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Erlangized Fluid Queues with ApplicationTo Uncontrolled Fire PerimeterDavid A. Stanford a , Guy Latouche b , Douglas G. Woolford a , DennisBoychuk c & Alek Hunchak aa Department of Statistical & Actuarial Sciences , The University ofWestern Ontario , London, Ontario, Canadab Département d’Informatique , Université Libre de Bruxelles ,Brussels, Belgiumc The Ontario Ministry of Natural Resources , Sault Ste. Marie ,Ontario, CanadaPublished online: 16 Feb 2007.

To cite this article: David A. Stanford , Guy Latouche , Douglas G. Woolford , Dennis Boychuk & AlekHunchak (2005) Erlangized Fluid Queues with Application To Uncontrolled Fire Perimeter, StochasticModels, 21:2-3, 631-642, DOI: 10.1081/STM-200056242

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Stochastic Models, 21:631–642, 2005Copyright © Taylor & Francis, Inc.ISSN: 1532-6349 print/1532-4214 onlineDOI: 10.1081/STM-200056242

ERLANGIZED FLUID QUEUES WITH APPLICATION TOUNCONTROLLED FIRE PERIMETER

David A. Stanford � Department of Statistical & Actuarial Sciences, The Universityof Western Ontario, London, Ontario, Canada

Guy Latouche � Département d’Informatique, Université Libre de Bruxelles,Brussels, Belgium

Douglas G. Woolford � Department of Statistical & Actuarial Sciences,The University of Western Ontario, London, Ontario, Canada

Dennis Boychuk � The Ontario Ministry of Natural Resources, Sault Ste. Marie,Ontario, Canada

Alek Hunchak � Department of Statistical & Actuarial Sciences, The University ofWestern Ontario, London, Ontario, Canada

� The present paper develops an “Erlangization” method for fluid queues. It then appliesthe results we derive to a forestry problem: the evolution of an uncontrolled fire perimeter overtime. Specifically, we focus on the probability of containing a fire prior to reaching a randomlydistributed, finite time horizon. Transitions to lower non-zero levels are also investigated. Apreliminary model is introduced to demonstrate the potential of the application, and numericalresults are given for illustrative purposes.

Keywords Erlangization; Fire line; Fluid queue; Forest fire; Markov process;Uncontrolled fire perimeter.

Mathematics Subject Classification Primary 65C40, 65F30; Secondary 60K40.

1. INTRODUCTION

When a forest fire is discovered, various suppression methods may beemployed if extinction of the fire is desired. However, prior to extinction,the primary goal of wildfire fighting is containment of the fire. This

Received October 2004; Accepted February 2005Address correspondence to David A. Stanford, Department of Statistical and Actuarial Sciences,

The University of Western Ontario, London, Ontario, Canada; E-mail: stanford@stats.uwo.ca

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632 Stanford et al.

is commonly done via the creation of a “fire line,” built primarily byground crews. A fire line removes the available fuel from around the fire’sperimeter, obstructing growth beyond the fire line. Once a fire has beencompletely surrounded by fire lines, the fire is said to be fully contained.

The present paper is concerned with developing the “Erlangization”method, by which the finite-time probability of interest is approximated bythe corresponding probability at an Erlang-distributed horizon, in a fluid-queue context. We then turn our focus to a pertinent application of fluidqueues: the fire containment problem. We present a fluid queue model forthe evolution of uncontrolled fire perimeter over time. We are interestedin the probability of fully containing the fire (i.e. reaching level 0) prior tosome finite time, as well as the probabilities of transiting to lower, non-zerolevels. The “Erlangization” approximation amounts to determining theprobability in question occurring by the random horizon, where the meantime to the horizon is set equal the time point of interest t . For Erlang-l distributed horizons, a sequence of approximations is obtained thatconverges asymptotically to the true finite-time probability (see Asmussenet al.[2] and Stanford et al.[9]).

