On Modelling, Control and Management of a Refugee Exodus Using Theory of Mass-Servicing Systems

Copyright lE> IFAC Conflict Management and Resolution in Regions of Long Confronted Nations (SWIIS 2000), Ohrid, Republic of Macedonia, 2000

ON MODELLING, CONTROL AND MANAGEMENT OF A REFUGEE EXODUS USING THEORY OF MASS·SERVICING SYSTEMS

Zoran Gacovski and Gjorgi Hristovski

Ministry ofDefence. Orce Nikolov bb, Skopje. Republic ofMacedonia Fax.: ++389-91-119-572; E-mail: [email protected]

Abstract: In the spring 1999, due to the Kosovo crisis in its closest neighbourhood., Republic of Macedonia was exposed to a large surge of refugees. Around 400,000 refugees have crossed the border in a period of 30 days. This situation has overloaded the state resources in many fields, and especially in food, water and power supply, transport, communications, environment preservation and medical treaunent. In this paper we give short presentation of all efforts that were made by our Government, and we explain the way of modelling and analysis of those processes via massive servicing systems (queuing theory) as well as network planning. This analysis can be useful everywhere when the similar overloads appear in such a short period and territory. Copyright @ 2000 1FAC. Keywords: Mass-servicing systems; queuing theory; simulation modelling; war exodus; refugee flow management; Petri-nets.

Table 1, where a comparison is given what a similar event would mean to USA, Germany and France In addition to the ongoing transitional restructuring should their respective population would have been crisis and wars in the countries of the former S.F.R of suddenly increased by as it happened to Macedonia Yugoslavia (Dimirovski, 1999), last year during Kosovo crisis some 379.523 people were forced to In response with regard to the humanitarian tragedy move into Republic of Macedonia, one way or of the population of Kosovo, the RM. Ministry for another, as a result of these crisis events. Out of this Urbanism and Construction, assisted by the NATO number, some 92.100 departed to third countries, forces in Macedonia, have built nine camps in order 154.989 were accommodated in families in the RM., to accommodate refugee citizens from Yugoslavia. 112.434 were accommodated within the camps built These camps were: Stenkovec 1 and 2, Ccegrane 1 and some 20.000 those citizens of the nowadays F.R and 2, Bojane, Neprossteno, Radusa, Senokos and of Yugoslavia are estimated to be not registered at all. Blace. Certainly, both expected and unexpected difficulties appeared during the execution process of Within a month time, almost all of a sudden, the RM. this humanitarian action by Macedonian faced the social and certain political pressure of an Government. At this occasion, in particular, we increase of its population by 15% approximately. The would like to mention the following difficulties: actual impact of this pressure may be inferred from. transportation problems; 1. INfRODUCTION

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food, water and power supply; cross-passing the border and entering without ID documents; political activities of the refugees in and outside the camps, some of which were illegal; organised and unorganised criminal activities; health problems (danger of epidemics, daily health prevention; about 100 deaths, and approximately 600 new born); ecological problems (water pollution, devastation of the near forests); problems with humanitarian commodities, (expired reserves, smuggling, illegal trade);

2. SOME ELEMENTS OF QUEUE THEORY The phenomena of crowd and congestion are important factors for planning of crossroads, airports, material flow, production processes, computer architecture etc., i.e. all situations where we wait for service, response or failure to occur. Queuing theory (Kendall, 1951; Morse, 1958) deals with problems, which involve queuing (or waiting). In essence all queuing systems can be broken into individual sub-systems consisting of entities queuing for some activity (as shown in Fig. 2 below). Typically we can talk of the individual sub-system as dealing with customers queuing for service. To analyse this sub-system we need information relating to:

Table 1. A comparison presentation of the load effiects onRof Macedorua caused b)y therefu gee fl ow Estimated Almost as COUNTRY Current Population Population the PopuIncrease lation in for 14.77% the sate of 1,945,932 287,423 Macedonia Iceland 253,250,000 37,406,175 USA Canada 81,591,000 12,051,361 Bel~ium Gennanv 57,981,000 8,564,057 Guinea France

- arrival process - how customers arrive e.g. singly or in groups; how the arrivals are distributed in time (e.g. what is the probability distribution of time between successive arrivals (the inter-arrival time distribution);

I

Arrival

The above detail depicts the actual exodus phenomenon that took place and the magnitude of the refugee flow into our country. In Figure 1, there is given a graphical representation of the time series of this refugee flow during the period April-June inclusive.

H

Queue

H

Service

H

Output

Fig. 2 Components of a queue model - service mechanism - a description of the resources needed for service to begin; how long the service win take (the service time distribution); the number of servers available and whether the servers are in series (each server has a separate queue) or in parallel (one queue for all servers).

