# Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior

Authors: Ushio Sumita; Jia-Ping Huang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

The class of counting processes constitutes a significant part of applied probability. The classic counting processes include Poisson processes, nonhomogeneous
Poisson processes, and renewal processes. More sophisticated counting processes,
including Markov renewal processes, Markov modulated Poisson processes, age-dependent counting processes, and the like, have been developed for accommodating
a wider range of applications. These counting processes seem to be quite different
on the surface, forcing one to understand each of them separately. The purpose
of this paper is to develop a unified multivariate counting process, enabling one to
express all of the above examples using its components, and to introduce new counting processes. The dynamic behavior of the unified multivariate counting process is
analyzed, and its asymptotic behavior ast→∞ is established. As an application,
a manufacturing system with certain maintenance policies is considered, where the
optimal maintenance policy for minimizing the total cost is obtained numerically.


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## Body

## 1. Introduction

A stochastic process{N(t):t≥0} is called a counting process when N(t) is nonnegative, right continuous and monotone nondecreasing with N(0)=0. The classic counting processes of importance include a Poisson process, a nonhomogeneous Poisson process (NHPP) and a renewal process. More sophisticated counting processes have been developed in order to accommodate a wider range of applications. A Markov renewal process, for example, extends an ordinary renewal process in that the interarrival time between two successive arrivals has a probability distribution depending on the state transition of the underlying Markov chain, see, for example, Pyke [1, 2]. In Masuda and Sumita [3], the number of entries of a semi-Markov process into a subset of the state space is analyzed, while a Markov-modulated Poisson process (MMPP) is introduced by Neuts [4] where jumps of the MMPP occur according to a Poisson process with intensity λi whenever the underlying Markov-chain is in state i.The MMPP is generalized subsequently into several different directions. Lucantoni et al. [5] develop a Markovian arrival process (MAP), where a Markov-chain defined on 𝒥=G∪B with G∩B=ϕ,G≠ϕ, and B≠ϕ is replaced to a state i∈G as soon as it enters a state j∈B with probability p̃ji and the counting process describes the number of such replacements occurred in [0,t]. If we set G={0G,1G,…,JG} and B={0B,1B,…,JB}, and an absorbing state jB∈B can be reached only from its counter part jG∈G with instantaneous replacement back to jG itself, the resulting MAP becomes an MMPP. Another direction for generalizing the MMPP is to replace the underlying Markov-chain by a semi-Markov process, which we call a semi-Markov-modulated Poisson process (SMMPP). To the best knowledge of the authors, the SMMPP is first addressed by Dshalalow [6], where a systematic approach for dealing with modulated random measures is proposed in a more abstract way. Such modulated random measures include the MMPP and the SMMPP as special cases. An application of the SMMPP to queueing theory is discussed in Dshalalow and Russell [7]. More recently, in a series of papers by Agarwal et al. [8] and Dshalalow [9, 10], the original approach above has been further extended and refined. The SMMPP is studied in detail independently by Özekici and Soyer [11].In the SMMPP, the counting process under consideration is modulated according to state transitions of the underlying semi-Markov process. A further generalization may be possible by considering a counting process whose arrival intensity depends on not only the current state of the semi-Markov process but also the current age of the process. This line of research is originated by Sumita and Shanthikumar [12] where an age-dependent counting process generated from a renewal process is studied. Here, items arrive according to an NHPP which is interrupted and reset at random epochs governed by a renewal process.All of these counting processes discussed above seem to be quite different on the surface, forcing one to understand each of them separately. The purpose of this paper is to develop a unified multivariate counting process which would contain all of the above examples as special cases. In this regard, we consider a system where items arrive according to an NHPP. This arrival stream is interrupted from time to time where the interruptions are governed by a finite semi-Markov processJ(t) on 𝒥={0,1,2,…,J}. Whenever a state transition of the semi-Markov process occurs from i to j, the intensity function of the NHPP is switched from λi(x) to λj(x) with an initial value reset to λj(0). In other words, the arrivals of items are generated by the NHPP with λi(x) when the semi-Markov process is in state i with x denoting the time since the last entry into state i. Of particular interest in analysis of such systems are the multivariate counting processes M̲(t)=[Mi(t)]i∈𝒥 and N̲̲(t)=[Nij(t)]i,j∈𝒥 where Mi(t) counts the cumulative number of items that have arrived in [0,t] while the semi-Markov process is in state i and Nij(t) represents the cumulative number of the state transitions of the semi-Markov process from i to j in [0,t]. The joint multivariate counting process [M̲(t),N̲̲(t)] enables one to unify many existing counting processes in that they can be derived in terms of the components of [M̲(t),N̲̲(t)]. Because of this reason, hereafter, we call [M̲(t),N̲̲(t)] the unified multivariate counting process. The dynamic behavior of [M̲(t),N̲̲(t)] is captured through analysis of the underlying Laplace transform generating functions, yielding asymptotic expansions of the means and the variances of the partial sums of its components as t→∞.Applications of the unified multivariate counting process can be found, for example, in modern communication networks. One may consider a high-speed communication link for transmitting video signals between two locations. Video sequences are transmitted as streams of binary data that vary over time in traffic intensity according to the level of movement, the frequency of scene changes, and the level of transmission quality. Consequently, efficient transmission of video traffic can be achieved through variable bit rate coding. In this coding scheme, data packets are not generated at a constant rate from the original sequence, but rather at varying rates. By doing so, one achieves less fluctuation in transmission quality level and, at the same time, transmission capacity can be freed up whenever possible. As in Maglaris et al. [13], such a mechanism may be implemented by using multimode encoders where each mode reflects a certain level of data compression, and the change between modes is governed by the underlying video sequence according to buffer occupancy levels. A system of this sort can be described in the above framework with M(t)=∑i∈𝒥Mi(t) representing the number of packet arrivals at the origination site and Nij(t) describing the number of the encoder changes in [0,t]. The state of the underlying semi-Markov process at time t then corresponds to the current mode of the encoder. Other types of applications include system reliability models where the semi-Markov process describes the status of the system under consideration while the interruptions correspond to system failures and replacements. A cost function associated with such a system may then be constructed from the unified multivariate counting processes [M̲(t),N̲̲(t)]. In this paper, a manufacturing system with certain maintenance policies is considered, where the unified multivariate counting process enables one to determine numerically the optimal maintenance policy for minimizing the total cost.The structure of this paper is as follows. In Section2, key transform results of various existing counting processes are summarized. Detailed description of the unified multivariate counting process is provided in Section 3 and its dynamic behavior is analyzed in Section 4 by examining the probabilistic flow of the underlying stochastic processes and deriving transform results involving Laplace transform generating functions. Section 5 is devoted to derivation of the existing counting processes of Section 2 in terms of the components of the unified multivariate counting process. Asymptotic analysis is provided in Section 6, yielding asymptotic expansions of the means and the variances of the partial sums of its components. An application is discussed in Section 7, where a manufacturing system with certain maintenance policies is considered and the optimal maintenance policy for minimizing the total cost is obtained numerically. Some concluding remarks are given in Section 8.Throughout the paper, matrices and vectors are indicated by double underlines (A̲̲,b̲̲, etc.) and underlines (X̲,y̲, etc.) respectively. The vector with all components equal to 1 is denoted by 1̲ and the identity matrix is written as I̲̲. A submatrix of a̲̲ is defined by a̲̲GB=[aij]i∈G,j∈B.


## 2. Various Counting Processes of Interest

In this section, we summarize key transform results of various counting processes of interest, which can be expressed in terms of the components of the unified multivariate counting process proposed in this paper as we will see. We begin the discussion with one of the most classical arrival processes, the Poisson process.


### 2.1. Poisson Process

Poisson process of intensityλ is characterized by a sequence of independently and identically distributed (i.i.d.) exponential random variables (Xj)j=1∞ with common probability density function (p.d.f.) fX(x)=λe-λx. Let Sn=∑j=1nXj. Then, the associated Poisson process {N(t):t≥0} is defined as a counting process satisfying
(2.1)N(t)=n⟺Sn≤t<Sn+1.
If a system has an exponential lifetime of mean λ-1 and is renewed instantaneously upon failure, Xj represents the lifetime of the jth renewal cycle. The Poisson process {N(t):t≥0} then counts the number of failures that have occurred by time t.Letpn(t)=P[N(t)=n∣N(0)=0] and define the probability generating function (p.g.f.) π(v,t) by
(2.2)π(v,t)=E[vN(t)]=∑n=0∞pn(t)vn.
It can be seen, see, for example, Karlin and Taylor [14], that
(2.3)ddtpn(t)=-λpn(t)+λpn-1(t)
where pn(t)=0 for n<0. Multiplying vn on both sides of (2.3) and summing from 0 to ∞, one then finds that
(2.4)∂∂tπ(v,t)=-λ(1-v)π(v,t).
Since pn(0)=δ{n=0} where δ{P}=1 if statement P is true and δ{P}=0 otherwise, one has π(v,0)=1. Equation (2.4) can then be solved as
(2.5)π(v,t)=e-λt(1-v);pn(t)=e-λt(λt)nn!.


### 2.2. Nonhomogeneous Poisson Process (NHPP)

An NHPP{M(t):t≥0} differs from a Poisson process in that the failure intensity of the system is given as a function of time t. Accordingly, (2.3) should be rewritten as
(2.6)ddtpm(t)=-λ(t)pm(t)+λ(t)pm-1(t).
By taking the generating function of  (2.6), one finds that
(2.7)∂∂tπ(u,t)=-λ(t)(1-u)π(u,t).
With L(t)=∫0tλ(y)dy, this equation can be solved as
(2.8)π(u,t)=e-L(t)(1-u);pm(t)=e-L(t)L(t)mm!.
The reader is referred to Ross [15] for further discussions of NHPPs.


### 2.3. Markov-Modulated Poisson Process (MMPP)

Let{J(t):t≥0} be a Markov-chain in continuous time on 𝒥={0,…,J} governed by a transition rate matrix ν̲̲=[νij]. Let λ̲⊤=[λ0,…,λJ] and define the associated diagonal matrix λ̲̲D=[δ{i=j}λi]. An MMPP {M(t):t≥0} characterized by (ν̲̲,λ̲̲D) is a pure jump process where jumps of M(t) occur according to a Poisson process with intensity λi whenever the Markov-chain J(t) is in state i.Letνi=∑j∈𝒥νij and define ν̲̲D=[δ{i=j}νi]. The infinitesimal generator Q̲̲ associated with the Markov-chain J(t) is then given by
(2.9)Q̲̲=-ν̲̲D+ν̲̲.
For i,j∈𝒥, let
(2.10)p̲̲(k,t)=[pij(k,t)];pij(k,t)=P[M(t)=k,J(t)=j∣J(0)=i,M(0)=0],
and define the associated matrix generating function π̲̲(u,t) by
(2.11)π̲̲(u,t)=∑k=0∞‍p̲̲(k,t)uk.
It can be seen that
(2.12)∂∂tpij(k,t)=-(λj+νj)pij(k,t)+∑r∈𝒥‍pir(k,t)νrj+λjpij(k-1,t).
In matrix notation, this can be rewritten as
(2.13)∂∂tp̲̲(k,t)=-p̲̲(k,t){λ̲̲D+ν̲̲D-ν̲̲}+p̲̲(k-1,t)λ̲̲D.
By taking the generating function of (2.13) together with (2.9), one sees that
(2.14)∂∂tπ̲̲(u,t)=π̲̲(u,t){Q̲̲-(1-u)λ̲̲D}.
Since M(0)=0, one has π̲̲(u,0)=I̲̲, where I̲̲=[δ{i=j}] is the identity matrix, so that the above differential equation can be solved as
(2.15)π̲̲(u,t)=e{Q̲̲-(1-u)λ̲̲D}t=∑k=0∞tkk!{Q̲̲-(1-u)λ̲̲D}k,
where A̲̲0=defI̲̲ for any square matrix A̲̲. It should be noted that π̲̲(1,t)=eQ̲̲t, which is the transition probability matrix of J(t) as it should be. By taking the Laplace transform of both sides of (2.15), π̲̲̂(u,s)=∫0∞e-stπ̲̲(u,t) is given by
(2.16)π̲̲̂(u,s)={sI̲̲-Q̲̲+(1-u)λ̲̲D}-1.In general, the interarrival times generated by an MMPP are not independent nor identically distributed. In multimedia computer and communication networks, data packets are mingled together with voice and image packets generated from analogue sources. Since arrival patterns of such packets differ from each other, MMPPs have provided useful means to model arrival processes of packets in multimedia computer and communication networks, see, for example, Heffes and Lucantoni [16] and Sriram and Whitt [17]. Burman and Smith [18], and Knessl et al. [19] studied a single server queuing system with an MMPP arrival process and general i.i.d. service times. Neuts et al. [20] established characterization theorems for an MMPP to be a renewal process in terms of lumpability of the underlying Markov-chain J(t). The reader is referred to Neuts [4] for further discussions of MMPP.An MMPP can be extended by replacing the underlying Markov-chain in continuous time by a semi-Markov process as discussed in Section1. This process is denoted by SMMPP. To the best knowledge of the authors, the SMMPP is first addressed by Dshalalow [6], where a systematic approach for dealing with modulated random measures is proposed in a more abstract way. Such modulated random measures include the MMPP and the SMMPP as special cases. An application of the SMMPP to queueing theory is addressed in Dshalalow and Russell [7]. More recently, the original approach above has been further extended and refined in a series of papers by Agarwal et al. [8] and Dshalalow [9, 10]. The SMMPP is studied in detail independently by Özekici and Soyer [11] including transient characterizations and ergodic analysis. Both MMPP and SMMPP will be proven to be expressible in terms of the components of the unified multivariate counting process proposed in this paper.


### 2.4. Renewal Process

Renewal processes can be considered as a generalization of Poisson processes in that a sequence of i.i.d. exponential random variables are replaced by that of any i.i.d. nonnegative random variables with common distribution functionA(x). The resulting counting process {N(t):t≥0} is still characterized by (2.1). Let pn(t)=P[N(t)=n∣N(0)=0] as before. One then sees that
(2.17)pn(t)=A(n)(t)-A(n+1)(t),
where A(n)(t) denotes the n-fold convolution of A(x) with itself, that is,  A(n+1)(t)=∫0tA(n)(t-x)dA(x) and A(0)(t)=U(t) which is the step function defined as U(t)=1 for t≥0 and U(t)=0 else.Letπn(s)=∫0∞e-stpn(t)dt and α(s)=∫0∞e-stdA(t). By taking the Laplace transform of both sides of (2.17), it follows that
(2.18)πn(s)=1-α(s)sα(s)n.
By taking the generating function of the above equation with π(v,s)=∑n=0∞πn(s)vn, one has
(2.19)π(v,s)=1-α(s)s11-vα(s).
The reader is referred to Cox [21], or Karlin and Taylor [14] for further discussions of renewal processes.


### 2.5. Markov Renewal Process (MRP)

An MRP is an extension of an ordinary renewal process in that, in the interval[0,t), the former describes the recurrence statistics for intermingling classes of epochs of an underlying semi-Markov process, whereas the latter counts the number of recurrences for a single recurrent class of epochs. More specifically, let {J(t):t≥0} be a semi-Markov process on 𝒥={0,…,J} governed by a matrix p.d.f. a̲̲(x) where a̲̲(x)≥0̲̲ and ∫0∞a̲̲(x)dx=A̲̲0 is a stochastic matrix which is assumed to be ergodic. Let ε(ℓ) be a recurrent class consisting of the entries of the semi-Markov process to state ℓ for ℓ∈𝒥, and define Ñℓr(t) to be a counting process describing the number of recurrences for ε(r) given that there was an epoch of ε(ℓ) at time t=0. Then Ñ̲ℓ(t)=[Ñℓ0(t),…,ÑℓJ(t)] is called an MRP.The study of MRPs can be traced back to early 1960s represented by the two original papers by Pyke [1, 2], followed by Keilson [22, 23], Keilson and Wishart [24, 25], Çinlar [26, 27] and McLean and Neuts [28]. Since then, the area attracted many researchers and a survey paper by Çinlar [29] in 1975 already included more than 70 leading references. The study has been largely focused on the matrix renewal function H̲̲(t)=[Hℓr(t)] with Hℓr(t)=E[Ñℓr(t)], the associated matrix renewal density, and the limit theorems. For example, one has the following result concerning the Laplace transform of H̲̲(t) by Keilson [23]:
(2.20)ℒ{H̲̲(t)}=1sα̲̲(s)[I̲̲-α̲̲(s)]-1,
where α̲̲(s) is the Laplace transform of a̲̲(t). The unified multivariate counting process of this paper contains an MRP as a special case and provides more information based on dynamic analysis of the underlying probabilistic flows.


