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

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 as 𝑡→∞ 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.

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 𝜆_𝑖 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 𝑝̃_𝑗𝑖 and the counting process describes the number of such replacements occurred in [0,𝑡]. If we set 𝐺={0_𝐺,1_𝐺,…,𝐽_𝐺} and 𝐵={0_𝐵,1_𝐵,…,𝐽_𝐵}, and an absorbing state 𝑗_𝐵∈𝐵 can be reached only from its counter part 𝑗_𝐺∈𝐺 with instantaneous replacement back to 𝑗_𝐺 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 process 𝐽(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 𝜆_𝑖(𝑥) to 𝜆_𝑗(𝑥) with an initial value reset to 𝜆_𝑗(0). In other words, the arrivals of items are generated by the NHPP with 𝜆_𝑖(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 𝑀(𝑡)=[𝑀_𝑖(𝑡)]_𝑖∈𝒥 and 𝑁(𝑡)=[𝑁_𝑖𝑗(𝑡)]_{𝑖,𝑗∈𝒥} where 𝑀_𝑖(𝑡) counts the cumulative number of items that have arrived in [0,𝑡] while the semi-Markov process is in state 𝑖 and 𝑁_𝑖𝑗(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 [𝑀(𝑡),𝑁(𝑡)] enables one to unify many existing counting processes in that they can be derived in terms of the components of [𝑀(𝑡),𝑁(𝑡)]. Because of this reason, hereafter, we call [𝑀(𝑡),𝑁(𝑡)] the unified multivariate counting process. The dynamic behavior of [𝑀(𝑡),𝑁(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∈𝒥𝑀_𝑖(𝑡) representing the number of packet arrivals at the origination site and 𝑁_𝑖𝑗(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 [𝑀(𝑡),𝑁(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 Section 2, 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 (𝐴,𝑏, etc.) and underlines (𝑋,𝑦, etc.) respectively. The vector with all components equal to 1 is denoted by 1 and the identity matrix is written as 𝐼. A submatrix of 𝑎 is defined by 𝑎_𝐺𝐵=[𝑎_𝑖𝑗]_{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 intensity 𝜆 is characterized by a sequence of independently and identically distributed (i.i.d.) exponential random variables (𝑋_𝑗)^∞𝑗=1 with common probability density function (p.d.f.) 𝑓_𝑋(𝑥)=𝜆𝑒^−𝜆𝑥. Let 𝑆_𝑛=∑^𝑛𝑗=1𝑋_𝑗. Then, the associated Poisson process {𝑁(𝑡)∶𝑡≥0} is defined as a counting process satisfying 𝑁(𝑡)=𝑛⟺𝑆_𝑛≤𝑡<𝑆_𝑛+1.(2.1) If a system has an exponential lifetime of mean 𝜆⁻¹ and is renewed instantaneously upon failure, 𝑋_𝑗 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.

Let 𝑝_𝑛(𝑡)=P[𝑁(𝑡)=𝑛∣𝑁(0)=0] and define the probability generating function (p.g.f.) 𝜋(𝑣,𝑡) by 𝜋(𝑣,𝑡)=E𝑣^𝑁(𝑡)=^∞_𝑛=0𝑝_𝑛(𝑡)𝑣^𝑛.(2.2) It can be seen, see, for example, Karlin and Taylor [14], that 𝑑𝑑𝑡𝑝^𝑛(𝑡)=−𝜆𝑝_𝑛(𝑡)+𝜆𝑝_𝑛−1(𝑡)(2.3) where 𝑝_𝑛(𝑡)=0 for 𝑛<0. Multiplying 𝑣^𝑛 on both sides of (2.3) and summing from 0 to ∞, one then finds that 𝜕𝜕𝑡𝜋(𝑣,𝑡)=−𝜆(1−𝑣)𝜋(𝑣,𝑡).(2.4) Since 𝑝_𝑛(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 𝜋(𝑣,𝑡)=𝑒^{−𝜆𝑡(1−𝑣)};𝑝_𝑛(𝑡)=𝑒^−𝜆𝑡(𝜆𝑡)^𝑛𝑛!.(2.5)

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 𝑑𝑑𝑡𝑝^𝑚(𝑡)=−𝜆(𝑡)𝑝_𝑚(𝑡)+𝜆(𝑡)𝑝_𝑚−1(𝑡).(2.6) By taking the generating function of (2.6), one finds that 𝜕𝜕𝑡𝜋(𝑢,𝑡)=−𝜆(𝑡)(1−𝑢)𝜋(𝑢,𝑡).(2.7) With 𝐿(𝑡)=∫^𝑡0𝜆(𝑦)𝑑𝑦, this equation can be solved as 𝜋(𝑢,𝑡)=𝑒^{−𝐿(𝑡)(1−𝑢)};𝑝_𝑚(𝑡)=𝑒^{−𝐿(𝑡)}𝐿(𝑡)^𝑚𝑚!.(2.8) 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 𝜈=[𝜈_𝑖𝑗]. Let 𝜆^⊤=[𝜆₀,…,𝜆_𝐽] and define the associated diagonal matrix 𝜆_𝐷=[𝛿_{𝑖=𝑗}𝜆_𝑖]. An MMPP {𝑀(𝑡)∶𝑡≥0} characterized by (𝜈,𝜆_𝐷) is a pure jump process where jumps of 𝑀(𝑡) occur according to a Poisson process with intensity 𝜆_𝑖 whenever the Markov-chain 𝐽(𝑡) is in state 𝑖.

