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STATIONARY MARKOVIAN ARRIVAL PROCESSES: RESULTS AND OPEN PROBLEMS

Published online by Cambridge University Press:  22 February 2022

AZAM ASANJARANI*
Affiliation:
Department of Mathematics, The University of Auckland, Auckland1142, New Zealand
YONI NAZARATHY
Affiliation:
School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, Australia; e-mail: y.nazarathy@uq.edu.au.
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Abstract

We consider two classes of irreducible Markovian arrival processes specified by the matrices C and D: the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) hold. We also carry out some numerical experiments that fail to disprove this conjecture. Importantly, modelling folklore has often treated MMPPs as “bursty”, and implicitly assumed that (III) holds. However, to the best of our knowledge, proving this is still an open problem.

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Research Article
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© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Point processes on the line, generated by transitions of continuous-time Markov chains (CTMCs), have been studied intensely by the applied probability community over the past few decades under the umbrella of matrix analytic methods (MAM) [Reference Latouche and Ramaswami19]. These have been applied to teletraffic [Reference Akar, Oguz and Sohraby1], business networks [Reference Herbertsson and Rootzén17], social operations research [Reference Xing, Li, Bi, Wilamowska-Korsak and Zhang29] and biological systems [Reference Olsson and Hössjer25]. The typical model referred to as the Markovian arrival process (MAP) is comprised of a finite-state irreducible CTMC, which generates events at selected instances of state change or according to Poisson processes modulated by the CTMC. MAPs are dense in the class of point processes so that they can essentially approximate any point process [Reference Asmussen and Koole5]. Yet, at the same time, they are analytically tractable and may often be incorporated effectively within more complex stochastic models [Reference Neuts24].

In general, treating point processes as stationary often yields a useful mathematical perspective that matches scenarios when there is no known dependence on time. In describing a point process, we use $N(t)$ to denote the number of events during the interval $[0,t]$ , and further use the sequence $\{T_{n}\}$ to denote the sequence of interarrival times. Two notions of stationarity are useful in this respect. A point process is time-stationary if the distribution of the number of events within a given set is invariant to time translations. That is, for a given subset of the timeline, ${\cal T} \subset {\mathbb R}$ , the distribution of the number of events during ${\cal T}$ is the same as the distribution of the number of events during the shifted set $\{u+t ~|~ u \in {\cal T}\}$ for any time shift t.

A point process is event-stationary if the joint distribution of $T_{k_{1}},\ldots ,T_{k_{n}}$ is the same as that of $T_{k_{1}+\ell },\ldots ,T_{k_{n}+\ell }$ for any integer sequence of indices $k_{1}, \ldots ,k_{n}$ and any integer shift $\ell $ . For a given model of a point process, one may often consider either the event-stationary or the time-stationary case. The probability laws of both cases agree in the case of the Poisson process; however, this is not true in general. For MAPs, time-stationarity and event-stationarity are easily characterized by the initial distribution of the background CTMC. Starting it at its stationary distribution yields time-stationarity, and starting at the stationary distribution of an associated discrete-time Markov chain yields event-stationarity. This associated discrete-time Markov chain, with transition probability matrix P, defined later (see Section 2), records the state at every time point $T_{n}$ and is indexed by the (discrete) time index, n.

A common way to parametrise MAPs is by considering the generator Q of a finite-state irreducible CTMC and setting $Q= C + D$ . The off-diagonal elements of the matrix C determine state transitions without event counts, while the diagonal elements of C are negative. The matrix D determines state transitions associated with event counts. Such parametrisation hints at considering two special cases: Markov-modulated Poisson processes (MMPPs) arising from a diagonal matrix D, and Markov-switched Poisson processes (MSPPs) arising from a diagonal matrix C. We note that a single probability law of a MAP can be represented in multiple ways, and the diagonal D or the diagonal C representations for MMPP or respectively MSPP are only one of many forms.

MMPPs are a widely used class of processes in modelling and are a typical example of a Cox process, also known as a doubly stochastic Poisson process [Reference Grandel, Dold and Eckman11, Reference Tang, Prabhu and Pacheco28]. For a detailed outline of a variety of classic MMPP results, see the work of Fischer and Meier-Hellstern [Reference Fischer and Meier-Hellstern10] and the references therein. MSPPs were introduced by Liu and Neuts [Reference Liu and Neuts20] and to date, have not been very popular for modelling. However, the duality of diagonal D versus diagonal C motivates us to consider and contrast both these processes. We also note that hyper-exponential renewal processes are special cases of MSPPs as well as Markovian transition counting processes (MTCPs). The latter is an alternative model for the MMPPs [Reference Asanjarani, Nazarathy, van Do, Takahashi, Yue and Nguyen3].

There are two key random variables associated with the given MAP, which we denote as $T_{1}^{\boldsymbol {\pi }}$ and $T_{1}^{\boldsymbol {\alpha }}$ . These are distributed as the first interarrival time, $T_{1}$ , in the time-stationary and event-stationary versions, respectively. We note that ${\boldsymbol {\pi }}$ and ${\boldsymbol {\alpha }}$ notionally indicate the nature of the initial distribution of the MAP, and make sense when considering Proposition 2.2 later.

Our focus in this paper is on second-order properties of MMPPs and MSPPs and related traits. Consider the squared coefficient of variation and the limiting index of dispersion of counts given by

(1.1) $$ \begin{align} c^{2} = \frac{\mathrm{Var}(T_{1}^{\boldsymbol{\alpha}})}{\mathbb{E}^{2}\,[ T_{1}^{\boldsymbol{\alpha}}]} \quad \mbox{and} \quad d^{2} = \lim_{t \to \infty} \frac{\mathrm{Var}(N(t))}{\mathbb{E}[N(t)]}, \end{align} $$

respectively.

From a modelling perspective, MMPPs are often used for bursty traffic. In fact, in certain cases, modelling folklore has assumed that for MMPPs, $c^{2} \ge 1$ (see for example, [Reference Heindl16]). The $c^{2} \ge 1$ assumption is perhaps used without direct proof due to the fact that $d^{2} \ge 1$ is straightforward to verify, and there is a similarity between these measures (for example, for a renewal process, $c^{2} = d^{2}$ ). However, as we highlight in this paper, determining such “burstiness” properties is not straightforward.

