Renewal reward process 4 Renewal Reward Processes Let fN(t);t 0gbe a renewal process with i. (Proc. Feb 1, 2023 · The cumulative reward by the time t ≥ 0 is the random variable W t ≔ ∑ i ≥ 1 X i 1 {T i ≤ t}, which is measurable because X is separable [5]. The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. An Up- and Downcrossing Technique. Viewed 864 times 1 $\begingroup$ Consider a train station to which Renewal process was a generalization of Poisson process * ( ) +. Let the process R(t) denote the accumulated reward till time t, when the reward accrual is continuous The renewal reward processes in that paper had inﬁnite-variance inter-renewal times but ﬁnite-variance rewards. When the renewal process is a Poisson process, the renewal reward process becomes a compound Poisson process. total reward earned by time t . In words: the rate at which rewards are earned is equal to the expected reward over a \cycle" divided by an expected \cycle length Renewal Reward Processes Renewal Reward Processes Consider a renewal process fN(t);t 0ghaving inter-arrival times fX 1;X 2;:::gand suppose that each time a renewal occurs we receive a reward. We are interested in the cumulative reward process given by \[C(t) = \sum_{i=1} ^{N(t)} R_i. The formulation provides a more powerful, general approach to the fluctuation analysis of bead-motor assays begun by Svoboda et al. 4: Look at the expected reward per cycle divided by the expected length of a cycle. In the fuzzy random renewal reward process, both the inter- 7. We have seen that the distribution of the first renewal epoch has no effect on the time or ensemble-average behavior of a renewal process (other than the ensemble dependence on time for an arithmetic process). Usually, the rewards at the renewal times are assumed to be positive. Thus, assuming that an item is replaced by a new (identical) one on each failure, the classical problem in this area is to estimate the mean number of spares that are needed for a long term operation of a technical system or to assess the probability of The simplest example of renewal-reward process has unit rewards and corresponds to the counting renewal process t → Nt:= P i≥11{Ti≤t}. (Euro J Oper Res 169:189-201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell's theorem in random fuzzy sense. Feb 1, 2023 · In the absence of disorder, the total reward over time defines a so-called renewal–reward process [31], so that in fact we deal with renewal–reward processes in random environments. We deﬂne an associated continuous-time renewal-reward stochastic process by setting R(t) · NX(t) n=1 Rn; t ‚ 0: The stochastic process fR(t) : t ‚ 0g is the renewal-reward process. We can do a surprising amount The stochastic process fR(t) : t ‚ 0g is the renewal-reward process. Our approach exploits subadditivity properties of renewal models, which in the absence of disorder lead to LDPs for the total reward under optimal hypotheses Jan 18, 2022 · For maintainers, it plays an important guiding role in engineering practice. The stochastic process t ↦ W t is the so-called renewal–reward process or compound renewal process, which plays an important role in applications [1], [2], [3], [4]. We assume the reward rate to be equal to the age of the process at any time t, and. 3 Finite State the renewal process governed by F. Consider a periodic-review inventory system for which the demands for a Mar 14, 2003 · Renewal Theory. So far, two basic types of renewal reward processes with only random parameters or Markov renewal processes are a class of random processes in probability and statistics that generalize the class of Markov jump processes. There are various names of the renewal reward process in the literature, e. Nat … In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. Renewal-reward Processes. 7, we have discussed a compound Poisson process. $R(t)$ models a rate at which the process is accumulating a reward. Importantly, this formulation lets one derive a set of formulae that relate the long-time slopes 更新奖赏过程（Renewal Reward Process）是复合泊松过程的一个推广。