The Down/Up crossing Properties of Weighted Markov Branching Processes

YANYUN LI ∗, Central South University

JUNPING LI ∗∗, Central South University

111${}^{*}\,$Postal address: School of Mathematics and Statistics, Central
South University, Changsha, 410083, China. E-mail address:
2310962168@qq.com222${}^{**}\,$Postal address: School of Mathematics and
Statistics, Central South University, Changsha, 410083, China. E-mail address:
<EMAIL_ADDRESS>
###### Abstract

We consider the down/up crossing property of weighted Markov branching
processes . The joint probability distribution of multi crossing numbers of
such processes are obtained by using a new method. In particular, for Markov
branching processes, the probability distribution of death number is given and
for bulk-arrival queueing model, the joint probability distributions of
service number and arrival number is also given.

Keywords: Weighted Markov branching process; Down crossing; Up crossing; joint
probability distribution.

AMS 2000 SUBJECT CLASSIFICATION: PRIMARY 60J27
AMS 2000 SUBJECT CLASSIFICATION:  SECONDARY 60J35

1\. Introduction

The ordinary Markov branching processes (MBPs) play an important role in the
classical field of stochastic processes. Some related references are
Harris[10], Athreya and Ney[5], Asmussen and Hering[3].

The basic property governing the evolution of an MBP is the branching
property, different particles act independently when giving birth or death. In
most realistic situation, however, the independence property is unlikely to be
realized. Indeed, particles usually interact with each other. This is the main
reason why there always have been an increasing effort to generalise the
ordinary branching processes to more general branching models (see, for
instance, Athreya and Jagers [2]). Models like this were first studied by
Sevast’yanov [17]. Some authors, including Vatutin [19], Li $\&$ Chen[11] and
Li, Chen $\&$ Pakes [13] considered the branching process with state-
independent immigration. Moreover, Li $\&$ Liu [14] added state-independent
migration to the above branching process. Yamazato [20] investigated a
branching process with immigration which only occurs at state zero. Being
viewed as an extension of Yamazato’s model, Chen [16] discussed a more general
branching process with or without resurrection. For the further discussion of
this model, see Chen [6] [7], Chen, Li $\&$ Ramesh[8] and Chen, Pollett, Zhang
$\&$ Li [9] considered weighted Markov branching process. Within this
structure, Chen, Li and Ramesh [8] considered the uniqueness and extinction of
Weighted Markov Branching Processes, which is the further consideration of
branching models discussed in Chen[7].

Since zero is the absorbing state in the branching structure, it is nature and
also important to consider the crossing property of such process, i.e., for
some fixed $m(\neq 0)$, what is the probability distribution of $m$-range
crossing number for the process until its extinction? In particular, the
$-1$-range crossing number is just the death number for the process until its
extinction. The aim of this paper is to consider such problems for weighted
Markov branching processes.

In order to start our discussion, we first define our model by specifying the
infinitesimal generator, i.e., the so-called $Q$-matrix. Throughout this
paper, let $\mathbb{Z}_{+}=\\{0,1,2,\cdots\\}$.

######  Definition 1..1.

A $Q$-matrix $Q=(q_{ij};\ i,j\in\mathbb{Z}_{+})$ is called a weighted
branching $Q$-matrix ( henceforth referred to as a WMB $Q$-matrix), if

$\displaystyle q_{ij}=\begin{cases}w_{i}b_{j-i+1},\ if\ i\geq 1,j\geq i-1\\\
0,\ \ \ \ \ \mbox{otherwise}\end{cases}$ (1.1)

where

$\displaystyle b_{j}\geq 0\ (j\neq 1),\ 0<-b_{1}=\sum_{j\neq 1}b_{j}<\infty,\
w_{i}>0\ (i\geq 1).$ (1.2)

######  Definition 1..2.

A weighted Markov branching process ( henceforth referred to as a WMBP) is a
continuous-time Markov chain with state space $\mathbb{Z}_{+}$ whose
transition function $P(t)=(p_{ij}(t);$ $i,j\in\mathbb{Z}_{+})$ satisfies

$\displaystyle p^{\prime}_{ij}(t)=\sum_{k=0}^{\infty}p_{ik}(t)q_{kj},\ i\geq
0,\ j\geq 1,\ t\geq 0,$ (1.3)

where $Q=(q_{ij};\ i,j\in\mathbb{Z}_{+})$ is defined in (1.1)-(1.2).

Chen, Li $\&$ Ramesh (2005)[8] derived the regularity conditions for WMBPs in
terms of the death rate $b_{0}$ and birth rate
$m_{b}=\sum_{j=1}^{\infty}jb_{j+1}$. Therefore, we assume the process is
regular in this paper.

2\. Preliminaries

Let $\mathbb{N}\subset\mathbb{Z}_{+}$ be a finite subset with
$1\notin\mathbb{N}$ and $b_{k}>0$ for all $k\in\mathbb{N}$. The number of
elements in $\mathbb{N}$ is denoted by $N$, i.e., $N=|\mathbb{N}|$. We
consider the $(\mathbb{N}-1)$-range crossing number of the process until its
extinction, i.e., the probability distribution of the $N$-dimensional random
vector $(N_{i};\ i\in\mathbb{N})$, where $N_{i}$ is the $(i-1)$-range crossing
number of the process until its extinction.

In order to begin our discussion, define

$\displaystyle B(u)=\sum_{j=0}^{\infty}b_{j}u^{j}$ (2.1)

and

$\displaystyle
B_{\mathbb{N}}(u,\emph{{v}})=\sum_{j\in\mathbb{N}}b_{j}u^{j}v_{j},\ \ \
\bar{B}_{\mathbb{N}}(u)=\sum_{j\in\mathbb{N}^{c}}b_{j}u^{j}$ (2.2)

where $\emph{{v}}=(v_{j};\ j\in\mathbb{N})$. It is obvious that $B(u),\
\bar{B}_{\mathbb{N}}(u)$ are well defined at least on $[0,1]$, and
$B_{\mathbb{N}}(u,\emph{{v}})$ is well defined at least on $[0,1]^{N+1}$.

The following lemma is due to mathematical analysis and thus the proof is
omitted here.

