ON A CLASS OF RENEWAL QUEUEING AND RISK PROCESSES

Transcription

1 ON A CLASS OF RENEWAL QUEUEING AND RISK ROCESSES K.K.Thap a, M.J.Jacob b a Departent of Statstcs, SNMC, M.G.Unversty, Kerala INDIA b Departent of Matheatcs, NITC Calcut, Kerala - INDIA

2 UROSE:- In ths paper, we nvestgate how queung theory have been appled to derve results for a Sparre Andersen rs odel n whch the dstrbuton of nter-cla te s hyper Exponental. Desgn/Methodology/Approach:- We explot the dualty results between the queueng theory and rs processes to derve expressons for ultate run probablty and oents of te of run n ths renewal rs odel. ractcal plcatons:- The thee of ths paper s to stress connecton between queung theory and rs process. Fndngs:- (1).Ths paper derves explct expresson for the Laplace transfors of the dle/watng te dstrbutons n GI/G/1 odel and the ultate run probablty s obtaned. (2). The relaton between the te of run and busy perod n M/G/1 queung syste s used to derve the expected te of run. THAMI & JACOB JENA

3 MODEL Let T 1, T 2...be a sequence of..d rando varables. T has a Hyper Erlang dstrbuton wth 1 λ t ( λ) t e =1 ( 1)! gt ()= p, t 0. Wth Laplace transfor λ gˆ( )= p ( ) θ =1 λ + θ, 1, 2... are non-negatve ntegers, λ 1, λ 2... are postve nubers. THAMI & JACOB JENA

4 WHY Hyper Erlang? Hyper Erlang odel s sutable for analytc analyss and general enough to capture the statstcs of the rando te varables of nterest The hyper Erlang dstrbuton can be used to approxate the dstrbuton of any nonnegatve rando varable Dstrbutons such as the exponental odel, the Erlang odel and the hyper exponental are specal cases of hyper Erlang dstrbuton. THAMI & JACOB JENA

5 ULTIMATE RUIN? N(t) be the nuber of cla arrval upto te t. u 0, ntal reserve. reus flow at the rate c per unt te. The rs reserve process The cla surplus process N() t Rt ()= u ct X + N() t N() t Aggregate cla X s coprsed of a cla =1 nuber process {N(t)} whose nter cla tes are as HE =1 =1 St ()= X ct THAMI & JACOB JENA

6 The cla aounts X 1, X 2,..., ndependent of N(t), wth dstrbuton F(x) and ean 1. Y=cT, the nter cla revenue rando varable wth θ y A(y)={Y y} and Laplace transfor, θ 0 θ ( )= x x θ e df( x) 0 The probablty of ultate run ψ ( u)= { nfr()<0} t t 0 The probablty of run before te τ Te of run ψ ( u, τ )= { nf R()<0} t 0 t τ τ ( u)=nf{ t 0: R( t)<0} =nf{ t 0: S()> t u} aˆ( )= e da( y) M = sup{ St ( )} axu wth nfnte te probablty. 0 t< THAMI & JACOB JENA

7 Agan the ultate run probablty ψ()= u {()< τ u }= { M > u} Let r s a unque postve soluton of the equaton aˆ( θ ) x ( θ )=1 We call ths soluton as Lundberg s exponent. THAMI & JACOB JENA

8 = c ρ ρ Dualty Queung & Run GI/G/1 - a sngle server renewal queung dscplne. T 1, T 2,... be the nter-arrval tes wth dstrbuton G(t). X 1, X 2,... be the servce tes of custoers wth dstrbuton F(x). The traffc ntensty of the queue ρ = E( X ) E( T ) In run,ths rato s the avg aount of clas/unt te. N() t 1 Run theory have the property X ρ, t t =1 reu loadng factor = c η ρ ρ Always try to ensure η > 0. THAMI & JACOB JENA

9 If η > 0, then M < and ψ(u) < 1. Denote Z n = X n -T n S n = Z 1 +Z Z n, S 0 = 0 Let H(z) = { Z n z } and E(z) exsts. Basc process underlyng the queung s the rando wal { S n }. THAMI & JACOB JENA

10 W n watng te of the n th custoer I n - dle perod, just ternates upon the arrval of the n th custoer. Then W n+1 = (W n + Z n+1 ) +, I n+1 = (W n + Z n+1 ) -. Let M n = ax{ 0, S 1, S 2,...}. M n the ax. aggregate loss n run theory. THAMI & JACOB JENA

11 Faous nown result W n = D M n ρ < 1, W n converges to a r v W, M n converges to M as n, {W > u} = ψ(u) = {M > u}. Survval probablty δ(u) = {M u}. Laplace transfor of M M ˆ ( θ )= θδ ˆ ( θ ) THAMI & JACOB JENA

12 Defne the R.V. N = n{ n > 0: S n > 0 } and N = n{n > 0: S n < 0 } Defne the Ladder heght dstrbuton n (0, ) + G ( x)= { N = n, S x} n and Gn ( x)= { N = n, S x} n ( -, 0 ) N N We express the dstrbuton of H n ters of the ladder heght dstrbutons G + n and G n THAMI & JACOB JENA