简单来说，我们希望研究的是简单来说，我们希望研究的是 R(t) = \sum_{i = 1}^{N(t)} r_i Nov 1, 2022 · The simplest example of renewal-reward process has unit rewards and corresponds to the counting renewal process t 7→ N t := P i ≥ 1 1 { T i ≤ t } . ( ) is asymptotically normally distributed with mean ⁄ and variance ⁄ . Modified 2 years, 5 months ago. Hence, if we start observing a renewal process at some arbitrarily large time t, then the observed renewal process is the equilibrium renewal process. Let a reward be earned at rate 1 whnen-ver the computer is working. Tijms (Chapter 2: Renewal-Reward Processes). Woo/Multivariate renewal-reward process 2 In this paper, we assume that there are time lags added to the original arrival times of renewal process. Nov 4, 2023 · Renewal reward process: In Chap. In nth renewal du- Apr 24, 2022 · Renewal reward processes can be used to derive some asymptotic results for the age processes of a standard renewal process So, suppose that we have a renewal process with interarrival sequence $ \bs{X} $, arrival sequence $ \bs{T} $, and counting process $ \bs{N} $. This paper studies a type of renewal reward processes with random inter-arrival Numereous studies have been done about a renewal reward process with a discrete interference of chance because of their practical and theoretical importance. g. Fluctuation analysis allows one to gain insight into the Renewal Reward Process • If X <¯ ∞ or E[Rn] < ∞, then lim t→∞ 1 t Z t 0 R(τ)dτ = E[Rn] X¯, with probability 1. References L. Sep 1, 2007 · Based on the concepts, this paper addresses two processes--fuzzy random renewal process and fuzzy random renewal reward process. USA 91(25):11782, 1994). Jan 17, 2023 · This one is an example from the book A First Course in Stochastic Models by H. 2 Renewal process A random point process = ft ngfor which the interarrival times fX ngform an i. Motivated by the above, let fN(t);t >0gbe a renewal process with interarrival times X n, n >1, and denote the time of the n-th renewal by S n = X 1 + + X n. Jun 22, 2017 · Renewal reward process is used to measure the cumulative occasional rewards up to some given time. 1 showed that if a renewal-reward process has an expected inter-renewal interval $\overline{X}$ and an expected inter-renewal reward $\mathrm{E}\left[R_{n}\right]$, then the time-average reward is $\mathrm{E}\left[R_{n}\right] / \overline{X}$ with probability 1. A renewal process is an idealized stochastic model for events that occur randomly in time (generically called renewals or arrivals). Poisson Arrivals See Time Averages. This paper studies a type of 7. The basic mathematical assumption is that the times between the successive arrivals are independent and identically distributed. 4. Renewal-reward or cumulative processes are a useful extension of the renewal process that allows the association of an additional random variable to account for the stepping of the motor. Sep 22, 2022 · I understand this is a standard renewal-reward process as defined in Wikipedia, where the rewards depend on the arrival times. Ask Question Asked 5 years, 8 months ago. t n is then called the nth renewal epoch and F(x) = P(X x) denotes the common interarrival time distribution. This paper studies a type of renewal reward processes with random inter-arrival times and uncertain rewards from the point of view of first hitting time. Operational Nov 25, 2009 · In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. They obtain the chance distribution of the considered process and derive the reward rate. The rate of the renewal process is de ned as def= 1=E(X) which is justi ed by Nov 25, 2009 · The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. Jan 1, 2005 · Recently, Zhao et al. Natl. 1 The renewal function is nite for all t 0 i. process and fuzzy random renewal reward process. Renewal reward theorem applies to a reward process R(t) that accrues reward continuously over a renewal duration. 6. Three kinds of maintenance policies- fuzzy age replacement policy, fuzzy block replacement policy and fuzzy inspection We use the special notation H0;N0; etc. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. The total reward in a renewal duration X n remains R Jun 29, 2022 · On the other hand, Yao and Zhou propose uncertain random renewal reward process, in which the interarrival times and the rewards are assumed to be random variables and uncertain variables, respectively. Our emphasis is on sample-path methods. H(t) < 1: Proof: Using lemma A:13 there is a > 0 such that for any t 0 and all k 2 N A common criteria function for the selection of decision variables in renewal reward processes is the leading term in the asymptotic expansion of the expected cost per unit time, namely the ratio of expected cost per cycle to expected cycle length. Example 1 The pair $(X, R)$ is sometimes called a renewal reward process. It is important to investigate the characteristics of renewal reward process. A renewal--reward process with dependent components and heavy-tailed interarrival times is investigated, and an asymptotic expansion as $t\to\infty$ for the Feb 17, 2011 · Renewal-reward processes are used to provide a framework for the mathematical description of single-molecule bead-motor assays for processive motor proteins. d. That is, a reward is earned each time the previous m data values are all distinct. Fuzzy elementary renewal theorem and fuzzy renewal reward theorem are developed and shown