ON A CLASS OF RENEWAL QUEUEING AND RISK PROCESSES

ON A CLASS OF RENEWAL ON A CLASS OF RENEWAL QUEUEING AND RISK QUEUEING AND RISK

PROCESSESPROCESSES

K.K.Thampia, M.J.Jacobb

aDepartment of Statistics, SNMC, M.G.University, Kerala – INDIA

bDepartment of Mathematics, NITCCalicut, Kerala - INDIA

THAMPI & JACOB JENA 2009THAMPI & JACOB JENA 2009 22

PURPOSE:- In this paper, we investigate how queuing theory have been applied to derive results for a SparreAndersen risk model in which the distribution of inter-claim time is hyper Exponential.Design/Methodology/Approach:- We exploit the duality results between the queueing theory and risk processes to derive expressions for ultimate ruin probability and moments of time of ruin in this renewal risk model.Practical implications:- The theme of this paper is to stress connection between queuing theory and risk process.Findings:-

(1).This paper derives explicit expression for the Laplacetransforms of the idle/waiting time distributions in GI/G/1 model and the ultimate ruin probability is obtained. (2). The relation between the time of ruin and busy periodin M/G/1 queuing system is used to derive the expected time of ruin.


MODELMODELLet T1, T2 . . .be a sequence of i.i.d random variables.Ti has a Hyper Erlang distribution with

With Laplace transform

P, k1, k2 . . . are non-negative integers, λ1, λ2 . . . are positive numbers.

1

=1

( )( ) = , 0.( 1)!

k k k tP i i i ii i

ii i

k t eg t p tk

λλ − −

≥−∑

=1

ˆ ( ) = ( )P

ki i ii

i i i

kg pk

λθλ θ+∑


WHY Hyper WHY Hyper ErlangErlang??Hyper Erlang model is suitable for analytic analysis and general enough to capture the statistics of the random time variables of interest The hyper Erlang distribution can be used to approximate the distribution of any non-negative random variableDistributions such as the exponential model, the Erlang model and the hyper exponential are special cases of hyper Erlang distribution.


ULTIMATE RUIN?ULTIMATE RUIN?N(t) be the number of claim arrival upto time t.u ≥ 0 , initial reserve. Premiums flow at the rate c per unit time.The risk reserve process

The claim surplus process

Aggregate claim is comprised of a claim number process {N(t)} whose inter claim times are as HE

( )

=1( ) =

N t

ii

R t u ct X+ −∑( )

=1

( ) =N t

ii

S t X ct−∑( )

=1

N t

ii

X∑


The claim amounts X1, X2, . . ., independent of N(t), with distribution F(x) and mean m1.Y=cT, the inter claim revenue random variable with A(y)=P{Y≤y} and Laplace transform ,

The probability of ultimate ruin The probability of ruin before time τ

Time of ruin

maximum with infinite time probablity.

0ˆ( ) = ( )ya e dA yθθ

∞ −∫

0( ) = ( )x

xm e dF xθθ∞

∫

0( ) = { ( ) < 0}inf

tu P R tψ

≥

0( , ) = { ( ) < 0}inf

tu P R t

τψ τ

≤ ≤

( ) = inf{ 0 : ( ) < 0}u t R tτ ≥ = inf{ 0 : ( ) > }t S t u≥

0 <= { ( )}sup

tM S t

≤ ∞


Again the ultimate ruin probability

Let r is a unique positive solution of the equation

We call this solution as Lundberg’s exponent.

( ) = { ( ) < }= { > }u P u P M uψ τ ∞

ˆ( ) ( ) = 1xa mθ θ


Duality Duality ––Queuing & RuinQueuing & RuinGI/G/1 - a single server renewal queuing discipline.T1, T2, . . . be the inter-arrival times with distribution G(t).X1, X2, . . . be the service times of customers with distribution F(x).The traffic intensity of the queue In ruin,this ratio is the avg amount of claims/unit time.Ruin theory have the propertyPremium loading factor Always try to ensure η > 0.

( )=( )

E XE T

ρ

( )

=1

1 ,N t

ii

X tt

ρ→ →∞∑

= c ρηρ−

= c ρηρ−


If η > 0, then M < ∞ and ψ(u) < 1.

Denote Zn = Xn - Tn

Sn = Z1+Z2+ . . . +Zn , S0 = 0

Let H(z) = P{ Zn ≤ z } and E(z) exists.

