Hiroshi Shimamoto, Hiroshima University, Japan

Fumitaka Kurauchi, Kyoto University, Japan Jan-Dirk Schmöcker, Tokyo Institute Of Technology, Japan

Michael G H Bell, Imperial College London, UK Received: October 2007 Accepted: January 2008

Evaluating Critical Lines and Stations Considering

the Impact of the Consequence Using Transit

Assignment Model -Case Study of London’s

Underground Network

Hiroshi Shimamoto

Fumitaka Kurauchi

Jan-Dirk Schmöcker

Michael G H Bell

As the problem of full transit vehicles is encountered daily by

passengers in most of the big cities, previous research evaluated the

consequence of overcrowding in terms of on-board crowding and

passengers not being able to board with full vehicles. The impact of

overcrowding in the real world is, however, not necessarily

proportional to these numbers. This paper attempts to specify the

critical lines and stations of a network by considering the number of

passengers failing to board and attempting to evaluate its impact on

service quality and safety risks. The hypothesis is that larger stations

with wider platforms can often cope better with overcrowding than

smaller stations. Therefore a station size dependent satisfaction function is proposed, which takes values from 0 to 1. The method is

applied to London’s underground network with a number of scenarios

which show critical stations in the network if delays occur.

1. Introduction

The problem of full transit vehicles is encountered daily by

passengers in most of the big cities of the world. For example, many commuters during the morning peak-hours fail to board the transits in

London’s underground. This overcrowding leads not only to journey

unreliability through having to wait for emptier services but also to stress,

delays through passengers trying to push on overcrowded trains and even health and safety risks. The costs related to crowding and delays of its

2

public transportation network are estimated at £230m per year for the

City of London alone (Oxford Economic Forecasting, 2003). In London crowding will occur during the morning peak-hour at any working day

even if all services run as scheduled, however extreme congestion

leading to significant numbers of passengers not being able to board will

only occur if transits are delayed. This is however frequently the case; London Underground Limited publishes on their homepage that they

cancel in average 5 % of all service almost every day and the rate is even

higher during the morning peak period (London Underground Limited, 2006). Moreover, more than half of the services are cancelled if a major

accident occurred.

The U.K.’s Rail Safety and Standards Board (RSSB, 2005) discussed how to control the crowding on the platform and recommended to take

measures with following steps; (1) measures to exclude passengers from

certain parts of the stations, (2) closing pedestrian access to the station,

(3) keep trains away from busy platforms, (4) keep trains away from stations which are overcrowded, (5) suspension of trains services, and (6)

evacuation of station. However, if the service frequency is reduced or the

station is closed, there is a possibility that passengers change their route choice behaviour and therefore that the location of overcrowding

stations/platforms varies. Hence, it is very important for operators to

comprehend the influence of reducing frequency of a certain line or that of closing a certain station: i.e. which line or station is critical against

reduction of service level.

The authors have evaluated the effect of overcrowding on public

transit and its mitigation measures using the transit assignment model described in for example, Shimamoto et al, 2006. In this and other papers,

only the consequence (for example, how many passengers failed to

board) is evaluated. The impact of overcrowding in the real world is, however, not necessarily proportional to the number of passengers who

fail to board. For instance, suppose the same number of passengers fail to

board at a major station with a large platform and at a minor station with

a small platform. Then, even if those who fail to board affect the service little (in terms of safety hazard or delay) at the major station, they might

affect the service much more at the minor station. Lam et al (1999)

investigated the effects of crowding at the Light Rail Transit (LRT) station platforms in Hong Kong. They classified five Levels of Service

(LOS) on LRT platform crowding and examine the crowding effects of

passengers discomfort at the vehicles and the platforms throughout the SP survey. They focused on how passengers perceive the congested level,

