# Limit Distribution of Inventory Level of Perishable Inventory Model

Authors: Hailing Dong; Guochao Jiang
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

This paper studies a perishable inventory model, which assumes that each perishable
item has finite lifetime, and only one item is consumed each time. The lifetimes of
perishable items are independent random variables with the general distribution and so are
the consumption internal. Under this assumption, by using backward equations and limit
distribution of Markov skeleton processes, this paper obtains the existence conditions and
the explicit expression of the limit distribution of the inventory level of perishable inventory
model.


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## Body

## 1. Introduction

Perishable goods are common in our daily life. In this paper, perishable goods refer to the items that have finite lifetime, like putrescible foods, easily-expired medicines, volatile liquids, and so on. Perishable inventory model can be widely used in blood banks, chemical and food industry.In the past few decades, researchers have paid much attention to perishable inventory model. The inventory problem of perishable items was first studied by Whitin [1] who considered fashion goods perishing at the end of a prescribed storage period. Ghare and Schrader [2] proposed an inventory model, in which the rate of perishable is a constant, and the consumption internals have exponential distribution. Based on the inventory model proposed by Ghare and Schrader [2], series of studies are carried out (see Raafat [3], Goyal and Giri [4], and their references). Recently, Li et al. [5] considered some factors, like demand, deteriorating rate, price discount, allow shortage or not, inflation, time value of money, and so on, as important factors in the perishable inventory study, then they divided current perishable items inventory study literatures into two categories from the perspective of study scope and reviewed the literature for each category. Karmakar and Choudhury [6] focused on the modeling of perishable items with shortages and reviewed the corresponding inventory models. Other representative works can be seen in [7–15].An interesting and important study of perishable inventory model is about the inventory level process. Ravichandran [16] obtained the explicit expression of the stationary distribution of the inventory level in operating the (S,s) policy, with positive lead time and poisson demand. Chiu [17] developed the expected inventory level to determine a best (Q,r) ordering policy under a positive order lead time and fixed life perishability. Liu and Yang [18] analyzed an (s,S) continuous review model and obtains the matrix-geometric solutions for the steady-state probability distribution of the inventory level, with finite lifetimes and positive lead times. Sivakumar [19] obtained the joint probability distribution of the inventory level and the number of demands in the orbit, where the life time of each items is assumed to be exponential. Other related papers can be seen in [20–22] and so on.In order to facilitate the mathematical treatment, most of these papers assume that the lifetime of item or consumption internal equals to constant or has exponential distribution, so that the inventory level of perishable inventory model can be reduced to a Markov process. However, in practice, the lifetime of item or consumption interval is not necessarily exponential, but a wide range of distribution. In this case, the inventory level of perishable inventory model is not and hardly been converted into a Markov process, which leads to a bottleneck on the mathematical treatment. To the best of our knowledge, no previous studies obtained the existence conditions of the limit distribution of the inventory level. Thus, we intend to work at it.Markov skeleton process provides an effective solution to the problem. Markov skeleton processes which are proposed by Hou et al. [23] in 1997 are more extensive than Markov processes. Markov skeleton process has been in-depth studied (representative works, see [24–27]). This paper proves that the inventory level of perishable inventory model is a positive recurrent Doob skeleton process which is a special case of Markov skeleton processes. Hence, by applying backward equations and limit distribution of Markov skeleton processes, this paper obtains the existence conditions and the explicit expression of the limit distribution of the inventory level. Moreover, this paper obtains the probability of the inventory level greater than 0 and the probability of the inventory level less than or equal to 0, which can then be used for the evaluation of inventory system performance.This paper is organized as follows. Section2 introduces Markov skeleton processes and presents its backward equations and limit distribution. Section 3 introduces a perishable inventory model and applies Markov skeleton processes approach to study the limit distribution of inventory level process.


## 2. Markov Skeleton Processes

In this section, we introduce Markov skeleton processes and present its backward equations and limit distribution.


