Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes

1.   Introduction

Reflected diffusion processes are used extensively in various applied fields, such as mathematical biology, queueing theory, finance and neuroscience. In particular, diffusion processes with a reflection condition at the origin arise as diffusion limits of a number of classical birth-death processes in population dynamics and as a heavy-traffic approximation for queueing systems (see, for instance, Giorno et al. ^[1], Ward and Glynn ^[2], Di Crescenzo et al. ^[3,4]). Reflected diffusion processes are also applied in economics and finance for modeling regulated markets, interest rates and stochastic volatility (cf., for instance, Linetsky ^[5], Veestraeten ^[6]). Moreover, in neuronal models, the membrane potential evolution can be described by focusing the attention on the diffusion processes confined by a lower reflecting boundary that can be interpreted as the neuronal reversal hyperpolarization potential (cf. Lánský and Ditlevsen ^[7], Buonocore et al. ^[8,9], D'Onofrio et al. ^[10]). References to other applications of reflected diffusion processes in neuroscience, in population dynamics, in economics, in finance and in queueing systems can be found in Di Crescenzo et al. ^[11], Giorno and Nobile ^[12,13], Mishura and Yurchenko-Tytarenko ^[14]).

In various types of instances, first-passage time (FPT) problems are invoked to describe events such as extinction in population dynamics, busy period in queueing systems and firing times in neuronal modeling (cf., for instance, Ricciardi et a. ^[15], Masoliver and Perelló ^[16], Bo et al. ^[17], Abundo and Pirozzi ^[18], Giorno and Nobile ^[19]).

In several applications, it is useful to consider diffusion processes with linear infinitesimal drift and linear infinitesimal variance, having state-space $[0, +\infty)$ with a zero-flux condition at the zero-state. This class incorporates the Wiener, Ornstein-Uhlenbeck and Feller diffusion processes with reflection in the zero-state. Such processes are widely used in the modeling queueing systems under assumptions of heavy-traffic and in the description of populations growth in a random environment. In these contexts, the number of customers or individuals is bound to take non negative values, so that a reflection condition in the zero-state must thus be imposed. In particular, in queueing systems a reflecting condition in the zero-state is required because new customers can access the system if it is empty; moreover, in the population dynamics the reflection in zero allows to include immigration effects. In other contexts, the reflecting boundary can refer to a non-zero state and can be also time-dependent (cf., for instance, ^[8,9]). In the present paper, we restrict the attention to the Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in $[0, +\infty)$ with a reflection in the zero-state.

1.1.   Plan of the paper

In Section 2, we briefly review some background results on the time-inhomogeneous and time-homogeneous diffusion processes restricted to the interval $[0, +\infty)$, with a reflecting or a zero-flux condition in the zero-state. In this case, a time-inhomogeneous diffusion process is characterized by time-dependent infinitesimal moments, whereas the infinitesimal moments of a time-homogeneous diffusion process are not time-dependent.

In Sections 3–5, we consider the time-inhomogeneous Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in $[0, +\infty)$ (cf. Table 1).

We determine closed form expressions for the transition probability density function (pdf) and for the related conditional moments of the first and the second order in the following cases:

● for the TNH-RW process $X(t)$ with $\beta(t) = \gamma\, \sigma^2(t)$, where $\gamma\in{\mathbb R}$;

● for the TNH-ROU process $Y(t)$ with $\beta(t) = \gamma\, \sigma^2(t)\, e^{-A(t|0)}$, where $\gamma\in\mathbb{R}$ and $A(t|0) = \int_0^t\alpha(u)\, du$;

● for the TNH-RF process $Z(t)$ with $\beta(t) = \xi\, r(t)$, where $\xi > 0$.

For these processes, we analyze the asymptotic behavior of the transition densities. Moreover, in Section 5, for $\beta(t) = r(t)/2$ and for $\beta(t) = 3r(t)/2$ some relationships between the transition pdf for Wiener, Ornstein-Uhlenbeck and Feller processes are proved.

In Sections 3–5, we also take into account the time-homogeneous Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in $[0, +\infty)$ (see Table 2). For these processes, we study:

● the asymptotic transition pdf (steady-state density);

● the transition pdf for the TH-RW and for the TH-RF processes;

● the Laplace transform (LT) of the transition pdf ($\beta\in\mathbb{R}$) and the transition pdf ($\beta = 0$) for the TH-ROU process;

● the asymptotic behavior of the FPT moments for boundaries near to zero-state and for large boundaries.

Moreover, the mean, the coefficient of variation and the skewness of FPT for the TH-RW, TH-ROU and TH-RF processes are analyzed for various choices of parameters making use of Mathematica.

In Section 6, we consider some examples of the TNH-RW, TNH-ROU and TNH-RF diffusion processes useful to modeling queueing systems in heavy-traffic conditions; for these processes, we assume that the infinitesimal drift and the infinitesimal variance are time-dependent and include periodic functions. Several numerical computations with Mathematica are performed to analyze the conditional averages and the conditional variances and their asymptotic behaviors for some choices of the periodic functions and of the parameters.

This paper is dedicated to the memory of Patricia Román Román. Her untimely death leaves a great hole in our scientific community and an even greater hole in our hearts.

2.   Background results

In this section, we briefly review some results on the diffusion processes that will be used in the next sections to analyze Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in $[0, +\infty)$, with a zero-flux condition at the zero-state.

Let $D(t)$ be a time-inhomogeneous diffusion (TNH-RD) process with infinitesimal drift $\zeta_1(x, t)$ and the infinitesimal variance $\zeta_2(x, t)$, restricted to interval $[0, +\infty)$, with $+\infty$ unattainable end-point and a zero-flux condition in the zero-state. The transition pdf $r_D(x, t|x_0, t_0)$ of $D(t)$ can be obtained as the solution of the Fokker-Planck equation (cf. Dynkin ^[20])

with the initial delta condition $\lim_{t\downarrow t_0} r_D(x, t|x_0, t_0) = \delta(x-x_0)$ and the condition:

We note that (2.1) can be re-written as

where

represents probability flux (or current) of $D(t)$. Eq. (2.2) is the zero-flux or reflecting condition in the zero-state and corresponds to requiring that $\int_0^{+\infty} r_D(x, t|x_0, t_0)\, dx = 1$ for all $t \geq t_0$.