Basic process underlying the queuing is the random walk { Sn }.


Wn – waiting time of the nth customer

In- idle period, just terminates upon the arrival of the nth customer.

Then Wn+1 = (Wn + Zn+1)+ , In+1 = (Wn + Zn+1)- .

Let Mn = max{ 0, S1 , S2 , . . .}.

Mn – the max. aggregate loss in ruin theory.


Famous known result

ρ < 1, Wn converges to a r v W, Mn

converges to M as n → ∞, P{W > u} = ψ(u) = P{M > u}.

Survival probability δ(u) = P{M ≤ u}.

Laplace transform of M ˆˆ ( ) = ( )M θ θδ θ

=D

n nW M


Define the R.V. N = min{ n > 0: Sn > 0 } and = min{n > 0: Sn < 0 }

Define the Ladder height distribution in (0, ∞)

and in ( -∞, 0 )

We express the distribution of H in terms of the ladder height distributions and

N

( ) = { = , }n NG x P N n S x+ ≤

( ) = { = , }n NG x P N n S x− ≤

nG+nG−


Transforms of first ascending and descending ladder epochs

Wiener-Hopf factorization

In – Total idle period, Wn – Waiting time.

( , ) = { }, ( , ) = { }i Si SN NN NE e E e ωωχ γ ω γ χ γ ω γ

1 { } = [1 { }][1 { }]i Si Z i SN Nn N NE e E e E e ωω ωγ γ γ− − −

( , I ) =(I , )D

d dn n n nW W

( , ) =(I , )D

n n n n nM S M W−


Main ResultsMain ResultsQueue interarrival times Ti have distribution G(t),

service time Xi have density HEr

Zn = Xn - Tn, α = E(Z). Φ1(ω) and Φ2(ω) be the Ch. fn. of Tn and Xn.

1 2( ) = ( ) ( )φ ω φ ω φ ω−

1=1

= ( ) ( )P

km m mm

m m m

kpk i

λ φ ωλ ω

−−∑

=1

='̂(0)

P ii

i

p

fλρ −

∑


ResultResult--11For the random walk induced by the above ch. fn.

are the roots of the equation

of which j=1,2, . . .P.

1 1

=1

(1 )( , ) = 1 [1 ]

Nm m j

j m m

k kk i

λ λ ξχ γ ω

λ ω− −

− −−∏

1=1=1=1=

1 1=1

( ) ( ) ( ) ( )

( , ) =( (1 ) )

P PP kk k jm mm m m m m j j

jmmj m

N

jj

k i p k k i

k i

λ ω γ λ λ ω φ ω

χ γ ωλ ξ ω

− − − −

− −

∑∏ ∏

∏

= ( ), = 1, 2,...j j j Nξ ξ γ

1 11 1=1

ˆ ( (1 )) = 1(1 (1 ))

Pm

kmm

m m

p g kkk

γ λ ξλ ξλ

−− −

∑

< 1jξ


ResultResult--22For the queue GI/ HEr(ki, λi)/ 1

1 1

=1 =11 1

{ } = (1 ) , < 1lim

1

P Ni W kn m

n m jm m

j

kiE e ik k

ω λω ρωλ λξ

→∞−

−−

∏ ∏

1

1 1 1 1=1 =11 1

1=1=1=1=

( ) ( ) ( )1

, > 1( ) ( ) ( ) ( )

P Nkm m m

m j jP PP kk k jm m

m m m m m j jjmm

j m

k ik i kk

k i p k k i

λ ωλ α ω λλ ξ

ρλ ω λ λ ω φ ω

−

× +−

+ − +

∏ ∏

∑∏ ∏I{ } =lim

i n

nE e ω

→∞


ResultResult--33For the dual queue HEr(ki, λi)/G/1

1

1 1 1 1 1=1=1 =11 1

1 1=1=1=1=

( ) (1 ) ( )1

{ } =lim( ) ( ) ( ) ( )

P NPkm m mm

i W mm jm jnP PP kn k k jm m

m m m m m j jjmm

j m

k p ik i kk

E ek i p k k i

ω

λ ωλ ρ ω λλ λ ξ

λ ω λ λ ω φ ω

−

→∞

− +−

+ − +

∑∏ ∏

∑∏ ∏

I 1 1

=1 =11 1

{ } = (1 ) ( )lim

1

P Ni kn m

n m jm m

j

kiE e ik k

ω λωωλ λξ

→∞−

−−

∏ ∏


Applications to Risk Applications to Risk Renewal risk model of the form (HEr(ki, λi), G, c)

then

where ξj are the solutions of the equation

with ξj < 1, j = 1. . . P.