### 2.6. Number of Entries of a Semi-Markov Process into a Subset of the State Space (NESMPS)

Another type of counting processes associated with a semi-Markov process on𝒥={0,…,J} governed by a matrix p.d.f. a̲̲(x) is studied in Masuda and Sumita [3], where the state space 𝒥 is decomposed into a set of good states G(≠ϕ) and a set of bad states B(≠ϕ) satisfying 𝒥=G∪B and G∩B=ϕ. The counting process NGB(t) is then defined to describe the number of entries of J(t) into B by time t.WhileNGB(t) is a special case of MRPs, the detailed analysis is provided in [3], yielding much more information. More specifically, let X(t) be the age process associated with J(t), that is,
(2.21)X(t)=t-sup{τ:J(t)∣τ-τ+≠0,0<τ≤t},
where f(x)∣x-x+=f(x+)-f(x-), and define
(2.22)F̲̲n(x,t)=[Fn:ij(x,t)],
where
(2.23)Fn:ij(x,t)=P[X(t)≤x,NGB(t)=n,J(t)=j∣X(0)=NGB(0)=0,J(0)=i].
One then has
(2.24)f̲̲n(x,t)=∂∂xF̲̲n(x,t).
The associated matrix Laplace transform generating function can then be defined as
(2.25)φ̲̲(v,w,s)=∑n=0∞vn∬0∞e-wt-stf̲̲n(x,t)dxdt.
It has been shown in Masuda and Sumita [3] that
(2.26)φ̲̲(v,w,s)=1w+s·γ̲̲0(s){I̲̲-vβ̲̲(s)}-1{I̲̲-α̲̲D(w+s)}.
Here, with α̲̲(s)=∫0∞e-sta̲̲(t)dt and α̲̲CD(s)=[αij(s)]i∈C,j∈D for C,D⊂𝒥, the following notation is employed:(2.27)α̲̲(s)=[α̲̲GG(s)α̲̲GB(s)α̲̲BG(s)α̲̲BB(s)];α̲̲D(s)=[δ{i=j}αi(s)]withαi(s)=∑j∈𝒥αij(s);(2.28)χ̲̲G(s)={I̲̲GG-α̲̲GG(s)}-1;χ̲̲B(s)={I̲̲BB-α̲̲BB(s)}-1;(2.29)β̲̲(s)=[0̲̲BB0̲̲BGα̲̲GBχ̲̲B(s)α̲̲GBχ̲̲B(s)α̲̲BGχ̲̲G(s)];(2.30)γ̲̲0(s)=[α̲̲BBχ̲̲B(s)χ̲̲B(s)α̲̲BGχ̲̲G(s)0̲̲GBα̲̲GGχ̲̲G(s)]+I̲̲.
As we will see, the unified multivariate counting process proposed in this paper enables one to deal with multidimensional generalization of NESMPSs as a special case.


### 2.7. Markovian Arrival Process (MAP)

As for Poisson processes, a renewal process requires interarrival times to form a sequence of i.i.d. nonnegative random variables. As we have seen, a class of MMPPs enables one to avoid this requirement by introducing different Poisson arrival rates depending on the state of the underlying Markov chain. An alternative way to avoid this i.i.d. requirement is to adapt a class of MAPs, originally introduced by Lucantoni et al. [5]. We discuss here a slightly generalized version of MAPs in that a set of absorbing states is not necessarily a singleton set.Let{J*(t):t≥0} be an absorbing Markov-chain on 𝒥*=G∪B with G≠ϕ,B≠ϕ and G∩B=ϕ, where all states in B are absorbing. Without loss of generality, we assume that G={0,…,m}, and B={m+1,…,m+K}. For notational convenience, the following transition rate matrices are introduced.(2.31)ν̲̲GG*=[νij]i,j∈G;ν̲̲GB*=[νir]i∈G,r∈B.
The entire transition rate matrix ν̲̲* governing J*(t) is then given by
(2.32)ν̲̲*=[ν̲̲GG*ν̲̲GB*0̲̲0̲̲].A replacement Markov-chain{J(t):t≥0} on G is now generated from {J*(t):t≥0}. Starting from a state in G, the process J(t) coincides with J*(t) within G. As soon as J*(t) reaches state r∈B, it is instantaneously replaced at state j∈G with probability p̃rj and the process continues. Let
(2.33)C̲̲=ν̲̲GG*,D̲̲=ν̲̲GB*p̲̲̃BG,
where p̲̲̃BG=[p̃rj]r∈B,j∈G. Then the transition rate matrix ν̲̲ and the infinitesimal generator Q̲̲ of J(t) are given as
(2.34)ν̲̲=C̲̲+D̲̲;Q̲̲=-ν̲̲D+ν̲̲,
where
(2.35)ν̲̲D=C̲̲D+D̲̲D,
with
(2.36)C̲̲D=[δ{i=j}ci];ci=∑j∈Gcij,D̲̲D=[δ{i=j}di];di=∑j∈Gdij.
Let p̲̲(t) be the transition probability matrix of J̲̲(t) with its Laplace transform defined by π̲̲(s)=∫0∞e-stp̲̲(t)dt. From the Kolmogorov forward equation (d/dt)p̲̲(t)=p̲̲(t)Q̲̲ with p̲̲(0)=I̲̲, one has
(2.37)π̲̲(s)={sI̲̲-Q̲̲}-1.Let{NMAP(t):t≥0} be the counting process keeping the record of the number of replacements in [0,t) and define
(2.38)fij(k,t)=P[NMAP(t)=k,J(t)=j∣J(0)=i],i,j∈G.
By analyzing the probabilistic flow at state j at time t+Δ, it can be seen that
(2.39)fij(k,t+Δ)=fij(k,t){1-∑ℓ∈G(cjℓ+djℓ)Δ}+∑ℓ∈Gfiℓ(k,t)cℓjΔ+∑ℓ∈Gfiℓ(k-1,t)dℓjΔ+o(Δ).
It then follows that
(2.40)∂∂tfij(k,t)=-fij(k,t)∑ℓ∈G(cjℓ+djℓ)+∑ℓ∈Gfiℓ(k,t)cℓj+∑ℓ∈Gfiℓ(k-1,t)dℓj.
In matrix notation, the above equation can be rewritten as
(2.41)∂∂tf̲̲(k,t)=-f̲̲(k,t)ν̲̲D+f̲̲(k,t)C̲̲+f̲̲(k-1,t)D̲̲.
We now introduce the following matrix Laplace transform generating function:
(2.42)φ̲̲(v,s)=[φij(v,s)];φij(v,s)=∫0∞e-st∑k=0∞fij(k,t)vkdt.
Taking the Laplace transform with respect to t and the generating function with respect to k of both sides of (2.41), one has
(2.43)φ̲̲(v,s){sI̲̲+ν̲̲D-C̲̲-vD̲̲}=I̲̲,
which can be solved as
(2.44)φ̲̲(v,s)={sI̲̲-Q̲̲+(1-v)D̲̲}-1.
We note from (2.37) and (2.44) that φ̲̲(1,s)=π̲̲(s) as it should be.It may be worth noting that an MMPP is a special case of an MAP, which can be seen in the following manner. Let an MAP be defined onG∪B where G={0G,…,JG} and B={0B,…,JB}. Transitions within G is governed by ν̲̲. An entry into jB∈B is possible only from jG∈G. When this occurs, the Markov process is immediately replaced at the entering state jG. The counting process for the number of such replacements then has the Laplace transform generating function φ̲̲(v,s) given in (2.44) where D̲̲ is replaced by λ̲̲D, which coincides with π̲̲̂(u,s) of (2.16) for MMPPs, proving the claim.


### 2.8. Age-Dependent Counting Process Generated from a Renewal Process (ACPGRP)

An age-dependent counting process generated from a renewal process has been introduced and studied by Sumita and Shanthikumar [12], where items arrive according to an NHPP which is interrupted and reset at random epochs governed by a renewal process. More specifically, let {N(t):t≥0} be a renewal process associated with a sequence of i.i.d. nonnegative random variables (Xj)j=1∞ with common p.d.f. a(x). The age process X(t) is then defined by
(2.45)X(t)=t-sup{τ:N(t)∣τ-τ+=1,0<τ≤t}.
In other words, X(t) is the elapsed time since the last renewal. We next consider an NHPP Z(x) governed by an intensity function λ(x). If we define
(2.46)L(x)=∫0xλ(y)dy,
one has
(2.47)g(x,k)=P[Z(x)=k]=exp(-L(x))L(x)kk!,k=0,1,2,….
Of interest, then, is a counting process {M(t):t≥0} characterized by
(2.48)P[M(t+Δ)-M(t)=1∣M(t)=m,N(t)=1,X(t)=x]=λ(x)Δ+o(Δ),(m,n,x)∈S,Δ>0.
Here S=ℤ+×ℤ+×ℝ+ where ℤ+ is the set of nonnegative integers, and ℝ+ is the set of nonnegative real numbers.Since the counting processM(t) is governed by the intensity function depending on the age process X(t) of the renewal process N(t), it is necessary to analyze the trivariate process [M(t),N(t),X(t)]. Let the multivariate transform of [M(t),N(t),X(t)] be defined by
(2.49)φ(u,v,w,s)=∫0∞e-stE[uM(t)vN(t)e-wX(t)]dt.
It has been shown in Sumita and Shanthikumar [12] that
(2.50)φ(u,v,w,s)=β*(u,w+s)1-vβ(u,s),
where we define, for m≥0 with A̅(t)=∫t∞a(x)dx,(2.51)bm(t)=a(t)g(t,m);bm*(t)=A̅(t)g(t,m),βm(s)=∫0∞e-stbm(t)dt;βm*(s)=∫0∞e-stbm*(t)dt,β(u,s)=∑m=0∞βm(s)um;β*(u,s)=∑m=0∞βm*(s)um.
The Laplace transform generating function of M(t) defined by
(2.52)π(u,s)=∫0∞e-stE[uM(t)]dt
is then given by π(u,s)=φ(u,1,0,s), so that one has from (2.50)
(2.53)π(u,s)=β*(u,s)1-β(u,s).This class of counting processes denoted by ACPGRP may be extended where the underlying renewal process is replaced by an MMPP or an SMMPP. We define the former as a class of age-dependent counting processes governed by an MMPP, denoted by Markov-modulated age-dependent nonhomogeneous Poisson process (MMANHPP), and the latter as a class of age-dependent counting processes governed by an SMMPP, denoted by semi-Markov-modulated age-dependent nonhomogeneous Poisson process (SMMANHPP). The two extended classes are new and become special cases of the unified counting process proposed in this paper as we will see.All of the counting processes discussed in this section are summarized in Figure1.Figure 1
Various counting processes.


## 2.1. Poisson Process

Poisson process of intensityλ is characterized by a sequence of independently and identically distributed (i.i.d.) exponential random variables (Xj)j=1∞ with common probability density function (p.d.f.) fX(x)=λe-λx. Let Sn=∑j=1nXj. Then, the associated Poisson process {N(t):t≥0} is defined as a counting process satisfying
(2.1)N(t)=n⟺Sn≤t<Sn+1.
If a system has an exponential lifetime of mean λ-1 and is renewed instantaneously upon failure, Xj represents the lifetime of the jth renewal cycle. The Poisson process {N(t):t≥0} then counts the number of failures that have occurred by time t.Letpn(t)=P[N(t)=n∣N(0)=0] and define the probability generating function (p.g.f.) π(v,t) by
(2.2)π(v,t)=E[vN(t)]=∑n=0∞pn(t)vn.
It can be seen, see, for example, Karlin and Taylor [14], that
(2.3)ddtpn(t)=-λpn(t)+λpn-1(t)
where pn(t)=0 for n<0. Multiplying vn on both sides of (2.3) and summing from 0 to ∞, one then finds that
(2.4)∂∂tπ(v,t)=-λ(1-v)π(v,t).
Since pn(0)=δ{n=0} where δ{P}=1 if statement P is true and δ{P}=0 otherwise, one has π(v,0)=1. Equation (2.4) can then be solved as
(2.5)π(v,t)=e-λt(1-v);pn(t)=e-λt(λt)nn!.


## 2.2. Nonhomogeneous Poisson Process (NHPP)

An NHPP{M(t):t≥0} differs from a Poisson process in that the failure intensity of the system is given as a function of time t. Accordingly, (2.3) should be rewritten as
(2.6)ddtpm(t)=-λ(t)pm(t)+λ(t)pm-1(t).
By taking the generating function of  (2.6), one finds that
(2.7)∂∂tπ(u,t)=-λ(t)(1-u)π(u,t).
With L(t)=∫0tλ(y)dy, this equation can be solved as
(2.8)π(u,t)=e-L(t)(1-u);pm(t)=e-L(t)L(t)mm!.
The reader is referred to Ross [15] for further discussions of NHPPs.


## 2.3. Markov-Modulated Poisson Process (MMPP)

Let{J(t):t≥0} be a Markov-chain in continuous time on 𝒥={0,…,J} governed by a transition rate matrix ν̲̲=[νij]. Let λ̲⊤=[λ0,…,λJ] and define the associated diagonal matrix λ̲̲D=[δ{i=j}λi]. An MMPP {M(t):t≥0} characterized by (ν̲̲,λ̲̲D) is a pure jump process where jumps of M(t) occur according to a Poisson process with intensity λi whenever the Markov-chain J(t) is in state i.Letνi=∑j∈𝒥νij and define ν̲̲D=[δ{i=j}νi]. The infinitesimal generator Q̲̲ associated with the Markov-chain J(t) is then given by
(2.9)Q̲̲=-ν̲̲D+ν̲̲.
For i,j∈𝒥, let
(2.10)p̲̲(k,t)=[pij(k,t)];pij(k,t)=P[M(t)=k,J(t)=j∣J(0)=i,M(0)=0],
and define the associated matrix generating function π̲̲(u,t) by
(2.11)π̲̲(u,t)=∑k=0∞‍p̲̲(k,t)uk.
It can be seen that
(2.12)∂∂tpij(k,t)=-(λj+νj)pij(k,t)+∑r∈𝒥‍pir(k,t)νrj+λjpij(k-1,t).
In matrix notation, this can be rewritten as
(2.13)∂∂tp̲̲(k,t)=-p̲̲(k,t){λ̲̲D+ν̲̲D-ν̲̲}+p̲̲(k-1,t)λ̲̲D.
By taking the generating function of (2.13) together with (2.9), one sees that
(2.14)∂∂tπ̲̲(u,t)=π̲̲(u,t){Q̲̲-(1-u)λ̲̲D}.
Since M(0)=0, one has π̲̲(u,0)=I̲̲, where I̲̲=[δ{i=j}] is the identity matrix, so that the above differential equation can be solved as
(2.15)π̲̲(u,t)=e{Q̲̲-(1-u)λ̲̲D}t=∑k=0∞tkk!{Q̲̲-(1-u)λ̲̲D}k,
where A̲̲0=defI̲̲ for any square matrix A̲̲. It should be noted that π̲̲(1,t)=eQ̲̲t, which is the transition probability matrix of J(t) as it should be. By taking the Laplace transform of both sides of (2.15), π̲̲̂(u,s)=∫0∞e-stπ̲̲(u,t) is given by
(2.16)π̲̲̂(u,s)={sI̲̲-Q̲̲+(1-u)λ̲̲D}-1.In general, the interarrival times generated by an MMPP are not independent nor identically distributed. In multimedia computer and communication networks, data packets are mingled together with voice and image packets generated from analogue sources. Since arrival patterns of such packets differ from each other, MMPPs have provided useful means to model arrival processes of packets in multimedia computer and communication networks, see, for example, Heffes and Lucantoni [16] and Sriram and Whitt [17]. Burman and Smith [18], and Knessl et al. [19] studied a single server queuing system with an MMPP arrival process and general i.i.d. service times. Neuts et al. [20] established characterization theorems for an MMPP to be a renewal process in terms of lumpability of the underlying Markov-chain J(t). The reader is referred to Neuts [4] for further discussions of MMPP.An MMPP can be extended by replacing the underlying Markov-chain in continuous time by a semi-Markov process as discussed in Section1. This process is denoted by SMMPP. To the best knowledge of the authors, the SMMPP is first addressed by Dshalalow [6], where a systematic approach for dealing with modulated random measures is proposed in a more abstract way. Such modulated random measures include the MMPP and the SMMPP as special cases. An application of the SMMPP to queueing theory is addressed in Dshalalow and Russell [7]. More recently, the original approach above has been further extended and refined in a series of papers by Agarwal et al. [8] and Dshalalow [9, 10]. The SMMPP is studied in detail independently by Özekici and Soyer [11] including transient characterizations and ergodic analysis. Both MMPP and SMMPP will be proven to be expressible in terms of the components of the unified multivariate counting process proposed in this paper.