Let 𝜈_𝑖=∑_𝑗∈𝒥𝜈_𝑖𝑗 and define 𝜈_𝐷=[𝛿_{𝑖=𝑗}𝜈_𝑖]. The infinitesimal generator 𝑄 associated with the Markov-chain 𝐽(𝑡) is then given by 𝑄=−𝜈_𝐷+𝜈.(2.9) For 𝑖,𝑗∈𝒥, let 𝑝𝑝(𝑘,𝑡)=𝑝_𝑖𝑗(𝑘,𝑡)_𝑖𝑗;(𝑘,𝑡)=P[𝑀(𝑡)=k,𝐽(𝑡)=𝑗∣𝐽(0)=𝑖,𝑀(0)=0],(2.10) and define the associated matrix generating function 𝜋(𝑢,𝑡) by 𝜋(𝑢,𝑡)=^∞_𝑘=00𝑥0200𝑑𝑝(𝑘,𝑡)𝑢^𝑘.(2.11) It can be seen that 𝜕𝜕𝑡𝑝^𝑖𝑗(𝑘,𝑡)=−𝜆_𝑗+𝜈_𝑗𝑝_𝑖𝑗(𝑘,𝑡)+_𝑟∈𝒥0𝑥0200𝑑𝑝_𝑖𝑟(𝑘,𝑡)𝜈_𝑟𝑗+𝜆_𝑗𝑝_𝑖𝑗(𝑘−1,𝑡).(2.12) In matrix notation, this can be rewritten as 𝜕𝜕𝑡𝑝(𝑘,𝑡)=−𝑝(𝑘,𝑡)𝜆_𝐷+𝜈_𝐷−𝜈+𝑝(𝑘−1,𝑡)𝜆_𝐷.(2.13) By taking the generating function of (2.13) together with (2.9), one sees that 𝜕𝜕𝑡𝜋(𝑢,𝑡)=𝜋(𝑢,𝑡)𝑄−(1−𝑢)𝜆_𝐷.(2.14) Since 𝑀(0)=0, one has 𝜋(𝑢,0)=𝐼, where 𝐼=[𝛿_{𝑖=𝑗}] is the identity matrix, so that the above differential equation can be solved as 𝜋(𝑢,𝑡)=𝑒^{𝑄−(1−𝑢)𝜆𝐷𝑡}=^∞_𝑘=0𝑡^𝑘𝑘!𝑄−(1−𝑢)𝜆_𝐷^𝑘,(2.15) where 𝐴=^0def𝐼 for any square matrix 𝐴. It should be noted that 𝜋(1,𝑡)=𝑒^𝑄𝑡, which is the transition probability matrix of 𝐽(𝑡) as it should be. By taking the Laplace transform of both sides of (2.15), 𝜋(𝑢,𝑠)=∫^∞0𝑒^−𝑠𝑡𝜋(𝑢,𝑡) is given by 𝜋(𝑢,𝑠)=𝑠𝐼−𝑄+(1−𝑢)𝜆_𝐷⁻¹.(2.16)

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 Section 1. 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 function 𝐴(𝑥). The resulting counting process {𝑁(𝑡)∶𝑡≥0} is still characterized by (2.1). Let 𝑝_𝑛(𝑡)=P[𝑁(𝑡)=𝑛∣𝑁(0)=0] as before. One then sees that 𝑝_𝑛(𝑡)=𝐴^(𝑛)(𝑡)−𝐴^(𝑛+1)(𝑡),(2.17) where 𝐴^(𝑛)(𝑡) denotes the 𝑛-fold convolution of 𝐴(𝑥) with itself, that is, 𝐴^(𝑛+1)(𝑡)=∫^𝑡0𝐴^(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.

Let 𝜋_𝑛(𝑠)=∫^∞0𝑒^−𝑠𝑡𝑝_𝑛(𝑡)𝑑𝑡 and 𝛼(𝑠)=∫^∞0𝑒^−𝑠𝑡𝑑𝐴(𝑡). By taking the Laplace transform of both sides of (2.17), it follows that 𝜋_𝑛(𝑠)=1−𝛼(𝑠)𝑠𝛼(𝑠)^𝑛.(2.18) By taking the generating function of the above equation with 𝜋(𝑣,𝑠)=∑^∞𝑛=0𝜋_𝑛(𝑠)𝑣^𝑛, one has 𝜋(𝑣,𝑠)=1−𝛼(𝑠)𝑠1−𝑣1𝛼(𝑠).(2.19) 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. 𝑎(𝑥) where 𝑎(𝑥)≥0 and ∫^∞0𝑎(𝑥)𝑑𝑥=𝐴₀ 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 𝑁_ℓ𝑟(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 𝑁_ℓ(𝑡)=[𝑁_ℓ0(𝑡),…,𝑁_ℓ𝐽(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 𝐻(𝑡)=[𝐻_ℓ𝑟(𝑡)] with 𝐻_ℓ𝑟(𝑡)=E[𝑁_ℓ𝑟(𝑡)], the associated matrix renewal density, and the limit theorems. For example, one has the following result concerning the Laplace transform of 𝐻(𝑡) by Keilson [23]: ℒ𝐻(𝑡)=1𝑠𝛼(𝑠)𝐼−𝛼(𝑠)⁻¹,(2.20) where 𝛼(𝑠) is the Laplace transform of 𝑎(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 on 𝒥={0,…,𝐽} governed by a matrix p.d.f. 𝑎(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 𝑁_𝐺𝐵(𝑡) is then defined to describe the number of entries of 𝐽(𝑡) into 𝐵 by time 𝑡.

While 𝑁_𝐺𝐵(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, 𝑋(𝑡)=𝑡−sup𝜏∶𝐽(𝑡)∣^𝜏+𝜏−≠0,0<𝜏≤𝑡,(2.21) where 𝑓(𝑥)∣^𝑥+𝑥−=𝑓(𝑥+)−𝑓(𝑥−), and define 𝐹_𝑛(𝑥,𝑡)=𝐹_{𝑛∶𝑖𝑗}(𝑥,𝑡),(2.22) where 𝐹_{𝑛∶𝑖𝑗}(𝑥,𝑡)=P𝑋(𝑡)≤𝑥,𝑁_𝐺𝐵(𝑡)=𝑛,𝐽(𝑡)=𝑗∣𝑋(0)=𝑁_𝐺𝐵(0)=0,𝐽(0)=𝑖.(2.23) One then has 𝑓_𝑛(𝑥,𝑡)=𝜕𝜕𝑥𝐹^𝑛(𝑥,𝑡).(2.24) The associated matrix Laplace transform generating function can then be defined as 𝜑(𝑣,𝑤,𝑠)=^∞_𝑛=0𝑣^𝑛^∞0𝑒^{−𝑤𝑡−𝑠𝑡}𝑓_𝑛(𝑥,𝑡)𝑑𝑥𝑑𝑡.(2.25) It has been shown in Masuda and Sumita [3] that 𝜑(𝑣,𝑤,𝑠)=1𝑤+𝑠⋅𝛾₀(𝑠)𝐼−𝑣𝛽(𝑠)⁻¹𝐼−𝛼_𝐷(𝑤+𝑠).(2.26) Here, with 𝛼(𝑠)=∫^∞0𝑒^−𝑠𝑡𝑎(𝑡)𝑑𝑡 and 𝛼_𝐶𝐷(𝑠)=[𝛼_𝑖𝑗(𝑠)]_{𝑖∈𝐶,𝑗∈𝐷} for 𝐶,𝐷⊂𝒥, the following notation is employed:

