THE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES

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1 Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages The website: Scietif Reseach of the Istitute of Mathemats ad Compute Sciece THE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES Mhal Matalyck Tacaa Romaiuk Istitute of Mathemats ad Compute Sciece Czestochowa Uivesity of Techology Abstact. I the peset pape the aalysis of models fo claim pocessig i isuace compaies whe the total umbe of isuace cotacts may be a fuctio of time is caied out. Closed by the stuctue queueig etwoks with bouded time of claims stay i the queues of pocessig systems seves as models fo claim pocessig.. Itoductio The aalysis of some mathematal models fo eqitype ad multi-type claims pocessig was caied out i the pape [] aleady. Let a isuace compay cosists of a cetal depatmet ad siste compaies. Evey advaced claim passes two stages of pocessig - the estimatio stage i ay of siste compaies ad the stage of paymet i the cetal depatmet. Assume that the total time fo waitig of a isue who advace a claim i the queue fo ith siste compay ad the time he eeds fo applatio i the othe siste compay is distibuted accodig to the epoetial ule with paamete ν. The isue who is ot seved i the ith siste compay advaces a claim i the th siste compay with pobability qi i. The queueig etwok with bouded time fo claim waitig i the queues of pocessig systems seves as the pobabilist model fo claim pocessig i this case. Let us descibe such a etwok. Coside a closed queueig etwok whh cosists of + pocessig systems S 0 S... S i whh K eqitype claims culate. The system S i cosists of m i idetal pocessig lies the time of pocessig i each lie is distibuted accodig to the epoetial ule with aveage µ i 0. Besides suppose that the time of stay of claims i the queue of ith pocessig system is a vaiate whh is distibuted accodig to the epoetial ule with a paamete ν i 0. The claims fo pocessig ae chose accodig to the FIFO disciplie. The claim pocessig of whh i the system S i is fiished passes to the queue of the system S 0 with pobability p i ad the claim the waitig time of whh is elapsed passes to the queue of the system S with pobability 0. I the q i

2 44 M. Matalyck T. Romaiuk geeal case the mates of tasitios P pi qi Q i 0 ae ot idetal ad they ae the mates of tasitio pobabilities of ieducible Makov chai. The vecto k ( k ( k (... k ( whee k i ( is a umbe of claims i ( 0 t the system S i at the momet of time t i 0 foms ( + -dimesioal Makov pocess with cotiuous time ad fiite umbe of states. Obviously 0 k ( K k i ( whee K is a umbe of claims i the system se the system is closed. I [] it is detemied that the desity of pobability distibutio of the elative k0( k( k ( vaiables vecto ξ (... satisfies the Kolmogoov-Fokke- K K K -Plac equatio to ( Ο ε whee ε K Whee: p( ε t ( Ai ( p( + ( Bi ( p( 0 i 0 i 0 i A ( [ µ p mi( l + ( ν q u( ] ( i B ( ii 0 [ µ i mi( l + ( ν i i u( Bi ( µ i pi mi( l i ( i li νiqiu( i li ] ( i p p i i i q q i i i p + i pii i i q + qii i u( is a Heavyside fuctio. As it was show i [] fom the equatio ( it follows that to the same ( (... ( whee accuacy the compoets of the vecto ( ki( i( M i 0 ca be detemied fom the diffee- K ki( i( M K tial equatios set ( 0 t

3 The Aalysis of Some Models fo Claim Pocessig i Isuace Compaies 45 di ( Ai ( ( dt 0 i [ µ p mi( l ( + ( ( ν q u( ( ] (3 i 0 The equatios set (3 ca be obtaied fom the equatio ( if oe pefoms the aveage-out opeatio o its left ad ight pats i.e. oe should itegate its both pats i the age fom 0 to by each compoet i i 0 ad multiply the itegable fuctio o a coespodig compoet i additio. The the itegal fom Ο ( ε gives us the epessio of the ode Ο ( ε. The ight pats of the equatios (3 ae piecewise discotiuous fuctios. Usig the decompositio of the phase space oe ca detemie a eplit fom of the set (3 i the domais of cotiuity of its ight pat whee 0 di ( 0 [ µ pil+ ( ( ν qi ] + µ pi ( i 0 dt Ω0 ( Ω ( Ω ( { : l < ( } Ω ( { : 0 ( l} 0 ae o-ovelappig sets of the ides of the vecto ( compoets. The above descibed queueig etwok may be used as a geealized model of the claims pocessig i a isuace compay descibed i []. Let a isuace compay cocluded K equitype isuace cotacts with isues. Let m i compay employees (estimatos ae occupied with claims estimatio ad m compay employees ae occupied with claims paymet. Assume that a pobability of claim advacig i the ith siste compay o the iteval of time [ t t + t] equals to µ 0( p0i t + ο( whee µ 0( t is a piecewise costat fuctio with two itevals of costacy whh chaacteize the itesity of claims ety: µ µ 0( µ 0 0 t [0 T / ] t ( T Claims pocessig times by the estimatos i the ith siste compay ad claims pocessig times by the estimatos i the cetal depatmet ae distibuted accodig to the epoetial ule with itesities µ ad µ coespodigly. Besides the total time of the isue stay who advaces a claim i the queue of the ith siste compay ad the time he eeds fo applatio i the othe siste compay ae also distibuted accodig to the epoetial ule with othe paame- / T ] i