how they can be applied to maintenance policies. Within the framework of … Expand Renewal-reward processes are used to provide a framework for the mathematical description of single-molecule bead-motor assays for processive motor proteins. d random variables. We assume that the R n, n 1, are iid r. Sheldon M. We denote by R n, the reward earned at the time of the n-th renewal. Ask Question Asked 7 years, 1 month ago. The Formula of Little. The stochastic process {Y(t),t > 0} is called a renewal reward process. random variable pairs. interarrival times fX i;i 1g. Observe that Fe is the limiting distribution of the age and the excess time for the renewal process with common inter-renewal distribution F. This carries over to reward functions with almost no change. Suppose we start observing a renewal process at some arbitrary time t. We can do a surprising amount To explore the inspection paradox in the context of a renewal-reward process, we obtain asymptotic expressions for the mean and distribution function of the reward associated with the spread (total life) of the process. R i may be Sep 16, 1999 · Abstract The renewal reward process describes the cumulative reward related to the renewals of a system. In Section 3 we t will consider processes of the form Y(t) J V(s)ds , where V is a real-valued Here, we will present some basic results in renewal theory such as the elementary renewal theorem and the inspection paradox (Section 1), and the renewal reward theorem (Section 2). Glynn and Whitt [7] investigated the connection between LDPs of the inverse processes t → Nt and i → Ti, providing a full LDP for Nt under the Cram´er condition. ABSTRACT In this note, a renewal reward process is dealt with, and the asymptotic be havior of the expected total reward when time tends to infinity is studied in the We \earn" reward Rn at time Sn. Nonetheless, passwords are often stolen by cyber criminals due to inferior user security awareness. Given the eruption of Internet of Things (IoT) systems, it has become critical for connected devices to have persistent security. In the fuzzy random renewal process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is Feb 4, 2020 · Therefore, the renewal reward process is a generalization of the marked Poisson process. If we let R(t) = NX(t) n=1 R n then R(t Apr 23, 2022 · We will return to the asymptotic behavior of the alternating renewal process in the next section on renewal reward processes. In the fuzzy random renewal pro-cess, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is presented. In this paper, both the inter-renewal times and the rewards are allowed to have inﬁnite variance. Ross, in Introduction to Probability Models (Tenth Edition), 2010 Definition 7. Now suppose that at the time of each renewal a reward is received; we denote by R n the reward received at the end of the Finiteness of the mathematical expectation of the renewal-reward process is proved. In this case the cumulative reward process N(t) is the number of customers arrived by time t, and by Theorem 17, N(t)=ta!:s: E(N)=E(T) =: the long run rate of arrival: (C) Consider now the reward-renewal process with inter-arrival times N i, and the associated rewards S Apr 24, 2022 · Many quantities of interest in the study of renewal processes can be described by a special type of integral equation known as a renewal equation. Lemma 1. As a continuation of the work [39], in this paper we discuss a renewal reward process with fuzzy random interarrival times and rewards under the independence with t-norms (>-independence), which induces the (generalized) t-norm-based extension principle for the operations of fuzzy realizations of fuzzy random variables; and we derive a new Nov 12, 2020 · The renewal reward process describes the cumulative reward related to the renewals of a system. May 22, 2022 · There are many situations in which, along with a renewal counting process $\{N(t) ; t>0\}$, there is another randomly varying function of time, called a reward function $\{R(t) ; t>0\}$. Renewal reward process is used to measure the cumulative occasional rewards up to some given time. These results also yield a simplified demonstration of the elementary renewal-reward theorem. Formally, Lemma1. The Pollaczek – Khintchine Formula. 