Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance

Tülay Yazır; Aslı Bektaş Kamışlık; Tahir Khaniyev; Zulfiye Hanalioglu

doi:10.1515/dema-2025-0153

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Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance

Tülay Yazır , Aslı Bektaş Kamışlık , Tahir Khaniyev and Zulfiye Hanalioglu

Published/Copyright: July 30, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This study investigates the renewal reward process under the assumption that the random variables describing the discrete interference of chance follow a generalized gamma distribution. A moment-based approximation method is employed to derive novel results for the renewal function, enabling an approximation of the ergodic distribution of the process. Furthermore, the limiting distribution of the ergodic distribution is also derived. The theoretical findings are illustrated through a specific example in which the demand random variable η 1 is represented by a third-order Erlang distribution with parameter θ = 1 .

Keywords: moment-based approximation; ergodic distribution; renewal reward process; generalized gamma distribution

MSC 2010: 60K05; 60K15

1 Introduction

In practical applications, the most commonly utilized models for random recurrent events are renewal processes, renewal reward processes, and random walk processes. These processes can be used to model and solve a number of interesting problems in diverse fields, for example, in establishing the optimal maintenance schedule of a system [1], modeling new purchases following the expiration of the warranty period in warranty analysis [2–5], modeling the time between consecutive demands in inventory models [6], planning of supply chain [7], continuous sampling plans [8], and modeling sequential risks in risk theory [9]. For these stochastic processes, many properties and results are presented in the literature [10–17]. Despite the significance of these studies, mathematical challenges are encountered when calculating exact formulas for the ergodic distributions of the processes involved in the research. To address real-world problems effectively, it is preferable to utilize expressions that do not involve complex mathematical structures. To overcome this challenge, studies in the literature have focused on deriving asymptotic expansions and approximate formulas for key characteristics of the processes. For example, one of the most well-known asymptotic expansions for the renewal function was proposed by Feller [11]. Feller proposed the following asymptotic expansion for the renewal function U(x)

U ( x ) = E ( N ( x ) ) = x μ 1 + μ 2 2 μ 1 2 + o ( 1 ) ,

where N ( x ) = inf { n ≥ 1 : Y n > x } , x ≥ 0 , is renewal process generated by the i.i.d. random variables X i , and

Y n = ∑ i = 1 n X i , n = 1 , 2 , …

Feller’s work has facilitated the derivation of various asymptotic expansions and approximation formulas for different characteristics of renewal, renewal reward, and random walk processes [18–21]. In addition to these studies, Brown and Solomon [22] obtain the approximation Var ( C ( t ) ) = c t + d + o ( 1 ) , where { C ( t ) , t ≥ 0 } is a renewal reward process. Csenki [23] derived an asymptotic representation of the form C ( t ) = ξ t + η + o ( 1 ) , t → ∞ for the renewal reward process C ( t ) with retrospective reward structure.

In this study, a semi-Markovian renewal reward process with generalized gamma-distributed interference of chance is investigated, and an approximation formula for the ergodic distribution of the process is derived. The literature contains a substantial body of research on similar processes and their different characteristics [24–32].

Studies carried out in the literature so far have enabled the derivation of significant mathematical results; however, the existing literature has some critical limitations that could be restrictive in various practical applications. First, in many studies conducted in the literature, the asymptotic expansion method has been utilized as a mathematical tool. The most significant limitation of this approach is that, in the results obtained through the asymptotic expansion method, terms beyond the first two or three are typically represented using big-O or little-o notation due to their diminishing significance compared to the first few terms. In asymptotic expansions, although the initial terms are more significant, the lack of a precise functional form for the way the remaining terms approach zero leads to a substantial loss of information. To address this gap, we derived a new approximation formula for the ergodic distribution of the process X ( t ) , which clearly illustrates how the remaining terms approach zero.

Moreover, the renewal function produced by the demand random variables serves as the primary tool for obtaining the characteristics of the process analyzed in this study. The existing literature generally assumes that the distribution function F ( t ) of the demand random variable underlying the renewal process is known. For many classes of distributions, it is not possible to obtain the renewal function. In cases where the renewal function can be derived, its mathematical structure is often too complex for practical use. However, in many applications, calculating the moments of the random variables is relatively straightforward. Therefore, the approximation formulas obtained in this study are derived solely by calculating the first three moments of demand random variables, without the need for a closed-form expression of the distribution function.

Finally, in many studies, specific distributions have been used for the random variables representing interference of chances. On the other hand, generalized gamma distribution is a flexible probability distribution that extends the gamma distribution to include an additional shape parameter. The generalized gamma distribution can take the form of different distributions such as the Weibull, Rayleigh, Maxwell, Nakagami, Frechet, Gumbel, and gamma depending on its parameters. A useful and important property of the generalized gamma distribution is that many common and interesting probability distributions occur as special cases or limits [33]. Therefore, this study addresses a situation where the interference of chance random variables belongs to a very broad class of distributions instead of a specific one.

