the dierence between the average delay per vehicle without the pre-signal, d(normal), and the average bus delay with the pre-signal category B, d(CB). This can be expressed as d CB d normal ÿ d CB
29
1 d CB rm gp ÿ rp 2
29
or
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Jianping Wu and N. Hounsell
5. THE BENEFIT AND DISBENEFITS TO NON-PRIORITY VEHICLES IN PRE-SIGNALISED JUNCTIONS
This section develops equations to estimate the impact of pre-signals on non-priority trac. 5.1. Non-priority vehicle delay in pre-signal For non-priority vehicles, the total delay consists of two parts, the delay at the pre-signal and that at the main signal. There is no dierence on delay of non-priority vehicles at pre-signal between categories A and B. As shown in Fig. 9, line AC is the arrival of non-priority vehicles and area ABD the total non-priority vehicles delay at the pre-signal, Dn(P), which has the form 1 Dn P rm rp d 2
30
5.2. The delay and delay savings of non-priority vehicles with pre-signal category A 5.2.1. Non-priority vehicle delay in main signal with pre-signal category A. As shown in Fig. 6, the area BCDE is the total delay at the main signal for all non-priority vehicles and buses those arriving during time gp . Therefore, the total non-priority vehicle delay at the main signal with presignal category A, Dn CA2, has the form Dn CA2 BCDE
C d C d gp dB
With consideration of eqn (18), it becomes Dn CA2
d frm gp C 2d 2rm gp C rp C gm ÿ gp d dB rm g2p 2dB g 31 2 d dB C d gp dB
5.2.2. Total non-priority vehicle delay with pre-signal category A. The total non-priority vehicle delay with pre-signal category A, Dn CA , is the sum of delays at the pre-signal and main signal. Dn CA Dn P Dn CA2 With consideration of eqn (30) and eqn (31), this becomes Dn CA
1 frm C 3 3d r2p C2 2C rp rm gp C 2d dB rm gp C2 d 2dB g 2C d dB C d gp dB 32
Fig. 9. The cumulative arrival/departure of non-priority vehicles at the pre-signal.
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5.2.3. Average non-priority vehicle delay with pre-signal category A. The average non-priority vehicle delay with pre-signal category A, dn CA, has the general form dn CA
Dn CA C d
Substitution of eqn (32) yields dn CA
1 frm C2 2d r2p C 2C rp rm gp d dB rm gp C 2dB g 2C d dB C d gp dB
33
5.2.4. Delay savings of non-priority vehicles with pre-signal category A. The delay savings of nonpriority vehicles with pre-signal category A, d CA, is de®ned as the dierence between the delay without pre-signal and that with pre-signal category A. The general form is dn CA d normal ÿ dn CA Consideration of eqns (13) and (33) yields dn CA ÿ
r2p gm d dB 2C d dB C d gp dB
34
As we can see, eqn (34) gives a negative value. This means that the average delay for non-priority vehicle has a small increase with pre-signal category A. The reason for this can be explained by comparing Figs 10 and 11. Assuming all buses and non-priority vehicles arrive randomly, an average trac queue at the main signal will have the pattern as shown in Fig. 10. With pre-signal category A, buses arriving during time rp are given priority to access the bus advance area. Therefore, these buses will take the front positions of the queue at the main signal. The trac queue at the main signal will have the pattern as shown in Fig. 11. More non-priority vehicles will locate to the rear part of the trac queue compared to the without pre-signal case. There is a reduction in average delay to buses (as shown in eqn (24)) and, meanwhile, an increase of average delay to non-priority vehicles with pre-signal category A (as shown in eqn (34)). However, the total vehicle (including buses and non-priority vehicles) delay with pre-signal category A in the approach remains the same. This hypothesis can be veri®ed as below. Without the pre-signal, the total vehicle (including buses and non-priority vehicles) delay, TD, is 1 TD rm C d dB 2
35
With pre-signal category A, the total delay, TD CA, will be the sum of total bus delay and total non-priority vehicle delay. This can be shown that
Fig. 10. Vehicle locations without pre-signal.
Fig. 11. Vehicle locations with pre-signal category A.