3

but the transit operators would also concern the congestion level since

they have to take some measures as congestion gets worse. Based on these backgrounds, this paper evaluates the effect of

congestion using the reliability analysis technique, in which the critical

lines and stations are specified considering the scenarios with reduction

of service frequency of each line. In the context of network reliability, several researches have been carried out to address several concerns over

the reliability of the transportation network. In particular, in the field of

network analysis against disasters the main aim has been to evaluate the robustness/vulnerability of the network or identify critical components of

the network. For instance, Du and Nicholson (1997) evaluated the

connectivity reliability of degradable transport networks in which the connectivity reliability is inferred to the probability of all OD pairs to

still be connected after different possible road closure patterns. Bell

(1999) proposed a game theoretic based approach between the “evil

entity” aiming to degrade the network so as to maximise the total travel time and the travellers re-routing to minimise their travel time. The result

of this game will be the most critical link in the network. D’Este and

Taylor (2003) defined the concept of node vulnerability whose accessibility index decreases significantly with a small number of links

degraded. Kurauchi et al. (2007) evaluated the road network and

identified the critical link from the network capacity indicator; i.e. the network is vulnerable if a minor link degradation reduces the network

capacity, and such links are defined as critical.

In addition, this paper not only considers the consequence of

overcrowding in terms of failing-to board but also attempts to evaluate its impact on service quality and safety risks. The transit assignment

model is outlined in the next section. Section 3 proposes the idea of

identifying critical lines and stations and shows their evaluation criterion. Then, using these criteria, two types of concept of reliability for the

urban public transit are proposed. The method is then applied to

London’s Underground network in Section 4. Finally, Section 5

concludes this paper.

2. Outline of the Transit Assignment Model

2.1. Common lines problem

One of the key differences between road transport and

(frequency-based) public transit is the waiting time at each platform.

4

Therefore, passengers might wait a long time to get on a line of the

shortest travel time: i.e. they cannot always get to their destination earliest when they take a line of the shortest travel time. In case of no

information on the platform as to the arrival of the next vehicle comes,

the fastest way to get to their destination is known to take the first

vehicle to come among their “attractive” lines. This issue is referred as the common lines problem, which is defined as the problem to find the

set of optimal attractive lines among all lines bringing the travellers

closer to their destination (Chiriqui and Robilland, 1975). Suppose passengers arrive randomly and vehicles also arrive statistically

independent with given exponentially distributed headways, the

probability that passengers choose line i, pi, and the expected travel time, T, are calculated as follows:

Kk k

ii

f

fp (1)

Kk k

Kk kk

f

tfT

1 (2)

where K, tk and fk respectively represent the set of attractive lines,

in-vehicle time on transit line k and frequency on transit line k.

As a result, the optimal route between origin and destination is not a single route any more. Nguyen and Pallottino (1988) defined attractive

lines as hyperpath. The hyperpath connecting an origin r to a destination

s is defined as sets of stops, arcs and arc transition probabilities Hp=(Ip, Ap, p), where Hp is a hyperpath connecting r to s, if:

Hp is acyclic with at least one arc;

node r has no predecessors and s no successors;

for every node iIp-{r,s}, there is a path from r to s traversing i, and if i is not a destination, then i has at most one immediate successor;

the vector, p, contains the arc split probabilities, which satisfies

1 iOUTa

ap , pIi (3)

And

5

0ap , pAa . (4)

where OUT(i) are the sets of arcs leading out of node i. The arc split

probabilities are discussed in Section 2.3 after the introduction of the

fail-to-board probabilities.

2.2. Fail-to-board probability

In public transit, the number of passengers on board cannot exceed

the capacity of vehicles and passengers already on board have an

absolute priority over passengers who want to board. To reflect this, Bell et al.(2002) proposed a Capacity-Constrained transit assignment

model(CapCon), introducing the “fail-to-board probability”, qkl, which is

the probability that passengers who want to board at platform k to line l

can not get onto the vehicle due to overcrowding. The number of passengers travelling between stations can be represented as below:

klklkl demandboardingklboardonline xqxx 1 , LlUk l , (5)

where xa denotes the flow of arc a. linekl, on-boardkl, boarding-demandkl respectively denote the line arc, on-board arc, boarding arc of platform k

of line l. L and Ul represent the set of transit lines and set of platforms on

line l, respectively.

In the calculation, qkl is determined so that kllinex does not exceed

the vehicle capacity;

LlUk

x

xcapq l

demandboarding

boardonl

kl

kl

kl

, 1,min,0max1: (6)

Where

lll zfcap (7)

and capl and zl respectively denotes the capacity of line l and that of

vehicle on line l. Then, Kurauchi et. al.(2003) proposed a transit

assignment model (CapCon-CL) by incorporating the idea of “Common Lines” into the CapCon model. In CapCon-CL, a fail-to-board node is

6

introduced between stop node (platform) and boarding node (see Figure

1). At this node, passengers may fail to board with the probability of qkl, and those who can not board the vehicle are hypothetically transferred to

their destination through a dummy arc.