4 46 M. Matalyck T. Romaiuk te ν i.e. a isue who is ot seved i the ith siste compay with pobability q advaces a claim i the th siste compay i. i The compay state at the momet of time t may be descibed by the vecto k ( k ( k (... k ( whee k i ( ad k ( ae the umbe of claims whh ( t ae i the ith siste compay ad i the cetal depatmet coespodigly. The compay pefomace (aveage iputs of the compay o the itevals of time [ 0 T / ] ( T / T ] coespodigly may be descibed by the fuctioal [; ] W ( T W ( T m... m T dii ( + Eili T 0 whee: d i Å i - cost coeffiets. We ae iteested i the poblem of detemiatio of estimatos umbe o the itevals of time [ 0 T / ] ad ( T / T ] whh miimizes the aveage iputs (4 ude esttios o the aveage claims umbe K i ( whh ae o the diffeet pocessig stages. Natually the closed queueig etwok with bouded time of claims stay i the queues whh cosists of the cetal pocessig system S (cetal depatme outlyig pocessig systems S S... S (siste compaies ad the system S 0 whh coespods to the eteal eviomet (souce of claims ety may seve as a pobabilist model of claims pocessig m 0 K. Tasitios pobabilities betwee systems ae as follows: p 0i 0 p p 0 i p 0 i othe cases; q 0 i q i q 0 i othe i cases. i i dt i (4. Aalysis of the geealized model The peset model may be geealized o the case of the multy-type claims whe thei total umbe does ot deped o time. Let the total umbe of isuace cotacts cocluded to the momet of time t t [0 T ] be defied by c a fuctio K ( whee K c ( is a umbe of cotacts of the type c c c. Suppose that a isuace compay cosists of siste compaies whh geeally speakig may diffe i sets of claim types whh they ca seve as well as i umbe of employees. Assume that the pobability of the type c claim

5 The Aalysis of Some Models fo Claim Pocessig i Isuace Compaies 47 advaced i the umbe i siste compay o the iteval of time [ t t + t] is µ t + ο( µ ( p t + ο( whee µ ( is a itesity of filig of 0( 0c 0 t 0c t the type c claim p0 c. Evey claim advaced i the ith siste compay may be i two opeatig steps: the stage of estimatio ad the stage of paymet. Let m compay s specialists (estimatos of the ith siste compay be occupied with estimatio of the type c claims ad let the time of claim pocessig be distibuted accodig to the epoetial ule with the aveage value µ c. The claim whh passed the estimatio stage i the ith siste compay comes i to the paymet depatmet of the same siste compay whee it is pocessed by oe of m i cashies ad the time equied to the claim paymet by evey cashie is distibuted accodig to the epoetial ule with the aveage value µ as well. Besides assume that the time of waitig of a isue who advace the type c claim i the ith siste compay ad the time he eeds fo applatio i the othe siste compay is bouded by a vaiate whh is distibuted accodig to the epoetial ule with paamete ν c. That is the isue who is ot seved i the ith siste compay advaces the type c claim i the siste compay umbe with pobability q c ad this siste compay estimates the claims of the such type i c. The state of the isuace compay at the momet of time t may be descibed by the vecto ( k ( k (... k ( k (... k ( k (... k ( k ( k( t whee k ( is a umbe of type c claims whh ae i the estimatio stage i the ith siste compay at the momet of time t c ; k i ( is a umbe claims whh ae i the paymet stage i the ith siste compay at the momet of time t k0 ( k ( is a umbe of cotacts whh do c ot eed advacig of the claim at the momet of time t (isued accidet did ot occu. The compay s aveage loss fom oe isue o the iteval of time [ T T ] may be defied by a fuctioal [] T W ( T T m... m ( d t El t dt T T ( + ( (5 c T