11 1 1 1

1 1 1=1=1 =11 1

=1 =1 =1

( ) ( ) ( )1ˆ( ) =

ˆ( ) ( ) ( ) ( )

P NPk jm m m

jm jj jP PP

k k km m mm m m j j m m

m j mj m

pk k k sc k mk c c

sp k k cs f s k cs

λ λ λλλ λ ξ

δλ λ λ

−

≠

− −−

− − −

∑∏ ∏

∑ ∏ ∏

1 1

1 1=1

ˆ ( (1 )) = 1(1 (1 ))

Pm

kmm

m m

p kfk ck

λ ξλ ξλ

−− −

∑


Dual risk model of the form where the interarrival of claims follow any arbitrary distribution and claim size distribution is hyper Erlang(ki , λi), then


under the condition that ξj < 1, j = 1. . . P.

1 1

=1 =11 1

( )ˆ( ) = ( )

1

kP Nmm m

m j

j

k s ks ss k

λ λδλ

ξ

+

+−

∏ ∏

(G, HEr(ki, λi), c),

1 11 1=1

ˆ ( (1 )) = 1(1 (1 ))

Pm

kmm

m m

p g ckkk

λ ξλ ξλ

−− −

∑


Some Explicit ResultsSome Explicit ResultsRisk processes with claim amount distribution is Erlang (n, β) and inter-occurrence of claims are hyper Erlang(ki , λi), then

Taking inverse Laplace transform


with |ξj| < 1, j = 1, 2, . . .P.

=1

1( )ˆ( ) = ( )(1 )

n ni

i i

sss s

ξβδβ ξ

−+− +∏

(1 )

=1=1

(1 )( ) = 1

nnu jn i

iji i jj i

u e β ξ ξφ ξ

ξ ξ− −

≠

−−

−∑ ∏

=1= ( )

(1 )

Pkn i i i

ii i i

kpk c

λξλ β ξ+ −∑


Time of RuinTime of RuinN(t) denotes the number of claim arrivals upto the time of ruin, assume c =1, N(t) will be

Again

The overshoot above the level u of the random walk {Sn} be

=1 =1

( ) = inf{ : < 0}n n

i ii i

N t n u T X+ −∑ ∑( )

=1( ) =

N t

ii

u Tτ ∑

( ( )) ( ) <( ) =

( ) =S u u if u

Y uif u

τ ττ

+ − ∞⎧⎨ ∞ ∞⎩


Assume the inter-arrival of claims are Hyper Erlang(ki, λi) with claim size distribution 1/β. The risk process of the type (HEr(ki, λi ), M, 1)

Dual queueing M/HEr(ki , λi)/1

Ti be the inter-arrival time, and Xj be the service time of the customer in the busy period.

V(u) be the duration of busy period, I(u) the busy period that follows the busy period.


Duality arguments between queueing and risk process give V(u) is distributed as Y+(u) and I(u) are identically distributedBut in queue, u is not a factor, V(0) = V and I(0) = I are the busy period and idle period of the regular queueing setup.Y+(0) is distributed as I and τ(0) = τ is distibuted as V.V(u) is distributed as in M/HEr(ki , λi)/1 queueing system:

IMPORTANT: the service time of the first customer starts in T1+u and the service times of all customers are distributed as Hyper Erlang

( )u uτ +

( )u uτ +


TWO CASES ARISE:

, ruin occurs with probability 1.Busy period is finite.The Laplace transform, where is the Laplacetransform of the busy period in M/HEr(ki , λi)/1 queueing system.

