Topic area C6 Paper no. 929
OPTIMAL SCHEDULING OF CONNECTIONS IN RAILWAY SYSTEMS
Paper prepared for the 8th WCTR, Antwerp, July, 1998 revised version
Drs. Rob M.P. Goverde Delft University of Technology Faculty of Civil Engineering and Geo Sciences Transportation Planning and Traffic Engineering Section
OPTIMAL SCHEDULING OF CONNECTIONS IN RAILWAY SYSTEMS Rob M.P. Goverde1
ABSTRACT Complex railway networks typically consist of widely diffused OD-pairs by which it is impossible to provide direct trips only. Therefore, passengers often have to change trains at transfer stations. Guaranteeing scheduled connections is complicated by delays during operation. To improve and maintain the quality of transfers requires optimal scheduling and control of connections at transfer stations. For this, feeder and distributing lines are identified that have to be synchronized to give good connections. Here, synchronization is the mutual adjustment of the arrival and departure times of individual trains of connecting lines at a transfer station. Optimal synchronization is obtained when the average waiting time of the transferring passengers is as small as possible. This paper deals with the computation of optimal buffer times in timetables at scheduled connections. These connection buffer times reduce arrival delays of trains at transfer stations during operation and improves the reliability of the connections. The optimal buffer time is computed by minimization of the total expected transfer waiting time of passengers at a transfer station. Both the uncertainty in the trip times between stations and the operational control of connections is explicitly incorporated. Additionally, several transfer quality measures are derived from the obtained expressions for the optimal connection buffer time.
1 1.1
INTRODUCTION Background
A major cause of travel disutility in public transport systems is the time a passenger needs to travel from origin to destination. Travel time can be partitioned in the components access time, waiting time at boarding stop, trip time(s), transfer (waiting) time(s), and egress time. Passengers judge these various components differently. Travel disutility can thus be expressed by weighted travel time, i.e., all travel time components are multiplied by a factor (a weight) to take account of their relative importance to the travel disutility. Generally, a passenger weighs the travel time components outside the vehicle more heavily than the trip time. Various studies show that the transfer time weighs substantially higher than the other components (Van der Waard, 1989; Van Goeverden et al., 1990). Passengers who have the alternative to travel by car weigh the transfer time even higher. 1
Contact: Rob M.P. Goverde, Delft University of Technology, Faculty of Civil Engineering and Geo Sciences, Transportation Planning and Traffic Engineering Section, Stevinweg 1, P.O. Box 5048, NL-2600 GA Delft, The Netherlands, phone: +31.15.2783178, fax: +31.15.783179, e-mail:
[email protected]
1
Policies to reduce the (weighted) transfer time therefore result in a considerable reduction of the travel disutility. Moreover, improvement of public transport quality with respect to car-owners is additionally stimulated by the increasing congestion on the Dutch highways and from an environmental point of view. The increasing attention in multimodal transport systems is another stimulus to focus on developing policies for reducing transfer disutility. The mutual adjustment of the various transport services - both intermodal and unimodal - is here crucial. This paper uses railway terminology. However, the considered concepts are applicable to all modes of public transport with scheduled services. This includes unimodal networks of bus services, tram services, and metro services, as well as multimodal public transport network services. In fact, this paper considers a case study of scheduling an intermodal connection between a feeder bus line and a connecting railway line.
1.2
Synchronization
The aim of this paper is the reduction of transfer disutility using scheduling and dispatching policies. These policies can be applied to existing transport systems without demanding changes of the system itself. Also strategic policies - like infrastructural adaptations - may give considerable improvements. However, these latter policies imply structural changes of the transport system (mostly) accompanied with high costs. Transfer disutility primarily depends on the following conflicting factors: • the connection time (transfer time and transfer waiting time), and • the probability of missing a connection. Both large connection times and missing a connection imply large mean waiting times of passengers at a transfer station. However, a small connection time implies a higher probability of missing the connection. Above factors can be influenced by synchronizing connecting trains at a transfer station. Synchronization is the amount of mutual adjustment of individual trains from different lines at transfer stations. Synchronization of connections is complicated by delays during operation. Synchronization management deals with the scheduling and control of connections at transfer stations taking delays into account. This implies both static and dynamic policies (Knoppers and Muller, 1995). In short, static policies refer to designing timetables in such a way that small delays of feeder trains do not endanger scheduled connections. Dynamic policies refer to controlling connections during operation. The majority of relevant studies assume the train (or bus) schedule for the individual lines to be fixed, i.e., all trip times and stopping times are fixed from the starting station to the terminal station. The transfer optimization problem is then defined as finding optimal offset times at the starting stations for each line such that the resulting transfer waiting times at points where lines meet is minimized, see for instance (Bookbinder and Désilets, 1992). A few studies also take variations in trip times into account. The negative aspect of delay transfers from feeder trains to connecting trains is not considered in these studies.