3 Finite State Answer to For a renewal reward process consider Wn = (R1 + R2 + Science; Advanced Physics; Advanced Physics questions and answers; For a renewal reward process consider Wn = (R1 + R2 + · · · + Rn ) / ( X1 + X2 + · · · + Xn) , where Wn represents the average reward earned during the first n cycles. • A regenerative process deﬁnes a renewal-reward process, which is a complex problem. R i may depend on the ith interarrival time X i, but (X i;R i) are i. Bibliographic Notes. Similarly, as we have de ned it in the previous lecture, the renewal rewards process R(t) satis es lim t!1 R(t) t = E[W] E[D]. The analytic expressions of the chance distribution and the expected value of the Renewal reward process is used to measure the cumulative occasional rewards up to some given time. Abstract : The paper considers a process in which rewards are being earned and for which there exist time points at involves the analysis of renewal processes with costs and rewards. sequence is called a renewal process. This condition was later relaxed by Sep 1, 2007 · Based on the concepts, this paper addresses two processes—fuzzy random renewal process and fuzzy random renewal reward process. More precisely, we suppose that the inter-renewal times are in the domain of attraction of a stable to the enzyme’s chemical changes. We assume the reward rate to be equal to the age of the process at any time t, and R(t) = Zt 0 A(u)du. Remark 4. Key Words and Phrases: Renewal process, renewal function, renewal-reward process, mathe-matical expectation Nov 6, 2019 · Limit theorem and elementary renewal theorem are related to the long-run average rate of an event in a renewal process, while renewal reward process is related to the long-run average reward or cost in a renewal process. This delayed renewal process exhibits stationary Apr 23, 2022 · In a delayed renewal process only the first arrival time is changed. 1 Introduction Recall that a renewal process is an arrival process in which the interarrival intervals are positive,1 independent and identically distributed (IID) random variables (rv’s). Let R i, i = 1;2;:::be i. The compound process R(t) = X N(t) i=1 R i is called a renewal reward process. In the first section of this paper, we will prove the analogue of Blackwell's theorem for a renewal reward process. Renewal processes (since they are arrival processes) can be speciﬁed in three standard ways, ﬁrst, Let us now transform the renewal process into a delayed renewal reward process by supposing that a reward of 1 is earned at time n, n ⩾ m, if the values X n-m + 1, …, X n are all distinct. ,The proposed uncertain random renewal reward process proved useful for the optimization of maintenance strategy with maintenance limited time for a new type of aircraft components, which provides scientific support for aircraft maintenance decision-making for civil Oct 5, 2021 · Similarly, we associate with each down interval a reward which depends on the length of it through some function. To determine the average value of the age of a renewal process, consider the following gradual reward process. This means that {X(t+u),t ≥ 0} given S n = u is conditionally independent of {X(t),0 ≤ t ≤ u} and has the same distibution as {X(t),t ≥ 0}. This paper studies a type of renewal reward processes with random inter-arrival times and In this note, a renewal reward process is dealt with, and the asymptotic behavior of the expected total reward when time tends to infinity is studied in the case of independent and non-identically distributed random variables. The chance distribution of the renewal reward process is obtained, and the reward rate is derived. Sep 27, 2007 · Popova and Wu considered a renewal reward process with fuzzy random interarrival times and rewards, focusing their attention on the long-run average fuzzy reward per unit time [22]. To address these issues, in this paper, we develop a . Example (Regenerative Process) • A stochastic process {X(t),t ≥ 0} is regenerative if there are random times S1,S2, at which the process restarts. i. Renewal equations almost always arise by conditioning on the time of the first arrival and by using the defining property of a renewal process—the fact that the process restarts at each arrival time, independently of the past. Modified 5 years, 7 months ago. The results in this chapter are well known, so the references are not provided. point t. Rabehasaina and J. If the sequence of nonnegative random variables {X 1, X 2, …}is independent and identically distributed, then the counting process {N(t),t ≥ 0} is said to be a renewal process. Renewal Theory and Its Applications. \] Renewal Reward Process • If X <¯ ∞ or E[Rn] < ∞, then lim t→∞ 1 t Z t 0 R(τ)dτ = E[Rn] X¯, with probability 1. Finiteness of the mathematical expectation of the renewal-reward process is proved. The random variable R(t) is the cumulative reward earned up to time t; see Section 3. Jan 19, 2018 · Renewal theory has an industrial origin describing the numbers of replacements that are performed while the repairable item is operating. Finally, we will give a in-depth analysis of waiting times and the inspection paradox. Then, the observed renewal process is the equilibrium renewal