Uncertain Random Alternating Renewal Process ... - Semantic Scholar
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 23, NO. 5, OCTOBER 2015
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Uncertain Random Alternating Renewal Process With Application to Interval Availability Kai Yao and Jinwu Gao
Abstract—As a mixture of a random variable and an uncertain variable, an uncertain random variable is a tool to deal with the indeterminacy quantities involving randomness and human uncertainty. This paper aims at proposing a concept of an uncertain random alternating renewal process to model a repairable system with random on-times and uncertain off-times. An alternating renewal theorem is proved, which gives the limit chance distribution of the interval availability of the uncertain random alternating renewal system. Index Terms—Alternating renewal process, renewal process, uncertain random variable, uncertainty theory.
I. INTRODUCTION HE alternating renewal process, as one of the most popular processes in renewal theory, is used to model systems ON and OFF alternately for some indeterminacy time. Traditionally, the process is assumed to behave randomly, and parameters such as interarrival times or system lifetimes are usually regarded as random variables. An important application of alternating renewal process is to model the interval availability of a repairable system; see [1] or [2] for details. In order to model a quantity with unclear bounds, Zadeh [3] proposed a concept of fuzzy set via membership function. Then fuzzy renewal process was proposed, of which the interarrival times and other parameters are fuzzy variables instead of random variables. Zhao and Liu [4] proved a fuzzy elementary renewal theorem for a type of fuzzy renewal process, Hong [5] discussed a fuzzy renewal reward process whose interarrival times and rewards were fuzzy variables with triangular norm, and Li [6] verified the alternating renewal theorem for a type of alternating renewal process with convex fuzzy variables. As a mixture of fuzzy variable and random variable, concepts of fuzzy random variable and random fuzzy variable were proposed by Kwakernaak [7], [8] and Liu [9], respectively. Then, parameters of a renewal process are characterized as fuzzy random variables or random fuzzy variables in some literature. For example, Shen et al. [10] proposed a type of alternating renewal process with random fuzzy variables. See [11]–[16].
T
Manuscript received February 2, 2014; revised June 25, 2014; accepted August 15, 2014. Date of publication September 26, 2014; date of current version October 2, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61403360 and Grant 61374082. (Corresponding author: Jinwu Gao.) K. Yao is with the School of Management, University of Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]). J. Gao is with the School of Information, Renmin University of China, Beijing 100872, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2014.2360551
In our daily life, we sometimes meet with a situation where we have no historical data about the event when making decisions. In this case, we have to rely on the experts for their belief degree about the possible outcome. In order to deal with the belief degree, some theories have been founded such as Dempster–Shafer theory (see [17] and [18]), prospect theory (see [19]), and rough set theory (see [20]). In order to deal with the uncertainty arising from the belief degree, an uncertainty theory was founded by Liu [21] and refined by Liu [22] based on normality, duality, subadditivity, and product axioms. In order to model the evolution of uncertain phenomenon, Liu [23] proposed a concept of uncertain process. Meanwhile, Liu [23] designed an uncertain renewal process as an important example, of which interarrival times are uncertain variables instead of random variables or fuzzy variables. Furthermore, Liu [22] proposed an uncertain renewal reward process whose interarrival times and rewards are both uncertain variables. After that, Yao