As previously stated, this article aims to derive a moment-based approximation for the ergodic distribution of the renewal reward process under the assumption of generalized gamma-distributed interference of chance. Another objective of this study is to obtain weak convergence of ergodic distribution of the process under suitable assumptions. Our approximation procedure is based on the findings of Kambo et al. [34]. In this study, the approximation formula for the renewal function given with (1.1) will be referred as to Kambo sense approximation.

(1.1) U ( t ) = t μ 1 + ( μ 2 − 2 μ 1 2 ) ( 1 − e s 0 t ) 2 μ 1 .

Here, s 0 = 6 μ 1 ( μ 2 − 2 μ 1 2 ) 3 μ 2 2 − 2 μ 1 μ 3 , μ k = E ( η 1 k ) ; k = 1 , 2 , 3 and η 1 is the underlying random variable of the renewal process. Note that approximation (1.1) is is based only on the on the first three moments of demand random variable η 1 .

The remainder of the article is organized as follows. In Section 2, we introduce the preliminaries and provide a mathematical model of the process. Moreover, Section 2 presents the ergodicity of the process X ( t ) and derives exact expressions for the ergodic distribution. Section 3 develops moment-based approximations for the ergodic distribution of the process. In Section 4, we provide a specific example, illustrating the approximation method by assuming the demand random variable follows a third-order Erlang distribution and the interference of chance random variable follows a generalized gamma distribution. Finally, Section 5 concludes with a summary of the findings and potential directions for future work.

To establish the foundation for our study, we first present the necessary preliminaries, including key properties related to the renewal reward process and the moment-based approximation method.

2 Preliminaries

It is first necessary to introduce the essential notations and provide a mathematical explanation of this model before embarking upon an analysis of the main problem.

2.1 The model

Let X ( t ) represent the inventory level in a depot at time t . It is assumed that X ( t ) at time t = 0 is given by λ z > 0 , for arbitrarily chosen positive constant λ . Let η i , i = 1 , 2 , 3 … represent demand random variables. We suppose that the system receives a random demand of magnitude η i at each random time T i . Therefore, the initial inventory level of λ z in the system decreases by η i at each random time T i . Let ξ i be a random variable representing the interarrival times between consecutive demands. X ( t ) represent the inventory level in a depot at time t . It is assumed that X ( t ) at time t = 0 is given by λ z > 0 , for arbitrarily chosen positive constant λ . Let η i , i = 1 , 2 , 3 … represents demand random variables. We suppose that the system receives a random demand of magnitude η i at each random time T i . Therefore, the initial inventory level of λ z in the system decreases by η i at each random time T i . Let ξ i be a random variable representing the interarrival times between consecutive demands. In this case, we can define T n as follows: T n = ∑ i = 1 n ξ i . Hence, the change in inventory within the system during the first period will be as follows:

X ( T 1 ) ≡ X 1 = λ z − η 1 , … , X ( T n ) ≡ X n = λ z − ∑ i = 1 n η i .

The decrease in inventory continues until a random time τ 1 . The inventory level first falls below zero at time τ 1 , and this marks the end of the system’s first period. As soon as the inventory level in the system falls below zero, a stock quantity of λ ζ 1 is immediately added to the system. Thus, the system begins its second period with a stock quantity of λ ζ 1 and proceeds in a similar manner to the first period. The stock quantity, initially set at λ ζ 1 , continues to decrease until it falls below zero at a random time τ 2 .

2.2 Mathematical construction of the process

Let { ξ n } , { η n } , and { ζ n } , for n ≥ 1 , be three sequences of random variables defined on a probability space ( Ω , ℱ , P ) , with the variables in each sequence being independent and identically distributed. Suppose that { ξ n } , { η n } , and { ζ n } take only non-negative values. Denote the distribution functions of { ξ n } , { η n } , and { ζ n } by Φ ξ ( t ) , F η ( x ) , and F ζ ( z ) , as follows, respectively:

Φ ξ ( t ) ≡ P { ξ 1 ≤ t } , t ≥ 0 , F η ( x ) ≡ P { η 1 ≤ x } , x ≥ 0 , F ζ ( z ) ≡ P { ζ 1 ≤ z } , z ≥ 0 .

Define the renewal sequences { T n } and { Y n } as follows:

T n = ∑ i = 1 n ξ i , Y n = ∑ i = 1 n η i , n ≥ 1 , T 0 = Y 0 = 0 , n = 1 , 2 , …

Let { N n } , for n ≥ 0 , be a sequence of integer-valued random variables defined as follows:

N 0 = 0 , N 1 = N 1 ( λ z ) = inf { k ≥ 1 : λ z − Y k < 0 } ,

N n + 1 ≡ N n + 1 ( λ ζ n ) = inf { k ≥ N n + 1 : λ ζ n − ( Y k − Y N n ) < 0 } , n ≥ 1 .

