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STOCHASTIC MODELS Vol. 19, No. 1, pp. 113–124, 2003
Matrix-Exponential Distributions: Calculus and Interpretations via Flows Mogens Bladt1,* and Marcel F. Neuts2 1
2
IIMAS-UNAY, Mexico University of Arizona, Tucson, Arizona, USA
ABSTRACT By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace– Stieltjes transforms on ½0; 1Þ: That approach includes and generalizes the formalism of PHdistributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP). Key Words: Matrix-exponential distributions; Phase-type distributions; Flow models; Rational Laplace – Stieltjes transforms; Rational arrival process; Matrix-analytic methods; Renewal theory; Risk theory.
*Correspondence: Mogens Bladt, IIMAS-UNAY, A.P. 20-726, 01000 Mexico, DF; E-mail:
[email protected]. 113 DOI: 10.1081/STM-120018141 Copyright q 2003 by Marcel Dekker, Inc.
1532-6349 (Print); 1532-4214 (Online) www.dekker.com
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1.
INTRODUCTION
The relation of the familiar phase-type distributions to finite Markov chains is useful in analyzing probability models. Using finite Markov chains as modules, we can replace many analytic derivations by probabilistic constructions. In this paper, we show how, for matrix-exponential distributions, something similar can be done by using simple deterministic flow models. Matrix-exponential distributions are probability distributions with support on ½0; 1Þ and with rational Laplace transforms. The class of matrix-exponential distributions contains all phase-type distributions, the distributions of absorption times for finite statespace Markov jump process with any number of transient states and one absorbing state. Many results valid for phase-type distributions also hold for matrix-exponential distributions. In Ref.[2] a number of such results from renewal and queueing theory are proved by transform methods and matrix algebra. Also, no counter examples on such extensions are known although many potential extensions still lack rigorous proofs. The authors believe that flow arguments yields these extensions almost immediately. That is illustrated for the risk-reserve model of Asmussen and Bladt.[4] The original derivations were based entirely on probabilistic arguments and are not extendable to transform methods. That example serves as a test case for the power of the flow arguments. Given a rational function or an equivalent matrix-exponential form, there are no feasible procedures to ascertain whether these correspond to probability distributions. Because of their relation to Markov jump processes, that issue does not arise for phase-type distributions. On the other hand, an excessive number of parameters may be involved in representing a phase-type distribution, while highly parsimonious matrix-exponential representations can be given for probability distributions with rational transforms. Also dimensionality is an important issue in applications, and generally a matrix-exponential representation of a phasetype distributionswillbe of lower order than the corresponding phase-type representation,see Ref.[2] for details. It may hence be beneficial, at least in the numerical phase, to represent phase-type distributions by matrix-exponentials of lower order. For further properties and bibliography of matrix-exponential distributions we refer to Ref.[2]. The remainder of the paper is organized as follows. In Section 2 we relate flow models to matrix-exponential distributions. Section 3 is a basic tutorial on matrix-exponential renewal theory, where we establish some known results with flow arguments. In Section 4 we prove an extension of the risk-reserve model of Asmussen and Bladt[4] to cases where claims are matrix-exponentially distributed rather than of phase-type. Finally in Section 5 we show how the flow model may also be used for interpreting Rational Arrival Processes (RAP) of Ref.[3].
2.
MATRIX-EXPONENTIAL DISTRIBUTIONS AND FLOWS
If the function f(x) defined by f ðxÞ ¼ a expðSxÞs 0 ;
for
x$0
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Matrix-Exponential Distributions
115
is a (possibly defective) probability density, then the corresponding distribution F(·) is called a matrix-exponential distribution with representation MEða; S; s 0 Þ: As shown in Ref.[2], it is always possible to choose s 0 ¼ 2Se; where e is a column vector of ones. We shall do so in this paper. Henceforth, we refer to the representation MEða; SÞ; since s 0 is implicit. As also shown in Ref.[2], a and S can always be chosen to have real elements only. The following is an example of a matrix-exponential distribution that is not of phase type (since its density has positive zeros, see Ref.[8] for criteria for a given distribution to be of phase-type).
Example 2.1.
Consider the probability density
1 f ðxÞ ¼ 1 þ 2 ð1 2 cosð2pxÞÞ exp ð2xÞ; 4p with Laplace transform f^ ðsÞ ¼
1 þ 4p 2 : s 3 þ 3s 2 þ ð3 þ 4p 2 Þs þ 1 þ 4p 2
That distribution has a representation MEða; SÞ with 0
a ¼ ð1 0 0Þ;
0 21 2 4p 2
B S¼B @3 2
1 þ 4p 2
2
26
2
25
For further details we refer to Ref.[2].