## 2.4. Renewal Process

Renewal processes can be considered as a generalization of Poisson processes in that a sequence of i.i.d. exponential random variables are replaced by that of any i.i.d. nonnegative random variables with common distribution functionA(x). The resulting counting process {N(t):t≥0} is still characterized by (2.1). Let pn(t)=P[N(t)=n∣N(0)=0] as before. One then sees that
(2.17)pn(t)=A(n)(t)-A(n+1)(t),
where A(n)(t) denotes the n-fold convolution of A(x) with itself, that is,  A(n+1)(t)=∫0tA(n)(t-x)dA(x) and A(0)(t)=U(t) which is the step function defined as U(t)=1 for t≥0 and U(t)=0 else.Letπn(s)=∫0∞e-stpn(t)dt and α(s)=∫0∞e-stdA(t). By taking the Laplace transform of both sides of (2.17), it follows that
(2.18)πn(s)=1-α(s)sα(s)n.
By taking the generating function of the above equation with π(v,s)=∑n=0∞πn(s)vn, one has
(2.19)π(v,s)=1-α(s)s11-vα(s).
The reader is referred to Cox [21], or Karlin and Taylor [14] for further discussions of renewal processes.


## 2.5. Markov Renewal Process (MRP)

An MRP is an extension of an ordinary renewal process in that, in the interval[0,t), the former describes the recurrence statistics for intermingling classes of epochs of an underlying semi-Markov process, whereas the latter counts the number of recurrences for a single recurrent class of epochs. More specifically, let {J(t):t≥0} be a semi-Markov process on 𝒥={0,…,J} governed by a matrix p.d.f. a̲̲(x) where a̲̲(x)≥0̲̲ and ∫0∞a̲̲(x)dx=A̲̲0 is a stochastic matrix which is assumed to be ergodic. Let ε(ℓ) be a recurrent class consisting of the entries of the semi-Markov process to state ℓ for ℓ∈𝒥, and define Ñℓr(t) to be a counting process describing the number of recurrences for ε(r) given that there was an epoch of ε(ℓ) at time t=0. Then Ñ̲ℓ(t)=[Ñℓ0(t),…,ÑℓJ(t)] is called an MRP.The study of MRPs can be traced back to early 1960s represented by the two original papers by Pyke [1, 2], followed by Keilson [22, 23], Keilson and Wishart [24, 25], Çinlar [26, 27] and McLean and Neuts [28]. Since then, the area attracted many researchers and a survey paper by Çinlar [29] in 1975 already included more than 70 leading references. The study has been largely focused on the matrix renewal function H̲̲(t)=[Hℓr(t)] with Hℓr(t)=E[Ñℓr(t)], the associated matrix renewal density, and the limit theorems. For example, one has the following result concerning the Laplace transform of H̲̲(t) by Keilson [23]:
(2.20)ℒ{H̲̲(t)}=1sα̲̲(s)[I̲̲-α̲̲(s)]-1,
where α̲̲(s) is the Laplace transform of a̲̲(t). The unified multivariate counting process of this paper contains an MRP as a special case and provides more information based on dynamic analysis of the underlying probabilistic flows.


## 2.6. Number of Entries of a Semi-Markov Process into a Subset of the State Space (NESMPS)

Another type of counting processes associated with a semi-Markov process on𝒥={0,…,J} governed by a matrix p.d.f. a̲̲(x) is studied in Masuda and Sumita [3], where the state space 𝒥 is decomposed into a set of good states G(≠ϕ) and a set of bad states B(≠ϕ) satisfying 𝒥=G∪B and G∩B=ϕ. The counting process NGB(t) is then defined to describe the number of entries of J(t) into B by time t.WhileNGB(t) is a special case of MRPs, the detailed analysis is provided in [3], yielding much more information. More specifically, let X(t) be the age process associated with J(t), that is,
(2.21)X(t)=t-sup{τ:J(t)∣τ-τ+≠0,0<τ≤t},
where f(x)∣x-x+=f(x+)-f(x-), and define
(2.22)F̲̲n(x,t)=[Fn:ij(x,t)],
where
(2.23)Fn:ij(x,t)=P[X(t)≤x,NGB(t)=n,J(t)=j∣X(0)=NGB(0)=0,J(0)=i].
One then has
(2.24)f̲̲n(x,t)=∂∂xF̲̲n(x,t).
The associated matrix Laplace transform generating function can then be defined as
(2.25)φ̲̲(v,w,s)=∑n=0∞vn∬0∞e-wt-stf̲̲n(x,t)dxdt.
It has been shown in Masuda and Sumita [3] that
(2.26)φ̲̲(v,w,s)=1w+s·γ̲̲0(s){I̲̲-vβ̲̲(s)}-1{I̲̲-α̲̲D(w+s)}.
Here, with α̲̲(s)=∫0∞e-sta̲̲(t)dt and α̲̲CD(s)=[αij(s)]i∈C,j∈D for C,D⊂𝒥, the following notation is employed:(2.27)α̲̲(s)=[α̲̲GG(s)α̲̲GB(s)α̲̲BG(s)α̲̲BB(s)];α̲̲D(s)=[δ{i=j}αi(s)]withαi(s)=∑j∈𝒥αij(s);(2.28)χ̲̲G(s)={I̲̲GG-α̲̲GG(s)}-1;χ̲̲B(s)={I̲̲BB-α̲̲BB(s)}-1;(2.29)β̲̲(s)=[0̲̲BB0̲̲BGα̲̲GBχ̲̲B(s)α̲̲GBχ̲̲B(s)α̲̲BGχ̲̲G(s)];(2.30)γ̲̲0(s)=[α̲̲BBχ̲̲B(s)χ̲̲B(s)α̲̲BGχ̲̲G(s)0̲̲GBα̲̲GGχ̲̲G(s)]+I̲̲.
As we will see, the unified multivariate counting process proposed in this paper enables one to deal with multidimensional generalization of NESMPSs as a special case.


## 2.7. Markovian Arrival Process (MAP)

As for Poisson processes, a renewal process requires interarrival times to form a sequence of i.i.d. nonnegative random variables. As we have seen, a class of MMPPs enables one to avoid this requirement by introducing different Poisson arrival rates depending on the state of the underlying Markov chain. An alternative way to avoid this i.i.d. requirement is to adapt a class of MAPs, originally introduced by Lucantoni et al. [5]. We discuss here a slightly generalized version of MAPs in that a set of absorbing states is not necessarily a singleton set.Let{J*(t):t≥0} be an absorbing Markov-chain on 𝒥*=G∪B with G≠ϕ,B≠ϕ and G∩B=ϕ, where all states in B are absorbing. Without loss of generality, we assume that G={0,…,m}, and B={m+1,…,m+K}. For notational convenience, the following transition rate matrices are introduced.(2.31)ν̲̲GG*=[νij]i,j∈G;ν̲̲GB*=[νir]i∈G,r∈B.
The entire transition rate matrix ν̲̲* governing J*(t) is then given by
(2.32)ν̲̲*=[ν̲̲GG*ν̲̲GB*0̲̲0̲̲].A replacement Markov-chain{J(t):t≥0} on G is now generated from {J*(t):t≥0}. Starting from a state in G, the process J(t) coincides with J*(t) within G. As soon as J*(t) reaches state r∈B, it is instantaneously replaced at state j∈G with probability p̃rj and the process continues. Let
(2.33)C̲̲=ν̲̲GG*,D̲̲=ν̲̲GB*p̲̲̃BG,
where p̲̲̃BG=[p̃rj]r∈B,j∈G. Then the transition rate matrix ν̲̲ and the infinitesimal generator Q̲̲ of J(t) are given as
(2.34)ν̲̲=C̲̲+D̲̲;Q̲̲=-ν̲̲D+ν̲̲,
where
(2.35)ν̲̲D=C̲̲D+D̲̲D,
with
(2.36)C̲̲D=[δ{i=j}ci];ci=∑j∈Gcij,D̲̲D=[δ{i=j}di];di=∑j∈Gdij.
Let p̲̲(t) be the transition probability matrix of J̲̲(t) with its Laplace transform defined by π̲̲(s)=∫0∞e-stp̲̲(t)dt. From the Kolmogorov forward equation (d/dt)p̲̲(t)=p̲̲(t)Q̲̲ with p̲̲(0)=I̲̲, one has
(2.37)π̲̲(s)={sI̲̲-Q̲̲}-1.Let{NMAP(t):t≥0} be the counting process keeping the record of the number of replacements in [0,t) and define
(2.38)fij(k,t)=P[NMAP(t)=k,J(t)=j∣J(0)=i],i,j∈G.
By analyzing the probabilistic flow at state j at time t+Δ, it can be seen that
(2.39)fij(k,t+Δ)=fij(k,t){1-∑ℓ∈G(cjℓ+djℓ)Δ}+∑ℓ∈Gfiℓ(k,t)cℓjΔ+∑ℓ∈Gfiℓ(k-1,t)dℓjΔ+o(Δ).
It then follows that
(2.40)∂∂tfij(k,t)=-fij(k,t)∑ℓ∈G(cjℓ+djℓ)+∑ℓ∈Gfiℓ(k,t)cℓj+∑ℓ∈Gfiℓ(k-1,t)dℓj.
In matrix notation, the above equation can be rewritten as
(2.41)∂∂tf̲̲(k,t)=-f̲̲(k,t)ν̲̲D+f̲̲(k,t)C̲̲+f̲̲(k-1,t)D̲̲.
We now introduce the following matrix Laplace transform generating function:
(2.42)φ̲̲(v,s)=[φij(v,s)];φij(v,s)=∫0∞e-st∑k=0∞fij(k,t)vkdt.
Taking the Laplace transform with respect to t and the generating function with respect to k of both sides of (2.41), one has
(2.43)φ̲̲(v,s){sI̲̲+ν̲̲D-C̲̲-vD̲̲}=I̲̲,
which can be solved as
(2.44)φ̲̲(v,s)={sI̲̲-Q̲̲+(1-v)D̲̲}-1.
We note from (2.37) and (2.44) that φ̲̲(1,s)=π̲̲(s) as it should be.It may be worth noting that an MMPP is a special case of an MAP, which can be seen in the following manner. Let an MAP be defined onG∪B where G={0G,…,JG} and B={0B,…,JB}. Transitions within G is governed by ν̲̲. An entry into jB∈B is possible only from jG∈G. When this occurs, the Markov process is immediately replaced at the entering state jG. The counting process for the number of such replacements then has the Laplace transform generating function φ̲̲(v,s) given in (2.44) where D̲̲ is replaced by λ̲̲D, which coincides with π̲̲̂(u,s) of (2.16) for MMPPs, proving the claim.


## 2.8. Age-Dependent Counting Process Generated from a Renewal Process (ACPGRP)

An age-dependent counting process generated from a renewal process has been introduced and studied by Sumita and Shanthikumar [12], where items arrive according to an NHPP which is interrupted and reset at random epochs governed by a renewal process. More specifically, let {N(t):t≥0} be a renewal process associated with a sequence of i.i.d. nonnegative random variables (Xj)j=1∞ with common p.d.f. a(x). The age process X(t) is then defined by
(2.45)X(t)=t-sup{τ:N(t)∣τ-τ+=1,0<τ≤t}.
In other words, X(t) is the elapsed time since the last renewal. We next consider an NHPP Z(x) governed by an intensity function λ(x). If we define
(2.46)L(x)=∫0xλ(y)dy,
one has
(2.47)g(x,k)=P[Z(x)=k]=exp(-L(x))L(x)kk!,k=0,1,2,….
Of interest, then, is a counting process {M(t):t≥0} characterized by
(2.48)P[M(t+Δ)-M(t)=1∣M(t)=m,N(t)=1,X(t)=x]=λ(x)Δ+o(Δ),(m,n,x)∈S,Δ>0.
Here S=ℤ+×ℤ+×ℝ+ where ℤ+ is the set of nonnegative integers, and ℝ+ is the set of nonnegative real numbers.Since the counting processM(t) is governed by the intensity function depending on the age process X(t) of the renewal process N(t), it is necessary to analyze the trivariate process [M(t),N(t),X(t)]. Let the multivariate transform of [M(t),N(t),X(t)] be defined by
(2.49)φ(u,v,w,s)=∫0∞e-stE[uM(t)vN(t)e-wX(t)]dt.
It has been shown in Sumita and Shanthikumar [12] that
(2.50)φ(u,v,w,s)=β*(u,w+s)1-vβ(u,s),
where we define, for m≥0 with A̅(t)=∫t∞a(x)dx,(2.51)bm(t)=a(t)g(t,m);bm*(t)=A̅(t)g(t,m),βm(s)=∫0∞e-stbm(t)dt;βm*(s)=∫0∞e-stbm*(t)dt,β(u,s)=∑m=0∞βm(s)um;β*(u,s)=∑m=0∞βm*(s)um.
The Laplace transform generating function of M(t) defined by
(2.52)π(u,s)=∫0∞e-stE[uM(t)]dt
is then given by π(u,s)=φ(u,1,0,s), so that one has from (2.50)
(2.53)π(u,s)=β*(u,s)1-β(u,s).This class of counting processes denoted by ACPGRP may be extended where the underlying renewal process is replaced by an MMPP or an SMMPP. We define the former as a class of age-dependent counting processes governed by an MMPP, denoted by Markov-modulated age-dependent nonhomogeneous Poisson process (MMANHPP), and the latter as a class of age-dependent counting processes governed by an SMMPP, denoted by semi-Markov-modulated age-dependent nonhomogeneous Poisson process (SMMANHPP). The two extended classes are new and become special cases of the unified counting process proposed in this paper as we will see.All of the counting processes discussed in this section are summarized in Figure1.Figure 1
Various counting processes.