2
An alternative approach is the problem of synchronizing the departure and arrival times of connected trains at and between all transfer stations. The synchronization problem is now defined as finding optimal connection buffer times at transfer stations. Also stopping times are variable in the sense that optimal stopping buffer times, induced by the connection buffer times, are introduced. The buffer times are optimized taking train arrival delays explicitly into account. The buffer times then compensate for train arrival delays during operation (Meng, 1991; Knoppers and Muller, 1995). Also incorporated is the possibility for connecting trains to wait for a delayed feeder train (synchronization control) to secure planned transfers. This paper derives explicit expressions for the optimal connection buffer time and the corresponding minimum expected transfer waiting time. Also transfer quality measures are defined in which the stochastic behaviour of the arrival times is incorporated explicitly. Section 2 considers the concepts of buffer times and synchronization control margins, and also gives some basic assumptions. Section 3 models the transfer waiting time. Section 4 derives expressions for the optimal connection buffer time and the corresponding optimum expected transfer waiting time for exponential arrival delays. Section 5 incorporates the effect of early arrivals on the optimal connection buffer time. Section 6 defines transfer quality measures. All concepts are illustrated by means of a case study in Section 7.
2 2.1
SCHEDULING AND CONTROL OF CONNECTIONS Introduction
This paper considers railway lines as directed services between transfer stations. A train of a line i is also referred to as train i, although in fact successive trains of a particular line may physically be not the same. A timetable with fixed cycle time is assumed, i.e., the pattern of scheduled arrival and departure times of all trains is fixed modulo the cycle time. The (constant) scheduled interdeparture time hi between two successive trains on line i is thus a divider of the cycle time.
2.2
Connection Buffer Time
Consider the (single) connection ij from line i to line j at a particular transfer station. The connection time corresponding to a connection ij is the interval between the arrival time Ai of the feeder train i and the departure time Dj of the connecting train j, t ijc = D j − Ai . It is assumed that a minimum connection time is defined in which the passengers should be able to change trains. The actual transfer times - consisting of alighting time, walking time (including possible orientation), and boarding time - depend on individual walking speed and acquaintance with the station, the relative position of the arrival and departure platform (cross-platform, two platforms apart, etc.), the geography of the station (platform lengths, distances between platforms, widths of corridors and door-ways, presence of escalators, 3
etc.), and the pedestrian flows and densities in the station. The minimum connection time is thus determined in such a way that all transferring passengers are able to succesfully transfer with high probability. Above the minimum connection time an extra amount of time is scheduled to compensate for small arrival delays of the feeder train. This connection buffer time thus reduces the probability of missing a connection but increases the scheduled connection time and hence also the (total) scheduled travel time. Solving this dilemma is the subject of this paper. Resuming, the scheduled connection time with respect to feeder line i and connecting line j is given as t ijc = t ijc,min + rijc , where t ijc,min denotes the minimum connection time and rijc the connection buffer time.
2.3
Synchronization Control Margin
During operation a connecting train may wait for delayed transferring passengers if a feeder train arrives behind schedule. The resulting synchronization control time is the additional time above the scheduled departure time that the connecting train waits at the platform to secure the connection for the delayed transferring passengers. Figure 1 Buffer times and synchronization control time space/situation time delayed feeder-train 1
connecting train 2
A1 A2
t12c, min
t 2s, min
r12c
r2s
D2
D2 s2
0 ≤ p1 < r12c
r12c ≤ p1 ≤ r12c + s2
p1 > r12c + s2
eliminated arrival delay by connection buffer time
synchronization control time
dissolved connection
4
Large departure delays may endanger connections at following stations or perturb train movements on junctions such as level crossings, or splitting and emerging sections. To avoid this, the so-called synchronization control margin defines a maximum admissible synchronization control time to each (connecting) train j. If a feeder train is that much delayed that synchronization control would result in exceeding the synchronization control margin then the connection is dissolved and the train may leave as scheduled. The synchronization control margin of a train j is denoted as s j . Figure 1 illustrates the principle of the connection buffer time and the synchronization control margin in the case of a punctual connecting train and a delayed feeder train (Goverde, 1997). This figure also shows the minimum stopping time t s,min and a stopping j s buffer time r j of the connecting train j. This paper assumes that the synchronization control margins are known and concentrates on deriving optimal connection buffer times with respect to these given synchronization control margins.
