© Hindawi Publishing Corp.
ON FUNCTIONALS OF A MARKED POISSON PROCESS
OBSERVED BY A RENEWAL PROCESS
JEWGENI H. DSHALALOW and JEAN-BAPTISTE BACOT
(Received 22 May 2000)
Abstract.We study the functionals of a Poisson marked processΠobserved by a re-newal process. A sequence of observations continues untilΠcrosses some fixed level at one of the observation epochs (the first passage time). In various stochastic models appli-cations (such as queueing withN-policy combined with multiple vacations), it is necessary to operate with the value ofΠprior to the first passage time, or prior to the first pas-sage time plus some random time. We obtain a time-dependent solution to this problem in a closed form, in terms of its Laplace transform. Many results are directly applicable to the time-dependent analysis of queues and other stochastic models via semi-regenerative techniques.
2000 Mathematics Subject Classification. 60K10, 60K15, 60K25.
1. Introduction. This paper investigates a class of functionals of a Poisson marked processΠobserved by a renewal processτ (and thereby forming a marked renewal process). It is of interest to see the processΠcross some fixed threshold which, how-ever, may be observed only later byτ. The value ofΠat the first passage time, as observed byτ, is known as the first excess level. We are targeting joint functionals of the Poisson processΠ, prior to the first passage time, and of the first excess level as well as the value ofΠ, between the first passage time and some other instant generated by a given random variable.
Studies on crossing level analysis were earlier rendered in Dshalalow [5,6,7,8], but they were not applied to time-dependent processes. In the present paper, when ex-amining such functionals, we arrive at compact analytic formulas of time-dependent discrete-valued processes, in terms of Laplace transforms (promising to yield closed forms also for their continuous-valued compounds). Many results are directly appli-cable to time-dependent analysis of semi-regenerative processes occurring in queues and other stochastic models.
2. First passage time of a basic delayed marked renewal process. In this section, for consistency, we give some preliminaries on a class of marked delayed renewal processes pertinent to the upcoming sections. The below results are a modification of those in Dshalalow [6,7,8] and therefore somewhat new.
All stochastic processes are considered on a probability space(Ω,Ᏺ,P). Let
(A,τ)= ∞
n=0
Xnετn (2.1)
be a delayed marked renewal process with position dependent marking whereεais a point mass. We assume that the marksX0,X1,...are nonnegative integer-valued. The point processτ= {τ0,τ1,...}is placed on the real axis and it is related to the following random variables and functionals.
The sequence{χn=τn−τn−1,Xn; n=1,...}is a sequence of i.i.d. two-variate ran-dom variables with the joint common transformation
γ(z,θ)=EzX1e−θχ1, |z|<1,Re(θ)≥0, (2.2)
marginal probability distribution function (PDF)V=V(t),t≥0, ofχ1, and its Laplace-Stieltjes transform (LST)
γ(θ)=γ(1,θ). (2.3)
Let
µ(z,θ)=EzX0e−θχ0, |z|<1,Re(θ)≥0, (2.4) with marginal PDFV0=V0(t),t≥0, ofχ0, its LST
µ(θ)=µ(1,θ), (2.5)
and probability generating function (pgf) ofX0,
m(z)=µ(z,0). (2.6)
With the notationAk=X0+···+Xk, we have{Ak}as the marginal delayed (discrete-valued) renewal process of(A,τ).
Now, letNbe a fixed positive integer, which(A,τ)is supposed to cross at one of the epochsτ0,τ1,...,called the first passage time. More formally, letν=inf{k=0,1,...: Ak≥N}be known, [6], as the termination index. Thenτνis the first passage time and Aνis the first excess level. We are interested in the functional
whereNis the above control threshold of the termination index, pre-first excess level, first excess level, and the first passage time, respectively. The value of the pre-first excess level may be of independent interest, as it informs us one step ahead of some “catastrophe.”
In what follows, we will need the following transformation:
Dpf (w)=(1−w)
p≥0
wpf (p), |w|<1, (2.8)
wherefis an integrable Borel measurable function. (Notice that whilepis a dummy variable, it is being used in the notationDpfor convenience.)
With Taylor-like functional
Ᏸk w(·)=
lim w→0
1 k!
∂k ∂wk
1
1−w(·), k≥0,
0, k <0, (2.9)
we can restoref subject toDp
Ᏸk w
Dpf (w)=f (k). (2.10)
The subscriptw inᏰ serves alsothe same purpose asp. For various special cases throughout the remainder of this paper, we notice a few elementary properties of the operatorᏰpy(see Dshalalow [8])
Ᏸpyis a linear operator with fixed points at every constant function. (2.11)
For any functionh, analytic at zero,
Ᏸpyykh(y)=
0, p < k,
Ᏸpy−kh(y), p≥k. (2.12)
Now, given the process(A,τ), we introduce the auxiliary sequence of random variables
νp=infk:Ak> p:p=0,1,... of whichνN−1=ν, (2.13) and apply the transformation Dp tothe sequence {L(ξ,x,z,θ;p+1)}p. Then, L(ξ,x,z,θ;N)can be traced back by means of the operatorᏰN−1
w . All this is subject to the following theorem.