A related property is that $T_{1}^{\boldsymbol {\alpha }}$ exhibits a decreasing hazard rate (DHR), where for a random variable with probability distribution function (PDF) $f(t)$ and cumulative distribution function (CDF) $F(t)$ , the hazard rate is

$$ \begin{align*} h(t)=\frac{f(t)}{1-F(t)}. \end{align*} $$

A further related property is the stochastic order (see Ch. VII of [Reference Asmussen, Rozovskii and Yor4]) $T_{1}^{\boldsymbol {\pi }} \ge _{\mbox {st}} T_{1}^{\boldsymbol {\alpha }}$ . We denote these properties as follows:

(I) $$ \begin{align} & \kern-22.5pt d^{2} \ge 1, \kern20pt \end{align} $$
(II) $$ \begin{align} & T_{1}^{\boldsymbol{\alpha}} \,\, \text{exhibits DHR}, \end{align} $$
(III) $$ \begin{align} & \kern-22.8pt c^{2} \ge 1, \kern20pt \end{align} $$
(IV) $$ \begin{align} & \kern-30pt T_{1}^{\boldsymbol{\pi}} \ge_{\mbox{st}} T_{1}^{\boldsymbol{\alpha}}. \end{align} $$

All these properties are related and in this paper, we highlight the relationships between (I), (II), (III) and (IV) and establish the following. For MSPPs and MMPPs of order $2$ , properties (I)–(IV) hold. For general MMPPs, it is known that (I) holds; however, a counter-example shows that (II) does not hold and we conjecture that (III) and (IV) hold. We note that (I) differs from (II)–(IV) as $d^{2}$ is not directly related to the random variable $T_{1}$ .

Our interest in this class of problems stemmed from relationships between different types of MAPs; as discussed by Nazarathy and Weiss [Reference Nazarathy and Weiss23], and Asanjarani and Nazarathy [Reference Asanjarani, Nazarathy, van Do, Takahashi, Yue and Nguyen3]. Once it became evident that $c^{2} \ge 1$ for MMPPs is an open problem even though it is acknowledged as a modelling fact in folklore, we searched for alternative proof avenues. This led to the stochastic order in (IV) as well as to considering DHR properties.

The remainder of the paper is structured as follows. In Section 2, we present preliminaries, focusing on the relationships between (I)–(IV) as well as defining MMPPs and MSPPs. In Section 3, we present our main results and some conjectures about MMPPs. We conclude the paper in Section 4.

2 Preliminaries

First, consider (I)–(IV) and their relationships. To establish (III), there are several possible avenues based on (I), (II) and (IV). We now explain these relationships.

  • Using (I): First, from the theory of simple point processes on the line, note the relationship between $d^{2}$ and $c^{2}$ :

    $$ \begin{align*} d^{2} = c^{2}\bigg(1+2\sum_{j=1}^{\infty} \frac{\mathrm{Cov}(T_{0}^{\boldsymbol{\alpha}},T_{j}^{\boldsymbol{\alpha}})}{\mathrm{Var}(T_{0}^{\boldsymbol{\alpha}})}\bigg), \end{align*} $$
    (for more details, see [Reference Gusella12]). However, the autocorrelation structure is typically intractable, and hence does not yield results. If we were focusing on a renewal process where $T_{i}$ and $T_{j}$ are independent for $i \neq j$ , then this immediately shows that $d^{2} = c^{2}$ . However, our focus is broader, and hence (I) (indicating that $d^{2} \ge 1$ ) does not appear to yield a path for (III).
  • Using (II): An alternative way is to consider (II) and use the fact that for any DHR random variable, we have $c^{2} \ge 1$ (see [Reference Stoyan and Daley27]). Hence, if (II) holds, then (III) holds.

  • Using (IV): We have the following lemma, implying that (III) is a consequence of the stochastic order (IV).

Lemma 2.1. Consider a simple nontransient point process on the line, and let $T_{1}^{\boldsymbol {\pi }}$ and $T_{1}^{\boldsymbol {\alpha }}$ represent the first interarrival time in the time-stationary case and the event-stationary case, respectively. Then, $c^{2} \ge 1$ if and only if $\mathbb {E}[T_{1}^{\boldsymbol {\pi }}] \geq \mathbb {E}[T_{1}^{\boldsymbol {\alpha }}]$ .

Proof. From point process theory (see for example, (3.4.17) of [Reference Daley and Vere-Jones9]),

$$ \begin{align*} \mathbb{E}[T_{1}^{\boldsymbol{\pi}}]=\tfrac{1}{2}\lambda^{*} \mathbb{E}[(T_{1}^{\boldsymbol{\alpha}})^{2}], \end{align*} $$

where

$$ \begin{align*} \lambda^{*} = \lim_{t \to \infty} \frac{E[N(t)]}{t} = \frac{1}{\mathbb{E}[T_{1}^{\boldsymbol{\alpha}}]}. \end{align*} $$

Now,

$$ \begin{align*} c^{2} =\frac{\mathbb{E}[(T_{1}^{\boldsymbol{\alpha}})^{2}]-(\mathbb{E}[T_{1}^{\boldsymbol{\alpha}}])^{2}}{(\mathbb{E}[T_{1}^{\boldsymbol{\alpha}}])^{2}} = 2 \frac{\mathbb{E}[T_{1}^{\boldsymbol{\pi}}]}{\mathbb{E}[T_{1}^{\boldsymbol{\alpha}}]} - 1, \end{align*} $$

and we obtain the result.

MAPs: We now describe the MAP. A MAP of order p (MAP $_{p}$ ) is generated by a two-dimensional Markov process $\{(N(t), X(t)); t \geq 0\}$ on state space $\{0,1, 2, \ldots \}\times \{1,2, \ldots , p\}$ . The counting process $N(\cdot )$ counts the number of “events” in $[0,t]$ with $\mathbb P(N(0)=0)=1$ . The phase process $X(\cdot )$ is an irreducible CTMC with state space $\{1, \ldots , p\}$ , initial distribution $\boldsymbol {\eta }$ and generator matrix Q. A MAP is characterized by parameters $(\boldsymbol {\eta }, C,D)$ , where the matrix C has negative diagonal elements and nonnegative off-diagonal elements, and records the rates of phase transitions which are not associated with an event. The matrix D has nonnegative elements and describes the rates of phase transitions which are accompanied with an event (an increase of $N(t)$ by 1). Moreover, we have $Q=C+D$ . More details are in the books by Asmussen [Reference Asmussen, Rozovskii and Yor4, Ch. XI] and He [Reference He13, Ch. 2].

MAPs are attractive due to the tractability of many of their properties, including distribution functions, generating functions, and moments of both counting process $N(t)$ and the sequence of interarrival times $\{T_{n}\}$ .

Since Q is assumed irreducible and finite, it has a unique stationary distribution $\boldsymbol {\pi }$ satisfying $\boldsymbol {\pi } Q = \mathbf {0}^{\prime }$ and $\boldsymbol {\pi } \mathbf {1} = 1$ , where $\mathbf {0}^{\prime }$ is a row vector of 0’s and $\mathbf {1}$ is a column vector of 1’s. Note that from $Q {\mathbf 1} = \mathbf {0}^{\prime }$ , we have $-C {\mathbf 1}=D {\mathbf 1}$ . Of further interest is the discrete-time Markov chain with irreducible stochastic matrix $P = (-C)^{-1}D$ and stationary distribution $\boldsymbol {\alpha }$ , where $\boldsymbol {\alpha } P=\boldsymbol {\alpha }$ and $\boldsymbol {\alpha } \mathbf {1}=1$ .