Platform 1

Platform 2

O

Alight

Stop

Fail(B)

Fail(B) Board

Board

Access D

emand

Alight

Alight

Alight

Boardin

g

Demand

Boarding

Dem

and

Boarding

Boarding

D

Line

Line

Line

Line

On-Board Wait

On-Board Wait

Egress

Egress

WalkFailure

Failure

Failure

Figure 1. Network Description

2.3. Arc Split Probability

Arc split probability at stop nodes and fail-to-board nodes (see Figure

1) are defined as below:

1. Stop nodes

Arc split probability at stop nodes is proportionate to the line frequency

as below:

)(,,)( iOUTaSiFf ppipalap (8)

where,

7

)(

)(

iOUTa

alip

p

fF (9)

and fl(a), Sp represents for a frequency of line l and sets of stop nodes on hyperpath p, respectively.

2. Fail-to-board nodes

Passengers are split according to the fail to board probability at fail-to-board nodes as below:

si

si

apDaifq

Daifq1 , pEi (10)

where Ep and Ds are the sets of fail-to-board nodes and sets of failure arcs

destined to s, respectively.

2.4. The Cost of Hyperpath

Passengers might consider not only the travel time but also how crowded the vehicle or platform is when they determine their routes: i.e.

they might take longer but less congested routes instead of the shortest

but heavily crowded route. To reflect this, the cost of hyperpath p is assumed as below.

p

kp

pp Fk

k

Sk

kpkp

Aa

aapap qWTcg

1ln (11)

)(0

)(

)(

)(1

otherwise

BAa

WAa

LAa

a

(12)

where LA, WA and BA represents for Line arc, Walking arc and Boarding demand arc respectively (see Figure 1).

The first and the second terms represent the travel time for the arcs

and the expected waiting time at stop nodes respectively. ap and ap denote the probabilities that traffic traverses arc a and the probability of

traversing node i respectively. and denote ratios of the value of waiting and walking time to the value of in-vehicle time, respectively.

Further denotes the boarding resistance which is added every time

8

passengers board. WTip is the expected waiting time at stop node i of

hyperpath p, which can be calculated as follows:

ipip FWT 1 (13)

iOUTa

alip

p

fF (9)

The third term in Equation (11) represents the cost associated with

the risk of failing to board. is the user’s risk averseness and with this parameter, probability of delay through failing to board is converted into

a time value. If passengers are absolutely risk averse, and are not

considering normal expected travel time and waiting time; when =0 passengers do not care about the risk of failing to board or failing to

access the platform. Note that because Equation (11) is separable with

given fail-to-board probabilities, Bellman’s principle can be applied to find the minimum-cost hyperpath (see Appendix A).

2.5. Mathematical Formulation

Let us assume that passengers use a hyperpath of minimum cost as

defined in Equation (11) where the cost of a hyperpath is a function of

the failure-to-board probability for each transit line on each platform. The fail-to-board probability depends on boarding demand, passengers

already on board, and the transit line capacity, which in turn depends on

the fail-to-board probability. Therefore, the transit assignment model can

be formulated as a following fixed point problem which defines the equilibrium hyperpath flow vector y* and fail-to-board probability vector

q*:

0, *** qyuy , 0qyu **, , y (14)

0, *** qyvq , 0qyv **, , 1q0 (15)

Where

***** ,, rspp mgu qyqy (16)

9

klklkl demandboardinghboardonlkl xqxcapv 1, **

qy ,

LlUk l , (17)

and and hkl respectively denotes set of feasible hyperpath and

fail-to-board node of line l on platform k. As u denotes a vector of cost difference between gp(y*,q*) and

the minimum cost from the origin of the hyperpath p(r) to the destination

(s), *

rsm , Equation (14) represents the user equilibrium condition. Also,

as v denotes the vector of vacancies on the line arc on line l from

platform k, Equation (15) represents the capacity constraint condition. The existence of a fixed point is intuitive since any excess

demand simply implies non-zero failures to board. However, because of

the non-linear relationship in Equation (15), there is a possibility of

multiple fix points. This fixed point problem can be solved by combining the Method of Successive Averages (MSA) and absorbing Markov chains

(Kurauchi et.al. , 2003).

