Fuzzy random renewal reward process and its applications
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Author's personal copy Information Sciences 179 (2009) 4057–4069
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Fuzzy random renewal reward process and its applications Shuming Wang *, Junzo Watada 1 Graduate School of Information, Production and Systems, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Fukuoka, Japan
a r t i c l e
i n f o
Article history: Received 8 April 2008 Received in revised form 6 August 2009 Accepted 8 August 2009
Keywords: Renewal process Renewal reward theorem Fuzzy random variable Archimedean t-norm >-Independence
a b s t r a c t This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the >-independence associated with any continuous Archimedean t-norm >. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are >-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. 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. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Renewal reward processes are an important sort of renewal models and have a wide range of real-life applications (see [9,20]). Stochastic renewal theory, based on probability theory, is well developed [1,2,8,31,32]. It is well-known that in stochastic renewal reward processes, the interarrival times and rewards are assumed to be independent and identically distributed (i.i.d.) random variables, and the stochastic renewal reward theorem is one of the most significant results in this area. On the other hand, in order to deal with vague or fuzzy uncertainty in renewal processes, several researchers recently investigated fuzzy renewal processes in which the interarrival times and rewards are assumed to be imprecise and are characterized by fuzzy variables. For example, Zhao and Liu [41] discussed a fuzzy renewal process generated by a sequence of i.i.d. positive fuzzy variables and obtained a fuzzy elementary renewal theorem and a fuzzy renewal reward theorem, respectively. Hong [11] discussed a renewal process in which interarrival times and rewards are depicted by L—R fuzzy numbers under t-norm-based fuzzy operations. In practical applications, randomness and fuzziness often coexist in a single process and thus are required to be considered simultaneously. In such cases, uncertainty cannot be handled in a satisfactory manner by using only either random variables or fuzzy numbers; we therefore need to combine the two and turn to a new tool to deal with this twofold uncertain process. The fuzzy random variable was introduced by Kwakernaak [15,16] in 1978 to study randomness and fuzziness at the same time, and it was defined as a function from a probability space to a collection of fuzzy numbers with certain measurability requirements. Later on, some variants and extensions were developed by other researchers for different purposes; see for example, Kruse and Meyer [14], Liu and Liu [21], López-Diaz and Gil [25], Luhandjula [26], and Puri and Ralescu [29]. Based on the concept of the fuzzy random variable, some renewal processes in fuzzy random environments have been discussed in the literature. For instance, Hwang [12] investigated a renewal process in which the interarrival times are assumed * Corresponding author. Mobile: +81 80 3228 1030; fax: +81 93 692 5179. E-mail addresses: [email protected] (S. Wang), [email protected] (J. Watada). 1 Mobile: +81 90 3464 4929; fax: +81 93 692 5179. 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.08.016
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as i.i.d. fuzzy random variables, and proved an almost sure convergence theorem with the probability measure for the renewal rate. Modeling the rewards as i.i.d. fuzzy random variables, Popova and Wu [30] studied a fuzzy random renewal reward process and derived a theorem for the long-run average reward in the form of a level-wise convergence with the probability of one. Zhao and Tang [42] derived some other properties of fuzzy random renewal processes, and obtained a Blackwell’s renewal theorem and a Smith’s key renewal theorem for fuzzy random interarrival times. Furthermore, Li et al. [17] introduced the fuzzy random variable into delayed renewal processes, and discussed a fuzzy random delayed renewal process as well as a fuzzy random equilibrium renewal process which is a special case of the former. In all studies on fuzzy random renewal processes mentioned above, the operations of fuzzy realizations of fuzzy random variables which are fuzzy numbers or fuzzy variables, are based on an independence associated with the minimum t-norm (Min-independence), or the extension principle with minimum t-norm. Nevertheless, a number of practical applications in the past two decades show that the classical extension principle (or Min-independence) is not always the optimal way to combine fuzzy numbers (for example, in image processing [4,6], measurement theory [34,37], fuzzy control [3,28], type-2 fuzzy system [19,27], and artificial neural networks [24]). The operations associated with different kinds of t-norms may be required for fuzzy numbers in different specific situations and applications. A more general extension principle makes use of a general t-norm operator. Such a generalized extension principle yields different operations for fuzzy numbers or fuzzy variables, in accordance with different t-norms. Recently, several studies have been reported which focus on such tnorm-based operations of fuzzy numbers (see [5,7,10,13]), and fuzzy random variables (see [34,35,38,39]). In particular, Wang et al. [39] have discussed a fuzzy random renewal process under the t-norm-based extension principle, and proved a fuzzy random elementary renewal theorem for the long-run expected renewal rate. 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 fuzzy random renewal reward theorem for the long-run expected average reward. In contrast with the fuzzy random renewal processes in [12,17,30,42] whose results hold only for the minimum t-norm, the renewal reward theorem obtained in this paper can be applied to more general situations using the class of continuous Archimedean t-norms, such as the product t-norm, Dombi t-norm, and Yager t-norm. Additionally, the results obtained in this paper well degenerates to the classical renewal reward theorem in the stochastic process. The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries on >-independent fuzzy variables and fuzzy random variables. Section 3 discusses a fuzzy random renewal reward process, and derives a fuzzy random renewal reward theorem. In Section 4, we provide two applications to further explain how to use the fuzzy random renewal reward theorem obtained above. Finally, our conclusions are drawn in Section 5.
2. Preliminaries 2.1. >-independent fuzzy variables Given a universe C, let Pos be a set function defined on the power set PðCÞ of C. The set function Pos is said to be a possibility measure if it satisfies the following conditions (Pos1) Posð;Þ ¼ 0, and PosðCÞ ¼ 1; S (Pos2) Posð i2I Ai Þ ¼ supi2I PosðAi Þ for any subclass fAi ji 2 Ig of PðCÞ, where I is an arbitrary index set. The triplet ðC; PðCÞ; PosÞ is called a possibility space. Based on possibility measure, a self-dual set function Cr, named credibility measure (see [18]), is defined as
1 ð1 þ PosðAÞ PosðAc ÞÞ; 2
CrðAÞ ¼
A 2 PðCÞ;
ð1Þ
where Ac is the complement of A. Let R be the set of real numbers. A function Y : C ! R is said to be a fuzzy variable defined on C, and the possibility distribution lY of Y is defined by lY ðtÞ ¼ PosfY ¼ tg; t 2 R, which is the possibility of event fY ¼ tg. A fuzzy variable Y is said to be positive almost surely, if CrfY 6 0g ¼ 0. Definition 1 [18]. Let Y be a fuzzy variable. The expected value of Y is defined as
E½Y ¼
Z 0
1
CrfY P rgdr
Z
0
CrfY 6 rgdr
ð2Þ
1
provided that one of the two integrals is finite. Particularly, for nonnegative fuzzy variable Y; E½Y ¼
R1 0
CrfY P rgdr.
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Example 1. Assume that Y ¼ ða; b; cÞ is a triangular fuzzy variable, whose possibility distribution is
8 > < ðx aÞ=ðb aÞ; if a 6 x 6 b; lY ðxÞ ¼ ðc xÞ=ðc bÞ; if b 6 x 6 c; > : 0; otherwise: From (1) and (2), we can compute the expected value of Y is
E½Y ¼
a þ 2b þ c : 4
Now, we recall the >-independence of fuzzy variables. A triangular norm (t-norm for short) is a function > : ½0; 12 ! ½0; 1 such that for any x; y; z 2 ½0; 1 the following four axioms are satisfied [13]: (T1) (T2) (T3) (T4)
Commutativity: >ðx; yÞ ¼ >ðy; xÞ. Associativity: >ðx; >ðy; zÞÞ ¼ Tð>ðx; yÞ; zÞ. Monotonicity: >ðx; yÞ 6 >ðx; zÞ whenever y 6 z. Boundary condition: >ðx; 1Þ ¼ x.
The associativity (T2) allows us to extend each t-norm > in a unique way to an n-ary operation in the usual way by induction, defining for each n-tuple ðx1 ; x2 ; . . . ; xn Þ 2 ½0; 1n
>nk¼1 xk ¼ > >n1 k¼1 xk ; xn ¼ >ðx1 ; x2 ; . . . ; xn Þ: A t-norm > is said to be Archimedean if >ðx; xÞ < x for all x 2 ð0; 1Þ. It is easy to check that the minimum t-norm is not Archimedean. Moreover, from [33], every continuous Archimedean t-norm > can be represented by a continuous and strictly decreasing function f : ½0; 1 ! ½0; 1 with f ð1Þ ¼ 0 and
>ðx1 ; . . . ; xn Þ ¼ f ½1 ðf ðx1 Þ þ þ f ðxn ÞÞ; for all xi 2 ð0; 1Þ; 1 6 i 6 n, where f
( f ½1 ðyÞ ¼
½1
ð3Þ
is the pseudo-inverse of f, defined by
f 1 ðyÞ; if y 2 ½0; f ð0Þ; 0;
if y 2 ðf ð0Þ; 1Þ:
The function f is called the additive generator of >. Example 2. Some common continuous Archimedean t-norms are given as below [13]. (1) Yager t-norm ðk 2 ð0; 1ÞÞ:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k >Yk ðx; yÞ ¼ max 1 ð1 xÞk þ ð1 yÞk ; 0 with additive generator fkY ðxÞ ¼ ð1 xÞk . (2) Dombi t-norm ðk 2 ð0; 1ÞÞ:
>Dk ðx; yÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kffi k 1x k 1þ þ 1y y x
with additive generator fkD ðxÞ ¼ ðð1 xÞ=xÞk . (3) Product t-norm: >P ðx; yÞ ¼ xy with additive generator f P ¼ log. Definition 2 [5] . Let > be a t-norm. A family of fuzzy variables fY i ; i 2 Ig is called >-independent if for any subset fi1 ; i2 ; . . . ; in g I with n P 2,
PosfY ik 2 Bk ; k ¼ 1; 2; . . . ; ng ¼ >nk¼1 PosfY ik 2 Bk g;
ð4Þ
for any subsets B1 ; B2 ; . . . ; Bn of R. Particularly, fuzzy variables Y 1 ; . . . ; Y n are >-independent if
PosfY k 2 Bk ; k ¼ 1; 2; . . . ; ng ¼ >nk¼1 PosfY k 2 Bk g;
ð5Þ
for any subsets B1 ; B2 ; . . . ; Bn of R. Furthermore, we say two families of fuzzy variables fY i ; i 2 Ig and fZ j ; j 2 Jg are mutually >independent if for any fi1 ; . . . ; in g I and fj1 ; . . . ; jm g J with n; m P 1, fuzzy vectors ðY i1 ; . . . ; Y in Þ and ðZ j1 ; . . . ; Z jm Þ are >independent.