### 2.1. Definition of Markov Skeleton Processes

Definition 2.1 (see [26]).
A stochastic processX={X(t,ω),0≤t<∞)} which takes values on a polish space (E,ℰ) is called a Markov skeleton process if there exists a sequence of optional stopping times {τn}n≥0, satisfying (i)
τn↑+∞ with τ0=0, and for each n≥0, τn<∞;(ii)
for alln=0,1,…, τn+1=τn+θτn·τ1;(iii)
for everyτn and any bounded ℰ[0,∞)-measurable function f defined on E[0,∞)(2.1)E[f(X(τn+⋅))∣FτnX]=E[f(X(τn+⋅))∣X(τn)]P-a.s.,
where Ωτn=(ω:τn(ω)<∞), and ℱτnX={A:∀t≥0,A∩(ω:τn≤t)∈ℱtX} is the σ-algebra on Ωτn. {τn}n=0∞ is called skeleton time sequence of the Markov skeleton process X. Furthermore, if on Ωτn(2.2)E[f(X(τn+⋅))∣FτnX]=E[(f(X(τn+⋅))∣X(τn))]=EX(τn)[f(X(⋅))]
P-a.s. holds, where Ex(·) denotes the expectation corresponding to P(·∣X(0)=x), then X is called a time homogeneous Markov skeleton process.Definition 2.2 (see [26]).
A time homogeneous Markov skeleton processX={X(t,ω),≤t<∞}→{X(t,ω),0≤t<∞} is called normal, if there exists a function h(x,t,A) on E×R+×ε, such that (i)
for fixedx and t, h(x,t,·) is a finite measure on ε,(ii)
for fixedA∈ε,h(·,·,A) is ε×ℬ(R+) measurable function on E×R+,(iii)
for anyt≥0,A∈ε,(2.3)h(X(τn),t,A)=P{X(τn+t)∈A,τn+1-τn>t∣X(τn)}P-a.s.


### 2.2. Backward Equations of Markov Skeleton Processes

Theorem 2.3 (see [26]).
Suppose thatX={X(t);t≥0} is a normal Markov skeleton process with {τn}n=0∞ as its skeleton time sequence, then for any x∈E,t≥0,A∈ε,
(2.4)p(x,t,A)=h(x,t,A)+∫E∫0t∑n=1∞q(n)(x,ds,dy)h(y,t-s,A).
Thus, p(x,t,A) is a minimal nonnegative solution to the following nonnegative equation system: ∀x∈E,t≥0,A∈ε,
(2.5)p(x,t,A)=h(x,t,A)+∫E∫0tq(x,ds,dy)p(y,t-s,A).
Formula (2.5) is called the backward equations of Markov skeleton processes.


### 2.3. Limit Distribution of Markov Skeleton Processes

Definition 2.4 (see [27]).
Suppose thatX(t) is a normal Markov skeleton process with {τn}n=0∞ as its skeleton time sequence. If there exists probability measure π(·) on (E,ℰ), such that for any A∈ℰ,
(2.6)P(X(τ1)∈A∣X(0)=x,τ1=s)=P(X(τ1)∈A)=π(A),
then X(t) is called a Doob skeleton process, π(·) is called the characteristic measure of X(t), and {τn,n=1,2,…} is the Doob skeleton time sequence of X(t).
For anyn∈N,t≥0,A∈ℰ,
(2.7)q(n)(X(τm),t,A)=P{X(τm+n)∈A,τm+n≤t∣X(τm)}
and q(1)(x,t,A) is abbreviated to q(x,t,A),
(2.8)p(x,t,A)=P{X(t)∈A∣X(0)=x},F(x,t)=P(τ1≤t∣X(0)=x),∀x∈E,t≥0,F(t)=∫EF(x,t)π(dx),t≥0,h(t,A)=∫Eh(x,t,A)π(dx),m=∫0∞tdF(t).Definition 2.5 (see [27]).
Suppose thatX(t) is a Doob skeleton processes. If m<∞ and for any x∈E, F(x,0)≡0, F(x,∞)≡1, then X(t) is called a positive recurrent Doob skeleton process.Theorem 2.6 (see [27]).
Suppose thatX(t) is a positive recurrent Doob skeleton process. If F(t) is not lattice distribution, then for ∀A∈ℰ, the limit distribution p(·) of X(t) exists,
(2.9)p(A)=lim⁡t→∞p(x,t,A)=∫0∞h(t,A)dtm,
and p(·) is a probability distribution in (E,ℰ).


