International Journal of Mathematics and Mathematical Sciences

1. Introduction

A stochastic process is called a counting process when is nonnegative, right continuous and monotone nondecreasing with . 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 whenever the underlying Markov-chain is in state .

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 with , and is replaced to a state as soon as it enters a state with probability and the counting process describes the number of such replacements occurred in . If we set and , and an absorbing state can be reached only from its counter part with instantaneous replacement back to 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 on . Whenever a state transition of the semi-Markov process occurs from to , the intensity function of the NHPP is switched from to with an initial value reset to . In other words, the arrivals of items are generated by the NHPP with when the semi-Markov process is in state with denoting the time since the last entry into state . Of particular interest in analysis of such systems are the multivariate counting processes and where counts the cumulative number of items that have arrived in while the semi-Markov process is in state and represents the cumulative number of the state transitions of the semi-Markov process from to in . The joint multivariate counting process enables one to unify many existing counting processes in that they can be derived in terms of the components of . Because of this reason, hereafter, we call the unified multivariate counting process. The dynamic behavior of 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 .

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 representing the number of packet arrivals at the origination site and describing the number of the encoder changes in . The state of the underlying semi-Markov process at time 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 . 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 (, etc.) and underlines (, etc.) respectively. The vector with all components equal to is denoted by and the identity matrix is written as . A submatrix of is defined by .

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 is characterized by a sequence of independently and identically distributed (i.i.d.) exponential random variables with common probability density function (p.d.f.) . Let . Then, the associated Poisson process is defined as a counting process satisfying If a system has an exponential lifetime of mean and is renewed instantaneously upon failure, represents the lifetime of the th renewal cycle. The Poisson process then counts the number of failures that have occurred by time .

Let and define the probability generating function (p.g.f.) by It can be seen, see, for example, Karlin and Taylor [14], that where for . Multiplying on both sides of (2.3) and summing from to , one then finds that Since where if statement is true and otherwise, one has . Equation (2.4) can then be solved as

2.2. Nonhomogeneous Poisson Process (NHPP)

An NHPP differs from a Poisson process in that the failure intensity of the system is given as a function of time . Accordingly, (2.3) should be rewritten as By taking the generating function of (2.6), one finds that With , this equation can be solved as The reader is referred to Ross [15] for further discussions of NHPPs.

2.3. Markov-Modulated Poisson Process (MMPP)

Let be a Markov-chain in continuous time on governed by a transition rate matrix . Let and define the associated diagonal matrix . An MMPP characterized by is a pure jump process where jumps of occur according to a Poisson process with intensity whenever the Markov-chain is in state .

Let and define . The infinitesimal generator associated with the Markov-chain is then given by For , let and define the associated matrix generating function by It can be seen that In matrix notation, this can be rewritten as By taking the generating function of (2.13) together with (2.9), one sees that Since , one has , where is the identity matrix, so that the above differential equation can be solved as where for any square matrix . It should be noted that , which is the transition probability matrix of as it should be. By taking the Laplace transform of both sides of (2.15), is given by

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 . 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 . The resulting counting process is still characterized by (2.1). Let as before. One then sees that where denotes the -fold convolution of with itself, that is, and which is the step function defined as for and else.

Let and . By taking the Laplace transform of both sides of (2.17), it follows that By taking the generating function of the above equation with , one has 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 , 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 be a semi-Markov process on governed by a matrix p.d.f. where and is a stochastic matrix which is assumed to be ergodic. Let be a recurrent class consisting of the entries of the semi-Markov process to state for , and define to be a counting process describing the number of recurrences for given that there was an epoch of at time . Then 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 with , the associated matrix renewal density, and the limit theorems. For example, one has the following result concerning the Laplace transform of by Keilson [23]: where is the Laplace transform of . 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 governed by a matrix p.d.f. is studied in Masuda and Sumita [3], where the state space is decomposed into a set of good states and a set of bad states satisfying and . The counting process is then defined to describe the number of entries of into by time .