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The generalized extension principle can be induced by >-independence. In fact, for >-independent fuzzy variables Y k ; 1 6 k 6 m with possibility distributions lk ; 1 6 k 6 m, and a function g : Rm ! R, the possibility distribution of gðY 1 ; Y 2 ; . . . ; Y m Þ is determined via the possibility distributions l1 ; l2 ; . . . ; lm as
lgðY 1 ;Y 2 ;...;Y m Þ ðxÞ ¼ PosfgðY 1 ; Y 2 ; . . . ; Y m Þ ¼ xg ¼
sup x¼gðx1 ;x2 ;...;xm Þ x1 ;x2 ;...xm 2R
>m k¼1 lk ðxk Þ;
ð6Þ
where > can be any general t-norm. This is the (generalized) extension principle associated with t-norm. 2.2. Fuzzy random variables Definition 3 [21]. Suppose that ðX; R; PrÞ is a probability space, Fv is a collection of fuzzy variables defined on possibility space ðC; PðCÞ; PosÞ. A fuzzy random variable is a map n : X ! Fv such that for any Borel subset B of R; PosfnðxÞ 2 Bg is a measurable function of x. Suppose n is a fuzzy random variable on X, from the above definition, we know for each x 2 X; nðxÞ is a fuzzy variable. Further, a fuzzy random variable n is said to be positive if for almost every x, fuzzy variable nðxÞ is positive almost surely. Example 3. Let X be a random variable on probability space ðX; R; PrÞ. Define that for every x 2 X; nðxÞ ¼ ðXðxÞ 2; XðxÞ þ 2; XðxÞ þ 6Þ which is a triangular fuzzy variable defined on some possibility space ðC; PðCÞ; PosÞ. Then, n is a (triangular) fuzzy random variable. To a fuzzy random variable n on X, for each x 2 X, the expected value of the fuzzy variable nðxÞ, denoted by E½nðxÞ, has been proved to be a measurable function of x (see Theorem 2 in [21]), i.e., it is a random variable. Based on such fact, the expected value of the fuzzy random variable n is defined as the mathematical expectation of the random variable E½nðxÞ. Definition 4 [21]. Let n be a fuzzy random variable given on a probability space ðX; R; PrÞ. The expected value of n is defined as
E½n ¼
Z Z X
1
Crf nðxÞ P r g dr
0
Z
0
Crf nðxÞ 6 r g dr PrðdxÞ:
ð7Þ
1
Example 4. Consider the triangular fuzzy random variable n defined in Example 3. Suppose the X is a discrete random variable, which takes values X 1 ¼ 3 with probability 0.2, and X 2 ¼ 6 with probability 0.8. Try to calculate the expected value of n. From the distribution of random variable X, we know the fuzzy random variable n takes on fuzzy variables nðX 1 Þ ¼ ð1; 5; 9Þ with probability 0.2, and nðX 2 Þ ¼ ð4; 8; 12Þ with probability 0.8. Further, we need to compute the expected values of fuzzy ¼ 5, and E½nðX 2 Þ ¼ 4þ28þ12 ¼ 8. Finally, by definition (7), variables nðX 1 Þ and nðX 1 Þ, respectively. That is E½nðX 1 Þ ¼ 1þ25þ9 4 4 the expected value of n is E½n ¼ E½nðX 1 Þ 0:2 þ E½nðX 2 Þ 0:8 ¼ 7:4. Definition 5 [22]. Let n be a fuzzy random variable, and B a Borel subset of R. The mean chance of an event n 2 B is defined as
Chfn 2 Bg ¼
Z
CrfnðxÞ 2 BgPrðdxÞ:
ð8Þ
X
Example 5. For the triangular random variable n defined in Example 4, now we shall find the mean chance of fuzzy random event fn 6 9g. Recall that fuzzy random variable n takes on fuzzy variables nðX 1 Þ ¼ ð1; 5; 9Þ with probability 0.2, and nðX 2 Þ ¼ ð4; 8; 12Þ with probability 0.8, by the definition, we can compute that CrfnðX 1 Þ 6 9g ¼ 1, and CrfnðX 2 Þ 6 9g ¼ 0:625. From (8), we can calculate Chfn 6 9g ¼ 1 0:2 þ 0:625 0:8 ¼ 0:7. Since Cr has self-duality, the mean chance measure Ch also has been proved to be self-dual (see [22]). What’s more, making use of the mean chance measure, the expected value (7) is equivalent to the following form (see [22]):
E½n ¼
Z
1
Chf n P r g dr
0
Z
0
Chf n 6 r g dr:
ð9Þ
1
In addition, the continuity properties on mean chance function can be found in [40]. A sequence of fuzzy random variables fnn g is said to be uniformly and essentially bounded if there is two real numbers bL and bU such that for each k ¼ 1; 2; . . ., we have ChfbL 6 nn 6 bU g ¼ 1. Moreover, we have the following convergence mode for a sequence of fuzzy random variables. Definition 6 [23]. A sequence fnn g of fuzzy random variables is said to converge in chance Ch to a fuzzy random variable n, Ch denoted nn ! n, if for every e > 0,
lim Chfjnn nj P eg ¼ 0:
n!1
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3. Fuzzy random renewal reward process Let nn ; n ¼ 1; 2; . . . be a sequence of fuzzy random variables defined on a probability space ðX; R; PrÞ. For each n, we denote nn as the interarrival time between the ðn 1Þth and the nth event. Define
S0 ¼ 0;
Sn ¼
n X
nk ; n P 1:
k¼1
It is clear that Sn is the time when the nth renewal occurs, and for any given x 2 X and integer n, Sn ðxÞ ¼ n1 ðxÞ þ þ nn ðxÞ is a fuzzy variable. Let NðtÞ denote the total number of the events that have occurred by time t. Then we have
NðtÞ ¼ maxfnj0 < Sn 6 tg: For any x 2 X; NðtÞðxÞ ¼ maxfnjSn ðxÞ 6 tg is a nonnegative integer-valued fuzzy variable on the possibility space ðC; PðCÞ; PosÞ. That is, NðtÞðxÞðcÞ is a nonnegative integer for any c 2 C; x 2 X and t > 0. We call NðtÞ a fuzzy random renewal variable, and the process fNðtÞ; t > 0g a fuzzy random renewal process. On the basis of the above renewal process fNðtÞ; t > 0g with fuzzy random interarrival times nn ; n P 1, suppose each time a renewal occurs we receive a reward which is a positive fuzzy random variable. We denote gn the reward earned at each time of the nth renewal. Let CðtÞ represent the total reward earned by time t, then we have
CðtÞ ¼
NðtÞ X
gk ;
ð10Þ
k¼1
where NðtÞ is the fuzzy random renewal variable. In this paper, we discuss issues within the following conditions: A1. > can be any continuous Archimedean t-norm with additive generator f. The interarrival times fnn g and rewards fgn g are positive fuzzy random variables, and for almost every x 2 X; fnn ðxÞg and fgn ðxÞg are >-independent fuzzy variable sequences, respectively. A2. P is a nonnegative unimodal real function with Pð0Þ ¼ 1 and PðrÞ ¼ 0, where r can be any positive real number. The support of P is denoted by N. A3. The convex hull of composition function f P in the support N of P satisfies: coðf PÞN ðrÞ > 0 for any nonzero r 2 N. The condition A1 ensures that the operations of fuzzy realizations of fuzzy random interarrival times and rewards are determined by the generalized extension principle associated with any continuous Archimedean t-norm >. In condition A2, the function P is utilized to construct the positive distributions of the interarrival times and rewards. In fact, we note that the interarrival times and rewards between two fuzzy random events are positive fuzzy random variables, hence, making use of the function P in condition A2, we can characterize the interarrival times by fuzzy random variables nk ; k ¼ 1; 2; . . . with distributions lnk ðxÞ ðxÞ ¼ PI ðx U k ðxÞÞ; U k P a almost sure, and characterize the rewards by fuzzy random variables gk ; k ¼ 1; 2; . . . with distributions lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ; V k P b almost surely, respectively, where U k and V k are random variables, and PI and PR are different functions with PI ðaÞ ¼ 0 and PR ðbÞ ¼ 0; a; b > 0. Condition A3 is essentially a convexity condition on the continuous Archimedean t-norm > and the distributions of the interarrival times nk and rewards gk . Here, we introduce briefly the convex hull of a function: Let E be a subset of R. The convex hull of function g on E, denoted coðgÞE , is defined as
coðgÞE ðzÞ ¼ inf
( n X
) kk ðf PÞðxk Þ
ðz 2 EÞ;
ð11Þ
k¼1
P where the infimum is taken over all representations of z as a (finite) convex combination nk¼1 kk xk of points of E. From Tiel [36], we know coðgÞE is the largest convex function hðxÞ such that hðxÞ 6 gðxÞ; x 2 E. Within the convexity condition A3, an important result (Lemma 1) has been derived in [39], which also bases the results of fuzzy random renewal reward process in this work. Assuming that > is a continuous Archimedean t-norm, we have the following result for >-independent and identically distributed fuzzy variables. Lemma 1 [39]. Suppose that fY k g is a sequence of >-independent fuzzy variables with identical possibility distribution P. If coðf PÞðrÞ > 0 for any nonzero r 2 N, then
lim l1nW n ðzÞ ¼
n!1
1; z ¼ 0; 0;
otherwise;
where W n ¼ Y 1 þ Y 2 þ þ Y n .
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Lemma 2. Assume fgk g is a sequence of fuzzy random rewards with lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ for almost every x 2 X, where V k ; k ¼ 1; 2; . . . are i.i.d. random variables with finite expected values. Then we have
g1 þ þ gn n
Ch
! E½V 1 :
ð12Þ
Proof. We denote X k ¼ gk V k . Hence, for any x 2 X, fuzzy variables X k ðxÞ ¼ gk ðxÞ V k ðxÞ have the same possibility distributions lX k ðxÞ ðxÞ ¼ PR ðxÞ for k ¼ 1; 2; . . . We first claim that
X 1 þ þ X n Ch ! 0: n
ð13Þ
From the assumption A2 on PR and Lemma 5 in Hong and Ro [10], for any x 2 X, we know the possibility distribution lX 1 ðxÞþþXn ðxÞ is nonincreasing on Rþ , and nondecreasing on R . Therefore, for any x 2 X and > 0, we have
_ X 1 ðxÞ þ þ X n ðxÞ X 1 ðxÞ þ þ X n ðxÞ P 6 Pos X 1 ðxÞ þ þ X n ðxÞ P Pos Cr 6 n n n _ X 1 ðxÞ þ þ X n ðxÞ X 1 ðxÞ þ þ X n ðxÞ ¼ ¼ : ¼ Pos Pos n n
Therefore, Lemma 1 implies
X 1 ðxÞ þ þ X n ðxÞ P ¼ 0: lim Cr n!1 n Using the dominated convergence theorem, we have
Z X 1 ðxÞ þ þ X n ðxÞ X 1 þ þ X n P PrðdxÞ ¼ 0: P ¼ lim Ch lim Cr n!1 n!1 n n X
This proves our claim. On the other hand, since V k ; k ¼ 1; 2; . . . are i.i.d. random variables with finite expected values, by the strong law of large numbers for random variables, we get
V 1 þ þ V n a:s: ! E½V 1 : n Noting that for any
ð14Þ
> 0,
) ( ) ) ( g þ þ g V 1 þ þ V n X 1 þ þ X n 1 n : Ch E½V 1 P 6 Ch E½V 1 P P 2 þ Ch n n n 2 (
Combining (13) with (14) implies
g1 þ þ gn n
Ch
! E½V 1 :
The lemma is proved.