3
TRANSFER WAITING TIME
The transfer waiting time is the time that the transferring passengers have to wait on the actual departure of the connecting train. It is assumed that early arrivals do not occur and that the minimum connection time reflects the actual necessary transfer time (without waiting time). The transfer waiting time is now directly related to the actual arrival time of the feeder train. That is, if the feeder train has an arrival delay pi then the arrival time of the transferring passengers in the connecting train has a delay pi as well. The transfer waiting time is thus a function of the arrival delay pi of the feeder train i. If the feeder train arrives on time or within the connection buffer time then the transferring passengers have to wait for the scheduled departure time. For intermediate arrival delays of the feeder train the connecting train may wait to secure the connection, and leaves right after the arrival of the transferring passengers. If the feeder train has an arrival delay such that securing the connection results in exceeding the synchronization control margin then the connection is dissolved and the transferring passengers miss the connection. The passengers then have to wait on the next train of the connecting line, which has a scheduled departure time Dj+hj . The transfer waiting time can thus be given as rijc − pi wij ( pi ) = rijc + h j − pi 0
if 0 ≤ pi ≤ rijc if rijc + s j < pi ≤ rijc + h j otherwise,
(1)
see Figure 2. The transfer waiting time depends on the parameters rijc , s j , and hj . The above derivation of the transfer waiting time assumes that if the transferring passengers miss a connection they are able to catch the next train of the connecting line. This assumption is by no means restrictive. If this assumption appears to be violated either one of the following situations is likely responsible:
5
Figure 2 The transfer waiting time with respect to connection ij wij
0
rijc
rijc + s j
rijc + h j
pi
• the interdeparture time hj of the connecting line is too small: scheduling of a connection to a high-frequency line is unnecessary, since the mean transfer waiting time without scheduling is ½ hj; • the scheduled trip time of the feeder train from the preceding transfer station is computed too tight: the computation of trip times is beyond the scope of this paper. A second assumption is that transferring passengers who miss the connection wait for the next train on the same connecting line and do not leave the station in a train of another line. For scheduling purposes this assumption is tolerable. In practice passengers who for instance miss a connection to an intercity train might take a stop train in the same direction but this clearly should not be incorporated in the derivation of optimal connections in a timetable.
4 4.1
OPTIMAL CONNECTION BUFFER TIME Exponential Arrival Delays
The applicability of exponential arrival delays in railway systems is motivated by German studies (Meng, 1991; Schwanhäußer, 1974; Weigand, 1981), where this stochastic behaviour has been studied intensely. Meng (1991) explicitly gives a number of examples where the hypothesis of exponential distributed arrival delays of intercity trains at German transfer stations is tested by a chi-square test. Recently, Delft University of Technology started a research project on the stochastic behaviour of trip times and arrival delays of trains on the Netherlands railway network. Arrival delays are modelled as an exponential distribution, i.e., it is assumed that the arrival delay pi of a train of a particular line i has a density function f i ( pi ) = λe − λpi ,
pi ≥ 0,
(2)
for a given parameter λ > 0 , see Figure 3. This distribution is also referred to as the negative-exponential distribution. The parameter λ is the reciprocal of the mean delay. For example, λ = 0.5 implies a mean delay of 2 minutes. 6
Figure 3 The exponential density ( λ = 0.5 ) 0.5
0.4
0.3
0.2
0.1
0 0
5
10 arrival delay p (min)
15
20
i
The exponential density is a decreasing function which implies that small arrival delays are more common than larger delays, which seems very realistic. Note, that trip times usually follow a bell-shaped distribution with an exponential-like tail, such as the normal distribution, the log-normal distribution or the Weibull distribution (with shape parameter larger than 1). Scheduling a trip time truncates the trip time density, and the resulting arrival delays behave like the exponential density (if the scheduled trip time is not taken too tight).