Refugees

- queue discipline - how, from the set of customers waiting for service, do we choose the one to be served next, e.g. FIFO (first-in-first-out), LIFO (Iastin-first-out), randomly. Note that an integral ingredient to queuing situations is the uncertainty in, for example, inter-arrival times 100000 and service times. This means that probability and statistics are needed to analyse queuing situations. In - Y APL.-RIL --+---MA --+-------+----+ tenus of the analysis of queuing situations the types JULY of questions in which we are interested, are typically concerned with measure of system perfonnance and might include: waiting time, service time, average Fig. 1 The time series representation of refugee flow length of the queue, occupation of the server, related Now, we address the issues of using the theory of probabilities etc. These are questions that need to be answered so that management can evaluate mass-servicing systems, the queue theory, and the theory of Petri-nets in dealing with the management alternatives in an attempt to control the situation, such as: number of servers, buffer size etc. of this refugee flow.

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I

In order to get answers to questions of concern, there are two basic approaches: analytic methods of queuing theory (fonnula based) and simulation (computer based). In the case of simple, one-server, queuing systems analytical approach is feasible. However, complex queuing systems are analysed almost exclusively by using simulation (Dimirovski and co-authors, 1994, 1996).

respectively, then the overall arrival process remains exponential, with parameter 1.1+1.2. (This extends to more than two arrival streams, too). Splitting is when a exponential process with rate A is split into two streams, where each customer is assigned to stream one with probability PI, and stream two with probability Pz= I-PI, independent of the other customers, the resulting processes remains exponential with rates PIA and PzA respectively. In the examples of some queues, which are given bellow, we are using only exponential distributions for both of arrival and service times.

At many problems of queuing theory, the arrival distribution is not constant, and most often it is an exponential distribution :

Petri nets are concept developed for modelling of discrete-event, concurrent, conflict and synchronous processes and systems, and enable appropriate modelling of two-level task-orineted control system architecture for dicrete-event and hybrid processes to be controlled (Dimirovski and co-authors, 1996) according to the principle of increasing precision with decreasing intelligence (Saridis, 1989). Next section presents more detail within the present context of simulation modelling of queuing discreteevent processes to be controlled.

where 1>0. The middle value of the exponential distribution is 1/1., and standard deviation is also 1/1.. (So this value is treated as middle time of arrival/service). Two-parameter frequency distribution, which is usually being applied in the queuing theory, is called Gamma or ErIang distribution (Morse, 1958). Constant and exponential distributions can be treated as special cases of this distribution. If we take that particular time arrivals are k independent times, every with exponential distribution, than Gamma distribution can be written as:

3. SIMULATION VIA PETRI NETS A Petri net (Zurawski and Zhou, 1994) is a graphical net-like structure that is consisting of two types of nodes: states and transitions, which are connected with directed arcs. The state dynamics of the Petrinet is given with its marking vector, i.e. the number of tokens in each state. The physical meaning of the transition activation is: all enabled transitions are "fired" and the system is transferred from one to another state, i.e. from one to another marking vector. In the next moment, there are new-enabled transitions, which will "fire", and this sequential process continues until there exist enabled transitions.

F(x) =~f"'tk-Ie-A1dt=_I-rAr(k) (k-l)!o (k-l)! where for the gamma function r A>.(k), there exist inb the literature tabulated values, which depend on k and Ax=z. The middle value of the Gamma distribution is k/A, because the middle value of k independent arrival times is equal to k-times of middle time of one particular arrival. Standard deviation of the Gamma distribution is ..JkIA, because of the fact that the deviation ofk independent arrival times is equal to ktimes of deviation of one particular arrival, which is (1/1.)2.

Today's simulation models use standard software elements: random number generation, transformation to a stochastic distribution, incrementing of simulation time, adding and removing of events, analysis of stochastic data, report creating etc. Our simulation of queue is based on Petri-net simulation (Dimirovski and Gacovski, 1996; IIiev and coauthors, 1996) with given input and output matrices (connections of states and transitions) and initial marking vector. We can receive than: the average number of tokens in all states during simulation, the average time when particular state is empty (without

We can easy prove that the negative exponential distribution is a special case of Gamma distribution, which is obtained for k= 1 and for k=oc, we can obtain the constant distribution. Exponential processes can be combined and split. Combining is if a queue is fed by two independent exponential processes with rates AI and 1.2

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tokens) and current marking vector. With these parameters, for a given queue, we can propose: arrival and service distributions, number of servers, system capacity, priority definition etc. We will make use and simulate two of the most investigated queues: MlM/I and MIM/k.

we multiply this value with the average number of tokens in S2 (percent of time when the system is active) we will obtain the average service time for each customer. We have made two simulations (for different values of p) and the results for queue parameters are given in the table above.