Moreoever, let inf ( ∅ ) = + ∞ be stipulated. Define the times τ n at which the inventory level falls below zero as follows:

τ 0 = 0 , τ 1 = τ 1 ( λ z ) = T N 1 = ∑ i = 1 N 1 ( λ z ) ξ i ; τ n = T N n = ∑ i = 1 N n ξ i ; for n ≥ 2 .

Next, define the counting process ν ( t ) as follows:

ν ( t ) = max { n ≥ 0 : T n ≤ t } , t > 0 .

In this formulation, ν ( t ) counts the number of events that have occurred by time t .

Finally, the considered process can be defined as follows:

(2.1) X ( t ) = λ ζ n − ( Y ν ( t ) − Y N n ) ; for τ n ≤ t < τ n + 1 , n = 0 , 1 , 2 , … ,

where Y ν ( τ n ) = Y N n , ζ 0 = z > 0 , and λ > 0 . A sample path of the process X ( t ) is presented in Figure 1.

$Figure 1 A trajectory of the process X ( t ) X\left(t) .$

Figure 1

A trajectory of the process X ( t ) .

Note that λ ζ n denote the initial stock level at the beginning of the n th period. We assume that the random variables ζ n follow the generalized gamma distribution with the following p.d.f.:

f ζ ( z ; a , d , p ) = p a d Γ d p z d − 1 exp − z a p , z ≥ 0 , a , d , p > 0 ,

where Γ ( z ) is gamma function and is given by the integral:

Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t , Re ( z ) > 0 .

The aim of this study is to propose new approximation formulas for the ergodic distribution of the process X ( t ) defined by (2.1). To achieve this goal, propositions related to the ergodicity of the process will be presented in Subsection 2.3.

2.3 The ergodicity of the process X ( t ) and exact expressions for the ergodic distribution

The calculation of finite-dimensional distributions of the process X ( t ) is highly intricate due to its complex mathematical structure. The calculation of stationary characteristics may resolve this difficulty, which is why we will investigate the ergodic distribution of the process X ( t ) . To this end, we will first demonstrate that the process is ergodic under certain conditions. Let Q X ( x ; λ ) denote the ergodic distribution function of the process X ( t ) , defined as follows:

Q X ( x ; λ ) ≡ lim t → ∞ P { X ( t ) ≤ x } , x ≥ 0 .

We will use the following propositions to establish the ergodicity of the process X ( t ) .

Proposition 2.1

[35] To establish the ergodicity of the process X ( t ) , we impose the following conditions on the initial sequences of random variables { ξ n } , { η n } , and { ζ n } , for n ≥ 1 :

0 < E ( ξ 1 ) < ∞ ;
E ( η 1 ) > 0 and η 1 is non-arithmetic;
E ( η 1 k ) = μ k < ∞ for k = 1 , 2 , 3 ;
ζ 1 follows a generalized gamma distribution with the following pdf:
f ζ ( z ; a , d , p ) = p a d Γ d p z d − 1 exp − z a p , z ≥ 0 , a , d , p > 0 .

Under these conditions, the process X ( t ) is ergodic.

Proposition 2.2

[35] Assume the conditions in Proposition 2.1 be satisfied. Then for every bounded and measurable function f ( x ) (where f : ( 0 , ∞ ) → R ) the following relationship holds with probability 1:

lim t → ∞ 1 t ∫ 0 t f ( X ( u ) ) d u = ∫ 0 ∞ ∫ 0 ∞ f ( v ) [ U η ( λ z ) − U η ( λ z − v ) ] d F ζ ( z ) d v ∫ 0 ∞ U η ( λ z ) d F ζ ( z ) ,

where U η ( x ) denotes the renewal function generated by the random variables { η n } , n = 1 , 2 , … . The renewal function is defined as follows:

U η ( x ) = ∑ n = 0 ∞ F η * n ( x ) ,

where F η * n ( x ) represents the n-fold convolution of the distribution function F η ( x ) .

Corollary (2.1) provides the exact expression for ergodic distribution Q X ( x ; λ ) .

Corollary 2.1

Assume the conditions of Proposition 2.1 be satisfied. Then the exact formula for the ergodic distribution Q X ( x ; λ ) of the process X ( t ) can be expressed as follows:

(2.2) Q X ( x ; λ ) = 1 − E ( U η ( λ ζ 1 − x ) ) E ( U η ( λ ζ 1 ) ) , x ≥ 0 ,

where, U η ( x ) is the renewal function associated with the sequence { η n } and E ( U η ( x ) ) is the expected value of U η ( x ) .