######  Lemma 2..1.

Suppose that $\\{f_{\textbf{k}};\ \textbf{k}\in\mathbb{Z}_{+}^{N}\\}$ is a
sequence on $\mathbb{Z}_{+}^{N}$,
$F(\textbf{v})=\sum_{\textbf{k}\in\mathbb{Z}_{+}^{N}}\textbf{v}^{\textbf{k}}$
is the generating function of $\\{f_{\textbf{k}};\
\textbf{k}\in\mathbb{Z}_{+}^{N}\\}$. Then for any $j\in\mathbb{Z}_{+}$,

$\displaystyle
F^{j}(\textbf{v})=\sum_{\textbf{l}\in\mathbb{Z}_{+}^{N}}f^{*(j)}_{\textbf{l}}\textbf{v}^{\textbf{l}}$
(2.3)

where

$\displaystyle f^{*(0)}_{\textbf{0}}=1,\ f^{*(0)}_{\textbf{l}}=0\
(\textbf{l}\neq\textbf{0}),\ \
f^{*(j)}_{\textbf{l}}=\sum_{\textbf{k}^{(1)}+\cdots+\textbf{k}^{(j)}=\textbf{l}}f_{\textbf{k}^{(1)}}\cdots
f_{\textbf{k}^{(j)}},\ (j\geq 1)$

is the $j$’th convolution of $\\{f_{\textbf{k}};\
\textbf{k}\in\mathbb{Z}_{+}^{N}\\}$.

The function $\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\emph{{v}})$ will plays
a key role in our discussion. The following theorem reveals its properties.

######  Theorem 2..2.

(i) For any $\textbf{v}\in[0,1]^{N+1}$,

$\displaystyle\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\textbf{v})=0$ (2.4)

possesses at most $2$ roots in $[0,1]$. The minimal nonnegative root of
$\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\textbf{v})=0$ is denoted by
$\rho(\textbf{v})$, then $\rho(\textbf{v})\leq\rho$, where $\rho$ is the
minimal nonnegative root of $B(u)=0$.

(ii) $\rho(\textbf{v})\in C^{\infty}([0,1)^{N})$ and $\rho(\textbf{v})$ can be
expanded as a multivariate Taylor sieris

$\displaystyle\rho(\textbf{v})=\sum_{\textbf{k}\in\mathbb{Z}_{+}^{N}}\rho_{\textbf{k}}\textbf{v}^{\textbf{k}}.$
(2.5)

where $\rho_{\textbf{k}}\geq 0,\forall\ \textbf{k}\in\mathbb{Z}_{+}^{N}$.

######  Proof 2..3.

Note that $0\leq B_{\mathbb{N}}(u,\emph{{0}})\leq
B_{\mathbb{N}}(u,\emph{{v}})\leq B_{\mathbb{N}}(u,\emph{{1}})$, we know that

$\displaystyle\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\emph{{v}})\leq B(u).$

(i) follows from Li and Chen [12]. Next to prove (ii). Without loss of
generality, suppose that $\mathbb{N}=\\{2,3\\}$ and thus

$\displaystyle B_{\mathbb{N}}(u,\emph{{v}})=B_{\mathbb{N}}(u,v_{2},v_{3}).$

Denote $f(u,v_{2},v_{3})=B_{\mathbb{N}}(u,v_{2},v_{3})$ and
$g(u)=\bar{B}_{\mathbb{N}}(u)$. Then,
$g(\rho(v_{2},v_{3}))+f(\rho(v_{2},v_{3}),v_{2},v_{3})\equiv 0$ and hence

$\displaystyle\begin{cases}g^{\prime}_{u}(\rho(v_{2},v_{3}))\cdot\rho^{\prime}_{v_{2}}(v_{2},v_{3})+f^{\prime}_{u}(\rho(v_{2},v_{3}),v_{2},v_{3})\cdot\rho^{\prime}_{v_{2}}(v_{2},v_{3})+f^{\prime}_{v_{2}}(\rho(v_{2},v_{3}),v_{2},v_{3})\equiv
0\\\
g^{\prime}_{u}(\rho(v_{2},v_{3}))\cdot\rho^{\prime}_{v_{3}}(v_{2},v_{3})+f^{\prime}_{u}(\rho(v_{2},v_{3}),v_{2},v_{3})\cdot\rho^{\prime}_{v_{3}}(v_{2},v_{3})+f^{\prime}_{v_{3}}(\rho(v_{2},v_{3}),v_{2},v_{3})\equiv
0\end{cases}$

which implies that $\rho^{\prime}_{v_{2}}$ and $\rho^{\prime}_{v_{3}}$ are
well defined in $[0,1)^{2}$ since

$\displaystyle
g^{\prime}_{u}(\rho(v_{2},v_{3}))+f^{\prime}_{u}(\rho(v_{2},v_{3}),v_{2},v_{3})<g^{\prime}_{u}(\rho)+f^{\prime}_{u}(\rho,1,1)=B^{\prime}(\rho)\leq
0$

Similarly, by induction recursion, we know that $\rho(v_{2},v_{3})\in
C^{\infty}([0,1)^{2})$. Now suppose that

$\displaystyle\rho(\emph{{v}})=\sum_{\emph{{k}}\in\mathbb{Z}_{+}^{N}}\rho_{\emph{{k}}}\emph{{v}}^{\emph{{k}}}.$

Substituting above expression into (2.4) yields

$\displaystyle 0$ $\displaystyle\equiv$
$\displaystyle\bar{B}_{\mathbb{N}}(\rho(\emph{{v}}))+B_{\mathbb{N}}(\rho(\emph{{v}}),\emph{{v}})$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{N}^{c}}b_{j}(\rho(\emph{{v}}))^{j}+\sum_{j\in\mathbb{N}}b_{j}(\rho(\emph{{v}}))^{j}v_{j}$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{N}^{c}}b_{j}\sum_{\emph{{l}}\geq\emph{{0}}}\rho^{*(j)}_{\emph{{l}}}\emph{{v}}^{\emph{{l}}}+\sum_{j\in\mathbb{N}}b_{j}\sum_{\emph{{l}}\geq\emph{{0}}}\rho^{*(j)}_{\emph{{l}}}\emph{{v}}^{\emph{{l}}+\emph{{e}}_{j}}$
$\displaystyle=$
$\displaystyle\sum_{\emph{{l}}\geq\emph{{0}}}(\sum_{j\in\mathbb{N}^{c}}b_{j}\rho^{*(j)}_{\emph{{l}}})\emph{{v}}^{\emph{{l}}}+\sum_{j\in\mathbb{N}}b_{j}\sum_{\emph{{l}}\geq\emph{{0}}}\rho^{*(j)}_{\emph{{l}}}\emph{{v}}^{\emph{{l}}+\emph{{e}}_{j}}.$