## 3. Unified Multivariate Counting Process[M̲(t),N̲̲(t)]

In this section, we propose a stochastic process representing the unified multivariate counting process discussed in Section1, which would contain all of the counting processes given in Section 2 as special cases. More specifically, we consider a system where items arrive according to an NHPP. This arrival stream is governed by a finite semi-Markov process on 𝒥={0,…,J} in that the intensity function of the NHPP depends on the current state of the semi-Markov process. That is, when the semi-Markov process is in state i with the current dwell time of x, items arrive according to a Poisson process with intensity λi(x). If the semi-Markov process switches its state from i to j, the intensity function λi(x) is interrupted, the intensity function at state j is reset to λj(0), and the arrival process resumes. Of particular interest would be the multivariate counting processes M̲(t)⊤=[M0(t),…,MJ(t)] and N̲̲(t)=[Nij(t)] with Mi(t) and Nij(t) counting the number of items that have arrived in state i by time t and the number of transitions of the semi-Markov process from state i to state j by time t respectively. The two counting processes M̲(t) and N̲̲(t) enable one to introduce a variety of interesting performance indicators as we will see.Formally, let{J(t):t≥0} be a semi-Markov process on 𝒥={0,…,J} governed by a matrix cumulative distribution function (c.d.f.)
(3.1)A̲̲(x)=[Aij(x)],
which is assumed to be absolutely continuous with the matrix probability density function (p.d.f.)
(3.2)a̲̲(x)=[aij(x)]=ddxA̲̲(x).
It should be noted that, if we define Ai(x) and A̅i(x) by
(3.3)Ai(x)=∑j∈𝒥Aij(x);A̅i(x)=1-Ai(x),
then Ai(x) is an ordinary c.d.f. and A̅i(x) is the corresponding survival function. The hazard rate functions associated with the semi-Markov process are then defined as
(3.4)ηij(x)=aij(x)A̅i(x),i,j∈𝒥.For notational convenience, the transition epochs of the semi-Markov process are denoted byτn,n≥0, with τ0=0. The age process X(t) associated with the semi-Markov process is then defined as
(3.5)X(t)=t-max{τn:0≤τn≤t}.
At time t with J(t)=i and X(t)=x, the intensity function of the NHPP is given by λi(x). For the cumulative arrival intensity function Li(x) in state i, one has
(3.6)Li(x)=∫0xλi(y)dy.
The probability of observing k arrivals in state i within the current age of x can then be obtained as
(3.7)gi(x,k)=e-Li(x)Li(x)kk!,k=0,1,2,…,i∈𝒥.Of interest are the multivariate counting processes(3.8)M̲(t)⊤=[M0(t),…,MJ(t)],
where Mi(t) represents the total number of items that have arrived by time t while the semi-Markov process stayed in state i, and
(3.9)N̲̲(t)=[Nij(t)],
with Nij(t) denoting the number of transitions of the semi-Markov process from state i to state j by time t. It is obvious that Ni(t)=def∑ℓ∈𝒥Nℓi(t) denotes the number of entries into state i by time t. The initial state is not included in N•i(t) for any i∈𝒥. In other words, if J(0)=i, N•i(t) remains 0 until the first return of the semi-Markov process to state i. In the next section, we will analyze the dynamic behavior of [M̲(t),N̲̲(t)], yielding the relevant Laplace transform generating functions. In the subsequent section, all of the counting processes discussed in Section 2 would be expressed in terms of M̲(t) and N̲̲(t), thereby providing a unified approach for studying various counting processes. The associated asymptotic behavior as t→∞ would be also discussed.


## 4. Dynamic Analysis of[M̲(t),N̲̲(t)]

The purpose of this section is to examine the dynamic behavior of the multivariate stochastic process[M̲(t),N̲̲(t)] introduced in Section 3 by observing its probabilistic flow in the state space. Figure 2 depicts a typical sample path of the multivariate process.Figure 2
Typical sample path of[M̲(t),N̲̲(t)].Since[M̲(t),N̲̲(t)] is not Markov, we employ the method of supplementary variables. More specifically, the multivariate stochastic process [M̲(t),N̲̲(t),X(t),J(t)] is considered. This multivariate stochastic process is Markov and has the state space S=ℤ+J+1×ℤ+(J+1)×(J+1)×ℝ+×𝒥, where ℤ+J+1 and ℤ+(J+1)×(J+1) are the set of (J+1) and (J+1)×(J+1) dimensional nonnegative integer vectors and matrices respectively, ℝ+ is the set of nonnegative real numbers and 𝒥={0,…,J}. Let Fij(m̲,n̲̲,x,t) be the joint probability distribution of [M̲(t),N̲̲(t),X(t),J(t)] defined by
(4.1)Fij(m̲,n̲̲,x,t)=P[M̲(t)=m̲,N̲̲(t)=n̲̲,X(t)≤x,J(t)=j∣M̲(0)=0̲,N̲̲(0)=0̲̲,J(0)=i].In order to assure the differentiability ofFij(m̲,n̲̲,x,t) with respect to x, we assume that X(0) has an absolutely continuous initial distribution function D(x) with p.d.f. d(x)=(d/dx)D(x). (If X(0)=0 with probability 1, we consider a sequence of initial distribution functions {Dk(x)}k=1∞ satisfying Dk(x)→U(x) as k→∞ where U(x)=1 for x≥0 and U(x)=0 otherwise. The desired results can be obtained through this limiting process.) One can then define
(4.2)fij(m̲,n̲̲,x,t)=∂∂xFij(m̲,n̲̲,x,t).
By interpreting the probabilistic flow of the multivariate process [M̲(t),N̲̲(t),X(t),J(t)] in its state space, one can establish the following equations:
(4.3)fij(m̲,n̲̲,x,t)=δ{i=j}δ{m̲=mi1̲i}δ{n̲̲=0̲̲}d(x-t)A̅i(x)A̅i(x-t)gi(t,mi)+(1-δ{n̲̲=0̲̲})∑k=0mjfij(m̲-k1̲j,n̲̲,0+,t-x)A̅j(x)gj(x,k);(4.4)fij(m̲,n̲̲,0+,t)=(1-δ{n̲̲=0̲̲})∑ℓ∈𝒥∫0∞fiℓ(m̲,n̲̲-1̲̲ℓj,x,t)ηℓj(x)dx;(4.5)fij(m̲,n̲̲,x,0)=δ{i=j}δ{m̲=0̲}δ{n̲̲=0̲̲}d(x),
where 1̲i is the column vector whose ith element is equal to 1 with all other elements being 0, 1̲̲ij=1̲i1̲j⊤ and fij(m̲,n̲̲,0+,t)=0forN̲̲≤0̲̲.The first term of the right-hand side of (4.3) represents the case that J(t) has not left the initial state J(0)=i by time t[δ{i=j}=1andδ{n̲̲=0̲̲}=1] and there have been mi items arrived during time t[δ{m̲=mi1̲i}=1], provided that J(0)=i  and  X(0)=x-t. The second term corresponds to the case that J(t) made at least one transition from J(0)=i by time t[δ{n̲̲=0̲̲}=0], the multivariate process [M̲(t),N̲̲(t),X(t),J(t)] just entered the state [m̲-k1̲j,n̲̲,0+,j] at time t-x, J(t) remained in state j until time t with X(t)=x, and there have been k items arrived during the current age x, provided that J(0)=i and X(0)=x-t. For the multivariate process at [m̲,n̲̲,0+,j] at time t, it must be at [m̲,n̲̲-1̲̲ℓj,x,ℓ] followed by a transition from ℓ to j at time t which increases Nℓj(t) by one, explaining (4.4). Equations (4.5) merely describe the initial condition that [M̲(0),N̲̲(0),X(0),J(0)]=[0̲,0̲̲,x,i].In what follows, the dynamic behavior of the multivariate process[M̲(t),N̲̲(t),X(t),J(t)] would be captured by establishing the associated Laplace transform generating functions based on (4.3), (4.4) and (4.5). For notational convenience, the following matrix functions are employed:
(4.6)b̲̲k(t)=[bk:ij(t)];bk:ij(t)=aij(t)gi(t,k),(4.7)b̲̲k:D*(t)=[δ{i=j}bk:i*(t)]=[bk:0*(t)⋱bk:J*(t)];bk:i*(t)=A̅i(t)gi(t,k),(4.8)r̲̲k*(t)=[rk:ij*(t)];rk:ij*(t)=gi(t,k)∫0∞d(x-t)aij(x)A̅i(x-t)dx,(4.9)β̲̲k(s)=[βk:ij(s)];βk:ij(s)=∫0∞e-stbk:ij(t)dt,(4.10)β̲̲(u̲,s)=[βij(ui,s)];βij(ui,s)=∑k=0∞βk:ij(s)uik,(4.11)β̲̲k:D*(s)=[βk:0*(s)⋱βk:J*(s)];βk:i*(s)=∫0∞e-stbk:i*(t)dt,(4.12)β̲̲D*(u̲,s)=[β0*(u0,s)⋱βJ*(uJ,s)];βi*(ui,s)=∑k=0∞βk:i*(s)uik,(4.13)ρ̲̲k*(s)=[ρk:ij*(s)];ρk:ij*(s)=∫0∞e-strk:ij*(t)dt,(4.14)ρ̲̲*(u̲,s)=[ρij*(ui,s)];ρij*(ui,s)=∑k=0∞ρk:ij*(s)uik,(4.15)ξ̲̲(m̲,n̲̲,0+,s)=[ξij(m̲,n̲̲,0+,s)];ξij(m̲,n̲̲,0+,s)=∫0∞e-stfij(m̲,n̲̲,0+,t)dt,(4.16)ξ̲̲̂(u̲,v̲̲,0+,s)=[ξ̂ij(u̲,v̲̲,0+,s)];ξ̂ij(u̲,v̲̲,0+,s)=∑m̲∈ℤ+J+1∑n̲̲∈ℤ+(J+1)×(J+1)∖{0̲̲}ξij(m̲,n̲̲,0+,s)u̲m̲v̲̲n̲̲,(4.17)φ̲̲(m̲,n̲̲,w,s)=[φij(m̲,n̲̲,w,s)];φij(m̲,n̲̲,w,s)=∫0∞∫0∞e-wxe-stfij(m̲,n̲̲,x,t)dtdx,(4.18)φ̲̲̂(u̲,v̲̲,w,s)=[φ̂ij(u̲,v̲̲,w,s)];φ̂ij(u̲,v̲̲,w,s)=∑m̲∈ℤ+J+1∑n̲̲∈ℤ+(J+1)×(J+1)φij(m̲,n̲̲,w,s)u̲m̲v̲̲n̲̲,
where u̲m̲v̲̲n̲̲=∏i∈𝒥uimi∏(i,j)∈𝒥×𝒥∖{(0,0)}vijnij. We are now in a position to prove the main theorem of this section.Theorem.
LetX(0)=0. Then:
(4.19)ξ̲̲̂(u̲,v̲̲,0+,s)=β̲̲̃(u̲,v̲̲,s){I̲̲-β̲̲̃(u̲,v̲̲,s)}-1;(4.20)φ̲̲̂(u̲,v̲̲,w,s)={I̲̲-β̲̲̃(u̲,v̲̲,s)}-1β̲̲D*(u̲,w+s),
where β̲̲̃(u̲,v̲̲,s)=[vij·βij(ui,s)].Proof.
First, we assume thatX(0) has a p.d.f. d(x). Substituting (4.3) and (4.5) into (4.4), one sees that
(4.21)fij(m̲,n̲̲,0+,t)=(1-δ{n̲̲=0̲̲})×{∑ℓ∈𝒥∫0∞δ{i=ℓ}δ{m̲=mi1̲i}δ{n̲̲=1̲̲ℓj}d(x-t)A̅i(x)A̅i(x-t)gi(t,mi)ηℓj(x)dx+∑ℓ∈𝒥(1-δ{n̲̲=1̲̲ℓj})∫0∞∑k=0mℓfiℓ(m̲-k1̲ℓ,n̲̲-1̲̲ℓj,0+,t-x)×A̅ℓ(x)gℓ(x,k)ηℓj(x)dx{∑ℓ∈𝒥∫0∞δ{i=ℓ}}}=(1-δ{n̲̲=0̲̲}){δ{m̲=mi1̲i}δ{n̲̲=1̲̲ij}gi(t,mi)∫0∞d(x-t)aij(x)A̅i(x-t)dx+∑ℓ∈𝒥(1-δ{n̲̲=1̲̲ℓj})∑k=0mℓ∫0∞fiℓ(m̲-k1̲ℓ,n̲̲-1̲̲ℓj,0+,t-x)×aℓj(x)gℓ(x,k)dx{∑ℓ∈𝒥∫0∞δ{i=ℓ}}}.
Consequently, one sees from (4.6) and (4.8) that
(4.22)fij(m̲,n̲̲,0+,t)=(1-δ{n̲̲=0̲})×{δ{m̲=mi1̲i}δ{n̲̲=1̲̲ij}rmi:ij*(t)+∑ℓ∈𝒥(1-δ{n̲̲=1̲̲ℓj})×∑k=0mℓ∫0∞fiℓ(m̲-k1̲ℓ,n̲̲-1̲̲ℓj,0+,t-x)bk:ℓj(x)dx}.
where aij(x)=A̅i(x)ηij(x) is employed from (3.4). By taking the Laplace transform of both sides of (4.22) with respect to t, it follows that
(4.23)ξij(m̲,n̲̲,0+,s)=(1-δ{n̲̲=0̲̲})×{δ{m̲=mi1̲i}δ{n̲̲=1̲̲ij}ρmi:ij*(s)+∑ℓ∈𝒥(1-δ{n̲̲=1̲̲ℓj})∑k=0mℓξiℓ(m̲-k1̲ℓ,n̲̲-1̲̲ℓj,0+,s)βk:ℓj(s)}.
By taking the multivariate generating function of (4.23) with respect to m̲ and n̲̲, it can be seen that
(4.24)ξ̂ij(u̲,v̲̲,0+,s)=∑m̲∈ℤ+J+1∑n̲̲∈ℤ+(J+1)×(J+1)∖{0̲̲}ξij(m̲,n̲̲,0+,s)u̲m̲v̲̲n̲̲=∑m̲∈ℤ+J+1∑n̲̲∈ℤ+(J+1)×(J+1)∖{0̲̲}{δ{m̲=mi1̲i}δ{n̲̲=1̲̲ij}ρmi:ij*(s)u̲m̲v̲̲n̲̲+∑ℓ∈𝒥(1-δ{n̲̲=1̲̲ℓj})∑k=0mℓξiℓ(m̲-k1̲ℓ,n̲̲-1̲̲ℓj,0+,s)βk:ℓj(s)u̲m̲v̲̲n̲̲}.
It then follows from (4.10), (4.14), (4.16) and the discrete convolution property that
(4.25)ξ̂ij(u̲,v̲̲,0+,s)=∑mi=0∞uimivijρmi:ij*(s)+∑ℓ∈𝒥(vℓj∑m̲∈ℤ+J+1∑k=0mℓ∑n̲̲∈ℤ+(J+1)×(J+1)∖{0̲̲}ξiℓ(m̲-k1̲ℓ,n̲̲,0+,s)βk:ℓj(s)u̲m̲v̲̲n̲̲)=vijρij*(ui,s)+∑ℓ∈𝒥vℓjξ̂iℓ(u̲,v̲̲,0+,s)βℓj(uℓ,s).
The last expression can be rewritten in matrix form, and one has
(4.26)ξ̲̲̂(u̲,v̲̲,0+,s)=ρ̲̲̃*(u̲,v̲̲,s)+ξ̲̲̂(u̲,v̲̲,0+,s)β̲̲̃(u̲,v̲̲,s),
which can be solved for ξ̲̲̂(u̲,v̲̲,0+,s) as
(4.27)ξ̲̲̂(u̲,v̲̲,0+,s)=ρ̲̲̃*(u̲,v̲̲,s){I̲̲-β̲̲̃(u̲,v̲̲,s)}-1,
where ρ̲̲̃*(u̲,v̲̲,s)=[vij·ρij*(ui,s)].
Next, we introduce the following double Laplace transform:
(4.28)εk:i(w,s)=∬0∞e-wxe-std(x-t)A̅i(x)A̅i(x-t)gi(t,k)dtdx,
and the associated diagonal matrix
(4.29)ε̲̲D(u̲,w,s)=[ε0*(u0,w,s)⋱εJ*(uJ,w,s)];εi(ui,w,s)=∑k=0∞εk:i(w,s)uik.
By taking the double Laplace transform of (4.3), one sees from (4.7) and (4.28) that
(4.30)φij(m̲,n̲̲,w,s)=δ{i=j}δ{m̲=mi1̲i}δ{n̲̲=0̲̲}εmi:i(w,s)+(1-δ{n̲̲=0̲̲})∑k=0mjξij(m̲-k1̲j,n̲̲,0+,s)βk:j*(w+s).
By taking the double generating function, this then leads to
(4.31)φ̂ij(u̲,v̲̲,w,s)=∑m̲∈ℤ+J+1∑n̲̲∈ℤ+(J+1)×(J+1)φij(m̲,n̲̲,w,s)u̲m̲v̲̲n̲̲=δ{i=j}∑mi=0∞uimiεmi:i(w,s)+∑m̲∈ℤ+J+1∑n̲̲∈ℤ+(J+1)×(J+1)∖{0̲̲}(∑k=0mjξij(m̲-k1̲j,n̲̲,0+,s)βk:j*(w+s)u̲m̲v̲̲n̲̲)=δ{i=j}εi(ui,w,s)+ξ̂ij(u̲,v̲̲,0+,s)βj*(uj,w+s).
By rewriting the last expression in matrix form, it can be seen that
(4.32)φ̲̲̂(u̲,v̲̲,w,s)=ε̲̲D(u̲,w,s)+ξ̲̲̂(u̲,v̲̲,0+,s)β̲̲D*(u̲,w+s).
We now consider the limiting processD(x)→U(x). For the p.d.f., this limiting process becomes d(x)→δ(x) where δ(x) is the Delta function defined as a unit function for convolution, that is,  f(x)=∫f(x-τ)δ(τ)dτ for any integrable function f. Accordingly, it can be seen from (4.8) that rk:ij*(t)→bk:ij(t). This in turn implies from (4.28) that εk:i(w,s)→βk:i*(w+s). Consequently, it follows in matrix form that ρ̲̲̃*(u̲,v̲̲,s)→β̲̲̃(u̲,v̲̲,s) and ε̲̲D(u̲,w,s)→β̲̲D*(u̲,w+s). From (4.27), it can be readily seen that ξ̲̲̂(u̲,v̲̲,0+,s)→β̲̲̃(u̲,v̲̲,s){I̲̲-β̲̲̃(u̲,v̲̲,s)}-1, proving (4.19). One also sees from (4.28) that
(4.33)φ̲̲̂(u̲,v̲̲,w,s)→β̲̲D*(u̲,w+s)+ξ̲̲̂(u̲,v̲̲,0+,s)β̲̲D*(u̲,w+s)=[I̲̲+β̲̲̃(u̲,v̲̲,s){I̲̲-β̲̲̃(u̲,v̲̲,s)}-1]β̲̲D*(u̲,w+s)=[I̲̲-β̲̲̃(u̲,v̲̲,s)+β̲̲̃(u̲,v̲̲,s)]{I̲̲-β̲̲̃(u̲,v̲̲,s)}-1β̲̲D*(u̲,w+s)={I̲̲-β̲̲̃(u̲,v̲̲,s)}-1β̲̲D*(u̲,w+s),
which proves (4.20), completing the proof.In the next section, it will be shown that all the transform results obtained in Section 2 can be derived from Theorem 4.1.