𝛼(𝑠)=𝛼^𝐺𝐺𝛼(𝑠)𝛼_𝐺𝐵(𝑠)_𝐵𝐺𝛼(𝑠)𝛼_𝐵𝐵(𝑠)_𝐷;𝜒(𝑠)=𝛿_{𝑖=𝑗}𝛼_𝑖(𝑠)with𝛼_𝑖(𝑠)=∑_𝑗∈𝒥𝛼_𝑖𝑗(𝑠);(2.27)_𝐺(𝑠)=𝛽𝐼_𝐺𝐺−𝛼_𝐺𝐺(𝑠)⁻¹;𝜒_𝐵(𝑠)=𝐼_𝐵𝐵−𝛼_𝐵𝐵(𝑠)⁻¹;(2.28)(𝑠)=0^𝐵𝐵0_𝐵𝐺𝛾𝛼_𝐺𝐵𝜒_𝐵(𝑠)𝛼_𝐺𝐵𝜒_𝐵(𝑠)𝛼_𝐵𝐺𝜒_𝐺(𝑠);(2.29)₀(𝑠)=𝛼^𝐵𝐵0𝜒_𝐵(𝑠)𝜒_𝐵(𝑠)𝛼_𝐵𝐺𝜒_𝐺(𝑠)_𝐺𝐵𝛼_𝐺𝐺𝜒_𝐺(𝑠)+𝐼.(2.30) 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.

𝜈^∗𝐺𝐺=𝜈_𝑖𝑗_{𝑖,𝑗∈𝐺};𝜈^∗𝐺𝐵=𝜈_𝑖𝑟_{𝑖∈𝐺,𝑟∈𝐵}.(2.31) The entire transition rate matrix 𝜈^∗ governing 𝐽^∗(𝑡) is then given by 𝜈^∗=𝜈^∗𝐺𝐺𝜈^∗𝐺𝐵00.(2.32)

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 𝑝̃_𝑟𝑗 and the process continues. Let 𝐶=𝜈^∗𝐺𝐺,𝐷=𝜈^∗𝐺𝐵𝑝̃_𝐵𝐺,(2.33) where 𝑝̃_𝐵𝐺=[𝑝̃_𝑟𝑗]_{𝑟∈𝐵,𝑗∈𝐺}. Then the transition rate matrix 𝜈 and the infinitesimal generator 𝑄 of 𝐽(𝑡) are given as 𝜈=𝐶+𝐷;𝑄=−𝜈_𝐷+𝜈,(2.34) where 𝜈_𝐷=𝐶_𝐷+𝐷_𝐷,(2.35) with 𝐷𝐶_𝐷𝐷=𝛿_{𝑖=𝑗}𝑐_𝑖;𝑐_𝑖=_𝑗∈𝐺𝑐_𝑖𝑗,=𝛿_{𝑖=𝑗}𝑑_𝑖;𝑑_𝑖=_𝑗∈𝐺𝑑_𝑖𝑗.(2.36) Let 𝑝(𝑡) be the transition probability matrix of 𝐽(𝑡) with its Laplace transform defined by 𝜋(𝑠)=∫^∞0𝑒^−𝑠𝑡𝑝(𝑡)𝑑𝑡. From the Kolmogorov forward equation (𝑑/𝑑𝑡)𝑝(𝑡)=𝑝(𝑡)𝑄 with 𝑝(0)=𝐼, one has 𝜋(𝑠)=𝑠𝐼−𝑄⁻¹.(2.37)