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Jianping Wu and N. Hounsell
TD CA D CA Dn CA
36
Substituting eqns (19) and (32) into eqn (36)) yields 1 TD CA rm C d dB 2
37
The total vehicle delay in eqns (35) and (37) are the same. 5.3. The delay and delay savings of non-priority vehicles with pre-signal category B 5.3.1. Non-priority vehicle delay at the main signal with pre-signal category B. The area BFDE in Fig. 8 indicates the total delay of non-priority vehicles at the main signal with pre-signal category B, Dn CB2, which has the form: Dn CB2
d rm gp d rm gp gm C dB 2 d dB
38
Detailed derivation of eqn (38) can be found in Appendix B. 5.3.2. Total non-priority vehicle delay with pre-signal category B. The total non-priority vehicle delay with pre-signal category B, Dn CB, is the sum of delays at the pre-signal and the main signal. The general form of this is Dn CB Dn P Dn CB2 or Dn CB
C d rm d C dB 2 d dB
39
with consideration of eqns (30) and (38). 5.3.3. Average non-priority vehicle delay with pre-signal category B. vehicle delay with pre-signal category B, dn CB, is given by dn CB
The average non-priority
Dn CB C d
Substitution of eqn (39) yields dn CB
rm d C dB 2 d dB
40
5.3.4. Delay savings for non-priority vehicles with pre-signal category B. The delay savings for the non-priority vehicles with pre-signal category B, dn CB, is de®ned as the dierence in vehicle delay without pre-signal and that with pre-signal category B. This has a general form dn CB d normal ÿ dn CB This becomes dn CB ÿ With consideration of eqn (13) and eqn (40).
gm dB 2 d dB
41
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As we can see, eqn (41) gives a negative value. This means that there is an increase in average delay for non-priority vehicles with pre-signal category B. The reason for this is similar to that with pre-signal category A. In the case of pre-signal category B, all the buses queued in the bus lane pre-signal are discharged into the bus advance area during time gpB rp when the nonpriority vehicles are held by the pre-signal red light. Therefore, all the buses will take the front positions of the queue at the main signal. The trac queue will have a pattern as shown in Fig. 12. Although there are no non-priority vehicles which will be held to the next cycle at the main signal stop line, the change of the position of non-priority vehicles in the trac queue leads to a slight increase in their average vehicle delay. Unlike pre-signal category A, the total vehicle delay (including buses and non-priority vehicles) with pre-signal category B also increased because the buses arriving at the bus lane pre-signal during time rpB gp are not able to discharge into the bus advance area and leave the main signal stop line until green time gpB rp in the following signal cycle. This hypothesis is veri®ed as below. Without the pre-signal, the total delay has the form as shown in eqn (35). With pre-signal category B, the total delay, TD CB, is the sum of total bus delay and total non-priority vehicle delay. TD CB D CB Dn CB Substitution of eqns (27) and (39) yields gm C 2dB 1 TD CB rm C d dB gp C dB ÿ 2 d dB 2
42
The extra total vehicle delay with pre-signal category B per cycle, TD CB, is the dierence of the total vehicle delay without pre-signal and that with pre-signal category B. This is TD CB gp C dB ÿ
gm C 2dB 2 d dB
43
by comparing the eqns (42) and (35). 6. THE BUS ADVANCE AREA AND THE RELOCATED TRAFFIC QUEUE LENGTH
The size of bus advance area depends on the layout of the approach and can be estimated by the following equation. N mL BAA d dB rm H or L BAA
d dB rm H Nm
44
where L BAA is the length of the bus advance area and H the average vehicle queued headway (a default value of 5.5 m may be used when no data available) from the approach. Nm , d , dB , and rm are as de®ned earlier (also see Glossary in Appendix). The relocated trac queue of non-priority trac due to the pre-signal has its maximum length at the end of red time, rm , and may be estimated by the following equation.
Fig. 12. Vehicle locations with pre-signal category B.
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Jianping Wu and N. Hounsell
Q relocated
rm d H Np
45
where Q relocated is the estimated maximum queue length (m) backwards from the non-priority trac pre-signal stop line. rm , d , Np and H are de®ned as earlier (also see Glossary in Appendix). Equation (45) is useful to estimate the potential impact of the locally relocated trac queue to the upstream trac. 7. APPLICATION EXAMPLES
Three examples are presented here to demonstrate how to use the equations developed in this paper for signal time setting; estimations of delay and delay savings and the lengths of the relocated trac queue and of the bus advance area required when designing pre-signalised junctions. 7.1. The example approaches The example approach 1, as shown in Fig. 13, has the following data: Np 1; Nm 2; C 100 s; Sl 2000=3600; d 1000=3600, dB 60=3600. Assuming a bus has a pcu=2.0, the converted dB 120=3600 (pcu/s). The example approach 2, as shown in Fig. 14, has the following data: Np 2; Nm 2; C 100 s; Sl 2000=3600; d 1000=3600, dB 60=3600. Assuming a bus has a pcu=2.0, the converted dB 120=3600 (pcu/s). The example approach 3, as shown in Fig. 15, has the following data: Np 2; Nm 3; C 100 s; Sl 2000=3600; d 2000=3600, dB 60=3600. Assuming a bus has a pcu=2.0, the converted dB 120=3600 (pcu/s).
