Zhang Peipei Wang Juan

(College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China)

Abstract This paper focuses on discussing some basic properties of the Markov interacting branching processes with immigration which is a natural generalization of the ordinary Markov branching process. After obtaining some key preliminary results, we first obtain some elegant conditions regarding regularity and uniqueness. Then the extinction vector is obtained which is very easy to be calculated. The mean extinction time and the conditional mean extinction time are revealed. The mean explosion time and the total mean life time of the processes are also investigated and resolved.

Key words Branching process Immigration Regularity and uniqueness Extinction probability Mean extinction time Mean explosive time

1 Introduction

The ordinary Markov branching processes (MBPs) from one of the most important classes of stochastic processes. Standard references are, among many others, Harris [1], Athreya and Ney [2] and Asmussen and Hering [3]. The basic property which governs the evolution of an MBP is the branching property. That is that different particles act independently when giving birth or death. Although this basic assumption is extremely useful in analyzing properties of MBP which makes the theory and applications of MBP surprisingly fruitful, few practical models satisfy this assumption. Indeed, in practical cases, particles usually interact with each other. This may explain the reason why there always have been a great interest and an increasing effort to generalize the ordinary branching processes to more general branching models. Such efforts are clearly reflected in [4]. Many different interacting branching models which are generalizations of the ordinary MBP have been proposed and studied in [5-8]. This trend has significantly enriched the research of branching processes and bring fruitful results in the theory and applications of branching processes.

The main aim of this paper is to consider the generalized Markov interacting branching processes with immigration which is a natural generalization of the model considered in [9]. The criteria for regularity, extinction behaviors and properties for such model will be deeply investigated.

More specifically, we shall define our model by specifying the infinitesimal characteristic, i.e., the so-called q-matrix. Throughout this paper, letZ+={0,1,2,…} denote the set of nonnegative integers.

Definition 1.1Aq-matrixQ=(qij;i,j∈Z+) is called a generalized interacting branching processes with immigration (henceforth referred to as a GIBIq-matrix), if there exists a positive integerm≥1 such that

(1)

where

(2)

Since it is possibly a little bit less intuitive from the view point of probability, so we need re-parameterize our model by lettingbj=lpj(j≥m) andbm=-l，aj=kej(j≥m) andam=-k， which is very convenient. Then we have

(3)

where

(4)

indicating that our GIBIq-matrix is conservative.

Definition 1.2A continuous-time Markov chain on the state spaceZ+is called GIBI, if whose transition functionP(t)=(pij;i,j∈Z+) satisfies

P′(t)=P(t)Q,

(5)

whereQis a GIBIq-matrix given in (1)-(2).

Definition 1.3A GIBIq-matrixQis said to be slowly-exit if ∑∞i=m(1/ωi)=∞ or fast-exit if ∑∞i=m(1/ωi)<∞.

We can see that this general model is based on the two given sequences {bj;j≥0} in terms of the distribution {pj;j≥0} and the real valuea>0 and {ωj;j≥m} together with the positive integerm. With the sequence {bj;j≥0} being fixed, we can obtain many interesting and important models by varying the sequence {ωj;j≥m}. In fact, many important branching models which have been extensively considered and have important applications can be viewed as special cases of our general branching processes model. For example, if we letm=1 andωi=i(i≥1)， we then obtain the much-studied MBP to which huge publications exist. See [2] or [3]. The stoppedMX/M/1 queue is the casem=1 andωi≡1 (see [10,11]).And also some other special cases ofωihave also been addressed. For example, a family of interesting differential-integral equations together with with extinction behaviors have been revealed in [12] whenωi=iνwhereνis a non-negative real number. For arbitraryωi， Chen [13] obtained some uniqueness criteria. In addition if we letm=2 andωi= we then recover the collision branching process (CBP), detailed addressed in [14]. Ifm=2 together withωi=1， the model becomes a queueing model.