Observe the relation between the stationary distributions $\boldsymbol {\pi }$ and $\boldsymbol {\alpha }$ :

(2.1) $$ \begin{align} \boldsymbol{\alpha}=\frac{\boldsymbol{\pi} D}{\boldsymbol{\pi} D \mathbf{1}} \quad \text{and} \quad \boldsymbol{\pi}=\frac{\boldsymbol{\alpha} (-C)^{-1}}{\boldsymbol{\alpha} (-C)^{-1}\mathbf{1}}=\lambda^{*}\boldsymbol{\alpha} (-C)^{-1}, \end{align} $$

where $\lambda ^{*} = \boldsymbol {\pi } D {\mathbf 1} = - \boldsymbol {\pi } C {\mathbf 1}$ .

The following known proposition, as distilled from the literature [Reference Asmussen, Rozovskii and Yor4, Ch. XI], provides the key results on MAPs that we use in this paper. It shows that $T_{1}$ is a phase-type (PH) random variable with parameters $(\boldsymbol {\eta }, C)$ , for the initial distribution of the phase and the sub-generator matrix, respectively. It further shows that the initial distribution of the phase process may render the MAP as time-stationary or event-stationary. This proposition also motivates the notation $T_{1}^{\boldsymbol {\pi }}$ and $T_{1}^{\boldsymbol {\alpha }}$ .

Proposition 2.2. Consider a MAP with parameters ( $\boldsymbol {\eta }$ ,C,D), then for $t\ge 0$ ,

$$ \begin{align*} \mathbb P(T_{1}> t) = \boldsymbol{\eta} e^{C t} {\mathbf 1}. \end{align*} $$

Further, if $\boldsymbol {\eta } = \boldsymbol {\pi }$ , then the MAP is time-stationary, and if $\boldsymbol {\eta }=\boldsymbol {\alpha }$ , it is event-stationary, where $\boldsymbol {\pi }$ and $\boldsymbol {\alpha }$ are the stationary distributions of Q and P, respectively.

Note that for such a $PH(\boldsymbol {\eta }, C)$ random variable, the density $f(t)$ and the hazard rate $h(t)$ are respectively

$$ \begin{align*} f(t) = \boldsymbol{\eta} e^{C t} D {\mathbf 1} \quad \text{and} \quad h(t) = \frac{\boldsymbol{\eta}e^{Ct} D \mathbf{1}}{\boldsymbol{\eta}e^{Ct} \mathbf{1}}. \end{align*} $$

Further, to show DHR, the derivative of the hazard rate

$$ \begin{align*} h^{\prime}(t)= \frac{\boldsymbol{\eta} C e^{Ct} (-C) \mathbf{1} \, \boldsymbol{\eta}e^{Ct} \mathbf{1} - \boldsymbol{\eta} C e^{Ct} \mathbf{1}\, \boldsymbol{\eta}e^{Ct} (-C) \mathbf{1} }{(\boldsymbol{\eta}e^{Ct} \mathbf{1})^{2}}. \end{align*} $$

We now describe the second-order properties associated with each case.

Event-stationary case: For an event-stationary MAP (sometimes an event-stationary MAP is referred to as an interval-stationary MAP [Reference Fischer and Meier-Hellstern10]), where $\boldsymbol {\eta }=\boldsymbol {\alpha }$ , the (generic) interarrival time is phase-type distributed, $PH(\boldsymbol {\alpha }, C)$ , and thus the kth moment is

$$ \begin{align*} M_{k}=\mathbb{E}[T_{n}^{k}]=k! \boldsymbol{\alpha} (-C)^{-k}\mathbf{1} =(-1)^{k+1} \, k! \, \frac{1}{\lambda^{*}} \boldsymbol{\pi} (C^{-1})^{k-1} {\mathbf 1}, \end{align*} $$

with the first and second moments (here represented in terms of $\boldsymbol {\pi }$ and C)

$$ \begin{align*} M_{1}=\frac{1}{\lambda^{*}} \boldsymbol{\pi} {\mathbf 1} = \frac{1}{\lambda^{*}} \quad \text{and} \quad M_{2}=2 \frac{1}{\lambda^{*}} \boldsymbol{\pi} (-C)^{-1} {\mathbf 1}. \end{align*} $$

Hence, the squared coefficient of variation (SCV) of events (intervals) has the following simple formula:

(2.2) $$ \begin{align} c^{2}+1 = \frac{M_{2}}{M_{1}^{2}} = \frac{-2 \, (1/\lambda^{*}) \, \boldsymbol{\pi} \, C^{-1} {\mathbf 1}}{(1/\lambda^{*})^{2}} = 2 \boldsymbol{\pi} C {\mathbf 1} \boldsymbol{\pi} C^{-1} {\mathbf 1}. \end{align} $$

Time-stationary case: For the time-stationary MAP with parameters $(\boldsymbol {\eta }, C,D)$ (where $\boldsymbol {\eta }=\boldsymbol {\pi }$ ) (see [Reference Asmussen, Rozovskii and Yor4]),

(2.3) $$ \begin{align} \kern-32pt \mathbb{E}[N(t)]&= {\boldsymbol{\pi}} D \mathbf{1}\,t, \kern50pt \end{align} $$
(2.4) $$ \begin{align} \mathrm{Var}(N(t))&=({\boldsymbol{\pi}}D \mathbf{1}+2\, {\boldsymbol{\pi}}D D_{Q}^{\sharp} D \mathbf{1})\,t+o(t), \end{align} $$

where $o(t)/t$ vanishes as $t \to \infty $ , and $D_{Q}^{\sharp }$ is the deviation matrix associated with Q defined by the formula

$$ \begin{align*} D_{Q}^{\sharp}=\lim_{t\rightarrow\infty} D^{\sharp}_{Q}(t)=\int_{0}^{\infty}(e^{Qu}-\mathbf{1}{\boldsymbol{\pi}})\, du. \end{align*} $$

Note that in some sources [Reference Asmussen, Rozovskii and Yor4, Reference Narayana and Neuts21], the variance formula (2.4) is presented in terms of the matrix $Q^{-}:=(\mathbf {1}{\boldsymbol {\pi }} - Q)^{-1}$ . The relation between these two matrices is $Q^{-}=D_{Q}^{\sharp } +\mathbf {1}{\boldsymbol {\pi }}$ (see [Reference Coolen-Schrijner and Van Doorn8]).