4.2
Expected Transfer Waiting Time
With respect to (1) the expected (or mean) transfer waiting time wij is given as wij
= Ewij ( p) ∞
= ∫−∞ wij ( p) f i ( p)dp rijc
rijc + h j
0
rij s j
= ∫ [ rijc − p] f i ( p)dp + ∫ c + [rijc + h j − p] f i ( p)dp .
(3)
By substitution of the exponential delay density (2) and using partial integration, (3) can be expressed analytically as wij (rijc ; s j , h j , λ ) = rijc −
1 1 − λrijc 1 − λ ( rijc + h j ) 1 −λ (r c + s ) + e + e + (h j − s j − )e ij j . λ λ λ λ
(4)
Note, that the expected transfer waiting time can be considered as a function of the connection buffer time, the parameters s j , hj , and the exponential parameter λ . Figure 4 shows the expected transfer waiting time (4) as a function of the connection buffer time for various values of the synchronization control margin. For large buffer times (larger than 10 minutes) the influence of the synchronization control margin is neglectable: any
7
arrival delay can be absorbed by the buffer time. On the other hand, the effect of the synchronization control margin is considerable for small buffer times. The mutual adjustment of buffer time and control margin is an important planning aspect for reliable connections. From a passenger point of view the impact on waiting times is crucial, including waiting times for transferring passengers, through-passing passengers and starting passengers. Equation (4) is here an important planning tool. Note that with respect to waiting times induced by securing connections, the waiting time of through-passing passengers is simply the train stopping buffer time at the station, and the waiting time of starting passengers is the synchronization control time (both of the connecting train). Figure 4 The expected transfer waiting time wij ( rijc ; s j ) for (from upper to lower) s j = 0, ,5 and hj = 30 min, λ =0.5 30
w ij (min)
25
20 s =0 j
mean transfer waiting time
15
10
5 s =5
0 0
j
5
10
15
20
25
30
c
connection buffer timerij (min)
The expected transfer waiting time as shown in Figure 4 is a convex function in the connection buffer time (for these particular parameter values). The following lemma states that this holds in general. Note that the convexity property implies the existence of a unique minimum. Lemma 1 [Convexity of expected transfer waiting time] Let h j > s j ≥ 0 . Then the expected transfer waiting time (4) is a convex function of the connection buffer time rijc . Proof A standard result from optimization theory is that a (smooth) function of one variable is convex if and only if its second derivative is positive on its domain, see for instance Luenberger (1984). It is shown here that this holds for wij . The derivative of wij is dwij c ij
dr
( rijc ) = 1 − [1 + e
− λh j
+ ( λh j − λs j − 1)e
− λs j
]e
− λrijc
.
The second derivative of wij is obtained by differentiating (5). By rearranging it follows
8
(5)
d 2 wij (drijc ) 2
(rijc ) = λ2 (h j − s j )e
>0
− λ ( rijc + s j )
− λs
− λh
+ λ (1 − e j + e j )e
− λrijc
> 0.
(6)
>0
Note that 0 < e − x ≤ 1 for all x ≥ 0 by which follows that the second term in (6) is positive. From (6) it follows that wij is convex.
4.3
Optimal Connection Buffer Time
The optimal connection buffer time rijc* is that connection buffer time for which the expected transfer waiting time is minimized. The optimal connection buffer time is thus found by solving the minimization problem min wij (rijc ) . rijc ≥0
(7)
The following theorem gives the explicit solution to problem (7). Theorem 1 [Optimal connection buffer time] Let h j > s j ≥ 0 and the expected transfer waiting time be defined as (4). Then the minimization problem (7) has a unique solution and the optimal connection buffer time for which this minimum is obtained is given as rijc* ( s j ) = rijc* ( s j ; h j , λ ) =
1 − λh − λs ln[1 + e j + ( λh j − λs j − 1)e j ] . λ
(8)
Proof Lemma 1 states that wij is a convex function of rijc . Therefore, the minimum of wij with respect to rijc exists and is unique, i.e., there is only one local minimum which is also the global minimum. Moreover, it can be shown that the slope of wij is decreasing in rijc = 0 for all combinations of admissible parameter values. Therefore, the minimum is obtained for a positive buffer time by which constraint rijc ≥ 0 is always valid and can be neglected. The corresponding optimal connection buffer time (8) is now obtained by taking the derivative (5) equal to zero and solving this equation for rijc . Note that the argument of the logarithm is positive for all admissible parameter combinations by which (8) is well defined.