We next prove $\rho_{\emph{{l}}}\geq 0$ using mathematical induction respect
to $\emph{{l}}\cdot\emph{{1}}$. If $\emph{{l}}\cdot\emph{{1}}=0$, then
$\rho_{\emph{{0}}}=\rho(\emph{{0}})\geq 0$ since it is the minimal nonnegative
root of $\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\emph{{0}})=0$. If
$\emph{{l}}\cdot\emph{{1}}=1$, i.e., $\emph{{l}}=\emph{{e}}_{k}$ for some
$k\in\mathbb{N}$. Then,

$\displaystyle\sum_{j\in\mathbb{N}^{c}}b_{j}\rho^{*(j)}_{\emph{{e}}_{k}}+b_{k}\rho^{*(k)}_{\emph{{0}}}=0$

i.e.,

$\displaystyle\sum_{j\in\mathbb{N}^{c}}b_{j}j\rho^{j-1}_{\emph{{0}}}\rho_{\emph{{e}}_{k}}+b_{k}\rho^{k}_{\emph{{0}}}=0$

hence

$\displaystyle\rho_{\emph{{e}}_{k}}=-\frac{b_{k}\rho^{k}_{\emph{{0}}}}{\bar{B}^{\prime}_{\mathbb{N}}(\rho_{\emph{{0}}})}\geq
0,\ \ k\in\mathbb{N}$

since $\bar{B}^{\prime}_{\mathbb{N}}(\rho_{\emph{{0}}})<0$.

Assume $\rho_{\emph{{l}}}\geq 0$ for l satisfying
$\emph{{l}}\cdot\emph{{1}}\leq m$, then for
$\bar{\emph{{l}}}\cdot\emph{{1}}=m+1$, there exists l and $k\in\mathbb{N}$
such that $\bar{\emph{{l}}}=\emph{{l}}+\emph{{e}}_{k}$ and
$\emph{{l}}\cdot\emph{{1}}\leq m$, therefore,

$\displaystyle\sum_{j\in\mathbb{N}^{c}}b_{j}\rho^{*(j)}_{\emph{{l}}+\emph{{e}}_{k}}+b_{k}\rho^{*(k)}_{\emph{{l}}}=0$

i.e.,

$\displaystyle\sum_{j\in\mathbb{N}^{c}}b_{j}j\rho^{j-1}_{\emph{{0}}}\rho_{\emph{{l}}+\emph{{e}}_{k}}+\sum_{j\in\mathbb{N}^{c}\setminus\\{1\\}}b_{j}\sum_{\emph{{l}}^{(1)}+\cdots+\emph{{l}}^{(j)}=\emph{{l}}+\emph{{e}}_{k},\
\emph{{l}}^{(1)}\cdot\emph{{1}},\cdots,\emph{{l}}^{(j)}\cdot\emph{{1}}\leq
m}\rho_{\emph{{l}}^{(1)}}\cdots\rho_{\emph{{l}}^{(j)}}+b_{k}\rho^{*(k)}_{\emph{{l}}}=0.$

Hence

$\displaystyle\rho_{\bar{\emph{{l}}}}=\rho_{\emph{{l}}+\emph{{e}}_{k}}=-\frac{\sum_{j\in\mathbb{N}^{c}\setminus\\{1\\}}b_{j}\sum_{\emph{{l}}^{(1)}+\cdots+\emph{{l}}^{(j)}=\emph{{l}}+\emph{{e}}_{k},\
\emph{{l}}^{(1)}\cdot\emph{{1}},\cdots,\emph{{l}}^{(j)}\cdot\emph{{1}}\leq
m}\rho_{\emph{{l}}^{(1)}}\cdots\rho_{\emph{{l}}^{(j)}}+b_{k}\rho^{*(k)}_{\emph{{l}}}}{\bar{B}^{\prime}_{\mathbb{N}}(\rho_{\emph{{0}}})}\geq
0$

since $\bar{B}^{\prime}_{\mathbb{N}}(\rho_{\emph{{0}}})<0$. By mathematical
induction, we know that $\rho_{\emph{{l}}}\geq 0,\forall\
\emph{{l}}\in\mathbb{Z}_{+}^{N}$. $\Box$

3\. Down/up crossing property

In this section, we consider the down/up crossing properties of weighted
Markov branching process.

Let $\mathbb{N}\subset\mathbb{Z}_{+}$ be a finite subset with
$1\notin\mathbb{N}$ and $b_{k}>0$ for all $k\in\mathbb{N}$. $N=|\mathbb{N}|$
denotes the number of elements in $\mathbb{N}$.

Since our main purpose is to count the $(\mathbb{N}-1)$-range crossing
numbers, we use a new method to discuss such crossing numbers, i.e., consider
a new $Q$-matrix $\tilde{Q}=(q_{(i,\emph{{k}}),(j,\emph{{l}})};\
(i,\emph{{k}}),$ $(j,\emph{{l}})\in\mathbb{Z}_{+}^{N+1})$.

$\displaystyle q_{(i,\emph{{k}}),(j,\emph{{l}})}=\begin{cases}w_{i}b_{j-i+1},\
\ \ if\ i\geq 1,j\geq i-1,j-i+1\in\mathbb{N}^{c}\\\ w_{i}b_{j-i+1},\ \ \ if\
i\geq 1,\ j\geq i-1,\ \emph{{l}}=\emph{{k}}+\emph{{e}}_{j-i+1},\
j-i+1\in\mathbb{N}\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ otherwise.\end{cases}$ (3.1)

Therefore, $\tilde{Q}$ determines a $(N+1)$-dimensional Markov chain
$(X(t),\emph{{Y}}(t))$, where $X(t)$ is the weighted Markov branching process,
$\emph{{Y}}(t)=(Y_{k}(t);k\in\mathbb{N})$ (assume $Y_{k}(0)=0\
(k\in\mathbb{N})$) counts the $(\mathbb{N}-1)$-range crossing numbers until
time $t$. In particular,

(i) if $\mathbb{N}=\\{0\\}$ then $Y_{0}(t)$ counts the down crossing number
(i.e., the death number) of $\\{X(t):t\geq 0\\}$ until time $t$.

(ii) If $\mathbb{N}=\\{m\\}\ (m\geq 2)$, then $Y_{m}(t)$ counts the
$(m-1)$-range up crossing number of $\\{X(t):t\geq 0\\}$ until time $t$.