Expressions in closed form for $r_D(x, t|x_0, t_0)$ can be obtained only for some choices of the infinitesimal moments. For instance, if $D(t)$ is a TNH-RD process with space-state $[0, +\infty)$, having infinitesimal drift and infinitesimal variance

with the "prime" symbol denoting derivative with respect to the argument, $d\in \mathbb{R}$, $h_2(t)\neq 0$ and $h_1(t)/h_2(t)$ is a non-negative and monotonically increasing function, then $r_D(x, t|x_0, t_0)$ can be determined in closed form (cf. Buonocore et al. ^[8]):

where $f_D(x, t|x_0, t_0)$ and $F_D(x, t|x_0, t_0) = \int_{-\infty}^xf _D(z, t|x_0, t_0)\; dz$ are the transition pdf and probability distribution function of the corresponding unrestricted diffusion process with space-state $\mathbb{R}$, respectively. Specifically, for the diffusion process (2.3), $f_D(x, t|x_0, t_0)$ is a normal density with mean and variance

We remark that $r_D(x, t|x_0, t_0)$, given in (2.4), satisfies the Fokker-Planck equation (2.1), the initial delta condition, the zero-flux condition (2.2) and also $\int_0^{+\infty}r_D(x, t|x_0, t_0)\; dx = 1$. Moreover, under suitable conditions, the infinitesimal moments of the Wiener process and of the Ornstein-Uhlenbeck process satisfy (2.3); in these cases, the transition pdf $r_D(x, t|x_0, t_0)$ is obtainable from (2.4). Instead, for the Feller process the infinitesimal variance depends on $x$, so that (2.4) does not hold.

The knowledge of density $r_D(x, t|x_0, t_0)$ allows to evaluate the conditional moments:

2.1.   Time-homogeneous reflected diffusion process

For a time-homogeneous reflected diffusion (TH-RD) process $D(t)$, with state-space $[0, +\infty)$, one has $\zeta_1(x, t) = \zeta_1(x)$ and $\zeta_2(x, t) = \zeta_2(x)$ for all $t$. For the TH-RD process, the transition pdf $r_D(x, t|x_0, t_0) = r_D(x, t-t_0|x_0, 0) = r_D(x, t-t_0|x_0)$ and, in this case, we assume that $t_0 = 0$.

We denote by

the scale function and the speed density. These functions allow to classify the end-points of a diffusion process into regular, natural, exit, entrance and in attracting or nonattracting boundaries (cf. Karlin and Taylor ^[21]). Let

be the random variable that describes the FPT of $D(t)$ through $S$ starting from $D(0) = x_0\neq S$. We denote by $g_D(S, t|x_0) = dP\{T_D(S|x_0)\leq t\}/dt$ the FPT density and by $P_D(S|x_0) = \int_0^{+\infty}g_D(S, t|x_0)\, dt$ the ultimate FPT probability. When $P_D(S|x_0) = 1$, let

be the $n$-th FPT moment. The functions (2.6) allow to determine the FPT moments thanks to the Siegert formula (cf. Siegert ^[22]). Specifically, if $D(t)$ is a TH-RD process in $[0, +\infty)$, with $+\infty$ nonattracting end-point, for $n = 1, 2, \ldots$ it results:

● If $x_0 > S > 0$, one has $P_D(S|x_0) = 1$ and, if $\int_z^{+\infty}s_D(u)\, du$ converges, the FPT moments can be evaluated as:

Equation (2.7) holds also for $S = 0$ provided that the zero-state is a regular boundary.

● If $0\leq x_0 < S$, one has $P_D(S|x_0) = 1$ and the FPT moments can be iteratively computed as:

Making use of (2.8), in Giorno et al. ^[23] the following asymptotic results for the FPT moments are proved when the boundary $S$ moves indefinitely away from the zero-state and when $S$ is in the neighborhood of zero.

Remark 2.1. For the TH-RD process $D(t)$, we denote by

One has:

1. If ${\lim\limits_{x\uparrow +\infty}}\, k_1(x) = +\infty$ and ${\lim\limits_{x\uparrow +\infty}}\, k_2(x) = -\infty$, then

2. If ${\lim\limits_{x\downarrow 0}}\, k_1(x) = 0$ and ${\lim\limits_{x\downarrow 0}}\, k_2(x) = \nu$, with $-\infty < \nu < 1$, then

where

3.   Reflected Wiener process

Let $\{X(t), t\geq t_0\}$, $t_0\geq 0$, be a TNH-RW process, having infinitesimal drift and infinitesimal variance

respectively, with $\beta(t)\in\mathbb{R}$ and $\sigma(t) > 0$ continuous functions for all $t$. For the TNH-RW process, the results of Section 2 hold by choosing $\zeta_1(x, t) = \beta(t)$ and $\zeta_2(x, t) = \sigma^2(t)$. We denote by $r_X(x, t|x_0, t_0)$ the transition pdf of $X(t)$.

The reflected Wiener process can be used as the diffusion approximation of the queueing system $M/M/1$ in heavy traffic conditions (see Giorno et al. ^[1], Kingman ^[24], Harrison ^[25]). It plays also an important role in economics and in finance (cf. Veestraeten ^[6], Linetsky ^[26]).

Making use of the Fokker-Planck equation (2.1) with the boundary condition (2.2), for $x_0\geq 0$ one obtains the first two conditional moments of $X(t)$:

In the following proposition, we determine the transition pdf $r_X(x, t|x_0, t_0)$ in a special case.

Proposition 3.1. Let $X(t)$ be a TNH-RW process, having $\beta(t) = \gamma\, \sigma^2(t)$, with $\gamma\in\mathbb{R}$, and $\sigma(t) > 0$ in (3.1). One has:

where

with

and ${\rm Erf}(x) = (2/\sqrt{\pi})\int_0^xe^{-z^2}dz$ denoting the error function.

Proof. Comparing (2.3) with (3.1), one has $d\, h_1^{\prime}(t) = \beta(t)$, $h_2^{\prime}(t) = 0$ and $h_1^{\prime}(t)h_2(t) = \sigma^2(t)$ for all $t$, from which

Hence, for $\beta(t) = \gamma\, \sigma^2(t)$, Eq. (3.2) follows from (2.4) by setting $\gamma = d/c$. □

Under the assumption of Proposition 3.1, making use of (3.3) in (3.2), we have:

where ${\rm Erfc}(x) = 1-{\rm Erf}(x)$. We note that Eq. (3.5) for $t_0 = 0$ is in agreement with Eq. (29) in Molini et al. ^[27].

Corollary 3.1. Under the assumptions of Proposition 3.1, the following results hold:

● If $\beta(t) = 0$, one has:

● If $\beta(t) = \gamma\, \sigma^2(t)$, with $\gamma\neq 0$, one obtains:

Proof. It follows making use of (3.5) in (2.5) with $k = 1, 2$, respectively. □

Corollary 3.2. The TNH-RW process, having $A_1(t) = \gamma\, \sigma^2(t)$ and $A_2(t) = \sigma^2(t)$, with $\gamma\in\mathbb{R}$ and $\sigma(t) > 0$, admits the following asymptotic behaviors:

1. when $\gamma < 0$ and $\lim_{t\to +\infty}\sigma^2(t) = \sigma^2$ one has:

2. when $\gamma < 0$ and $\sigma^2(t)$ is a positive periodic function of period $Q$, one obtains:

Proof. Eqs. (3.8) and (3.9) follow taking the appropriate limits in (3.5). □

Under the assumptions of Corollary 3.2, the TNH-RW process $X(t)$ exhibits an exponential asymptotic behavior with mean and variance given by $(2\, |\gamma|)^{-1}$ and $(2\, |\gamma|)^{-2}$, respectively.