The Erlangization approach is theoretically appealing in that allthe expressions that are computed can be interpreted probabilistically.Furthermore, one is able to avoid the solution of differential equations andthe use of transform techniques. But there are computational benefits aswell for developing this approach. Results from Asmussen et al.[2] as wellas Stanford et al.[9], where the method has been applied to risk models,reveal that for the simplest case of exponentially-distributed horizons, theresults are typically within 10% of the true value, and never more than 20%away. On the negative side, the same computational experience reveals thatthe successive gains as l increases can be disappointing. Nonetheless, evenin these contexts, extrapolations have been used to improve the accuracysubstantially (see Asmussen et al.[2] and Stanford et al.[9]).

We identify a fairly general model as a “fluid queue” defined as follows:consider a Markov process ��(x)�x≥0 on a finite state space � , and associatea real number ri �= 0 to each state i in � : ri is the rate at which some fluidincreases or decreases when the Markov process is in state i . (We commentlater on the case where for some i , ri = 0 in a remark.) We define thepiece-wise linear function

F (x) = u +∫ x

0r�(v) dv,

where u is the fluid level at time zero. The fluid queue is defined as

�(x) = F (x) − min(0, F̃ (x)),

where F̃ (x) = min0≤v≤x F (v).

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Erlangized Fluid Queues for Uncontrolled Fire Perimeters 633

Potential fluid queue applications are wide-spread. In our forest firecase, a common probability of interest is the probability of fully containinga fire, that is, stopping further growth, by time t .

The first passage time of �(x) to zero was first treated in Barlowet al.[4] and London et al.[7]. Much of the work since then has been focusedon computational aspects, such as finding new and better algorithmicprocedures. A key paper in this development was Ramaswami[8]. At MAM4, da Silva Soares and Latouche[6] proposed efficient computationalalgorithms for the determination of key parameters which we employherein. One other paper relevant to the discussion is Badescu et al.[3],which determines the Laplace-Stieltjes transform of the time to ruin in afairly general risk model.

2. BACKGROUND ON THE ERLANGIZATION OF FLUID QUEUES

We start from a Markovian process ��(x) : x ≥ 0� which governs thesuccessive periods of increase and decrease in the fluid queue. As hasbeen shown in da Silva Soares and Latouche[6], it is easy to translatebetween systems with non-unit rates of change, and one in which ∀i , ri =±1. Similarly, Ahn & Ramaswami[1], Theorems 1 and 2, have identifiedthe simplicity with which one can determine transient results for non-unitchange systems. Relative to unit-change systems these entail multiplicationby diagonal matrices of rates of increase or decrease, or their inverses.Therefore, we shall presume the unit-rate situation, and leave to the readerthe pertinent modifications required for the more general case. The statesof this process, called “phases,” are grouped into two subsets: the set �+of phases with ri = 1 (of order k) and the set �− of phases with ri = −1 (oforder m); the infinitesimal generator of the phase process is partitioned as

� =[�++ �+−�−+ �−−

](1)

The amount of fluid in the system is the “level”, and da Silva Soares andLatouche[6] provide expressions for the joint stationary distribution of thelevel and the phase.

We superimpose a single, phase-distributed horizon with representation(�,H ) and stages �1, � � � , l� on this fluid queue, and construct the followingcontinuous time Markov chain for the composite process. There is anabsorbing state 0 once the horizon has been reached, and two sets of statesprior to that point in time: S1 = �1, � � � , l� × �+ comprising the compositestates when fluid is increasing; and S2 = �1, � � � , l� × �− for when fluid isdecreasing. Upon reaching the horizon, the fluid level remains constant;

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634 Stanford et al.

that is, it remains at its last level prior to absorption. For h = −Hel , thischain has infinitesimal generator Q given by

Q = 0 0 0h ⊗ ek H ⊕ �++ Il ⊗�+−h ⊗ em Il ⊗�−+ H ⊕ �−−

. (2)

where ⊗ denotes the Kronecker product and ⊕ the Kronecker sum.We rewrite (2) in terms of matrices Tij for transitions from the transient

states in Si to (or among) the transient states in Sj prior to reaching thehorizon:

Q = 0 0 0h ⊗ ek T11 T12

h ⊗ em T21 T22

.