(iii) If $\mathbb{N}=\\{0,m\\}\ (m\geq 2)$, then
$\emph{{Y}}(t)=(Y_{0}(t),Y_{m}(t))$ counts the death number and the
$(m-1)$-range up corssing number of $\\{X(t):t\geq 0\\}$ until time $t$.

Let
$P(t)=(p_{(i,\emph{{k}}),(j,\emph{{l}})}(t);(i,\emph{{k}}),(j,\emph{{l}})\in\mathbb{Z}_{+}^{N+1})$
denote the transition probability of $(X(t),\emph{{Y}}(t))$.

######  Lemma 3..1.

For $P(t)$, we have

$\displaystyle\sum_{(j,\textbf{l})\in\mathbb{Z}_{+}^{N+1}}p^{\prime}_{(i,\textbf{0}),(j,\textbf{l})}(t)u^{j}\textbf{v}^{\textbf{l}}$
(3.2) $\displaystyle=$
$\displaystyle[\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\textbf{v})]\cdot\sum_{j\geq
1,\textbf{k}\in\mathbb{Z}_{+}^{N}}p_{(i,\textbf{0}),(j,\textbf{l})}(t)w_{j}u^{j-1}\textbf{v}^{\textbf{l}}$

where $\bar{B}_{\mathbb{N}}(u),\ B_{\mathbb{N}}(u,\textbf{v})$ are defined in
$(\ref{eq2.2})$,
$\textbf{v}^{\textbf{l}}=\prod_{k\in\mathbb{N}}v_{k}^{l_{k}}$. Moreover,

$\displaystyle\sum_{(j,\textbf{l})\in\mathbb{Z}_{+}^{N+1}}p_{(i,\textbf{0}),(j,\textbf{l})}(t)u^{j}\textbf{v}^{\textbf{l}}-u^{i}$
(3.3) $\displaystyle=$
$\displaystyle[\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\textbf{v})]\cdot\sum_{j\geq
1,\textbf{k}\in\mathbb{Z}_{+}^{N}}(\int_{0}^{t}p_{(i,\textbf{0}),(j,\textbf{k})}(t)dt)\cdot
w_{j}u^{j-1}\textbf{v}^{\textbf{k}}.$

######  Proof 3..2.

It follows from Kolmogorov forward equation and some algebra. $\Box$

Let

$\displaystyle\tau=\inf\\{t\geq 0;\ X(t)=0\\}$ (3.4)

be the extinction time of $\\{X(t);t\geq 0\\}$.

The following theorem gives the probability generating function of
$(\mathbb{N}-1)$-crossing numbers conditioned on $\tau<\infty$.

######  Theorem 3..3.

Suppose that $\\{X(t);t\geq 0\\}$ is a weighted Markov branching process with
$X(0)=1$. Then the probability generating function $G(\textbf{v})$ of
$(\mathbb{N}-1)$-range crossing numbers conditioned on $\tau<\infty$ is given
by

$\displaystyle
G(\textbf{v})=\rho^{-1}\sum_{\textbf{l}\in\mathbb{Z}_{+}^{N}}\rho_{\textbf{l}}\textbf{v}^{\textbf{l}},\
\ \textbf{v}\in[0,1]^{N}$ (3.5)

where $\rho$ is the minimal nonnegative root of $B(u)=0$, $\rho_{\textbf{0}}$
is the minimal nonnegative root of $\bar{B}_{\mathbb{N}}(u)=0$ and
$\rho_{\textbf{l}}\ (\textbf{l}\neq\textbf{0})$ are given by the following
recursion.

$\displaystyle\rho_{\textbf{l}+\textbf{e}_{k}}=-\frac{\sum_{j\in\mathbb{N}^{c}\setminus\\{1\\}}b_{j}\sum_{\textbf{l}^{(1)}+\cdots+\textbf{l}^{(j)}=\textbf{l}+\textbf{e}_{k},\
\textbf{l}^{(1)}\cdot\textbf{1},\cdots,\textbf{l}^{(j)}\cdot\textbf{1}\leq\textbf{l}\cdot\textbf{1}}\rho_{\textbf{l}^{(1)}}\cdots\rho_{\textbf{l}^{(j)}}+b_{k}\rho^{*(k)}_{\textbf{l}}}{\bar{B}^{\prime}_{\mathbb{N}}(\rho_{\textbf{0}})},\
\ k\in\mathbb{N}.$ (3.6)

######  Proof 3..4.

Let $\tilde{Q}=(q_{(i,\emph{{k}}),(j,\emph{{l}})};\ (i,\emph{{k}}),$
$(j,\emph{{l}})\in\mathbb{Z}_{+}^{N+1})$ be the $Q$-matrix defined in (3.1)
and $(X(t),\emph{{Y}}(t))$ be the $\tilde{Q}$-process. Then
$\emph{{Y}}(t)=(Y_{k}(t);k\in\mathbb{N})$ counts the $(\mathbb{N}-1)$-range
crossing numbers until time $t$ and
$\emph{{Y}}(\tau)=(Y_{k}(\tau);k\in\mathbb{N})$ counts the
$(\mathbb{N}-1)$-range crossing numbers conditioned on $\tau<\infty$.

Let
$P(t)=(p_{(i,\emph{{k}}),(j,\emph{{l}})}(t);(i,\emph{{k}}),(j,\emph{{l}})\in\mathbb{Z}_{+}^{N+1})$
denote the transition probability of $(X(t),\emph{{Y}}(t))$. It follows from
Lemma 3..1 that

$\displaystyle\sum_{(j,\emph{{l}})\in\mathbb{Z}_{+}^{N+1}}p_{(1,\emph{{0}}),(j,\emph{{l}})}(t)u^{j}\emph{{v}}^{\emph{{l}}}-u$
(3.7) $\displaystyle=$
$\displaystyle[\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\emph{{v}})]\cdot\sum_{j\geq
1,\emph{{k}}\in\mathbb{Z}_{+}^{N}}(\int_{0}^{t}p_{(1,\emph{{0}}),(j,\emph{{k}})}(s)ds)\cdot
w_{j}u^{j-1}\emph{{v}}^{\emph{{k}}}.$

Letting $t\rightarrow\infty$ on both sides of (3.7) and noting that
$\lim_{t\rightarrow\infty}p_{(1,\emph{{0}}),(j,\emph{{l}})}(t)=0$ for $j\geq
1$ (since $(j,\emph{{l}})$ are transient states for $j\geq 1$) yield