## 2.1. Definition of Markov Skeleton Processes

Definition 2.1 (see [26]).
A stochastic processX={X(t,ω),0≤t<∞)} which takes values on a polish space (E,ℰ) is called a Markov skeleton process if there exists a sequence of optional stopping times {τn}n≥0, satisfying (i)
τn↑+∞ with τ0=0, and for each n≥0, τn<∞;(ii)
for alln=0,1,…, τn+1=τn+θτn·τ1;(iii)
for everyτn and any bounded ℰ[0,∞)-measurable function f defined on E[0,∞)(2.1)E[f(X(τn+⋅))∣FτnX]=E[f(X(τn+⋅))∣X(τn)]P-a.s.,
where Ωτn=(ω:τn(ω)<∞), and ℱτnX={A:∀t≥0,A∩(ω:τn≤t)∈ℱtX} is the σ-algebra on Ωτn. {τn}n=0∞ is called skeleton time sequence of the Markov skeleton process X. Furthermore, if on Ωτn(2.2)E[f(X(τn+⋅))∣FτnX]=E[(f(X(τn+⋅))∣X(τn))]=EX(τn)[f(X(⋅))]
P-a.s. holds, where Ex(·) denotes the expectation corresponding to P(·∣X(0)=x), then X is called a time homogeneous Markov skeleton process.Definition 2.2 (see [26]).
A time homogeneous Markov skeleton processX={X(t,ω),≤t<∞}→{X(t,ω),0≤t<∞} is called normal, if there exists a function h(x,t,A) on E×R+×ε, such that (i)
for fixedx and t, h(x,t,·) is a finite measure on ε,(ii)
for fixedA∈ε,h(·,·,A) is ε×ℬ(R+) measurable function on E×R+,(iii)
for anyt≥0,A∈ε,(2.3)h(X(τn),t,A)=P{X(τn+t)∈A,τn+1-τn>t∣X(τn)}P-a.s.


## 2.2. Backward Equations of Markov Skeleton Processes

Theorem 2.3 (see [26]).
Suppose thatX={X(t);t≥0} is a normal Markov skeleton process with {τn}n=0∞ as its skeleton time sequence, then for any x∈E,t≥0,A∈ε,
(2.4)p(x,t,A)=h(x,t,A)+∫E∫0t∑n=1∞q(n)(x,ds,dy)h(y,t-s,A).
Thus, p(x,t,A) is a minimal nonnegative solution to the following nonnegative equation system: ∀x∈E,t≥0,A∈ε,
(2.5)p(x,t,A)=h(x,t,A)+∫E∫0tq(x,ds,dy)p(y,t-s,A).
Formula (2.5) is called the backward equations of Markov skeleton processes.


## 2.3. Limit Distribution of Markov Skeleton Processes

Definition 2.4 (see [27]).
Suppose thatX(t) is a normal Markov skeleton process with {τn}n=0∞ as its skeleton time sequence. If there exists probability measure π(·) on (E,ℰ), such that for any A∈ℰ,
(2.6)P(X(τ1)∈A∣X(0)=x,τ1=s)=P(X(τ1)∈A)=π(A),
then X(t) is called a Doob skeleton process, π(·) is called the characteristic measure of X(t), and {τn,n=1,2,…} is the Doob skeleton time sequence of X(t).
For anyn∈N,t≥0,A∈ℰ,
(2.7)q(n)(X(τm),t,A)=P{X(τm+n)∈A,τm+n≤t∣X(τm)}
and q(1)(x,t,A) is abbreviated to q(x,t,A),
(2.8)p(x,t,A)=P{X(t)∈A∣X(0)=x},F(x,t)=P(τ1≤t∣X(0)=x),∀x∈E,t≥0,F(t)=∫EF(x,t)π(dx),t≥0,h(t,A)=∫Eh(x,t,A)π(dx),m=∫0∞tdF(t).Definition 2.5 (see [27]).
Suppose thatX(t) is a Doob skeleton processes. If m<∞ and for any x∈E, F(x,0)≡0, F(x,∞)≡1, then X(t) is called a positive recurrent Doob skeleton process.Theorem 2.6 (see [27]).
Suppose thatX(t) is a positive recurrent Doob skeleton process. If F(t) is not lattice distribution, then for ∀A∈ℰ, the limit distribution p(·) of X(t) exists,
(2.9)p(A)=lim⁡t→∞p(x,t,A)=∫0∞h(t,A)dtm,
and p(·) is a probability distribution in (E,ℰ).


## 3. Limit Distribution of Inventory Level of Perishable Inventory Model

The perishable inventory model studied in this paper has been proposed and investigated in [26], which obtained the backward equations of the inventory level of this model. Different from [26], this paper study the limit distribution of the inventory level.