Theorem2.1. The functionalLsatisfies the following formula:
L(ξ,x,z,θ;N)=µ(z,θ)−ᏰN−1 w
µ(wz,θ)−ξµ(wxz,θ)γ(z,θ)−γ(wz,θ) 1−ξγ(wxz,θ)
, (2.14)
whereL,µ,γ, andᏰare defined in (2.2), (2.4), (2.7), and (2.9), respectively. Proof. First, observe that
(which is due to[6,8] or it can also be easily verified), whereA−1=0 and1Ais the indicator function of a setA. Then, from (2.15),
DpLξ,x,z,θ;p+1(w)=Λξ,xw,z,θ−Λξ,x,zw,θ
= ∞
n=0 ξnΛ
n(xw,z,θ)− ∞
n=0 ξnΛ
n(x,wz,θ), (2.16)
where
Λn(x,z,θ)=ExAn−1zAne−θτn. (2.17) Clearly,
Λ0(x,z,θ)=EzA0e−θτ0=µ(z,θ), (2.18) while
Λn(x,z,θ)=E(xz)An−1zXne−θτn−1e−θχn. (2.19) The latter easily reduces to
µ(xz,θ)γn−1(xz,θ)γ(z,θ), (2.20)
which leads to
Λξ,x,z,θ=µ(z,θ)+µ(xz,θ)γ(z,θ)ξ1−ξγ(xz,θ)1 , (2.21)
and then to
DpLξ,x,z,θ;p+1(w)=µ(z,θ)−µ(wz,θ)+ξµ(wxz,θ)γ(z,θ)1−ξγ(wxz,θ)−γ(wz,θ). (2.22)
The rest is due to(2.11).
Of course, the parameterNin the functionalL(ξ,x,z,θ;N)can be dropped through-out the rest of the paper, as it was used only temporarily.
We make use of the following special cases.
Case1. Omitting the pre-first excess level, that is, settingx=1, simplifies (2.14)
Lξ,1,z,θ=µ(z,θ)−1−ξγ(z,θ)ᏰN−1 w
µ(wz,θ)
1−ξγ(wz,θ)
. (2.23)
Case2. Let
Π= ∞
j=1
Ujεrj (2.24)
be a stationary compound Poisson process of intensityλand witha(z)as the common pgf of marksU1,U2,...(arriving batches) and let
Π0=X0ε0+Π (2.25)
be “observed” byτat epochsτ0,τ1,...,sothatτ0=0. Then, with
we have
(A,τ)= ∞
n=0
Xnετn (2.27)
as the delayed marked renewal process embedded inΠ0. Abbreviating
λ(z,θ)=λ−λa(z)+θ, (2.28)
we have by standard probability arguments thatγ(z,θ)defined in (2.2) and (2.3) is
γ(z,θ)=γλ(z,θ). (2.29)
Then, fromTheorem 2.1, we have the following corollary.
Corollary2.2. Under the assumptions (2.24), (2.25), (2.26), and (2.27), the func-tionalL(defined in (2.7)) reduces to
Lξ,x,z,θ=m(z)−ᏰN−1 w
m(wz)−ξm(wxz)γ
λ(z,θ)−γλ(wz,θ) 1−ξγλ(wxz,θ)
, (2.30)
wherem,γ, andᏰare subject to (2.6), (2.9), (2.28), and (2.29), respectively. In partic-ular, withX0=ia.s. orm(z)=ziand properties (2.12), (2.30) reduce to
L(ξ,x,z,θ)=ξ(xyz)iᏰN−i−1 w
γλ(yz,θ)−γλ(wyz,θ) 1−ξγλ(wxyz,θ)
. (2.31)
On the other hand, settingx=1in (2.30) (also, in agreement with the special case of (2.23) forτ0=0) yields
Lξ,1,z,θ=m(z)−1−ξγλ(z,θ)ᏰN−1 w
m(wz) 1−ξγλ(wz,θ)
. (2.32)
A further special case of (2.32) (frequently encountered in applications) is whenX0=i a.s. By (2.12),
Lξ,1,z,θ=zi−zi1−ξγλ(z,θ)ᏰN−i−1 w
1 1−ξγλ(wz,θ)
. (2.33)
Notice that formulas (2.32) and (2.33) (that are previously known from Dshalalow [6]) can be applied to queuing models of type MX/G/1 with server vacations and N-policy. Formula (2.33) will then represent the functional of the queuing process (initiated at leveli) at the end of a vacation period and the first passage time on a vacation period[0,τν]generated by multiple vacations.
frequently takes place when studying various classes of semi-regenerative processes, such as those in queuing. Under the assumptions ofCase 2, we target the functional
Hξ,x,y,z,θ=
∞
0 e
−θtEξνxAν−1yAνzNt1{
τν≤t}1{τν+Σ>t}dt, (3.1)
where1A is the indicator function of a setA, Σis a nonnegative random variable, independent ofτ andΠ0, with the PDFS and LSTσ, andNt=Π0([0,t]). Note that if Σ= ∞, then1{τν+Σ>t}degenerates to1 andS=σ=0.