## 5. Derivation of the Special Cases from the Unified Counting Process

We are now in a position to demonstrate the fact that the proposed multivariate counting process introduced in Section3 and analysed in Section 4 indeed unifies the existing counting processes discussed in Section 2. We will do so by deriving the transform results of Section 2 from Theorem 4.1.


### 5.1. Derivation of Poisson Process

LetN(t) be a Poisson process with intensity λ as discussed in Section 2.1. From (2.5), one sees that π̂(v,s)=∫0∞e-stE[vN(t)]dt=∫0∞e-stπ(v,t)dt is given by
(5.1)π̂(v,s)=1s+λ(1-v).For the unified counting process, we consider a single state Markov-chain in continuous time where only the number of self transitions in(0,t] is counted. More specifically, let 𝒥={0}, N̲̲(t)=[N00(t)], u0=1, v00=v, w=0, λ0(t)=0 and a00(x)=λe-λx. We note from (3.7) that λ0(t)=0 implies g0(x,k)=δ{k=0} so that bk:00(t)=δ{k=0}a00(t) from (4.6). This then implies β00(1,s)=λ/(s+λ). Similarly, since A̅0(x)=e-λx, one has β0*(1,s)=1/(s+λ). It then follows from Theorem 4.1 that
(5.2)φ̂0(1,v,0+,s)=11-vβ00(1,s)β0*(1,s)=1s+λ(1-v).
Hence, from (5.1) and (5.2), one concludes that π̂(v,s)=φ̂0(1,v,0+,s).


### 5.2. Derivation of NHPP

LetM(t) be an NHPP of Section 2.2 characterized by a time dependent intensity function λ(t). It can be seen from (2.8) that π̂(u,s)=∫0∞e-stE[uM(t)]dt=∫0∞e-stπ(u,t)dt is given by
(5.3)π̂(u,s)=∫0∞e-ste-L(t)(1-u)dt.In order to see thatM(t) can be viewed as a special case of the proposed multivariate counting process, we first consider a single state semi-Markov process where the dwell time in the state is deterministic given by T. The marginal counting process M̲(t)=[M0(t)] then converges in distribution to M(t) as T→∞ as we show next.Let𝒥={0}, u0=u, v00=1, w=0, and λ0(x)=λ(x). It then follows that a00(x)=δ(x-t) and therefore bk:00(t)=δ(t-T)g0(t,k) from (4.6). This in turn implies that
(5.4)β00(u,s)=∫0∞e-stδ(t-T)e-L0(t)(1-u)dt=e-{sT+L0(T)(1-u)}.
Let U(t) be the step function defined by U(t)=0 if t<0 and U(t)=1 otherwise. Since A̅00(t)=1-U(t-T)=δ{0≤t<T}, one sees that
(5.5)β0*(u,s)=∫0∞e-stδ{0≤t<T}e-L0(t)(1-u)dt=∫0Te-ste-L0(t)(1-u)dt.
Theorem 4.1 together with (5.4) and (5.5) then leads to
(5.6)φ̂00(u,1,0+,s)=11-β00(u,s)β0*(u,s)=∫0Te-ste-L0(t)(1-u)dt1-e-{sT+L0(T)(1-u)}.
Now it can be readily seen that φ̂00(u,1,0+,s) in (5.6) converges to π̂(u,s) in (5.3) as T→∞, proving the claim.


### 5.3. Derivation of MMPP

LetM(t) be an MMPP of Section 2.3 characterized by (ν̲̲,λ̲̲D). We show that the Laplace transform generating function π̂(u,s)=∫0∞e-stE[uM(t)]dt given in (2.16) can be derived as a special case of Theorem 4.1.For the unified multivariate counting process, let𝒥={0,…,J}, M(t)=∑i∈𝒥Mi(t), u̲=u1̲, v̲̲=1̲̲, w=0, λi(t)=λi, and a̲̲(x)=[e-νixνij]=e-ν̲̲Dxν̲̲. From (3.3) one sees that A̲̲̅D(x)=I̲̲-A̲̲D=[δ{i=j}{1-∑j∈𝒥Aij(x)}]=[δ{i=j}e-νix]=e-ν̲̲Dx. Therefore, one sees from (4.6), (4.9) and (4.10) that
(5.7)β̲̲(u1̲,s)=∑k=0∞∫0∞e-stg̲̲D(t,k)a̲̲(t)dtuk1̲={sI̲̲+ν̲̲D+(1-u)λ̲̲D}-1ν̲̲
and similarly one has, from (4.7), (4.11) and (4.12),
(5.8)β̲̲D*(u1̲,s)=∑k=0∞∫0∞e-stA̲̲̅D(t)g̲̲D(t,k)dtuk1̲={sI̲̲+ν̲̲D+(1-u)λ̲̲D}-1.
It then follows from Theorem 4.1, (5.7)and (5.8) that
(5.9)φ̲̲̂(u1̲,1̲̲,0+,s)={I̲̲-β̲̲̃(u1̲,1̲̲,s)}-1β̲̲D*(u1̲,s)={I̲̲-β̲̲(u1̲,s)}-1β̲̲D*(u1̲,s)={sI̲̲+ν̲̲D+(1-u)λ̲̲D-ν̲̲}-1={sI̲̲-Q̲̲+(1-u)λ̲̲D}-1,
which coincides with (2.16) as expected.


### 5.4. Derivation of Renewal Process

In order to demonstrate that a renewal process is a special case of the unified multivariate counting process, we follow the line of the arguments for the case of Poisson processes. Namely, let𝒥={0}, N(t)=N0(t), u0=1, v00=v, w=0, a0(x)=a(x) and A̅0(x)=1-∫0xa(y)dy. From Theorem 4.1, one has
(5.10)φ0(1,v,0+,s)=11-vβ(1,s)β*(1,s)=11-v∫0∞e-sta(t)dt×∫0∞e-stA̅(t)dt=11-vα(s)·1-α(s)s,
which agrees with (2.19).


### 5.5. Derivation of MRP

LetÑ̲ℓ(t)=[Ñℓ0(t),…,ÑℓJ(t)] be an MRP discussed in Section 2.5. We recall that Ñℓr(t) describes the number of entries of the semi-Markov process into state r in (0,t] given that J(0)=ℓ. For the unified multivariate counting process, Nij(t) counts the number of transitions from state i to state j in (0,t]. Hence, one has Ñℓr(t)=∑i∈𝒥Nir(t) provided that J(0)=ℓ. Accordingly, we set v̲̲=[ṽ01̲,…,ṽr1̲,…,ṽJ1̲], that is,  vir=ṽr for all i∈𝒥. With w=0+, u̲=1̲, λℓ(t)=0 for all ℓ∈𝒥, one has βℓr(1,s)=αℓr(s) and βℓ*={1-αℓ(s)}/s from (4.6), (4.7) and (4.9) through (4.12), where αℓ(s)=∑r∈𝒥αℓr(s). Let ṽ̲̲D=[δ{ℓ=r}ṽr]. It then follows from Theorem 4.1 that
(5.11)φ̲̲̂(1̲,v̲̲,0+,s)={I̲̲-[ṽrαℓr(s)]}-1×[δ{ℓ=r}1-αℓ(s)s]={I̲̲-α̲̲(s)ṽ̲̲D}-1×[δ{ℓ=r}1-αℓ(s)s].
It should be noted that, with ṽ̲=[ṽ0,…,ṽJ]⊤ and ṽ̲Ñ̲(t)=∏r∈𝒥ṽrÑℓr(t), the ℓ-r element of φ̲̲̂(1̲,v̲̲,0+,s) in (5.11) can be written as
(5.12)φ̂ℓr(1̲,v̲̲,0+,s)=ℒ{E[ṽ̲Ñ̲(t),J(t)=r∣J(0)=ℓ]}.We now focus onÑℓr(t) for ℓ=0,1,…,J. In doing so, let ṽ̲̲D:j=def[1̲0,…,ṽj1̲j,…,1̲J] and define
(5.13)ψ̲(ṽr,s)=def[ℒ{E[ṽrÑr(t)∣J(0)=0]}⋮ℒ{E[ṽrÑr(t)∣J(0)=J]}].
It then follows from (5.11) through (5.13) that
(5.14)ψ̲(ṽr,s)=φ̲̲̂(1̲,ṽ̲̲D:r,0+,s)1̲={I̲̲-α̲̲(s)ṽ̲̲D:r}-1×[δ{ℓ=r}1-αℓ(s)s]×1̲,
that is, one has
(5.15)ψ̲(ṽr,s)=1s{I̲̲-α̲̲(s)ṽ̲̲D:r}-1×{1̲-α̲̲(s)1̲}.We recall thatHℓr(t)=E[Ñℓr(t)], which can be obtained by differencing ψ̲(ṽr,s) with respect to ṽr at ṽr=1. More formally, one has
(5.16)∂∂ṽrψ̲(ṽr,s)|ṽr=1=[ℒ{E[Ñr(t)∣J(0)=0]}⋮ℒ{E[Ñr(t)∣J(0)=J]}],
which is the rth column of ℒ{H̲̲(t)} given in (2.20). By noting that
(5.17)ddx{I̲̲-f̲̲(x)}-1={I̲̲-f̲̲(x)}-1{ddxf̲̲(x)}{I̲̲-f̲̲(x)}-1,
one sees from (5.15) that
(5.18)∂∂ṽjψ̲(ṽj,s)|ṽj=1=1s{I̲̲-α̲̲(s)}-1[α0j(s)0̲̲⋮0̲̲αJj(s)]{I̲̲-α̲̲(s)}-1{1̲-α̲̲(s)1̲}.
Since {I̲̲-α̲̲(s)}-1=∑k=0∞α̲̲(s)k and α̲̲0(s)=I̲̲, it can be seen that
(5.19){I̲̲-α̲̲(s)}-1{1̲-α̲̲(s)1̲}=∑k=0∞α̲̲(s)k{1̲-α̲̲(s)1̲}=1̲.
Substituting (5.19) into (5.18), one then concludes that
(5.20)∂∂ṽjψ̲(ṽj,s)|ṽj=1=1s{I̲̲-α̲̲(s)}-1[α0j(s)⋮αJj(s)].
This in turn implies that
(5.21)[∂∂ṽ0ψ̲(ṽ0,s)|ṽ0=1,…,∂∂ṽJψ̲(ṽJ,s)|ṽJ=1]=1s{I̲̲-α̲̲(s)}-1α̲̲(s),
which agrees with ℒ{H̲̲(t)} of (2.20), completing the derivation.


### 5.6. Derivation of NESMPS

As in Section2.6, let the state space 𝒥 of J(t) be decomposed into a set of good states G(≠ϕ) and a set of bad states B(≠ϕ) satisfying J=G∪B and G∩B=ϕ. The counting process NGB(t) of Section 2.6 describes the number of entries of J(t) into B by time t. The Laplace transform generating function of the joint probability of NGB(t), the age process X(t) and J(t) is given in (2.26).In the context of the unified multivariate counting process[M̲(t),N̲̲(t)] discussed in Section 3, one expects to have
(5.22)NGB(t)=∑i∈G∑j∈BNij(t).
In order to prove (5.22) formally, we set
(5.23)u̲=1̲;v̲̲=[1̲̲BB1̲̲BGv1̲̲GB1̲̲GG].
From (3.7), (4.6), (4.9) and (4.10), one has βij(1,s)=αij(s) so that
(5.24)β̲̲̃(1̲,v̲̲,s)=[α̲̲BB(s)α̲̲BG(s)vα̲̲GB(s)α̲̲GG(s)],
where β̲̲̃(u̲,v̲̲,s) is as given in Theorem 4.1. Similarly, it can be seen from (3.7), (4.7), (4.11) and (4.12) that βi*(1,w+s)={1-αi(w+s)}/(w+s) and hence
(5.25)β̲̲D*(1̲,w+s)=1w+s{I̲̲-α̲̲D(w+s)}.
Substituting (5.24) and (5.25) into (4.20), it then follows that
(5.26)φ̲̲̂(1̲,v̲̲,w,s)=1w+s{I̲̲-β̲̲̃(1̲,v̲̲,s)}-1{I̲̲-α̲̲D(w+s)}.By comparing (2.26) with (5.26), (5.22) holds true if and only if
(5.27)γ̲̲0(s){I̲̲-vβ̲̲(s)}-1={I̲̲-β̲̲̃(1̲,v̲̲,s)}-1.
In what follows, we prove that (5.27) indeed holds true. From (5.24), one sees that
(5.28)I̲̲-β̲̲̃(1̲,v̲̲,s)=[χ̲̲B-1(s)-α̲̲BG(s)-vα̲̲GB(s)χ̲̲G-1(s)],
where χ̲̲G(s) and χ̲̲B(s) are as given in (2.28). We define the inverse matrix of   (5.28) by
(5.29){I̲̲-β̲̲̃(1̲,v̲̲,s)}-1=[C̲̲BB(v,s)C̲̲BG(v,s)C̲̲GB(v,s)C̲̲GG(v,s)].
Since the multiplication of the two matrices in (5.28) and (5.29) yields the identity matrix, it follows that(5.30)χ̲̲B-1(s)C̲̲BB(v,s)-α̲̲BG(s)C̲̲GB(v,s)=I̲̲BBχ̲̲B-1(s)C̲̲BG(v,s)-α̲̲BG(s)C̲̲GG(v,s)=I̲̲BG-vα̲̲GB(s)C̲̲BB(v,s)+χ̲̲G-1(s)C̲̲GB(v,s)=I̲̲GB-vα̲̲GB(s)C̲̲BG(v,s)+χ̲̲G-1(s)C̲̲GG(v,s)=I̲̲GG.
Solving the above equations for C̲̲••(v,s), one has(5.31)C̲̲BB(v,s)=χ̲̲B(s)+vχ̲̲B(s)α̲̲BG(s)χ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s),(5.32)C̲̲BG(v,s)=χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1,(5.33)C̲̲GB(v,s)=vχ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s),(5.34)C̲̲GG(v,s)=χ̲̲G(s){I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1.We next turn our attention to the left-hand side of (5.27). From (2.29), one sees that
(5.35)I̲̲-vβ̲̲(s)=[I̲̲BB0̲̲BG-vα̲̲GB(s)χ̲̲B(s)I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)].
As before, we define the inverse matrix of (5.35) as
(5.36){I̲̲-vβ̲̲(s)}-1=[D̲̲BB(v,s)D̲̲BG(v,s)D̲̲GB(v,s)D̲̲GG(v,s)].
Multiplying the two matrices in (5.35) and (5.36) then yields
(5.37)D̲̲BB(v,s)=I̲̲BBD̲̲BG(v,s)=0̲̲BG-vα̲̲GB(s)χ̲̲B(s)D̲̲BB(v,s)+{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}D̲̲GB(v,s)=0̲̲GB-vα̲̲GB(s)χ̲̲B(s)D̲̲BG(v,s)+{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}D̲̲GG(v,s)=I̲̲GG,
which in turn can be solved for D̲̲••(v,s) as(5.38)D̲̲BB(v,s)=I̲̲BB,(5.39)D̲̲BG(v,s)=0̲̲BG,(5.40)D̲̲GB(v,s)=v{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s),(5.41)D̲̲GG(v,s)={I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1.
Let the left-hand side matrix of (5.27) be described as
(5.42)γ̲̲0(s){I̲̲-vβ̲̲(s)}-1=[Z̲̲BB(v,s)Z̲̲BG(v,s)Z̲̲GB(v,s)Z̲̲GG(v,s)].
From (2.30) and (5.36) through (5.41), one sees that
(5.43)Z̲̲BB(v,s)=I̲̲BB+α̲̲BB(s)χ̲̲B(s)+vχ̲̲B(s)α̲̲BG(s)χ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s).
From (2.28), one easily sees that χ̲̲B(s)=I̲̲BB+α̲̲BB(s)χ̲̲B(s), and hence Z̲̲BB(v,s)=C̲̲BB(v,s) from (5.31). The fact that Z̲̲BG(v,s)=C̲̲BG(v,s) is straightforward from (5.32). With χ̲̲G(s)=I̲̲GG+α̲̲GG(s)χ̲̲G(s), one has Z̲̲GB(v,s)=C̲̲GB(v,s) from (5.33) and Z̲̲GG(v,s)=C̲̲GG(v,s) from (5.34), completing the proof for (5.27).