### 3.1. Perishable Inventory Model

First, we present the details of the perishable inventory model as follows (see [26]). (1)
Assume that lifetimes of inventory commodities are i.i.d random variables, with a common distribution functionF(t), where F(t) is continuous and satisfies ∫0∞tdF(t)=μ1.(2)
Sell one item each time, and the sale times of each item are i.i.d random variables, with a common distribution functionG(t), where G(t) is continuous and satisfies ∫0∞tdG(t)=μ2. Assume that the sale times are also independent of the commodities' lifetimes.(3)
The maximum capacity of the warehouse is a fixed valueSmax. When the inventory level becomes Smin(Smin<0) (i.e., the quantity of out of stock arrives Smin), new commodities are replenished to increase the inventory level until it reaches Smax.LetS(t) denote inventory level at time t. When F(t) and G(t) are not exponential distributions, {S(t),t≥0} is not a Markov process. In this case, we introduce supplementary variables as follows: θ(t) denotes the lifetime of the item in stock at time t, and θ̂(t) denotes the time interval between the last sale before t and time t.As one item is consumed and the other item perishes at the same time is a rare event, so we don't consider this case and supposeF(t) and G(t) are continuous distribution. Let τn denote the nth discontinuous point of (S(t),θ(t),θ̂(t)), that is, one item is consumed or perishes at τn. At τn, (S(t),θ(t),θ̂(t)) has Markov property, so by Definition 2.1, (S(t),θ(t),θ̂(t)) is a Markov skeleton process with τn as its Markov skeleton time sequence.


### 3.2. Limit Distribution of Inventory Level

In this subsection, we obtain the limit distribution of inventory level.Suppose thatT0=0, and Tn denotes the nth times when the process (S(t),θ(t),θ̂(t)) returns to state (Smax,0,0). Let (3.1)Tn+1=Tn+θTn⋅T1,n=1,2,…,Y0=T1,Yi=Ti+1-Ti,i≥1,
then Yi is the replenishment interval. By Definition 2.2, (3.2)hSmax⁡,j(t)=P{S(t)=j,t≤T1∣S(0)=Smax⁡,θ(0)=θ̂(0)=0}.
Let M(t) denote the distribution function of Yi, and M denote the expectation of Yi, then, (3.3)M=E{T1∣S(0)=Smax⁡,θ(0)=θ̂(0)=0}=∫0∞P{T1≥t∣S(0)=Smax⁡,θ(0)=θ̂(0)=0}dt=∫0∞∑j=Smin⁡+1Smax⁡hj(t)dt.Theorem 3.1.
Ifμ1<+∞, μ2<+∞, S(t) is a positive recurrent Doob skeleton process with {Tn} as its Doob skeleton time sequence.Proof.
AsTn denotes the beginning of the nth replenishment and Tn- denotes the moment before the nth replenishment, we have S(Tn-)=Smin, S(Tn)=Smax, then S(t) satisfies Markov property at Tn, which assures that S(t) is a Markov skeleton process. At the beginning of every replenishment, S(Tn)≡Smax, so we have
(3.4)P(S(T1)=j∣S(0)=i,T1=s)=δ0(j-Smax⁡).
Thus, S(t) is Doob skeleton process by Definition 2.4. If μ1<+∞, μ2<+∞, we obtain M=E[Yi]<+∞. Therefore, S(t) is a positive recurrent Doob skeleton process with {Tn} as its Doob skeleton time sequence.Theorem 3.2.
Ifμ1<+∞, μ2<+∞, and F(x), G(x) are not lattice distribution, then, for j∈{Smin+1,Smin+2,…,Smax},
(3.5)pj(t)=lim⁡t→∞p(S(t)=j∣S(0)=i)=∫0∞hj(t)dtM=∫0∞hj(t)dt∫0∞[∑j=Smin⁡+1Smax⁡hj(t)]dt;
for A={1,2,…,Smax},
(3.6)p(A)=lim⁡t→∞p(S(t)∈A∣S(0)=i)=∫0∞h(t,A)dtM=∫0∞[∑j=1Smax⁡hj(t)]dt∫0∞[∑j=Smin⁡+1Smax⁡hj(t)]dt;
for B={Smin+1,Smin+2,…,0},
(3.7)p(B)=lim⁡t→∞p(S(t)∈B∣S(0)=i)=∫0∞h(t,B)dtM=∫0∞[∑j=Smin⁡+10hj(t)]dt∫0∞[∑j=Smin⁡+1Smax⁡hj(t)]dt,
and p(·) is a probability distribution in (E,ℰ).Proof.
IfF(x), G(x) are not lattice distribution, then M(t) is not lattice distribution. According to Theorems 2.6, 3.1, and formula (3.3), we get formulas (3.5)–(3.7). Thus, the proof of the theorem is completed.