$\displaystyle\sum_{\emph{{l}}\in\mathbb{Z}_{+}^{N}}a_{\emph{{l}}}\emph{{v}}^{\emph{{l}}}-u=[\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,\emph{{v}})]\cdot\sum_{j\geq
1,\emph{{k}}\in\mathbb{Z}_{+}^{N}}(\int_{0}^{\infty}p_{(1,\emph{{0}}),(j,\emph{{k}})}(t)dt)\cdot
w_{j}u^{j-1}\emph{{v}}^{\emph{{k}}}.$ (3.8)

where
$a_{\emph{{l}}}=\lim_{t\rightarrow\infty}p_{(1,\emph{{0}}),(0,\emph{{l}})}(t)$.
By Theorem 2..2, let $u=\rho(\emph{{v}})$ in (3.8), we obtain that

$\displaystyle\sum_{\emph{{l}}\in\mathbb{Z}_{+}^{N}}a_{\emph{{l}}}\emph{{v}}^{\emph{{l}}}-\rho(\emph{{v}})\equiv
0$

which implying $a_{\emph{{l}}}=\rho_{\emph{{l}}}\
(\emph{{l}}\in\mathbb{Z}_{+}^{N})$.

Finally,

$\displaystyle P(\emph{{Y}}(\tau)=\emph{{l}}|\tau<\infty)$ $\displaystyle=$
$\displaystyle\frac{P(\emph{{Y}}(\tau)=\emph{{l}},\tau<\infty)}{P(\tau<\infty)}$
$\displaystyle=$
$\displaystyle\rho^{-1}\lim_{t\rightarrow\infty}P(\emph{{Y}}(t)=\emph{{l}},\tau<t)$
$\displaystyle=$
$\displaystyle\rho^{-1}\lim_{t\rightarrow\infty}p_{(1,\emph{{0}}),(0,\emph{{l}})}(t)$
$\displaystyle=$ $\displaystyle\rho^{-1}\rho_{\emph{{l}}}.$

The proof is complete. $\Box$

######  Remark 3..5.

Theorem 3..3 gives the probability generating function of
$(\mathbb{N}-1)$-crossing numbers conditioned on $\tau<\infty$. Therefore, the
joint probability distribution of $(\mathbb{N}-1)$-crossing numbers
$\emph{{Y}}(\tau)$ conditioned on $\tau<\infty$

$\displaystyle P(\emph{{Y}}(\tau)=\emph{{l}}\
|\tau<\infty)=\rho^{-1}\rho_{\emph{{l}}},\ \ \emph{{l}}\in\mathbb{Z}_{+}^{N}.$

If $X(t)$ starts from $X(0)=i(>1)$, then the probability generating function
of $(\mathbb{N}-1)$-crossing numbers conditioned on $\tau<\infty$ is

$\displaystyle G_{i}(\emph{{v}})=[G(\emph{{v}})]^{i}$

and the joint probability distribution of $(\mathbb{N}-1)$-crossing numbers
$\emph{{Y}}(\tau)$ conditioned on $\tau<\infty$

$\displaystyle P(\emph{{Y}}(\tau)=\emph{{l}}\
|\tau<\infty)=\rho^{-i}\rho^{*(i)}_{\emph{{l}}},\ \
\emph{{l}}\in\mathbb{Z}_{+}^{N}.$

As direct consequences of Theorem 3..3 and Remark 3..5, the following
corollaries gives the probability distribution of death number and
$(m-1)$-range up crossing number conditioned on $\tau<\infty$ for fixed $m>1$.

######  Corollary 3..6.

Suppose that $\\{X(t);t\geq 0\\}$ is a weighted Markov branching process with
$X(0)=1$. Then the probability generating function $G(v)$ of death number
conditioned on $\tau<\infty$ is given by

$\displaystyle G(v)=\rho^{-1}\sum_{k=0}^{\infty}\rho_{k}v^{k}$

and hence the probability distribution of death number $Y_{0}(\tau)$ is given
by

$\displaystyle P(Y_{0}(\tau)=k\ |\tau<\infty)=\rho^{-1}\rho_{k},\ \ k\geq 0$

where $\rho_{0}=0$ and $\rho_{k}\ (k\geq 1)$ are given by the following
recursion.

$\displaystyle\rho_{1}=-b_{1}^{-1}b_{0},$
$\displaystyle\rho_{k+1}=-b_{1}^{-1}\sum_{j=2}^{k+1}b_{j}\rho^{*(j)}_{k+1},\ \
k\geq 1.$

######  Proof 3..7.

Note that $\mathbb{N}=\\{0\\}$, $\rho_{0}=0$ is the minimal nonnegative root
of

$\bar{B}_{\mathbb{N}}(u)=\sum_{j=1}^{\infty}b_{j}u^{j}=0$

and

$\displaystyle\sum_{j=2}^{\infty}b_{j}\sum_{l_{1}+\cdots+l_{j}=k+1,\
l_{1},\cdots,l_{j}\leq
k}\rho_{l_{1}}\cdots\rho_{l_{j}}=\sum_{j=2}^{k+1}b_{j}\rho^{*(j)}_{k+1}.$

By Theorem 3..3, we immediately obtain the result. $\Box$

######  Remark 3..8.

By checking the proof of Theorem 3..3, we find that the joint probability
distribution of crossing numbers conditioned on $\tau<\infty$ does not depend
on $\\{w_{i};\ i\geq 1\\}$, therefore, we have

(i) For Markov branching process, the probability distribution of death number
$Y_{0}(\tau)$ conditioned on $\tau<\infty$ is given by

$\displaystyle P(Y_{0}(\tau)=k\ |\tau<\infty)=\rho^{-1}\rho_{k},\ \ k\geq 0$

where $\rho_{k}\ (k\geq 0)$ are given in Corollary 3..6.

(ii) For $M^{X}/M/1$ queueing process, the probability distribution of service
number $Y_{0}(\tau)$ in a busy period is given by

$\displaystyle P(Y_{0}(\tau)=k\ |\tau<\infty)=\rho^{-1}\rho_{k},\ \ k\geq 0$

where $\rho_{k}\ (k\geq 0)$ are given in Corollary 3..6.