and Li [24] proposed an uncertain alternating renewal process, and Zhang et al. [25] proposed an uncertain delayed renewal process. Inspired by the concept of fuzzy random variable, Liu [26] proposed an uncertain random variable to deal with a system behaving randomly and uncertainly. It can be regarded as a mixture of an uncertain variable and a random variable. Concepts of chance distribution, expected value, and variance were also provided by Liu [26] to describe an uncertain random variable. Following that, Liu [27] gave an operational law of uncertain random variables, and Yao and Gao [28] proved the law of large numbers for a sequence of uncertain random variables. In addition, Gao and Yao [29] presented a concept of uncertain random renewal process to model the sudden jumps in an uncertain random system. In our everyday life, some uncertain random systems are usually in two states: ON and OFF. They are initially ON and remain ON for some time; then, they go OFF and remain OFF for some time alternately. In this paper, we propose an uncertain random alternating renewal process to model such a system of which the on-times and off-times are uncertain random variables. The emphasis of this paper is put on the limit chance distribution of the interval availability, i.e., the ratio of the total on-times to the total time. The rest of this paper is organized as follows. In Section II, we review some concepts and properties about uncertain variable and uncertain random variable. After that, a type of uncertain random alternating renewal process is presented in Section III, and its total on-times and off-times are bounded by some inequalities. Then, an alternating renewal theorem is proved in Section IV, which gives the limit chance distribution of the interval availability of a system. After that, an application
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 23, NO. 5, OCTOBER 2015
of the alternating renewal theorem to system reliability is given in Section V. Finally, some remarks are made in Section VI. II. PRELIMINARY In this section, we introduce some basic concepts about uncertain variable and uncertain random variable, which are used throughout the paper. A. Uncertain Variable Uncertainty theory is a branch of axiomatic mathematics to deal with human uncertainty arising from the belief degree. First, a concept of uncertain measure was proposed to indicate the belief degree on the possible events. Definition 1 (see [21]): Let L be an σ-algebra on a nonempty set Γ. A set function M : L → [0, 1] is called an uncertain measure, if it satisfies the following axioms. Axiom 1 (Normality Axiom): M{Γ} = 1, for the universal set Γ. Axiom 2 (Duality Axiom): M{Λ} + M{Λc } = 1, for any event Λ. Axiom 3 (Subadditivity Axiom): For every countable sequence of events Λ1 , Λ2 , . . . , we have ∞ ∞ Λi ≤ M {Λi } . M i=1
i=1
Axiom 4 (Product Axiom): Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . . , n. Then, the product uncertain measure M on the product σ-algebra satisfies n n M Λk = Mk {Λk }. i=1
k =1
An uncertain variable is a measurable function from an uncertain space to real numbers, just as a random variable is on a probability space. Definition 2 (see [21]): An uncertain variable ξ is a measurable function from an uncertainty space (Γ, L, M) to the set of real numbers , i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ | ξ(γ) ∈ B} is an event. Definition 3 (see [21]): Let ξ be an uncertain variable. Then, its uncertainty distribution is defined by Φ(x) = M{ξ ≤ x} for any real number x. Definition 4 (see [30]): The uncertain variables {ξi } are said to be independent if ∞ ∞ M (ξi ∈ Bi ) = M{ξi ∈ Bi } i=1
i=1
for any Borel sets {Bi } of real numbers. The inverse function Φ−1 is called the inverse uncertainty distribution of ξ if it exists and is unique for each α ∈ (0, 1). For a sequence of independent and identically distributed (i.i.d.)
uncertain variables {ξi } with a common uncertainty distribution Φ, we have ∞ ∞ −1 (ξi ≤ Φ (α)) = M{ξi ≤ Φ−1 (α)} = α M i=1
i=1