6 48 M. Matalyck T. Romaiuk whee: k( ( M m l ( coeffiets d E have cost meaig c. We ae iteested i the poblem of detemiatio o the iteval of time T ] of the estimatos ad cashies umbe whh miimize the aveage [ T loss (5 ude esttio o the aveage umbe of claims ( whh ae i the vaious opeatig steps c. Usually the queues of isues occus as a ule i the estimatio stages so we will solve the followig poblem: W ( T T m... m m T T ( dt > l T T T T T T T T i ( dt l T T T T T T c i mi ( dt ( dt c (6 A closed by the stuctue queueig etwok may seve as a model of the descibed pocess ad the total umbe of diffeet-type claims i it is descibed by the fuctio of time. The etwok cosists of + systems S S S... S 0 S S the system S 0 coespods to the eteal eviomet (the claim... S... is ot advaced ad the time of claim waitig i the systems S c queues is bouded by a epoetial vaiate. The tasitio poba- bilities betwee the etwok s systems ae p 0 p pi0 0 c. Besides the followig claim tasitios fom the systems queues ae c possible: q 0 i c q 0 i the othe cases. Seve disciplies i the etwok s systems ae FIFO. The othe paametes ae descibed befoe. Usig the method descibed i [] it is detemied that the desity of po- k( babilities distibutio of the vecto elative vaiables ξ ( satisfies to ( Ο ε ( whee ε ( t to the diffeetial equatio i the patial deivatives of the secod ode: s i

7 The Aalysis of Some Models fo Claim Pocessig i Isuace Compaies 49 p( t c ε ( ( A ( p( + ( Bs ( p( K ( + p( c s s + (7 whee: A ( s [ ν s q s ( s + µ 0 s u( s s s s + µ s p s mi( s l s ] + (8 B ( s [ ν sq s ( s ( + µ s 0 ( u( s s ( s s ( + µ ( s p s mi( s ( l s ( ] + B ( ν q ( ( ( u( ( ( p mi( ( l ( s s µ s µ ( 0 0 i m l ( c q q s s p p s s q 0 s q s s s c i c i othe cases i c s 0 i othe cases q i s c c s p s p i s c c The equatio (7 whe K cost coide with the well-kow Kolmogoov-Fokke-Plac equatio fo the desity of pobabilities of -dimesioal Makov pocess. Usig the Gaussia appoimatio method fo the equatio (7

8 50 M. Matalyck T. Romaiuk oe ca obtai the usual diffeetial equatios set fo the compoets of the vecto ( ad the solutio of the poblem (6. 3. Eample Coside the case whe a isuace compay whh cosists of two siste compaies sets up a equitype cotacts. Fo the solutio of the poblem (6 it is ecessay to fid compoets of the vecto ( ( ( ( ( (. They satisfy the equatios set whh follows fom (8 ( ( ν 0( ( 0( ( + ( ν + lν l ν + µ 0( ε ( ( ν ( µ ( + µ l ε ( µ + ( ( ν µ 0( ( µ 0( ( ( ν + lν ν l + µ 0( ε ( ( ν ( µ ( + µ l ε ( µ ν ν 0 0 ( 0 ( 0 ( ( ( 0 0 ( ( ( + ( + a I the case whe the itesity µ 0( is piecewise costat ad esi( b + d a e b d ae costats o evey iteval of itesity costacy of omig flow ude defied iitial coditios we obtai that all ( have the followig type ( [ m k α si( b + m β cos( b + m γ e ] λ 5 0 i c. zp zp zp zp p z k 5 c λ t zp zpk The fuctioal W ( T T m m m m g m+ g0 is a liea fuctio of c. Resttios of the optimizatio poblems ae liea as well: c m hk m+ h 0 k 4. So i the cosideed case the poblem (6 is the liea pogammig poblem. Eample. Let a isuace compay whh cosists of two siste compaies sets up the equitype isuace cotacts ad let the total umbe of cotacts be descibed by the fuctio of time 3si(πt / fuctioig is descibed by the followig paametes: t [0364] ad its µ ( 0 µ µ 0 0 t [08] t (8364]

9 E The Aalysis of Some Models fo Claim Pocessig i Isuace Compaies 5 µ µ µ 5 µ 5 µ 70 µ 80 E 0 E 0 E 0 d 5 d 5 d d 0 p ν 0.3 ν 0.4 Solvig the poblem (6 o evey iteval of itesity costacy µ 0( t we obtai that estimatos ad cashie should opeate i the fist siste compay o the iteval of time [ 08] ad estimatos ad cashie - i the secod siste compay. 5 estimatos ad cashie should opeate i the fist siste compay o the iteval of time (8364] ad 4 estimatos ad cashie - i the secod siste compay. Refeeces [] Matalycki M. Romaiuk T. O some mathematal poblems of claims pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Scieces Czestochowa Uivesity of Techology 003 ( [] Matalytski M. Romaiuk T. Asimptot aalysis of closed queueig etwok ad its applatio Wiestik GUP 004 (i Russia.