### 5.7. Derivation of MAP

We recall that an MAP is constructed from an absorbing Markov-chainJ*(t) in continuous time on 𝒥=G∪B, with B being a set of absorbing states, governed by ν̲̲* defined in (2.32). A replacement Markov-chain J(t) is then generated from J*(t), where J(t) coincides with J*(t) within G starting from a state in G. As soon as J*(t) reaches state r∈B, it is instantaneously replaced at state j∈G with probability p̃rj and the process continues.In order to deduce an MAP from the unified multivariate counting process[M̲(t),N̲̲(t)] as a special case, we start from (4.20) in Theorem 4.1, where the underlying semi-Markov process is now the replacement Marcov chain J(t) discussed above. This replacement Marcov chain is defined on G governed by ν̲̲=C̲̲+D̲̲ with C̲̲=[cij]=[νij]i,j∈G and D̲̲=[dij]=[∑r∈Bνirp̃rj]i,j∈G as in (2.33). We note that βij(1,s)=αij(s) from (3.7), (4.6), (4.9) and (4.10), and βi*(1,s)={1-αi(s)}/s from (3.7), (4.7), (4.11) and (4.12). Hence, it follows that
(5.44)φ̲̲̂(1̲,v̲̲,0+,s)={I̲̲-[vij·αij(s)]}-1[δ{i=j}·1-αi(s)s].
Let ν̲̲D=C̲̲D+D̲̲D as in (2.35). Since J(t) is a Markov chain, the dwell time in state i is independent of the next state and is exponentially distributed with parameter νi=ci+di. Consequently, one has
(5.45)αi(s)=νis+νi;αij(s)=cij+dijνiαi(s)=cij+dijs+νi.
Substituting (5.45) into (5.44) and noting [δ{i=j}·((1-αi(s))/s)]-1=sI̲̲+ν̲̲D, it follows that
(5.46)φ̲̲̂(1̲,v̲̲,0+,s)={sI̲̲-Q̲̲+C̲̲+D̲̲-[vij(cij+dij)]}-1,
where Q̲̲ in (2.34) is employed.As it stands, the Laplace transform generating function of (5.46) describes the matrix counting process N̲̲(t)=[Nij(t)] where vij corresponds to Nij(t). For NMAP(t) of Section 2.7, it is only necessary to count the number of replacements in (0,t]. Given that state j∈G is visited from the current state i∈G, this move is direct within G with probability cij/(cij+dij), and such a move involves replacement with probability dij/(cij+dij). Accordingly, we set
(5.47)vij=cijcij+dij+dijcij+dijv.
Substitution of (5.47) into (5.46) then leads to
(5.48)φ̲̲̂(1̲,v̲̲,0+,s)={sI̲̲-Q̲̲+(1-v)D̲̲}-1,
which coincides with φ̲̲(v,s) of (2.44), as expected.


### 5.8. Derivation of ACPGRP

The age-dependent counting process of Sumita and Shanthikumar [12] has been extended in this paper where the underlying renewal process is replaced by a semi-Markov process with state dependent nonhomogeneous hazard functions. Accordingly, the original model can be treated as a special case of the unified multivariate counting process proposed in this paper by setting 𝒥={0}, M(t)=M0(t), N(t)=N00(t), X(t)=X0(t), u0=u, v00=v. With this specification, from Theorem 4.1, one sees that
(5.49)φ̂00(u,v,w,s)=β*(u,w+s)1-vβ(u,s).
It then follows that
(5.50)π(u,s)=φ̂00(u,1,0,s)=β*(u,s)1-β(u,s),
which coincides with Equation (2.53).


## 5.1. Derivation of Poisson Process

LetN(t) be a Poisson process with intensity λ as discussed in Section 2.1. From (2.5), one sees that π̂(v,s)=∫0∞e-stE[vN(t)]dt=∫0∞e-stπ(v,t)dt is given by
(5.1)π̂(v,s)=1s+λ(1-v).For the unified counting process, we consider a single state Markov-chain in continuous time where only the number of self transitions in(0,t] is counted. More specifically, let 𝒥={0}, N̲̲(t)=[N00(t)], u0=1, v00=v, w=0, λ0(t)=0 and a00(x)=λe-λx. We note from (3.7) that λ0(t)=0 implies g0(x,k)=δ{k=0} so that bk:00(t)=δ{k=0}a00(t) from (4.6). This then implies β00(1,s)=λ/(s+λ). Similarly, since A̅0(x)=e-λx, one has β0*(1,s)=1/(s+λ). It then follows from Theorem 4.1 that
(5.2)φ̂0(1,v,0+,s)=11-vβ00(1,s)β0*(1,s)=1s+λ(1-v).
Hence, from (5.1) and (5.2), one concludes that π̂(v,s)=φ̂0(1,v,0+,s).


## 5.2. Derivation of NHPP

LetM(t) be an NHPP of Section 2.2 characterized by a time dependent intensity function λ(t). It can be seen from (2.8) that π̂(u,s)=∫0∞e-stE[uM(t)]dt=∫0∞e-stπ(u,t)dt is given by
(5.3)π̂(u,s)=∫0∞e-ste-L(t)(1-u)dt.In order to see thatM(t) can be viewed as a special case of the proposed multivariate counting process, we first consider a single state semi-Markov process where the dwell time in the state is deterministic given by T. The marginal counting process M̲(t)=[M0(t)] then converges in distribution to M(t) as T→∞ as we show next.Let𝒥={0}, u0=u, v00=1, w=0, and λ0(x)=λ(x). It then follows that a00(x)=δ(x-t) and therefore bk:00(t)=δ(t-T)g0(t,k) from (4.6). This in turn implies that
(5.4)β00(u,s)=∫0∞e-stδ(t-T)e-L0(t)(1-u)dt=e-{sT+L0(T)(1-u)}.
Let U(t) be the step function defined by U(t)=0 if t<0 and U(t)=1 otherwise. Since A̅00(t)=1-U(t-T)=δ{0≤t<T}, one sees that
(5.5)β0*(u,s)=∫0∞e-stδ{0≤t<T}e-L0(t)(1-u)dt=∫0Te-ste-L0(t)(1-u)dt.
Theorem 4.1 together with (5.4) and (5.5) then leads to
(5.6)φ̂00(u,1,0+,s)=11-β00(u,s)β0*(u,s)=∫0Te-ste-L0(t)(1-u)dt1-e-{sT+L0(T)(1-u)}.
Now it can be readily seen that φ̂00(u,1,0+,s) in (5.6) converges to π̂(u,s) in (5.3) as T→∞, proving the claim.


## 5.3. Derivation of MMPP

LetM(t) be an MMPP of Section 2.3 characterized by (ν̲̲,λ̲̲D). We show that the Laplace transform generating function π̂(u,s)=∫0∞e-stE[uM(t)]dt given in (2.16) can be derived as a special case of Theorem 4.1.For the unified multivariate counting process, let𝒥={0,…,J}, M(t)=∑i∈𝒥Mi(t), u̲=u1̲, v̲̲=1̲̲, w=0, λi(t)=λi, and a̲̲(x)=[e-νixνij]=e-ν̲̲Dxν̲̲. From (3.3) one sees that A̲̲̅D(x)=I̲̲-A̲̲D=[δ{i=j}{1-∑j∈𝒥Aij(x)}]=[δ{i=j}e-νix]=e-ν̲̲Dx. Therefore, one sees from (4.6), (4.9) and (4.10) that
(5.7)β̲̲(u1̲,s)=∑k=0∞∫0∞e-stg̲̲D(t,k)a̲̲(t)dtuk1̲={sI̲̲+ν̲̲D+(1-u)λ̲̲D}-1ν̲̲
and similarly one has, from (4.7), (4.11) and (4.12),
(5.8)β̲̲D*(u1̲,s)=∑k=0∞∫0∞e-stA̲̲̅D(t)g̲̲D(t,k)dtuk1̲={sI̲̲+ν̲̲D+(1-u)λ̲̲D}-1.
It then follows from Theorem 4.1, (5.7)and (5.8) that
(5.9)φ̲̲̂(u1̲,1̲̲,0+,s)={I̲̲-β̲̲̃(u1̲,1̲̲,s)}-1β̲̲D*(u1̲,s)={I̲̲-β̲̲(u1̲,s)}-1β̲̲D*(u1̲,s)={sI̲̲+ν̲̲D+(1-u)λ̲̲D-ν̲̲}-1={sI̲̲-Q̲̲+(1-u)λ̲̲D}-1,
which coincides with (2.16) as expected.


## 5.4. Derivation of Renewal Process

In order to demonstrate that a renewal process is a special case of the unified multivariate counting process, we follow the line of the arguments for the case of Poisson processes. Namely, let𝒥={0}, N(t)=N0(t), u0=1, v00=v, w=0, a0(x)=a(x) and A̅0(x)=1-∫0xa(y)dy. From Theorem 4.1, one has
(5.10)φ0(1,v,0+,s)=11-vβ(1,s)β*(1,s)=11-v∫0∞e-sta(t)dt×∫0∞e-stA̅(t)dt=11-vα(s)·1-α(s)s,
which agrees with (2.19).


## 5.5. Derivation of MRP

LetÑ̲ℓ(t)=[Ñℓ0(t),…,ÑℓJ(t)] be an MRP discussed in Section 2.5. We recall that Ñℓr(t) describes the number of entries of the semi-Markov process into state r in (0,t] given that J(0)=ℓ. For the unified multivariate counting process, Nij(t) counts the number of transitions from state i to state j in (0,t]. Hence, one has Ñℓr(t)=∑i∈𝒥Nir(t) provided that J(0)=ℓ. Accordingly, we set v̲̲=[ṽ01̲,…,ṽr1̲,…,ṽJ1̲], that is,  vir=ṽr for all i∈𝒥. With w=0+, u̲=1̲, λℓ(t)=0 for all ℓ∈𝒥, one has βℓr(1,s)=αℓr(s) and βℓ*={1-αℓ(s)}/s from (4.6), (4.7) and (4.9) through (4.12), where αℓ(s)=∑r∈𝒥αℓr(s). Let ṽ̲̲D=[δ{ℓ=r}ṽr]. It then follows from Theorem 4.1 that
(5.11)φ̲̲̂(1̲,v̲̲,0+,s)={I̲̲-[ṽrαℓr(s)]}-1×[δ{ℓ=r}1-αℓ(s)s]={I̲̲-α̲̲(s)ṽ̲̲D}-1×[δ{ℓ=r}1-αℓ(s)s].
It should be noted that, with ṽ̲=[ṽ0,…,ṽJ]⊤ and ṽ̲Ñ̲(t)=∏r∈𝒥ṽrÑℓr(t), the ℓ-r element of φ̲̲̂(1̲,v̲̲,0+,s) in (5.11) can be written as
(5.12)φ̂ℓr(1̲,v̲̲,0+,s)=ℒ{E[ṽ̲Ñ̲(t),J(t)=r∣J(0)=ℓ]}.We now focus onÑℓr(t) for ℓ=0,1,…,J. In doing so, let ṽ̲̲D:j=def[1̲0,…,ṽj1̲j,…,1̲J] and define
(5.13)ψ̲(ṽr,s)=def[ℒ{E[ṽrÑr(t)∣J(0)=0]}⋮ℒ{E[ṽrÑr(t)∣J(0)=J]}].
It then follows from (5.11) through (5.13) that
(5.14)ψ̲(ṽr,s)=φ̲̲̂(1̲,ṽ̲̲D:r,0+,s)1̲={I̲̲-α̲̲(s)ṽ̲̲D:r}-1×[δ{ℓ=r}1-αℓ(s)s]×1̲,
that is, one has
(5.15)ψ̲(ṽr,s)=1s{I̲̲-α̲̲(s)ṽ̲̲D:r}-1×{1̲-α̲̲(s)1̲}.We recall thatHℓr(t)=E[Ñℓr(t)], which can be obtained by differencing ψ̲(ṽr,s) with respect to ṽr at ṽr=1. More formally, one has
(5.16)∂∂ṽrψ̲(ṽr,s)|ṽr=1=[ℒ{E[Ñr(t)∣J(0)=0]}⋮ℒ{E[Ñr(t)∣J(0)=J]}],
which is the rth column of ℒ{H̲̲(t)} given in (2.20). By noting that
(5.17)ddx{I̲̲-f̲̲(x)}-1={I̲̲-f̲̲(x)}-1{ddxf̲̲(x)}{I̲̲-f̲̲(x)}-1,
one sees from (5.15) that
(5.18)∂∂ṽjψ̲(ṽj,s)|ṽj=1=1s{I̲̲-α̲̲(s)}-1[α0j(s)0̲̲⋮0̲̲αJj(s)]{I̲̲-α̲̲(s)}-1{1̲-α̲̲(s)1̲}.
Since {I̲̲-α̲̲(s)}-1=∑k=0∞α̲̲(s)k and α̲̲0(s)=I̲̲, it can be seen that
(5.19){I̲̲-α̲̲(s)}-1{1̲-α̲̲(s)1̲}=∑k=0∞α̲̲(s)k{1̲-α̲̲(s)1̲}=1̲.
Substituting (5.19) into (5.18), one then concludes that
(5.20)∂∂ṽjψ̲(ṽj,s)|ṽj=1=1s{I̲̲-α̲̲(s)}-1[α0j(s)⋮αJj(s)].
This in turn implies that
(5.21)[∂∂ṽ0ψ̲(ṽ0,s)|ṽ0=1,…,∂∂ṽJψ̲(ṽJ,s)|ṽJ=1]=1s{I̲̲-α̲̲(s)}-1α̲̲(s),
which agrees with ℒ{H̲̲(t)} of (2.20), completing the derivation.


