PART IV MODELS FOR ANALYSING SECURITY RISKS AND POLICY IMPLICATIONS
4.2 Analysis and Results
The network generated has 159 nodes, a size much smaller than databases generated by previous studies such as for power grids, the Internet or the air travel network. In a network of such a small size, it is difficult to observe well-defined features of the common network models (Dunne et al., 2002). Never-theless, the behaviour of the network can still be identified by examining the different properties attached to it. Among these, the degree distribution of the model is a property of particular interest. Basically, a node degree denotes the number of connections each node is linked to. However, due to the fact that
more than one service may provide a path between two ports, it makes more sense to consider as degree the number of neighbours that a port has. The resulting degree distribution (shown in Figure 3) can be approximated by a very strong power law equal to P(k) = 87.3k−1.6, which could be indicative of an underlying scale-free network.
Regarding the remaining complex network properties of the model, it was found to have an average path between any two nodes of approximately six stops, a clustering coefficient of 0.0278, and a network diameter (maximum number of stops between any two nodes) of 28. Further tests can be run in order to determine the busy nodes on the network. The table below presents a selection of the most heavily used nodes under different definitions of heavy use. (Opt.Paths in the table refer to the number of optimum paths between any two ports in the network that the examined port is a part of.)
Figure 5: Degree Distribution of the Liner Shipping Network between Europe and North America
Station Neighbours Links Opt. Paths
Antwerp 15 152 5239
Bremerhaven 7 124 903
Charleston 12 174 3661
Felixstowe 7 35 216
Halifax 7 47 1585
Station Neighbours Links Opt. Paths
Hamburg 7 78 387
Le Havre 11 112 1891
Manzanillo 10 54 3900
Miami 6 54 2092
Montreal 10 64 1653
New York 12 144 2745
Rotterdam 14 156 5371
Table 1: Critical Nodes and Under Various Definitions of Network Vulnerability
Simulations of informed intentional attacks using these results targeted the busiest nodes and assessed the impact of each action on the network. After each individual attack on a node, the state of the network was reassessed in order to identify the most vulnerable node that would also constitute the next target.
Further analysis can be performed to evaluate the impact of various events on the network as a whole. Our algorithms are capable of determining how container shipments would have to be rerouted to account for the defective node, by identifying a new minimum cost path given the current situation.
Through this procedure, optimal container routes and points of transhipment are recalculated, and the resulting state of the network is compared to the original one before the events. As such, shipment reroutings necessary to avoid currently infeasible paths are identified. Using these results, we can get an estimate of the additional load borne by different parts of the network in its current state by calculating the changes in the number of container routes passing through each node.
The figure below provides a visualization of this process. Indicated by the arrow is the port of Singapore, which is closed due to an imaginary attack, while the circles in bold are the indirectly affected ports that will face the highest extra transhipment load so that containers will reach their destinations without being handled in the affected port of Singapore. As shown in the figure below, the most heavily affected ports are Long Beach, Shanghai and Pusan, with a lot more lying in Europe and Far East that are affected to a smaller but not negligible extent. The wide distribution of the indirectly affected nodes illustrates the global impact of the closure of Singapore.
It is worth mentioning that our process at this early stage of the project does not take into account the processing capacity of the ports, and assumes that
indirectly affected ports will be able to process the additional load. The repercussions would be even wider if, more realistically, capacity is taken into account. Modelling capacity is one of the immediate goals of this project.
Figure 6: Visualization of Impact of Network Events
5 C O N C L U S I O N A N D F U T U R E R E S E A R C H
This chapter started by providing a brief introduction to the complex network theory and its potential applications for modelling liner shipping networks.
For the purpose of this chapter, we only modelled part of the global shipping network, namely the trans-Atlantic network, and have limited the analysis to system robustness and reliability against node failure. Nevertheless, we also mentioned current and future efforts to model the global maritime container transportation network and related intermodal links.
We have collected a database from one of the world’s busiest shipping networks and modelled them as if belonging to one of the standard types of complex network for the purpose of robustness against both random and targeted node failure. Analysis of the network properties has shown that it relates closely to generic scale-free networks with an average path of approx-imately six port stops. Simulation of both random and intentional attacks has revealed that the most critical nodes are not necessarily the busiest ones, and that some ports may be more heavily affected than others, with impacts stretching to ports located beyond the trans-Atlantic network studied in this paper.
More analysis is needed to fully understand the structure, network proper-ties and robustness of the global shipping network, but the study reported here can shed some light on how complex networks theory can be as useful for the analysis of shipping and intermodal routes as it is for other real world net-works. One of the immediate goals of this project is to remove various assump-tions made so far in the interests of greater realism. The database of maritime routes will be extended to cover all of the currently existing liner routes, with ship scheduling also taken into account. Finally, port parameters like TEU
storage and number of quay cranes will also be added to the model in order to obtain estimates of handling and storage capacities.