In order for the fluid queue to reach zero, presuming the fluid isincreasing at the start, the process must first return to its initial level x . Theprobability of this occurring is clearly independent of the actual level. Letus call the instant in time when this return occurs �, and define the matrix by

ij = P �� < ∞, J� = j ∈ S2 |X0 = x , J0 = i ∈ S1�

where Jt denotes the phase at time t , and Xt the fluid level. Similarly, definethe matrix G(x) as the matrix of first passage probabilities to reach level0 from level x , assuming one starts in a decreasing state. Defining to bethis first passage time, i.e. = inf�t > 0 : X (t) = 0�, we write

Gij(x) = P � < ∞, J = j ∈ S2 |X0 = x , J0 = i ∈ S2�

We are now ready to state and prove the following theorems.

Theorem 2.1. The matrix of first passage probabilities G(x) satisfies

G(x) = exp(Ux) (3)

where U is given by

U = T22 + T21 (4)

= H ⊕ �−− + (Il⊗�−+) . (5)

Proof. The relation (3) stems from arguments relating to the process of“downward records” in da Silva Soares and Latouche[6]; see in particular

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Erlangized Fluid Queues for Uncontrolled Fire Perimeters 635

equations (9) and (12) there. As pointed out by da Silva Soares andLatouche[6], it is also obtained by adapting Theorem 3.2 in Ramaswami[8].

Equation (4) here is precisely equation (13) in da Silva Soares andLatouche[6], but we repeat the argument here. Transitions among states inS2 occur directly via the matrix T22, and indirectly if one shifts to the statesin S1 at any particular level, but subsequently manage to return to the samelevel. The transition rates from S2 to S1 are given by T21, and the likelihoodsof subsequently returning to S2 at whatever level we left are contained inthe matrix . Combining these facts yields (4), and substitution of therelevant elements yields (5).

Theorem 2.2. The matrix of probabilities satisfies the equation

=∫ ∞

0exp((H ⊕ �++)y)(Il ⊗�+−) exp(Uy)dy. (6)

Proof. Equation (12) in da Silva Soares and Latouche[6] states that

=∫ ∞

0exp(T11y)T12G(y)dy. (7)

The probabilistic explanation is as follows. By conditioning on the theamount y of fluid accumulated during the first increase, we see that onemust eventually dissipate the same amount in order to return to our initiallevel, and necessarily one will be in a decreasing state when this occurs.The stated relation (6) follows directly after substituting for (3) and the T1j

matrices, j = 1, 2.

Remark 1. The matrix is actually determined in da Silva Soares andLatouche[6] not using the integral equation (6), but rather as the northeastcorner of the matrix of first passage times G in a specially constructed quasibirth and death process, with blocks A0,A1, and A2 given by

A0 =[(1/2)I 0

0 0

], A1 =

[(1/2)P11 0

P21 0

], A2 =

[0 (1/2)P12

0 P22

](8)

where the matrices Pij are uniformized equivalents of Tij , i , j = 1, 2 (seeequation (17) there for further details). This approach first appeared inRamaswami[8].

Remark 2. The case where ri = 0 for some state or states i requiresspecial treatment. One could imagine reconstructing (2) along lines that

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636 Stanford et al.

allow for a third set of states in which the fluid level is not changing.Denoting this set by S0, one would then need to condition on whetherthe transition from S2 took us directly to S1 or first to S0, and so on, inorder to arrive at an analogue to (4). Rather than examine this here,we leave it to the interested reader to develop. We observe that Beanet al.[5] have considered the case where ri = 0. We also note that as far asdetermining the stationary distribution is concerned, the assumption thatri is not null can be relaxed fairly easily; see for instance da Silva Soares andLatouche[6].

We are now ready to state the probability of reaching various levels ofinterest prior to the horizon being reached.

3. TRANSITIONS OF INTEREST

In all of the results to follow, we fix the initial fluid level to be X0 = u.

3.1. Transitions to Level Zero

Since the primary goal of fire suppression is containment, our primaryinterest is in the chance of reaching level 0, as it represents the fire being“fully contained.” The following result can be stated:

Theorem 3.1.1. Assume that the system can be presumed to be in an increasingstate at the start, with the various relative likelihoods given by initial probabilityvector �0. The probability �(u,H ) of reaching level 0 prior to the horizon is given by

�(u,H ) = (�⊗ �0) exp�Uu�e . (9)

Proof. The stated result is readily obtained by conditioning on the initialstate in S1 and the state in S2 at the moment of return to the initial level.A similar result applies in the case that the process is initially decreasing,using a similar argument, yielding the following theorem.