3.1.   Time-homogeneous reflected Wiener process

For the TH-RW process $X(t)$, in (3.1) we set $\beta(t) = \beta$ and $\sigma^2(t) = \sigma^2$, with $\beta\in\mathbb{R}$ and $\sigma > 0$. The scale function and the speed density, defined in (2.6), are:

respectively. When $\beta > 0$ ($\beta\leq 0$) the end-point $+\infty$ is an attracting (nonattracting) natural boundary. By choosing $\beta(t) = \beta$, $\sigma^2(t) = \sigma^2$ and $\gamma = \beta/\sigma^2$, from Eq. (3.5) one has:

and from (3.6) and (3.7) one obtains the first two conditional moments. Eq. (3.11) is in agreement with the expression given in Cox and Miller ^[28] obtained by using the method of images. We note that, if $\beta < 0$, the TH-RW process $X(t)$ admits an asymptotic behavior and the steady-state density is given in (3.8) with $\gamma = \beta/\sigma^2$. Hence, for $\beta < 0$ the steady-state density of the TH-RW process $X(t)$ is exponential with mean $E(X) = \sigma^2/(2\, |\beta|)$ and variance ${\rm Var}(X) = \sigma^4/(4\, \beta^2)$.

In Figure 1, we consider a TH-RW process $X(t)$ in $[0, +\infty)$, having $A_1 = \beta$ and $A_2 = \sigma^2$, with $\beta = -0.6$, $\sigma = 1$, $x_0 = 5$ and $t_0 = 0$. Making use of Algorithm 4.2 in Buonocore et al. ^[29], we obtain a random sample of $N = 5\cdot 10^4$ observations of $X(t)$. Then, we compare the histogram of the random sample with the transition pdf (3.11) as function of $x$ $(x\geq 0)$ for $t = 3$ on the left and $t = 5$ on the right. As shown in Figure 1, the histograms fit the exact transition densities (3.11) reasonably well.

3.2.   FPT for TH-RW process

For the TH-RW process, if $0\leq S < x_0$ the FPT through $S$ starting from $x_0$ is not affected to reflecting boundary in the zero-state; in this case, if $\beta < 0$ the ultimate FPT probability $P_X(S|x_0) = 1$ and from (2.7) the FPT mean is $t_1^{(X)}(S|x_0) = (S-x_0)/\beta$. Moreover, if $0\leq x_0 < S$ the probability $P_X(S|x_0) = 1$ and making use of (2.8) one obtains:

When $0\leq x_0 < S$, from (3.12) one has:

● $t_1^{(X)}(S|x_0)$ decreases as $\beta$ increases and

● for $\beta\leq 0$, $t_1^{(X)}(S|x_0)$ decreases as $\sigma^2$ increases; moreover, one has

In Figure 2, the FPT mean (3.12) of the TH-RW process is plotted for $x_0 = 5$, $S = 10$ and for different choices of $\beta$ and $\sigma^2$.

Making use of (3.10) in Remark 2.1, one can derive some asymptotic behaviors for the FPT moments of the TH-RW process.

Remark 3.1. For the TH-RW process $X(t)$, when $n = 1, 2, \ldots$ one has:

1. ${\lim\limits_{S\uparrow+\infty}{ t_n^{(X)}(S|0)\over [t_1^{(X)}(S|0)]^n} = n!}, \qquad \beta < 0,$

2. ${ \lim\limits_{S\downarrow 0}{ t_n^{(X)}(S|0)\over [t_1^{(X)}(S|0)]^n} = {(-1)^n\, E_{2n}\over (2n-1)!!}},$

where

denote the Euler numbers.

Then, from Remark 3.1 it follows:

● for $S\uparrow+\infty$ one has:

so that for $\beta < 0$ the FPT density $g_X(S, t|0)$ of the TH-RW process exhibits an exponential asymptotic behavior for large boundaries;

● for $S\downarrow 0$ results:

In particular, one has:

In Tables 3–5, $t_1^{(X)}(S|0)$ is computed via (3.12), whereas $t_2^{(X)}(S|0)$ and $t_3^{(X)}(S|0)$ are numerically evaluated making use of the Siegert formula (2.8), with the scale function and the speed density given in (3.10). In particular, in Table 3, the mean $t_1^{(X)}(S|0)$, the variance ${\rm Var}^{(X)}(S|0)$, the coefficient of variation ${\rm Cv}^{(X)}(S|0)$ and the skewness $\Sigma^{(X)}(S|0)$ of the FPT are listed for $\beta = -0.1$, $\sigma = 1$ and some choices of $S > 0$. In agreement with the exponential asymptotic behavior, being $\beta < 0$, the coefficient of variation and the skewness approach to 1 and to 2, respectively, as $S$ increases.

Moreover, in Tables 4 and 5, for the TH-RW process with $\beta = -0.1, 0.1$ and $\sigma = 1$, we compare the FPT moments $t_2^{(X)}(S|0)$ and $t_3^{(X)}(S|0)$ with the approximate values $m_2^{(X)}(S|0)$ and $m_3^{(X)}(S|0)$, respectively. We note that the goodness of the approximations improves as the boundary $S$ approaches the reflecting zero-state.

4.   Reflected Ornstein-Uhlenbeck process

Let $\{Y(t), t\geq t_0\}$, $t_0\geq 0$, be a TNH-ROU process, having infinitesimal drift and infinitesimal variance

with $\alpha(t)\in\mathbb{R}$, $\beta(t)\in\mathbb{R}$ and $\sigma(t) > 0$ continuous functions for all $t$. For the TNH-ROU process, the results of Section 2 hold by choosing $\zeta_1(x, t) = \alpha(t)\, x+\beta(t)$ and $\zeta_2(x, t) = \sigma^2(t)$. Note that when $\alpha(t) = 0$ for all $t$, the process $Y(t)$ identifies with the TNH-RW process $X(t)$ with infinitesimal moments (3.1). We denote by $r_Y(x, t|x_0, t_0)$ the transition pdf of $Y(t)$.

The reflected Ornstein-Uhlenbeck process arises as a diffusion approximation for population dynamics and for queueing systems (cf. Giorno et al. ^[1,30], Ward and Glynn ^[31]). Furthermore, the membrane potential evolution in neuronal diffusion models can be described by focusing the attention on the Ornstein-Uhlenbeck process confined by a lower reflecting boundary (cf., for instance, Buonocore et al. ^[9]). The reflected Ornstein-Uhlenbeck process can be also applied to the regulated financial market (see, Linetsky ^[26], Nie and Linetsky ^[32]).