13 Transfors of frst ascendng and descendng ladder epochs N ωs S N N N χ( γ, ω)= E{ γ e }, χ( γ, ω)= E{ γ e ω } Wener-Hopf factorzaton ωz S n N ωs N N N 1 γee { }=[1 E{ γ e }][1 E{ γ e ω }] I n Total dle perod, W n Watng te. d d ( W,I )=(I, W ) ( M S, M )=(I, W ) D n n n n D n n n n n THAMI & JACOB JENA

14 Man Results Queue nterarrval tes T have dstrbuton G(t), servce te X have densty HE r Z n = X n -T n, α = E(Z). Φ 1 (ω) and Φ 2 (ω) be the Ch. fn. of T n and X n. φ( ω)= φ ( ω) φ ( ω) ρ = p =1 λ fˆ'(0) 1 2 λ = ( ) φ ( ω) p =1 λ ω 1 THAMI & JACOB JENA

15 Result-1 For the rando wal nduced by the above ch. fn. N 1 1 j χγω (, )=1 [1 ] χγω (, )= =1 j=1 λ λ (1 ξ ) λ ω ξ j = ξ j( γ), j =1,2,... N are the roots of the equaton p γ g ˆ( 1λ1(1 ξ))=1 of whch ξ <1 j=1,2,.... j j j =1 j=1 j= N ( λ ω) γ p ( λ ) ( λ ω) φ ( ω) λ j=1 =1 1 1 (1 (1 ξ )) λ j ( λ (1 ξ ) ω) 1 1 j THAMI & JACOB JENA

16 Result-2 For the queue GI/ HE r (, λ )/ 1 N ωw ω 1λ n 1 { }= (1 ), <1 lee ρ n =1 λ ω j=1 1λ1 1 ξ j I l { n Ee ω }= n =1 λ ω N 1 ( ) ( 1λ1) αω ( 1λ1+ ) =1 1λ1 j=1 1 ξ j j + j j + =1 j=1 j=, ρ >1 ( λ ω) p ( λ ) ( λ ω) φ ( ω) 1 THAMI & JACOB JENA

17 Result-3 For the dual queue HE r (, λ )/G/1 N 1 λ p 1λ1 ρ1ω 1λ1+ ωw =1 1 1 =1 j=1 1 n j + p j j + =1 =1 j=1 j= n lee { }= ω ( ) (1 ) ( ) λ λ ξ ( λ ω) ( λ ) ( λ ω) φ ( ω ) j 1 1 N ωi ω 1λ n 1 lee { }= (1 ) ( ) n =1 λ ω j=1 1λ1 1 ξ j THAMI & JACOB JENA

18 Applcatons to Rs Renewal rs odel of the for (HE r (, λ ), G, c) then N 1 j ( c ) ( 1λ1 1 ) ( ) =1 1λ1 j=1 λj c j=1 c 1 ξ j p( λ) ( jλj cs) f( s) ( λ cs) =1 j=1 =1 j ˆ( δ s )= ˆ where ξ j are the solutons of the equaton p 1λ 1 fˆ( (1 ξ ))=1 =1 1λ1 (1 (1 ξ )) c λ wth ξ j < 1, j = λ p λ λ s THAMI & JACOB JENA

19 Dual rs odel of the for where the nterarrval of clas follow any arbtrary dstrbuton and cla sze dstrbuton s hyper Erlang(, λ ), then N ( ) 1 1 ˆ( )= λ + s λ s ( ) δ =1 j=1 where ξ j are the solutons of the equaton s (G, HEr(, λ), c), s 1λ1+ 1 ξ j p λ λ =1 1 1 (1 (1 )) ξ gc ˆ( λ (1 ξ))=1 1 1 under the condton that ξ j < 1, j = THAMI & JACOB JENA

20 Soe Explct Results Rs processes wth cla aount dstrbuton s Erlang (n, β) and nter-occurrence of clas are hyper Erlang(, λ ), then n n ( β + s) 1 ξ ˆ( )= ( δ s ) s =1 β(1 ξ ) + s Tang nverse Laplace transfor n n (1 ) (1 ) n u ξ j φ( u)=1 ξ e β ξ ξ ξ =1 j=1 j j where ξ j are the solutons of the equaton ξ λ c n = p ( ) =1 λ + β(1 ξ) wth ξ j < 1, j = 1, 2,.... THAMI & JACOB JENA

21 Te of Run N(t) denotes the nuber of cla arrvals upto the te of run, assue c =1, N(t) wll be Agan Nt ()=nf{ nu : + T X <0} τ ( u) = N ( t) =1 T n =1 =1 The overshoot above the level u of the rando wal {S n } be + S( τ( u)) u f τ( u)< Y ( u)= f τ ( u)= n THAMI & JACOB JENA

22 Assue the nter-arrval of clas are Hyper Erlang(, λ ) wth cla sze dstrbuton 1/β. The rs process of the type (HEr(, λ ), M, 1) Dual queueng M/HEr(, λ )/1 T be the nter-arrval te, and X j be the servce te of the custoer n the busy perod. V(u) be the duraton of busy perod, I(u) the busy perod that follows the busy perod. THAMI & JACOB JENA