3.1. MIMI1 Queue

3.2. MIMIK Queue

The simplest massive servicing system is MlM/I queue, which is defined via appropriate exponential stochastic rates of arrival and service, given with A. and lJ., one server with infinite capacity, and FIFO service priority. In the system following variables can be defined:

This system is characterised with the exponential arrival distribution A. and service distribution lJ., and it is consisting of one queue, k service units, and FIFO service priority. The well known fonnulas given below apply: 1. System service factor:

1. System service factor:

p=A. I klJ.,

p=A. I lJ.,

2. Probability for empty system:

2. Probabilities of the pennanent states in the system: Po=I-p, PI=POP, ... , Pi=popi ,

Po=l/(LpD/n!+ Ilk! LpD/k"0k) 3. Probability for n objects in the system Pn :

3. Average number of customers in the system: T= p/(l-p),

4. Average number of customers that are servicing: S=A. I lJ.,

4. Average number of customers that are servicing:

S=p,

5. Average number of customers in the

5. Average number of customers in the queue: Q=T-S=p2/(l-p),

system: T=Q+S,

6. Average waiting time: W=T/lJ.,

6. Average waiting time: W=Q/A. ,

7. Average time spent in system: W·=W/(I-po).

7. Average time spent in system: W·=T/A..

This system can be modelled using the Petri-net formalism given on Fig. 3. Transitions 1 and 3 have stochastic behaviour' with known exponential distributions, and transitions 2 and 4 are instantaneous. The perfonnance evaluation can be done on the following way described in the sequel. Fig. 3. MlM/I queue. The average value of number of tokens in S2 gives the average time when the system is active, Le. when it serves (S), while average number of tokens in S4 gives the average time when the system is empty (PO). The average number of tokens in SI gives the average number of objects, which are waiting for service, while the sum of tokens in SI and S2 gives the average number of customers in the system.

Table 2. Petri-net based simulated evaluation of ~I model and attainable perfonnance

p=113 p=213

When the simulation time is divided with the number of served objects (number of tokens in Ss) we obtain the average time that object spends in the system. If

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T P T P

Po

Q

s

T

W

W

0.66 0.657 0.33 0.31

0.166 0.156 1.33 1.287

0.33 0.342 0.66 0.689

05 0.49 2 1.976

2 1.9 8 7.9

6 5.8 12 11.47

This system can be modelled with the Petri-net model given on Figure 4. Transition 1 has exponential stochastic distribution with middle value A, and transitions 5, 6 and 7 have middle value ~. Transitions 2, 3 and 4 are instantaneous.

Table 3. Petri-net based simulated evaluation of the MIM/k model and attainable perfonnance p=113

T

p=213

T

P

P

Po

0

S

0.717 0.72 0.513 0.54

0.050 0.051 0.754 0.747

1 1.092 2 2.09

T 1.05 1.14 2.754 2.838

W

W

0.151 0.14 1.132 1.094

3.151 3.43~

4.132 4.261

4. CONCLUSIONS

The phenomena of crowd and congestion are important factors for planning of crossroads, airports, material flow, production processes, computer architecture etc., i.e. all situations where we wait for service, response or failure to occur. The exodus refugee flow is a complex stochastic, discrete-event process that should be controlled and managed, but it is most difficult to develop appropriate strategies. The queuing theory and Petri-net based discreteevent simulation technique seems promising. In terms of the analysis of queuing situations, the types of questions in which we were interested are typically concerned with measure of system performance and might include: waiting time, service time, average length of the queue, occupation of the server, related probabilities etc. Petri nets represent a mathematical formalism that has been developed by Karl Adam Petri within the context of communication among and with automata /computers. However, it may well be used for simulation modelling of all sorts of discrete-event, concurrent, conflict and asynchronous processes.

Fig.4. MIM/k queue.

The perfonnance evaluation can implemented as follows: The average value of sum of tokens in 82, 83 and 84 gives the average time when the system is active, i.e. when it serves (S), while average sum of tokens in Ss, S6 and S7 gives the average time when the system is empty (Po). The average number of tokens in SI gives the average number of objects which are waiting for service, while the sum of tokens in S2, S3 and S4 gives the average number of customers in the system. It is interested to note that for small values of p, if the first server is busy, the customer will be served on the second or on next free server.

We presented in this paper, our simulation of queue analysis based on Petri-net simulation with given input and output matrices (connections of states and transitions) and initial marking vector. We have received than: the average number of tokens in all states during simulation, the average time when particular state is empty (without tokens) and current marking vector. With these parameters, for a given queue, we can propose: arrival and service distributions, number of servers, system capacity, priority definition etc. We have simulated in this paper, two of the most investigated queues, MlM/l and MIM/k, which find application to approximate control and management of exodus refugee flows.

This model enables calculation of parameters for each server separately, and after dividing with the number of channels we obtain the average load of the syst.;:m. We have made two simulations (for different values of p) and the results for queue parameters are given in the Table 3 bellow.

It is believed that more adequate simulation based methodology would emerge by employing fuzzy

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