Emergency Reaction of a Resource Redistribution System
to Shock Events Depending on its Topology
Vasily Lubashevskiy, Taro Kanno, and Kazuo Furuta
Graduate School of Engineering, The University of Tokyo, Japan

1Introduction

In recent years the problem of disaster mitigation has
attracted much attention. We consider this problem fora socio-technological system comprising many cities connected with one another via transport network. The system is supposed to be comparable in size with a prefecture
in Japan or a State in US. It has been partly affected bya large scale disaster, e.g., earthquake, tsunami, volcano
eruption, etc.; the evacuation process and the transport
network destruction cause increased demand for the vital
resources (pure water, medical drugs, warm clothes, fuel,and so on). As a result, in some part of the system thedemand becomes higher than the normal amount of suchlifelines and the situation becomes critical. So there should
be some mechanism of resource redistribution enabling to
mitigate the disaster aftermath and to supply the affected
or involved regions with the required resources at least up
to the level of their stable functioning.
There are currently a family of relative notions, suchas resilience, flexibility, survivability, agility, used in describing the disaster mitigation (for review see, e.g., [1]).
Following [2] the term resilience has various definitions andit can be used differently. In particular, first, the system
performs under abnormal and external influence [2] andthe pre-disaster functioning cannot be implemented because of the system partial destruction [3]. Second, thesystem behavior is determined by contribution of a large
number of its elements or/and subsystems different in sizeand structure. The second feature enables us to classify
the mitigation of disaster consequences in these systems as
an emergent phenomenon. A similar point of view can befound in discussing helicopter system functioning [4] andgeneral problems in reliability engineering [5]. Relative
discussion with respect the seismic resilience can be foundin [6].
The choice of a prefecture and state as characteristic
examples of a given system is based on the following.
First, such administrative units interact with one anotheras whole entities. Second, internal resources available insuch structure are great enough for mitigating disasters
individually. Third, being complex in structure and containing large number of subsystems differece of scales insuch a unit governs the functioning of its elements as wellas their interaction directly.
The purpose of the present work is to analyze the emergency reaction of the system depending on the topologyof resource stock network. We examine two types of thistopology, the centralized one, where the stocks are locatedat some main nodes of the system, and the distributed one,
where the stocks with low capacity are located at all thenodes. To simulate the dynamics of the resource redistri-bution we adopt the following assumptions. The intensityof the required resource redistribution is limited by therate of uploading source materials from stocks to vehicles.
This limitation is caused by the fact that the number ofuploading places and the rate of special uploading vehicles
is not high. Resources are taken from the nearest to the affected territory node, which should send requiring amountof resorces, only minimal volume of vital resources could
be kept.
2Characteristic stages

To quantify the states of the system elements let usintroduce the following characteristics of the cities. The
first one is the critical amount Qc,i of resources in city irequired for its residents and evacuees to survive. Thenstands for the current amount of the vital
the value Qaffiresources in the affected city i, with the inequality
Qaff
i < Qc,ibeing assumed to hold. In addition we will use the value
Qi to describe the current resource amount in city i. To
compare the state criticality of the cities with one another
we also introduce the quantity
θi =Qi − Qc,i,Qc,iwhich may be regarded as the measure of the proximity to
the critical state ascribed to one resident in city i.
From our point of view there could be singled out threecharacteristic stages in the resource redistribution. The
first one is the emergency reaction stage. The system is in
this stage while there are cities where Qi < Qc,i and, thus,
the lives of their residents are in danger. Therefore thecrucial point of the emergency reaction stage is its fastest
implementation.
The main goal of the second stage is to form a new supplying network instead of the previous one broken by thedisaster and to localize the area involved into the resource
redistribution process. During the third stage the formed
networks have to provide external resource flow to the affected region to recover the broken parts of the system.
3Implementation algorithm of the emergency reaction stage