Theorem 3.1.2. Assume that the system can be presumed to be in a decreasingstate at the start, with initial probability vector 0. The probability �(u,H ) ofreaching level 0 prior to the horizon being reached is given by

�(u,H ) = (�⊗ 0) exp�Uu�e . (10)

3.2. Transitions to Lower Non-zero Levels

Transitions to non-zero levels are also of interest, as a decision makeralso needs to know the potential of a fire starting from level u decreasing

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Erlangized Fluid Queues for Uncontrolled Fire Perimeters 637

to a certain size. The foregoing two theorems readily generalize to give thefollowing results for transitions to lower non-zero levels:

Corollary 3.2.1. Assume the system is in, respectively, the set of increasing statesat the start with initial probability vector �0, or the set of decreasing states at thestart with initial probability vector 0. The probability �(u, x ,H ) of reaching levelx where 0 < x < u prior to reaching the horizon in these two cases is given by,respectively,

�(u, x ,H ) = (�⊗ �0) exp�U (u − x)�e , (11)

�(u, x ,H ) = (�⊗ 0) exp�U (u − x)�e . (12)

4. A MODEL FOR UNCONTROLLED FIRE LINES

There are limited resources available to be deployed when fightingseveral forest fires. The primary goal of this modelling is to provide fire linemanagers with a tool to aid their resource allocation decisions. Specifically,given different resource allocations we want to provide “best estimates” ofwhere fires would be in 24 or 48 hours.

A fluid queue model requires linear rates of fluid increase anddecrease. Clearly a forest fire does not grow in area linearly with time;the larger a fire is, the faster it can spread. However, it is reasonable toassume, at least as a very good approximation, that the perimeter of thefire will grow linearly over time. In the absence of any intervention by firefighting authorities, the perimeter is entirely uncontrolled. The quantity of“uncontrolled fire perimeter” (defined as the amount of the fire perimeterthat is not contained by a constructed fire line) is a standard performancemeasure used to describe the fire. Its amount and precise rates of changewill be largely dependent upon environmental factors such as wind, rain,ground level moisture, etc.

Conversely, upon intervention, the fire crews would be working linearlyover time on the construction and maintenance of a fire line. The rates atwhich this fire line can be constructed are largely dependent on the sameenvironmental conditions that affect fire growth, as well as the amountof intervention (number of ground crews, airborne support, etc). In“favourable” conditions with sufficient resources, fire crews build the linerapidly relative to the fire’s growth, decreasing the amount of uncontrolledperimeter. In “unfavourable” conditions, the growth in the fire perimeterwould exceed the rate of fire line creation, increasing the amount ofuncontrolled perimeter. In fact, in some extremely unfavourable states, itis necessary for fire crews to halt fire line construction altogether.

Thus, when considering the construction of a fire line, the twocompeting linear processes (fire growth and rate of fire line construction)

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638 Stanford et al.

justify the use of the fluid queue framework to model uncontrolled fireperimeter.

Our initial goal is to demonstrate the capability of a fluid queuemodel in terms of calculating probabilities of interest while maintaining a“reasonable” number of parameters for statistical testing against real data.To do this, we introduce the following preliminary “toy model,” comprisedof four states. Noting that the presence or absence of wind is generallyaccepted by foresters to be a highly influential measure, our model is basedsolely on two factors: wind conditions and an indication of relative progressin fire line construction. The idea is that if the state is initially windy, onemust first wait for the wind to die down before an attempt can be madeto construct fire line. Initially, the construction of fire line is expected tobe slow, but eventually one gains the upper hand, and subsequent fire lineconstruction proceeds at a faster pace, subject to disruption due to otherbouts of windy weather, and a small likelihood of losing ground once again.The corresponding states are defined as follows:

WN: windy, not held.CN: calm, not held.WH: windy, half held.CH: calm, half held.