Let {𝑁_MAP(𝑡)∶𝑡≥0} be the counting process keeping the record of the number of replacements in [0,𝑡) and define 𝑓_𝑖𝑗(𝑘,𝑡)=𝑃𝑁_MAP(𝑡)=𝑘,𝐽(𝑡)=𝑗∣𝐽(0)=𝑖,𝑖,𝑗∈𝐺.(2.38) By analyzing the probabilistic flow at state 𝑗 at time 𝑡+Δ, it can be seen that 𝑓_𝑖𝑗(𝑘,𝑡+Δ)=+𝑓_𝑖𝑗(_ℓ∈𝐺𝑘,𝑡)1−_ℓ∈𝐺𝑐_𝑗ℓ+𝑑_𝑗ℓΔ+_ℓ∈𝐺𝑓_𝑖ℓ(𝑘,𝑡)𝑐_ℓ𝑗Δ𝑓_𝑖ℓ(𝑘−1,𝑡)𝑑_ℓ𝑗Δ+𝑜(Δ).(2.39) It then follows that 𝜕𝜕𝑡𝑓^𝑖𝑗(𝑘,𝑡)=−𝑓_𝑖𝑗(𝑘,𝑡)_ℓ∈𝐺𝑐_𝑗ℓ+𝑑_𝑗ℓ+_ℓ∈𝐺𝑓_𝑖ℓ(𝑘,𝑡)𝑐_ℓ𝑗+_ℓ∈𝐺𝑓_𝑖ℓ(𝑘−1,𝑡)𝑑_ℓ𝑗.(2.40) In matrix notation, the above equation can be rewritten as 𝜕𝜕𝑡𝑓(𝑘,𝑡)=−𝑓(𝑘,𝑡)𝜈_𝐷+𝑓(𝑘,𝑡)𝐶+𝑓(𝑘−1,𝑡)𝐷.(2.41) We now introduce the following matrix Laplace transform generating function: 𝜑(𝑣,𝑠)=𝜑_𝑖𝑗(𝑣,𝑠);𝜑_𝑖𝑗(𝑣,𝑠)=^∞0𝑒^{∞−𝑠𝑡}_𝑘=0𝑓_𝑖𝑗(𝑘,𝑡)𝑣^𝑘𝑑𝑡.(2.42) Taking the Laplace transform with respect to 𝑡 and the generating function with respect to 𝑘 of both sides of (2.41), one has 𝜑(𝑣,𝑠)𝑠𝐼+𝜈_𝐷−𝐶−𝑣𝐷=𝐼,(2.43) which can be solved as 𝜑(𝑣,𝑠)=𝑠𝐼−𝑄+(1−𝑣)𝐷⁻¹.(2.44) We note from (2.37) and (2.44) that 𝜑(1,𝑠)=𝜋(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 on 𝐺∪𝐵 where 𝐺={0_𝐺,…,𝐽_𝐺} and 𝐵={0_𝐵,…,𝐽_𝐵}. Transitions within 𝐺 is governed by 𝜈. An entry into 𝑗_𝐵∈𝐵 is possible only from 𝑗_𝐺∈𝐺. When this occurs, the Markov process is immediately replaced at the entering state 𝑗_𝐺. The counting process for the number of such replacements then has the Laplace transform generating function 𝜑(𝑣,𝑠) given in (2.44) where 𝐷 is replaced by 𝜆_𝐷, which coincides with 𝜋(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 (𝑋_𝑗)^∞𝑗=1 with common p.d.f. 𝑎(𝑥). The age process 𝑋(𝑡) is then defined by 𝑋(𝑡)=𝑡−sup𝜏∶𝑁(𝑡)∣^𝜏+𝜏−=1,0<𝜏≤𝑡.(2.45) 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 𝐿(𝑥)=^𝑥0𝜆(𝑦)𝑑𝑦,(2.46) one has 𝑔(𝑥,𝑘)=P[𝑍(𝑥)=𝑘]=exp(−𝐿(𝑥))𝐿(𝑥)^𝑘𝑘!,𝑘=0,1,2,….(2.47) Of interest, then, is a counting process {𝑀(𝑡)∶𝑡≥0} characterized by P[𝑀=(𝑡𝜆+(𝑥Δ))Δ−+𝑀𝑜((Δ𝑡))=,1(𝑚∣𝑀,𝑛,(𝑥𝑡))=∈𝑚𝑆,,𝑁Δ>(𝑡)0=.1,𝑋(𝑡)=𝑥](2.48) 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 process 𝑀(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 𝜑(𝑢,𝑣,𝑤,𝑠)=^∞0𝑒^−𝑠𝑡E𝑢^𝑀(𝑡)𝑣^𝑁(𝑡)𝑒^{−𝑤𝑋(𝑡)}𝑑𝑡.(2.49) It has been shown in Sumita and Shanthikumar [12] that 𝜑(𝑢,𝑣,𝑤,𝑠)=𝛽^∗(𝑢,𝑤+𝑠)1−𝑣𝛽(𝑢,𝑠),(2.50) where we define, for 𝑚≥0 with 𝐴(𝑡)=∫^∞𝑡𝑎(𝑥)𝑑𝑥,

𝛽𝑏_𝑚(𝑡)=𝑎(𝑡)𝑔(𝑡,𝑚);𝑏^∗𝑚(𝑡)=𝐴(𝑡)𝑔(𝑡,𝑚),_𝑚𝛽((𝑠𝑢),=𝑠)=^∞0𝑒^−𝑠𝑡_∞𝑏_𝑚(𝑡)𝑑𝑡;𝛽^∗𝑚(𝑠)=^∞0𝑒^−𝑠𝑡𝑏^∗𝑚(𝑡)𝑑𝑡,_𝑚₌₀𝛽_𝑚(𝑠)𝑢^𝑚;𝛽^∗(𝑢,𝑠)=^∞_𝑚₌₀𝛽^∗𝑚(𝑠)𝑢^𝑚.(2.51) The Laplace transform generating function of 𝑀(𝑡) defined by 𝜋(𝑢,𝑠)=^∞0𝑒^−𝑠𝑡E𝑢^𝑀(𝑡)𝑑𝑡(2.52) is then given by 𝜋(𝑢,𝑠)=𝜑(𝑢,1,0,𝑠), so that one has from (2.50) 𝜋(𝑢,𝑠)=𝛽^∗(𝑢,𝑠)1−𝛽(𝑢,𝑠).(2.53)

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 Figure 1.

Figure 1 

Various counting processes.

3. Unified Multivariate Counting Process [𝐌(𝐭),𝐍(𝐭)]

In this section, we propose a stochastic process representing the unified multivariate counting process discussed in Section 1, 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 𝜆_𝑖(𝑥). If the semi-Markov process switches its state from 𝑖 to 𝑗, the intensity function 𝜆_𝑖(𝑥) is interrupted, the intensity function at state 𝑗 is reset to 𝜆_𝑗(0), and the arrival process resumes. Of particular interest would be the multivariate counting processes 𝑀(𝑡)^⊤=[𝑀₀(𝑡),…,𝑀_𝐽(𝑡)] and 𝑁(𝑡)=[𝑁_𝑖𝑗(𝑡)] with 𝑀_𝑖(𝑡) and 𝑁_𝑖𝑗(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 𝑀(𝑡) and 𝑁(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.) 𝐴(𝑥)=𝐴_𝑖𝑗(𝑥),(3.1) which is assumed to be absolutely continuous with the matrix probability density function (p.d.f.) 𝑎(𝑥)=𝑎_𝑖𝑗(𝑥)=𝑑𝑑𝑥𝐴(𝑥).(3.2) It should be noted that, if we define 𝐴_𝑖(𝑥) and 𝐴_𝑖(𝑥) by 𝐴_𝑖(𝑥)=_𝑗∈𝒥𝐴_𝑖𝑗(𝑥);𝐴_𝑖(𝑥)=1−𝐴_𝑖(𝑥),(3.3) then 𝐴_𝑖(𝑥) is an ordinary c.d.f. and 𝐴_𝑖(𝑥) is the corresponding survival function. The hazard rate functions associated with the semi-Markov process are then defined as 𝜂_𝑖𝑗(𝑥)=𝑎^𝑖𝑗(𝑥)𝐴_𝑖(𝑥),𝑖,𝑗∈𝒥.(3.4)