23 Dualty arguents between queueng and rs process gve V(u) s dstrbuted as τ ( u) + u Y + (u) and I(u) are dentcally dstrbuted But n queue, u s not a factor, V(0) = V and I(0) = I are the busy perod and dle perod of the regular queueng setup. Y + (0) s dstrbuted as I and τ(0) = τ s dstbuted as V. V(u) s dstrbuted as τ ( u) + u n M/HEr(, λ )/1 queueng syste: IMORTANT: the servce te of the frst custoer starts n T 1 +u and the servce tes of all custoers are dstrbuted as Hyper Erlang THAMI & JACOB JENA

24 TWO CASES ARISE: p λ β <1 =1, run occurs wth probablty 1. Busy perod s fnte. The Laplace transfor, where V ˆ( θ ) s the Laplace transfor of the busy perod n M/HEr(, λ )/1 queueng syste. ( (1 ) ˆ ˆ u V( )) V ( )= ˆ θ θ+ β θ θ e V( θ ) u There fore, EV { ( u)}= u + 1 β =1 =1 p λ p λ THAMI & JACOB JENA

25 Vu ( )= τ ( u) + u gves E{()}= τ u p ( βu + 1 λ p 1 β λ =1 =1 CASE II p λ β >1 =1 the probablty of run s less than 1 To get an expresson for te of run n ths case, obtan the Esscher transfor of G and F Transfored rs process has nter-arrval te θt e g() t dstrbuton g t θ ()= g ˆ( θ ) THAMI & JACOB JENA

26 rt f ( x)=( ) e β θ θ eβ g() θt gr ()= t gr ˆ( ) ( ) x Cla sze dstrbuton Wth Laplace transfors and f ( x)=( ) e β θ θ β θ ( ) x f θ ( x)= e θ x gˆ ()= s θ x f( x) ( θ ) gˆ( θ + s) gs ˆ( ) r s the Lundberg exponent rt e g() t gr ()= t gr ˆ( ) Consder the rs process wth dstrbuton n whch nter-cla dstrbuton rt e g() t gr ()= t gr ˆ( ) THAMI & JACOB JENA

27 β =1 p >1 λ Wth Laplace transfor gˆ ( θ )= r λ ( ) p =1 λ + θ + r λ p ( ) =1 λ + r Cla sze dstrbuton s exp(β-r) Assue β =1 p λ >1 THAMI & JACOB JENA

28 Laplace transfor of the te of run τ ( u) r u( β ( β r) Vˆ ( ) ˆ ( )=(1 ) r θ τ ˆ u θ e V r( θ) β where Vˆ ( θ ) = r =1 λ p ( ) ˆ λ + θ + β V r ( θ )( β r) λ p ( ) =1 λ + r THAMI & JACOB JENA

29 δ 1, the frst oent of the busy perod n M/HEr/1 queue wth arrval rate β-r d δ ˆ 1 = V r ( θ) θ =0 dθ p λ + 1 ( ) λ λ + r = λ ( ) ( ) ( ) + r + r =1 λ p p β r =1 λ =1 λ λ + 1 THAMI & JACOB JENA

30 Tang the frst dervatve of (7.10) w.r.t θ at θ = 0 Then r E{ τ( u) : τ < } = δ1(1 + u( β r))(1 ) e β E{() τ u τ < }= δ (1 + u( β r)) 1 ru THAMI & JACOB JENA

31 EXAMLES If the nter-arrval of clas s Er (n, λ) and cla sze s β n(1 + u( β r)) E{() τ u τ < }= ( λ+ r ) ( β r ) If hyper exponental (p, λ ) E{() τ u τ < }= λ p 2 =1 ( λ + r) λ ( ) ( β ) =1 λ r =1 λ p r p + ( λ + r) 2 THAMI & JACOB JENA

32 CONCLUSION Several results of queung theory da/storage processes can be effectvely appled to rs theory by slghtly changng the arguents used. Now t s wdely accepted that the odelng deas used n queung theory has relevance n rs theory also. Our wor s an attept n ths drecton. THAMI & JACOB JENA

33 Asussen, S (2000), "Run robabltes", World Scentfc, Sngapore, NJ, London, Hong Kong. Borovov, A.A.,(1976)"Stochastc rocesses n Queueng Theory", Sprnger-Verlag, New Yor, Hedelberg, Berln Cheng,Y.,Qhe,T.(2003), "The oents of the surplus before run and defct at run n the Erlang(2) rs process," North Aercan Actuaral Journal, Vol. l7, No. 1, pp Dcson,D.C.M., Hpp,C.(1998),"Run probabltes for Erlang(2) rs processes". Insurance: Matheatcs and Econocs 22, Frostg, E.(2004) "Upper bounds on the expected te of run and on the expected recovery te", Adv. Appl. rob., 36, rabhu, N.U.(1998),"Stochastc Storage rocesses, Queues, Insurance Rs and Das"(second edn.) Sprnger, New Yor, Hedelberg, Berln. THAMI & JACOB JENA

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