######  Corollary 3..9.

Suppose that $\\{X(t);t\geq 0\\}$ is a weighted Markov branching process with
$X(0)=1$ and $m(>1)$ is fixed. Then the probability generating function $G(v)$
of $(m-1)$-range up-crossing number conditioned on $\tau<\infty$ is given by

$\displaystyle G(v)=\rho^{-1}\sum_{k=0}^{\infty}\rho_{k}v^{k}$

and hence the probability distribution of $(m-1)$-range up-crossing number
$Y_{m}(\tau)$ conditioned on $\tau<\infty$ is given by

$\displaystyle P(Y_{0}(\tau)=k\ |\tau<\infty)=\rho^{-1}\rho_{k},\ \ k\geq 0$

where $\rho_{0}$ is the minimal nonnegative root of $B_{m}(u)=\sum_{j\neq
m}b_{j}u^{j}=0$ and $\rho_{k}\ (k\geq 1)$ are given by the following
recursion.

$\displaystyle\rho_{1}=-\frac{b_{m}\rho_{0}^{m}}{B^{\prime}_{m}(\rho_{0})},$
$\displaystyle\rho_{k+1}=-\frac{\sum_{i\neq
1,m}b_{i}\sum_{j_{1}+\cdots+j_{i}=k+1;\ j_{1},\cdots,j_{i}\leq
k}\rho_{j_{1}}\cdots\rho_{j_{i}}+b_{m}\rho^{*(m)}_{k}}{B^{\prime}_{m}(\rho_{0})},\
\ k\geq 1.$

######  Proof 3..10.

It follows directly from Theorem 3..3 with $\mathbb{N}=\\{m\\}$. $\Box$

######  Theorem 3..11.

Suppose that $\\{X(t);t\geq 0\\}$ is a weighted Markov branching process with
$X(0)=1$, $\rho=1$ (i.e., $B^{\prime}(1)\leq 0$), $m\neq 1$, $Y_{m}(\tau)$ is
the $(m-1)$-range crossing number. Then

$\displaystyle E[Y_{m}(\tau)]=\sum_{k=0}^{\infty}k\rho_{k}$

and

$\displaystyle
Var[(Y_{m}(\tau)]=\sum_{k=0}^{\infty}k^{2}\rho_{k}-(\sum_{k=0}^{\infty}k\rho_{k})^{2}$

where $\rho_{k}(k\geq 0)$ are given in Corollary 3..6 (in the case $m=0$) and
in Corollary 3..9 (in the case $m>1$).

######  Proof 3..12.

Since $\rho=1$, we know that $G(v)$ is the probability generating function of
$Y_{m}(\tau)$, therefore,

$\displaystyle E[Y_{m}(\tau)]=G^{\prime}(1^{-}),\ \
Var[(Y_{m}(\tau)]=G^{{}^{\prime\prime}}{(1^{-})}+G^{\prime}(1^{-})-[G^{\prime}(1^{-})]^{2}.$

The results follow from Corollaries 3..6 and 3..9. $\Box$

Theorem 3..3 gives the joint probability distribution of
$(\mathbb{N}-1)$-range crossing numbers conditioned on $\tau<\infty$. Now, we
consider the case $\tau=\infty$.

Let $m\in\mathbb{Z}_{+}$ with $b_{m}>0$ and $\tilde{Q}_{m}=(q_{(i,k),(j,l)};\
(i,k),(j,l)\in\mathbb{Z}_{+}^{2})$ be a $Q$-matrix defined by (3.1) with
$\mathbb{N}=\\{m\\}$. Suppose that $(X(t),Y_{m}(t))$ is the
$\tilde{Q}_{m}$-process, where $X(t)$ is the weighted Markov branching
process, $Y_{m}(t)$ counts the $(m-1)$-range crossing number until time $t$.
$P(t)=(p_{(i,k),(j,l)};\ (i,k),(j,l)\in\mathbb{Z}_{+}^{2})$ is the
$\tilde{Q}_{m}$-function.

######  Theorem 3..13.

Suppose that $(X(t),Y_{m}(t))$ is the $\tilde{Q}_{m}$-process with
$(X(0),Y(0))=(1,0)$ and $\rho<1$. Then

$\displaystyle P(Y_{m}(\infty)=\infty|\tau=\infty)=1.$ (3.9)

######  Proof 3..14.

It follows from Lemma 3..1 with $\mathbb{N}=\\{m\\}$ that for any
$u,v\in[0,1]$,

$\displaystyle\sum_{(j,k)\in\mathbb{Z}_{+}^{2}}p_{(1,0),(j,k)}(t)u^{j}v^{k}-u=B_{m}(u,v)\cdot\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}(\int_{0}^{t}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}u^{j-1}v^{k}\ \ \ \ \ \ \ \ $

where $B_{m}(u,v)=\sum_{i\neq m}b_{i}u^{i}+b_{m}u^{m}v$. i.e.,

$\displaystyle\sum_{k=0}^{\infty}p_{(1,0),(0,k)}(t)v^{k}+\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}p_{(1,0),(j,k)}(t)u^{j}v^{k}-u$
(3.10) $\displaystyle=$ $\displaystyle
B_{m}(u,v)\cdot\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}(\int_{0}^{t}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}u^{j-1}v^{k}.$

Letting $t\rightarrow\infty$ in the above equality and noting
$\lim_{t\rightarrow\infty}\sum_{k=0}^{\infty}p_{(1,0),(0,k)}(t)v^{k}=\rho(v)$
yield

$\displaystyle\rho(v)-u=B_{m}(u,v)\cdot\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}(\int_{0}^{\infty}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}u^{j-1}v^{k},\ \ \ \forall\ u,v\in[0,1)$

where $\rho(v)$, the minimal nonnegative root of $B_{m}(u,v)=0$ for fixed
$v\in[0,1]$, is given in Corollary 3..9. Letting $u\uparrow 1$ and using
monotone convergence theorem yield

$\displaystyle
1-\rho(v)=b_{m}(1-v)\cdot\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}(\int_{0}^{\infty}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}v^{k},\ \ \ \forall\ v\in[0,1)$

which implies

$\displaystyle\sum_{j=1}^{\infty}(\int_{0}^{\infty}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}=b_{m}^{-1}\cdot(1-\sum_{i=0}^{k}\rho_{i}),\ \ \ k\geq 0$ (3.11)

and

$\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}(\int_{0}^{\infty}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}v^{k}<\infty,\ \ \ \ \forall\ v\in[0,1).$

On the other hand, letting $u\uparrow 1$ in (3.10) yields

$\displaystyle
1-\sum_{k=0}^{\infty}p_{(1,0),(0,k)}(t)v^{k}-\sum_{k=0}^{\infty}P(Y_{m}(t)=k,\tau>t)v^{k}$
$\displaystyle=$ $\displaystyle
b_{m}(1-v)\cdot\sum_{k=0}^{\infty}\sum_{j=1}^{\infty}(\int_{0}^{t}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}v^{k}$