## 5.6. Derivation of NESMPS

As in Section2.6, let the state space 𝒥 of J(t) be decomposed into a set of good states G(≠ϕ) and a set of bad states B(≠ϕ) satisfying J=G∪B and G∩B=ϕ. The counting process NGB(t) of Section 2.6 describes the number of entries of J(t) into B by time t. The Laplace transform generating function of the joint probability of NGB(t), the age process X(t) and J(t) is given in (2.26).In the context of the unified multivariate counting process[M̲(t),N̲̲(t)] discussed in Section 3, one expects to have
(5.22)NGB(t)=∑i∈G∑j∈BNij(t).
In order to prove (5.22) formally, we set
(5.23)u̲=1̲;v̲̲=[1̲̲BB1̲̲BGv1̲̲GB1̲̲GG].
From (3.7), (4.6), (4.9) and (4.10), one has βij(1,s)=αij(s) so that
(5.24)β̲̲̃(1̲,v̲̲,s)=[α̲̲BB(s)α̲̲BG(s)vα̲̲GB(s)α̲̲GG(s)],
where β̲̲̃(u̲,v̲̲,s) is as given in Theorem 4.1. Similarly, it can be seen from (3.7), (4.7), (4.11) and (4.12) that βi*(1,w+s)={1-αi(w+s)}/(w+s) and hence
(5.25)β̲̲D*(1̲,w+s)=1w+s{I̲̲-α̲̲D(w+s)}.
Substituting (5.24) and (5.25) into (4.20), it then follows that
(5.26)φ̲̲̂(1̲,v̲̲,w,s)=1w+s{I̲̲-β̲̲̃(1̲,v̲̲,s)}-1{I̲̲-α̲̲D(w+s)}.By comparing (2.26) with (5.26), (5.22) holds true if and only if
(5.27)γ̲̲0(s){I̲̲-vβ̲̲(s)}-1={I̲̲-β̲̲̃(1̲,v̲̲,s)}-1.
In what follows, we prove that (5.27) indeed holds true. From (5.24), one sees that
(5.28)I̲̲-β̲̲̃(1̲,v̲̲,s)=[χ̲̲B-1(s)-α̲̲BG(s)-vα̲̲GB(s)χ̲̲G-1(s)],
where χ̲̲G(s) and χ̲̲B(s) are as given in (2.28). We define the inverse matrix of   (5.28) by
(5.29){I̲̲-β̲̲̃(1̲,v̲̲,s)}-1=[C̲̲BB(v,s)C̲̲BG(v,s)C̲̲GB(v,s)C̲̲GG(v,s)].
Since the multiplication of the two matrices in (5.28) and (5.29) yields the identity matrix, it follows that(5.30)χ̲̲B-1(s)C̲̲BB(v,s)-α̲̲BG(s)C̲̲GB(v,s)=I̲̲BBχ̲̲B-1(s)C̲̲BG(v,s)-α̲̲BG(s)C̲̲GG(v,s)=I̲̲BG-vα̲̲GB(s)C̲̲BB(v,s)+χ̲̲G-1(s)C̲̲GB(v,s)=I̲̲GB-vα̲̲GB(s)C̲̲BG(v,s)+χ̲̲G-1(s)C̲̲GG(v,s)=I̲̲GG.
Solving the above equations for C̲̲••(v,s), one has(5.31)C̲̲BB(v,s)=χ̲̲B(s)+vχ̲̲B(s)α̲̲BG(s)χ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s),(5.32)C̲̲BG(v,s)=χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1,(5.33)C̲̲GB(v,s)=vχ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s),(5.34)C̲̲GG(v,s)=χ̲̲G(s){I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1.We next turn our attention to the left-hand side of (5.27). From (2.29), one sees that
(5.35)I̲̲-vβ̲̲(s)=[I̲̲BB0̲̲BG-vα̲̲GB(s)χ̲̲B(s)I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)].
As before, we define the inverse matrix of (5.35) as
(5.36){I̲̲-vβ̲̲(s)}-1=[D̲̲BB(v,s)D̲̲BG(v,s)D̲̲GB(v,s)D̲̲GG(v,s)].
Multiplying the two matrices in (5.35) and (5.36) then yields
(5.37)D̲̲BB(v,s)=I̲̲BBD̲̲BG(v,s)=0̲̲BG-vα̲̲GB(s)χ̲̲B(s)D̲̲BB(v,s)+{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}D̲̲GB(v,s)=0̲̲GB-vα̲̲GB(s)χ̲̲B(s)D̲̲BG(v,s)+{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}D̲̲GG(v,s)=I̲̲GG,
which in turn can be solved for D̲̲••(v,s) as(5.38)D̲̲BB(v,s)=I̲̲BB,(5.39)D̲̲BG(v,s)=0̲̲BG,(5.40)D̲̲GB(v,s)=v{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s),(5.41)D̲̲GG(v,s)={I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1.
Let the left-hand side matrix of (5.27) be described as
(5.42)γ̲̲0(s){I̲̲-vβ̲̲(s)}-1=[Z̲̲BB(v,s)Z̲̲BG(v,s)Z̲̲GB(v,s)Z̲̲GG(v,s)].
From (2.30) and (5.36) through (5.41), one sees that
(5.43)Z̲̲BB(v,s)=I̲̲BB+α̲̲BB(s)χ̲̲B(s)+vχ̲̲B(s)α̲̲BG(s)χ̲̲G(s)×{I̲̲GG-vα̲̲GB(s)χ̲̲B(s)α̲̲BG(s)χ̲̲G(s)}-1α̲̲GB(s)χ̲̲B(s).
From (2.28), one easily sees that χ̲̲B(s)=I̲̲BB+α̲̲BB(s)χ̲̲B(s), and hence Z̲̲BB(v,s)=C̲̲BB(v,s) from (5.31). The fact that Z̲̲BG(v,s)=C̲̲BG(v,s) is straightforward from (5.32). With χ̲̲G(s)=I̲̲GG+α̲̲GG(s)χ̲̲G(s), one has Z̲̲GB(v,s)=C̲̲GB(v,s) from (5.33) and Z̲̲GG(v,s)=C̲̲GG(v,s) from (5.34), completing the proof for (5.27).


## 5.7. Derivation of MAP

We recall that an MAP is constructed from an absorbing Markov-chainJ*(t) in continuous time on 𝒥=G∪B, with B being a set of absorbing states, governed by ν̲̲* defined in (2.32). A replacement Markov-chain J(t) is then generated from J*(t), where J(t) coincides with J*(t) within G starting from a state in G. As soon as J*(t) reaches state r∈B, it is instantaneously replaced at state j∈G with probability p̃rj and the process continues.In order to deduce an MAP from the unified multivariate counting process[M̲(t),N̲̲(t)] as a special case, we start from (4.20) in Theorem 4.1, where the underlying semi-Markov process is now the replacement Marcov chain J(t) discussed above. This replacement Marcov chain is defined on G governed by ν̲̲=C̲̲+D̲̲ with C̲̲=[cij]=[νij]i,j∈G and D̲̲=[dij]=[∑r∈Bνirp̃rj]i,j∈G as in (2.33). We note that βij(1,s)=αij(s) from (3.7), (4.6), (4.9) and (4.10), and βi*(1,s)={1-αi(s)}/s from (3.7), (4.7), (4.11) and (4.12). Hence, it follows that
(5.44)φ̲̲̂(1̲,v̲̲,0+,s)={I̲̲-[vij·αij(s)]}-1[δ{i=j}·1-αi(s)s].
Let ν̲̲D=C̲̲D+D̲̲D as in (2.35). Since J(t) is a Markov chain, the dwell time in state i is independent of the next state and is exponentially distributed with parameter νi=ci+di. Consequently, one has
(5.45)αi(s)=νis+νi;αij(s)=cij+dijνiαi(s)=cij+dijs+νi.
Substituting (5.45) into (5.44) and noting [δ{i=j}·((1-αi(s))/s)]-1=sI̲̲+ν̲̲D, it follows that
(5.46)φ̲̲̂(1̲,v̲̲,0+,s)={sI̲̲-Q̲̲+C̲̲+D̲̲-[vij(cij+dij)]}-1,
where Q̲̲ in (2.34) is employed.As it stands, the Laplace transform generating function of (5.46) describes the matrix counting process N̲̲(t)=[Nij(t)] where vij corresponds to Nij(t). For NMAP(t) of Section 2.7, it is only necessary to count the number of replacements in (0,t]. Given that state j∈G is visited from the current state i∈G, this move is direct within G with probability cij/(cij+dij), and such a move involves replacement with probability dij/(cij+dij). Accordingly, we set
(5.47)vij=cijcij+dij+dijcij+dijv.
Substitution of (5.47) into (5.46) then leads to
(5.48)φ̲̲̂(1̲,v̲̲,0+,s)={sI̲̲-Q̲̲+(1-v)D̲̲}-1,
which coincides with φ̲̲(v,s) of (2.44), as expected.


## 5.8. Derivation of ACPGRP

The age-dependent counting process of Sumita and Shanthikumar [12] has been extended in this paper where the underlying renewal process is replaced by a semi-Markov process with state dependent nonhomogeneous hazard functions. Accordingly, the original model can be treated as a special case of the unified multivariate counting process proposed in this paper by setting 𝒥={0}, M(t)=M0(t), N(t)=N00(t), X(t)=X0(t), u0=u, v00=v. With this specification, from Theorem 4.1, one sees that
(5.49)φ̂00(u,v,w,s)=β*(u,w+s)1-vβ(u,s).
It then follows that
(5.50)π(u,s)=φ̂00(u,1,0,s)=β*(u,s)1-β(u,s),
which coincides with Equation (2.53).