For notational convenience, the transition epochs of the semi-Markov process are denoted by 𝜏_𝑛,𝑛≥0, with 𝜏₀=0. The age process 𝑋(𝑡) associated with the semi-Markov process is then defined as 𝑋(𝑡)=𝑡−max𝜏_𝑛∶0≤𝜏_𝑛≤𝑡.(3.5) At time 𝑡 with 𝐽(𝑡)=𝑖 and 𝑋(𝑡)=𝑥, the intensity function of the NHPP is given by 𝜆_𝑖(𝑥). For the cumulative arrival intensity function 𝐿_𝑖(𝑥) in state 𝑖, one has 𝐿_𝑖(𝑥)=^𝑥0𝜆_𝑖(𝑦)𝑑𝑦.(3.6) The probability of observing 𝑘 arrivals in state 𝑖 within the current age of 𝑥 can then be obtained as 𝑔_𝑖(𝑥,𝑘)=𝑒^{−𝐿𝑖(𝑥)}𝐿_𝑖(𝑥)^𝑘𝑘!,𝑘=0,1,2,…,𝑖∈𝒥.(3.7)

Of interest are the multivariate counting processes 𝑀(𝑡)^⊤=𝑀₀(𝑡),…,𝑀_𝐽(𝑡),(3.8) where 𝑀_𝑖(𝑡) represents the total number of items that have arrived by time 𝑡 while the semi-Markov process stayed in state 𝑖, and 𝑁(𝑡)=𝑁_𝑖𝑗(𝑡),(3.9) with 𝑁_𝑖𝑗(𝑡) denoting the number of transitions of the semi-Markov process from state 𝑖 to state 𝑗 by time 𝑡. It is obvious that 𝑁_𝑖(𝑡)=^def∑_ℓ∈𝒥𝑁_ℓ𝑖(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 [𝑀(𝑡),𝑁(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 𝑀(𝑡) and 𝑁(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 [𝐌(𝐭),𝐍(𝐭)]

The purpose of this section is to examine the dynamic behavior of the multivariate stochastic process [𝑀(𝑡),𝑁(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 [𝑀(𝑡),𝑁(𝑡)].

Since [𝑀(𝑡),𝑁(𝑡)] is not Markov, we employ the method of supplementary variables. More specifically, the multivariate stochastic process [𝑀(𝑡),𝑁(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 𝐹_𝑖𝑗(𝑚,𝑛,𝑥,𝑡) be the joint probability distribution of [𝑀(𝑡),𝑁(𝑡),𝑋(𝑡),𝐽(𝑡)] defined by 𝐹_𝑖𝑗𝑚,𝑛,𝑥,𝑡=P𝑀(𝑡)=𝑚,𝑁(𝑡)=𝑛,𝑋(𝑡)≤𝑥,𝐽(𝑡)=𝑗∣𝑀(0)=0,𝑁(0)=0,𝐽(0)=𝑖.(4.1)

In order to assure the differentiability of 𝐹_𝑖𝑗(𝑚,𝑛,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 {𝐷_𝑘(𝑥)}^∞𝑘=1 satisfying 𝐷_𝑘(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 𝑓_𝑖𝑗𝑚,𝑛,𝑥,𝑡=𝜕𝜕𝑥𝐹^𝑖𝑗𝑚,𝑛,𝑥,𝑡.(4.2) By interpreting the probabilistic flow of the multivariate process [𝑀(𝑡),𝑁(𝑡),𝑋(𝑡),𝐽(𝑡)] in its state space, one can establish the following equations: 𝑓_𝑖𝑗𝑚,𝑛,𝑥,𝑡=𝛿_{𝑖=𝑗}𝛿_{{𝑚=𝑚𝑖1𝑖}}𝛿_{𝑛=0}𝑑(𝑥−𝑡)𝐴_𝑖(𝑥)𝐴_𝑖(𝑥−𝑡)𝑔^𝑖+1−𝛿_{𝑛=0}𝑡,𝑚_𝑖𝑓^𝑚𝑗_𝑘=0𝑓_𝑖𝑗𝑚−𝑘1_𝑗,𝑛,0+,𝑡−𝑥𝐴_𝑗(𝑥)𝑔_𝑗(𝑥,𝑘);(4.3)_𝑖𝑗𝑓𝑚,𝑛,0+,𝑡=1−𝛿_{𝑛=0}_ℓ∈𝒥^∞0𝑓_𝑖ℓ𝑚,𝑛−1_ℓ𝑗,𝑥,𝑡𝜂_ℓ𝑗(𝑥)𝑑𝑥;(4.4)_𝑖𝑗𝑚,𝑛,𝑥,0=𝛿_{𝑖=𝑗}𝛿_{𝑚=0}𝛿_{𝑛=0}𝑑(𝑥),(4.5) where 1_𝑖 is the column vector whose 𝑖th element is equal to 1 with all other elements being 0, 1_𝑖𝑗=1_𝑖1^⊤𝑗 and 𝑓_𝑖𝑗(𝑚,𝑛,0+,𝑡)=0for𝑁≤0.