The necessity of the renewal function for the calculation of the precise formula of the ergodic distribution of the process X ( t ) is evident from equation (2.2). Exact expressions for the renewal function can only be derived when the sequence of random variables follows simple distributions, such as exponential distribution, and Erlang distribution. For many distributions, finding exact and closed-form expressions for the renewal function is highly challenging, and the resulting expressions are often complex. Consequently, numerous studies have been conducted using various approaches. These studies have led to the derivation of asymptotic expansions and bounds [11,36–38]. The findings of this study build upon the work of Kambo et al. [34], where a moment-based approximation for the renewal function is proposed. More specifically, let η 1 , η 2 , … , η n represent independent and identically distributed (i.i.d.) non-negative random variables with a common distribution. Define Y 0 = 0 and Y n = η 1 + η 2 + … + η n for n ≥ 1 . The following random variable,

N ( t ) = sup { n : Y n ≤ t } ,

was introduced by Smith [10] for t ≥ 1 with Y 0 = 0 . The work of Kambo et al. [34] focuses on the well-known renewal function M ( t ) = E ( N ( t ) ) for t ≥ 0 . The approximation proposed by Kambo et al. [34] is presented in Proposition 2.3 as follows:

Proposition 2.3

[34] Let Y 0 = 0 and Y n = η 1 + η 2 + … + η n , for n ≥ 1 . Furthermore, assume that the first three raw moments of η i exist and are known. The renewal function M ( t ) = E ( N ( t ) ) can then be approximated as follows:

(2.3) M ( t ) ≈ t μ 1 + μ 2 − 2 μ 1 2 2 μ 1 2 ( 1 − e s 0 t ) ,

where

s 0 = 6 μ 1 ( μ 2 − 2 μ 1 2 ) 3 μ 2 2 − 2 μ 1 μ 3 .

Note that the approximation (2.3) is valid only when s 0 < 0 , [34]. In addition, the sequence { η n } , n = 1 , 2 , … consists of demand random variables where μ n = E ( η 1 n ) , n = 1 , 2 , 3 .

In this study, for the validity of the approximation method employed, it is necessary that the condition s 0 < 0 be satisfied for the random variables generating the renewal function (in this process, this corresponds to the demand random variables). The condition that s 0 is nonpositive, is satisfied by many standard distributions, including the uniform, gamma, Erlang, lognormal, E k − 1 , k (for k ≥ 2 ), and Weibull distributions. In addition, s 0 is non-positive for truncated normal distribution and inverse Gaussian distribution under the specified conditions [34].

Consider that if we define

Y n = ∑ i = 1 n η i ,

the renewal process generated by the demand random variable can be expressed as follows:

N η ( z ) = inf { n ≥ 1 : Y n > z } , z ≥ 0 .

Let U η ( z ) = E ( N η ( z ) ) . This expression is presented by Feller [11] as follows:

U η ( z ) = ∑ n = 0 ∞ F η * n ( z ) ,

where F η * n ( z ) represents the n -fold convolution of F η for F η ( x ) ≡ P { η 1 ≤ x } . The following relationship between N ( z ) = sup { n : Y n ≤ z } given by Smith [10] and N η ( z ) = inf { n ≥ 1 : Y n > z } given by Feller [11] is well established, holding with probability 1:

N η ( z ) = N ( z ) + 1 ,

and

U η ( z ) ≡ E ( N η ( z ) ) = E ( N ( z ) ) + 1 .

Thus, on the basis of the results of Kambo’s findings, presented in Proposition 2.3 with approximation (2.3), we derive the following approximation for the renewal function:

(2.4) U η ( z ) ≈ z μ 1 + μ 2 2 μ 1 2 − e s 0 z ( μ 2 − 2 μ 1 2 ) 2 μ 1 2 ,

where s 0 = 6 μ 1 ( μ 2 − 2 μ 1 2 ) 3 μ 2 2 − 2 μ 1 μ 3 and μ n = E ( η 1 n ) , n = 1 , 2 , 3 .

Let us now derive new approximation formulas for the ergodic distribution of the process.

3 Moment-based approximations for the ergodic distribution of the process X ( t )

The main aim of this section is to obtain moment-based approximation for the ergodic distribution of the process X ( t ) . In the previous sections of our study, exact formulas for the ergodic distribution of the process X ( t ) have been derived. However, the practical application of these exact expressions for the ergodic distribution presents numerous challenges. The primary difficulty lies in the fact that the exact form of Q X ( x ; λ ) has a mathematically complex structure. A practical way to overcome this difficulty is to obtain an asymptotic expansion for Q X ( x , λ ) , as λ → ∞ . In this section, an approximation for Q X ( x ; λ ) has been derived based on the moments of demand random variables η n and based on interference of chance random variables ζ n . However, before applying the approximation, it is necessary to standardize the ergodic process X ( t ) . Let us define the process Y ( t ) , which is a linear transformation of the X ( t ) as follows:

Y ( t ) ≡ X ( t ) λ .

In this case for each y ∈ ( 0 , ∞ ) , the ergodic distribution function Q Y ( y ; λ ) of the process Y ( t ) is given by:

(3.1) Q Y ( y ; λ ) ≡ lim t → ∞ P { Y ( t ) ≤ y } = lim t → ∞ P X ( t ) λ ≤ y = lim t → ∞ P { X ( t ) ≤ λ y } = 1 − E ( U η ( λ ζ 1 − λ y ) ) E ( U η ( λ ζ 1 ) ) .

In this study, approximation (2.4) will be employed for the renewal function generated by demand random variables. In addition, it will be assumed that the p.d.f. of the random variables representing the interference of chance follows the generalized gamma distribution provided below.