h
Theorem 1. Suppose ðn1 ; g1 Þ; ðn2 ; g2 Þ; . . . is a sequence of pairs of fuzzy random interarrival times and rewards, where lnk ðxÞ ðxÞ ¼ PI ðx U k ðxÞÞ; U k ðxÞ P a, and lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ; V k ðxÞ P b, for almost every x 2 X; k ¼ 1; 2; . . . ; NðtÞ is the fuzzy random renewal variable, and CðtÞ is the total reward. If fU k g and fV k g are i.i.d. random variable sequences with finite expected values, respectively, then
CðtÞ Ch ! E½V 1 : NðtÞ Proof. We note that for any x 2 X; NðtÞðxÞ ¼ maxfnjn1 ðxÞ þ þ nn ðxÞ 6 tg, the following result has been proved in [39]:
1 Cr ! 0; NðtÞðxÞ for almost every x 2 X. Cr For any x 2 X, we know CðtÞðxÞ ¼ g1 ðxÞ þ þ gNðtÞðxÞ ðxÞ. Now, we prove CðtÞðxÞ=NðtÞðxÞ ! E½V 1 for almost every x 2 X. Recall that the convergence in credibility is equivalent to the convergence almost uniformly, it suffices to prove
CðtÞðxÞ a:u: ! E½V 1 NðtÞðxÞ
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almost surely w.r.t. x 2 X. On one hand, from the proof of Lemma 2 we have
g1 ðxÞ þ þ gn ðxÞ n
Cr
! E½V 1 ;
for almost every x 2 X, which is equivalent to
g1 ðxÞ þ þ gn ðxÞ a:u: n That is, for any
! E½V 1 :
> 0 and d > 0, there exist A 2 PðCÞ with CrðAÞ < d=2 and a positive integer M such that for all c 2 C n A,
g1 ðxÞðcÞ þ þ gn ðxÞðcÞ E½V 1 < n whenever n P M. On the other hand, we note that
1 1 Cr a:u: ! 0 () ! 0: NðtÞðxÞ NðtÞðxÞ Hence, for the above positive integer M, there exist B 2 PðCÞ with CrðBÞ < d=2 and a positive real number tM such that for all c 2 C n B,
1 1 6 ; NðtÞðxÞðcÞ M whenever t P t M . Combining the above, for any > 0 and d > 0, there exists a positive real number tM such that for all c 2 C n A [ B, where CrðA [ BÞ < d, we have
NðtÞðxÞðcÞ P M;
g1 ðxÞðcÞ þ þ gNðtÞðxÞðcÞ ðxÞðcÞ and E½V 1 < NðtÞðxÞðcÞ
a:u:
provided t P t M . As a consequence, for almost every x 2 X; CðtÞðxÞ=NðtÞðxÞ ! E½V 1 . That is almost surely, for every
>0
CðtÞðxÞ lim Cr E½V 1 P ¼ 0: t!1 NðtÞðxÞ Furthermore, applying the dominated convergence theorem, we have
Z CðtÞ CðtÞðxÞ lim Ch lim Cr E½V 1 P ¼ E½V 1 P PrðdxÞ ¼ 0; t!1 NðtÞ NðtÞðxÞ X t!1 which proves the required result. h Theorem 2. Under the same conditions as in Theorem 1, we have
CðtÞ Ch E½V 1 ! : t E½U 1 Proof. By Theorem 1, we have the result
CðtÞ Ch ! E½V 1 : NðtÞ
ð15Þ
In addition, the following result has been derived in [39]:
NðtÞ Ch 1 ! : t E½U 1 Furthermore, we can get the following equivalent events:
CðtÞ E½V 1 CðtÞ NðtÞ E½V 1 CðtÞ NðtÞ NðtÞ E½V 1 () () t E½U 1 NðtÞ t E½U 1 NðtÞ t t
NðtÞ E½V 1 NðtÞ 1 CðtÞ 1 CðtÞ NðtÞ 1 : þ E½V 1 E½V 1 þ E½V 1 þ E½V 1 () t E½U 1 t E½U 1 NðtÞ E½U 1 NðtÞ t E½U 1
ð16Þ
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Without losing any generality, let 0 < < 1. We can obtain that
CðtÞ E½V 1 P 6 Ch NðtÞ 1 CðtÞ E½V 1 P þ Ch CðtÞ E½V 1 P E½U 1 Ch NðtÞ t t E½U 1 E½U 1 NðtÞ 3 3 NðtÞ 1 P þ Ch t E½U 1 3E½V 1 CðtÞ CðtÞ NðtÞ 1 E½U 1 P 6 Ch E½V 1 P E½V 1 P þ Ch þ Ch t E½U 1 NðtÞ NðtÞ 3 3 3 NðtÞ 1 P : þ Ch t E½U 1 3E½V 1 On one hand, by (15), we have
CðtÞ Ch E½V 1 P ! 0ðt ! 1Þ; and NðtÞ 3 CðtÞ E½U 1 E½V 1 P ! 0ðt ! 1Þ: Ch NðtÞ 3 On the other hand, from (16), we have
NðtÞ 1 P Ch ! 0ðt ! 1Þ; and t E½U 1 3 NðtÞ 1 P ! 0ðt ! 1Þ: Ch t E½U 1 3E½V 1 As a consequence, we obtain
CðtÞ E½V 1 P ¼ 0: lim Ch t!1 t E½U 1 The theorem is proved. h Theorem 3 (Fuzzy random renewal reward theorem). Suppose ðn1 ; g1 Þ; ðn2 ; g2 Þ; . . . is a sequence of pairs of fuzzy random interarrival times and rewards, where fgk g is uniformly essentially bounded, lnk ðxÞ ðxÞ ¼ PI ðx U k ðxÞÞ with U k P a þ h almost surely, and lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ, for almost every x 2 X; k ¼ 1; 2; . . . ; NðtÞ is the fuzzy random renewal variable, and CðtÞ is the total reward. If fU k g and fV k g are i.i.d. random variable sequences with finite expected values, respectively, and fnk ðxÞg and fgk ðxÞg are mutually >-independent for almost every x 2 X, then
lim
t!1
E½CðtÞ E½V 1 ¼ : t E½U 1
ð17Þ
Proof. From the supposition, rewards fgk g is a sequence of uniformly and essentially bounded fuzzy random variables, i.e., there exist bL and bU such that for each k ¼ 1; 2; . . ., we have ChfbL 6 gk 6 bU g ¼ 1. It follows from the self-duality of Ch that Chfgk R ½bL ; bU g ¼ 0. Therefore, by the definition of the mean chance, the following equalities
CrfbL 6 gk ðxÞ 6 bU g ¼ 1;
k ¼ 1; 2; . . .
ð18Þ
hold for almost every x 2 X. On the other hand, since U k P a þ h almost surely, we have
Crfnk ðxÞ P hg ¼ 1;
k ¼ 1; 2; . . . ;
ð19Þ
for almost every x 2 X. In the following, we first prove
CðtÞ Ch P r 6 IbU Pr ðrÞ; t h
ð20Þ
for any t; r > 0, where Ifg is the indicator function of event fg. Since the sequences fnk ðxÞg and fgk ðxÞg are mutually >-independent almost surely, for any t; r > 0 and almost every x, we have
(NðtÞðxÞ ) ( ) ! n X X CðtÞðxÞ Cr gk ðxÞ P rt ¼ sup Pos gk ðxÞ P rt \ fNðtÞðxÞ ¼ ng P r 6 Pos t n k¼1 k¼1 ( ) ! n X gk ðxÞ P rt ; PosfNðtÞðxÞ ¼ ng : ¼ sup > Pos n
k¼1
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Furthermore, from (18), we can deduce
CðtÞðxÞ Cr P r 6 sup >ðPosfn bU P rtg; PosfNðtÞðxÞ ¼ ngÞ ¼ PosfNðtÞðxÞ bU P rtg ¼ PosfNðtÞðxÞ P ½rt=bU g t n ¼ PosfS½rt=bU ðxÞ 6 tg; where ½ is the smallest integer larger than or equal to . Recall the facts in (19), we have
PosfS½rt=bU ðxÞ 6 tg 6 Pos h 6
t ½rt=bU
bU 6 Pos r 6 ¼ IbU Pr ðrÞ: h h
As a consequence, we have for any t; r > 0, the following inequality
CðtÞðxÞ Cr P r 6 IbU Pr ðrÞ t h holds almost surely w.r.t. x 2 X. Integrating w.r.t. x on the above inequality, we obtain (20). Since CðtÞ=t is a positive fuzzy random variable for any t > 0, by (7) we have
E½CðtÞ ¼ t
Z
1
Ch
0
CðtÞ P r dr: t
According to Theorem 2, we know
CðtÞ Ch E½V 1 ! ; t E½U 1 it follows from Theorem 4 in [23] that
lim Ch
t!1
CðtÞ E½V 1 P r ¼ Ch Pr ; t E½U 1
ð21Þ
for almost every r 2 R. Noting that
Z
1
0
IbU Pr ðrÞdr ¼ h
bU < 1; h
combining (20) with (21), the Lebesgue’s dominated convergence theorem implies
E½CðtÞ lim ¼ t!1 t
Z 0
1
CðtÞ E½V 1 lim Ch P r dr ¼ : t!1 t E½U 1
The proof of the theorem is complete. h Remark 1. In Theorem 3, if fðnk ; gk Þg becomes a sequence of random variable pairs, then the result (17) degenerates to that of the stochastic renewal reward theorem. In fact, since each pair of nk and gk become random variables, for any given x 2 X, both of the fuzzy variables nk ðxÞ and gk ðxÞ degenerate to crisp numbers. Hence the possibility distributions of nk ðxÞ and gk ðxÞ degenerates to
lnk ðxÞ ðrÞ ¼ PI ðr U k ðxÞÞ ¼
1; if r ¼ U k ðxÞ; 0; otherwise;
and
lgk ðxÞ ðrÞ ¼ PR ðr V k ðxÞÞ ¼
1; if r ¼ V k ðxÞ 0; otherwise;
respectively, which implies nk ; gk ¼ U k ; V k for k ¼ 1; 2; . . . Therefore, the result (17) of Theorem 3 becomes
lim
t!1
E½CðtÞ E½g1 ¼ ; t E½n1
which is the right result of stochastic renewal reward theorem (see [32]). Furthermore, if both of the fuzzy random interarrival times and rewards have symmetric distributions, then we can obtain the following result which is the same as that of the stochastic renewal reward theorem. Corollary 1. Under the condition of Theorem 3, furthermore, if for the distributions of interarrival times and rewards, the PI and PR are symmetric with respect to zero. Then we have
lim
t!1
E½CðtÞ E½g1 ¼ : t E½n1
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Proof. Since PI and PR are symmetric with respect to zero, respectively, we know that for every x 2 X, ln1 ðxÞ and lg1 ðxÞ are symmetric with respect to U 1 ðxÞ and V 1 ðxÞ, respectively, and we can calculate that
Z
1
Crfn1 ðxÞ P rgdr ¼ U 1 ðxÞ;
0
and
Z
1
0
Crfg1 ðxÞ P rgdr ¼ V 1 ðxÞ:
Integrating with respect to x on the above equalities, we obtain
E½n1 ¼
Z Z X
and
E½g1 ¼
1
ð22Þ
0
Z Z X
Crfn1 ðxÞ P rgdr PrðdxÞ ¼ E½U 1 ;
1
0
Crfg1 ðxÞ P rgdr PrðdxÞ ¼ E½V 1 ;
ð23Þ
respectively. It follows from Theorem 3 and (22) and (23) that
lim
t!1
E½CðtÞ E½g1 ¼ : t E½n1
The proof of the corollary is complete. h