1 C C: A
A
Consider a system of m doubly infinite containers of a liquid, each with a zero mark. Their initial contents are a1 ; . . .; am (positive or negative relative to the zero mark.) One more container, numbered m þ 1; has an initial content amþ1 satisfying 0 # amþ1 , 1: The algebraic sum a1 þ · · ·am þ amþ1 ¼ 1: Liquid flows from container i to container j, with 1 # i; j # m; i – j; at a constant rate Sij. That means that if container i holds vi(t) at time t then an amount vi(t)Sijdt flows from i to j during ½t; t þ dtÞ: For i, i – m þ 1; there is a flow at a constant rate s0i from container i to container m þ 1: We define Sii by setting X Sii ¼ 2 Sih 2 s0i ;
for
1 # i # m:
ð1Þ
h–i
Define S ¼ ðSij Þi;j¼1;...;m : Let the components of the vector vðtÞ ¼ ðv1 ðtÞ; . . .; vm ðtÞÞ be the amounts of liquid in the first m containers. We emphasize that the components of v(t) are deterministic functions of t. Let a ¼ ða1 ; . . .; am Þ:
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Proposition 2.1.
For t $ 0; the vector v(t) satisfies
v0 ðtÞ ¼ vðtÞS;
vð0Þ ¼ a:
This entails that vðtÞ ¼ a expðStÞ:
Keeping track of inflow and outflow, we obtain that for 1 # i # m; 2 3 X X vi ðt þ dtÞ ¼ vi ðtÞ41 2 Sih dt 2 s0i dt5 þ vj ðtÞSji dt:
Proof.
h–i
j–i
The first equation of the proposition now follows by (1) by letting dt tend to zero; the second is obvious. A We define the following crucial property: Definition 2.1. A flow (a, S) is valid if and only if the content vmþ1 ðtÞ of container m þ 1 is non-decreasing and tends to 1 as t ! 1: Valid flows obviously exist. The representation (a, S) of a matrix-exponential distribution determines a valid flow. In particular, if a is nonnegative and S is a subgenerator, then (a, S) is a valid flow.
Theorem 2.1. Let Y , MEða; SÞ: Consider the valid flow generated by (a, S). Let x [ ½0; 1 and T(x) be the time from flow is initiated until container m þ 1 reaches level x. Then Y , TðUÞ;
Proof.
U , Uniform½0; 1:
The net inflow to container m þ 1 during ½t; t þ dtÞ is
I t dt ¼
m X
vk ðtÞs0k dt:
k¼1
When U , Uniform½0; 1 the probability that U is contained in the interval ½vmþ1 ðtÞ; vmþ1 þ I t dtÞ is simply Itdt. Hence the density g of T(U) is given by gðtÞdt ¼ I t dt ¼
m X
vk ðtÞs0k dt ¼ ae St s 0 dt:
k¼1
Hence TðUÞ , Y: We shall prove the following well known property with a flow argument.
A
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Matrix-Exponential Distributions
Proposition 2.2. is given by
117
The distribution function F of a ME(a,S) distributed random variable
FðxÞ ¼ 1 2 a expðSxÞe; where e is the column vector of ones.
Proof. For the valid flow corresponding to MEða; SÞ the containers 1; . . .; m will contain a expðSxÞ by time x. The total content of containers 1; . . .; m by this time is hence a expðSxÞe: Since the content in all containers sum to 1, the content in container m þ 1 by time x is thus 1 2 a expðSxÞe: Hence TðUÞ # x if and only if U # 1 2 a expðSxÞe; and FðxÞ ¼ PðTðUÞ # xÞ ¼ PðU # 1 2 a expðSxÞeÞ ¼ 1 2 a expðSxÞe: A 3.