(3.2) f ζ ( z ; a = 1 , d , p ) = p Γ ( d ⁄ p ) z d − 1 exp ( − z p ) , z ≥ 0 , d , p > 0 .

In this section, two propositions and their proofs will be provided before obtaining the moment-based approximation for the ergodic distribution of the standardized process Y ( t ) .

Proposition 3.1

Let the conditions of Propositions 2.1 and 2.3 be satisfied. Moreover, let us define

(3.3) E ( U η ( λ ζ 1 − λ y ) ) ≡ ∫ z = y ∞ U η ( λ z − λ y ) f ζ ( z ) d z .

Let y > 0 , z ≥ 0 , d , p > 0 , and λ is sufficiently large. Then an approximation in the sense of Kambo for E ( ( U η ( λ ζ 1 − λ y ) ) ) can be given as follows:

(3.4) E ( ( U η ( λ ζ 1 − λ y ) ) ) ≈ d 1 μ 1 p Γ d + 1 p , y p − y Γ d p , y p λ + d 1 c 1 p Γ d p , y p + d 1 c 2 y d − 1 s 0 ( y d − 1 e − y p ) λ − 1 .

Here,

c 1 = μ 2 2 μ 1 2 , c 2 = μ 2 − 2 μ 1 2 , d 1 = p Γ ( d ⁄ p ) , , s 0 = 6 μ 1 ( μ 2 − 2 μ 1 2 ) 3 μ 2 2 − 2 μ 1 μ 3 .

Moreover, Γ ( s , x ) is upper incomplete gamma function defined with following integral:

(3.5) Γ ( s , x ) = ∫ x ∞ t s − 1 exp ( − t ) d t

for Re ( s ) > 0 , x ≥ 0 .

Proof

According to Proposition 2.3, the following Kambo sense approximation can be written, when λ is sufficiently large:

(3.6) U η ( λ z − λ y ) ≈ λ ( z − y ) μ 1 + c 1 − c 2 e s 0 ( λ ( z − y ) ) , y < z .

Then we obtain the following approximation for E ( ( U η ( λ ζ 1 − λ y ) ) ) :

(3.7) E ( ( U η ( λ ζ 1 − λ y ) ) ) ≡ ∫ z = y ∞ U η ( λ z − λ y ) f ζ ( z ) d z ≈ ∫ z = y ∞ d 1 λ ( z − y ) μ 1 + c 1 − c 2 e s 0 ( λ ( z − y ) ) z d − 1 e − z p d z = ∫ t = 0 ∞ d 1 λ t μ 1 + c 1 − c 2 e s 0 λ t ( y + t ) e − ( t + y ) p d t = I 1 ( y ) + I 2 ( y ) − I 3 ( y ) .

For each y ≥ 0 , z ≥ 0 , and d , p > 0 , we have:

(3.8) I 1 ( y ) ≡ d 1 μ 1 ∫ t = 0 ∞ λ t ( t + y ) d − 1 e − ( t + y ) p d t = d 1 λ p μ 1 ∫ u = y p ∞ u d − p + 1 p e − u d u − ∫ u = y p ∞ y u d − p p e − u d u = d 1 μ 1 p Γ d + 1 p , y p − y Γ d p , y p λ .

(3.9) I 2 ( y ) ≡ d 1 c 1 ∫ t = 0 ∞ ( t + y ) d − 1 e − ( t + y ) p d t = d 1 c 1 p ∫ u = y p ∞ u d − p p e − u d u = d 1 c 1 p Γ d p , y p .

On the other hand for y > 0 and s 0 < 0 , we have:

(3.10) I 3 ( y ) ≡ c 2 d 1 ∫ t = 0 ∞ e s 0 λ t ( t + y ) d − 1 e − ( t + y ) p d t ≈ d 1 c 2 y d − 1 e − y p ∫ t = 0 ∞ e s 0 λ t d t = − d 1 c 2 y d − 1 e − y p s 0 λ − 1 .

Substituting (3.8), (3.9), and (3.10) into the (3.7), desired result holds.□

Proposition 3.2

Let the conditions of Propositions 2.1 and 2.3 be satisfied. For each y > 0 , z ≥ 0 , and d , p > 0 let us define

(3.11) E ( U η ( λ ζ 1 ) ) ≡ ∫ z = 0 ∞ U η ( λ z ) f ζ ( z ) d z .

Then, for λ is sufficiently large, an approximation for E ( ( U η ( λ ζ 1 ) ) ) is obtained as follows:

(3.12) E ( U η ( λ ζ 1 ) ) ≈ d 1 μ 1 p Γ d + 1 p λ + d 1 c 1 p Γ ( d p ) + d 1 c 2 ( − 1 ) d Γ ( d ) ( s 0 ) d λ − d .