Figure 5 shows the optimal connection buffer time (8) as a function of the synchronization control margin. It is clearly a decreasing function: the larger a synchronization control time is allowed to be, the less connection buffer time is required. Equation (8) gives for a given synchronization control margin (and given parameters) the optimal connection buffer time with respect to minimum transfer waiting time. It can be used directly for planning reliable connections if the synchronization control margin is determined by for instance operator policy (an IC-train only waits for another IC-train for 2 minutes, and does not wait on other (lower priority) trains or based on experience or by extensive calculations.
9
Figure 5 The optimal connection buffer time rijc* ( s j ) for λ =0.5, and h j =30, 15 and 10 min 6
5 h =30
r ijc (min)
j
4
h =15 j
connection buffer time
3
2
h =10 j
1
0 0
5
10
15
synchronizationcontrol time s (min) j
The following theorem gives a remarkable result concerning the optimum expected transfer waiting time. Theorem 2 [Minimum expected transfer waiting time] Let h j > s j ≥ 0 and wij be defined as (4). Then the minimum expected transfer waiting time, i.e., the solution to (7), equals the optimal connection buffer time (8), wij (rijc* ) = rijc* .
(9)
Proof The proof follows immediately by evaluating the expected transfer waiting time (4) in the optimal connection buffer time (8). Recall from basic mathematics that exp(-ln(x))=1/x. It then follows 1 1 − λh j 1 − λs + e + (h j − s j − )e j 1 1 − λh − λs λ λ λ + ln[1 + e j + (λh j − λs j − 1)e j ] − wij (rijc* ) = − λh j − λs j λ λ 1+ e + (λh j − λs j − 1)e =
1 1 1 − λh − λs + ln[1 + e j + (λh j − λs j − 1)e j ] − λ λ λ
= rijc* .
Theorem 2 states that equation (8) defines both the optimal connection buffer time and the minimum expected transfer waiting time. Note that Theorem 2 does not implie that the connection buffer time equals its corresponding expected transfer waiting time for any arbitrary connection buffer time. In fact the fixed-point equation wij (rijc ) = rijc only holds for rijc = rijc* which can be proven by simply solving this fixed-point equation for rijc .
10
5
PUNCTUALITY AND EARLY ARRIVALS
If a high percentage of trains arrives (slightly) before the scheduled arrival time then the amount of on-time trains, i.e., trains that are not behind schedule, is relative large with respect to the delayed trains. If early arrivals are assumed to correspond to zero delay then the arrival delay density can be modelled as q i f i ( pi ) = − λpi (1 − q i ) λe
if pi = 0 if pi > 0,
(10)
where qi is the punctuality probability, i.e., the probablity that trains have zero delay. Analogous to (3), the expected transfer waiting time is now given as ∞
wij = q i rijc + (1 − q i ) lim ∫ wij ( p)λe − λp dp . ε ↓0
ε
The second term can be computed using partial integration and subsequently taking the limit of ε to zero from above. The analog to (4) then is wij (rijc ; s j , h j , λ ) = rijc + (1 − qi )( −
1 1 − λrijc 1 − λ ( rijc + h j ) 1 −λ (r c +s ) + e + e + (h j − s j − )e ij j ) . (11) λ λ λ λ
The expected transfer waiting time (11) is again a convex function which can be proven in the same way as the proof of Lemma 1. The minimization problem (7) with the expected transfer waiting time given as (11) has then again a unique solution. Theorem 3 Let h j > s j ≥ 0 and the expected transfer waiting time be defined as (11). Then the minimization problem (7) has a unique solution and the optimal connection buffer time for which this minimum is obtained is given as +
−λh − λs 1 rijc* ( s j ) = rijc* ( s j ; h j , λ ) = ln[(1− qi )(1+ e j + (λh j − λs j − 1)e j )] , λ
(12)
where (x)+=max(0,x). Proof The proof is analog to the proof of Theorem 1. Since (11) is a convex function, the minimum of (7) exists and is unique. The optimal connection buffer time is obtained by computing the derivative of (11), taking this derivative to zero and solving for the connection buffer time.