CALCULATION WITH FLOWS: RENEWAL THEORY
In the theory of phase type distributions, we construct the Markov chain with infinitesimal generator Q* ¼ S þ s 0 a: As is shown in Ref.[7], after possible deletion of superfluous phases, that generator can always be made irreducible. The generator Q* is useful in deriving renewal theoretical results by simple arguments. We construct a piece-wise deterministic random process, called flow with restarts to play the analogous role in renewal theory with ME-distributions. In fact, this is a special case of a RAP (see Ref.[3]). Let U 1 ; U 2 ; . . . be a sequence of i.i.d. random variables uniformly distributed over the interval ½0; 1: Let X n ¼ TðU n Þ be the corresponding flow times of a valid flow corresponding to some matrix-exponential distribution MEða; SÞ; where T(·) is defined in Theorem 2.1. Then the renewal processes with matrix-exponential interarrival times may be modeled by piecing together the sequence of terminating flows. This creates a piecewise deterministic Markov process (see Refs.[5,6]), where the jumps corresponding to arrivals happen when the flow system refills to initial amounts a. Consider a renewal processes represented by a flow with restarts. Let the content of container i at time t be Ui(t), and define zi ðtÞ ¼ EðU i ðtÞÞ: The m-row-vectors U(t) and z(t) have components Ui(t) and zi(t) respectively. Here, as in the rest of the paper, o(dt) terms are deliberately suppressed.
Lemma 3.1.
The renewal density f(t) can expressed as follows:
fðtÞ ¼ zðtÞs0 :
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Proof.
fðtÞdt ¼ Pðarrival in ½t; t þ dtÞÞ ¼ EðPðarrival in ½t; t þ dtÞjUðtÞÞÞ 0 ¼ E@
X
1 U k ðtÞs0k dtA
k
¼
X zk ðtÞs0k dt:
A
k
Lemma 3.2.
For t $ 0; the vector z(t) is given by
zðtÞ ¼ a exp½ðS þ s 0 aÞt:
We give two proofs the first of which uses a renewal argument. The second exploits infinitesimal properties of flows similar to arguments for phase-type distributions using transition rates.
Proof 1. On the event where the original flow started at time 0 is still current at time t, the expected content vector is a expðStÞ: On the complementary event that there have been flow restarts in ½0; tÞ; conditioning on the time u of the first restart, the contribution to the expectation vector z(t) is given by Z t a exp½Sðt 2 uÞs 0 zðuÞdu: 0
Therefore, zðtÞ ¼ a expðStÞ þ
Z
t
a exp½Sðt 2 uÞs 0 zðuÞdu: 0
Consider the matrix V(t) defined by Z t VðtÞ ¼ expðStÞ þ exp½Sðt 2 uÞs 0 zðuÞdu: 0
Premultiplying by expð2StÞ and differentiating leads to the linear system of differential equations V 0 ðtÞ ¼ SVðtÞ þ s 0 zðtÞ;
with
Vð0Þ ¼ a:
Replacing z(t) by aV(t), and integrating we obtain the stated result.
A
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Matrix-Exponential Distributions
Proof 2. that
119
Fix a time t. On the event A where there are no arrivals in ½t; t þ dtÞ; we have
EðU k ðt þ dtÞ1A jUðtÞÞ ¼ U k ðtÞ þ
X
U j ðtÞSjk dt 2
j–k
X
U k ðtÞSkj dt 2 U k ðtÞs0k dt;
j–k
as the development of Uk(t) on A is deterministic and the content of container k at time t þ dt is the content at time t plus the net inflow. Taking expectation, EðU k ðt þ dtÞ1A Þ ¼ zk ðtÞ þ
X
zj ðtÞSjk dt 2
j–k
X zk ðtÞSkj dt 2 zk ðtÞs0k dt: j–k
On the complementary event, AC, there will be a jump somewhere in ½t; t þ dtÞ: Here we know that the conditional expectation EðU k ðt þ dtÞjA C Þ ¼ ak þ oðdtÞ: Thus zk ðt þ dtÞ ¼ Eð1A U k ðt þ dtÞ þ 1A C U k ðt þ dtÞÞ ¼ Eð1A U k ðt þ dtÞjUðtÞÞ þ EðU k ðt þ dtÞjA C ÞPðA C Þ X X X ¼ zk ðtÞ þ zj ðtÞSjk dt 2 zk ðtÞSkj dt 2 zk ðtÞs0k dt þ ak zj ðtÞs0j dt: j–k
P
j–k
j
2 s0i ; we have X X zj ðtÞSjk dt þ zi ðtÞs0i dtak : zk ðt þ dtÞ ¼ zk ðtÞ þ
Then, since Sii ¼ 2
j–i Sij
j
i
This leads to the differential equations X z0k ðtÞ ¼ ðSjk þ s0j ak Þzj ðtÞ; j
for 1 # k # m; whose solution is
A
0
zðtÞ ¼ a exp½ðS þ s aÞt; Theorem 3.1.