Notice that the weather can be either windy or calm, and that therelative progress in fire containment is either not held or half held. Our“increasing” states are those in which the uncontrolled fire perimeter isgrowing. These states are WN, CN and WH. The only “decreasing” state isCH. The corresponding transition diagram is

WH

WN CN

CH

�

1/2

1/5

1/4

1/5

1/2

��

��

�

�

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Erlangized Fluid Queues for Uncontrolled Fire Perimeters 639

which leads to the following rate matrix:

� =

−0. 50 0. 50 0 00. 20 −0. 45 0 0. 25� 0 −(0. 50 + �) 0. 500 0 0. 20 −0. 20

(13)

We interpret the transitions as follows: for any seven day time period,it is windy on average for two days before returning to a calm state for anaverage of five days. We also see that, while in a calm not held state, it ismore likely to move to a calm, half held state, than back to a windy notheld state. And finally, we include a variable transition rate, �, that “feedsback” into the windy, not held state.

Remark. We note that one could elaborate substantially on theaforementioned model. Its primary purpose here is to demonstrate thepotential application of the fluid queue framework. The given transitionrates are “ballpark” estimates, based on expert opinions. As indicatedearlier, further work, using appropriate statistical methods, needs tobe done to determine which factors need to be reflected in our statedescription, and to obtain statistically valid parameter estimates for them.

5. NUMERICAL RESULTS

In this section we include two tables of probabilities calculated forour given model. These probabilities were calculated using the statisticalsoftware package R (an implementation of the S language). Computationtimes are fast – a matter of seconds for all of the tabular data displayedherein.

Probabilities �(u,H ) of full containment from initial level u prior toreaching the horizon H are given in Tables 1 and 2. Both tables displayprobabilities for feedback rates � = 0 or 1/10; and initial levels u = 1, 10or 100. Horizons are Erlang-l distributed, where l = 1, 2 or 3, scaled tohave mean duration 2/a (the Erlang-2 being our benchmark case). Table 1displays probabilities corresponding to the system starting in the WN state,while Table 2 corresponds to an initial state of CN.

Globally, the probabilities behave as expected: The probability of fullcontainment increases as either the mean horizon length 2/a increases,or as the initial size u decreases. Increasing the feedback rate � has theopposite effect. And finally, starting the process in the CN state, rather thanin the WN state, increases the probability of full containment.

We also comment on the behaviour of the probabilities resulting froman Erlang-1 (i.e. exponential) horizon. The exponential distribution hasgreater variability when compared to higher order Erlangs with equal

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640 Stanford et al.

TABLE 1 Probability of Transitions to 0, �(u,H ) Starting in WN State, Comparable Erlang-lHorizons, l = 1, 2, 3

Initial state � u 2/a Erlang(1, a2 ) Erlang(2, a) Erlang(3, 3a

2 )

WN 0.0 1 1 0.02635 0.02047 0.01780WN 0.0 1 10 0.40068 0.44741 0.46773WN 0.0 1 100 0.87796 0.95442 0.97583WN 0.0 1 1000 0.98635 0.99930 0.99994

WN 0.0 10 1 0.00008 0.00000 0.00000WN 0.0 10 10 0.17048 0.16018 0.15276WN 0.0 10 100 0.79290 0.90333 0.94308WN 0.0 10 1000 0.97608 0.99842 0.99985

WN 0.0 100 1 0.00000 0.00000 0.00000WN 0.0 100 10 0.00003 0.00000 0.00000WN 0.0 100 100 0.28619 0.29012 0.28411WN 0.0 100 1000 0.87908 0.97078 0.99182

WN 0.1 1 1 0.02616 0.02040 0.01776WN 0.1 1 10 0.37628 0.42150 0.44144WN 0.1 1 100 0.84238 0.91687 0.93925WN 0.1 1 1000 0.97953 0.99689 0.99850

WN 0.1 10 1 0.00008 0.00000 0.00000WN 0.1 10 10 0.14390 0.13706 0.13221WN 0.1 10 100 0.72577 0.82654 0.86475WN 0.1 10 1000 0.96266 0.99365 0.99726

WN 0.1 100 1 0.00000 0.00000 0.00000WN 0.1 100 10 0.00001 0.00000 0.00000WN 0.1 100 100 0.16355 0.14037 0.12373WN 0.1 100 1000 0.80910 0.92225 0.96003

means. This results in an increased likelihood of very late, or very earlyhorizons for the Exponential case. Thus, for short time periods, the longer-tailed exponential has a greater probability of full containment. Conversely,for long time periods, the exponential has a greater likelihood of an earlyhorizon, decreasing the chance of full containment when compared tohigher order Erlang horizons.