An alternative to compute the optimal connection buffer time is by minimizing the mean transfer waiting time corresponding to the distribution of the feeder train arrival times instead of its arrival delays (Knoppers and Muller, 1995). In this way the transfer waiting times corresponding to the early arriving feeder trains are taken into account more accurate. However, in railway systems early train arrivals are discouraged. Early arriving trains may perturb scheduled train movements at junctions and result in higher capacity utilization of
11
station platform tracks. Trains that are ahead of their schedule have to slow down which is also advantegeous for economical reasons. Modelling early arriving trains as punctual also results in a seperation of punctuality management and synchronization management. Note that also in bus systems punctuality management is an important issue. A bus that runs ahead of its schedule will meet small amounts of boarding passengers at the stops and as a result of the smaller stopping times the bus will get more and more ahead of its schedule. As a contrast, the following bus will meet more and more boarding passengers at the stops and will thus get increasingly delayed. In railway systems early departures are prohibited.
6
TRANSFER QUALITY MEASURES
Several measures for transfer quality can be defined using the above developed theory. Note that it is impossible to realize the optimal connection buffer time for each transfer in a timetable. The actual scheduled buffer time is the result of a decision process of weighing various conflicting interests. These are for instance • the dependencies of arrival and departure times on (the waiting times of) different passenger groups (through-passing passengers and passengers of various transfer relations); • the interdependencies of arrival and departure times at and from a single station due to minimum headways resulting from safety criteria and conflicting movements of train routes through stations; • the interdependencies of arrival and departure times between various stations; • priority differences of various train services; • station tracks capacity utilization; • train capacity utilization; • stability of scheduled train circulation times; • railway operator policy. In the sequel rijc denotes the actual connection buffer time for connection ij, whereas the superscript * denotes the optimal value. The first measure is the mean connection time loss, Q1 = wij (rijc ) − wij (rijc* ) = wij (rijc ) − rijc* . Note that the second equality follows from Theorem 2. The measure Q1 is hence the additional travel time of transferring passengers from train i to train j on the average if the implemented connection buffer time is not the optimal one. Alternatively, Q1 can also be viewed as the mean connection time improvement if the optimal connection buffer time will be implemented. Note that both smaller and larger connection buffer times than the optimal one result in an increase of mean transfer waiting time. Q1 is hence a nonnegative real number. It is zero if the actual connection buffer time equals the optimal connection buffer time, and positive otherwise. The unit of Q1 equals the time unit of (4) and (8), or (11) and (12) respectively. An efficient measure to compare several transfers is the relative mean connection time loss defined as the ratio
12
Q2 =
wij (rijc ) − wij (rijc* ) wij (rijc* )
=
wij (rijc ) − rijc* rijc*
.
Q2 measures the average connection time loss with respect to the optimal optimal connection buffer time. It is zero if the actual buffer time is optimal and a positive fraction otherwise. Q2⋅100% is the percentage of connection time loss with respect to the optimal connection buffer time. A slightly different measure to Q2 is the relative connection time improvement possibility with respect to the actual connection buffer time Q3 =
wij (rijc ) − wij (rijc* ) wij (rijc )
=
wij (rijc ) − rijc* wij (rijc )
.
Q3⋅100% is the percentage of possible connection time improvement with respect to the actual connection buffer time. If the trip time distribution of the feeder line is known then the percentage of trips for which transferring passengers miss their connection can be computed. Let F be the trip time cumulative distribution function. Then the miss probability is Q4 = 1 − F (t ir + rijc ) , where t ir is the scheduled running time (trip time) of train i. Note that the probability that trains arrive within the connection buffer time is the cumulative distribution function value of the sum of the trip time and the connection buffer time. Q4⋅100% is the miss percentage.