Here,

c 1 = μ 2 2 μ 1 2 , c 2 = μ 2 − 2 μ 1 2 , d 1 = p Γ ( d ⁄ p ) , s 0 = 6 μ 1 ( μ 2 − 2 μ 1 2 ) 3 μ 2 2 − 2 μ 1 μ 3 ,

and Γ ( z ) is gamma function.

Proof

According to Proposition 2.3 following Kambo sense approximation can be written, when λ is sufficiently large:

(3.13) U η ( λ z ) ≈ λ z μ 1 + c 1 − c 2 e s 0 ( λ z ) .

Then by using the Taylor series expansion of e − z p , we obtain following approximation for E ( U η ( λ ζ 1 ) ) as follows:

(3.14)□ E ( U η ( λ ζ 1 ) ) ≡ ∫ z = 0 ∞ U η ( λ z ) f ζ ( z ) d z ≈ d 1 ∫ z = 0 ∞ λ z d μ 1 e − z p d z + d 1 ∫ z = 0 ∞ c 1 z d − 1 e − z p d z − d 1 c 2 ∫ z = 0 ∞ e s 0 ( λ z ) z d − 1 e − z p d z ≈ d 1 λ p μ 1 ∫ u = 0 ∞ u d + 1 p − 1 e − u d u + d 1 c 1 p ∫ u = 0 ∞ u d p − 1 e − u d u + d 1 c 2 ( − 1 ) d s 0 ∫ u = 0 ∞ e − u u n − 1 d u λ − d = d 1 μ 1 p Γ d + 1 p λ + d 1 c 1 p Γ ( d p ) + d 1 c 2 ( − 1 ) d Γ ( d ) ( s 0 ) d λ − d .

The primary objective of this study is to derive an approximation for the ergodic distribution of the standardized process Y ( t ) . By utilizing Propositions 3.1 and 3.2, we now present the following theorem, which represents the central purpose of this research.

Theorem 3.1

Let the conditions of Propositions 2.1 and 2.3 be satisfied. Then for each y > 0 , the following approximation is obtained for ergodic the distribution Q Y ( y ; λ ) of the process Y ( t ) , assuming that λ is sufficiently large:

(3.15) Q Y ( y ; λ ) ≈ 1 − A 1 ( y ) B 1 + B 2 A 1 ( y ) − B 1 A 2 ( y ) B 1 2 λ − 1 + B 1 B 2 A 2 ( y ) − B 2 2 A 1 ( y ) − A 3 ( y ) B 1 2 B 1 3 λ − 2 + B 3 A 1 ( y ) B 1 2 λ − ( d + 1 ) .

Here,

A 1 ( y ) = d 1 μ 1 p Γ d + 1 p , y p − d 1 y μ 1 p Γ d p , y p ; A 2 ( y ) = d 1 c 1 p Γ d p , y p ; A 3 ( y ) = − c 2 d 1 y d − 1 e − y p s 0 ; B 1 = d 1 μ 1 p Γ d + 1 p ; B 2 = d 1 c 1 p Γ d p ; B 3 = d 1 c 2 ( − 1 ) d Γ ( d ) s 0 d .

Proof

The exact formula for ergodic distribution of the process X ( t ) is given with (3.1); moreover, approximations for E ( U η ( λ ζ 1 − λ y ) ) and E ( U η ( λ ζ 1 ) ) were given with (3.4) and (3.12), respectively. Taking into account that

Q Y ( y ; λ ) = 1 − E ( U η ( λ ζ 1 − λ y ) ) E ( U η ( λ ζ 1 ) ) ,

and Taylor series expansion, for all y > 0 , we have

(3.16)□ Q Y ( y ; λ ) ≈ 1 − A 1 ( y ) λ + A 2 ( y ) + A 3 ( y ) λ − 1 B 1 λ + B 2 + B 3 λ − d = [ B 1 − A 1 ( y ) ] λ + [ B 2 − A 2 ( y ) ] − A 3 ( y ) λ − 1 + B 3 λ − d λ B 1 1 + B 2 B 1 λ − 1 + B 3 B 1 λ − ( d + 1 ) ≈ 1 − A 1 ( y ) B 1 + B 2 − A 2 ( y ) B 1 λ − 1 − A 3 ( y ) B 1 λ − 2 − B 3 B 1 λ − ( d + 1 ) ⋅ 1 − B 2 B 1 λ − 1 + B 2 2 B 1 2 λ − 2 − B 3 B 1 λ − ( d + 1 ) ≈ 1 − A 1 ( y ) B 1 + B 2 A 1 ( y ) − B 1 A 2 ( y ) B 1 2 λ − 1 + B 1 B 2 A 2 ( y ) − B 2 2 A 1 ( y ) − A 3 ( y ) B 1 2 B 1 3 λ − 2 + B 3 A 1 ( y ) B 1 2 λ − ( d + 1 ) .

By using Theorem 3.1, we derive an approximation formula for the ergodic distribution of the process Y ( t ) under specified conditions. An additional notable property of the process Y ( t ) is its limiting distribution. Previous studies suggest that the distribution function of the interference of the chance random variable serves as the principal determinant of the limiting distribution. Corollary 3.1 serves as the weak convergence for ergodic distribution and provides limit distribution.