4. Applications In this section, in order to explain the utility of the fuzzy random renewal reward theorem (Theorem 3), we provide two examples for the application of Theorem 3 to a multi-service system and a replacement problem, respectively. 4.1. Computing the average reward in a service system Consider a multi-service station. Assume that there are 6 kinds of services provided by the system and that customers come for the service i at probability pi , where pi ¼ 1=6; i ¼ 1; 2; . . . ; 6. The customers are served independently of each other, and service time T i (min) provided by service i and the associated cost C i ($) of service i in the station are assumed to be positive triangular fuzzy variables as shown in Table 1. Here, for each service i, the service time T i and the cost C i are assumed to be >-independent. Taking the t-norm > as a Yager t-norm >Yk ; k > 1, we shall calculate the long-term expected rewards earned by this multi-service station. Let nk be the interarrival time between the ðk 1Þth and kth customers requesting services, and gk be the reward gained from the kth customer, k ¼ 1; 2; . . . Under the above assumptions, we know the service requested by the kth customer and the associated reward are stochastic, while the service time and the cost of each service in the station are fuzzy, and are characterized by triangular fuzzy variable T i and C i , respectively, for i ¼ 1; 2; . . . ; 6. Taking all the scenarios into account, the interarrival times fnk g and rewards fgk g can be considered as two sequences of fuzzy random variables. The distributions of the interarrival times nk and rewards gk , for k ¼ 1; 2; . . . can be presented as follows:
nk and
gk
T1 p1
T2 p2
T3 p3
T4 p4
T5 p5
T6 ; p6
C1
C2
C3
C4
C5
C6
p1
p2
p3
p4
p5
p6
;
respectively. Table 1 Distributions of the service time and cost. Service i
Service time T i (M)
1 2 3 4 5 6
T1 T2 T3 T4 T5 T6
¼ ð2; 3; 5Þ ¼ ð3; 4; 6Þ ¼ ð4; 5; 7Þ ¼ ð5; 6; 8Þ ¼ ð6; 7; 9Þ ¼ ð7; 8; 10Þ
Cost C i ($) C1 C2 C3 C4 C5 C6
¼ ð10; 50; 60Þ ¼ ð30; 70; 80Þ ¼ ð20; 60; 70Þ ¼ ð40; 80; 90Þ ¼ ð60; 100; 110Þ ¼ ð50; 90; 100Þ
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The total service time Sn for the first n customers is calculated by Sn ¼ n1 þ þ nn , and the total number NðtÞ of customers who have been served and the total reward CðtÞ by time t, are given by
NðtÞ ¼ maxfn > 0j0 < Sn 6 tg; and
CðtÞ ¼ g1 þ g2 þ þ gNðtÞ ; respectively. Given the above distributions of fuzzy random interarrival times fnk g and rewards fgk g, without losing any generality, we assign values of two i.i.d. random variable sequences fU k g and fV k g on probability space X as shown in Table 2. Hence, the distributions of each pair of nk and gk for k ¼ 1; 2; . . . can be rewritten as
nk ðxÞ ¼ ðU k ðxÞ 1; U k ðxÞ; U k ðxÞ þ 2Þ ðminÞ; and
gk ðxÞ ¼ ðV k ðxÞ 40; V k ðxÞ; V k ðxÞ þ 10Þ ð$Þ; for any x 2 X. Therefore, we can find the possibility function PI ¼ ð1; 0; 2Þ. Furthermore, since
8 > < x þ 1; if x 2 ½1; 0; PI ðxÞ ¼ 2x ; if x 2 ½0; 2; 2 > : 0; otherwise; we have
8 k > < ðxÞ ; if x 2 ½1; 0; x k fkY PI ðxÞ ¼ ; if x 2 ½0; 2; > : 2 1; otherwise; which is a convex function on [1, 2]. Thus, we have coðfkY PI ÞðxÞ ¼ fkY PI ðxÞ > 0 for any nonzero x 2 ½1; 2. Similarly, we can get coðfkY PR ÞðxÞ > 0 for any nonzero x 2 ½40; 10. As a consequence, by Theorem 3, we can calculate that the average reward earned per minute by this service station in the long run is
P6 E½CðtÞ p V 1 ð xi Þ : lim ¼13:6 ð$Þ: ¼ P6i¼1 i t!1 t i¼1 pi U 1 ðxi Þ 4.2. Calculating the average cost of replacement We now address the calculation of the long-term expected average cost of element replacement in a machine. Suppose that the lifetimes nk of some kind of elements are positive triangular fuzzy random variables with distributions as follows:
nk ¼ ðU k 3; U k ; U k þ 3Þ ðmonthÞ;
k ¼ 1; 2; . . . ;
ð24Þ
where U k are i.i.d. uniformly distributed random variables with U k Uð4; 6Þ. Each time we replace a broken element with a new one, we have to pay for it. The costs gk ; k ¼ 1; 2; . . . of the element are also assumed to be positive triangular fuzzy random variables that have the distributions given below:
gk ¼ ðV k 20; V k ; V k þ 40Þ ð$Þ; k ¼ 1; 2; . . . ;
ð25Þ
where V k are i.i.d. random variables with V k Uð30; 90Þ. The fuzzy realizations of the fuzzy random lifetimes and costs, i.e., fnk ðxÞg and fgk ðxÞg, are two sequences of >-independent fuzzy variables, respectively. Here, we take t-norm > as the product t-norm with the additive generator f P ¼ log. Moreover, we assume that the costs and the lifetimes of the elements are
Table 2 Distributions of the random parameters U k and V k . Service i
Realization U k ðxi Þ
Realization V k ðxi Þ
Probability
1 2 3 4 5 6
U k ðx1 Þ ¼ 3 U k ðx2 Þ ¼ 4 U k ðx3 Þ ¼ 5 U k ðx4 Þ ¼ 6 U k ðx5 Þ ¼ 7 U k ðx6 Þ ¼ 8
V k ðx1 Þ ¼ 50 V k ðx2 Þ ¼ 70 V k ðx3 Þ ¼ 60 V k ðx4 Þ ¼ 80 V k ðx5 Þ ¼ 100 V k ðx6 Þ ¼ 90
1/6 1/6 1/6 1/6 1/6 1/6
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mutually independent. Similar to the previous example, the total number of elements that have been replaced by time t, denoted NðtÞ, can be expressed by
NðtÞ ¼ maxfn > 0j0 < n1 þ n2 þ þ nn 6 tg; and the total cost of the replacement by time t, denoted CðtÞ, can be given by
CðtÞ ¼ g1 þ g2 þ þ gNðtÞ : Now we utilize Theorem 3 to compute the long-run expected average cost per month in this replacement process. We note that the lifetimes and the costs are positive triangular fuzzy random variables, and according to (24) and (25), for any x 2 X,
PI ¼ nk ðxÞ U k ðxÞ ¼ ð3; 0; 3Þ and
PR ¼ gk ðxÞ V k ðxÞ ¼ ð20; 0; 40Þ; for k ¼ 1; 2; . . . are the distributions of triangular fuzzy variables which are bounded and unimodal. Furthermore, we can calculate that
8 xþ3 > < 3 ; if x 2 ½3; 0; PI ðxÞ ¼ 3x ; if x 2 ½0; 3; 3 > : 0; otherwise; and we have
8 x > < log 3 þ 1 ; if x 2 ½3; 0; f P PI ðxÞ ¼ log 3x þ 1 ; if x 2 ½0; 3; > : 1; otherwise; which is a convex function on [3, 3] and therefore satisfies coðf P PI ÞðxÞ > 0 for any nonzero x 2 ½3; 3. By the same reasoning, we can work out that the composition function f P PR is convex and takes on positive values for any nonzero x 2 ½20; 40. Finally, by (24) and (25) again, we know that gk for k ¼ 1; 2; . . . are uniformly and essentially bounded fuzzy random variables, and the random parameters U k ; k ¼ 1; 2; . . . of nk ensure that U k P 3 almost surely. As a consequence, from Theorem 3, we obtain
lim
t!1
E½CðtÞ E½V 1 ¼ ¼ 12 ð$Þ: t E½U 1
That is, the long-run expected average cost of the replacement is 12 $ per month. Remark 2. From the application examples in both Sections 4.1 and 4.2, we see that, in order to apply the fuzzy random renewal reward theorem obtained in this paper, it is required to verify the convexity condition A3, i.e., coðf PI ÞðxÞ > 0 and coðf PR ÞðyÞ > 0 for any nonzero x 2 NPI and nonzero y 2 NPR , respectively. So in the application of our fuzzy random renewal reward theorem, we should select the suitable continuous Archimedean t-norms which satisfy the convexity condition A3 by considering the combination of the additive generator function f of the t-norm > and the distributions of the interarrival times and rewards. In other words, the condition A3 severs as a criterion of the t-norm-selection.
5. Concluding remarks In this paper, under the >-independence associated with continuous Archimedean t-norms, we studied a fuzzy random renewal reward process, and obtained the following new results. (i) We proved some limit theorems for fuzzy random renewal rewards (Theorems 1 and 2). (ii) We derived a fuzzy random renewal reward theorem (Theorem 3) for the long-run expected reward per unit time of the renewal reward process. In addition, we provided two applications, one involving a multi-service system and the other an element-replacement problem, respectively, so as to illustrate the use of our fuzzy random renewal reward theorem. In our future work, there are several issues which should be further discussed based on the current results. First of all, making use of the results obtained in this paper, additional other renewal processes in the fuzzy random environment can be studied, such as delayed renewal reward process and alternating renewal reward process. Furthermore, under the condition of >-independence discussed in this paper, we can also investigate renewal risk processes by considering fuzzy random interest. Finally, the application examples given in this work mainly focus on the calculation of the long-term expected average reward by using the fuzzy random reward renewal theorem obtained which we obtained; in our further studies, we will combine the results in this paper with intelligent computation approaches to solve some practical optimization problems, such as optimal replacement policy and inventory problems.