The renewal density f(t) corresponding to F(·) is given by
fðtÞ ¼ a exp½ðS þ s 0 aÞts 0 : Proof.
Combine Lemmas 3.1 and 3.2.
A
As for phase-type distributions we can exploit the flow interpretation to obtain further properties for renewal processes with matrix-exponential interarrival times. For example we easily get:
Theorem 3.2. The residual lifetime by time x, Vx, which is defined as the time from x until the next renewal, has a matrix-exponential distribution with representation ða expððS þ s 0 ÞxÞ; SÞ:
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Proof. If the contents of the m containers are U(x) by time x and the next renewal takes place in ½x þ y; x þ y þ dyÞ; then from x to x þ y the flow moves deterministically, so the contents of the containers by time x þ y is UðxÞ expðSyÞ: Thus the probability of a renewal in ½x þ y; x þ y þ dyÞ given U(x), is UðxÞ expðSyÞs 0 dy: Hence PðV x [ ½y; y þ dyÞÞ ¼ EðPðV x [ ½y; y þ dyÞjUðxÞÞÞ ¼ EðUðxÞ expðSyÞs 0 dyÞ ¼ zðxÞ expðSyÞs 0 dy; and the result follows from Lemma 3.2. 4.
A
AN EXTENSION OF A MODEL FROM RISK-THEORY
One of the more complex derivations using phase-type distributions is found in Ref.[4], and is an example where the transform methods used in Ref.[2] do not apply as the ruin probability is obtained through solving two systems of differential equations, of which the first is non-linear. For phase-type distributions the arguments leading to the differential equations are entirely probabilistic, i.e. use the interpretation of the underlying Markov jump processes as building blocks. We summarize the model description of Asmussen and Bladt.[4] The risk reserve at time t of an insurance fund is Rt, with R0 ¼ u . 0: The reserve grows at a rate p(r) which depends on the current reserve r, i.e. between claims the reserve develops deterministically as ðd=dtÞRt ¼ pðRt Þ: The reserve decreases by jumps equal in size to the arriving claims. It is assumed that the successive claim sizes are independent with the common distribution B(·). In Ref.[4], B(·) is a phase-type distribution; here, we assume that B(·) is matrix-exponential MEða; SÞ and is represented by the corresponding valid flow. As in Section 3 of Ref.[4] we assume that claims arrive according to a homogeneous Poisson process of rate b. The extension to arrival processes like the Markov modulated Poisson process (see Refs.[1,4]) or MAPs is entirely similar and follows the steps in Section 4 of Ref.[4]. The extension to RAPs is more involved as it requires the notion of a marked RAP and is currently under investigation. To visualize how flows enter our derivations, at every claim epoch, we generate a copy of the deterministic, but randomly terminating flow. The amount of the claim corresponds to the duration of the flow. Corresponding to each point in the downward jump of a claim, there is a well-defined vector of contents in the flow model. By ni(u), we denote the expected amount of liquid in the container i at the first downcrossing of level u, conditional on the reserve at time 0 being u. The following is a generalization of Proposition 2 in Ref.[4] to the present setting. Our argument is entirely similar but is adapted to flow models rather than to phase-type distributions. Theorem 4.1. For u . 0 and 1 # i # m; the quantities ni(u) satisfy the system of nonlinear differential equations ! m m X X 2pðuÞn 0i ðuÞ ¼ bai þ ni ðuÞ pðuÞnj ðuÞs0j 2 b þ pðuÞnj ðuÞSji : ð2Þ j¼1
j¼1
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Matrix-Exponential Distributions
121
Proof. Figure 1 may help in following this and the next proof. We do the accounting of all possible occurrences in ½0; dtÞ: With probability bdt, there is a claim in that interval. If so, an immediate downcrossing of u occurs and the content of i at that time is ai. With the complimentary probability 1 2 bdt; there is no arrival in ½0; dtÞ and the reserve rises to u þ pðuÞdt: For there to be a downcrossing of u, there must be a downcrossing of the level u þ pðuÞdt: Consider the claim that results in the first downcrossing of u þ pðuÞdt: There are two alternatives: Either (case (1)) the flow model for that claim terminates in ½u; u þ pðuÞdtÞ or (case (2)) it continues and also crosses below u. Consider case (1). Let U 1k denote the content of container k upon downcrossing level u þ pðuÞdt and let U 1 ¼ ðU 11 ; . . .; U 1m Þ: Then the probability of case (1) to happen is Pðdowncross stops in ðu;u þ pðuÞdtÞ ¼ EðPðdowncross stops in ðu;u þ pðuÞdtjU 1 ÞÞ 0 1 X U 1k s0k pðuÞdtA ¼ E@ k
¼
X
nk ðu þ pðuÞdtÞs0k pðuÞdt:
k
Conditional on that event, the expected content of container i upon downcrossing level u is simply ni(u) as the process essentially starts over at level u. In case (2) we calculate on
Figure 1. The evolution of a risk-reserve process. The right hand side axis corresponds to the virtual time. As indicated by “Flow initiates” and “Flow terminates,” claims are thought to be generated by randomly stopped flows.