Corollary 3.1

Let the conditions of Theorem 3.1 be satisfied. In this case, for each y > 0 , the ergodic distribution function Q Y ( y ; λ ) converges to the function π ( y ) weakly as λ → ∞ . That is,

Q Y ( y ; λ ) → π ( y ) ,

where π ( y ) is given as follows:

(3.17) π ( y ) = 1 − Γ d + 1 p , y p − y Γ d p , y p Γ d + 1 p .

Proof

Since y > 0 , z ≥ 0 and d , p > 0 convergence of Q Y ( y ; λ ) follows easily from (3.16) as λ → ∞ .□

To further explore the relationship between the ergodic distribution and the underlying renewal process, we examine the connection between the limit distribution of the process Y ( t ) and the residual waiting time of the renewal process generated by the interference random variables ζ n . This relationship is formalized in the following corollary.

Corollary 3.2

Let the conditions of Theorem 3.1 be satisfied. Moreover, let us denote the limit distribution for the residual waiting time of the renewal process generated by the interference of chance random variables ζ n , n = 1 , 2 , … with Π 0 ( y ) . Then Π 0 ( y ) defined as follows:

Π 0 ( y ) = 1 m 1 ∫ 0 y ( 1 − F ζ ( z ) ) d z ,

where F ζ 1 ( t ) = P ( ζ 1 ≤ t ) and m 1 = E ( ζ 1 ) . In this case, the limit distribution π ( y ) obtained in Theorem 3.1 and the residual waiting time of the renewal process generated by the interference of chance random variables Π 0 ( y ) coincide, i.e.,

(3.18) π ( y ) = Π 0 ( y ) .

Proof

Since the random variable ζ 1 follows a generalized Gamma distribution, the first moment of the ζ 1 and the tail of its distribution function are given as follows, respectively:

(3.19) m 1 = E ( ζ 1 ) = Γ d + 1 p Γ d p , 1 − F ζ 1 ( y ) = Γ d p , y p Γ d p .

Taking into account that

Π 0 ( y ) = 1 m 1 ∫ 0 y ( 1 − F ζ 1 ( z ) ) d z ,

the probability density function of Π 0 ( y ) can be written as follows:

(3.20) ∂ ( Π 0 ( y ) ) ∂ y = 1 − F ζ 1 ( y ) m 1 = Γ d p , y p Γ d + 1 p .

On the other hand, for all a > 0 and x ≥ 0 , we have

(3.21) ∂ ∂ x Γ ( a , x ) = − x a − 1 e − x ( [39]; p. 262 ) .

By using (3.21), we obtain probability density function of π ( y ) as follows:

(3.22) ∂ ∂ y π ( y ) = ∂ ∂ y 1 − Γ d + 1 p , y p − y Γ d p , y p Γ d + 1 p = Γ d p , y p Γ d + 1 p .

Since (3.20) and (3.22) are equal, we can conclude that probability densities of π ( y ) and Π 0 ( y ) are equal, π ( y ) and Π 0 ( y ) are coincide.□

To illustrate the practical application of the theoretical framework, we now provide a concrete example that demonstrates the derivation and approximation of the ergodic distribution of the process Y ( t ) .

4 Example

In this section, we provide a specific example to examine the approximation of the ergodic distribution of the process Y ( t ) . In this example, we assume that the demand random variable η 1 follows a third-order Erlang distribution with parameter θ = 1 , ( η 1 ∼ Erlang ( n = 3 , θ = 1 ) ) . In other words, the probability density function of η 1 is given by

f η ( x ) = x 2 2 e − x , x ≥ 0 .

For a third-order Erlang distribution with parameter θ = 1 , the moments μ n = E ( X n ) , n = 1 , 2 , 3 can be derived as follows:

μ 1 = E ( η 1 ) = 3 , μ 2 = E ( η 1 2 ) = 12 , μ 3 = E ( η 1 3 ) = 60 .

Then by substituting these values into the expression for s 0 and simplify, we obtain:

s 0 = 6 μ 1 ( μ 2 − 2 μ 1 2 ) 3 μ 2 2 − 2 μ 1 μ 3 = − 3 2 .

In addition, let us assume that the distribution of the random variable ζ 1 is a specific case of the generalized gamma distribution with parameters p = 1 and d = 2 . In this case,

f ζ ( z ; d = 2 , p = 1 ) = 1 Γ ( 2 ) z e − z = z e − z .

Under these assumptions, the coefficients of the approximation for the ergodic distribution, given by (3.15), are calculated as follows, respectively:

(4.1) 1 − A 1 ( y ) B 1 = 1 + 0.5 y Γ ( 2 , y ) − 0.5 Γ ( 3 , y ) .