Applying (2.3) and (2.4), we can write $d^{2}$ in terms of MAP parameters as

(2.5) $$ \begin{align} d^{2}= 1+ \frac{2}{\lambda^{*}}\, {\boldsymbol{\pi}}D D_{Q}^{\sharp} D \mathbf{1}. \end{align} $$

MMPP: A MAP with a diagonal matrix D is an MMPP. MMPPs correspond to doubly stochastic Poisson processes (also known as Cox processes), where the modulating process is driven by a CTMC. MMPPs have been used extensively in stochastic modelling and analysis [Reference Fischer and Meier-Hellstern10]. The parameters of an MMPP $_{p}$ are $D= \text { diag}(\lambda _{i})$ , where $\lambda _{i} \geq 0$ for $i=1, \ldots , p$ , and $C=Q-D$ . For MMPPs, (2.3) and (2.4) can be simplified by using the relations:

$$ \begin{align*} \lambda^{*}=\boldsymbol{\pi} D \mathbf{1}=\sum_{i=1}^{p} {\pi}_{i} \lambda_{i}, \quad D \mathbf{1}=(\lambda_{1}, \ldots, \lambda_{p})^{\prime}, \quad \alpha_{i}=\frac{ \lambda_{i}}{\lambda^{*}} \pi_{i}, \end{align*} $$

where the last equation comes from (2.1).

MSPP: A MAP with a diagonal matrix C is an MSPP [Reference Artalejo, Gómez-Corral and He2, Reference He13]. For MSPP $_{p}$ , events switch between p Poisson processes, each with rate $D_{11}, \ldots , D_{pp}$ , where each phase transition also incurs an event. This is in contrast to MMPPs where the transition of the phase process never generates events. We denote the diagonal elements of C with $-\gamma _{i}$ and hence,

$$ \begin{align*} \lambda^{*} =\boldsymbol{\pi}(-C) \mathbf{1}=\sum_{i=1}^{p} \pi_{i} \gamma_{i}, \quad (-C) \mathbf{1}=(\gamma_{1}, \ldots, \gamma_{p})^{\prime} \quad \text{and} \quad \alpha_{i}=\frac{ \gamma_{i}}{\lambda^{*}} \pi_{i}. \end{align*} $$

Note that analysing MSPPs is considerably easier than MMPPs, because a diagonal C is much easier to handle than a nondiagonal C, and in an (irreducible) MMPP, C must be nondiagonal.

Properties (I)–(IV) for MAPs: Using the results above, for any irreducible MAP with matrices C and D, the main properties (I)–(IV) of this paper can be formulated as

(2.6) $$ \begin{align} \kern-118pt\mathrm{(I)}\qquad {\boldsymbol{\pi}}D D_{Q}^{\sharp} D \mathbf{1} \ge 0, \kern30pt\end{align} $$
(2.7) $$ \begin{align} \mathrm{(II)}\qquad \boldsymbol{\alpha} C e^{Ct} (-C) \mathbf{1} \,\, \boldsymbol{\alpha}e^{Ct} \mathbf{1} + (\boldsymbol{\alpha} C e^{Ct} \mathbf{1})^{2} \leq 0 \quad \text{for all} \, t \ge 0, \end{align} $$
(2.8) $$ \begin{align} \kern-119pt\mathrm{(III)}\qquad \boldsymbol{\pi} C {\mathbf 1} \boldsymbol{\pi} C^{-1} {\mathbf 1} \ge 1, \kern30pt\end{align} $$
(2.9) $$ \begin{align} \kern-80pt\mathrm{(IV)}\qquad \boldsymbol{\pi} e^{C t} \mathbf{1} \ge \boldsymbol{\alpha} e^{C t} \mathbf{1} \quad \text{for all} \, t \ge 0, \kern15pt\end{align} $$

respectively, where $D_{Q}^{\sharp }$ is the deviation matrix defined by the formula

$$ \begin{align*} D_{Q}^{\sharp}=\int_{0}^{\infty}(e^{Qu}-\mathbf{1}{\boldsymbol{\pi}})\, du. \end{align*} $$

3 Main results

We now show that an MSPP and an MMPP $_{2}$ satisfy (I)–(IV). Establishing (I), $d^{2} \ge 1$ , is not a difficult task for both MMPPs and MSPPs.

Proposition 3.1. An MMPP and an MSPP exhibit $d^{2} \ge 1$ .

Proof. This is a well-known result that for all doubly stochastic Poisson processes (Cox processes), $d^{2} \ge 1$ [Reference Kingman18, Ch. 6]. So, we have the proof for MMPPs.

The fact that for a given MMPP, we have $d^{2}\geq 1$ and using (2.5) results in

(3.1) $$ \begin{align} {\boldsymbol{\pi}}D D_{Q}^{\sharp} D \mathbf{1} \geq 0 \quad \text{for any diagonal nonnegative matrix }D\text{.} \end{align} $$

However, all MAPs satisfy $ {\boldsymbol {\pi }}D D_{Q}^{\sharp } D \mathbf {1}={\boldsymbol {\pi }}(-C) D_{Q}^{\sharp } (-C) \mathbf {1}$ . Since for an MSPP, $-C$ is a diagonal nonnegative matrix, from (3.1), we have (2.6).

It is not difficult to show that for (II), DHR holds for MSPP.

Proposition 3.2. For an MSPP, the hazard rate of the stationary interarrival time is nonincreasing.

Proof. By denoting $C=\text {diag}(-\gamma _{i})$ and $\mathbf {u}=e^{Ct} \mathbf {1}$ , the left-hand side of (2.7) can be written as

$$ \begin{align*} -\bigg(\sum\limits_{i=1}^{p}\alpha_{i} \gamma_{i}^{2} u_{i}\bigg)\bigg(\sum_{i=1}^{p} \alpha_{i} u_{i}\bigg)+\bigg(\sum_{i=1}^{p} \alpha_{i} \gamma_{i} u_{i}\bigg)^{2}. \end{align*} $$

Denoting $v_{i}=\alpha _{i} u_{i}$ , setting $q_{i}=v_{i}/\ (\sum _{i=1}^{p} v_{i})$ and dividing by $(\sum _{i=1}^{p} v_{i})^{2}$ results in

$$ \begin{align*} -\sum\limits_{i=1}^{p} \gamma_{i}^{2} q_{i}+\bigg(\sum\limits_{i=1}^{p} \gamma_{i} q_{i}\bigg)^{2}. \end{align*} $$

The above expression can be viewed as the negative of the variance of a random variable that takes values $\gamma _{i}$ with probability $q_{i}$ . Therefore, we have (2.7).

However, somewhat surprisingly, MMPPs do not necessarily possess DHR. An exception is MMPP $_{2}$ , as shown in Proposition 3.7; however, DHR for higher-order MMPPs does not always hold. The gist of the following example was communicated to us by Milkós Telek and Illés Horváth.