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_{𝑛=0}=1] and there have been 𝑚_𝑖 items arrived during time 𝑡[𝛿_{{𝑚=𝑚𝑖1𝑖}}=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_{𝑛=0}=0], the multivariate process [𝑀(𝑡),𝑁(𝑡),𝑋(𝑡),𝐽(𝑡)] just entered the state [𝑚−𝑘1_𝑗,𝑛,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 [𝑚,𝑛,0+,𝑗] at time 𝑡, it must be at [𝑚,𝑛−1_ℓ𝑗,𝑥,ℓ] followed by a transition from ℓ to 𝑗 at time 𝑡 which increases 𝑁_ℓ𝑗(t) by one, explaining (4.4). Equations (4.5) merely describe the initial condition that [𝑀(0),𝑁(0),𝑋(0),𝐽(0)]=[0,0,𝑥,𝑖].

In what follows, the dynamic behavior of the multivariate process [𝑀(𝑡),𝑁(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: b𝑏_𝑘(𝑡)=𝑏_{𝑘∶𝑖𝑗}(𝑡);𝑏_{𝑘∶𝑖𝑗}(𝑡)=𝑎_𝑖𝑗(𝑡)𝑔_𝑖(𝑡,𝑘),(4.6)_{∗𝑘∶𝐷}(𝑡)=𝛿_{𝑖=𝑗}𝑏^{∗𝑘∶𝑖}(𝑡)=⎡⎢⎢⎣𝑏^∗𝑘∶0(𝑡)⋱𝑏_{∗𝑘∶𝐽}(𝑡)⎤⎥⎥⎦;𝑏^{∗𝑘∶𝑖}𝑟(𝑡)=𝐴_𝑖(𝑡)𝑔_𝑖(𝑡,𝑘),(4.7)_∗𝑘(𝑡)=𝑟^{∗𝑘∶𝑖𝑗}(𝑡);𝑟^{∗𝑘∶𝑖𝑗}(𝑡)=𝑔_𝑖(𝑡,𝑘)^∞0𝑑(𝑥−𝑡)𝑎^𝑖𝑗(𝑥)𝛽𝐴_𝑖(𝑥−𝑡)𝑑𝑥,(4.8)_𝑘𝛽(𝑠)=𝛽_{𝑘∶𝑖𝑗}(𝑠);𝛽_{𝑘∶𝑖𝑗}(𝑠)=^∞0𝑒^−𝑠𝑡𝑏_{𝑘∶𝑖𝑗}(𝑡)𝑑𝑡,(4.9)𝛽𝑢,𝑠=𝛽_𝑖𝑗𝑢_𝑖,𝑠;𝛽_𝑖𝑗𝑢_𝑖,𝑠=^∞_𝑘=0𝛽_{𝑘∶𝑖𝑗}(𝑠)𝑢^𝑘𝑖,(4.10)_{∗𝑘∶𝐷}(𝑠)=⎡⎢⎢⎣𝛽^∗𝑘∶0(𝑠)⋱𝛽_{∗𝑘∶𝐽}(𝑠)⎤⎥⎥⎦;𝛽^{∗𝑘∶𝑖}𝛽(𝑠)=^∞0𝑒^−𝑠𝑡𝑏^{∗𝑘∶𝑖}(𝑡)𝑑𝑡,(4.11)_∗𝐷𝑢,𝑠=⎡⎢⎢⎣𝛽^∗0𝑢₀,𝑠⋱𝛽_∗𝐽𝑢_𝐽,𝑠⎤⎥⎥⎦;𝛽^∗𝑖𝜌𝑢_𝑖,𝑠=^∞_𝑘=0𝛽^{∗𝑘∶𝑖}(𝑠)𝑢^𝑘𝑖,(4.12)_∗𝑘𝜌(𝑠)=𝜌^{∗𝑘∶𝑖𝑗}(𝑠);𝜌^{∗𝑘∶𝑖𝑗}(𝑠)=^∞0𝑒^−𝑠𝑡𝑟^{∗𝑘∶𝑖𝑗}(𝑡)𝑑𝑡,(4.13)_∗𝑢,𝑠=𝜌^∗𝑖𝑗𝑢_𝑖,𝑠;𝜌^∗𝑖𝑗u_𝑖,𝑠=^∞_𝑘=0𝜌^{∗𝑘∶𝑖𝑗}(𝑠)𝑢^𝑘𝑖,(4.14) where 𝑢^𝑚𝑣^𝑛=∏_𝑖∈𝒥𝑢^𝑚𝑖_𝑖∏_{(𝑖,𝑗)∈𝒥×𝒥⧵{(0,0)}}𝑣^𝑛𝑖𝑗_𝑖𝑗. We are now in a position to prove the main theorem of this section.Theorem. Let 𝑋(0)=0. Then: 𝜉̂𝜑𝑢,𝑣,0+,𝑠=𝛽̃𝑢,𝑣,𝑠𝐼−𝛽̃(𝑢,𝑣,𝑠)⁻¹;(4.19)𝑢,𝑣,𝑤,𝑠=𝐼−𝛽̃(𝑢,𝑣,𝑠)⁻¹𝛽^∗𝐷𝑢,𝑤+𝑠,(4.20) where 𝛽̃(𝑢,𝑣,𝑠)=[𝑣_𝑖𝑗⋅𝛽_𝑖𝑗(𝑢_𝑖,𝑠)].