3. Method of Evaluation

3.1. Concept of the Critical Lines and the Critical Stations

In this study, the critical lines are defined as lines whose frequency

reduction causes a severe impact on the whole network. Firstly, the

impact is measured in terms of total number of passengers failing to board in the whole network. Therefore, the critical lines are formulated

as below where NF are the number of passengers failing to board and s is

the scenario indicator:

Find s such that )(max sNFtotals

(18)

Ll Uk

kltotal

l

sNFsNF )()( (19)

)()()( sqsxsNF kldemandboardingkl kl , LlUk l , (20)

As will be explained in the case study in Section 4, the frequency of

10

only one line is reduced in each scenario. Equation (19) shows the

number of passengers failing to board in the network and Equation (20) the number of passengers failing-to-board at each platform.

The critical stations are defined as stations whose platforms are

overcrowded due to the reduction of a certain line frequency. However, since our analysis is based on platform-overcrowding, the criticality of a

certain station is defined not as the sum of the fail-to-board passengers at

each platform belonging to the station but as the maximum among all platforms at this station:

)(max)( sNFsNF iPi

mm

(21)

where Pm represents the set of platforms at station m. Then, the critical

stations based on consequence are formulated as below:

For each scenario s, find m such that )(max sNFmm

(22)

where m is the station indicator. Note that critical stations are defined with each scenario (reduction of a certain line), and may not lie on the

line of reduced frequency.

3.2. Alternative criterion for the Impact of Overcrowding

Since the impact of overcrowding is not necessarily equal to the

number of passengers failing-to-board as described in Section 1 an alternative impact measure is developed in this paper. The impact of

overcrowding might also depend on for example (1) the size of platforms

or stations, (2) the number of exits or entrances and where they are, (3)

the potential impact on anther interchange line, and (4) the congestion mitigation possibilities provided through platform assistants, platform

markings and platform screen doors.

To consider these matters, a satisfaction function for the service operation is proposed. This satisfaction function will take the value of 1

when there is no congestion (fully satisfied), and should converge to 0 as

the number of passenger failing-to-board increases (not at all satisfied). The satisfaction criteria should be determined according to the above

four criteria. If the operator wants to emphasise the impact of crowding

on network delays one might give a higher weight to stations with large

11

interchanges to emphasise that the overcrowding of one line potential

also leads to overcrowding on other lines serving the same station. Similarly, if one wants to evaluate health and safety risks through

overcrowding the traffic engineer will be less concerned about some

overcrowding at large stations with several exits and large number of

station staff than if the same amount of overcrowding would occur at a small station. This second evaluation scenario considering health and

safety risks is taken as a case study in the following. Equation (23) and

Figure 2 express that the same amount of overcrowding is less acceptable at a smaller station than at a larger station.

0)(

10)(exp01.01

1

15)(exp01.01

1

0)(1

sNF

MSmsNF

MSmsNF

sNF

sIm

m

m

m

m

(23)

where MS stands for the set of major stations (definition of major stations

will be explained in Section 4). The relationship between the number of

fail-to-board passengers (NFm(s)) and the satisfaction function is then illustrated in Figure 2. Using the satisfaction function, the critical stations

based on the impact is formulated as below:

For each scenario s, find m such that )(min sImm

(24)

where m represents for a station indicator. Of course this example is used

to show the proposed methodology, and careful calibration of parameters should be made for the actual application.

12

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40 50 60 70 80 90 100110120130140150

I m(s

)

NFm(s)

Major Station Minor Station

Figure 2. Satisfaction function

4. Case Study with London Underground Network

4.1. Study Area and Data Collection

The proposed method is applied to the central part of London

Underground network as shown in Figure 3, which consists of 56 stations

and 14 lines. Following data were made available from Transport for London: travel time between stations, estimated walking time between

different platforms, service frequency and the capacity of each line. The

major stations are marked with red circle in Figure 3. The OD matrix of the whole network from 7:45 am to 8:45 am for passengers entering the

inner zone estimated by Schmöcker (2006) is used. The parameters ,

and , in the hyperpath cost function (Eq. 3), are calibrated to minimise

RMS errors from observed passengers’ flow of each line to =2.5, =1,

=3, =150 (Shimamoto et al., 2006).