Example 3.3. Set

(3.2) $$ \begin{align} Q =\left( \begin{array}{cccc} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0\\ 0 & 0 & -1 & 1\\ 1& 0&0 & -1 \end{array} \right) \quad \text{and} \quad D =\left( \begin{array}{cccc}\displaystyle 0.01 & 0 &\, \,0 & \,\,\,\,0 \\ 0 & 0.01 &\, \,0 &\, \,\,\,0\\ 0 & 0 & \,\,1 & \,\,\,\,0\\ 0& 0&\,\,0 & \,\,\,\,1 \end{array} \right). \end{align} $$

As shown in Figure 1, the hazard rate function for an MMPP with the above matrices is not monotone. Hence, at least for general MMPPs, trying to show (III), $c^{2}\ge 1$ , via hazard rates is not a viable avenue.

Figure 1 Nonmonotone hazard rate for an MMPP ${}_{4}$ .

Since hazard rates do not appear to be a viable path for establishing (III) for MMPPs, an alternative may be to consider the stochastic order (IV). Starting with MSPPs, we see that this property holds.

Proposition 3.4. For an MSPP, $T_{1}^{\boldsymbol {\pi }} \ge _{\mbox {st}} T_{1}^{\boldsymbol {\alpha }}$ .

Proof. Using inequality (2.9), we claim

$$ \begin{align*} (\boldsymbol{\pi} - \boldsymbol{\alpha})\,e^{C t} \mathbf{1} \ge 0 \quad \text{for all}\, t \ge 0. \end{align*} $$

Without loss of generality, we assume that there is an order $0<\gamma _{1}\leq \gamma _{2} \leq \cdots \leq \gamma _{p}$ (with $\gamma _{i} \neq \gamma _{j}$ for some $i,j$ ) for diagonal elements of the matrix $(-C)$ . There is a possibility that for $1 < p^{\prime } < p$ , $0=\gamma _{1}=\gamma _{2} = \cdots =\gamma _{p^{\prime }-1}$ and $0<\gamma _{p^{\prime }}$ ; however, in the rest of the proof, we assume that $p^{\prime }=1$ , meaning that all $\gamma _{i}$ are strictly positive. Adapting to the case of $p^{\prime }>1$ is straightforward.

Now, $\{\lambda ^{*}-\gamma _{i}\}_{i=1, \ldots , p}$ is a nonincreasing sequence and therefore in the sequence $\{\pi _{i}-\alpha _{i}\}=\{\pi _{i}(\lambda ^{*}-\gamma _{i})/\lambda ^{*}\}$ , when an element $\pi _{k}-\alpha _{k}$ is negative, all the elements $\pi _{i}-\alpha _{i}$ for $i\geq k$ are also negative. Moreover, both $\boldsymbol {\pi }$ and $\boldsymbol {\alpha }$ are probability vectors, so $(\boldsymbol {\pi }-\boldsymbol {\alpha })\mathbf {1}=\sum _{i}(\pi _{i}-\alpha _{i})=0$ . Therefore, at least the first element in the sequence $\{\pi _{i}-\alpha _{i}\}=\{\pi _{i}(\lambda ^{*}-\gamma _{i})/\lambda ^{*}\}$ is positive. Hence, there exists an index $1 < k \leq p$ such that $\pi _{i}-\alpha _{i}$ for $i=1, \ldots , k-1$ is nonnegative and for $i=k,\ldots , p$ is negative. Therefore,

$$ \begin{align*} ({\boldsymbol \pi} - {\boldsymbol\alpha}) \,e^{Ct} \mathbf{1} &=\underbrace{\sum_{i=1}^{k-1}({\pi_{i}}-{ \alpha_{i}})\,e^{-\gamma_{i} t}}_{\text{nonnegative}}+\underbrace{\sum_{i=k}^{p}({\pi_{i}}-{ \alpha_{i}})e^{-\gamma_{i} t}}_{\text{negative}}\\ & \geq e^{-\gamma_{k-1} t}\sum_{i=1}^{k-1}({\pi_{i}}-{ \alpha_{i}}) +e^{-\gamma_{k} t} \sum_{i=k}^{p}({\pi_{i}}-{ \alpha_{i}})\\ & = L (e^{-\gamma_{k-1} t}- e^{-\gamma_{k} t}) \geq 0\,, \end{align*} $$

where $L=\sum _{i=1}^{k-1}({\pi _{i}}-{ \alpha _{i}})=\sum _{i=k}^{p}({ \alpha _{i}}-\pi _{i})$ . Here the inequalities are a consequence of the ordering of $\gamma _{i}$ .

Hence, via Lemma 2.1 or alternatively via the DHR property in Proposition 3.2, we have the following result.

Corollary 3.5. For an MSPP, $c^{2} \ge 1$ .

In fact, for MSPPs, this is an easy result and it can also be proved independently by using the Cauchy–Schwarz inequality. Further, we can find an upper bound for $c^{2}$ as follows.

Proposition 3.6. An MSPP with diagonal matrix $C=-\text { diag} (\gamma _{i})$ satisfies

$$ \begin{align*} 1 \leq c^{2} \leq 2\,\frac{\kappa^{2}}{\gamma^{2}}-1, \end{align*} $$

where $\kappa =(\min {\gamma _{i}}+\max {\gamma _{i}})/2$ and $\gamma =\sqrt {(\min {\gamma _{i}})(\max {\gamma _{i}})}$ .

Proof. From (2.2), we have $c^{2} + 1= 2\boldsymbol {\pi } C {\mathbf 1} \boldsymbol {\pi } C^{-1} {\mathbf 1}$ . For $C=-\text { diag}(\gamma _{i})$ , we have $C^{-1}=-\text {diag}(1/\gamma _{i})$ and so,

(3.3) $$ \begin{align} \frac{c^{2} +1}{2}=\boldsymbol{\pi} C {\mathbf 1} \boldsymbol{\pi} C^{-1} {\mathbf 1} =\bigg(\sum_{i=1}^{p}\pi_{i} \gamma_{i}\bigg)\bigg(\sum_{i=1}^{p}\pi_{i} \frac{1}{\gamma_{i}}\bigg). \end{align} $$

From the Cauchy–Schwarz inequality and the Kantorovich inequality [Reference Steele26],

(3.4) $$ \begin{align} 1=\bigg(\sum_{i=1}^{p}\pi_{i} \,\gamma_{i}^{1/2}\, \frac{1}{\gamma_{i}^{1/2}}\bigg)^{2} \leq \bigg(\sum_{i=1}^{p}\pi_{i} \gamma_{i}\bigg)\bigg(\sum_{i=1}^{p}\pi_{i} \frac{1}{\gamma_{i}}\bigg) \le \frac{\kappa^{2}}{\gamma^{2}}. \end{align} $$

Combination of (3.3) and (3.4) results in

$$ \begin{align*} 1 \leq \frac{c^{2} +1}{2} \leq \frac{\kappa^{2}}{\gamma^{2}}, \end{align*} $$

which completes the proof.