Recalling the Fokker-Planck equation (2.1), with the boundary condition (2.2), for $x_0\geq 0$ one has the first two conditional moments of $Y(t)$:

being

We now determine the transition pdf $r_Y(x, t|x_0, t_0)$ in a special case.

Proposition 4.1. Let $Y(t)$ be a TNH-ROU process, having $\beta(t) = \gamma\, \sigma^2(t)\, e^{-A(t|0)}$, with $\gamma\in\mathbb{R}$, $\sigma(t) > 0$ and $A(t|t_0)$ given in (4.2). One has:

where

with

Proof. Comparing (2.3) with (4.1), one has

for all $t\geq 0$, from which

with $c\neq 0$. Then, if $\beta(t) = \gamma\, \sigma^2(t)\, e^{-A(t|0)}$, Eq. (4.3) follows from (2.4) by setting $\gamma = d/c$. □

Under the assumptions of Proposition 4.1, making use of (4.4) in (4.3), for $x\geq 0, x_0\geq 0$ we have:

where we have set:

Corollary 4.1. Under the assumptions of Proposition 4.1, the following results hold:

● when $\beta(t) = 0$, one obtains

● when $\beta(t) = \gamma\, \sigma^2(t)\, e^{-A(t|0)}$, with $\gamma\neq 0$, one has

with $H(t|x_0, t_0)$ given in (4.7).

Proof. It follows making use of (4.6) in (2.5) for $k = 1, 2$, respectively. □

We note that if $\alpha(t) = 0$ for all $t$, the conditional moments (4.8) and (4.9) identify with conditional moments (3.6) and (3.7) of the TNH-RW process with $\beta(t) = \gamma\, \sigma^2(t)$, being $H(t|x_0, t_0) = x_0+\gamma \, V_X(t|t_0)$ in (4.7).

Corollary 4.2. For the TNH-ROU process, having $B_1(x) = \alpha\, x$ and $B_2(t) = \sigma^2(t)$, with $\alpha\in\mathbb{R}$ and $\sigma(t) > 0$, the following asymptotic behaviors hold:

1. when $\alpha < 0$ and $\lim_{t\to +\infty} \sigma^2(t) = \sigma^2$ one has:

and the first two asymptotic moments are

2. when $\alpha < 0$ and $\sigma^2(t)$ is a positive periodic function of period $Q$, one obtains:

with

and the first two asymptotic moments are

Proof. Eqs. (4.10) and (4.11) follow from (4.6) by setting $\alpha(t) = \alpha$ and $\gamma = 0$. When $\alpha < 0$, in the case 1. one has $\lim_{t\to +\infty} V_Y(t|t_0) = \sigma^2/(2\, |\alpha|)$, whereas in the case 2. it results $\lim_{n\to +\infty} V_Y(t+nQ|t_0) = \omega_1(t)$. □

4.1.   Time-homogeneous reflected Ornstein-Uhlenbeck process

For the TH-ROU process $Y(t)$, in (4.1) we set $\alpha(t) = \alpha$, $\beta(t) = \beta$, $\sigma^2(t) = \sigma^2$, with $\alpha\neq 0$, $\beta\in\mathbb{R}$ and $\sigma > 0$. The scale function and the speed density, defined in (2.6), are:

respectively. When $\alpha > 0$ ($\alpha < 0$) the end-point $+\infty$ is an attracting (nonattracting) natural boundary.

Proposition 4.2. Let $r_{\lambda}^{(Y)}(x|x_0) = \int_0^{+\infty}e^{-\lambda\, t}r_Y(x, t|x_0)\, dt$ $(\operatorname{Re} \lambda > 0)$ be the LT of the transition pdf of the TH-ROU process $Y(t)$. One has

where $x_0\vee x = \max(x_0, x)$, $x_0\wedge x = \min(x_0, x)$, $D_{\nu}(z)$ is the parabolic cylinder function defined as

and

is the Kummer's confluent hypergeometric function, with $(a)_0 = 1$ and $(a)_n = a(a+1)\cdots(a+n-1)$ for $n = 1, 2, \ldots$

Proof. The proof is given in A. □

Eq. (4.13) can be used to analyze the asymptotic behavior of the TH-ROU process.

Corollary 4.3. For $\alpha < 0$, the TH-ROU process $Y(t)$ admits an asymptotic behavior. The steady-state density is

and the asymptotic moments of the first and second order are:

Proof. If $\alpha < 0$, from (4.13) one has:

Since (cf. Gradshteyn and Ryzhik ^[33], p. 1030, no. 9.251 and no. 9.254)

from (4.16) one obtains (4.14). The asymptotic moments (4.15) follow from (4.14). □

Eq. (4.14) is in agreement with the result of the Proposition 1 in Ward and Glinn ^[31]; moreover, for $\beta = 0$ Eq. (4.14) identifies with (4.10).

The inverse LT of (4.13) is obtainable only for $\beta = 0$; in this case, the following result holds.

Proposition 4.3. For the TH-ROU process $Y(t)$, having $B_1(x) = \alpha\, x$ and $B_2 = \sigma^2$, with $\alpha\neq 0$ and $\sigma > 0$, one has:

Proof. It follows from (4.6) with $\gamma = 0$, by setting $t_0 = 0$, $A(t|0) = e^{\alpha\, t}$ and $V_Y(t|0) = \sigma^2(e^{2\alpha t}-1)/\alpha$. Alternatively, Eq. (4.18) can be obtained from (4.13) and (A.4) with $\beta = 0$, by noting that $r_{\lambda}^{(Y)}(x|x_0) = f_{\lambda}^{(Y)}(x|x_0)+f_{\lambda}^{(Y)}(x|-x_0)$. □

In Figure 3, we consider a TH-ROU process $Y(t)$ in $[0, +\infty)$, having $B_1(x) = \alpha\, x+\beta$ and $B_2 = \sigma^2$, with $\alpha = -0.5$, $\beta = 0$, $\sigma = 1$, $x_0 = 5$ and $t_0 = 0$. Using the Algorithm 4.1 in Buonocore et al. ^[29], we obtain a random sample of $N = 5\cdot 10^4$ observations of $Y(t)$. Then, we compare the histogram of the random sample with the transition pdf (4.18) as function of $x$ $(x\geq 0)$ for $t = 2$ on the left and $t = 4$ on the right. Figure 3 shows the good agreement between the histograms obtained via simulation and the exact density (4.18).

Under the assumption of Proposition 4.3, the first two conditional moments of the TH-ROU process can be obtained from (4.8).