### 3.3. The Explicit Expression ofhj(t)

Next, we intend to give the explicit expression ofhj(t) by applying backward equations of Markov skeleton processes.By formula (3.4), we have π(dx)=δ0(x-Smax). Then, (3.8)hj(t)=∑i=0∞hij(t)δ0(i-Smax⁡)=hSmax⁡,j(t),
where hSmax,j(t) is defined in (3.2).Letτ̂0=0, τ̂n=τn∧T1,n=1,2,…, where τn denotes the nth discontinuous point of (S(t),θ1(t),θ2(t)), then (3.9)τ̂n↑T1,n↑+∞.According to Theorem3.1, (S(t),θ1(t),θ2(t),t<T1) is a Markov skeleton process with τ̂n as its skeleton time sequence.Fori>0, let (3.10)ĥ(i,θ,θ̂,j,A,Â,t)=P{S(t)=j,θ(t)∈A,θ̂(t)∈Â,t<τ̂1∣S(0)=i,θ(0)=θ,θ̂(0)=θ̂};q̂(i,θ,θ̂,ds,j,A,Â)=P{S(ds)=j,θ(ds)∈A,θ̂(ds)∈Â∣S(0)=i,θ(0)=θ,θ̂(0)=θ̂};p̂(i,θ,θ̂,j,A,Â,t)=P{S(t)=j,θ(t)∈A,θ̂(t)∈Â,t<T1∣S(0)=i,θ(0)=θ,θ̂(0)=θ̂}.
Thus, hSmax,j(t) can be expressed as follows: (3.11)hSmax⁡,j(t)=p̂(Smax⁡,0,0,j,t).Lemma 3.3.
WhenF(t)andG(t) are continuous, we have
(3.12)ĥ(Smax⁡,0,0,j,t)={0,j≠Smax⁡;(1-F(t))Smax⁡(1-G(t)),j=Smax⁡,q̂(Smax⁡,0,0,ds,j)={CSmax⁡1(1-F(s))Smax⁡-1(1-G(s))F(ds),j=Smax⁡-1,oneitemperishes,andnoitemisconsumed;CSmax⁡1(1-F(s))Smax⁡G(ds),j=Smax⁡-1,oneitemisconsumed,andnoitemperishes;0,j≠Smax⁡-1.Proof.
By the definition ofĥ(Smax,0,0,j,t), there is no state transition of S(t) up to t. If j≠Smax, ĥ(Smax,0,0,j,t)=0. If j=Smax, which means that no item is consumed or perishes up to t, then ĥ(Smax,0,0,j,t)=(1-F(t))Smax(1-G(t)).
By the definition ofq̂(Smax,0,0,ds,j), S(t) will transfer from state Smax to state j at ds. As F(t),G(t) are continuous, when j≠Smax-1, q̂(Smax,0,0,ds,j)=0.
If one item perishes at timeds, and no item is consumed up to s, then j=Smax-1, q̂(Smax,0,0,ds,j)=CSmax1(1-F(s))Smax-1(1-G(s))F(ds).
If one item is consumed at timeds, and no item perishes up to s, then j=Smax-1, q̂(Smax,0,0,ds,j)=CSmax1(1-F(s))SmaxG(ds).According to Theorem2.3 and Lemma 3.3, we have the following.Theorem 3.4.
p̂(Smax,0,0,j,t) is the minimal nonnegative solution to the following nonnegative linear equation,
(3.13)p̂(Smax⁡,0,0,j,t)=δSmax⁡,j(1-F(t))Smax⁡(1-G(t))+∫0tCSmax⁡1(1-F(s))Smax⁡-1(1-G(s))F(ds)p̂(Smax⁡-1,s,s,j,t-s)+∫0tCSmax⁡1(1-F(s))Smax⁡G(ds)p̂(Smax⁡-1,s,0,j,t-s).Thus, combining formulas (3.8), (3.11), and Theorem 3.4, the explicit expression of hj(t) is obtained; hj(t)=p̂(Smax,0,0,j,t) is the minimal nonnegative solution to formula (3.13).