For MMPPs, we do not have general proofs for properties (III) and (IV). Still, for MMPPs of order two (MMPP $_{2}$ ), things are easier and we are able to show that all properties (I)–(IV) hold.

Proposition 3.7. For an MMPP $_{2}$ , $c^{2}\geq 1$ and $d^{2}\geq 1$ , $h(t)$ is DHR and the stochastic order $T_{1}^{\boldsymbol {\pi }} \ge _{\mbox {st}} T_{1}^{\boldsymbol {\alpha }}$ holds. Further, $c^{2} = 1$ and $d^{2}=1$ if and only if $\lambda _{1} = \lambda _{2}$ .

Proof. Consider an MMPP $_{2}$ with parameters

(3.5) $$ \begin{align} C=\left(\begin{array}{@{}cc@{}} -\sigma_{1}-\lambda_{1} & \sigma_{1}\\ \sigma_{2} & -\sigma_{2}-\lambda_{2} \end{array}\right) \quad \text{and} \quad D=\left(\begin{array}{@{}cc@{}} \lambda_{1}&0\\ 0& \lambda_{2} \end{array}\right). \end{align} $$

Then, $\boldsymbol {\pi }=(\sigma _{2},\, \sigma _{1})/(\sigma _{1}+\sigma _{2})$ . As in [Reference Heffes and Lucantoni15], evaluation of the transient deviation matrix through (for example) Laplace transform inversion yields

$$ \begin{align*} \frac{\mathrm{Var}(N(t))}{\mathbb{E}[N(t)]}&=1+\frac{2\sigma_{1}\sigma_{2}(\lambda_{1}-\lambda_{2})^{2}}{(\sigma_{1}+\sigma_{2})^{2}(\lambda_{1}\sigma_{2}+\lambda_{2}\sigma_{1})}\\ &\quad-\frac{2\sigma_{1}\sigma_{2}(\lambda_{1}-\lambda_{2})^{2}}{(\sigma_{1}+\sigma_{2})^{3}(\lambda_{1}\sigma_{2}+\lambda_{2}\sigma_{1})t}(1-e^{-(\sigma_{1}+\sigma_{2})t}). \end{align*} $$

Therefore, from (1.1),

$$ \begin{align*} d^{2}=1+ \frac{2\sigma_{1}\sigma_{2}(\lambda_{1}-\lambda_{2})^{2}}{(\sigma_{1}+\sigma_{2})^{2}(\lambda_{1}\sigma_{2}+\lambda_{2}\sigma_{1})}\geq 1. \end{align*} $$

Further, explicit computation yields

$$ \begin{align*} c^{2} = 1 + \frac{2\sigma_{1}\sigma_{2}(\lambda_{1}-\lambda_{2})^{2}}{(\sigma_{1}+\sigma_{2})^{2}(\lambda_{2}\sigma_{1}+\lambda_{1}(\lambda_{2}+\sigma_{2}))} \geq 1. \end{align*} $$

Thus, it is evident that the MMPP $_{2}$ has $d^{2}>1$ and $c^{2}> 1$ as long as $\lambda _{1} \neq \lambda _{2}$ , and $d^{2}=c^{2}=1$ when $\lambda _{1} = \lambda _{2}$ .

For DHR and the stochastic order, first we note that for an MMPP $_{2}$ with the above parameters, $\boldsymbol {\alpha }=(\sigma _{2} \lambda _{1}, \sigma _{1} \lambda _{2})/(\sigma _{1} \lambda _{2}+\sigma _{2} \lambda _{1})$ . By setting $a=\sigma _{2} \lambda _{1} +\lambda _{2}(\sigma _{1}+\lambda _{1})>0$ and $b= \sigma _{1} +\sigma _{2} + \lambda _{1}+\lambda _{2}>0$ , after some simplification, the left-hand side of (2.7) is given by

$$ \begin{align*} - \frac{a e^{-bt}\sigma_{1} \sigma_{2}(\lambda_{1}-\lambda_{2})^{2}}{(\sigma_{2} \lambda_{1}+\sigma_{1}\lambda_{2})^{2}}, \end{align*} $$

which is strictly negative for $\lambda _{1} \neq \lambda _{2}$ and is zero for $\lambda _{1} = \lambda _{2}$ . For the stochastic order, by evaluating the matrix exponential and simplifying,

$$ \begin{align*} (\boldsymbol{\pi} - \boldsymbol{\alpha})e^{C t} \mathbf{1}= \frac{e^{-(b+\sqrt{b^{2}-4a} )/2 }(e^{\,t\sqrt{b^{2}-4a}}-1)\sigma_{1} \sigma_{2} (\lambda_{1}-\lambda_{2})^{2} }{(\sigma_{1}+\sigma_{2}) (\sigma_{1} \lambda_{2}+\sigma_{2} \lambda_{1}) \sqrt{b^{2}-4a}}, \end{align*} $$

which is strictly positive for $\lambda _{1} \neq \lambda _{2}$ and is zero for $\lambda _{1} = \lambda _{2}$ .

We note that $c^{2}>1$ in Proposition 3.7 may also be derived by considering a canonical form representation of second-order MAPs as presented by Bodrog et al. [Reference Bodrog, Heindl, Horváth and Telek7]. In that paper, they suggested two canonical form representations for MAPs of order 2. MMPPs fit the first canonical form which is represented in Definition 1 of [Reference Bodrog, Heindl, Horváth and Telek7] via,

(3.6) $$ \begin{align} \tilde{C}=\left[\begin{array}{@{}cc@{}}-\tilde{\lambda}_{1} & (1-\tilde{a}) \tilde{\lambda}_{1} \\ 0 & -\tilde{\lambda}_{2}\end{array}\right], \quad \tilde{D}=\left[\begin{array}{@{}cc@{}}\tilde{a} \tilde{\lambda}_{1} & 0 \\ (1-\tilde{b}) \tilde{\lambda}_{2} & \tilde{b} \tilde{\lambda}_{2}\end{array}\right]. \end{align} $$

We use tildes so as not to confuse the notation of Bodrog et al. [Reference Bodrog, Heindl, Horváth and Telek7] with the notation of our current paper. Given a representation of an MMPP $_{2}$ as in (3.5), one may use a similarity transform jointly for C and D to find an equivalent canonical form representation. In this case, a similarity transformation matrix which satisfies $B C = \tilde {C} B$ and $B D = \tilde {D} B$ can be represented as

$$ \begin{align*} B = \left[\begin{matrix} b_{1} & 1-b_{1} \\ 1-b_{2} & b_{2} \end{matrix}\right]. \end{align*} $$