The basic aim of this paper is to fill this gap by analyzing these very general models. Since the cases ofm=1 or 2 have been extensively considered, we assume in this paper thatm≥3， although most of the conclusions obtained in this paper apply perfectly well for casesm=1 orm=2. Note also that we shall assume thatmis the smallest positive integer such that all states {m,m+1,…} communicate, i.e.,G={m,m+1,…} is an irreducible class for the GIBIq-matrixQ.

It should be emphasized that although we use the terminology of interacting branching processes to denote our general model for the reason of convenience, its application is far beyond branching processes and queueing models. In fact, by choosing suitablemandωi， we could model many other natural phenomena. In fact, this is an important impulse for us to study these general models.

The structure of this paper is as follows. In the following Section 2, we present a few preliminary results which play an important role in our later analysis. In Section 3, we focus our discussions on uniqueness and regularity of our processes. In particular, several elegant regularity and uniqueness criteria are obtained. Section 4 is devoted to the study of extinction behavior and hitting times. Some important conclusions regarding extinction probabilities and mean extinction times and explosive times are also given. A couple of examples are then provided in Section 5.

2 Preliminaries

We first define the generating functions of the known sequence {pj;j≥0} and {ej;j≥0} asR(z)=∑∞j=0pjzjandT(z)=∑∞j=0ejzj. LetB(z)=l(R(z)-zm)，A(z)=k(T(z)-zm). By the definition, we could know that GIBI is completely determined by these sequences. So we could define their occurrence functionsB(z) andA(z) as follows:

(iii)B(z) can be expressed as

(6)

withρ0,ρk≥0(k≥1) and

(7)

Moreover, if {m,m+1,…} is irreducible forQ， thenρ1>0.

Definition 2.1From now on, we assume that allσ0,σ1,…,σm-1are distinct.

Note that this is purely a technical assumption. Indeed, it is easy to see how to modify the conclusions if this assumption is violated.

We define that

(8)

Sinceρ0-∑∞k=1ρkzkis analytic inside the unit diskDand |ρ0-∑∞k=1ρkzk|≥ρ0(1-|z|)>0 for all |z|<1，G(z) must also be analytic insideD. HenceG(s)， viewed as a real function ofs， can be expanded as a Taylor seriesG(s)=∑∞k=1gkskat least within the open interval |s|<1. We now have the following important conclusion.

Lemma 2.2The coefficients of the power series have the following properties:

(i) 0

(ii) IfB′(1)<0， then

(9)

and thusg∞:=limn→∞gn=0.

(iii) IfB′(1)=0， then ∑∞n=0gn=+∞ and the limitg∞:=limn→∞gnexists and is given by

(10)

In particular,g∞>0 if and only ifB″(1)<∞.

(iv) If 0

(11)

In particular,g∞>0 if and only ifB′(1)<∞.

ProofBy recognizing that (8) is precisely the basic identity of discrete renewal theory, we may exploit the theory of renewal sequences (see Kingman [17]). In the notation of Chapter 1 of Kingman,fk=ρk/ρ0(k≥1) andf∞=1-∑∞k=1fk， and the associated renewal sequence {un,n≥0} is given byuk=ρ0gk(k≥0). By the irreducibility of {m,m+1,…} and usingf1=ρ1/ρ0>0， we knowthat this renewal sequence is aperiodic and thatuk>0 for allk≥0 (see expression (1.1.15) of Kingman).

Now, (i) is true because it is simply a statement of the fact that 00， and a simple calculation shows that

f∞=(md-mb)/[ρ0(1-σ1)…(σm-1)].

Part (ii) then follows from Theorem 1.5 of Kingman.

Parts (iii) and (iv) are consequences of the Erd?s-Feller-Pollard theorem (Theorem 1.5 of Kingman): the limitg∞=limn→∞gn=ρ-10limn→∞unexists, and, becauseB′(1)≥0 implies thatf∞=0， it follows thatu∞=(∑∞k=1kfk)-1， equivalentlyg∞=(∑∞k=1kρk)-1， with the interpretation thatg∞=0 whenever the series is divergent. It remains to evaluateg∞. To this end, first assumeB′(1)=0， then by (8) we know that

then let’s take the derivative of both sides with respect tozand letz→1， we can know that

which is precisely (10).