Fig. 13. Example approach 1.
Fig. 14. Example approach 2.
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Fig. 15. Example approach 3.
7.2. Signal timing To set the signal times at the pre-signal and main signal, the following equations are used: . . . .
Green time in main signal: Equation (3) Red time in main signal: Equation (5) Green time in pre-signal: Equation (8) Red time in pre-signal: Equation (6)
Further, the signal times in bus lane with pre-signal category B are . Green time in bus lane pre-signal: Equation (11) . Red time in bus lane pre-signal: Equation (12) Table 1 summarises the signal times of the above three examples. 7.3. Bus delay and delay saving To estimate vehicle delay and delay savings with both pre-signal categories A and B, the following equations are used. . . . . . . . .
Average bus delay with pre-signal category A:eqn (21) Bus delay saving with pre-signal category A: eqn (24) Average bus delay with pre-signal category B: eqn (28) Bus delay saving with pre-signal category B: eqn (29) Average non-priority vehicle delay with pre-signal category A: eqn (33) Non-priority vehicle delay saving with pre-signal category A: eqn (34) Average non-priority vehicle delay with pre-signal category B: eqn (40) Non-priority vehicle delay saving with pre-signal category B: eqn (41)
Vehicle delay and delay savings with pre-signal categories A and B are calculated for all the three example approaches and listed in Tables 2±5. It is clear from these examples that the pre-signal category B is not bene®cial using ®xed time signalling. Active bus detection and signal response is likely to be needed to provide the required bene®ts. Table 1. Signal timings for the three examples Example approaches Example 1 Example 2 Example 3
rm (Seconds)
gm (Seconds)
rp (Seconds)
gp (Seconds)
72 72 65
28 28 35
36 54 32.5
64 46 67.5
For the pre-signal category B,rpB gp ,andgpB rp
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Jianping Wu and N. Hounsell Table 2. Bus delay and delay saving for pre-signal category A
Examples
One Two Three
Without pre-signal (sec/bus)
With pre-signal (sec/bus)
Bus delay saving (sec/bus)
Bus delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
34.50 32.55 30.80
1.50 3.45 1.70
4.2% 9.6% 5.2%
Table 3. Bus delay and delay saving for pre-signal category B Examples
One Two Three
Without pre-signal (sec/bus)
With pre-signal (sec/bus)
Bus delay saving (sec/bus)
Bus delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
86.00 68.00 82.50
ÿ50.00 ÿ32.00 ÿ50.00
ÿ139% ÿ89% ÿ154%
Table 4. Non-priority vehicle delay and delay saving for pre-signal category A Examples
One Two Three
Without pre-signal (sec/veh)
With pre-signal (sec/ veh)
Non-priority vehicle delay saving (sec/ veh)
Non-priority vehicle delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
36.18 36.41 32.60
ÿ0.18 ÿ0.41 ÿ0.10
ÿ0.50% ÿ1.14% ÿ0.30%
Table 5. Non-priority vehicle delay and delay saving for pre-signal category B Examples
One Two Three
Without pre-signal (sec/ veh)
With pre-signal (sec/ veh)
Non-priority vehicle delay saving (sec/ veh)
Non-priority vehicle delay saving in percentage (delay saving)/ (delay without pre-signal)
36.00 36.00 32.50
37.50 37.50 33.50
ÿ1.50 ÿ1.50 ÿ1.00
ÿ4.16% ÿ4.16% ÿ3.08%
Table 6. The relocated queue length of the three examples Examples
One Two Three
The maximum relocated queue length with pre-signals (Meters) 110 55 99
7.4. The relocated queue length at pre-signal The length of the relocated queue is critical sometimes for a short link with very close signal controlled junctions. Equation (45) can be used to estimate the maximum queue length of the relocated trac queue. Table 6 lists the results of the three examples with the assumption that H (average vehicle length) equals 5.5 m. There is no dierence in the relocated trac queue length of non-priority vehicles between pre-signal categories A and B.