7
CASE STUDY
Consider the intermodal connection of a bus service to a railway service. The actual trip times of the regional bus service from its starting point to the railway station (terminal station) have been collected by an on-board processor for a certain period of time. From this data the trip times of working days (Mondays till Fridays) are collected corresponding to 323 trips. From these trips 3 are cancelled since their schedule deviates from the timetable. Of these trips the off-peak hours (from 8:38 till 15:38) correspond to a reasonable homogeneous period with respect to the trip times. For this homogeneous period of the working days an optimal connection buffer time is computed. The relevant number of measured trips is 206. The published trip time in this homogeneous period is 45 minutes. Both the bus service and the train service have a headway of 30 minutes. It is assumed that trains do not wait for delayed busses, so the synchronization control margin is 0 minutes. Figure 6 shows the empirical trip time density along with a normal density fit. The assessed normal distribution has parameters µ = 43.07 and σ = 2.17. The scheduled trip time of 45 minutes corresponds to the 0.85th quantile which is 45.3 minutes. In general the 0.85th
13
quantile is assumed to be an efficient schedule time. It implies that 85% of the trips arrive on schedule. So no improvement on the scheduled trip time is necessary. Figure 6 The trip time empirical density and assessed normal density 0.2
0.15
0.1
0.05
0 0
10
20
30 40 trip times (min)
50
60
The arrival delays correspond to the trip times that exceed the scheduled trip time of 45 minutes. The arrival delay data is assembled by extracting the trip times of trips that are behind schedule and subtracting the scheduled trip time. This results in 41 delayed trips. Figure 7 shows the empirical arrival delay density and the exponential density fit. The assessed exponential parameter λ = 0.8159 is the maximum likelihood estimator. Figure 7 The arrival delay empirical and assessed exponential density 1
0.8
0.6
0.4
0.2
0 0
1
2 3 4 arrival delay (min)
5
6
The optimal connection buffer time can now be computed using (12). Note that the . minutes. The corresponding punctuality probability qi = 0.85. It follows that rijc* = 159 optimum mean transfer waiting time is 2.64 minutes. The optimal bus service departure time
14
from the starting point is thus scheduled as 45.3 + 1.6 = 46.9 minutes (trip time and connection buffer time) and additionally the minimum connection time before the train service departure time at the railway station. Figure 8 shows the quality measures as function of the connection buffer time. Figure 8 The transfer quality measures Q1 (mean connection time loss), Q2 (relative connection time loss), Q3 (relative connection time improvement possibility), and Q4 (miss probability) as function of the connection buffer time 30
0.16 0.14
Q
1
25
0.12 20 0.1
Q
2
15
0.08 0.06
10 0.04 5 0.02
Q
Q
3
0 0
5
10
15
connection buffer time
20 c r ij
25
4
0 0
30
5
10
15
20
25
30
c ij
(min)
connection buffer time r (min)
The connection buffer time reduces arrival delays from the bus service at the terminal station. From the (known) trip time distribution it can be computed what percentage of trips arrives within the buffer time. Let F be the normal cumulative distribution function with parameters µ = 43.07 and σ = 2.17. The quantile corresponding to 46.9 minutes trip time is F(46.9) = 0.96. This implies that 96% trips arrive within the buffer time. The miss probability Q4 = 1 - 0.96 = 0.04. The optimal connection buffer time thus results in a miss percentage of 4%. Note that without incorporating the connection buffer time the miss percentage is 15% corresponding to the scheduled trip time of the 85% quantile. Moreover, note that a larger connection buffer time than the optimal one results in a lower miss probability but a larger mean transfer waiting time.
8
CONCLUSIONS AND FUTURE RESEARCH
Stochastic trip times between stations result in arrival delays at a transfer station. Arrival delays are assumed to follow the exponential distribution. It is shown that the expected tranfer waiting time is then a convex function of the connection buffer time. The unique optimal connection buffer time is given explicitly. The method explicitly takes synchronization control into account. Here, a connecting train may wait for transferring passengers of a delayed feeder train during operation. The maximum admissible time for a train to wait is given by the synchronization control margin, and is assumed to be known for
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each train. The optimal buffer time is computed as a function of the synchronization control margin. Several transfer quality measures are defined that quantify the quality of a given connection buffer time with respect to the optimal connection buffer time. These measures are the mean connection time loss, the relative connection time loss, the relative connection time improvement, and the miss probability. All concepts are illustrated by a case study of an intermodal connection of a regional bus service to connecting services. Current research extends the method to derive optimal connection times for two-sided connections, all connections at a transfer station, and all connections in a service network. Here, the waiting time for through passengers is also taken into account as well as the number of passengers. ACKNOWLEDGEMENT - This publication is a result of the research programme Seamless Multimodal Mobility, carried out within TRAIL Research School for Transport, Infrastructure and Logistics, and financed by Delft University of Technology.
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