If 0

Lemma 2.3Let (pij(t);i,j∈Z+) and (φij(t);i,j∈Z+) be the minimal transition function and resolvent. Then for anyi≥mand |z|<1， we have

(12)

(13)

and

(14)

(15)

ProofWe suppose that 0≤z<1. The proof with -1

(16)

whereqj=-qjjand asn→∞,φ(n)ij(λ)↑φij(λ) for alli,j∈Z+.

Now we return to our GIBI q-matrixQonZ+and we still use {φij(λ);i,j∈Z+} to denote the corresponding Feller minimal resolvent. We first claim that for anyn≥0,i≥0 and 0

(17)

We assumei≥msince otherwise (17) is trivially true. We use mathematical induction onnto prove the conclusion. Firstly, the claim is trivially true forn=0.

Secondly, note that by (16), we can easily get that

(18)

and so (17) follows from induction principle. Next, letC(n+1)ij(λ)=φ(n+1)ij(λ)-φ(n)ij(λ)(n≥0)， we then haveC(n)ij(λ)≥0(n≥1) and

(19)

Using this notation, (18) can be rewritten as

(20)

Noting that by using (18) we may immediately obtain that for all 0≤z<1 andi≥m，

and hence

(21)

Lettingz=1 in (21) yields

(22)

Lettingn→∞ in (20) and using the above equality leads to the conclusion that for 0

However, we may find anε>0 such that for all 0<1-ε≤z<1 we haveB(z)≠0. Hence,

The monotone convergence theorem and dominated convergence theorem then yields

which in fact holds for all0

3 Regularity and Uniqueness

Theorem 3.1IfB′(1)≤0， thenQis regular (equivalently, the minimal process is honest).

ProofLet (φij(λ)) be the Feller minimalQ-resolvent. IfB′(1)≤0 and thus Lemma 2.1 implies thatB(z)>0 forzin [0, 1]. So from (15) we get that

Lettingz↑1 and from the limz→1A(z)=0 then yieldsλ∑∞j=0φij(λ)≥1， which implies that the equality holds for alli≥0 since we always haveλ∑∞j=0φij(da)≤1. We deduce that the minimal transition function is honest,and hence thatQis regular.

Theorem 3.2Suppose that ∑∞k=m(1/ωk)=+∞.

(i) IfB′(1)<∞， thenQis regular, implying that the minimal process is honest(and thus is the onlyQ-process).

(ii) IfB′(1)=∞， and ∑∞k=m(gk-m/ωk)<+∞， thenQis not regular, implying that the minimal process is dishonest.

(iii) IfB′(1)=+∞ and ∑∞k=m(gk-m/ωk)=+∞， thenQis regular.

ProofWe prove (ii) first. LetP(t)=(pij(t);i,j≥0) be the Feller minimalQ-function. Suppose the contrary is true:that this transition function is honest. Then, 1-∑m-1j=0pij(t)=∑∞k=mpik(t)(i≥m). Now lettingt→∞ and using (33) in Lemma 4.3 in advance, yields

On the other hand, if we denotef(s)=∑m-1j=0aijsj-sm， thenf′(1)=∑m-1j=1jaij-m<-1 and by Lemma 2.1 we knowf(s) has no zero in [0, 1). However, takingk=0 in expression (29) of Lemma 4.1 yieldsf(σ0)=0， a contradiction because we assumedB′(1)=+∞ implies 0<σ0<1 which, in turn, impliesf(σ0)>0 sincef′(1)<0 implies thatf(s)>0 for all 0≤s<1.