Therefore,

$\displaystyle\sum_{j=1}^{\infty}(\int_{0}^{t}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}=b_{m}^{-1}\cdot[1-\sum_{i=0}^{k}p_{(1,0),(0,i)}(t)-P(Y_{m}(t)\leq
k,\tau>t)],\ \ \ k\geq 0$

Letting $t\rightarrow\infty$ and using monotone convergence theorem yield

$\displaystyle\sum_{j=1}^{\infty}(\int_{0}^{\infty}p_{(1,0),(j,k)}(s)ds)\cdot
w_{j}=b_{m}^{-1}\cdot[1-\sum_{i=0}^{k}\rho_{i}-P(Y_{m}(\infty)\leq
k,\tau=\infty)],\ \ \ k\geq 0.$

Comparing the above equality with (3.11), we see that

$\displaystyle P(Y_{m}(\infty)\leq k,\tau=\infty)=0,\ \ \ k\geq 0.$

Hence,

$\displaystyle P(Y_{m}(\infty)=\infty,\tau=\infty)=1.$

The proof is complete. $\Box$

Finally, we give an example to illustrate the conclusions obtained above.

######  Example 3..15.

Let $X(0)=1$ and

$\displaystyle B(u)=\mu-(\mu+\lambda)u+\lambda u^{2}$

and $\mathbb{N}=\\{0,2\\}$. Then

$\displaystyle B_{\mathbb{N}}(u,y,z)=\mu y+\lambda zu^{2},\ \
\bar{B}_{\mathbb{N}}(u)=-(\mu+\lambda)u.$

Consider

$\displaystyle\bar{B}_{\mathbb{N}}(u)+B_{\mathbb{N}}(u,y,z)=\mu
y-(\mu+\lambda)u+\lambda zu^{2}=0.$ (3.12)

The minimal nonnegative root of (3.12) is

$\displaystyle\rho(y,z)=\frac{\mu+\lambda-\sqrt{(\mu+\lambda)^{2}-4\mu\lambda
zy}}{2\lambda z},\ \ \rho=\rho(1,1)=\frac{\mu}{\lambda}\vee 1.$

Using Tailor series
$\sqrt{1+x}=1+\frac{1}{2}x+\sum_{n=2}^{\infty}\frac{(2n-3)!!}{2^{n}n!}(-1)^{n-1}x^{n}$
yields

$\displaystyle\rho(y,z)$ $\displaystyle=$
$\displaystyle\frac{\mu+\lambda}{2\lambda z}\left[1-\sqrt{1-\frac{4\mu\lambda
zy}{(\mu+\lambda)^{2}}}\right]$ $\displaystyle=$
$\displaystyle\frac{\mu+\lambda}{2\lambda z}\left[1-(1-\frac{4\mu\lambda
zy}{2(\mu+\lambda)^{2}}+\sum_{n=2}^{\infty}\frac{(2n-3)!!}{2^{n}n!}(-1)^{n-1}(-\frac{4\mu\lambda
zy}{(\mu+\lambda)^{2}})^{n})\right]$ $\displaystyle=$
$\displaystyle\frac{\mu+\lambda}{2\lambda z}\left[\frac{4\mu\lambda
zy}{2(\mu+\lambda)^{2}}+\sum_{n=2}^{\infty}\frac{(2n-3)!!}{2^{n}n!}(\frac{4\mu\lambda
zy}{(\mu+\lambda)^{2}})^{n}\right]$ $\displaystyle=$ $\displaystyle\frac{\mu
y}{\mu+\lambda}+\sum_{n=2}^{\infty}\frac{(2n-3)!!2^{n-1}\mu^{n}\lambda^{n-1}}{n!(\mu+\lambda)^{2n-1}}z^{n-1}y^{n}.$

Therefore,

$\displaystyle\begin{cases}\rho_{1,0}=\frac{\mu}{\mu+\lambda},\\\
\rho_{n,n-1}=\frac{(2n-3)!!2^{n-1}\mu^{n}\lambda^{n-1}}{n!(\mu+\lambda)^{2n-1}},\
\ n\geq 2\\\ \rho_{n,m}=0,\ \ \rm{otherwise}\end{cases}$

and hence,

(i) if $\mu\geq\lambda$, then

$\displaystyle\begin{cases}P((Y_{0}(\tau),Y_{2}(\tau))=(1,0))=\frac{\mu}{\mu+\lambda},\\\
P((Y_{0}(\tau),Y_{2}(\tau))=(n,n-1))=\frac{(2n-3)!!2^{n-1}\mu^{n}\lambda^{n-1}}{n!(\mu+\lambda)^{2n-1}},\
\ n\geq 2\\\ P((Y_{0}(\tau),Y_{2}(\tau))=(n,m))=0,\ \
\rm{otherwise},\end{cases}$ $\displaystyle\begin{cases}P(Y_{0}(\tau)=0)=0,\\\
P(Y_{0}(\tau)=1)=\frac{\mu}{\mu+\lambda},\\\
P(Y_{0}(\tau)=n)=\frac{(2n-3)!!2^{n-1}\mu^{n}\lambda^{n-1}}{n!(\mu+\lambda)^{2n-1}},\
\ n\geq 2,\end{cases}$
$\displaystyle\begin{cases}P(Y_{2}(\tau)=0)=\frac{\mu}{\mu+\lambda},\\\
P(Y_{2}(\tau)=n)=\frac{(2n-1)!!2^{n}\mu^{n+1}\lambda^{n}}{(n+1)!(\mu+\lambda)^{2n+1}},\
\ n\geq 1.\end{cases}$

(ii) if $\mu<\lambda$, then

$\displaystyle\begin{cases}P((Y_{0}(\tau),Y_{2}(\tau))=(1,0)\
|\tau<\infty)=\frac{\lambda}{\mu+\lambda},\\\
P((Y_{0}(\tau),Y_{2}(\tau))=(n,n-1)\
|\tau<\infty)=\frac{(2n-3)!!2^{n-1}\mu^{n-1}\lambda^{n}}{n!(\mu+\lambda)^{2n-1}},\
\ n\geq 2\\\ P((Y_{0}(\tau),Y_{2}(\tau))=(n,m)\ |\tau<\infty)=0,\ \
\rm{otherwise},\end{cases}$ $\displaystyle\begin{cases}P(Y_{0}(\tau)=0\
|\tau<\infty)=0,\\\ P(Y_{0}(\tau)=1\
|\tau<\infty)=\frac{\lambda}{\mu+\lambda},\\\ P(Y_{0}(\tau)=n\
|\tau<\infty)=\frac{(2n-3)!!2^{n-1}\mu^{n-1}\lambda^{n}}{n!(\mu+\lambda)^{2n-1}},\
\ n\geq 2,\end{cases}$ $\displaystyle\begin{cases}P(Y_{2}(\tau)=0\
|\tau<\infty)=\frac{\lambda}{\mu+\lambda},\\\ P(Y_{2}(\tau)=n\
|\tau<\infty)=\frac{(2n-1)!!2^{n}\mu^{n}\lambda^{n+1}}{(n+1)!(\mu+\lambda)^{2n+1}},\
\ n\geq 1.\end{cases}$