process. A renewal reward process, also known as a cumulative renewal process, is similar to the compound Poisson process. • If non-arithmetic renewal process and r(z) is directly Rieman integrable, then lim t→∞ E[R(t)] = E[Rn] X¯. Glynn and Whitt [15] investigated renewal processes. In this paper the mathematical expectation of the renewal-reward process is investigated and some results for the renewal function are generalized to the mathematical expectation of the renewal-reward process. Thus, it's not surprising that the asymptotic behavior of a delayed renewal process is the same as the asymptotic behavior of the corresponding regular renewal process. Also, an inequality that gives us an upper bound for the renewal function is generalized for Aug 1, 2019 · is called an uncertain renewal reward process, where $N_t$ is an uncertain renewal process with interarrival times $\xi _i$ ’s. Applications and Special Cases With a clever definition of on and off , many stochastic processes can be turned into alternating renewal processes, leading in turn to interesting limits, via the basic limit theorem above. 1. Within the framework of uncerta This paper aims to propose a new type of uncertain random process, called uncertain random renewal reward process, in which the interarrival times and the rewards are assumed to be random variables and uncertain variables, respectively. Jan 9, 2006 · Vectorized code for simulation of renewal, stationary renewal, renewal reward, on-off processes. So far, two basic types of renewal reward processes with only random parameters or with only uncertain parameters have been proposed. This paper aims at proposing a new type of renewal reward process, which has uncertain interarrival times and random rewards in the framework of the chance theory The pair $(X, R)$ is sometimes called a renewal reward process. Password-based security measures provide an interface for users to effectively secure their personal data. Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Consider the renewal reward process where the reward associated with the interarrival time $ X_n $ is $\frac{1}{2} X_n^2$ for $ n \in \N $. Jun 6, 2022 · This article is a study of vector-valued renewal-reward processes on Rd. May be used for operations with constant piecewise (stair) functions or simulation of point Jun 1, 2019 · The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. Assuming that the time horizon of the maintenance policy approaches inﬁnity, simple asymptotic formulas have Apr 23, 2022 · The precise statement is different, depending on whether the renewal process is arithmetic or not. So far, two basic types of renewal reward processes with only random parameters or with only (B) The reward associate with the inter-arrival time X iis taken to be N i. \] May 22, 2022 · Arbitrary renewal-reward functions: non-arithmetic case; Theorem 4. times that the computer is working and shut down form an alternating renewal process. A Controlled Queue with Removable Server. Note that the consideration can be reward or cost in a renewal reward process. jj<1, the long run rate at which rewards are earned is given by lim t!1 R(t) t = E(R) E(X) = E(R) w:p:1:; (4) where (X;R) denotes a typical \cycle" (X j;R j); = fE(X)g 1 is the arrival rate for the renewal process. v. 3. Renewal reward theorem applies to a reward process R(t)that accrues reward continuously over a renewal duration. These delayed renewal epochs allow us to study the quantities related to inﬁnite server queues with correlated batch arrivals and multivariate Incurred But Not Re- Mar 16, 2020 · Suppose that customers arrive at a single-server system in accordance with a Poisson process with rate λ. • If arithemtic renewal process with span d, then lim n→∞ E[R(nd)] = E[Rn] X¯. Exercises. for the renewal process corresponding to A0 = 0; the probability with mass 1 in 0: This corresponds to the case, where S has a renewal at time 0 with probability 1: Theorem 1. We have seen that larger renewal intervals have a greater chance of containing t. 1. This paper aims at proposing a new type of renewal reward process, which has uncertain interarrival times and random rewards in the framework of the chance theory Jun 22, 2017 · Renewal reward process is used to measure the cumulative occasional rewards up to some given time. C. In this article, we derive the probability distribution of the total reward and its expected value. Observe that age is a linear increasing function of time in any renewal duration. Includes addition and integration of such processes. In an uncertain renewal reward process, it is not necessary that the uncertain interarrival times and the uncertain rewards are independent. Upon arriving a customer must pass through a door that leads to the server. Each component of the process has a fixed threshold. Abstract. RENEWAL PROCESSES 4. K. Aug 1, 2019 · The stochastic renewal process is a stochastic process, which counts the number of renewals that a stochastic system incurs. Recall that for an arithmetic renewal process, the interarrival times take values in a set of the form $ \{n d: n \in \N\} $ for some $ d \in (0, \infty) $, and the largest such $ d $ is the span of the distribution. May 5, 2019 · Renewal reward process. The random variable R(t) is the cumulative reward earned up to time t; see Section 7. 