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Table 7. The length of the bus advance area for the three examples Examples
Length of the bus advance area with pre-signals (Meters)
One Two Three
61.5 61.5 70.2
7.5. The size of bus advance area The size of bus advance area is an important element to design a pre-signal controlled junction. Equation (44) can be used to estimate the size of bus advance area. Table 7 lists the results of the three examples with the assumption that H (average vehicle queued headway) equals 5.5 m. There is no dierence in the size of bus advance area between presignal categories A and B. 8. CONCLUSIONS
This paper has illustrated how an analytical approach can be used for the pre-implementation evaluation of pre-signals. The approach covers issues of design, capacity, signal settings, delay and delay saving estimation for priority and non-priority trac, calculation of the relocated trac queue at pre-signal, and the design of the bus advance area. Examples have shown how delay savings to buses can be achieved with the category A pre-signal (where buses are unsignalised at the pre-signal) without signi®cant disbene®t to non-priority trac. Delay savings to buses are highest where there is a long red period at the non-priority trac pre-signal, which is possible when the proportion of green time at the main signal is low. However, category B pre-signals (where buses are signalised at the pre-signal) are shown to generally cause disbene®t to buses unless special bus detectors are installed to give signal priority to buses. In pre-signal category A, the bus delay saving is caused by the relocation of vehicles in the trac queue at the main signal, as shown by Fig. 11. There is a delay saving to buses and a small delay increase to non-priority vehicles. However, the total delay of the whole approach remains unchanged compared to without pre-signal. In pre-signal category B, although all buses can move to the front of the trac queue at the main signal, those buses arriving during time rpB gp are held at the bus lane pre-signal until the time gpB of the following cycle, extra bus delay therefore results. This also results in a overall delay increase for the approach as shown in eqn (43). The analytical approach has been based on two assumptions which are justi®ed for pre-implementation evaluation when an estimate of bene®ts is required. However, the approach excludes stochastic processes which can be potentially important for real time control where trac demand is at or near capacity. Pre-signal implementation may therefore require close attention to be paid to vehicle detection if the best performance is to be achieved. Overall, the approach developed in this paper for pre-signal timing, estimating delays, lengths of the relocated trac queue and bus advance area is considered to provide a good basis for evaluating pre-signal designs and could readily be extended to other signalling schemes of this nature.
REFERENCES Fox K., Montgomery F. and Shepherd S. (1995) Integrated ATT strategies for urban arteries; DRIVE II project PRIMAVERA: 2, Bus priority in SCOOT and SPOT using TIRIS. Trac Engineering and Control 36, 356±361. Hounsell N. and McDonald M. (1988) Evaluation of bus lanes. Contract Report 87. Transport and Road Research Laboratory, Department of Transport, Crowthorne, U.K. King G. N. (1983) Bus priority in LondonÐtechniques for the 1980's. Proceedings of Seminar K: Trac operation and management, PTRC 11th Annual Meeting. London. Oakes J. and Metzger D. (1995) Park view pre-signals in Uxbridge road, England. Trac Engineering and Control 36, 62±67. Oakes J., ThellMann A. M. and Kelly I. T. (1994) Innovative bus priority measures. Proceedings of Seminar J, Trac Management and Road Safety, 22nd PTRC European Transport Summer Annual Meeting, University of WARWICK, U.K., vol.381, pp.301±312. Roberts M. and Buchanan C. (1993) Bus priorityÐthe solution and west London demonstration project.Proceedings of Seminar C, Highways and Planning, 21st PTRC European Transport Summer Annual Meeting, vol.365, pp.49±57.
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Robertson D. I. and Vincent R. A. (1975) Bus priority in a network of ®xed-time signals. Laboratory Report, Transport and Road Research Laboratory, Department of Transport, Crowthorne, U.K. Tee A., Cuthbertson T. and Carson G. (1994) Public transport initiatives in Surrey. Trac Engineering and Control 35, 70± 73. Webster F. V. and Ellson P. B. (1965) Trac signals for high-speed roads.Road Research Technical Paper No. 74.Road Research Laboratory, HMSO, London.
APPENDIX A
A.1. Glossary and symbols
The terms used in this paper have the following de®nitions: Pre-signals: Bus advance area:
Trac signals at or near the end of bus lane to provide buses with priority access to the downstream junction. Signals apply to non-priority trac and may apply to priority trac count on strategy. The area between the end of a with-¯ow bus lane and the main junction stop line into which buses gain preferential access through the use of pre-signals.