Solving a system of equations for $b_{1}, b_{2}$ , and the parameters of the canonical form, we find four different equivalent solutions in the sense that the resulting canonical form agrees with the original MMPP $_{2}$ . One possible solution is

$$ \begin{align*}b_{1}=0 \quad \text{ and} \quad b_{2}= -\frac{\mathrm{r}-\lambda_{1}+\lambda_{2}+\sigma_{1}+\sigma_{2}}{2 \lambda_{1}-2 \lambda_{2}}. \end{align*} $$

Here, $r= \sqrt {b^{2}-4a}$ , where a and b are as in the proof of Proposition 3.7. With this similarity transform, the parameters of the canonical form (3.6) yielding MMPP $_{2}$ are

$$ \begin{align*} \tilde{ \lambda}_{1} = \frac{b-r}{2}, \quad \tilde{\lambda}_{2} = \frac{b+r}{2}, \quad \tilde{a} = \frac{\lambda_{2}(b+r)}{2a}, \quad \tilde{b} = \frac{\lambda_{1}(b-r)}{2a}. \end{align*} $$

Having this representation in hand, from the work of Bodrog et al. [Reference Bodrog, Heindl, Horváth and Telek7, Lemma 1], we observe that $c^{2}> 1$ .

Note that one can also represent MSPPs of order 2 in terms of their canonical representations [Reference Bodrog, Heindl, Horváth and Telek7]. However, for MSPP, the correlation coefficient can be both positive and negative, and thus one needs to choose between two alternative canonical representations.

3.1 Conjectures for MMPP

We embarked on this research due to the folklore assumption that for MMPP, $c^{2} \ge 1$ (III) (see for example, [Reference Heindl16]). To date, there is no known proof for an arbitrary irreducible MMPP. Still, we conjecture that both (III) and (IV) hold for MMPPs.

Conjecture 3.8. For an irreducible MMPP, $c^{2}\ge 1$ .

Conjecture 3.9. For an irreducible MMPP, $T_{1}^{\boldsymbol {\pi }} \ge _{\mbox {st}} T_{1}^{\boldsymbol {\alpha }}$ .

In an attempt to disprove these conjectures, we carried out a numerical experiment by generating random instances of MMPPs. Each instance is created by first generating a matrix Q where off-diagonal entries are generated from a uniform $(0,1)$ distribution and diagonal entries that ensure row sums are $0$ . We then generate a matrix D with diagonal elements that are exponentially distributed with rate $1$ . Such a $(Q,D)$ pair then implies ${\boldsymbol {\pi }}$ and ${\boldsymbol {\alpha }}$ using their definitions in Section 2. For each such MMPP, we calculate $\boldsymbol {\pi } C {\mathbf 1} \boldsymbol {\pi } C^{-1} {\mathbf 1} -1 $ as in (2.8) (for Conjecture 3.8) and $(\boldsymbol {\pi } - \boldsymbol {\alpha })e^{C t} \mathbf {1}$ as in (2.9) (for Conjecture 3.9), where we take $t \in \{0,0.2,0.4,\ldots ,9.8,10.0\}$ . Getting a strictly negative value on any of these would imply a counterexample for the associated conjecture.

We repeated this experiment for random MMPP instances of orders $3,4,5$ and $6$ with $10^{6}$ instances for each order. In all the cases, the calculated quantities were greater than $-10^{-15}$ . Note that in certain cases, the quantity associated with (IV) was negative and lying in the range $(-10^{-15},-10^{-16}]$ . We attribute these negative values to numerical error stemming from the calculation of the matrix exponential $e^{Ct}$ . We ran our experiments with the Julia programming language, V1.0 [Reference Nazarathy and Klok22]. The calculation time was approximately 1.5 hours on a desktop computer.

The purpose of carrying out such experiments is to try to disprove the conjectures; however, we failed to do so. Also note that when considering uniformly generated instances, it is possible that we miss corner cases that may disprove the conjectures. In trying to find some classes of such corner cases, we considered random cyclic Q matrices with nonzero entries similar to (3.2). We generated $10^{6}$ such (order $4$ ) examples, and all agreed with (III) and (IV) as well.

4 Conclusion

We have highlighted various related properties for point processes on the line and MAPs exhibiting diagonal matrices (C or D), in particular. Bodrog et al. [Reference Bodrog, Heindl, Horváth and Telek7] showed $c^{2} \ge 1$ for MMPPs of order $2$ , and here we have derived it alongside $T_{1}^{\boldsymbol {\pi }} \ge _{\mbox {st}} T_{1}^{\boldsymbol {\alpha }}$ using alternative means. Further, using a similar technique to our MSPP proof (proof of Proposition 3.2), we can also show properties for MMPPs with symmetric C matrices. However, for general MMPPs of order greater than $2$ , proving these properties remains an open problem.

We note that stepping outside of the matrix analytic paradigm and considering general Cox processes is also an option. In fact, since any Cox process can be approximated by an MMPP, one may formulate versions of conjectures $1$ and $2$ for Cox processes under suitable regularity conditions.

There is also a related branch of questions dealing with characterizing the Poisson process via $c^{2} = 1$ and considering when an MMPP is Poisson. For example, for the general class of MAPs, Bean and Green [Reference Bean and Green6] provide a condition for determining if a given MAP is Poisson. It is not hard to construct a MAP with $c^{2} = 1$ that is not Poisson. However, we conjecture that all MMPPs with $c^{2} = 1$ are Poisson. Yet, we do not have a proof. Further, we conjecture that for an MMPP, if $c^{2} = 1$ , then all $\lambda _{i}$ are equal (the converse is trivially true). We do not have a proof for orders beyond two. Related questions also hold for the more general Cox processes.

We also note that the MSPP class of processes generalizes hyper-exponential renewal processes, as well as a class of processes called a Markovian transition counting process (MTCP), as in the authors’ previous work [Reference Asanjarani, Nazarathy, van Do, Takahashi, Yue and Nguyen3]. Further, since the stationary interval interarrival times of MMPP processes are PH distributed, it is also of interest to consider related problems dealing with moments of phase-type distributions (see for example, [Reference He, Horváth, Horváth and Telek14] and the references therein).

Acknowledgements

Azam Asanjarani’s research is supported by the Faculty Research & Development Fund (FRDF), Faculty of Science, University of Auckland.Yoni Nazarathy is supported by DP180101602. We thank Peter Taylor, Qi-Ming He, Søren Asmussen, Milkós Telek, and Illés Horváth for useful discussions and insights.