2 Mathematical preliminaries Jun 1, 2018 · A renewal reward theorem is verified to show that the reward rate converges in distribution to an uncertain variable, which is highly related to the interarrival times and the expected values of random rewards. Finally, some application examples are provided to illustrate the utility of the result. Proof. We call the total reward earned in the time interval [0, t], t ≥ 0, an instantaneous alternating renewal reward process. The jumps of the process are assumed to be independent and identically distributed nonnegative random vectors with mutually dependent components, each of which may be either discrete or continuous (or a mixture of discrete and continuous components). Interrenewal distribution can be given as a function handle to an external random number generator. The process $ t \mapsto \int_0^t A_s \, ds $ for $ t \in [0, \infty) $ is a continuous reward process for this sequence of rewards, as defined in . However, each time someone passes through, the door becomes locked for the next t units of time. Or, equivalently, apply renewal reward processes, as in Section 7. Acad. 4 of Ross. We \earn" reward Rn at time Sn. Jewell [4] generalized the study of the fluctuations of a reward process imbedded in a renewal process. The characteristics of renewal process were the ratio value of total reward and its No headers. e. The general implications of refining this criteria function by adopting the first two terms in the asymptotic expansion as an objective function The following articles are merged in Scholar. This chapter introduces the stochastic renewal process, the stochastic renewal reward process, and the stochastic alternating renewal process. If G= F e, then the delayed renewal process is called the equilibrium renewal process. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. We will then discuss the key renewal theorem, and variants of the renewal process including reward renewal process, alternating renewal process, and delayed renewal process. This work extends the work in Zhao and Liu on fuzzy renewal process and renewal reward process from continuous case to more general case. . Also, an inequality that gives us an upper bound for the renewal function is generalized for the mathematical expectation of the renewal-reward process. It is a generalization of an ordinary renewal process. Sci. The total reward in a renewal duration X n remains R n as before, with the sequence((X n,R n) : n 2N) being iid. 6 of Ross. In the fuzzy random renewal process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is Nov 30, 2017 · Renewal Reward Process Problem. We noted that for a renewal process with inter-arrival times D n IID from the distribution of D, the number of arrivals until time t, denoted by N(t), satis es lim t!1 N(t) t = 1 E[D] with probability one. That is, X N(t)+1 tends to be larger than a ordinary renewal interval. Other classes of random processes, such as Markov chains and Poisson processes, can be derived as special cases among the class of Markov renewal processes, while Markov renewal processes are special cases among the more general class of renewal processes. Viewed 548 times 2 $\begingroup$ People arrive at a college May 22, 2022 · Delayed Renewal-reward Processes. cumulative process, jump process or doubly stochastic process . Their combined citations are counted only for the first article. Let Y e(t) denote the excess time for the (delayed) equilibrium renewal process Jan 15, 2018 · The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. R i may be Dec 1, 2021 · A renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time has been investigated, in order to provide some insight A process in which rewards are being earned and for which there exist time points at which the process begins anew is considered, that is, that there exists an embedded renewal process and an expression for the asymptotic mean reward earned during any time interval is obtained. Our first result is the strong law of large numbers for the delayed renewal process. brjkkq pyig elqm rsrrs iheyg vfelzem rfzbe esdae lji cst

Renewal reward process. random variable pairs.