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the event A that the processes crosses below the level u. Conditioning on the content U 1 of level u þ pðuÞdt we get that on A the content when downcrossing level u of container i, U 2i ; must satisfy X X EðU 2i 1A jU 1 Þ ¼ U 1i þ U 1j Sji pðuÞdt 2 U 1i Sij pðuÞdt 2 U 1i pðuÞdts0i ; j–i
j–i
by keeping track of the deterministic inflow and outflow over the interval ðu;u þ pðuÞdt: Taking expectations we obtain, X X EðU 2i 1A Þ ¼ ni ðu þ pðuÞdtÞ þ nj ðu þ pðuÞdtÞSji pðuÞdt 2 ni ðu þ pðuÞdtÞSij pðuÞdt j–i
j–i
2 ni ðu þ pðuÞdtÞpðuÞdts0i ¼ ni ðu þ pðuÞdtÞð1 þ Sii pðuÞdtÞ þ
X nj ðu þ pðuÞdtÞSji pðuÞdt j–i
¼ ni ðu þ pðuÞdtÞ þ
m X
nj ðu þ pðuÞdtÞSji pðuÞdt:
j¼1
Now adding all contributions to the expected content of i, we get that
ni ðuÞ ¼ai bdt þ ð1 2 bdtÞ
m X
nj ½u þ pðuÞdtðdji þ pðuÞdtSji Þ
j¼1
þ
m X
!
nj ðu þ pðuÞdtÞs0j ·pðuÞdt·ni ðuÞ
j¼1
;
ð3Þ
for 1 # i # m: Now using nj ½u þ pðuÞdt ¼ nj ðuÞ þ n0j ðuÞpðuÞdt and simplifying, we obtain Eq. (2). A Let now c (u) be the conditional probability that, starting at u, the reserve eventually downcrosses level 0. In terms of the functions ni(u), the ruin probability c (u) can be obtained as follows: P Theorem 4.2. The ruin probability is given by c ðuÞ ¼ m i¼1 li ðuÞ; where the functions lj(u) are the solutions to the system of differential equations m m X X l0i ðtÞ ¼ lj ðtÞni ðu 2 tÞs0j þ lj ðtÞSji ; ð4Þ i¼1
j¼1
for 1 # i # m and subject to li ð0Þ ¼ ni ðuÞ: Proof. Starting at the level u, we consider the first claim that results in a downcrossing of u. There are two alternatives. Either the first undershoot takes us all the way below 0, or the reserve remaining after that claim is u 2 u0 . 0; where u0 is the amount of undershoot caused by the claim. In the latter case, we repeat the procedure starting with reserve u 2 u0
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Matrix-Exponential Distributions
123
and obtain possibly a second point u 2 u00 ¼ u 2 u0 þ u0 2 u00 ; where u00 is the amount the first claim which downcrosses level u 2 u0 undershoots this level. Continuing this way we decompose a path leading to ruin according to the successive values of u0 ; u00 ; . . .: Let us refer to the independent variable of the process generated by u0 ; u00 ; . . . as virtual time to distinguish between it and the real time of the risk process (see Fig. 1). We piece together the various flows that correspond to the successive undershoots. In the risk model with initial reserve u, ruin occurs if and only if that concatenation of undershoots lasts at least until virtual time u. For t $ 0; let Uj(t) be the content of container j at virtual time t and lj(t) be the expected amount of container j. Let UðtÞ ¼ ðU 1 ðtÞ; . . .; U m ðtÞÞ: Let A denote the event that the current flow by virtual time t continues through ½t; t þ dtÞ: Then EðU i ðt þ dtÞjUðtÞÞ ¼ EðU i ðt þ dtÞ1A jUðtÞÞ þ EðU i jUðtÞ; A C ÞPðA C jUðtÞÞ:
ð5Þ
On A and conditional on U(t) we get through simple accounting that X X U j ðtÞSji dt 2 U i ðtÞSij dt 2 U i ðtÞs0i dt EðU i ðt þ dtÞ1A jUðtÞÞ ¼ U i ðtÞ þ j–i
¼
m X
j–i
ðdij þ Sji dtÞU j ðtÞ:
j¼1
For the complementary event, PðA C jUðtÞÞ ¼
X U j ðtÞs0j dt; j
while EðU i ðt þ dtÞjA C ; UðtÞÞ ¼ ni ðu 2 tÞ since virtual time t corresponds to level u 2 t of the reserve. Inserting these expressions into Eq. (5) and taking expectation yields