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Acknowledgments The authors are very grateful to the Editor-in-Chief and the anonymous referees for their insightful comments and suggestions which have contributed substantially to the improvement of this paper. This work was supported by the ‘‘Ambient SoC Global COE Program of Waseda University” of the Ministry of Education, Culture, Sports, Science and Technology, Japan, and by the Research Fellowships of the Japan Society for the Promotion of Science (JSPS) for Young Scientists. References [1] G. Allan, K. Oleq, S. Josef, Equivalences in strong limit theorems for renewal counting process, Statistics and Probability Letters 35 (4) (1997) 381–394. [2] G. Alsmeyer, Superposed and continuous renewal processes a Markov renewal approach, Stochastic Processes and Their Applications 61 (2) (1996) 311–322. [3] M. Benrejeb, A. Sakly, K.B. Othman, P. 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Information Sciences journal homepage: www.elsevier.com/locate/ins
Fuzzy random renewal reward process and its applications Shuming Wang *, Junzo Watada 1 Graduate School of Information, Production and Systems, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Fukuoka, Japan
a r t i c l e
i n f o
Article history: Received 8 April 2008 Received in revised form 6 August 2009 Accepted 8 August 2009
Keywords: Renewal process Renewal reward theorem Fuzzy random variable Archimedean t-norm >-Independence
a b s t r a c t This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the >-independence associated with any continuous Archimedean t-norm >. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are >-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. 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. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Renewal reward processes are an important sort of renewal models and have a wide range of real-life applications (see [9,20]). Stochastic renewal theory, based on probability theory, is well developed [1,2,8,31,32]. It is well-known that in stochastic renewal reward processes, the interarrival times and rewards are assumed to be independent and identically distributed (i.i.d.) random variables, and the stochastic renewal reward theorem is one of the most significant results in this area. On the other hand, in order to deal with vague or fuzzy uncertainty in renewal processes, several researchers recently investigated fuzzy renewal processes in which the interarrival times and rewards are assumed to be imprecise and are characterized by fuzzy variables. For example, Zhao and Liu [41] discussed a fuzzy renewal process generated by a sequence of i.i.d. positive fuzzy variables and obtained a fuzzy elementary renewal theorem and a fuzzy renewal reward theorem, respectively. Hong [11] discussed a renewal process in which interarrival times and rewards are depicted by L—R fuzzy numbers under t-norm-based fuzzy operations. In practical applications, randomness and fuzziness often coexist in a single process and thus are required to be considered simultaneously. In such cases, uncertainty cannot be handled in a satisfactory manner by using only either random variables or fuzzy numbers; we therefore need to combine the two and turn to a new tool to deal with this twofold uncertain process. The fuzzy random variable was introduced by Kwakernaak [15,16] in 1978 to study randomness and fuzziness at the same time, and it was defined as a function from a probability space to a collection of fuzzy numbers with certain measurability requirements. Later on, some variants and extensions were developed by other researchers for different purposes; see for example, Kruse and Meyer [14], Liu and Liu [21], López-Diaz and Gil [25], Luhandjula [26], and Puri and Ralescu [29]. Based on the concept of the fuzzy random variable, some renewal processes in fuzzy random environments have been discussed in the literature. For instance, Hwang [12] investigated a renewal process in which the interarrival times are assumed * Corresponding author. Mobile: +81 80 3228 1030; fax: +81 93 692 5179. E-mail addresses: [email protected] (S. Wang), [email protected] (J. Watada). 1 Mobile: +81 90 3464 4929; fax: +81 93 692 5179. 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.08.016
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as i.i.d. fuzzy random variables, and proved an almost sure convergence theorem with the probability measure for the renewal rate. Modeling the rewards as i.i.d. fuzzy random variables, Popova and Wu [30] studied a fuzzy random renewal reward process and derived a theorem for the long-run average reward in the form of a level-wise convergence with the probability of one. Zhao and Tang [42] derived some other properties of fuzzy random renewal processes, and obtained a Blackwell’s renewal theorem and a Smith’s key renewal theorem for fuzzy random interarrival times. Furthermore, Li et al. [17] introduced the fuzzy random variable into delayed renewal processes, and discussed a fuzzy random delayed renewal process as well as a fuzzy random equilibrium renewal process which is a special case of the former. In all studies on fuzzy random renewal processes mentioned above, the operations of fuzzy realizations of fuzzy random variables which are fuzzy numbers or fuzzy variables, are based on an independence associated with the minimum t-norm (Min-independence), or the extension principle with minimum t-norm. Nevertheless, a number of practical applications in the past two decades show that the classical extension principle (or Min-independence) is not always the optimal way to combine fuzzy numbers (for example, in image processing [4,6], measurement theory [34,37], fuzzy control [3,28], type-2 fuzzy system [19,27], and artificial neural networks [24]). The operations associated with different kinds of t-norms may be required for fuzzy numbers in different specific situations and applications. A more general extension principle makes use of a general t-norm operator. Such a generalized extension principle yields different operations for fuzzy numbers or fuzzy variables, in accordance with different t-norms. Recently, several studies have been reported which focus on such tnorm-based operations of fuzzy numbers (see [5,7,10,13]), and fuzzy random variables (see [34,35,38,39]). In particular, Wang et al. [39] have discussed a fuzzy random renewal process under the t-norm-based extension principle, and proved a fuzzy random elementary renewal theorem for the long-run expected renewal rate. 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 fuzzy random renewal reward theorem for the long-run expected average reward. In contrast with the fuzzy random renewal processes in [12,17,30,42] whose results hold only for the minimum t-norm, the renewal reward theorem obtained in this paper can be applied to more general situations using the class of continuous Archimedean t-norms, such as the product t-norm, Dombi t-norm, and Yager t-norm. Additionally, the results obtained in this paper well degenerates to the classical renewal reward theorem in the stochastic process. The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries on >-independent fuzzy variables and fuzzy random variables. Section 3 discusses a fuzzy random renewal reward process, and derives a fuzzy random renewal reward theorem. In Section 4, we provide two applications to further explain how to use the fuzzy random renewal reward theorem obtained above. Finally, our conclusions are drawn in Section 5.
2. Preliminaries 2.1. >-independent fuzzy variables Given a universe C, let Pos be a set function defined on the power set PðCÞ of C. The set function Pos is said to be a possibility measure if it satisfies the following conditions (Pos1) Posð;Þ ¼ 0, and PosðCÞ ¼ 1; S (Pos2) Posð i2I Ai Þ ¼ supi2I PosðAi Þ for any subclass fAi ji 2 Ig of PðCÞ, where I is an arbitrary index set. The triplet ðC; PðCÞ; PosÞ is called a possibility space. Based on possibility measure, a self-dual set function Cr, named credibility measure (see [18]), is defined as
1 ð1 þ PosðAÞ PosðAc ÞÞ; 2
CrðAÞ ¼
A 2 PðCÞ;
ð1Þ
where Ac is the complement of A. Let R be the set of real numbers. A function Y : C ! R is said to be a fuzzy variable defined on C, and the possibility distribution lY of Y is defined by lY ðtÞ ¼ PosfY ¼ tg; t 2 R, which is the possibility of event fY ¼ tg. A fuzzy variable Y is said to be positive almost surely, if CrfY 6 0g ¼ 0. Definition 1 [18]. Let Y be a fuzzy variable. The expected value of Y is defined as
E½Y ¼
Z 0
1
CrfY P rgdr
Z
0
CrfY 6 rgdr
ð2Þ
1
provided that one of the two integrals is finite. Particularly, for nonnegative fuzzy variable Y; E½Y ¼
R1 0
CrfY P rgdr.
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Example 1. Assume that Y ¼ ða; b; cÞ is a triangular fuzzy variable, whose possibility distribution is
8 > < ðx aÞ=ðb aÞ; if a 6 x 6 b; lY ðxÞ ¼ ðc xÞ=ðc bÞ; if b 6 x 6 c; > : 0; otherwise: From (1) and (2), we can compute the expected value of Y is
E½Y ¼
a þ 2b þ c : 4
Now, we recall the >-independence of fuzzy variables. A triangular norm (t-norm for short) is a function > : ½0; 12 ! ½0; 1 such that for any x; y; z 2 ½0; 1 the following four axioms are satisfied [13]: (T1) (T2) (T3) (T4)
Commutativity: >ðx; yÞ ¼ >ðy; xÞ. Associativity: >ðx; >ðy; zÞÞ ¼ Tð>ðx; yÞ; zÞ. Monotonicity: >ðx; yÞ 6 >ðx; zÞ whenever y 6 z. Boundary condition: >ðx; 1Þ ¼ x.
The associativity (T2) allows us to extend each t-norm > in a unique way to an n-ary operation in the usual way by induction, defining for each n-tuple ðx1 ; x2 ; . . . ; xn Þ 2 ½0; 1n
>nk¼1 xk ¼ > >n1 k¼1 xk ; xn ¼ >ðx1 ; x2 ; . . . ; xn Þ: A t-norm > is said to be Archimedean if >ðx; xÞ < x for all x 2 ð0; 1Þ. It is easy to check that the minimum t-norm is not Archimedean. Moreover, from [33], every continuous Archimedean t-norm > can be represented by a continuous and strictly decreasing function f : ½0; 1 ! ½0; 1 with f ð1Þ ¼ 0 and
>ðx1 ; . . . ; xn Þ ¼ f ½1 ðf ðx1 Þ þ þ f ðxn ÞÞ; for all xi 2 ð0; 1Þ; 1 6 i 6 n, where f
( f ½1 ðyÞ ¼
½1
ð3Þ
is the pseudo-inverse of f, defined by
f 1 ðyÞ; if y 2 ½0; f ð0Þ; 0;
if y 2 ðf ð0Þ; 1Þ:
The function f is called the additive generator of >. Example 2. Some common continuous Archimedean t-norms are given as below [13]. (1) Yager t-norm ðk 2 ð0; 1ÞÞ:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k >Yk ðx; yÞ ¼ max 1 ð1 xÞk þ ð1 yÞk ; 0 with additive generator fkY ðxÞ ¼ ð1 xÞk . (2) Dombi t-norm ðk 2 ð0; 1ÞÞ:
>Dk ðx; yÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kffi k 1x k 1þ þ 1y y x
with additive generator fkD ðxÞ ¼ ðð1 xÞ=xÞk . (3) Product t-norm: >P ðx; yÞ ¼ xy with additive generator f P ¼ log. Definition 2 [5] . Let > be a t-norm. A family of fuzzy variables fY i ; i 2 Ig is called >-independent if for any subset fi1 ; i2 ; . . . ; in g I with n P 2,
PosfY ik 2 Bk ; k ¼ 1; 2; . . . ; ng ¼ >nk¼1 PosfY ik 2 Bk g;
ð4Þ
for any subsets B1 ; B2 ; . . . ; Bn of R. Particularly, fuzzy variables Y 1 ; . . . ; Y n are >-independent if
PosfY k 2 Bk ; k ¼ 1; 2; . . . ; ng ¼ >nk¼1 PosfY k 2 Bk g;
ð5Þ
for any subsets B1 ; B2 ; . . . ; Bn of R. Furthermore, we say two families of fuzzy variables fY i ; i 2 Ig and fZ j ; j 2 Jg are mutually >independent if for any fi1 ; . . . ; in g I and fj1 ; . . . ; jm g J with n; m P 1, fuzzy vectors ðY i1 ; . . . ; Y in Þ and ðZ j1 ; . . . ; Z jm Þ are >independent.