6. CONCLUSIONS

We have introduced a model for uncontrolled forest fire perimeterbased on the application of Erlangization to fluid queues. The modelenables us to approximate the probabilities of fully containing the fire bytime periods of interest, as well as reaching lower, non-zero levels. Thenext step in the process is to perform statistical testing to identify usefulexplanatory variables among the various environmental factors (presumingthey exist). The subsequent step is develop a variety of models to determinewhich variables are most powerful in terms of their predictive properties, aswell as estimation of model parameters using data provided by the OntarioMinistry of Natural Resources and other possible sources.

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Erlangized Fluid Queues for Uncontrolled Fire Perimeters 641

TABLE 2 Probability of Transitions to 0, �(u,H ) Starting in CN State, Comparable Erlang-lHorizons, l = 1, 2, 3

Initial state � u 2/a Erlang(1, a2 ) Erlang(2, a) Erlang(3, 3a

2 )

CN 0.0 1 1 0.08636 0.08585 0.08567CN 0.0 1 10 0.51684 0.57757 0.60113CN 0.0 1 100 0.90662 0.96726 0.98296CN 0.0 1 1000 0.98963 0.99950 0.99996

CN 0.0 10 1 0.00027 0.00001 0.00000CN 0.0 10 10 0.21990 0.22261 0.22233CN 0.0 10 100 0.81878 0.92159 0.95577CN 0.0 10 1000 0.97933 0.99875 0.99988

CN 0.0 100 1 0.00000 0.00000 0.00000CN 0.0 100 10 0.00004 0.00000 0.00000CN 0.0 100 100 0.29553 0.30327 0.29977CN 0.0 100 1000 0.88200 0.97211 0.99235

CN 0.1 1 1 0.08592 0.08562 0.08550CN 0.1 1 10 0.49429 0.55268 0.57533CN 0.1 1 100 0.87872 0.93862 0.95544CN 0.1 1 1000 0.98442 0.99772 0.99890

CN 0.1 10 1 0.00027 0.00001 0.00000CN 0.1 10 10 0.18903 0.19317 0.19449CN 0.1 10 100 0.75707 0.85330 0.88754CN 0.1 10 1000 0.96746 0.99474 0.99775

CN 0.1 100 1 0.00000 0.00000 1.00000CN 0.1 100 10 0.00001 0.00000 0.00000CN 0.1 100 100 0.17061 0.14897 0.13297CN 0.1 100 1000 0.81313 0.92507 0.96187

The other direction for further research is theoretical. One wouldnaturally be interested in the probability of excursions to higher levels. Thistoo will have practical benefits. Decision-makers are not only interestedin the probability of containing a fire, but also in the chance that ismight grow beyond key pre-specified perimeters. One of the difficultiesin characterizing such excursions is that the dual of the matrix (thatwould provide the probability of returning to a given level in an increasingstate, when one is initially in a decreasing state) is level-dependent due tothe boundary at level 0. The behaviour of our process is obviously closelyrelated to the behaviour of fluid queues without boundaries at level 0, ashas been studied by Bean et al.[5]; see in particular Theorem 1 there. It isimportant, however, to distinguish between the level-independent natureof their probabilities and the level-dependent nature of ours.

ACKNOWLEDGMENTS

The work of David Stanford, Douglas Woolford and Alek Hunchak hasbeen assisted by Dr. Stanford’s NSERC Discovery Grant. We wish to thankthe referees whose comments have helped to improve the present work.

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REFERENCES

1. Ahn, S.; Ramaswami, V. Transient analysis of fluid models via elementary level crossingarguments. 2004, submitted for publication.

2. Asmussen, S.; Avram, F.; Usábel, M. Erlangian approximations for finite time ruin probabilities.ASTIN Bulletin 2002, 32, 267–281.

3. Badescu, A.; Breuer, L.; da Silva Soares, A.; Latouche, G.; Remiche, M.; Stanford, D.A. Riskprocesses analyzed as fluid queues. Scandinavian Actuarial Journal 2004, “prEview” site, postedSeptember 9, 1–16.

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