The consequences of real disasters practically cannot
been predicted. That is why a mechanism of the resource redistribution should be based on a certain algorithm rather than be of the form of a plan prepared beforehand. We will use a ”step-by-step” algorithm and thus
are to determine a certain time step ∆t. Let us specify it

as the time necessary to load up one vehicle. Then, theunit of the resource amount h should be set equal to thevehicle capacity. In this way we can ascribe to each stock
the number of volume units and number of units that canbe sent by one time step. Suppose that information about
current transport ways is known and we have a possibility
to send the vehicles to all the affected cities.
The algorithm uses the list {ik } of the cities orderedaccording to the value θi , which determines the city iworstwith the worst situation. Then we seek for the nearest city
j such that Qj > Qc,j +h and add the unit resource amounth to the queue ∆Qj→i to be sent to the affected citites at
the next system update event. Then we check whether
the virtual situation in the affected city iworst remains thesame and if so the procedure is repeated. Otherwise, thelist {ik } is updated and actually the same procedure isrepeated with respect to the new city iworst . The procedure runs until the load capacities of all the stocks areexhausted. When the number of planned parcels becomesequal to possible number of loaded vehicles per unit oftime, ∆t, parcels are sent, which is implemented in thesystem update procedure, t → t + ∆t and for the affectedcities
X∆Qj→i − r.c. ,Qi |t+∆t = Qi |t +jfor the involved cities
Qj |t+∆t = Qj |t −X∆Qj→i − r.c. ,ifor the not-involved cities
Qk |t+∆t = Qk |t − r.c. ,where the symbol ”r.c.” stands for the resource consumption. This implementation of the emergency reaction stage
is terminated when the condition Qj > Qc,j holds for all
the cities.
4The dynamics of the emergency reaction stage
depending on the resource network topology

The dynamics of the emergency reaction stage and theformation of the region involved into the corresponding resource redistribution process depends substantially on thedetails of the stock arrangement in the system including
their spatial distribution as well as their difference in capacity. In the present work we confine ourselves to thestudy of the simplest situation matching the limit cases.
Namely, we assume the resource stocks to be located in afew cities only so they capacities should be rather high or
the large number of low capacity stocks to be distributed
uniformly over the whole area under consideration.
Using the model described in the previous section wejustify the following scenario of the disaster mitigation.
When a few stocks are located at the main nodes of thesystem and have a large volume of resources, they individually determine the values {Qi } in cities located in large
parts of the area under consideration before and after thedisaster. They form a stable pattern of resource flow fromthe stocks to the affected region. However, when one ofthese stocks is damaged by the disaster, not only the region
affected directly but also all the cities supplied previously

by the damaged stock arrive under the critical conditions
because of resource consumption and the absence of local
stocks in their vicinity.
In the case of uniform distribution of relatively small
stocks over the whole area a stable quasi-centralized supplying network does not arise but such distribution is morestable with respect to different disaster consequences andis faster in implementation. In this case there can be singled out two regions different in properties. The first one
which may be categorized as involved region containing
cities with {Qi } gradually decreasing in the direction towards the affected region. The second one surrounding
the first region comprises the cities with pre-disaster conditions.
5Conclusion

This work has presented the mechanism of emergency
reaction of a resource redistribution system to a large scaledisaster which functioning can be regarded as an emergent
process. A certain concept of the full recovering of a region
after a shock event has been constructed and three mainstages of this recovering has been singled out.
The main attention of the work is focused on the initial stage called the emergency reaction stage. Its main
goal is to recover the affected region up to the minimal
survival conditions as fast as possible. A simple algorithm
implementing this stage is considered. To be specific andto demonstrate its efficiency two limit cases are studied
in detail. Namely, the first one is determined by a uniform distribution of relatively small resource stocks over
the area under consideration. The second case matches a
few large stocks located at main cities distant from one
another. The developed mathematical model justifies theresults meeting also the general argumentation, which enables us to expect the model to be efficient in studying
complex situations characterized multi-scale distributionsof resource stocks.
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