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P O RT E F F I C I E N C Y A N D T H E S TA B I L I T Y O F C O N TA I N E R L I N E R
S C H E D U L E S
Michael G.H. Bell, Khalid Bichou and Kevin Feldman Port Operations and Research Centre (PORTeC), Centre for Transport Studies,
Imperial College London, UK
Abstract
This chapter examines the stability of schedules, with particular reference to a container liner operating a regular service along a fixed route collecting containers at each port of call. There are, of course, many sources of random variation for an operation of this type, but only one is considered in this chapter, namely the arrival headways at the first port of call. It is assumed that there is no slack in the schedule, so an extension to the arrival headway at the first port allows more containers to arrive at the port (they are assumed to arrive at a uniform rate with no random variation), which then take longer to be loaded. This causes an extension to the departure headway which is longer than the initial extension to the arrival headway. A similar process occurs at subsequent ports of call, so that a small extension to the arrival headway at the first port of call becomes a rather larger extension to the arrival headway at the last port of call. If the schedule is resumed at the first port of call after a perturbation, the schedule may or may not re-establish itself at subsequent ports of call. It is shown in this chapter that the condition for this to occur is that the rate at which containers can be loaded must be at least twice the rate at which containers arrive at each port of call. If this condition does not apply, any perturbation will cause the schedule to break down irretrievably and the system is therefore not stable.
1 I N T R O D U C T I O N
This chapter examines the stability of schedules, with particular reference to a container liner operating a regular service along a fixed route collecting containers at each port of call. There are, of course, many sources of random variation for an operation of this type, but only one is considered here, namely the arrival headway which is the period between the arrivals of two consecutive ships at a given terminal.
This chapter draws upon the model developed by Newell and Potts (1964) which is one of the first models that analysed bus service reliability. The main assumptions are that the passenger arrival rate, the bus loading rate, scheduled headway and travel time between successive stops do not vary between stops
107
or buses. These assumptions remain in this chapter by replacing the passenger arrival rate by the container arrival rate to the quay, the bus loading rate by the ship loading rate, stops by ports of call and buses by ships. Further, Newell and Potts assumed that there was, at each stop, approximately the same number of passengers disembarking as there were passengers boarding. This process occurs simultaneously through the front and the back door of the bus.
We can use this assumption here, assuming that there are at least two quay cranes respectively loading containers onto the ship and unloading containers from the ship onto the quay. Further, in the initial model, the unloading rate exceeded the loading rate, which explains why the calculation of the bus departure times only took into account the loading time. Newell and Potts’s model has been successfully used to describe real-life situations with respect to bus scheduling.
The literature, to our knowledge, contains no study of the stability of schedules in the container shipping industry.
We are assuming that all containers which have arrived at the quay must be loaded onto the ship at the port and that containers can be loaded on any ship of the line. It is assumed that there is no slack in the schedule, so an extension to the arrival headway at the first port allows more containers to arrive at the port (they are assumed to arrive at a uniform rate with no random variation), which then take longer to be loaded. This causes an extension to the departure headway—the period between the departures of two consecutive ships from a given terminal—which is then longer than the initial extension to the arrival headway. A similar process occurs at subsequent ports of call, so that a small extension to the arrival headway at the first port of call becomes a rather larger disruption to the scheduled arrival headway at the last port of call. In this situation, the following ship will encounter fewer containers to pick up at the terminal and will thus spend less time at port than planned, meaning that it will leave the port prior to the scheduled time and thus catch up the leading ship. Because of this phenomenon and of the further assumption that ships do not overtake each other, ‘‘ship bunching’’ will occur.
An important assumption in our model is that the arrival rate of containers reflects the arrival rate at the quay crane, implying that the containers, once
‘‘arrived’’, are ready to be loaded onto the ship. Indeed, we are not considering that containers arrive from the hinterland into the yard and then to the quay but are considering a simple process where containers arrive continuously at the quay in order to be loaded onto the ship.
If the schedule is resumed at the first port of call after a perturbation, the schedule may or may not re-establish itself at subsequent ports of call. It is shown in this chapter that the condition for this to occur is that the rate at which containers can be loaded must be at least twice the rate at which containers arrive at each port of call. If this condition does not apply, any perturbation will cause the schedule to break down irretrievably and the system is therefore not stable.
The chapter goes on to derive analytical expressions for the variances for the arrival headways at the second and third ports of call. While expressions could be derived for subsequent ports, these become more complex along the route because of the growing complexity of the autocorrelation in the arrival head-ways. Simulation results show how the arrival headway variance grows explo-sively along the route, even where the system is stable. This variance can, however, be substantially reduced by increasing the rate at which containers are loaded. Thus the operating speed of quayside cranes is important for schedule stability.