## 6. Asymptotic Analysis

Let𝒜,ℳ and 𝒩 be arbitrary subsets of the state space 𝒥 of the underlying semi-Markov process, and define
(6.1)M𝒜(t)=∑i∈𝒜Mi(t);Nℳ𝒩(t)=∑i∈ℳ∑j∈𝒩Nij(t),
where M𝒜(t) describes the total number of items arrived in [0,t] according to the nonhomogeneous Poisson processes within 𝒜 and Nℳ𝒩(t) denotes the number of transitions from any state in ℳ to any state in 𝒩 occurred in [0,t]. Appropriate choice of 𝒜,ℳ and 𝒩 would then enable one to analyze processes of interest in a variety of applications. In the variable bit rate coding scheme for video transmission explained in Section 1, for example, one may be interested in the packet arrival stream for a specified mode of the encoder represented by M𝒜(t). In a reliability model, the underlying semi-Markov process may describe the state of a production machine. Minimal repair would take place whenever the system state is in 𝒜 at the cost of c, while complete overhaul would be forced at the cost of d if the machine state enters 𝒩⊂𝒥. The total maintenance cost S(t) is then given by S(t)=cM𝒜(t)+dN𝒥𝒩(t). A simplified version of this cost structure has been analyzed by Sumita and Shanthikumar [12] where the underlying semi-Markov process is merely a renewal process with 𝒥=𝒜=𝒩={1}. The purpose of this section is to study a more general cost structure specified by
(6.2)S(t)=cM𝒜(t)+dNℳ𝒩(t),
with focus on the Laplace transform generating function and the related moment asymptotic behaviors of M𝒜(t),Nℳ𝒩(t) and S(t) based on Theorem 4.1.For notational simplicity, we introduce the following vectors and matrices. Let𝒜,ℳ and 𝒩⊂𝒥 with their compliments defined respectively by 𝒜C=𝒥∖𝒜,ℳC=𝒥∖ℳ and 𝒩C=𝒥∖𝒩. The cardinality of a set A is denoted by ∣A∣.
(6.3)1̲=[1⋮1]∈ℝJ+1;1̲̲=[1⋯1⋮⋱⋮1⋯1]∈ℝ(J+1)×(J+1),(6.4)u̲(𝒜)=[ui]∈ℝJ+1withui={u,ifi∈𝒜,1,else;v̲̲(ℳ,𝒩)=[vij]∈ℝ(J+1)×(J+1)withvij={v,ifi∈ℳ,j∈𝒩,1,else.
Submatrices of A̲̲∈ℝ(J+1)×(J+1) are denoted by
(6.5)A̲̲𝒜•=[Aij]i∈𝒜,j∈𝒥∈ℝ∣𝒜∣×(J+1);A̲̲•𝒜=[Aij]i∈𝒥,j∈𝒜∈ℝ(J+1)×∣𝒜∣;A̲̲ℳ𝒩=[Aij]i∈ℳ,j∈𝒩∈ℝ∣ℳ∣×∣𝒩∣,
so that one has(6.6)A̲̲=[A̲̲ℳ𝒩A̲̲ℳ𝒩CA̲̲ℳC𝒩A̲̲ℳC𝒩C],
with understanding that the states are arranged appropriately.LetA̲̲k=∫0∞xka̲̲(x)dx, k=0,1,2,…. Throughout the paper, we assume that ∥A̲̲k∥<∞ for 0≤k≤2. In particular, one has A̲̲0=A̲̲(∞) which is stochastic. The Taylor expansion of the Laplace transform α̲̲(s) is then given by
(6.7)α̲̲(s)=A̲̲0-sA̲̲1+s22A̲̲2+o̲̲(s2).
Let e̲⊤ be the normalized left eigenvector of A̲̲0 associated with eigenvalue 1 so that e̲⊤A̲̲0=e̲⊤ and e̲⊤1̲=1.We recall from Theorem4.1 that
(6.8)φ̲̲̂(u̲,v̲̲,w,s)={I̲̲-β̲̲̃(u̲,v̲̲,s)}-1β̲̲D*(u̲,w+s),
where
(6.9)β̲̲̃(u̲,v̲̲,s)=[vij·βij(ui,s)].
From (3.6), (3.7), (4.6), (4.9) and (4.10), one sees that
(6.10)βij(u,s)=∫0∞e-ste-Li(t)(1-u)aij(t)dt.
Similarly, it follows from (3.6), (3.7), (4.7), (4.11) and (4.12) that
(6.11)βi*(u,s)=∫0∞e-ste-Li(t)(1-u)A̅i(t)dt.
The rth order partial derivatives of βij(u,s) and βi*(u,s) with respect to u at u=1 are then given by
(6.12)ζr:ij(s)=def(∂∂u)rβij(u,s)|u=1=∫0∞e-stLir(t)aij(t)dt,r=1,2;ζr:i*(s)=def(∂∂u)rβi*(u,s)|u=1=∫0∞e-stLir(t)A̅i(t)dt,r=1,2.
In matrix form,  (6.12) can be written as
(6.13)ζ̲̲r(s)=∫0∞e-stL̲̲Dr(t)a̲̲(t)dt,r=1,2;ζ̲̲r:D*(s)=∫0∞e-stL̲̲Dr(t)A̲̲̅D(t)dt,r=1,2.LetΦ̲̲r:k=∫0∞tkL̲̲Dr(t)a̲̲(t)dt and Φ̲̲r:D:k*=∫0∞tkL̲̲Dr(t)A̲̲̅D(t)dt, r=1,2. The Taylor expansion of ζ̲̲r(s) and ζ̲̲r:D*(s) is then given by
(6.14)ζ̲̲r(s)=Φ̲̲r:0-sΦ̲̲r:1+s22Φ̲̲r:2+o̲̲(s2),r=1,2;ζ̲̲r:D*(s)=Φ̲̲r:D:0*-sΦ̲̲r:D:1*+s22Φ̲̲r:D:2*+o̲̲(s2),r=1,2.In order to prove the main results of this section, the following theorem of Keilson [23] plays a key role.Theorem 6.1 (Keilson [23]).
Ass→0+, one has
(6.15){I̲̲-α̲̲(s)}-1=1sH̲̲1+H̲̲0+o̲̲(1),
where
(6.16)H̲̲1=1m1̲e̲⊤,m=e̲⊤A̲̲11̲,H̲̲0=H̲̲1(-A̲̲1+12A̲̲2H̲̲1)+(Z̲̲0-H̲̲1A̲̲1Z̲̲0)(A̲̲0-A̲̲1H̲̲1)+I̲̲,Z̲̲0=(I̲̲-A̲̲0+1̲·e̲⊤)-1.We are now in a position to prove the first key theorem of this section.Theorem.
Letp̲⊤(0) be an initial probability vector of the underlying semi-Markov process. As t→∞, one has
(6.17)E[M𝒜(t)]=p̲⊤(0){tP̲̲1+P̲̲0}1̲+o(1),E[Nℳ𝒩(t)]=p̲⊤(0){tQ̲̲1+Q̲̲0}1̲+o(1),
where
(6.18)P̲̲1=H̲̲1:•𝒜Φ̲̲1:0:𝒜•,P̲̲0=H̲̲0:•𝒜Φ̲̲1:0:𝒜•-H̲̲1:•𝒜Φ̲̲1:1:𝒜•+H̲̲1:•𝒜Φ̲̲1:D:0:𝒜•*,Q̲̲1=H̲̲1:•ℳ[A̲̲0:ℳ𝒩,0̲̲ℳ𝒩C],Q̲̲0=H̲̲0:•ℳ[A̲̲0:ℳ𝒩,0̲̲ℳ𝒩C]-H̲̲1:•ℳ[A̲̲1:ℳ𝒩,0̲̲ℳ𝒩C].Proof.
We first note from (6.8) together with (6.4) that
(6.19)ℒ{E[uM𝒜(t)]}=p̲⊤(0)φ̲̲̂(u̲(𝒜),1̲̲,0,s)1̲;ℒ{E[vNℳ𝒩(t)]}=p̲⊤(0)φ̲̲̂(1̲,v̲̲(ℳ,𝒩),0,s)1̲.
By taking the partial derivatives of (6.19) with respect to u at u=1 and v at v=1 respectively, one has
(6.20)ℒ{E[M𝒜(t)]}=p̲⊤(0){I̲̲-α̲̲(s)}-1{1s[ζ̲̲1:𝒜•(s)0̲̲𝒜C•]+[ζ̲̲1,D:𝒜•*(s)0̲̲𝒜C•]}1̲;ℒ{E[Nℳ𝒩(t)]}=1sp̲⊤(0){I̲̲-α̲̲(s)}-1[α̲̲ℳ𝒩(s)0̲̲ℳ𝒩C0̲̲ℳC𝒩0̲̲ℳC𝒩C]1̲.
Theorem 6.1 of Keilson [23] combined with (6.7), (6.13) then yields the Laplace transform expansions of (6.20), and the theorem follows by taking the inversion of the Laplace transform expansions.The next theorem can be shown in a similar manner by differentiating (6.19) twice with respect to u at u=1 and v at v=1 respectively, and proof is omitted.Theorem.
Ast→∞, one has
(6.21)E[M𝒜(t)(M𝒜(t)-1)]=p̲⊤(0){t2P̲̲12+t(2P̲̲1P̲̲0+2P̲̲̂0P̲̲1+P̲̲2)}1̲+o(t);E[Nℳ𝒩(t)(Nℳ𝒩(t)-1)]=p̲⊤(0){t2Q̲̲12+2t(Q̲̲1Q̲̲0+Q̲̲0Q̲̲1)}1̲+o(t),
where P̲̲̂0=H̲̲0:•𝒜Φ̲̲1:0:𝒜•-H̲̲1:•𝒜Φ̲̲1:1:𝒜•, P̲̲2=H̲̲1:•𝒜Φ̲̲2:0:𝒜• and other matrices are as defined in Theorem 6.2.Theorems 6.2 and  6.3 then lead to the following theorem providing the asymptotic expansions of Var[M𝒜(t)] and Var[Nℳ𝒩(t)].Theorem.
Ast→∞, one has
(6.22)Var[M𝒜(t)]=tp̲⊤(0)U̲̲01̲+o(t),Var[Nℳ𝒩(t)]=tp̲⊤(0)V̲̲01̲+o(t),
where
(6.23)U̲̲0=2P̲̲1P̲̲0+P̲̲1-P̲̲11̲·p̲⊤(0)P̲̲0+2P̲̲̂0P̲̲1-P̲̲0P̲̲1+P̲̲2;V̲̲0=2Q̲̲1Q̲̲0+Q̲̲1-Q̲̲11̲·p̲⊤(0)Q̲̲0+Q̲̲0Q̲̲1.Proof.
It can be readily seen that(6.24)Var[X]=E[X]2-E[X]2=E[X(X-1)]+E[X]-E[X]2.
Substituting the results of Theorems 6.2 and 6.3 into this equation, one sees that
(6.25)Var[M𝒜(t)]=p̲⊤(0){t2U̲̲1+tU̲̲0}1̲+o(t),Var[Nℳ𝒩(t)]=p̲⊤(0){t2V̲̲1+tV̲̲0}1̲+o(t),
where
(6.26)U̲̲1=P̲̲1(P̲̲1-1̲·p̲⊤(0)P̲̲1),U̲̲0=2P̲̲1P̲̲0+P̲̲1-P̲̲11̲·p̲⊤(0)P̲̲0+2P̲̲̂0P̲̲1-P̲̲01̲·p̲⊤(0)P̲̲1+P̲̲2;V̲̲1=Q̲̲1(Q̲̲1-1̲·p̲⊤(0)Q̲̲1),V̲̲0=2Q̲̲1Q̲̲0+Q̲̲1-Q̲̲11̲·p̲⊤(0)Q̲̲0+2Q̲̲0Q̲̲1-Q̲̲01̲·p̲⊤(0)Q̲̲1.
From Theorem 6.1, one has H̲̲1=(1/m)1̲e̲⊤ which is of rank one having identical rows. As can be seen from Theorem 6.2, both P̲̲1 and Q̲̲1 satisfy
(6.27)1̲·p̲⊤(0)P̲̲1=P̲̲1,1̲·p̲⊤(0)Q̲̲1=Q̲̲1,
so that U̲̲1=V̲̲1=0̲̲ and the theorem follows.The asymptotic behavior of E[S(t)] can be easily found from (6.2) and Theorem 6.2. The asymptotic expansion of Var[S(t)], however, requires a little precaution because it involves the joint expectation of M𝒜(t) and Nℳ𝒩(t). More specifically, one has
(6.28)Var[S(t)]=E[S(t)2]-E[S(t)]2=E[{cM𝒜(t)+dNℳ𝒩(t)}2]-E[cM𝒜(t)+dNℳ𝒩(t)]2=c2E[M𝒜2(t)]+2cdE[M𝒜(t)Nℳ𝒩(t)]+d2E[Nℳ𝒩2(t)]-c2E[M𝒜(t)]2-2cdE[M𝒜(t)]E[Nℳ𝒩(t)]-d2E[Nℳ𝒩(t)]2,
so that
(6.29)Var[S(t)]=c2Var[M𝒜(t)]+d2Var[Nℳ𝒩(t)]+2cdE[M𝒜(t)Nℳ𝒩(t)]-2cdE[M𝒜(t)]E[Nℳ𝒩(t)].
In order to evaluate E[M𝒜(t)Nℳ𝒩(t)], we note from (6.8) that
(6.30)ℒ{E[M𝒜(t)Nℳ𝒩(t)]}=p̲⊤(0)[∂2∂u∂vφ̲̲̂(u̲(𝒜),v̲̲(ℳ,𝒩),0+,s)|u=v=1]1̲.
The asymptotic expansion of E[M𝒜(t)Nℳ𝒩(t)] can then be obtained as for the previous theorems.Theorem.
Ast→∞, one has
(6.31)E[M𝒜(t)Nℳ𝒩(t)]=p̲⊤(0){t22T̲̲1+tT̲̲0}1̲+o(t),
where
(6.32)T̲̲1=P̲̲1Q̲̲1+Q̲̲1P̲̲1,T̲̲0=P̲̲̂0Q̲̲1+P̲̲1Q̲̲0+R̲̲+Q̲̲0P̲̲1+Q̲̲1P̲̲0,R̲̲=H̲̲1:•,(𝒜∩ℳ)[Φ̲̲1:0:(𝒜∩ℳ),𝒩,0̲̲(𝒜∩ℳ),𝒩C].Now the key theorem of this section is given from (6.29), Theorems 6.2, 6.4 and 6.5.Theorem.
Ast→∞, one has
(6.33)E[S(t)]=p̲⊤(0){t(cP̲̲1+dQ̲̲1)+cP̲̲0+dQ̲̲0}1̲+o(1),(6.34)Var[S(t)]=tp̲⊤(0)W̲̲01̲+o(t),
where W̲̲0=c2U̲̲0+d2V̲̲0+2cdT̲̲0-2cd(P̲̲11̲·p̲⊤(0)Q̲̲0+P̲̲01̲·p̲⊤(0)Q̲̲1).Proof.
Equation (6.33) follows trivially from Theorem 6.2. For (6.34), we note from Theorems 6.2, 6.4 and 6.5 together with (6.29) that
(6.35)Var[S(t)]=p̲⊤(0){t2W̲̲1+tW̲̲0}1̲+o(t),
where
(6.36)W̲̲1=cdT̲̲1-2cdP̲̲11̲·p̲⊤(0)Q̲̲1,W̲̲0=c2U̲̲0+d2V̲̲0+2cdT̲̲0-2cd(P̲̲11̲·p̲⊤(0)Q̲̲0+P̲̲01̲·p̲⊤(0)Q̲̲1).
From (6.27), the first term on the right-hand side of Equation (6.35) can be rewritten as
(6.37)cdp̲⊤(0)T̲̲11̲-2cdp̲⊤(0)P̲̲11̲·p̲⊤(0)Q̲̲11̲=cdp̲⊤(0)(P̲̲1Q̲̲1+Q̲̲1P̲̲1)1̲-2cd(p̲⊤(0)P̲̲11̲)(p̲⊤(0)Q̲̲11̲)=cd(p̲⊤(0)P̲̲1Q̲̲11̲+p̲⊤(0)Q̲̲1P̲̲11̲)-2cd(p̲⊤(0)P̲̲11̲)(p̲⊤(0)Q̲̲11̲)=cd{(p̲⊤(0)P̲̲11̲)(p̲⊤(0)Q̲̲11̲)+(p̲⊤(0)Q̲̲11̲)(p̲⊤(0)P̲̲11̲)}-2cd(p̲⊤(0)P̲̲11̲)(p̲⊤(0)Q̲̲11̲)=0,
completing the proof.


## 7. Dynamic Analysis of a Manufacturing System  for Determining Optimal Maintenance Policy

As an application of the unified multivariate counting process, in this section, we consider a manufacturing system with a certain maintenance policy, where the system starts anew at timet=0, and tends to generate product defects more often as time goes by. When the system reaches a certain state, the manufacturing system would be overhauled completely and the system returns to the fresh state. More specifically, let J(t) be a semi-Markov process on 𝒥={0,1,2,…,J} governed by A̲̲(x), describing the system state at time t where state 0 is the fresh state and state J is the maintenance state. When the system is in state j, 0≤j≤J-1, product defects are generated according to an NHPP with intensity λj(x). It is assumed that the system deteriorates monotonically and accordingly λj(x) increases as a function of both x and j. When the system reaches state J, the manufacturing operation is stopped and the system is overhauled completely. The maintenance time increases stochastically as a function of J. In other words, the further the maintenance is delayed, the longer the maintenance time would tend to be. Upon finishing the overhaul, the system is brought back to the fresh state 0. Of interest, then, is to determine the optimal maintenance policy concerning how to set J.In order to determine the optimal maintenance policy, it is necessary to define the objective function precisely. Letψd be the cost associated with each defect and let ψm be the cost for each of the maintenance operation. If we define two counting processes M(t) and N(t) as the total number of defects generated by time t and the number of the maintenance operations occurred by time t respectively, the total cost in [0,T] can be described as
(7.1)CJ(T)=ψd·E[M(T)]+ψm·E[N(T)].
Let N be the set of natural numbers. The optimal maintenance policy J* is then determined by
(7.2)CJ*(T)=minJ∈NCJ(T).In what follows, we present a numerical example by letting𝒥={0,1,…,J} for 1≤J≤9. For the underlying semi-Markov process, we define the matrix Laplace transform α̲̲(s) having IFR (Increasing Failure Rate) and DFR (Decreasing Failure Rate) dwell time distributions as described below, where the underlying parameters are set in such a way that the means of IFR dwell times are equal to those of DFR dwell times. By introducing matrices θ̲̲,θ̲̲̂ and p̲, for which the details are given in Table 1 along with other parameter values, we define
(7.3)α̲̲IFR(s)=[θiis+θii·θijs+∑j=0Jθij],α̲̲DFR(s)=[pi·θijs+∑j=0Jθij+(1-pi)·θ̂ijs+∑j=0Jθ̂ij],
where
(7.4)pi=((θii+∑j=0Jθij)/(θii·∑j=0Jθij)-1/∑j=0Jθ̂ij)(1/∑j=0Jθij-1/∑j=0Jθ̂ij),∑j=0Jθij≠∑j=0Jθ̂ij.Table 1
Parameters of the numerical example.
ParameterValueJ1,2,…,9λ̲(x)[…,3j2x2,…]⊤𝒜{0,1,…,J-1}ℳ{J-1}𝒩{J}M(t)∑i∈𝒜Mi(t)N(t)∑i∈ℳ,j∈𝒩Nij(t)Θ(i,j)exp(i-7.9)+exp(7.9-j)Θ̂(i,j)1exp(i+j-5)+12θ̲̲[Θ(0,0)Θ(0,1)0⋯00⋱⋱0⋮⋮0Θ(i,i)Θ(i,i+1)00⋱⋱⋱Θ(J,J+1)0⋯0Θ(J,J)]θ̲̲̂[Θ̂(0,0)Θ̂(0,1)0⋯00⋱⋱0⋮⋮0Θ̂(i,i)Θ̂(i,i+1)00⋱⋱⋱Θ̂(J,J+1)0⋯0Θ̂(J,J)]ψd10ψm1000T1000Based on Theorem6.2, the asymptotic behaviors of the mean of M(t) and N(t) per unit time with maintenance policy J=1,…,9 are depicted in Figures 3 and 4. One could see that both the mean of M(t) and N(t) per unit time converges to a positive value as time t increases. In order to determine the optimal maintenance policy, for J∈{1,2,…,9}, the corresponding total cost CJ(T) can be computed based on Theorem 6.6. Numerical results are shown in Table 2 and depicted in Figure 5. For the case of IFR, the optimal maintenance policy is at J*=6, while J*=5 for the DFR case, where the running period T is taken to be T=1000 hours.Table 2
Results of the problem of determining optimal maintenance policy.
Maintenance policyJTotal costMean ofN(t)IFR14558439.70504558.428122249788.36932249.679631226609.04591225.98394703523.0084700.58945423335.3246411.03846294670.0741246.59777337391.4590156.34058729034.3960113.872791319510.664495.4171DFR12855729.23992855.727721706004.84951705.808031055886.17261051.34924722164.3163651.57575663418.1912406.6856790848.6824258.815571110782.8637172.829981708966.7374125.876092544519.3472103.9733Mean ofM(t) per unit time.
(a)(b)Mean ofN(t) per unit time.
(a)(b)Optimal maintenance policy: IFR versus DFR withT = 1000.
(a)(b)


## 8. Concluding Remarks

In this paper, a unified multivariate counting process[M̲(t),N̲̲(t)] is proposed with nonhomogeneous Poisson processes lying on a finite semi-Markov process. Here the vector process M̲(t) counts the cumulative number of such nonhomogeneous Poisson arrivals at every state and the matrix process N̲̲(t) counts the cumulative number of state transitions of the semi-Markov process in [0,t]. This unified multivariate counting process contains many existing counting processes as special cases. The dynamic analysis of the unified multivariate counting process is given, demonstrating the fact that the existing counting processes can be treated as special cases of the unified multivariate counting process. The asymptotic behaviors of the mean and the variance of the unified multivariate counting process are analyzed. As an application, a manufacturing system with certain maintenance policies is considered. The unified multivariate counting process enables one to determine the optimal maintenance policy minimizing the total cost. Numerical examples are given with IFR and DFR dwell times of the underlying semi-Markov process. As for the future agenda, the impact of such distributional properties on the optimal maintenance policy would be pursued theoretically. Other possible theoretical extensions include: (1) analysis of the reward process associated with the unified multivariate counting process; and (2) exploration of further applications in the areas of modern communication networks and credit risk analysis such as CDOs (collateralized debt obligations) for  financial engineering.



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*Source: 219532-2009-09-22.xml*