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The generalized extension principle can be induced by >-independence. In fact, for >-independent fuzzy variables Y k ; 1 6 k 6 m with possibility distributions lk ; 1 6 k 6 m, and a function g : Rm ! R, the possibility distribution of gðY 1 ; Y 2 ; . . . ; Y m Þ is determined via the possibility distributions l1 ; l2 ; . . . ; lm as
lgðY 1 ;Y 2 ;...;Y m Þ ðxÞ ¼ PosfgðY 1 ; Y 2 ; . . . ; Y m Þ ¼ xg ¼
sup x¼gðx1 ;x2 ;...;xm Þ x1 ;x2 ;...xm 2R
>m k¼1 lk ðxk Þ;
ð6Þ
where > can be any general t-norm. This is the (generalized) extension principle associated with t-norm. 2.2. Fuzzy random variables Definition 3 [21]. Suppose that ðX; R; PrÞ is a probability space, Fv is a collection of fuzzy variables defined on possibility space ðC; PðCÞ; PosÞ. A fuzzy random variable is a map n : X ! Fv such that for any Borel subset B of R; PosfnðxÞ 2 Bg is a measurable function of x. Suppose n is a fuzzy random variable on X, from the above definition, we know for each x 2 X; nðxÞ is a fuzzy variable. Further, a fuzzy random variable n is said to be positive if for almost every x, fuzzy variable nðxÞ is positive almost surely. Example 3. Let X be a random variable on probability space ðX; R; PrÞ. Define that for every x 2 X; nðxÞ ¼ ðXðxÞ 2; XðxÞ þ 2; XðxÞ þ 6Þ which is a triangular fuzzy variable defined on some possibility space ðC; PðCÞ; PosÞ. Then, n is a (triangular) fuzzy random variable. To a fuzzy random variable n on X, for each x 2 X, the expected value of the fuzzy variable nðxÞ, denoted by E½nðxÞ, has been proved to be a measurable function of x (see Theorem 2 in [21]), i.e., it is a random variable. Based on such fact, the expected value of the fuzzy random variable n is defined as the mathematical expectation of the random variable E½nðxÞ. Definition 4 [21]. Let n be a fuzzy random variable given on a probability space ðX; R; PrÞ. The expected value of n is defined as
E½n ¼
Z Z X
1
Crf nðxÞ P r g dr
0
Z
0
Crf nðxÞ 6 r g dr PrðdxÞ:
ð7Þ
1
Example 4. Consider the triangular fuzzy random variable n defined in Example 3. Suppose the X is a discrete random variable, which takes values X 1 ¼ 3 with probability 0.2, and X 2 ¼ 6 with probability 0.8. Try to calculate the expected value of n. From the distribution of random variable X, we know the fuzzy random variable n takes on fuzzy variables nðX 1 Þ ¼ ð1; 5; 9Þ with probability 0.2, and nðX 2 Þ ¼ ð4; 8; 12Þ with probability 0.8. Further, we need to compute the expected values of fuzzy ¼ 5, and E½nðX 2 Þ ¼ 4þ28þ12 ¼ 8. Finally, by definition (7), variables nðX 1 Þ and nðX 1 Þ, respectively. That is E½nðX 1 Þ ¼ 1þ25þ9 4 4 the expected value of n is E½n ¼ E½nðX 1 Þ 0:2 þ E½nðX 2 Þ 0:8 ¼ 7:4. Definition 5 [22]. Let n be a fuzzy random variable, and B a Borel subset of R. The mean chance of an event n 2 B is defined as
Chfn 2 Bg ¼
Z
CrfnðxÞ 2 BgPrðdxÞ:
ð8Þ
X
Example 5. For the triangular random variable n defined in Example 4, now we shall find the mean chance of fuzzy random event fn 6 9g. Recall that fuzzy random variable n takes on fuzzy variables nðX 1 Þ ¼ ð1; 5; 9Þ with probability 0.2, and nðX 2 Þ ¼ ð4; 8; 12Þ with probability 0.8, by the definition, we can compute that CrfnðX 1 Þ 6 9g ¼ 1, and CrfnðX 2 Þ 6 9g ¼ 0:625. From (8), we can calculate Chfn 6 9g ¼ 1 0:2 þ 0:625 0:8 ¼ 0:7. Since Cr has self-duality, the mean chance measure Ch also has been proved to be self-dual (see [22]). What’s more, making use of the mean chance measure, the expected value (7) is equivalent to the following form (see [22]):
E½n ¼
Z
1
Chf n P r g dr
0
Z
0
Chf n 6 r g dr:
ð9Þ
1
In addition, the continuity properties on mean chance function can be found in [40]. A sequence of fuzzy random variables fnn g is said to be uniformly and essentially bounded if there is two real numbers bL and bU such that for each k ¼ 1; 2; . . ., we have ChfbL 6 nn 6 bU g ¼ 1. Moreover, we have the following convergence mode for a sequence of fuzzy random variables. Definition 6 [23]. A sequence fnn g of fuzzy random variables is said to converge in chance Ch to a fuzzy random variable n, Ch denoted nn ! n, if for every e > 0,
lim Chfjnn nj P eg ¼ 0:
n!1
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3. Fuzzy random renewal reward process Let nn ; n ¼ 1; 2; . . . be a sequence of fuzzy random variables defined on a probability space ðX; R; PrÞ. For each n, we denote nn as the interarrival time between the ðn 1Þth and the nth event. Define
S0 ¼ 0;
Sn ¼
n X
nk ; n P 1:
k¼1
It is clear that Sn is the time when the nth renewal occurs, and for any given x 2 X and integer n, Sn ðxÞ ¼ n1 ðxÞ þ þ nn ðxÞ is a fuzzy variable. Let NðtÞ denote the total number of the events that have occurred by time t. Then we have
NðtÞ ¼ maxfnj0 < Sn 6 tg: For any x 2 X; NðtÞðxÞ ¼ maxfnjSn ðxÞ 6 tg is a nonnegative integer-valued fuzzy variable on the possibility space ðC; PðCÞ; PosÞ. That is, NðtÞðxÞðcÞ is a nonnegative integer for any c 2 C; x 2 X and t > 0. We call NðtÞ a fuzzy random renewal variable, and the process fNðtÞ; t > 0g a fuzzy random renewal process. On the basis of the above renewal process fNðtÞ; t > 0g with fuzzy random interarrival times nn ; n P 1, suppose each time a renewal occurs we receive a reward which is a positive fuzzy random variable. We denote gn the reward earned at each time of the nth renewal. Let CðtÞ represent the total reward earned by time t, then we have
CðtÞ ¼
NðtÞ X
gk ;
ð10Þ
k¼1
where NðtÞ is the fuzzy random renewal variable. In this paper, we discuss issues within the following conditions: A1. > can be any continuous Archimedean t-norm with additive generator f. The interarrival times fnn g and rewards fgn g are positive fuzzy random variables, and for almost every x 2 X; fnn ðxÞg and fgn ðxÞg are >-independent fuzzy variable sequences, respectively. A2. P is a nonnegative unimodal real function with Pð0Þ ¼ 1 and PðrÞ ¼ 0, where r can be any positive real number. The support of P is denoted by N. A3. The convex hull of composition function f P in the support N of P satisfies: coðf PÞN ðrÞ > 0 for any nonzero r 2 N. The condition A1 ensures that the operations of fuzzy realizations of fuzzy random interarrival times and rewards are determined by the generalized extension principle associated with any continuous Archimedean t-norm >. In condition A2, the function P is utilized to construct the positive distributions of the interarrival times and rewards. In fact, we note that the interarrival times and rewards between two fuzzy random events are positive fuzzy random variables, hence, making use of the function P in condition A2, we can characterize the interarrival times by fuzzy random variables nk ; k ¼ 1; 2; . . . with distributions lnk ðxÞ ðxÞ ¼ PI ðx U k ðxÞÞ; U k P a almost sure, and characterize the rewards by fuzzy random variables gk ; k ¼ 1; 2; . . . with distributions lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ; V k P b almost surely, respectively, where U k and V k are random variables, and PI and PR are different functions with PI ðaÞ ¼ 0 and PR ðbÞ ¼ 0; a; b > 0. Condition A3 is essentially a convexity condition on the continuous Archimedean t-norm > and the distributions of the interarrival times nk and rewards gk . Here, we introduce briefly the convex hull of a function: Let E be a subset of R. The convex hull of function g on E, denoted coðgÞE , is defined as
coðgÞE ðzÞ ¼ inf
( n X
) kk ðf PÞðxk Þ
ðz 2 EÞ;
ð11Þ
k¼1
P where the infimum is taken over all representations of z as a (finite) convex combination nk¼1 kk xk of points of E. From Tiel [36], we know coðgÞE is the largest convex function hðxÞ such that hðxÞ 6 gðxÞ; x 2 E. Within the convexity condition A3, an important result (Lemma 1) has been derived in [39], which also bases the results of fuzzy random renewal reward process in this work. Assuming that > is a continuous Archimedean t-norm, we have the following result for >-independent and identically distributed fuzzy variables. Lemma 1 [39]. Suppose that fY k g is a sequence of >-independent fuzzy variables with identical possibility distribution P. If coðf PÞðrÞ > 0 for any nonzero r 2 N, then
lim l1nW n ðzÞ ¼
n!1
1; z ¼ 0; 0;
otherwise;
where W n ¼ Y 1 þ Y 2 þ þ Y n .
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Lemma 2. Assume fgk g is a sequence of fuzzy random rewards with lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ for almost every x 2 X, where V k ; k ¼ 1; 2; . . . are i.i.d. random variables with finite expected values. Then we have
g1 þ þ gn n
Ch
! E½V 1 :
ð12Þ
Proof. We denote X k ¼ gk V k . Hence, for any x 2 X, fuzzy variables X k ðxÞ ¼ gk ðxÞ V k ðxÞ have the same possibility distributions lX k ðxÞ ðxÞ ¼ PR ðxÞ for k ¼ 1; 2; . . . We first claim that
X 1 þ þ X n Ch ! 0: n
ð13Þ
From the assumption A2 on PR and Lemma 5 in Hong and Ro [10], for any x 2 X, we know the possibility distribution lX 1 ðxÞþþXn ðxÞ is nonincreasing on Rþ , and nondecreasing on R . Therefore, for any x 2 X and > 0, we have
_ X 1 ðxÞ þ þ X n ðxÞ X 1 ðxÞ þ þ X n ðxÞ P 6 Pos X 1 ðxÞ þ þ X n ðxÞ P Pos Cr 6 n n n _ X 1 ðxÞ þ þ X n ðxÞ X 1 ðxÞ þ þ X n ðxÞ ¼ ¼ : ¼ Pos Pos n n
Therefore, Lemma 1 implies
X 1 ðxÞ þ þ X n ðxÞ P ¼ 0: lim Cr n!1 n Using the dominated convergence theorem, we have
Z X 1 ðxÞ þ þ X n ðxÞ X 1 þ þ X n P PrðdxÞ ¼ 0: P ¼ lim Ch lim Cr n!1 n!1 n n X
This proves our claim. On the other hand, since V k ; k ¼ 1; 2; . . . are i.i.d. random variables with finite expected values, by the strong law of large numbers for random variables, we get
V 1 þ þ V n a:s: ! E½V 1 : n Noting that for any
ð14Þ
> 0,
) ( ) ) ( g þ þ g V 1 þ þ V n X 1 þ þ X n 1 n : Ch E½V 1 P 6 Ch E½V 1 P P 2 þ Ch n n n 2 (
Combining (13) with (14) implies
g1 þ þ gn n
Ch
! E½V 1 :
The lemma is proved.