(4.2) B 2 A 1 ( y ) − B 1 A 2 ( y ) B 1 2 = − 0.5 y Γ ( 2 , y ) − Γ ( 2 , y ) + 0.5 Γ ( 3 , y )

(4.3) B 1 B 2 A 2 ( y ) − B 2 2 A 1 ( y ) − A 3 ( y ) B 1 2 B 1 3 = 0.5 y Γ ( 2 , y ) + 6 y e − y + Γ ( 2 , y ) − 0.5 Γ ( 3 , y )

(4.4) B 3 A 1 ( y ) B 1 2 = 2 y Γ ( 2 , y ) − 2 Γ ( 3 , y ) .

By substituting the coefficients obtained above into the required places in the approximation (3.15), we obtain the following approximation for the ergodic distribution Q Y ( y ; λ ) :

(4.5) Q Y ( y ; λ ) ≈ 1 + 0.5 y Γ ( 2 , y ) − 0.5 Γ ( 3 , y ) + [ − 0.5 y Γ ( 2 , y ) − Γ ( 2 , y ) + 0.5 Γ ( 3 , y ) ] λ − 1 + [ 0.5 y Γ ( 2 , y ) + 6 y e − y + Γ ( 2 , y ) − 0.5 Γ ( 3 , y ) ] λ − 2 + [ 2 y Γ ( 2 , y ) − 2 Γ ( 3 , y ) ] λ − 3 .

Note that the upper incomplete gamma function, Γ ( s , y ) , has asymptotic approximations for s > 0 as y → 0 , which is given by

Γ ( s , y ) ≈ Γ ( s ) − y s s + O ( y s + 1 ) , for s > 0 , ([40]; p. 251) .

By applying this approximation for Γ ( 2 , y ) and Γ ( 3 , y ) , we obtain the following simplified form of Q Y ( y ; λ ) as y → 0 .

Q Y ( y ; λ ) ≈ 0.5 y − 0.25 y 2 + ( 0.5 y + 0.25 y 2 ) λ − 1 + ( 5.5 y − 6.25 y 2 ) λ − 2 + ( 2 y − y 2 − 4 ) λ − 3 .

On the other hand, the upper incomplete gamma function, Γ ( s , y ) , has the following asymptotic behavior for large y :

Γ ( s , y ) ≈ y s − 1 e − y 1 + s − 1 y + O ( y − 2 ) as y → ∞ ( [39]; p. 263 ) .

By using this for Γ ( 2 , y ) and Γ ( 3 , y ) , we obtain the following simplified form of Q Y ( y ; λ ) , as y → ∞

Q Y ( y ; λ ) ≈ 1 + ( − y e − y ) λ − 1 + ( 7 y e − y ) λ − 2 .

Having developed the moment-based approximation for the ergodic distribution, we now summarize the key findings and discuss potential avenues for future research.

5 Conclusion

In this study, we propose a novel approach for approximating the ergodic distribution of a renewal reward process with generalized gamma-distributed interference of chance. Building upon classical models such as renewal, renewal reward, and random walk processes, which have demonstrated wide applicability in maintenance scheduling, inventory management, and risk assessment, this work addresses several limitations found in the existing literature.

Previous studies commonly rely on asymptotic expansions, where terms beyond the initial few are expressed in big-O or little-o notation, leading to a significant loss of detail regarding the decay behavior of the remainder terms. By contrast, we derive a moment-based approximation for the ergodic distribution that explicitly characterizes the decay of the remainder terms, thereby enhancing both interpretability and practical applicability. Furthermore, our methodology circumvents the need for closed-form solutions for the distribution function of the renewal process, which are often complex or unattainable, by focusing only on the first three moments of the demand random variables.

In addition, by incorporating generalized gamma distributions for the interference of chance variables, our model extends to a broad range of practical distributions, including Weibull, Frechet, Maxwell, Rayleigh, and so on as special cases. This flexibility supports a wider application scope and provides a more versatile framework for modeling real-world stochastic processes. We also confirmed weak convergence of the ergodic distribution under suitable conditions, further validating our approximation.

Overall, our results provide a more accessible and accurate tool for practitioners, allowing for effective modeling of complex renewal reward processes in diverse fields without the need for intricate mathematical structures or restrictive distributional assumptions.

Future research may extend this moment-based approximation framework to the renewal reward process or semi-Markovian random walks where the interference of chance random variable ζ 1 follows heavy-tailed distributions, such as Pareto or Cauchy distributions. Such extensions would be particularly valuable in modeling extreme phenomena, including financial risk and environmental hazards.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. Tahir Khaniyev and Zulfiye Hanalioglu contributed to the construction of the model and assisted in the proofs of the theorems. Tülay Yazır and Aslı Bektaş Kamışlık applied the moment-based approach to the proposed model, carried out the necessary computations, and completed the proofs of the theorems. Aslı Bektaş Kamışlık prepared the manuscript with contributions from all co-authors.
Conflict of interest: The authors state no conflict of interest.

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Received: 2024-12-03

Revised: 2025-04-30

Accepted: 2025-06-02

Published Online: 2025-07-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2025-0153

Keywords for this article

moment-based approximation; ergodic distribution; renewal reward process; generalized gamma distribution

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