## 3.1. Perishable Inventory Model

First, we present the details of the perishable inventory model as follows (see [26]). (1)
Assume that lifetimes of inventory commodities are i.i.d random variables, with a common distribution functionF(t), where F(t) is continuous and satisfies ∫0∞tdF(t)=μ1.(2)
Sell one item each time, and the sale times of each item are i.i.d random variables, with a common distribution functionG(t), where G(t) is continuous and satisfies ∫0∞tdG(t)=μ2. Assume that the sale times are also independent of the commodities' lifetimes.(3)
The maximum capacity of the warehouse is a fixed valueSmax. When the inventory level becomes Smin(Smin<0) (i.e., the quantity of out of stock arrives Smin), new commodities are replenished to increase the inventory level until it reaches Smax.LetS(t) denote inventory level at time t. When F(t) and G(t) are not exponential distributions, {S(t),t≥0} is not a Markov process. In this case, we introduce supplementary variables as follows: θ(t) denotes the lifetime of the item in stock at time t, and θ̂(t) denotes the time interval between the last sale before t and time t.As one item is consumed and the other item perishes at the same time is a rare event, so we don't consider this case and supposeF(t) and G(t) are continuous distribution. Let τn denote the nth discontinuous point of (S(t),θ(t),θ̂(t)), that is, one item is consumed or perishes at τn. At τn, (S(t),θ(t),θ̂(t)) has Markov property, so by Definition 2.1, (S(t),θ(t),θ̂(t)) is a Markov skeleton process with τn as its Markov skeleton time sequence.


## 3.2. Limit Distribution of Inventory Level

In this subsection, we obtain the limit distribution of inventory level.Suppose thatT0=0, and Tn denotes the nth times when the process (S(t),θ(t),θ̂(t)) returns to state (Smax,0,0). Let (3.1)Tn+1=Tn+θTn⋅T1,n=1,2,…,Y0=T1,Yi=Ti+1-Ti,i≥1,
then Yi is the replenishment interval. By Definition 2.2, (3.2)hSmax⁡,j(t)=P{S(t)=j,t≤T1∣S(0)=Smax⁡,θ(0)=θ̂(0)=0}.
Let M(t) denote the distribution function of Yi, and M denote the expectation of Yi, then, (3.3)M=E{T1∣S(0)=Smax⁡,θ(0)=θ̂(0)=0}=∫0∞P{T1≥t∣S(0)=Smax⁡,θ(0)=θ̂(0)=0}dt=∫0∞∑j=Smin⁡+1Smax⁡hj(t)dt.Theorem 3.1.
Ifμ1<+∞, μ2<+∞, S(t) is a positive recurrent Doob skeleton process with {Tn} as its Doob skeleton time sequence.Proof.
AsTn denotes the beginning of the nth replenishment and Tn- denotes the moment before the nth replenishment, we have S(Tn-)=Smin, S(Tn)=Smax, then S(t) satisfies Markov property at Tn, which assures that S(t) is a Markov skeleton process. At the beginning of every replenishment, S(Tn)≡Smax, so we have
(3.4)P(S(T1)=j∣S(0)=i,T1=s)=δ0(j-Smax⁡).
Thus, S(t) is Doob skeleton process by Definition 2.4. If μ1<+∞, μ2<+∞, we obtain M=E[Yi]<+∞. Therefore, S(t) is a positive recurrent Doob skeleton process with {Tn} as its Doob skeleton time sequence.Theorem 3.2.
Ifμ1<+∞, μ2<+∞, and F(x), G(x) are not lattice distribution, then, for j∈{Smin+1,Smin+2,…,Smax},
(3.5)pj(t)=lim⁡t→∞p(S(t)=j∣S(0)=i)=∫0∞hj(t)dtM=∫0∞hj(t)dt∫0∞[∑j=Smin⁡+1Smax⁡hj(t)]dt;
for A={1,2,…,Smax},
(3.6)p(A)=lim⁡t→∞p(S(t)∈A∣S(0)=i)=∫0∞h(t,A)dtM=∫0∞[∑j=1Smax⁡hj(t)]dt∫0∞[∑j=Smin⁡+1Smax⁡hj(t)]dt;
for B={Smin+1,Smin+2,…,0},
(3.7)p(B)=lim⁡t→∞p(S(t)∈B∣S(0)=i)=∫0∞h(t,B)dtM=∫0∞[∑j=Smin⁡+10hj(t)]dt∫0∞[∑j=Smin⁡+1Smax⁡hj(t)]dt,
and p(·) is a probability distribution in (E,ℰ).Proof.
IfF(x), G(x) are not lattice distribution, then M(t) is not lattice distribution. According to Theorems 2.6, 3.1, and formula (3.3), we get formulas (3.5)–(3.7). Thus, the proof of the theorem is completed.