13

Northern North Picadilly North

Victoria North

Central East

District,

Hammersmith &

City East

Jubilee East

Northern (Bank)

SouthBakerloo South

Northern (Waterloo)

SouthVictoria South

District South

District West

Picadilly West

Central West

Hammersmith &

City West

Bakerloo WestMetropolitan

North

Jubilee North

Figure 3. Underground network of central London

Only one line is assumed to be delayed in each scenario. Therefore,

total 28 scenarios are considered as shown in Table 1. In each scenario, two types of case studies are conducted. In the first case, the impact of

minor delays in the service headway (10% increases) is estimated. Given

the statistics on LUL’s homepage this might well correspond to delays experienced daily during the morning peak. In the second case, major

delays corresponding to a reduction of 50% in line frequency are

considered.

4.2. Discussion

Table 1 summarises the result of calculation. First of all, the critical lines are identified based by the total number of passengers who failed to

board (NFtotal). From the table, Westbound Central Line (4/CeW) and

Southbound Victoria Line (26/ViS) seems to have a larger impact in the

case of minor delay. For the major delay, the relatively large values of NFtotal were obtained for scenarios 4/CeW, 10/DiW, 15/JuE, 16/JuN,

19/NoNL, 23/PiN, 24/PiS, 25/ViN and 26/ViN. Since the two lines which

were found to be critical in the minor delay scenarios were also critical in the case of major delay, these lines should be carefully managed and

monitored so as to maintain a good level of service.

The most critical station in each scenario is specified by the minimum

14

value of the evaluation criteria. It was found that the same station is

identified to be critical from both NFmax( )(max sNFmm

) as well as

Imin( )(min sImm

), as shown in the Table 1. Note that Victoria Station

was critical in the base case without delay (Scenario 0), and is generally

critical in most of the scenarios. There are very few scenarios in which the most critical station is neither Victoria nor along the delayed line (for

these cases the relevant stations are written in bold in Table 1). For

example, in Scenario 13/HCE (major delay on east-bound Hammersmith

& City Line) Tower Hill is the critical station even though it is not on the Hammersmith & City Line. As seen in Table 2, this is because the delay

encourages passengers at Liverpool and Aldgate to take the Circle Line.

This result implies the importance of evaluating the critical lines/ stations using network analysis techniques. Note further the complex network

impacts of delays on line loads. In Scenario 1 the minor delay scenario in

fact leads to more passengers failing to board than in the major delay scenario. The reason is that in the case of large delays more passengers

are encouraged to travel on different routes whereas the expectation of

only a minor frequency reduction is not enough to deter passengers from

their nominally shortest route.