First we prove the following lemma. Lemma3.1. The functionalhdefined as
hξ,x,y,z,θ= ∞
n=0
∞
0 e
−θtEξnxAn−1yAnzNt1{τ
n≤t}1{τn+Σ>t}
dt (3.2)
(withA−1:=0)satisfies the following formula:
hξ,x,y,z,θ=1−σ
λ(z,θ) λ(z,θ)
m(yz)+m(xyz) ξγ
λ(yz,θ) 1−ξγλ(xyz,θ)
,
z<1,xy ≤1,Re(θ)≥0,
(3.3)
wherem(z),λ(z,θ), andγ(θ)are due to (2.6), (2.28), and (2.29), respectively. Proof. (1) Forn >0,
hn(x,y,z,θ)=
∞
0 e −θtE1{
τn≤t}1{τn+Σ>t}E
xNτn−1yNτnzNt|τn−1,τn,Σdt
=
∞
0 e −θtE1{
τn≤t}1{τn+Σ>t}E
xNτn−1yNτnzNt|τn−1,τndt
=m(xyz)
∞
0 e −θtE1{
τn≤t}1{τn+Σ>t}eλτn−1[a(xyz)−1]
×eλχn[a(yz)−1]eλ(t−τn−1−χn)[a(z)−1]dt
=m(xyz)
∞
t=0
t
u=0
t−u
s=0
∞
ξ=t−u−se
−θteλu[a(xyz)−1]eλs[a(yz)−1]
×eλ(t−u−s)[a(z)−1]S(dξ)V (ds)V(n−1)∗(du)dt
=m(xyz)
∞
u=0e
λu[a(xyz)−1]−θu∞ s=0e
λs[a(yz)−1]−θs
×
∞
ξ=0
ξ
c=0e
λc[a(z)−1]−θcdc S(dξ)V(ds)V(n−1)∗(du)
=m(xyz)1−σ
λ(z,θ) λ(z,θ) γ
λ(yz,θ)γn−1λ(xyz),θ.
(3.4)
Note thatzeλ(z,θ)(s−c)is anL1-function for allz<1 and Re(θ)≥0 (as, by Schwarz lemma, it makesa(z)<1) and hence its integral on[s,∞)is convergent forz<1 and Re(θ)≥0.
(2) Forn=0, we easily verify that
h0x,y,z,θ=
∞
0 e −θtE1{
Σ>t}yN0zNtdt=m(yz)1−σ
λ(z,θ)
Finally, summing up allhn(x,y,z,θ)’s we have (3.3). Theorem3.2. The functional
Hξ,x,y,z,θ=
∞
0 e
−θtEξνxAν−1yAνzNt1{τ
ν≤t}1{τν+Σ>t}
dt (3.6)
satisfies the following formula:
Hξ,x,y,z,θ=1−σ
λ(z,θ) λ(z,θ) L
ξ,x,yz,θ, (3.7)
whereL(ξ,x,z,θ)=E[ξνxAν−1zAνe−θτν]satisfies (2.30).
Proof. As in the case ofL, we temporarily adhereNin notation ofH(ξ,x,y,z,θ;N). Obviously,
Hξ,x,y,z,θ;N= ∞
n=0
∞
0 e
−θtEξν1{
ν=n}xAn−1yAnzNt1{τn≤t}1{τν+Σ>t}dt, (3.8)
with A−1:=0. Now, we apply the transformation Dp tothe functionalH(ξ,x,y,z, θ;p+1), in notationDpH(ξ,x,y,z,θ;p+1)(w)by replacing, as in (2.13),ν by the auxiliary sequence{νp=inf{k:Ak> p}:p=0,1,...}of random variables in each of the functionalsH(ξ,x,y,z,θ;p+1)’s. Because of (2.15),
DpHξ,x,y,z,θ;p+1(w)
= ∞
n=0 ξn∞
0 e
−θtE(wx)An−1yAnzNt1{τ
n≤t}1{τn+Σ>t}
−ExAn−1(wy)AnzNt1{τ
n≤t}1{τn+Σ>t}
dt,
(3.9)
which yields
DpHξ,x,y,z,θ;p+1(w)=hξ,wx,y,z,θ−hξ,x,wy,z,θ, (3.10)
withhdefined in (3.2) o fLemma 3.1and hence, by (3.3), DpHξ,x,y,z,θ;p+1(w)
=1−σ
λ(z,θ) λ(z,θ)
m(yz)−m(wyz)+ξm(wxyz)γ
λ(yz,θ)−γλ(wyz,θ) 1−ξγλ(wxyz,θ)
. (3.11)
Finally, we restoreH(ξ,x,y,z,θ;N)by using the operatorᏰk
w fork=N−1 and its property (2.11).