Proof. First, we assume that 𝑋(0) has a p.d.f. 𝑑(𝑥). Substituting (4.3) and (4.5) into (4.4), one sees that 𝑓_𝑖𝑗𝑚,𝑛,0+,𝑡=×1−𝛿_{{𝑛=0}ℓ∈𝒥}^∞0𝛿_{𝑖=ℓ}𝛿_{{𝑚=𝑚𝑖1𝑖}}𝛿_{{𝑛=1ℓ𝑗}}𝑑(𝑥−𝑡)𝐴_𝑖(𝑥)𝐴_𝑖(𝑥−𝑡)𝑔^𝑖+𝑡,𝑚_𝑖𝜂_ℓ𝑗(𝑥)𝑑𝑥_ℓ∈𝒥1−𝛿^{𝑛=1_ℓ𝑗^}^∞0𝑚ℓ_𝑘=0𝑓_𝑖ℓ𝑚×−𝑘1_ℓ,𝑛−1_ℓ𝑗,0+,𝑡−𝑥=1−𝛿𝐴_ℓ(𝑥_{𝑛=0})𝑔_ℓ(𝑥,𝑘)𝜂_ℓ𝑗(𝑥)𝑑𝑥𝛿^{𝑚=𝑚_𝑖¹_𝑖^}𝛿_{{𝑛=1𝑖𝑗}}𝑔_𝑖𝑡,𝑚_𝑖^∞0𝑑(𝑥−𝑡)𝑎^𝑖𝑗(𝑥)+𝐴_𝑖(𝑥−𝑡)𝑑𝑥_ℓ∈𝒥1−𝛿^{𝑛=1_ℓ𝑗^}^𝑚ℓ_𝑘=0^∞0𝑓_𝑖ℓ𝑚×𝑎_ℓ𝑗−𝑘1_ℓ,𝑛−1_ℓ𝑗,0+,𝑡−𝑥(𝑥)𝑔_ℓ(𝑥,𝑘)𝑑𝑥.(4.21) Consequently, one sees from (4.6) and (4.8) that 𝑓_𝑖𝑗𝑚,𝑛,0+,𝑡=×1−𝛿𝛿_{𝑛=0}_{𝑚=𝑚𝑖1𝑖}𝛿_{𝑛=1𝑖𝑗}𝑟^∗𝑚_{𝑖∶𝑖𝑗}(𝑡)+_ℓ∈𝒥×_𝑚ℓ1−𝛿^{𝑛=1_ℓ𝑗^}_𝑘=0^∞0𝑓_𝑖ℓ𝑚−𝑘1_ℓ,𝑛−1_ℓ𝑗,0+,𝑡−𝑥𝑏_{𝑘∶ℓ𝑗}(𝑥)𝑑𝑥.(4.22) where 𝑎_𝑖𝑗(𝑥)=𝐴_𝑖(𝑥)𝜂_𝑖𝑗(𝑥) is employed from (3.4). By taking the Laplace transform of both sides of (4.22) with respect to 𝑡, it follows that 𝜉_𝑖𝑗𝑚,𝑛,0+,𝑠=×1−𝛿𝛿_{𝑛=0}_{𝑚=𝑚𝑖1𝑖}+𝛿_{𝑛=1𝑖𝑗}𝜌^∗𝑚_{𝑖∶𝑖𝑗}(𝑠)_ℓ∈𝒥1−𝛿^{𝑛=1_ℓ𝑗^}^𝑚ℓ_𝑘=0𝜉_𝑖ℓ𝑚−𝑘1_ℓ,𝑛−1_ℓ𝑗,0+,𝑠𝛽_{𝑘∶ℓ𝑗}(𝑠).(4.23) By taking the multivariate generating function of (4.23) with respect to 𝑚 and 𝑛, it can be seen that 𝜉̂_𝑖𝑗𝑢,𝑣,0+,𝑠=_{𝑚∈ℤ+𝐽+1}_{𝑛∈ℤ+(𝐽+1)×(𝐽+1)⧵{0}}𝜉_𝑖𝑗𝑚,𝑛,0+,𝑠𝑢^𝑚𝑣^𝑛=_{𝑚∈ℤ+𝐽+1𝑛∈ℤ+(𝐽+}_{1)×(𝐽+1)⧵{0}}𝛿^{{𝑚=𝑚𝑖1𝑖}}𝛿_{{𝑛=1𝑖𝑗}}𝜌^∗𝑚_{𝑖∶𝑖𝑗}(𝑠)u^𝑚𝑣^𝑛+_ℓ∈𝒥1−𝛿^{𝑛=1_ℓ𝑗^}^𝑚ℓ_𝑘=0𝜉_𝑖ℓ𝑚−𝑘1_ℓ,𝑛−1_ℓ𝑗,0+,𝑠𝛽_{𝑘∶ℓ𝑗}(𝑠)𝑢^𝑚𝑣^𝑛.(4.24) It then follows from (4.10), (4.14), (4.16) and the discrete convolution property that 𝜉̂_𝑖𝑗𝑢,𝑣,0+,𝑠=+^∞_𝑚𝑖=0𝑢^𝑚𝑖_𝑖𝑣_𝑖𝑗𝜌^∗𝑚_{𝑖∶𝑖𝑗}(𝑠)_ℓ∈𝒥⎛⎜⎜⎜⎝𝑣_ℓ𝑗_{𝑚∈ℤ+𝐽+1}^𝑚ℓ_𝑘=0_{𝑛∈ℤ+(𝐽+1)×(𝐽+1)⧵{0}}𝜉_𝑖ℓ𝑚−𝑘1_ℓ,𝑛,0+,𝑠𝛽_{𝑘∶ℓ𝑗}(𝑠)𝑢^𝑚𝑣^𝑛=𝑣_𝑖𝑗⎞⎟⎟⎟⎠𝜌^∗𝑖𝑗𝑢_𝑖,𝑠+_ℓ∈𝒥𝑣_ℓ𝑗𝜉̂_𝑖ℓ𝑢,𝑣,0+,𝑠𝛽_ℓ𝑗𝑢_ℓ,𝑠.(4.25) The last expression can be rewritten in matrix form, and one has 𝜉̂𝑢,𝑣,0+,𝑠=𝜌̃^∗𝑢,𝑣,𝑠+𝜉̂𝑢,𝑣,0+,𝑠𝛽̃𝑢,𝑣,𝑠,(4.26) which can be solved for 𝜉̂(𝑢,𝑣,0+,𝑠) as 𝜉̂𝑢,𝑣,0+,𝑠=𝜌̃^∗𝑢,𝑣,𝑠𝐼−𝛽̃(𝑢,𝑣,𝑠)⁻¹,(4.27) where 𝜌̃^∗(𝑢,𝑣,s)=[𝑣_𝑖𝑗⋅𝜌^∗𝑖𝑗(𝑢_𝑖,𝑠)].
Next, we introduce the following double Laplace transform:
𝜀_𝑘∶𝑖(𝑤,𝑠)=^∞0𝑒^−𝑤𝑥𝑒^−𝑠𝑡𝑑(𝑥−𝑡)𝐴_𝑖(𝑥)𝐴_𝑖(𝑥−𝑡)𝑔^𝑖(𝑡,𝑘)𝑑𝑡𝑑𝑥,(4.28) and the associated diagonal matrix 𝜀_𝐷𝑢,𝑤,𝑠=⎡⎢⎢⎣𝜀^∗0𝑢₀,𝑤,𝑠⋱𝜀_∗𝐽𝜀𝑢_𝐽,𝑤,𝑠_𝑖⎤⎥⎥⎦;𝑢_𝑖,𝑤,𝑠=^∞_𝑘=0𝜀_𝑘∶𝑖(𝑤,𝑠)𝑢^𝑘𝑖.(4.29) By taking the double Laplace transform of (4.3), one sees from (4.7) and (4.28) that 𝜑_𝑖𝑗𝑚,𝑛,𝑤,𝑠=+𝛿1_{𝑖=𝑗}−𝛿𝛿_{{𝑚=𝑚𝑖1𝑖}{𝑛=0}}𝛿_{𝑛=0}𝜀_{𝑚𝑖∶𝑖}(𝑤,𝑠)^𝑚𝑗_𝑘=0𝜉_𝑖𝑗𝑚−𝑘1_𝑗,𝑛,0+,𝑠𝛽^{∗𝑘∶𝑗}(𝑤+𝑠).(4.30) By taking the double generating function, this then leads to 𝜑_𝑖𝑗=𝑢,𝑣,𝑤,𝑠_{𝑚∈ℤ+𝐽+1}_𝑛∈ℤ+(_{𝐽+1)×(𝐽+1)}𝜑_𝑖𝑗𝑚,𝑛,𝑤,𝑠𝑢^𝑚𝑣^𝑛=𝛿_{∞{𝑖=𝑗}𝑚}_𝑖=0+𝑢^𝑚𝑖_𝑖𝜀_{𝑚𝑖∶𝑖}(𝑤,𝑠)_{𝑚∈ℤ+𝐽+1𝑛∈ℤ+(𝐽+1}_{)×(𝐽+1)⧵0}^𝑚𝑗_𝑘=0𝜉_𝑖𝑗𝑚−𝑘1_𝑗,𝑛,0+,𝑠𝛽^{∗𝑘∶𝑗}(𝑤+𝑠)𝑢^𝑚𝑣^𝑛=𝛿_{𝑖=𝑗}𝜀_𝑖𝑢_𝑖,𝑤,𝑠+𝜉̂_𝑖𝑗𝑢,𝑣,0+,𝑠𝛽^∗𝑗𝑢_𝑗,𝑤+𝑠.(4.31) By rewriting the last expression in matrix form, it can be seen that 𝜑𝑢,𝑣,𝑤,𝑠=𝜀_𝐷𝑢,𝑤,𝑠+𝜉̂𝑢,𝑣,0+,𝑠𝛽^∗𝐷𝑢,𝑤+𝑠.(4.32)
We now consider the limiting process 𝐷(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 𝑟^{∗𝑘∶𝑖𝑗}(𝑡)→𝑏_{𝑘∶𝑖𝑗}(𝑡). This in turn implies from (4.28) that 𝜀_𝑘∶𝑖(𝑤,𝑠)→𝛽^{∗𝑘∶𝑖}(𝑤+𝑠). Consequently, it follows in matrix form that 𝜌̃^∗(𝑢,𝑣,𝑠)→𝛽̃(𝑢,𝑣,𝑠) and 𝜀_𝐷(𝑢,𝑤,𝑠)→𝛽^∗𝐷(𝑢,𝑤+𝑠). From (4.27), it can be readily seen that 𝜉̂(𝑢,𝑣,0+,𝑠)→𝛽̃(𝑢,𝑣,𝑠){𝐼−𝛽̃(𝑢,𝑣,𝑠)}⁻¹, proving (4.19). One also sees from (4.28) that
𝜑𝑢,𝑣,𝑤,𝑠⟶=𝐼𝛽^∗𝐷𝑢,𝑤+𝑠+𝜉̂𝑢,𝑣,0+,𝑠𝛽^∗𝐷𝑢,𝑤+𝑠+𝛽̃𝑢,𝑣,𝑠𝐼−𝛽̃(𝑢,𝑣,𝑠)⁻¹𝛽^∗𝐷=𝐼𝑢,𝑤+𝑠=𝐼−𝛽̃𝑢,𝑣,𝑠+𝛽̃𝑢,𝑣,𝑠𝐼−𝛽̃(𝑢,𝑣,𝑠)⁻¹𝛽^∗𝐷𝑢,𝑤+𝑠−𝛽̃(𝑢,𝑣,𝑠)⁻¹𝛽^∗𝐷𝑢,𝑤+𝑠,(4.33) 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 Section 3 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

Let 𝑁(𝑡) be a Poisson process with intensity 𝜆 as discussed in Section 2.1. From (2.5), one sees that 𝜋(𝑣,𝑠)=∫^∞0𝑒^−𝑠𝑡E[𝑣^𝑁(𝑡)]𝑑𝑡=∫^∞0𝑒^−𝑠𝑡𝜋(𝑣,𝑡)𝑑𝑡 is given by 𝜋(𝑣,𝑠)=1𝑠+𝜆(1−𝑣).(5.1)

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}, 𝑁(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 𝑏_𝑘∶00(𝑡)=𝛿_{𝑘=0}𝑎₀₀(𝑡) from (4.6). This then implies 𝛽₀₀(1,𝑠)=𝜆/(𝑠+𝜆). Similarly, since 𝐴₀(𝑥)=𝑒^−𝜆𝑥, one has 𝛽^∗0(1,𝑠)=1/(𝑠+𝜆). It then follows from Theorem 4.1 that 𝜑₀(1,𝑣,0+,𝑠)=11−𝑣𝛽₀₀(1,𝑠)𝛽^∗