li ðt þ dtÞ ¼
m X
lj ðtÞ½dji þ Sji dt þ s0j dt·ni ðu 2 tÞ;
for
1 # i # m:
ð6Þ
j¼1
The differential equations (4) now follow immediately. We recall that, for valid flows, the sum of the contents at virtual time t of the urns i, 1 # i # m;P is precisely the probability that the flow has not terminated (cfr. Proposition 2.2). Therefore, m j¼1 lj ðuÞ is the probability that, at virtual time u a flow is active. That, precisely, is the ruin probability. A
5.
INTERPRETING RATIONAL ARRIVAL PROCESSES
Rational Arrival Processes was introduced in Ref.[3] as an extension of the Markovian Arrival Process (MAP). RAPs are point processes that allow for dependent matrixexponentially distributed inter-arrival times.
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In what follows we give a physical interpretation of RAPs. Let C and D be the coefficient matrices of a RAP together with the initial content vector a. Then, (see Ref.[3], Theorem 1.1) with s ¼ De; the joint density gn for the first n interarrival times is gn ðx1 ; . . .; xn Þ ¼ a expðCx1 ÞD expðCx2 ÞD. . .D expðCxn Þs: Now consider the following flow scheme. Initially, the containers 1; . . .; m contain levels a1 ; . . .; am : The flow rates between these containers are the elements of C. There is also be a net increasing inflow to container m þ 1: The flow from container i to container m þ 1 is s0i ¼ ð2CeÞi ¼ ðDeÞi : When a flow stops due to an arrival and the content vector of containers 1; . . .; m just prior to the arrival is b, then the containers are refilled to the content vector bD=ðbDe). This flow scheme provides a physical interpretation of the RAPs. To see this, consider the flows as just described. Then, g2 ðx1 ; x2 Þ ¼ g2 ðx2 jx1 Þg2 ðx1 Þ ¼
a expðCx1 ÞD expðCx2 Þs·a expðCx1 Þs a expðCx1 ÞDe
¼ a expðCx1 ÞD expðCx2 Þs: By induction, the joint density of the first n interarrival times in the flow scheme gn is gn ðx1 ; . . .; xn Þ ¼ a expðCx1 ÞD. . .D expðCxn Þs: By Theorem 1.1 of Ref.[3] the corresponding point process is a RAP with the given representation. Further applications of flow arguments in RAP’s, which will also involve marked RAP’s, will be presented in forthcoming articles. REFERENCES 1. 2.
3. 4. 5. 6. 7. 8.
Asmussen, S. Ruin Probabilities; World Scientific: Singapore, 2000. Asmussen, S.; Bladt, M. Renewal theory and queueing algorithms for matrixexponential distributions. In Matrix-Analytic Methods in Stochastic Models; Chakravarthy, S., Alfa, A.S., Eds.; Marcel Dekker: New York, 1996; 313– 341. Asmussen, S.; Bladt, M. Point processes with finite-dimensional conditional probabilities. Stoch. Proc. Appl. 1999, 82, 127 –142. Asmussen, S.; Bladt, M. Phase-type distributions and risk processes with statedependent premiums. Scand. Actuarial J. 1996, 1, 19– 36. Knight, F.B. Essays on Prediction Processes. IMS Lecture Notes series; The Institute of Mathematical Statistics: Hayward, CA, 1981; Vol. 1. Knight, F.B. Foundation of Prediction Processes; Clarendon Press: Oxford, 1992. Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach; Johns Hopkins: Baltimore, 1981. O’Cinneide. Characterization of phase-type distributions. Stoch. Models 1989, 6, 1– 57.
Received December 17, 2001 Accepted August 14, 2002