Next, we prove (i). The caseB′(1)≤0 is dealt with in Theorem 3.1, so we suppose that 0

then we may rewrite (15) fori=mas

Equating coefficients yields

(23)

for anyn≥m. Since ∑∞j=0λφmj(λ)≤1， Theorem 2.5.5 of Hunter [19] may be employed to deduce that

(24)

Since 00,∑∞j=0λφmj(λ)=1. Assume this fails. Then, there is aλ>0 with 1-λ∑∞j=0φmj(λ)>0. Lettingn→∞ in (23), and using (24), we obtain

Since

it follows that for a fixedλ>0 there is a constantδ>0 and an integerN>msuch that, for alln≥N,φmn(λ)≥δω-1n. This is a contradiction because ∑∞n=mω-1n=+∞. Thus,m∞j=0λφmj(λ)=1 for allλ>0. Now since {m,m+1,…} is an irreducible class, we obtain that ∑∞k=0λφik(λ)=1 for alli≥m. Also, since all states {0,1,…,m-1} are absorbing, the equality is trivially true fori≤m-1.

Finally, we prove (iii). The stated condition implies that the limit limn→∞ωnexists. Now if the sequence {ωk;k≥m} is eventually non-increasing or if it is eventually non-decreasing but the limit is finite, thenQis clearly regular because it is bounded. We may therefore only consider the case that for somen0,{ωn,n≥n0} is non-decreasing and limn→∞ωn=∞. For this latter case we can choose an integern1>n0such thatωn≥max{ωk:k≤n0} for {ωn:n≥n0} is non-decreasing. It follows from (23) and noting that the first term on the right-hand side of (23) is nonnegative, we get thatforn≥n1，

Hence, forn≥n1，

for allz∈[0,1), we see that

On lettingz→1 in the above expression, we find that the left-hand side tends to a finite value and thus so does the right-hand side. But the second term on the right-hand side certainly tends to a finite limit and therefore the first term must also tend to a finite value. However, this term is a product of two factors, the second of which tends to infinity because ∑∞k=0(gk/ωk+m)=∞. Thus, the first factor must tend to 0. We deduce thatλ∑∞j=0φmj(λ)=1(λ>0)， and henceλ∑∞j=0φij(λ)=1(λ>0)， for alli≥m， since {m,m+1,…} is irreducible. Finally, it is obvious thatλ∑∞j=0φij(λ)=1(λ>0) fori≤m-1 sineφij(λ)=λ-1δij.

A careful examination of the proof of Theorem 3.2 reveals that the truth of (ii) depends neither on the condition ∑∞k=m(1/ωk)=+∞， nor onB′(1)=∞. In other words, we have proved that, provided 0

Theorem 3.3If the GIBI q-matrixQis fast-exit, i.e., ∑∞k=m(1/ωk)<+∞， thenQis regular if and only ifB′(1)≤0.

ProofIn view of Theorem 3.1, we only need to prove thatQis not regular when 0

The previous three theorems establish regularity criteria. IfQis regular then there is only oneQ-process, namely the (honest) minimalQ-process. However, the converse may not be always true. Indeed, if ourQis not regular then, although there are infinitely many (even honest) transition functions, there may still exist only one GIBI because recall that our processes must satisfy the Kolmogorov forward equation. Therefore, in addition to the regularity criteria, we also need to establish criteria for uniqueness. Of course in this situation we only need to consider the cases that either 0

ProofWe only need to prove that under the stated conditions, the equation

Y(λ)(λI-Q)=0,0≤Y(λ)∈l,

(25)

has only the trivial solution for some (and hence for all)λ>0. Suppose the contrary, then (25) has a non-trivial solution {yn(λ):n≥0}. Writing out (25) we find that

(26)

and hence the convergence radius of the power seriesH(s):=∑∞n=mhnsn-mis strictly greater thanσ0. It follows that there is anε>0 such thatH(s) is well defined and finite on [0,σ0+ε). Thisεcan, of course, be chosen so small that it also satisfiesσ0+ε<1. LetY(s)=∑∞n=0ynsnand the definition ofA(s)， which is well defined and finite at least on [0,1). From (26), we obtain

H(s)B(s)=(1-A(s))Y(s).