There fore,

= 1< 1P i

ii

pβλ∑

ˆ( (1 ) ( ))ˆ ˆ( ) = ( )u VuV e Vθ β θ θθ θ− + −

ˆ( )V θ

=1

=1

{ ( )} =1

Pi

i iP

i

i i

puE V u

pλ

βλ

+

−

∑

∑


gives

CASE II

the probability of ruin is less than 1 To get an expression for time of ruin in this case, obtain the Esscher transform of G and FTransformed risk process has inter-arrival time distribution

( ) = ( )V u u uτ + =1

=1

( 1{ ( )} =

1

Pi

i iP

i

i i

p uE u

p

βλτβ

λ

+

−

∑

∑

=1> 1P i

ii

pβλ∑

( )( ) =ˆ( )

te g tg tg

θ

θ θ

−


Claim size distribution

With Laplace transformsand

r is the Lundberg exponent

Consider the risk process with distribution in which inter-claim distribution

( )( ) =( )

x

x

e f xf xm

θ

θ θ

ˆ ( )ˆ ( ) =ˆ( )

g sg sg sθθ +

( )( ) = ( ) xf x e β θθ β θ − −−

( )( ) = ( ) xf x e β θθ β θ − −−

( )( ) =ˆ( )

rt

re g tg t

g r

−

( )( ) =ˆ( )

rt

re g tg t

g r

−

( )( ) =ˆ( )

rt

re g tg t

g r

−


With Laplace transform

Claim size distribution is exp(β-r)

Assume

=1

=1

( )ˆ ( ) =

( )

Pki i

i ii i i

r Pki i i

ii i i

kpk rg

kpk r

λλ θθ

λλ

+ +

+

∑

∑

=1> 1P i

ii

pβλ∑

=1> 1P i

ii

pβλ∑


Laplace transform of the time of ruin ( )uτ

ˆ( ( ) ( )

=1

=1

ˆˆ ( ) = (1 ) ( )

( )ˆ ( )( )ˆwhere ( ) =( )

u r Vru r

P ki i iii

i i rr P ki i i

iii i

r e V

kpk V rV kp

k r

β β θτ θ θβ

λλ θ β θ βθ λ

λ

− − −−

+ + − −

+

∑

∑


δ1 , the first moment of the busy period in M/HEr/1 queue with arrival rate β-r

1 =0

1

=1

1

=1 =1

ˆ= ( ) |

( )=

( ) ( ) ( )

r

Pki i i i

i i i iP P

k ki i i i ii ii

i ii i i i i

d Vd

p kk r

k p kp rk r k r

θδ θθ

λλ λ

λ λβλ λ λ

+

+

+

− −+ +

∑

∑ ∑


Taking the first derivative of (7.10) w.r.t θ at θ = 0

Then

1{ ( ) : < } = (1 ( ))(1 ) rurE u u r eτ τ δ ββ

−∞ + − −

1{ ( ) | < } = (1 ( ))E u u rτ τ δ β∞ + −


EXAMPLESEXAMPLESIf the inter-arrival of claims is Er (n, λ) and claim size is β

If hyper exponential (pi , λi)

(1 ( )){ ( ) | < } =( ) ( )n u rE u

r rβτ τ

λ β+ −

∞+ − −

2=1

2=1 =1

( ){ ( ) | < } =( ) ( )

( )

Pi

ii i

P Pi i

i ii ii i

prE u

p r pr r

λλτ τ

λ λβλ λ

+∞

− −+ +

∑

∑ ∑


CONCLUSION

Several results of queuing theory dam/storage processes can be effectively applied to risk theory by slightly changing the arguments used. Now it is widely accepted that the modeling ideas used in queuing theory has relevance in risk theory also. Our work is an attempt in this direction.


Asmussen, S (2000), "Ruin Probabilities", World Scientific, Singapore, NJ, London, Hong Kong. Borovkov, A.A.,(1976)"Stochastic Processes in QueueingTheory", Springer-Verlag, New York, Heidelberg, Berlin Cheng,Y.,Qihe,T.(2003), "The moments of the surplus before ruin and deficit at ruin in the Erlang(2) risk process," North American Actuarial Journal, Vol. l7, No. 1, pp. 1-12. Dickson,D.C.M., Hipp,C.(1998),"Ruin probabilities for Erlang(2) risk processes". Insurance: Mathematics and Economics 22, 251-262. Frostig, E.(2004) "Upper bounds on the expected time of ruin and on the expected recovery time", Adv. Appl. Prob., 36, 377-397. Prabhu, N.U.(1998),"Stochastic Storage Processes, Queues, Insurance Risk and Dams"(second edn.) Springer, New York, Heidelberg, Berlin.

ON A CLASS OF RENEWAL QUEUEING AND RISK PROCESSES

Documents

Transcript of ON A CLASS OF RENEWAL QUEUEING AND RISK PROCESSES