Table 1. List of scenarios with the most critical station

Impact-based Impact-based

Scenario Line Name Remarks NF total NF max I min NF total NF max I min

1 Bakerloo NB 5694 143.7 0.007 Victoria 5541 150.3 0.004 Euston

2 Bakerloo SB 5614 122.3 0.028 Kings Cross 6428 127.4 0.020 Victoria

3 Central EB 5767 157.3 0.003 Victoria 7626 161.4 9.83×10-6 Queensway

4 Central WB 10464 215.4 4.43×10-8 Liverpool Street 16244 229.4 1.08×10

-8 Liverpool Street

5 Circle EB 5596 134.1 0.013 Victoria 5921 198.4 1.81×10-4 Victoria

6 Circle WB 5923 163.2 0.002 Victoria 6098 219.2 4.51×10-5 Victoria

7 District EB 5960 162.2 0.002 Victoria 9742 179.0 0.001 Victoria

8 District NB to Edgware Road 5674 146.2 0.006 Victoria 5686 143.2 0.007 Victoria

9 District WB from Temple 5694 146.7 0.006 Victoria 5715 160.4 0.002 Victoria

10 District WB 6390 174.3 0.001 Euston 12131 651.5 1.37×10-17 Bank-Monument

11 District SB to Edgware Road 6083 139.4 0.009 Victoria 7979 135.3 0.012 Victoria

12 District EB to Temple 6035 139.6 0.009 Euston 8159 157.1 0.003 Victoria

13 H & C EB 6074 136.2 0.011 Victoria 8445 148.9 3.40×10-5 Tower Hill

14 H & C WB 5685 134.8 0.012 Victoria 6757 141.3 0.008 Victoria

15 Jubilee EB 5560 142.3 0.008 Victoria 11318 151.9 0.004 Victoria

16 Jubilee NB 5675 148.6 0.005 Victoria 13653 337.2 1.72×10-17 London Bridge

17 Metropolitan EB 5681 142.8 0.007 Victoria 7184 132.2 0.015 Victoria

18 Metropolitan NB 5618 135.8 0.012 Victoria 5628 142.1 0.008 Victoria

19 Northern NB via London Bridge 6107 158.8 0.003 London Bridge 10038 240.9 1.06×10-5 London Bridge

20 Northern SB via London Bridge 5786 142.0 0.008 Victoria 9245 144.2 0.007 Victoria

21 Northern NB via Waterloo 5689 140.8 0.008 Victoria 5734 143.6 0.007 Victoria

22 Northern SB via Waterloo 6051 149.4 0.005 Kings Cross 9863 310.3 1.39×10-7 Euston

23 Piccadilly NB 5965 161.0 0.002 Victoria 11166 154.1 0.003 Victoria

24 Piccadilly SB 6458 159.5 0.002 Euston 12552 240.5 1.08×10-5 Euston

25 Victoria NB 6382 229.7 2.24×10-5 Victoria 13382 319.7 5.52×10

-8 Victoria

26 Victoria SB 7597 166.3 0.002 Euston 15119 146.8 0.006 Kings Cross

27 W & C From Waterloo to Bank 5715 167.8 0.001 Waterloo 5760 143.1 0.007 Victoria

28 W & C From Bank to Waterloo 5699 140.8 0.008 Victoria 5699 140.8 0.008 Victoria

- 5699 140.8 0.008 Victoria

Notations

H & C: Hammersmith & City EB: East bound NB: North bound

W & C: Waterloo & City EB: East bound SB: South bound

Consequence-based

Major delay

Critical Station

No Delay

Consequence-basedCritical Station

Minor delay

15

Table 2. Number of passengers travelling between stations

Line Scenario Kings Cross Farringdon Barbican Moorgate LiverpoolSt Aldgate

H & C delay 2798.87 2299.96 2114.22 2094.79 2603.35 2224.76

No delay 2850.78 2361.27 2178.76 2164.62 2510.21 2131.64

Difference -51.91 -61.31 -64.55 -69.84 93.14 93.12

H & C delay 10646.39 7694.86 6593.56 5238.39 1761.33

No delay 10592.38 7661.88 6562.45 5225.36 1761.33

Difference 54.01 32.98 31.11 13.03 0.00

Circle

Metropoltan

Finally, Figure 4 shows the number of failing passengers and the

evaluation criterion (based on satisfaction function) at Scenario 10/DiW (major delay on westbound District Line). Generally speaking, as the

number of passengers failing-to-board increase, the evaluation criterion

decreases. However, the evaluation criterion on Westminster station,

which is classified as a minor station, is relatively low even though only a few passengers are failing to board. This fact implies the importance of

evaluating the critical stations using proposed evaluation criteria

depending on the station size.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

100

200

300

400

500

600

700

BM EB BF

CS

TE

MH VI

TH LB SK WA EC EU

WM SS KC SJ GR

Eval

uat

ion

Cri

teri

on

Nu

mb

er o

f Fa

ilin

g P

asse

nge

rs

Station Code

Number of Failing Passengers Evaluation Criterion

Figure 4. Number of failing passengers and the evaluation criterion

16

5. Conclusion

This paper proposed a method to identify critical transit lines against

reduction of their frequency using an afore-constructed transit

assignment model. The line criticality is measured with total number of

failing to board passengers and the station criticality is measured with both the consequence and the impact of overcrowding. A satisfaction

function is proposed to evaluate the impact of overcrowding at each

platform. In the case study evaluating scenarios with the satisfaction function or in terms of number of passengers failing to board leads to the

same critical station. However, it is proposed that the evaluation with a

satisfaction function adds flexibility to the operator to analyse the impact of overcrowding. For example overcrowding at refurbished stations with

wide spaces might be less severe than at old stations with less space for

waiting passengers.