(27)

Now we see that bothH(s) andY(s) are well defined and finite at least on [0,σ0+ε)， and from the Lemma 2.1 we can know theA(s)<0 is always true. Lettings=σ0in (27) yieldsH(σ0)B(σ0)=(1-A(σ0))Y(σ0). That factsB(σ0)=0 andH(σ0)<∞， and 1-A(σ0)<0 imply thatY(σ0)=0， which in turn impliesyn≡0 , which leads to a contradiction.

We now summarize our main conclusions concerning uniqueness.

Theorem 3.5There is exactly one GIBI if any one of the following holds:

(i)B′(1)≤0 .

(ii)Qis slowly-exit and 0

4 Extinction

We now turn to considering the extinction behaviour and hitting times regarding GIBI.

Let {X(t),t≥0} be the corresponding GIBI which is the Feller minimal process and letP(t)=(pij(t);i,j∈Z+) denote its transition function. Define the extinction timesτkfork=0,1,…,m-1 as

It is clear thatτ=min{τ0,…,τm-1} is the overall extinction time. For alli>mandk≤m-1， define the individual extinction probabilities by

(28)

and the overall extinction probability byai=P(τ<∞|X(0)=i)=∑m-1k=0aik.

Lemma 4.1The quantities (aik,k=0,1,…,m-1) given in (28) satisfy the relationship

ai0+ai1σk+…+ai,m-1σm-1k=σik,k=0,1,…,m-1,

(29)

We now prove (29). By (13), we have

(30)

First assumeσ0=1. Then by Lemma 2.1(i), we knowB(s)>0 for alls∈(0,1) and hence for alls∈[0,1)，

Lettingt→∞ in the above equality ,and we have A(1)=0, then notingj≥mare transient yields ∑m-1j=0aij≥1. Since ∑m-1j=0aij≤1 is always true, we then have ∑m-1j=0aij=1， which shows that (29) holds fork=0.

Remark 4.2By Lemma 4.1, the extinction probability vectorATi=(ai0,ai1,…,aim-1) satisfies (for eachi) the matrix equationVAi=Bi， whereBTi=(σi0,…,σim-1)and the coefficient matrixVis independent ofiand is the well-known Vandermonde matrix and thus, if themzeros 1,σ0,…,σm-1ofB(z) are different as we have assumed, then the Vandermonde determinant ofVis non-zero and therefore the extinction probability vector can be uniquely determined. Note that if there does exist some zeros ofB(z) which are equal, then it is easy to see how to obtain the solution by some trivial amendments and thus our assumption that all zeros ofB(z) are different isindeed without loss any generality.

For anyi≥m， define

(31)

It is obvious thatGm(z)=G(z). By (30),Gi(z) is analytic inDand thus can be expanded as a Taylor series

Lemma 4.3Let (pij(t);i,j∈Z+) be the minimal transition function.

(i) For anyi,k≥2，

(32)

(33)

and hence

(34)

ProofIt follows from (13) that

Finally, suppose that ∑∞k=m(gk-m/ωk)<∞. From the remarks made preceding the statement of Lemma 4.3, we have

for anyi≥m. Therefore (32) implies (33). In particular, if ∑∞k=m(1/ωk)<∞， then ∑∞k=m(gk-m/ωk)<∞ since {gk} is bounded. Finally, (33) implies (34) is trivial and hence the proof is complete.

Theorem 4.4Let (pij(t),i,j∈Z+) be the Feller minimal Q-function.

(i) For anyi≥m， the extinction probabilities (aik,k=0,1,…,m-1) are given by

ai0+ai1σk+…+ai,m-1σm-1k=σik,k=0,1,…,m-1,

(35)

(ii) IfB′(1)≤0， thenai=∑m-1i=0aik=1， while if 0

ProofBy noting (28), we see that this theorem is nothing but re-statement of Lemma 4.1. Finally, the last conclusion can be easily obtained.