From (2.30), we have the following corollary.
Corollary3.3. Ifm(z)=zi, then formula (3.7) reduces to
Hξ,x,y,z,θ=1−σ
λ(z,θ) λ(z,θ) L
ξ,x,yz,θ, (3.12)
with the version ofLin the form
Lξ,x,yz,θ=ξ(xyz)iᏰN−i−1 w
γλ(yz,θ)−γλ(wyz,θ) 1−ξγλ(wxyz,θ)
In addition, forx=1,
Hξ,1,y,z,θ
=
∞
0 e
−θtEξνyAνzNt1{
τν≤t}1{τν+Σ>t}dt
=
1−σλ(z,θ)(yz)i−(yz)i1−ξγλ(yz,θ) λ(z,θ) ᏰNw−i−1
1
1−ξγλ(wxyz,θ)
, (3.14)
in which we identify
Lξ,1,z,θ=zi−zi1−ξγλ(z,θ)ᏰN−i−1 w
1 1−ξγλ(wz,θ)
. (3.15)
Corollary3.4. ForΣ= ∞a.s., as we previously noticed,S,σ, andbvanish. Thus, fromTheorem 3.2,
Gξ,x,y,z,θ=
∞
0 e
−θtEξνxAν−1yAνzNt1{τ ν≤t}
dt= 1
λ(z,θ)L
ξ,x,yz,θ, (3.16)
whereLsatisfies (2.30).
Corollary3.5. Takingξ=x=y=1in (2.30) and (3.16),
G(1,1,1,z,θ)=
∞
0 e
−θtEzNt1{τ ν≤t}
dt=λ(z,θ)1 L(1,1,z,θ), (3.17)
or in the form
G(1,1,1,z,θ)= 1
λ(z,θ)
m(z)−1−γλ(z,θ)ᏰN−1 w
m(wz)
1−v(wz,θ)
. (3.18)
ThenF(z,θ)=0∞e−θtE[zNt1{τν>t}]dtas
F(z,θ)=
∞
0 e
−θtEzNtdt−G(1,1,1,z,θ)
=λ(z,θ)1 1−γλ(z,θ)ᏰN−1 w
m(wz) 1−γλ(wz,θ)
,
(3.19)
or in the form
F(z,θ)=λ(z,θ)1 m(z)−L(1,1,z,θ). (3.20)
In particular, form(z)=zi,
F(z,θ)=λ(z,θ)zi 1−γλ(z,θ)ᏰN−i−1 w
1 1−γλ(wz,θ)
, (3.21)
or in the form
Corollary3.6. The functional
Γ(z,θ)=
∞
0 e
−θtEzNt1{τ ν+Σ>t}
dt (3.23)
equals
Γ(z,θ)=F(z,θ)+H(1,1,1,z,θ)
=
∞
0 e
−θtEzNt1{τ ν>t}
dt+
∞
0 e
−θtEzNt1{τ
ν≤t}1{τν+Σ>t}
dt
= 1
λ(z,θ)
m(z)−L(1,1z,θ)+1−σ
λ(z,θ)
λ(z,θ) L(1,1,z,θ)
=m(z)−σ
λ(z,θ)L(1,1,z,θ)
λ(z,θ) ,
(3.24)
or, whenm(z)=zi,
Γ(z,θ)=zi−σ
λ(z,θ)L(1,1,z,θ)
λ(z,θ) . (3.25)
Remark 3.7. Formula (3.25) takes place in queuing when, by using semi-regenerative techniques, one needs to evaluate the functional of the queue length on the first service cycle[0,τν+Σ1), which consists of[0,τν](and can be regarded as a “multiple vacation period”) and of the period(τν,τν+Σ1), whereΣ1is the duration of the first service. For this particular application, we need to setθ=0 and thereby haveL(1,1,z,θ)reduced to the marginal functional of the first excess level.
Acknowledgements. The authors are very grateful to Lajos Takács for his very valuable comments and many helpful discussions.
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Jewgeni H. Dshalalow: Applied Mathematics Program, Florida Institute of Technol-ogy, Melbourne, FL32901, USA
E-mail address:[email protected]