The symbols used in this paper are listed as below alphabetically with the Greek symbols appearing at the end: gp gm gpB Nm Np rm rp rpB Sl d dB
Eective green time at non-priority trac lane pre-signal Eective green time at main signal Eective green time at bus lane pre-signal Number of trac lanes in the bus advance area Number of trac lanes for non-priority trac at pre-signal stop line Eective red time at main signal Eective red time at non-priority trac lane pre-signal Eective red time at bus lane pre-signal Lane based saturation ¯ow (vehicles/s/lane) Rate of trac demand (vehicles/s) Rate of bus ¯ow (buses/s) Degree of saturation
APPENDIX B
B.1. The derivation of the main equations in the paper B.1.1. The area of ABEF in Figs 6. Figure B1 is the same as Fig. 6 with some added lines for equation derivation purposes.
Fig. B1. The cumulative arrival/departure at the main signal with pre-signal category A (same as Fig. 6).
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ABEF ABKF KEF
B1
1 1 ABKF rm ÿ rp rm rp dB 2rm ÿ rp rp dB 2 2
B2
1 KEF KE rp dB 2
B3
From Fig. B1, we have
Because, KE Sl Nm rp dB or rp dB Sl Nm
B4
rp dB 2 2Sl Nm
B5
Sl Nm
C d dB gm
B6
KEF
gm rp dB 2 2C d dB
B7
KE eqn (B3) becomes KEF From eqn (2) in the main body of the paper, we have
Therefore,
The area of ABEF therefore has the form ABEF
rp dB gm rp dB 2rm ÿ rp 2 c d dB
B8
B.1.2. The area of BCDE in Fig. 6. As shown in Fig. B1, the area of BCDE has the general form BCDE BCK LDF ÿ LDC ÿ KEF
B9
1 BCK rm ÿ rp rm d dB ÿ rp dB 2
B10
LDF
gm C d dB 2
B11
LDC
gm gm d dB 2
B12
The area of BCDE therefore has the form 1 gm rp dB 2 BCDE frm ÿ rp rm d dB ÿ rp dB gm C d dB ÿ g2m d dB ÿ g 2 C d dB
B13
by considering eqn (B10),(B11),(B12) and (B7). Or, in dierent format, BCDE
1 frm gp C 2d 2rm gp C rp C gm ÿ gp d dB rm g2p 2dB g 2C d dB
B14
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Fig. B2. The cumulative arrival/departure of buses at the bus lane pre-signal with pre-signal category B (same as Fig. 7).
Fig. B3. The cumulative arrival/departure at the main signal with pre-signal category B (same as Fig. 8).
B.1.3. The derivation of equation (27). As shown in Figs B2 and B3, the buses delay at the bus lane pre-signal equals the sum of the areas of ACJ and BED. The buses' delay at the main signal is the area of AJBEK, which can further break down to the areas of AJBL and LBEK. Therefore, the total bus delay with pre-signal category B is D CB ACJ BED AJBL LBEK
B15
As area ACJ plus area AJBL equals to area ACBL (see Fig. B3), eqn (B15) becomes D CB ACBL BED LBEK
B16
1 1 ACBL AC BLrp gp dB C dB rp 2 2
B17
1 1 1 BED gp ED gp gp dB g2p dB 2 2 2
B18
From Fig. B3, we have
Bus priority using pre-signals 1 1 1 LBEK BE LKC dB 2LK NEc dB 2 rm ÿ rp NEC dB 2 2 2
583 B19
Because NE Sl Nm C dB or NE
C dB Sl Nm
B20
eqn (B19) becomes 1 C dB LBEK 2 rm ÿ rp C dB 2 Sl Nm
B21
1 gm dB LBEK 2 rm ÿ rp C dB 2 d dB
B22
This further becomes
by considering eqn (B6). Therefore, eqn (16) has the form D CB
dB gm C dB C gp rp g2P 2C rm ÿ rp 2 d dB
B23
C dB 2rm gg ÿ rp 2
B24
or, in dierent format D CB
B.1.4. The derivation of equation (38). As shown in Fig. B3, the total non-priority vehicle delay at the main signal Dn CB2 has the form Dn CB2 BFDE
B25
Dn CB2 BFDE BFN NMDE ÿ FMD
B26
1 1 BFN rm ÿ rp FN rm ÿ rp rm d 2 2
B27
1 1 gm dB NMDE MD NEMN gm C d 2 2 d dB
B28
From Fig. B3, we have
by referring eqn (B20) and eqn (B6), and 1 1 FMD gm gm d g2m d 2 2
B29
d gm Bd f rm ÿ rp rm gm C ÿ g2m g 2 d dB
B30
d rm gp d rm gp gm C dB 2 d dB
B31
Therefore, Dn CB2 has the form Dn CB2 or, in a simpler format Dn CB2