## 3.3. The Explicit Expression ofhj(t)

Next, we intend to give the explicit expression ofhj(t) by applying backward equations of Markov skeleton processes.By formula (3.4), we have π(dx)=δ0(x-Smax). Then, (3.8)hj(t)=∑i=0∞hij(t)δ0(i-Smax⁡)=hSmax⁡,j(t),
where hSmax,j(t) is defined in (3.2).Letτ̂0=0, τ̂n=τn∧T1,n=1,2,…, where τn denotes the nth discontinuous point of (S(t),θ1(t),θ2(t)), then (3.9)τ̂n↑T1,n↑+∞.According to Theorem3.1, (S(t),θ1(t),θ2(t),t<T1) is a Markov skeleton process with τ̂n as its skeleton time sequence.Fori>0, let (3.10)ĥ(i,θ,θ̂,j,A,Â,t)=P{S(t)=j,θ(t)∈A,θ̂(t)∈Â,t<τ̂1∣S(0)=i,θ(0)=θ,θ̂(0)=θ̂};q̂(i,θ,θ̂,ds,j,A,Â)=P{S(ds)=j,θ(ds)∈A,θ̂(ds)∈Â∣S(0)=i,θ(0)=θ,θ̂(0)=θ̂};p̂(i,θ,θ̂,j,A,Â,t)=P{S(t)=j,θ(t)∈A,θ̂(t)∈Â,t<T1∣S(0)=i,θ(0)=θ,θ̂(0)=θ̂}.
Thus, hSmax,j(t) can be expressed as follows: (3.11)hSmax⁡,j(t)=p̂(Smax⁡,0,0,j,t).Lemma 3.3.
WhenF(t)andG(t) are continuous, we have
(3.12)ĥ(Smax⁡,0,0,j,t)={0,j≠Smax⁡;(1-F(t))Smax⁡(1-G(t)),j=Smax⁡,q̂(Smax⁡,0,0,ds,j)={CSmax⁡1(1-F(s))Smax⁡-1(1-G(s))F(ds),j=Smax⁡-1,oneitemperishes,andnoitemisconsumed;CSmax⁡1(1-F(s))Smax⁡G(ds),j=Smax⁡-1,oneitemisconsumed,andnoitemperishes;0,j≠Smax⁡-1.Proof.
By the definition ofĥ(Smax,0,0,j,t), there is no state transition of S(t) up to t. If j≠Smax, ĥ(Smax,0,0,j,t)=0. If j=Smax, which means that no item is consumed or perishes up to t, then ĥ(Smax,0,0,j,t)=(1-F(t))Smax(1-G(t)).
By the definition ofq̂(Smax,0,0,ds,j), S(t) will transfer from state Smax to state j at ds. As F(t),G(t) are continuous, when j≠Smax-1, q̂(Smax,0,0,ds,j)=0.
If one item perishes at timeds, and no item is consumed up to s, then j=Smax-1, q̂(Smax,0,0,ds,j)=CSmax1(1-F(s))Smax-1(1-G(s))F(ds).
If one item is consumed at timeds, and no item perishes up to s, then j=Smax-1, q̂(Smax,0,0,ds,j)=CSmax1(1-F(s))SmaxG(ds).According to Theorem2.3 and Lemma 3.3, we have the following.Theorem 3.4.
p̂(Smax,0,0,j,t) is the minimal nonnegative solution to the following nonnegative linear equation,
(3.13)p̂(Smax⁡,0,0,j,t)=δSmax⁡,j(1-F(t))Smax⁡(1-G(t))+∫0tCSmax⁡1(1-F(s))Smax⁡-1(1-G(s))F(ds)p̂(Smax⁡-1,s,s,j,t-s)+∫0tCSmax⁡1(1-F(s))Smax⁡G(ds)p̂(Smax⁡-1,s,0,j,t-s).Thus, combining formulas (3.8), (3.11), and Theorem 3.4, the explicit expression of hj(t) is obtained; hj(t)=p̂(Smax,0,0,j,t) is the minimal nonnegative solution to formula (3.13).



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*Source: 329531-2011-08-07.xml*