The definition of such a satisfaction function obviously depends on a good knowledge of the system and its facilities. In further work the shape

of the satisfaction for single stations should be discussed in more detail

with Transport for London. One way to model the impact of overcrowding at different platforms might be through the use of a

detailed pedestrian simulation model. Further note that this paper

assesses the critical lines and stations using a user equilibrium model, which implicitly assumes that passengers have perfect information as to

the frequency of each line in each scenario; i.e. they know which line is

degraded. If these delays appear very irregular this is obviously a strong

assumption. However passengers in London often have fairly good information about the delay situation on all lines through information

displayed at all stations. The scenarios discussed in this paper could

further be of interest to the transport operator who might want to know the impact of a longer term frequency reduction for example due to

construction works. Our scenario based analysis further illustrated the

often complex networks effects of delays. In some cases it might be

better for the operator to announce the delays are major instead of only minor in order to deter passengers from already overcrowded lines.

Finally, another interesting extension may be to evaluate the effect of

train arrival information suggested by the authors (Shimamoto et. al, 2005).

17

Acknowledgements

We would like to thank Transport for London for the provision of the

data used in the case study. Further we would like to thank Robin C d’A

Hirsch from the Railway Technology Strategy Centre at Imperial College

London for the discussion on impact of overcrowding.

Appendix A: Separable of the Cost Function

The separability of the generalised cost function (Equation (11)) can

be proven in the same way as in Kurauchi et. al. (2003). Let Vp denote

the set of all elementary paths in Hp, and al is 1 if arc a is included in l,

otherwise 0. Further, using arc split probabilities ap and if lp denotes the probability of choosing any particular elementary path l of hyperpath p, then clearly:

p

al

Aa

aplp

, pVl (25)

1 pVl

lp (26)

Let il be defined as 1 if elementary path l traverses node i. Then, the

probability that hyperpath p traverse node i, ip, can be calculated from

lp as follows:

pVl

lpilip , pIi (27)

Similarly ap, probability that traffic traverses arc a, i.e., arc transition probability, can be calculated as follows:

pVl

lpalap , pAa (28)

Using node and arc transition probabilities, the cost of hyperpath p

(Equation (11)) can be expressed as follows:

18

p pppVl Ek

kkl

Sk

kpkl

Aa

aalalp qWTcg 1ln (29)

Above result suggests that the cost for hyperpath p is a sum of the costs of the elementary paths weighted by their choice probabilities. Let g

’ip

denote the cost of the sub-hyperpath from i to s on hyperpath p, and

define lp(i) as:

ip

al

Aa

aplp i (30)

1 pVl

lp i (31)

Then, clearly,

ajai lpaplp , ipAa (32)

where i(a) and j(a) respectively are a head node and a tail node of arc a.

Using lp(i) , g’jp can be written as follows.

iOUTa

pajapiiFiiS

iOUTa

aapa

iOUTa

pajap

Ll

iiFiiSliaalp

Ll Ek

kkl

Sk

kkl

Aa

aalalpip

pp

pp

p ipipip

gqWTc

gqWTci

qWTcig

'1ln

'1ln

1ln'

,

(33)

where iS is 1 if node i is a stop node and 0 otherwise, iF is 1 if node i is a failure node and 0 otherwise, and a(i,l) denotes an arc on elementary

path l leading out from node i. Equation (33) gives the cost of hyperpath

from i to s as the sum of the cost at node i, which is either the expected

waiting time or the risk of failing to board, a cost for arcs connecting to other nodes weighted by the respective arc split probabilities, and a total

cost from node j to s also weighted by arc split probabilities. s the costs

of subsequent nodes are separable, Bellman’s principle shown by Equation (34) can be applied for finding the minimum cost hyperpath.

19

Eiifq

Siiff

f

sESIiifc

siif

saji

Ka

al

Ka

sajal

iOUTK

iOUTa

saja

is

i

*

*

*

*

1ln

1

min

min

0

(34)

Where I, S and E respectively denote set of nodes, set of stop nodes and set of fail-to-board nodes as illustrated in Figure 1.

Reference

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