4.2.   FPT for TH-ROU process

For the TH-ROU process, if $0\leq S < x_0$ the FPT through $S$ starting from $x_0$ is not affected to reflecting boundary in the zero-state; in this case, for $\alpha < 0$ the ultimate FPT probability $P_Y(S|x_0) = 1$ and from (2.7) the FPT mean is

where

Moreover, if $0\leq x_0 < S$ the probability $P_Y(S|x_0) = 1$ and making use of (2.8) one obtains:

where

When $0\leq x_0 < S$, from (4.20) one has:

● $t_1^{(Y)}(S|x_0)$ decreases as $\beta$ increases and

● for $\alpha < 0$, $t_1^{(Y)}(S|x_0)$ decreases as $\sigma^2$ increases; moreover, one has:

In Figures 4 and 5, the FPT mean (4.20) of the TH-ROU process is plotted for $x_0 = 5$, $S = 10$ and for different choices of $\beta$ and $\sigma^2$, with $\alpha = -0.02, 0.02$, respectively.

Making use of (4.12) in Remark 2.1, one can derive some asymptotic behaviors for the FPT moments of the TH-ROU process.

Remark 4.1. For the TH-ROU process $Y(t)$, when $n = 1, 2, \ldots$ one has:

1. ${ \lim\limits_{S\uparrow+\infty}{ t_n^{(Y)}(S|0)\over [t_1^{(Y)}(S|0)]^n} = n!}, \qquad \alpha < 0,$

2. ${ \lim\limits_{S\downarrow 0}{ t_n^{(Y)}(S|0)\over [t_1^{(Y)}(S|0)]^n} = {(-1)^n\, E_{2n}\over (2n-1)!!}, }$

where $E_0, E_1, \ldots$ are the Euler numbers.

Then, from Remark 4.1, it follows:

● for $S\uparrow+\infty$ one has:

so that for $\alpha < 0$ the FPT density $g_Y(S, t|0)$ of the TH-ROU process exhibits an exponential trend for large boundary;

● for $S\downarrow 0$ it results:

In particular, one has:

In Tables 6–9, $t_1^{(Y)}(S|0)$ is computed via (4.20), whereas $t_2^{(Y)}(S|0)$ and $t_3^{(Y)}(S|0)$ are numerically evaluated making use of (2.8) and (4.12). In particular, in Tables 6 and 7, the mean $t_1^{(Y)}(S|0)$, the variance ${\rm Var}^{(Y)}(S|0)$, the coefficient of variation ${\rm Cv}^{(Y)}(S|0)$ and the skewness $\Sigma^{(Y)}(S|0)$ of the FPT are listed for $\alpha = -0.02$, $\beta = -0.1, 0.1$, $\sigma = 1$ and some choices of $S > 0$. Being $\alpha < 0$, the coefficient of variation and the skewness approach to 1 and to 2, respectively, as $S$ increases.

Moreover, in Tables 8 and 9, for the TH-ROU process with $\alpha = -0.02$, $\beta = -0.1, 0.1$ and $\sigma = 1$ we compare the FPT moments $t_2^{(Y)}(S|0)$ and $t_3^{(Y)}(S|0)$ with the approximate values $m_2^{(Y)}(S|0)$ and $m_3^{(Y)}(S|0)$, respectively. From Tables 8 and 9, we note that the goodness of the approximations improves as the boundary $S$ approaches the reflecting zero-state.

5.   Feller process with a zero-flux condition in the zero-state

Let $\{Z(t), t\geq t_0\}$, $t_0\geq 0$, be a TNH-RF process, having the following infinitesimal drift and infinitesimal variance

with $\alpha(t)\in\mathbb{R}$, $\beta(t) > 0$ and $r(t) > 0$ continuous functions for all $t$, defined in the state-space $[0, +\infty)$ with a zero-flux condition in the zero-state. For the TNH-RF process, the results of Section 2 hold by choosing $\zeta_1(x, t) = \alpha(t)\, x+\beta(t)$ and $\zeta_2(x, t) = 2\, r(t)\, x$. We denote with $r_Z(x, t|x_0, t_0)$ the transition pdf of $Z(t)$.

Feller diffusion process is applied in population dynamics to model the growth of a population (cf. Ricciardi et al. ^[15], Giorno and Nobile ^[34]). This process is also used in queueing systems to describe the number of customers in a queue (cf. Di Crescenzo and Nobile ^[35]), in neurobiology to analyze the input-output behavior of single neurons (see, for instance, Giorno et al. ^[36], Ditlevsen and Lánský ^[37]), in mathematical finance to model asset prices, market indices, interest rates and stochastic volatility (see, Tian and Zhang ^[38], Cox et al. ^[39], Di Nardo and D'Onofrio ^[40]). We emphasize that in the mathematical finance, the Feller process is also known as Cox-Ingersoll-Ross (CIR) model.

Making use of Eq. (2.1), with the zero-flux condition in the zero-state (2.2), for $x_0\geq 0$ one obtains the conditional mean and the conditional variance of $Z(t)$:

with $A(t|t_0)$ defined in (4.2) and

We note that when $\alpha(t)\neq 0$, the conditional mean $M_Y(t|x_0, t_0)$ of the unrestricted Ornstein-Uhlenbeck process, given in (4.5), identifies with the conditional average of the Feller process $Z(t)$, given in (5.2). Similarly, when $\alpha(t) = 0$ for all $t$, the conditional average of the Feller process is equal to the conditional mean $M_X(t|x_0, t_0)$ of the unrestricted Wiener process, given in (3.4).

In the following proposition, we consider the transition pdf $r_Z(x, t|x_0, t_0)$ in a special case.

Proposition 5.1. Let $Z(t)$ be a TNH-RF process, with $\alpha(t)\in\mathbb{R}$, $r(t) > 0$ and $\beta(t) = \xi\, r(t)$, with $\xi > 0$, in (5.1). The following results hold:

where

is the modified Bessel function of the first kind, with $\Gamma(\nu)$ denoting the Euler gamma function. Moreover, one has

where $A(t|t_0)$ and $R(t|t_0)$ are defined in (4.2) and (5.3), respectively.

Proof. Eq. (5.4) follows as in Giorno and Nobile ^[34] and Masoliver ^[41]. Moreover, relations (5.5) are obtainable from (5.2). □

Since, for fixed $\nu$, when $z\to 0$ one has

the first formula in (5.4) follows from the second expression as $x_0\downarrow 0$.