References

Akar, N., Oguz, N. C. and Sohraby, K., “Matrix-geometric solutions of M/G/1-type Markov chains: a unifying generalized state space approach”, IEEE J. Sel. Areas Commun. 16 (1998) 626639; doi:10.1109/49.700901.CrossRefGoogle Scholar
Artalejo, J. R., Gómez-Corral, A. and He, Q. M., “Markovian arrivals in stochastic modelling: a survey and some new results”, SORT 34 (2010) 101144. https://ddd.uab.cat/pub/sort/sort_a2010m7-12v34n2/sort_a2010m7-12v34n2p101.pdf.Google Scholar
Asanjarani, A. and Nazarathy, Y., “A queueing approximation of MMPP/PH/1”, in: Queueing Theory and Network Applications (QTNA 2015), Volume 383 of Adv. Intell. Syst. Comput. (eds. van Do, T., Takahashi, Y., Yue, W. and Nguyen, V. H.), (Springer, Cham, 2016); doi:10.1007/978-3-319-22267-7_4.Google Scholar
Asmussen, S., “Applied probability and queues”, in: Stochastic Modelling and Applied Probability, Volume 51 of Appl. Math. (eds. Rozovskii, B. and Yor, M.), (Springer-Verlag, New York, 2003).Google Scholar
Asmussen, S. and Koole, G., “Marked point processes as limits of Markovian arrival streams”, J. Appl. Probab. 30 (1993) 365372; doi:10.2307/3214845.CrossRefGoogle Scholar
Bean, N. G. and Green, D. A., “When is a MAP Poisson?”, Math. Comput. Model. 31 (2000) 3146; doi:10.1016/S0895-7177(00)00070-4.CrossRefGoogle Scholar
Bodrog, L., Heindl, A., Horváth, G. and Telek, M., “A Markovian canonical form of second-order matrix-exponential processes”, European J. Oper. Res. 190 (2008) 459477; doi:10.1016/j.ejor.2007.06.020.CrossRefGoogle Scholar
Coolen-Schrijner, P. and Van Doorn, E. A., “The deviation matrix of a continuous-time Markov chain”, Probab. Engrg. Inform. Sci. 16 (2002) 351366; doi:10.1017/S0269964802163066.CrossRefGoogle Scholar
Daley, D. and Vere-Jones, D., An Introduction to the Theory of Point Processes, in: General Theory and Structure, Volume II of Probability and Its Applications Series (eds J. Gani, C. C. Heyde, P. Jagers, T. G. Kurtz), (Springer Science & Business Media, New York, 2007) ISBN:10.0387213376.Google Scholar
Fischer, W. and Meier-Hellstern, K., “The Markov-modulated Poisson process (MMPP) cookbook”, Perform. Eval. 18 (1993) 149171; doi:10.1016/0166-5316(93)90035-S.CrossRefGoogle Scholar
Grandel, J., Doubly Stochastic Poisson Processes, Volume 529 of Lecture Notes in Math. (eds. Dold, A. and Eckman, B.), (Springer-Verlag, Berlin–Heidelberg, 1976); doi:10.1007/BFb0077758.CrossRefGoogle Scholar
Gusella, R., “Characterizing the variability of arrival processes with indexes of dispersion”, IEEE J. Sel. Areas Commun. 9 (1991) 203211; doi:10.1109/49.68448.CrossRefGoogle Scholar
He, Q. M., Fundamentals of Matrix-Analytic Methods, (Springer, New York, 2014); doi:10.1007/978-1-4614-7330-5.CrossRefGoogle Scholar
He, Q. M., Horváth, G., Horváth, I. and Telek, M., “Moment bounds of PH distributions with infinite or finite support based on the steepest increase property”, Adv. in Appl. Probab. 51 (2019) 168183; doi:10.1017/apr.2019.7.CrossRefGoogle Scholar
Heffes, H. and Lucantoni, D., “A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance”, IEEE J. Sel. Areas Commun. 4 (1986) 856868; doi:10.1109/JSAC.1986.1146393.CrossRefGoogle Scholar
Heindl, A., “Decomposition of general tandem queueing networks with MMPP input”, Perform. Eval. 44 (2001) 523; available at https://acnpsearch.unibo.it/singlejournalindex/8052382.CrossRefGoogle Scholar
Herbertsson, A. and Rootzén, H., “Pricing kth-to-default swaps under default contagion: the matrix-analytic approach”, 2007, available at https://ssrn.com/abstract=962381; doi:10.2139/ssrn.962381.CrossRefGoogle Scholar
Kingman, J. F. C., Poisson Processes, Volume 3 of Oxford Studies in Probability (Oxford University Press, Oxford, 1993) ISBN:10.0198536933.Google Scholar
Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Mathematics, (SIAM, Philadelphia, PA, 1999); doi:10.1137/1.9780898719734.CrossRefGoogle Scholar
Liu, D. and Neuts, M. F., “Counter-examples involving Markovian arrival processes”, Stoch. Models 7 (1991) 499509; doi:10.1080/15326349108807202.Google Scholar
Narayana, S. and Neuts, M. F., “The first two moment matrices of the counts for the Markovian arrival process”, Comm. Statist. Stoch. Models 8 (1992) 459477; doi:10.1080/15326349208807234.CrossRefGoogle Scholar
Nazarathy, Y. and Klok, H., Statistics with Julia: Fundamentals for Data Science, Machine Learning and Artificial Intelligence, (Springer, Switzerland, 2021); doi:10.1007/978-3-030-70901-3.CrossRefGoogle Scholar
Nazarathy, Y. and Weiss, G., “The asymptotic variance rate of the output process of finite capacity birth-death queues”, Queueing Syst. 59 (2008) 135156; doi:10.1007/s11134-008-9079-4.CrossRefGoogle Scholar
Neuts, M. F., “A versatile Markovian point process”, J. Appl. Probab. 16 (1979) 764779; doi:10.2307/3213143.CrossRefGoogle Scholar
Olsson, F. and Hössjer, O., “Equilibrium distributions and simulation methods for age structured populations”, Math. Biosci. 268 (2015) 4551; doi:10.1016/j.mbs.2015.08.003.CrossRefGoogle ScholarPubMed
Steele, J. M., The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, (Cambridge University Press, Cambridge, 2004); doi:10.1017/CBO9780511817106.CrossRefGoogle Scholar
Stoyan, D. and Daley, D.J., Comparison Methods for Queues and Other Stochastic Models, (John Wiley & Sons Ltd, New York, 1983).Google Scholar
Tang, L. C., Prabhu, N. U. and Pacheco, A., Markov-Modulated Processes and Semiregenerative Phenomena, (World Scientific, Singapore, 2008). https://www.worldscientific.com/doi/abs/10.1142/6757.Google Scholar
Xing, Y., Li, L., Bi, Z., Wilamowska-Korsak, M. and Zhang, L., “Operations research (OR) in service industries: a comprehensive review”, Syst. Res. Behav. Sci. 30 (2013) 300353; doi:10.1002/sres.2185.CrossRefGoogle Scholar
Figure 0

Figure 1 Nonmonotone hazard rate for an MMPP${}_{4}$.