######  Example 3..16.

Let $X(0)=1$ and

$\displaystyle B(u)=2q-3pu+u^{3}.$

For $v\in[0,1)$, consider

$\displaystyle B(u)=2qv-3pu+u^{3}.$ (3.13)

Let $\rho(v)$ be the minimal nonnegative root of (3.13), obviously,
$\rho(0)=0$ and we can assume

$\rho(v)=\sum_{k=1}^{\infty}\rho_{k}v^{k}.$

then

$2qv-3p\rho(v)+\rho^{3}(v)\equiv 0,\ \ v\in[0,1).$

Hence,

$\displaystyle\rho_{1}=\frac{2q}{3p},\ \rho_{2}=0,$ (3.14)
$\displaystyle-p\rho^{(n+1)}(v)+\sum_{k=0}^{n}C_{n}^{k}(\sum_{i=0}^{k}C_{k}^{i}\rho^{(i)}(v)\rho^{(k-i)}(v))\rho^{(n-k+1)}(v)\equiv
0,\ \ n\geq 2.$

Letting $v=0$ in the above equality yields

$\displaystyle
p(n+1)!\rho_{n+1}=\sum_{k=2}^{n}C_{n}^{k}(\sum_{i=1}^{k-1}C_{k}^{i}i!(k-i)!\rho_{i}\rho_{k-i})(n-k+1)!\rho_{n-k+1},\
\ n\geq 2.$

i.e.,

$\displaystyle\rho_{n+1}=\frac{1}{p(n+1)}\cdot\sum_{k=2}^{n}(n-k+1)\rho_{n-k+1}\cdot(\sum_{i=1}^{k-1}\rho_{i}\rho_{k-i}),\
\ n\geq 2.$ (3.15)

Therefore,

$\displaystyle\begin{cases}P(Y_{0}(\tau)=0\ |\tau<\infty)=0,\\\
P(Y_{0}(\tau)=1|\tau<\infty)=\frac{2q}{3p},\\\
P(Y_{0}(\tau)=2|\tau<\infty)=0,\\\
P(Y_{0}(\tau)=n|\tau<\infty)=\rho^{-1}\rho_{n},\ \ n\geq 3\end{cases}$

where $\rho=\rho(1)$ is the minimal nonnegative root of $B(u)=0$ and
$\rho_{n}$ is given in (3.14) and (3.15).

## Acknowledgement

This work is substantially supported by the National Natural Sciences
Foundations of China (No. 11771452, No. 11971486).

## References

  * [1] Anderson W.(1991).  Continuous-Time Markov Chains: An Applications-Oriented Approach.  Springer-Verlag, New York.
  * [2] Asmussen S. and Jagers P.(1997). Classical and Mordern Branching Processes, Sptinger, Berlin.
  * [3] Asmussen S. and Hering H.(1983).  Branching Processes. Birkhauser, Boston.
  * [4] Athreya K.B.(1994).  Large Deviation Rates for Branching Processes–I. Single Type Case.  The Annals of Appl. Probab., 4(3):779-790.
  * [5] Athreya K.B. and Ney P.E.(1972).  Branching Processes. Springer, Berlin.
  * [6] Chen A.Y.(2002).  Uniqueness and extinction properties of generalised Markov branching processes.  J. Math. Anal. Appl., 274(2):482-494
  * [7] Chen A.Y.(2002).  Ergodicity and stability of generalised Markov branching processes with resurrection.  J. Appl. Probab., 39(4):786-803
  * [8] Chen A.Y. and Li J.P. and Ramesh N.(2005).  Uniqueness and extinction of weighted Markov branching processes.  Methodol. Comput. Appl. Probab., 7(4):489-516
  * [9] Chen A.Y. and Pollett P. and Li J.P. and Zhang H.J.(2007).  A remark on the uniqueness of weighted Markov branching processes.  J. Appl. Probab., 44(1):279-283
  * [10] Harris T.E.(1963).  The theory of branching processes. Springer, Berlin and Newyork.
  * [11] Li J.P. and Chen A.Y.(2006).  Markov branching processes with immigration and resurrection.  Markov Process. Related Fields, 12(1):139-168
  * [12] Li J.P. and Chen A.Y.(2008).  Decay property of stopped Markovian Bulk-arriving queues.  Adv. Appl. Probab., 40(1):95-121.
  * [13] Li J.P. and Chen A.Y. and Pakes A.G.(2012)  Asymptotic properties of the Markov branching process with immigration.  J. Theoret. Probab., 25(1):122-143.
  * [14] Li J.P. and Liu Z.M.(2011).  Markov branching processes with immigration-migration and resurrection.  Sci. China Math., 54(1):1043-1062.
  * [15] Liu J.N. and Zhang M.(2016).  Large deviation for supercritical branching processes with immigration.  Acta Mathematica Sinica, English Series. 32(8):893-900.
  * [16] Renshaw E. and Chen A.Y.(1997).  Birth-death processes with mass annihilation and state-dependent immigration.  Comm. Statist. Stochastic Models, 13(2):239-253.
  * [17] Sevastyanov B.A.(1949).  On certain types of Markov processes (in Russian).  Uspehi Mat. Nauk, 4, 194.
  * [18] Sun Q. and Zhang M.(2017).  Harmonic moments and large deviations for supercritical branching processes with immigration.  Frontiers of Mathematics in China, 12(5):1201–1220.
  * [19] Vatutin V.A.(1974).  Asymptotic behavior of the probability of the first degeneration for branching processes with immigration.  Teor. Verojatnost. i Primenen, 19(1):26-35.
  * [20] Yamazato M.(1975).  Some results on continuous time branching processes with state-dependent immigration.  J. Math. Soc. Japan, 27(3):479-496.