Corollary 5.1. The TNH-RF process $Z(t)$, having $C_1(x, t) = \alpha\, x+\xi\, r(t)$ and $C_2(x, t) = 2\, r(t)\, x$, with $\alpha\in\mathbb{R}$, $\xi > 0$ and $r(t) > 0$, admits the following asymptotic behaviors:

1. when $\alpha < 0$ and $\lim_{t\to +\infty}r(t) = r$, with $r > 0$, one has:

and the asymptotic moments are

2. when $\alpha < 0$ and $r(t)$ is a positive periodic function of period $Q$, one obtains:

with

and the asymptotic moments are

Proof. Eq. (5.7) follows from (5.4) making use of (5.6) and by noting that

Similarly, since $\lim_{n\to+\infty}\Bigl[e^{-A(t+nQ|t_0)}/R(t+nQ|t_0)\Bigr] = \omega_2(t)$, one can be obtain (5.8). □

We note that (5.7) is a gamma density of parameters $\xi$ and $r/|\alpha|$, that is a decreasing function of $x$ when $0 < \xi\leq 1$, whereas it has a single maximum in $x = r(\xi-1)/|\alpha|$ for $\xi > 1$. Similarly, (5.8) is a non-homogeneous gamma density.

In Propositions 5.2 and 5.3, we prove some relations between the transition pdf of TNH-RF process with $\beta(t) = r(t)/2$ and $\beta(t) = 3\, r(t)/2$ in (5.1) and the transition pdf of Wiener and of Ornstein-Uhlenbeck processes under suitable conditions on the zero-state and for specific choices of the infinitesimal moments.

Proposition 5.2. Let $Z(t)$ be a TNH-RF process with $C_1(x, t) = \alpha(t)\, x+r(t)/2$ and $C_2(x, t) = 2\, r(t)\, x$, where $\alpha(t)\in\mathbb{R}$ and $r(t) > 0$.

1. If $\alpha(t) = 0$ for all $t$, one has

where $r_X(x, t|x_0, t_0)$ denotes the transition pdf of the TNH-RW process with $A_1 = 0$ and $A_2(t) = r(t)/2$.

2. If $\alpha(t)$ is not always zero, it follows:

where $r_Y(x, t|x_0, t_0)$ denotes the transition pdf of the TNH-ROU process with $B_1(x, t) = \alpha(t)\, x/2$ and $B_2(t) = r(t)/2$.

Proof. For the TNH-RF process $Z(t)$, by setting $\beta(t) = r(t)/2$ in (5.4) and recalling that

for $x_0\geq 0, x > 0$ one has:

with $\widetilde R(t|t_0) = \int_{t_0}^tr(\theta)\, d\theta$ and with $A(t|t_0)$ and $R(t|t_0)$ given in (4.2) and (5.3), respectively. We now analyze separately the cases 1. and 2.

1. For the TNH-RW process $X(t)$, defined in (3.1), with $\beta(t) = 0$ and $\sigma^2(t) = r(t)/2$, one has $V_X(t|t_0) = \widetilde R(t|t_0)/2$, so that from (3.5) with $\gamma = 0$ one obtains:

for $x_0\geq 0$ and $x\geq 0$. Then, by comparing the first of (5.11) with (5.12), for $x_0\geq 0$ and $x > 0$ relation (5.9) follows.

2. In the TNH-ROU process $Y(t)$, defined in (4.1), we set $\beta(t) = 0$, $\sigma^2(t) = r(t)/2$ and we change $\alpha(t)$ into $\alpha(t)/2$; by virtue of (4.5) and (5.3), one has $V_Y(t|t_0) = R(t|t_0)e^{A(t|t_0)}/2$, so that from (4.6) with $\gamma = 0$ one obtains:

Hence, for $x_0\geq 0$ and $x > 0$, Eq. (5.10) follows by comparing the second of (5.11) with (5.13). □

Proposition 5.3. Let $Z(t)$ be a TNH-RF process with $C_1(t) = \alpha(t)\, x+3r(t)/2$ and $C_2(x, t) = 2\, r(t)\, x$, where $\alpha(t)\in\mathbb{R}$ and $r(t) > 0$.

1. If $\alpha(t) = 0$ for all $t$, one has

where $a_X(x, t|x_0, t_0)$ denotes the transition pdf of the inhomogeneous Wiener process with $A_1 = 0$ and $A_2(t) = r(t)/2$, restricted to $(0, +\infty)$ with an absorbing boundary in the zero-state.

2. If $\alpha(t)$ is not always zero, it follows:

where $a_Y(x, t|x_0, t_0)$ denotes the transition pdf of the inhomogeneous Ornstein-Uhlenbeck process with $B_1(x, t) = \alpha(t)\, x/2$ and $B_2(t) = r(t)/2$, restricted to $(0, +\infty)$ with an absorbing boundary in the zero-state.

Proof. For the TNH-RF process $Z(t)$, by setting $\beta(t) = 3r(t)/2$ in (5.4) and recalling that

for $x_0 > 0, x > 0$ one has:

with $\widetilde R(t|t_0) = \int_{t_0}^tr(\theta)\, d\theta$ and with $A(t|t_0)$ and $R(t|t_0)$ given in (4.2) and (5.3), respectively. We now take into account the cases 1. and 2.

1. For a time-inhomogeneous Wiener process with $A_1 = 0$ and $A_2(t) = r(t)/2$, for $x_0 > 0$ and $x > 0$ one has (cf. Giorno and Nobile ^[19])

Hence, Eq. (5.14) follows by comparing the first of (5.16) with (5.17).

2. For a time-inhomogeneous Ornstein-Uhlenbeck process with $B_1(x, t) = \alpha(t)\, x/2$ and $B_2(t) = r(t)/2$, one has (cf. Giorno and Nobile ^[19]):

Then, by comparing the second of (5.16) with (5.18), Eq. (5.15) follows. □

5.1.   Time-homogeneous Feller process

We consider the TH-RF process $Z(t)$, obtained from (5.1) by setting $\alpha(t) = \alpha$, $\beta(t) = \beta$, $r(t) = r$, with $\alpha\in\mathbb{R}$, $\beta > 0$, $r > 0$ and a zero-flux condition in the zero-state. From (2.6), for the TH-RF process $Z(t)$ one has:

As proved by Feller ^[42], the boundary 0 is regular for $0 < \beta < r$ and entrance for $\beta\geq r$. Furthermore, the end-point $+\infty$ is a nonattracting natural boundary for $\alpha\leq 0$ and an attracting natural boundary for $\alpha > 0$.

The transition pdf of the TH-RF process is obtainable from (5.4) by setting $\alpha(t) = \alpha$, $\beta(t) = \beta$, $r(t) = r$ and $\xi = \beta/r$. In particular,

● if $\alpha = 0$, one has:

● if $\alpha\neq 0$, one obtains:

The conditional mean and the conditional variance of the TH-RF process can be obtained from (5.5) with $\xi = \beta/r$. We note that if $\alpha < 0$, $\beta > 0$ and $r > 0$, the TH-RF process admits the steady-state density given in (5.7) with $\xi = \beta/r$.