h
Theorem 1. Suppose ðn1 ; g1 Þ; ðn2 ; g2 Þ; . . . is a sequence of pairs of fuzzy random interarrival times and rewards, where lnk ðxÞ ðxÞ ¼ PI ðx U k ðxÞÞ; U k ðxÞ P a, and lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ; V k ðxÞ P b, for almost every x 2 X; k ¼ 1; 2; . . . ; NðtÞ is the fuzzy random renewal variable, and CðtÞ is the total reward. If fU k g and fV k g are i.i.d. random variable sequences with finite expected values, respectively, then
CðtÞ Ch ! E½V 1 : NðtÞ Proof. We note that for any x 2 X; NðtÞðxÞ ¼ maxfnjn1 ðxÞ þ þ nn ðxÞ 6 tg, the following result has been proved in [39]:
1 Cr ! 0; NðtÞðxÞ for almost every x 2 X. Cr For any x 2 X, we know CðtÞðxÞ ¼ g1 ðxÞ þ þ gNðtÞðxÞ ðxÞ. Now, we prove CðtÞðxÞ=NðtÞðxÞ ! E½V 1 for almost every x 2 X. Recall that the convergence in credibility is equivalent to the convergence almost uniformly, it suffices to prove
CðtÞðxÞ a:u: ! E½V 1 NðtÞðxÞ
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almost surely w.r.t. x 2 X. On one hand, from the proof of Lemma 2 we have
g1 ðxÞ þ þ gn ðxÞ n
Cr
! E½V 1 ;
for almost every x 2 X, which is equivalent to
g1 ðxÞ þ þ gn ðxÞ a:u: n That is, for any
! E½V 1 :
> 0 and d > 0, there exist A 2 PðCÞ with CrðAÞ < d=2 and a positive integer M such that for all c 2 C n A,
g1 ðxÞðcÞ þ þ gn ðxÞðcÞ E½V 1 < n whenever n P M. On the other hand, we note that
1 1 Cr a:u: ! 0 () ! 0: NðtÞðxÞ NðtÞðxÞ Hence, for the above positive integer M, there exist B 2 PðCÞ with CrðBÞ < d=2 and a positive real number tM such that for all c 2 C n B,
1 1 6 ; NðtÞðxÞðcÞ M whenever t P t M . Combining the above, for any > 0 and d > 0, there exists a positive real number tM such that for all c 2 C n A [ B, where CrðA [ BÞ < d, we have
NðtÞðxÞðcÞ P M;
g1 ðxÞðcÞ þ þ gNðtÞðxÞðcÞ ðxÞðcÞ and E½V 1 < NðtÞðxÞðcÞ
a:u:
provided t P t M . As a consequence, for almost every x 2 X; CðtÞðxÞ=NðtÞðxÞ ! E½V 1 . That is almost surely, for every
>0
CðtÞðxÞ lim Cr E½V 1 P ¼ 0: t!1 NðtÞðxÞ Furthermore, applying the dominated convergence theorem, we have
Z CðtÞ CðtÞðxÞ lim Ch lim Cr E½V 1 P ¼ E½V 1 P PrðdxÞ ¼ 0; t!1 NðtÞ NðtÞðxÞ X t!1 which proves the required result. h Theorem 2. Under the same conditions as in Theorem 1, we have
CðtÞ Ch E½V 1 ! : t E½U 1 Proof. By Theorem 1, we have the result
CðtÞ Ch ! E½V 1 : NðtÞ
ð15Þ
In addition, the following result has been derived in [39]:
NðtÞ Ch 1 ! : t E½U 1 Furthermore, we can get the following equivalent events:
CðtÞ E½V 1 CðtÞ NðtÞ E½V 1 CðtÞ NðtÞ NðtÞ E½V 1 () () t E½U 1 NðtÞ t E½U 1 NðtÞ t t
NðtÞ E½V 1 NðtÞ 1 CðtÞ 1 CðtÞ NðtÞ 1 : þ E½V 1 E½V 1 þ E½V 1 þ E½V 1 () t E½U 1 t E½U 1 NðtÞ E½U 1 NðtÞ t E½U 1
ð16Þ
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Without losing any generality, let 0 < < 1. We can obtain that
CðtÞ E½V 1 P 6 Ch NðtÞ 1 CðtÞ E½V 1 P þ Ch CðtÞ E½V 1 P E½U 1 Ch NðtÞ t t E½U 1 E½U 1 NðtÞ 3 3 NðtÞ 1 P þ Ch t E½U 1 3E½V 1 CðtÞ CðtÞ NðtÞ 1 E½U 1 P 6 Ch E½V 1 P E½V 1 P þ Ch þ Ch t E½U 1 NðtÞ NðtÞ 3 3 3 NðtÞ 1 P : þ Ch t E½U 1 3E½V 1 On one hand, by (15), we have
CðtÞ Ch E½V 1 P ! 0ðt ! 1Þ; and NðtÞ 3 CðtÞ E½U 1 E½V 1 P ! 0ðt ! 1Þ: Ch NðtÞ 3 On the other hand, from (16), we have
NðtÞ 1 P Ch ! 0ðt ! 1Þ; and t E½U 1 3 NðtÞ 1 P ! 0ðt ! 1Þ: Ch t E½U 1 3E½V 1 As a consequence, we obtain
CðtÞ E½V 1 P ¼ 0: lim Ch t!1 t E½U 1 The theorem is proved. h Theorem 3 (Fuzzy random renewal reward theorem). Suppose ðn1 ; g1 Þ; ðn2 ; g2 Þ; . . . is a sequence of pairs of fuzzy random interarrival times and rewards, where fgk g is uniformly essentially bounded, lnk ðxÞ ðxÞ ¼ PI ðx U k ðxÞÞ with U k P a þ h almost surely, and lgk ðxÞ ðxÞ ¼ PR ðx V k ðxÞÞ, for almost every x 2 X; k ¼ 1; 2; . . . ; NðtÞ is the fuzzy random renewal variable, and CðtÞ is the total reward. If fU k g and fV k g are i.i.d. random variable sequences with finite expected values, respectively, and fnk ðxÞg and fgk ðxÞg are mutually >-independent for almost every x 2 X, then
lim
t!1
E½CðtÞ E½V 1 ¼ : t E½U 1
ð17Þ
Proof. From the supposition, rewards fgk g is a sequence of uniformly and essentially bounded fuzzy random variables, i.e., there exist bL and bU such that for each k ¼ 1; 2; . . ., we have ChfbL 6 gk 6 bU g ¼ 1. It follows from the self-duality of Ch that Chfgk R ½bL ; bU g ¼ 0. Therefore, by the definition of the mean chance, the following equalities
CrfbL 6 gk ðxÞ 6 bU g ¼ 1;
k ¼ 1; 2; . . .
ð18Þ
hold for almost every x 2 X. On the other hand, since U k P a þ h almost surely, we have
Crfnk ðxÞ P hg ¼ 1;
k ¼ 1; 2; . . . ;
ð19Þ
for almost every x 2 X. In the following, we first prove
CðtÞ Ch P r 6 IbU Pr ðrÞ; t h
ð20Þ
for any t; r > 0, where Ifg is the indicator function of event fg. Since the sequences fnk ðxÞg and fgk ðxÞg are mutually >-independent almost surely, for any t; r > 0 and almost every x, we have
(NðtÞðxÞ ) ( ) ! n X X CðtÞðxÞ Cr gk ðxÞ P rt ¼ sup Pos gk ðxÞ P rt \ fNðtÞðxÞ ¼ ng P r 6 Pos t n k¼1 k¼1 ( ) ! n X gk ðxÞ P rt ; PosfNðtÞðxÞ ¼ ng : ¼ sup > Pos n
k¼1
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Furthermore, from (18), we can deduce
CðtÞðxÞ Cr P r 6 sup >ðPosfn bU P rtg; PosfNðtÞðxÞ ¼ ngÞ ¼ PosfNðtÞðxÞ bU P rtg ¼ PosfNðtÞðxÞ P ½rt=bU g t n ¼ PosfS½rt=bU ðxÞ 6 tg; where ½ is the smallest integer larger than or equal to . Recall the facts in (19), we have
PosfS½rt=bU ðxÞ 6 tg 6 Pos h 6
t ½rt=bU
bU 6 Pos r 6 ¼ IbU Pr ðrÞ: h h
As a consequence, we have for any t; r > 0, the following inequality
CðtÞðxÞ Cr P r 6 IbU Pr ðrÞ t h holds almost surely w.r.t. x 2 X. Integrating w.r.t. x on the above inequality, we obtain (20). Since CðtÞ=t is a positive fuzzy random variable for any t > 0, by (7) we have
E½CðtÞ ¼ t
Z
1
Ch
0
CðtÞ P r dr: t
According to Theorem 2, we know
CðtÞ Ch E½V 1 ! ; t E½U 1 it follows from Theorem 4 in [23] that
lim Ch
t!1
CðtÞ E½V 1 P r ¼ Ch Pr ; t E½U 1
ð21Þ
for almost every r 2 R. Noting that
Z
1
0
IbU Pr ðrÞdr ¼ h
bU < 1; h
combining (20) with (21), the Lebesgue’s dominated convergence theorem implies
E½CðtÞ lim ¼ t!1 t
Z 0
1
CðtÞ E½V 1 lim Ch P r dr ¼ : t!1 t E½U 1
The proof of the theorem is complete. h Remark 1. In Theorem 3, if fðnk ; gk Þg becomes a sequence of random variable pairs, then the result (17) degenerates to that of the stochastic renewal reward theorem. In fact, since each pair of nk and gk become random variables, for any given x 2 X, both of the fuzzy variables nk ðxÞ and gk ðxÞ degenerate to crisp numbers. Hence the possibility distributions of nk ðxÞ and gk ðxÞ degenerates to
lnk ðxÞ ðrÞ ¼ PI ðr U k ðxÞÞ ¼
1; if r ¼ U k ðxÞ; 0; otherwise;
and
lgk ðxÞ ðrÞ ¼ PR ðr V k ðxÞÞ ¼
1; if r ¼ V k ðxÞ 0; otherwise;
respectively, which implies nk ; gk ¼ U k ; V k for k ¼ 1; 2; . . . Therefore, the result (17) of Theorem 3 becomes
lim
t!1
E½CðtÞ E½g1 ¼ ; t E½n1
which is the right result of stochastic renewal reward theorem (see [32]). Furthermore, if both of the fuzzy random interarrival times and rewards have symmetric distributions, then we can obtain the following result which is the same as that of the stochastic renewal reward theorem. Corollary 1. Under the condition of Theorem 3, furthermore, if for the distributions of interarrival times and rewards, the PI and PR are symmetric with respect to zero. Then we have
lim
t!1
E½CðtÞ E½g1 ¼ : t E½n1
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Proof. Since PI and PR are symmetric with respect to zero, respectively, we know that for every x 2 X, ln1 ðxÞ and lg1 ðxÞ are symmetric with respect to U 1 ðxÞ and V 1 ðxÞ, respectively, and we can calculate that
Z
1
Crfn1 ðxÞ P rgdr ¼ U 1 ðxÞ;
0
and
Z
1
0
Crfg1 ðxÞ P rgdr ¼ V 1 ðxÞ:
Integrating with respect to x on the above equalities, we obtain
E½n1 ¼
Z Z X
and
E½g1 ¼
1
ð22Þ
0
Z Z X
Crfn1 ðxÞ P rgdr PrðdxÞ ¼ E½U 1 ;
1
0
Crfg1 ðxÞ P rgdr PrðdxÞ ¼ E½V 1 ;
ð23Þ
respectively. It follows from Theorem 3 and (22) and (23) that
lim
t!1
E½CðtÞ E½g1 ¼ : t E½n1
The proof of the corollary is complete. h
4. Applications In this section, in order to explain the utility of the fuzzy random renewal reward theorem (Theorem 3), we provide two examples for the application of Theorem 3 to a multi-service system and a replacement problem, respectively. 4.1. Computing the average reward in a service system Consider a multi-service station. Assume that there are 6 kinds of services provided by the system and that customers come for the service i at probability pi , where pi ¼ 1=6; i ¼ 1; 2; . . . ; 6. The customers are served independently of each other, and service time T i (min) provided by service i and the associated cost C i ($) of service i in the station are assumed to be positive triangular fuzzy variables as shown in Table 1. Here, for each service i, the service time T i and the cost C i are assumed to be >-independent. Taking the t-norm > as a Yager t-norm >Yk ; k > 1, we shall calculate the long-term expected rewards earned by this multi-service station. Let nk be the interarrival time between the ðk 1Þth and kth customers requesting services, and gk be the reward gained from the kth customer, k ¼ 1; 2; . . . Under the above assumptions, we know the service requested by the kth customer and the associated reward are stochastic, while the service time and the cost of each service in the station are fuzzy, and are characterized by triangular fuzzy variable T i and C i , respectively, for i ¼ 1; 2; . . . ; 6. Taking all the scenarios into account, the interarrival times fnk g and rewards fgk g can be considered as two sequences of fuzzy random variables. The distributions of the interarrival times nk and rewards gk , for k ¼ 1; 2; . . . can be presented as follows:
nk and
gk
T1 p1
T2 p2
T3 p3
T4 p4
T5 p5
T6 ; p6
C1
C2
C3
C4
C5
C6
p1
p2
p3
p4
p5
p6
;
respectively. Table 1 Distributions of the service time and cost. Service i
Service time T i (M)
1 2 3 4 5 6
T1 T2 T3 T4 T5 T6
¼ ð2; 3; 5Þ ¼ ð3; 4; 6Þ ¼ ð4; 5; 7Þ ¼ ð5; 6; 8Þ ¼ ð6; 7; 9Þ ¼ ð7; 8; 10Þ
Cost C i ($) C1 C2 C3 C4 C5 C6
¼ ð10; 50; 60Þ ¼ ð30; 70; 80Þ ¼ ð20; 60; 70Þ ¼ ð40; 80; 90Þ ¼ ð60; 100; 110Þ ¼ ð50; 90; 100Þ
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The total service time Sn for the first n customers is calculated by Sn ¼ n1 þ þ nn , and the total number NðtÞ of customers who have been served and the total reward CðtÞ by time t, are given by
NðtÞ ¼ maxfn > 0j0 < Sn 6 tg; and
CðtÞ ¼ g1 þ g2 þ þ gNðtÞ ; respectively. Given the above distributions of fuzzy random interarrival times fnk g and rewards fgk g, without losing any generality, we assign values of two i.i.d. random variable sequences fU k g and fV k g on probability space X as shown in Table 2. Hence, the distributions of each pair of nk and gk for k ¼ 1; 2; . . . can be rewritten as
nk ðxÞ ¼ ðU k ðxÞ 1; U k ðxÞ; U k ðxÞ þ 2Þ ðminÞ; and
gk ðxÞ ¼ ðV k ðxÞ 40; V k ðxÞ; V k ðxÞ þ 10Þ ð$Þ; for any x 2 X. Therefore, we can find the possibility function PI ¼ ð1; 0; 2Þ. Furthermore, since
8 > < x þ 1; if x 2 ½1; 0; PI ðxÞ ¼ 2x ; if x 2 ½0; 2; 2 > : 0; otherwise; we have
8 k > < ðxÞ ; if x 2 ½1; 0; x k fkY PI ðxÞ ¼ ; if x 2 ½0; 2; > : 2 1; otherwise; which is a convex function on [1, 2]. Thus, we have coðfkY PI ÞðxÞ ¼ fkY PI ðxÞ > 0 for any nonzero x 2 ½1; 2. Similarly, we can get coðfkY PR ÞðxÞ > 0 for any nonzero x 2 ½40; 10. As a consequence, by Theorem 3, we can calculate that the average reward earned per minute by this service station in the long run is
P6 E½CðtÞ p V 1 ð xi Þ : lim ¼13:6 ð$Þ: ¼ P6i¼1 i t!1 t i¼1 pi U 1 ðxi Þ 4.2. Calculating the average cost of replacement We now address the calculation of the long-term expected average cost of element replacement in a machine. Suppose that the lifetimes nk of some kind of elements are positive triangular fuzzy random variables with distributions as follows:
nk ¼ ðU k 3; U k ; U k þ 3Þ ðmonthÞ;
k ¼ 1; 2; . . . ;
ð24Þ
where U k are i.i.d. uniformly distributed random variables with U k Uð4; 6Þ. Each time we replace a broken element with a new one, we have to pay for it. The costs gk ; k ¼ 1; 2; . . . of the element are also assumed to be positive triangular fuzzy random variables that have the distributions given below:
gk ¼ ðV k 20; V k ; V k þ 40Þ ð$Þ; k ¼ 1; 2; . . . ;
ð25Þ
where V k are i.i.d. random variables with V k Uð30; 90Þ. The fuzzy realizations of the fuzzy random lifetimes and costs, i.e., fnk ðxÞg and fgk ðxÞg, are two sequences of >-independent fuzzy variables, respectively. Here, we take t-norm > as the product t-norm with the additive generator f P ¼ log. Moreover, we assume that the costs and the lifetimes of the elements are
Table 2 Distributions of the random parameters U k and V k . Service i
Realization U k ðxi Þ
Realization V k ðxi Þ
Probability
1 2 3 4 5 6
U k ðx1 Þ ¼ 3 U k ðx2 Þ ¼ 4 U k ðx3 Þ ¼ 5 U k ðx4 Þ ¼ 6 U k ðx5 Þ ¼ 7 U k ðx6 Þ ¼ 8
V k ðx1 Þ ¼ 50 V k ðx2 Þ ¼ 70 V k ðx3 Þ ¼ 60 V k ðx4 Þ ¼ 80 V k ðx5 Þ ¼ 100 V k ðx6 Þ ¼ 90
1/6 1/6 1/6 1/6 1/6 1/6
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mutually independent. Similar to the previous example, the total number of elements that have been replaced by time t, denoted NðtÞ, can be expressed by
NðtÞ ¼ maxfn > 0j0 < n1 þ n2 þ þ nn 6 tg; and the total cost of the replacement by time t, denoted CðtÞ, can be given by
CðtÞ ¼ g1 þ g2 þ þ gNðtÞ : Now we utilize Theorem 3 to compute the long-run expected average cost per month in this replacement process. We note that the lifetimes and the costs are positive triangular fuzzy random variables, and according to (24) and (25), for any x 2 X,
PI ¼ nk ðxÞ U k ðxÞ ¼ ð3; 0; 3Þ and
PR ¼ gk ðxÞ V k ðxÞ ¼ ð20; 0; 40Þ; for k ¼ 1; 2; . . . are the distributions of triangular fuzzy variables which are bounded and unimodal. Furthermore, we can calculate that
8 xþ3 > < 3 ; if x 2 ½3; 0; PI ðxÞ ¼ 3x ; if x 2 ½0; 3; 3 > : 0; otherwise; and we have
8 x > < log 3 þ 1 ; if x 2 ½3; 0; f P PI ðxÞ ¼ log 3x þ 1 ; if x 2 ½0; 3; > : 1; otherwise; which is a convex function on [3, 3] and therefore satisfies coðf P PI ÞðxÞ > 0 for any nonzero x 2 ½3; 3. By the same reasoning, we can work out that the composition function f P PR is convex and takes on positive values for any nonzero x 2 ½20; 40. Finally, by (24) and (25) again, we know that gk for k ¼ 1; 2; . . . are uniformly and essentially bounded fuzzy random variables, and the random parameters U k ; k ¼ 1; 2; . . . of nk ensure that U k P 3 almost surely. As a consequence, from Theorem 3, we obtain
lim
t!1
E½CðtÞ E½V 1 ¼ ¼ 12 ð$Þ: t E½U 1
That is, the long-run expected average cost of the replacement is 12 $ per month. Remark 2. From the application examples in both Sections 4.1 and 4.2, we see that, in order to apply the fuzzy random renewal reward theorem obtained in this paper, it is required to verify the convexity condition A3, i.e., coðf PI ÞðxÞ > 0 and coðf PR ÞðyÞ > 0 for any nonzero x 2 NPI and nonzero y 2 NPR , respectively. So in the application of our fuzzy random renewal reward theorem, we should select the suitable continuous Archimedean t-norms which satisfy the convexity condition A3 by considering the combination of the additive generator function f of the t-norm > and the distributions of the interarrival times and rewards. In other words, the condition A3 severs as a criterion of the t-norm-selection.
5. Concluding remarks In this paper, under the >-independence associated with continuous Archimedean t-norms, we studied a fuzzy random renewal reward process, and obtained the following new results. (i) We proved some limit theorems for fuzzy random renewal rewards (Theorems 1 and 2). (ii) We derived a fuzzy random renewal reward theorem (Theorem 3) for the long-run expected reward per unit time of the renewal reward process. In addition, we provided two applications, one involving a multi-service system and the other an element-replacement problem, respectively, so as to illustrate the use of our fuzzy random renewal reward theorem. In our future work, there are several issues which should be further discussed based on the current results. First of all, making use of the results obtained in this paper, additional other renewal processes in the fuzzy random environment can be studied, such as delayed renewal reward process and alternating renewal reward process. Furthermore, under the condition of >-independence discussed in this paper, we can also investigate renewal risk processes by considering fuzzy random interest. Finally, the application examples given in this work mainly focus on the calculation of the long-term expected average reward by using the fuzzy random reward renewal theorem obtained which we obtained; in our further studies, we will combine the results in this paper with intelligent computation approaches to solve some practical optimization problems, such as optimal replacement policy and inventory problems.
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Acknowledgments The authors are very grateful to the Editor-in-Chief and the anonymous referees for their insightful comments and suggestions which have contributed substantially to the improvement of this paper. This work was supported by the ‘‘Ambient SoC Global COE Program of Waseda University” of the Ministry of Education, Culture, Sports, Science and Technology, Japan, and by the Research Fellowships of the Japan Society for the Promotion of Science (JSPS) for Young Scientists. References [1] G. Allan, K. Oleq, S. Josef, Equivalences in strong limit theorems for renewal counting process, Statistics and Probability Letters 35 (4) (1997) 381–394. [2] G. Alsmeyer, Superposed and continuous renewal processes a Markov renewal approach, Stochastic Processes and Their Applications 61 (2) (1996) 311–322. [3] M. Benrejeb, A. Sakly, K.B. Othman, P. 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