While is a special case of MRPs, the detailed analysis is provided in [3], yielding much more information. More specifically, let be the age process associated with , that is, where , and define where One then has The associated matrix Laplace transform generating function can then be defined as It has been shown in Masuda and Sumita [3] that Here, with and for , the following notation is employed:

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 be an absorbing Markov-chain on with and , where all states in are absorbing. Without loss of generality, we assume that , and . For notational convenience, the following transition rate matrices are introduced.

The entire transition rate matrix governing is then given by

A replacement Markov-chain on is now generated from . Starting from a state in , the process coincides with within . As soon as reaches state , it is instantaneously replaced at state with probability and the process continues. Let where . Then the transition rate matrix and the infinitesimal generator of are given as where with Let be the transition probability matrix of with its Laplace transform defined by . From the Kolmogorov forward equation with , one has

Let be the counting process keeping the record of the number of replacements in and define By analyzing the probabilistic flow at state at time , it can be seen that It then follows that In matrix notation, the above equation can be rewritten as We now introduce the following matrix Laplace transform generating function: Taking the Laplace transform with respect to and the generating function with respect to of both sides of (2.41), one has which can be solved as We note from (2.37) and (2.44) that 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 where and . Transitions within is governed by . An entry into is possible only from . When this occurs, the Markov process is immediately replaced at the entering state . The counting process for the number of such replacements then has the Laplace transform generating function given in (2.44) where is replaced by , which coincides with 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 be a renewal process associated with a sequence of i.i.d. nonnegative random variables with common p.d.f. . The age process is then defined by In other words, is the elapsed time since the last renewal. We next consider an NHPP governed by an intensity function . If we define one has Of interest, then, is a counting process characterized by Here where is the set of nonnegative integers, and is the set of nonnegative real numbers.

Since the counting process is governed by the intensity function depending on the age process of the renewal process , it is necessary to analyze the trivariate process . Let the multivariate transform of be defined by It has been shown in Sumita and Shanthikumar [12] that where we define, for with ,

The Laplace transform generating function of defined by is then given by , so that one has from (2.50)

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

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 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 with the current dwell time of , items arrive according to a Poisson process with intensity . If the semi-Markov process switches its state from to , the intensity function is interrupted, the intensity function at state is reset to , and the arrival process resumes. Of particular interest would be the multivariate counting processes and with and counting the number of items that have arrived in state by time and the number of transitions of the semi-Markov process from state to state by time respectively. The two counting processes and enable one to introduce a variety of interesting performance indicators as we will see.

Formally, let be a semi-Markov process on governed by a matrix cumulative distribution function (c.d.f.) which is assumed to be absolutely continuous with the matrix probability density function (p.d.f.) It should be noted that, if we define and by then is an ordinary c.d.f. and is the corresponding survival function. The hazard rate functions associated with the semi-Markov process are then defined as

For notational convenience, the transition epochs of the semi-Markov process are denoted by , with . The age process associated with the semi-Markov process is then defined as At time with and , the intensity function of the NHPP is given by . For the cumulative arrival intensity function in state , one has The probability of observing arrivals in state within the current age of can then be obtained as

Of interest are the multivariate counting processes where represents the total number of items that have arrived by time while the semi-Markov process stayed in state , and with denoting the number of transitions of the semi-Markov process from state to state by time . It is obvious that denotes the number of entries into state by time . The initial state is not included in for any . In other words, if , remains until the first return of the semi-Markov process to state . In the next section, we will analyze the dynamic behavior of , 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 and , thereby providing a unified approach for studying various counting processes. The associated asymptotic behavior as would be also discussed.

4. Dynamic Analysis of

The purpose of this section is to examine the dynamic behavior of the multivariate stochastic process 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 .

Since is not Markov, we employ the method of supplementary variables. More specifically, the multivariate stochastic process is considered. This multivariate stochastic process is Markov and has the state space , where and are the set of and dimensional nonnegative integer vectors and matrices respectively, is the set of nonnegative real numbers and . Let be the joint probability distribution of defined by