In Figure 6, we consider a TH-RF process $Z(t)$ in $[0, +\infty)$, having $C_1(x) = \alpha\, x+\beta$ and $C_2(x) = 2\, r\, x$, with $\alpha = 0$, $\beta = 0.25$, $r = 0.5$, $x_0 = 5$ and $t_0 = 0$. We compare the histogram of the random sample of $N = 5\cdot 10^4$ observations of $Z(t)$ with the transition pdf (5.20) as function of $x$ $(x\geq 0)$ for $t = 0.5$ on the left and $t = 1$ on the right. Instead, in Figure 7, we consider a TH-RF process $Z(t)$ in $[0, +\infty)$, with $\alpha = -0.5$, $\beta = 0.25$, $r = 0.5$, $x_0 = 5$ and $t_0 = 0$ and we compare the histogram of the random sample of $N = 5\cdot 10^4$ observations of $Z(t)$ with the transition pdf (5.21) as function of $x$ $(x\geq 0)$ for $t = 0.5$ on the left and $t = 1$ on the right. Figures 6 and 7 show the good agreement between the exact and simulated results.

5.2.   FPT moments for the TH-RF process

In Giorno and Nobile ^[43] the FPT problem through a state $S$ starting from $x_0$ for the TH-RF process has been considered. When $0 < S < x_0$ the ultimate FPT probability $P_Z(S|x_0) = 1$ if and only if $[\alpha < 0, \beta > 0]$ or $[\alpha = 0, 0 < \beta\leq r]$. Making use of (2.7), for $\alpha = 0$ and $0 < \beta\leq r$ one has that $t_1^{(Z)}(S|x_0)$ diverges, whereas if $\alpha < 0$ and $\beta > 0$ one obtains:

Moreover, if $x_0 > 0$ the ultimate FPT probability $P_Z(0|x_0) = 1$ if and only if $\alpha\leq 0$ and $0 < \beta < r$. From (2.7), for $\alpha = 0$ and $0 < \beta < r$ one has that $t_1^{(Z)}(0|x_0)$ diverges, whereas for $\alpha < 0$ and $0 < \beta < r$ (5.22) holds with $S = 0$.

Instead, when $0\leq x_0 < S$, one obtains $P_Z(S|x_0) = 1$ and from (2.8) it results

so that for $\alpha = 0$ one has $t_1^{(Z)}(S|x_0) = (S-x_0)/\beta$. Moreover, when $0\leq x_0 < S$, from (5.23) it follows:

● $t_1^{(Z)}(S|x_0)$ decreases as $\beta$ increases and

● $t_1^{(Z)}(S|x_0)$ decreases as $r$ increases and

In Figures 8 and 9, the FPT mean (5.23) of the TH-RF process is plotted for $x_0 = 5$, $S = 10$ and for different choices of $\beta$ and $r$, with $\alpha = -0.02, 0.02$, respectively.

Making use of (5.19) in Remark 2.1, one can derive the following asymptotic results for the FPT moments of the TH-RF process.

Remark 5.1. For the TH-RF process $Z(t)$, when $n = 1, 2, \ldots$ one has:

1. ${\lim\limits_{S\uparrow+\infty}{ t_n^{(Z)}(S|0)\over [t_1^{(Z)}(S|0)]^n} = n!}, \qquad \alpha < 0,$

2. ${\lim\limits_{S\downarrow 0}{ t_n^{(Z)}(S|0)\over [t_1^{(Z)}(S|0)]^n} = u_n}, \qquad 0 < \beta < r$

where

From Remark 5.1, the following asymptotic behaviors hold:

● for $S\uparrow +\infty$ one has

so that for $\alpha < 0$ the FPT density of TH-RF process exhibits an exponential trend for large boundary;

● for $S\downarrow 0$ it results:

In particular, one has:

In Tables 10–11, $t_1^{(Z)}(S|0)$ is computed via (5.23), whereas $t_2^{(Z)}(S|0)$ and $t_3^{(Z)}(S|0)$ are numerically evaluated making use of (2.8) and (5.19). In Table 10, the mean $t_1^{(Z)}(S|0)$, the variance ${\rm Var}^{(Z)}(S|0)$, the coefficient of variation ${\rm Cv}^{(Z)}(S|0)$ and the skewness $\Sigma^{(Z)}(S|0)$ of the FPT are listed for $\alpha = -0.02$, $\beta = 0.1$, $r = 0.5$ and some choices of $S > 0$. Being $\alpha < 0$, the coefficient of variation and the skewness approach to 1 and to 2, respectively, as $S$ increases.

Moreover, in Table 11, for the TH-RF process with $\alpha = -0.02$, $\beta = 0.1$ and $r = 0.5$, we compare the FPT moments $t_2^{(Z)}(S|0)$ and $t_3^{(Z)}(S|0)$ with the approximate values $m_2^{(Z)}(S|0)$ and $m_3^{(Z)}(S|0)$, respectively. We note that the goodness of the approximations improves as $S$ approaches zero.

6.   Applications of the results to queueing systems

The TNH-RW, TNH-ROU and TNH-RF diffusion processes can be seen as the continuous approximations of some time-inhomogeneous birth-death processes that modeling queueing systems in heavy-traffic conditions. Referring to the queueing systems, in this section we consider some examples in which the infinitesimal drifts and the infinitesimal variances are time-dependent and include periodic functions. The presence of periodicity in the infinitesimal moments express the existence of rush hours occurring on a daily basis. For the considered reflected processes, we apply the exact and asymptotic results obtained in Sections 3–5 to analyze the conditional averages and the conditional variances for some choices of the periodic functions and of the parameters.

Example 6.1. (Wiener model) We consider the TNH-RW process $X(t)$, having infinitesimal drift and infinitesimal variance $A_1(t) = \gamma\, \sigma^2(t)$ and $A_2(t) = \sigma^2(t)$, with $\gamma\in\mathbb{R}$ and $\sigma(t) > 0$. This process can be seen as the continuous approximation of the birth-death queueing system $N_1(t)$ with arrival and departure intensity functions $\lambda_n(t) = \lambda\sigma^2(t)/\varepsilon+\sigma^2(t)/(2\varepsilon^2)$ $(n = 0, 1, \ldots)$ and $\mu_n(t) = \mu\sigma^2(t)/\varepsilon+\sigma^2(t)/(2\varepsilon^2)$ $(n = 1, 2, \ldots)$, where $\lambda > 0$, $\mu > 0$ and $\varepsilon$ is a positive scaling parameter. Indeed, the scaled process $N_1(t)\, \varepsilon$ converges weakly to the diffusion process $X(t)$, having state-space $[0, +\infty)$, with infinitesimal moments (see, for instance, ^[1]):

with $\gamma = \lambda-\mu$. We assume that

where $\nu > 0$ is the average of the period function $\sigma^2(t)$ of period $Q$ and $c$ is the amplitude of the oscillations, with $0\leq c < 1$. These choices of parameters ensure that the infinitesimal variance is a positive function.