Theorem 4.5For the Feller minimal GIBI,Ei(τ) is finite for some (and then all)i≥mif and only ifB′(1)≤0 and in such case

More specifically,

(ii) if 0

ProofIt is easily seen from Theorem 4.5 and Lemma 2.1 that if 0

It is clear that when the extinction is not certain thenEi(τk)=+∞(i≥2,k=0,1). Under these circumstances it is natural to consider the conditional expected extinction times, given byEi(τk|τk<∞)=μik/aik， whereμik=Ei(τkI{τk<∞})(k=0,1).

Theorem 4.6The following statements hold for the Feller minimal GIBI starting in stateiwherei≥m.

Ei(τk|τk<∞)=a-1ik·μik,k≤m-1,i≥m,

whereμik=Ei(τkI{τk<∞}) and satisfy the linear equations

(36)

Ei(τk|τk<∞)=a-1ik·μik,k≤m-1,i≥m,

whereμiksatisfy the linear equations (36).

ProofWe prove (ii) first. Since 0

(37)

On integrating (37) and usingaik-pik(t)=P(t<τk<∞|X(0)=i)(k≤m-1)， we find that

Noting that |σj|σ0<1， lettingt→∞ and using the monotone convergence theorem and dominated convergence theorem and noting (32) gives (36). By Remark 4.2, we see thatS<∞ is equivalent to

Therefore, it can be easily seen by using some algebra thatμik<∞(k≤m-1) if and only ifS<∞.

Now we prove (i). SinceB′(1)≤0，P(τ<∞|X(0)=i)=1. It follows from Theorem 4.4 thatai=1， and so the ensuing honesty of the transition function allows us to deduce that ∑m-1k=0(aik-pik(t))=∑∞k=mpik(t). Noting |σj|<1 forj=1,…,m-1 we know that (36) still holds forj=1,…,m-1 in this case. Hence, we still have (35) withσ0=1. A similar argument yields the required conclusions. The proof is complete.

5 Explosion

We now turn our attention to evaluating the explosion probabilities and expected explosion times. In view of Theorem 3.1, we only need to consider the case that 0

Theorem 5.1The following statements hold for the minimal process starting in stateiwherei≥m.

(i) IfB′(1)≤0， thenai∞=0.

(38)

ProofIfB′(1)≤0， then the minimal process is honest and henceai∞=0. Under the conditions of (ii), it follows from Theorem 3.1 that the minimal process is dishonest and thuspi∞(t)=1-∑∞j=0pij(t)>0. Our expression forai∞follows on lettingt→∞ and using (34) together with Theorem 4.4. Next, we write

Integrating this with respect tot， and noting thatP(τ∞≤t|τ∞<∞,X(0)=i)=pi∞(t)/ai∞， yields (38).

Finally, we consider the following question. Suppose a GIBI starts at statei≥1， then before the final extinction or explosion, the process will “enjoy” its life in wandering over all the positive states. We are now interested in obtaining the overall mean holding time at each positive statek(≥m)， since it provides important and very useful information regarding the evolution behaviour of the GIBI. Let us agree to call them the global holding times. More specifically, letTkbe thetotal time spent in statek(≥m) and letμik=Ei(Tk)(i≥m). Then

denote the mean global holding time at statekbefore extinction or explosion. Clearlyμi=∑∞j=1μijis the mean total survival time of the GIBI when the process starts at statek(≥m). However, the solutions to this question have been implied by our previous work and thus we only need to summarize them here.

Theorem 5.2Suppose the Feller minimal GIBI starts at statei≥m.

(i) For anyk≥m， the mean global holding time at statekis always finite and given by

(ii) The mean total survival time of the GIBI is finite if and only ifSis convergent and, in which case, it is given by

ProofThe results follow directly from Theorem 4.3, particularly (32) and (33), and the proof of Theorem 4.5.