2 S TA B I L I T Y AT A S I N G L E T E R M I N A L
Following Newell and Potts (1964), and substituting ships for buses, terminals for stops and containers for bus passengers, consider a single terminal export-ing containers and define:
a= arrival rate of containers b= loading rate of containers s= ratio of arrival to loading rate
h = arrival headway of vessels (assumed to be uniform)
d(n)= nth departure headway (arrival headway at the next port of call) The nth departure headway is equal to the arrival headway minus the delay caused by the loading the leading ship plus the delay caused by loading the following ship, namely:
which implies that d = h, as one would expect. Further, it can be noted that equilibrium also implies that s<1 since the loading rate of containers must always exceed the arrival rate of containers at the quay. Subtracting (3) from (2) yields:
(4) (d(n)− d) = − ( s 1−s) (d
(n−1)− d).
This demonstrates that a small positive deviation from equilibrium departure headway leads to a subsequent negative deviation from equilibrium departure headway. Whether this is larger or smaller than the initial deviation depends
on whether s
1−sis greater than or less than 1. Stability therefore requires that s
1−s< 1, which in turn requires that s< 0.5. When s
1−s> 1, instability of the system will lead to ship bunching.
Consider a port where a ship calls regularly every day (or every 24 hours, so h = 24). If there is a one-hour deviation in the initial departure (d(0)= 25), subsequent departures will be affected as shown in Table 1 when s= 0.3 or s= 0.6:
d(n)
n h s = 0.3 s = 0.6
1 24 23.57 22.50
2 24 24.18 26.25
3 24 23.92 20.63
4 24 24.03 29.06
5 24 23.99 16.41
6 24 24.01 35.39
7 24 24.00 6.91
8 24 24.00 49.63
9 24 24.00 0.00
10 24 24.00 60.00
Table 1: Arrival and Departure Headways (h = 24, d(0)= 25)
Figure 1: Arrival and Departure Headways (h = 24, d(0)= 25)
When s= 0.3, the schedule eventually re-establishes itself. However, when s = 0.6, the schedule breaks down irretrievably and we get ship
‘‘bunching’’.
3 S TA B I L I T Y F O R T W O T E R M I N A L S
This analysis is now extended to the case of two ports, identified by subscripts 1 and 2.
d1(n)= 1 1−s1
h− s1
1−s1
d1(n−1)
d2(n)= 1 1−s2
d1(n)− s2
1−s2
d2(n−1)= 1
(1−s1)(1−s2) h− s1
(1−s1)(1−s2)d1(n−1)− s2
1−s2d2(n−1)
. Define
y =
[
h/((1h/(1−s1−)(1s1−)s2))]
, A =[
s1/((1s1−/(1s1−)(1s1−)s2)) 0 s2/(1−s2)]
and d(n)=
[
dd12(n)(n)]
.Hence:
(5) d(n)= y − Ad(n−1).
As before,
(6) (d(n)− d) = A(d(n−1)− d) with d =
[
dd12]
.Stability requires that the determinant of A is less than one, which in this case means that det(A) = s1s2
(1−s1)(1−s2) < 1. However, this is a necessary rather than a sufficient condition, as it may be possible for s1
1−s1
> 1, s2
1−s2
<1, and s1s2
(1−s1)(1−s2) <1. The departure headway from the first port is unstable and ship bunching will arise.
Table 2 shows the results for two ports in series, when at the first terminal h = 24 and d(0)= 25.
s1= 0.3, s2= 0.3 s1= 0.6, s2= 0.3
n h d(n)1 d(n)2 d(n)1 d(n)2
1 24 23.57 33.67 22.50 32.14
2 24 24.18 20.12 26.25 23.72
3 24 23.92 25.55 20.63 19.30
4 24 24.03 23.38 29.06 33.25
5 24 23.99 24.24 16.41 9.19
6 24 24.01 23.90 35.39 46.62
7 24 24.00 24.04 6.91 0.00
8 24 24.00 23.99 49.63 70.90
9 24 24.00 24.01 0.00 0.00
10 24 24.00 24.00 60.00 85.71
Table 2: Arrival and Departure Headways for Two Ports in Series
Figure 2: Arrival and Departure Headways for Two Ports in Series
It is evident that deviations from schedule are magnified as the ship pro-gresses from the first to the second terminal. However, when s1 = 0.3,s2= 0.3, operations gradually return to the schedule. When s1 = 0.6,s2 = 0.3, bunching arises although s1s2
(1−s1)(1−s2) = 0.6429 <1 (departure headways are set to zero when they would otherwise be negative, implying that ships cannot overtake each other).
4 S TA B I L I T Y F O R n T E R M I N A L S
The above argument for two ports generalizes for n terminals. In this case, stability would require si
(1−si) <1 for i = 1 . . . n.