In Figure 10, we suppose that $\sigma^2(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})]$ and that at the initial time $t_0 = 0$ the number of customers is $X(t_0) = x_0 = 5$. As proved in Corollary 3.2, the transition pdf of $X(t)$ admits the asymptotic exponential behavior (3.9) for $\gamma < 0$. In Figure 10, the conditional mean and the conditional variance, obtained from (3.7), and the related asymptotic behaviors are shown as function of $t$ for $\gamma = -0.5, -1$; the dotted lines indicate the corresponding asymptotic means and variances $(2\, |\gamma|)^{-1}$ and $(2\, |\gamma|)^{-2}$. $

Example 6.2. (Ornstein-Uhlenbeck model) We consider the TNH-ROU process $Y(t)$, having infinitesimal drift and infinitesimal variance $B_1(x, t) = \alpha\, x$ and $B_2(t) = \sigma^2(t)$, with $\alpha\in\mathbb{R}$ and $\sigma(t) > 0$. This process can be seen as the continuous approximation of the birth-death queueing system $N_2(t)$ with arrival and departure intensity functions $\lambda_n(t) = \lambda\, n+\sigma^2(t)/(2\varepsilon^2)$ $(n = 0, 1, \ldots)$ and $\mu_n(t) = \mu\, n+\sigma^2(t)/(2\varepsilon^2)$ $(n = 1, 2, \ldots)$, both depending on the number of customers in the systems, where $\lambda > 0$, $\mu > 0$ and $\varepsilon$ is a positive scaling parameter. Indeed, the scaled process $N_2(t)\, \varepsilon$ converges weakly to the diffusion process $X(t)$, having state-space $[0, +\infty)$, with infinitesimal moments (see, for instance, ^[1]):

where $\alpha = \lambda-\mu$. We assume that (6.1) holds.

As proved in Corollary 4.2, when $\alpha < 0$ the transition pdf of $Y(t)$ admits the asymptotic behavior (4.11) with

In Figure 11, the conditional mean and the conditional variance, obtained from (4.8), and the related asymptotic behaviors are shown as function of $t$ for $\sigma^2(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})]$, $t_0 = 0$, $x_0 = 5$ and $\alpha = -0.05, -0.1$; the dotted functions indicate the corresponding asymptotic mean $[2\omega_1(t)/\pi]^{1/2}$ and asymptotic variances $\omega_1(t)\, (1-2/\pi)$. $

Example 6.3. (Feller model) We consider the TNH-RF process $Z(t)$, having infinitesimal drift and infinitesimal variance $C_1(x, t) = \alpha\, x+\xi\, r(t)$ and $C_2(x, t) = 2\, r(t)\, x$, with $\alpha\in\mathbb{R}$, $\xi > 0$ and $r(t) > 0$. This process can be seen as the continuous approximation of the birth-death queueing system $N_3(t)$ with arrival and departure intensity functions $\lambda_n(t) = [\lambda+r(t)/\varepsilon]\, n+\xi\, r(t)/\varepsilon$ $(n = 0, 1, \ldots)$ and $\mu_n(t) = [\mu+r(t)/\varepsilon]\, n$ $(n = 1, 2, \ldots)$, both depending on the number of customers in the systems and on the positive scaling parameter $\varepsilon$. Indeed, the scaled process $N_3(t)\, \varepsilon$ converges weakly to a diffusion process, having state-space $[0, +\infty)$ (see, for instance, ^[34]) and one has:

where $\alpha = \lambda-\mu$. We assume that $r(t) = \sigma^2(t)$, with $\sigma^2(t)$ given in (6.1).

As proved in Corollary 5.1, when $\alpha < 0$ the transition pdf of $Z(t)$ admits the asymptotic behavior (5.8) with

In Figure 12, the conditional mean and the conditional variance, obtained from (5.5), and the related asymptotic behaviors are shown as function of $t$ for $r(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})]$, $t_0 = 0$, $x_0 = 5$, $\alpha = -0.08$ and $\xi = 1.0, 2.0$; the dotted functions indicate the corresponding asymptotic mean $\xi/\omega_2(t)$ and asymptotic variances $\xi/[\omega_2(t)]^2$. $

As highlighted in the Figures 10–12, the periodic intensity function (6.1), used in the infinitesimal drifts and in the infinitesimal variances of the reflected Wiener, Ornstein-Uhlenbeck and Feller models, affects the shapes of the conditional averages and of the conditional variances.

7.   Concluding remarks

For the Wiener, Ornstein Uhlenbeck and Feller processes, restricted to the interval $[0, +\infty)$, with reflecting or a zero-flux condition in the zero-state, we analyze the transition probability density functions and their asymptotic behaviors, paying particular attention to the time-inhomogeneous proportional cases and to the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Moreover, the FPT moments and their asymptotic behaviors are analyzed for the time-homogeneous cases. Finally, some applications of the obtained results to queueing systems are considered. Various numerical computations are performed with Mathematica to illustrate the role played by parameters.

A.   Proof of Proposition 4.2

Let $f_Y(x, t|x_0)$ be the transition pdf of the unrestricted time-homogeneous Ornstein-Uhlenbeck process with infinitesimal moments $B_1(x) = \alpha x+\beta$ and $B_2 = \sigma^2$. For $x, x_0\in{\mathbb R}$, we denote by

the probability current of the unrestricted Ornstein-Uhlenbeck process. The transition pdf $r_Y(x, t|x_0)$ of a time-homogeneous diffusion process restricted to $[0, +\infty)$, with zero reflecting boundary, satisfies the following integral equations (cf. Giorno et al. ^[44]):

Taking the LT in (A.2) one has:

where $f_{\lambda}^{(Y)}(x|x_0)$ is LT of $f_Y(x, t|x_0)$ and $j_{\lambda}^{(Y)}(0|x_0)$ is LT of $j_Y(0, t|x_0)$. Taking the LT in the first equation in (4.4), with $\alpha(t) = \alpha$, $\beta(t) = \beta$ and $\sigma^2(t) = \sigma^2$, we have:

with $x_0\wedge x = \min(x_0, x)$ and $x_0\vee x = \max(x_0, x)$. Moreover, taking the LT in Eq. (A.1) one has:

where ${\cal I}(x_0) = 1$ for $x_0 = 0$ and ${\cal I}(x_0) = 0$ for $x_0 > 0$. Therefore, making use of (A.4) and (A.5) in (A.3) and recalling the following relation

one obtains (4.13).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is partially supported by MIUR - PRIN 2017, project "Stochastic Models for Complex Systems", no. 2017JFFHSH. The authors are members of the research group GNCS of INdAM.

Conflict of interest

The authors declare there is no conflict of interest.