for any α ∈ (0, 1). Inverse uncertainty distribution plays a crucial role in operations of independent uncertain variables. Theorem 1 (see [22]): Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with uncertainty distributions Φ1 , Φ2 , . . . , Φn , respectively. If f (x1 , x2 , . . . , xn ) is strictly increasing with respect to x1 , x2 , . . . , xm and strictly decreasing with respect to xm +1 , xm +2 , . . . , xn , then ξ = f (ξ1 , ξ2 , . . . , ξn ) is an uncertain variable with an inverse uncertainty distribution −1 Φ−1 (α) = f Φ−1 1 (α), . . . , Φm (α),
−1 Φ−1 m +1 (1 − α), . . . , Φn (1 − α) . Definition 5 (see [21]): Let ξ be an uncertain variable. Then, the expected value of ξ is defined by +∞ 0 E[ξ] = M{ξ ≥ r}dr − M{ξ ≤ r}dr −∞
0
provided that at least one of the two integrals is finite. For an uncertain variable ξ with a regular uncertainty distribution Φ, we have +∞ 0 1 E[ξ] = (1 − Φ(r))dr − Φ(r)dr = Φ−1 (α)dα. −∞
0
0
B. Uncertain Random Variable Let (Γ, L, M) be an uncertainty space, and (Ω, A, Pr) be a probability space. Then (Γ, L, M) × (Ω, A, Pr) = (Γ × Ω, L × A, M × Pr) is called a chance space. Definition 6 (see [26]): Let (Γ, L, M) × (Ω, A, Pr) be a chance space, and Θ ∈ L × A be an uncertain random event. Then, the chance measure Ch of Θ is defined by 1 Ch{Θ} = Pr{ω ∈ Ω | M{γ ∈ Γ | (γ, ω) ∈ Θ} ≥ r}dr. 0
Liu [26] verified that the chance measure Ch satisfies normality, duality, and monotonicity properties, that is, 1) Ch{Γ × Ω} = 1; 2) Ch{Θ} + Ch{Θc } = 1 for any event Θ; and 3) Ch{Θ1 } ≤ Ch{Θ2 } for any events Θ1 and Θ2 with Θ1 ⊂ Θ2 . Besides, Hou [31] proved the subadditivity of chance measure, that is, ∞ ∞ Θi ≤ Ch{Θi } Ch i=1
i=1
for a sequence of events {Θi }. Definition 7 (see [26]): An uncertain random variable ξ is a measurable function from a chance space (Γ, L, M) × (Ω, A, Pr) to the set of real numbers, i.e., {ξ ∈ B} = {(γ, ω) | ξ(γ, ω) ∈ B} is an uncertain random event for any Borel set B.
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
Random variables and uncertain variables can be regarded as special cases of uncertain random variables. Let η be a random variables, and τ be an uncertain variable. Then, η + τ and η × τ are both uncertain random variables. Definition 8 (see [26]): Let ξ be an uncertain random variable. Then, its chance distribution is defined by Φ(x) = Ch {ξ ≤ x}
∀x ∈ .
The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain variable is just its uncertainty distribution. It follows from Definition 6 that the chance distribution of an uncertain random variable ξ is 1 Pr{ω ∈ Ω | M{γ ∈ Γ|ξ(γ, ω) ≤ x} ≥ r}dr. Ch{ξ ≤ x} = 0
As to a random variable η with a probability distribution Ψ, an uncertain variable τ , and a measurable function f , Liu [27] proved that f (η, τ ) has a chance distribution +∞ Φ(y) = F (x, y)dΨ(x) −∞
where F (x, y) is the uncertainty distribution of f (x, τ ). Theorem 2 (see [24], Law of Large Numbers): Let η1 , η2 , . . . be a sequence of i.i.d. random variables with a common probability distribution Ψ, τ1 , τ2 , . . . be a sequence of i.i.d. uncertain variables, and f (x, y) be a strictly monotone function. Define S0 = 0, and Sn = f (η1 , τ1 ) + · · · + f (ηn , τn ) ∀n ≥ 1.
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is an uncertain random event, i.e., for each t ∈ T , the function Xt is an uncertain random variable. For each fixed γ ∗ ∈ Γ, the function Xt (γ ∗ , ·) is a stochastic process. For each fixed ω ∗ ∈ Γ, the function Xt (·, ω ∗ ) is an uncertain process. Definition 11 (see [29]): Assume Xt is an uncertain random process on a chance space (Γ, L, M) × (Ω, A, Pr). Then, for each fixed γ ∗ ∈ Γ and ω ∗ ∈ Ω, the function Xt (γ ∗ , ω ∗ ) is called a sample path of the uncertain random process Xt . Definition 12 (see [29]): Let η1 , η2 , . . . be a sequence of i.i.d. random variables, τ1 , τ2 , . . . be a sequence of i.i.d. uncertain variables, and f (x, y) be a positive measurable function. Define S0 = 0 and Sn = f (η1 , τ1 ) + f (η2 , τ2 ) + · · · + f (ηn , τn ) for n ≥ 1. Then Nt = max{n | Sn ≤ t} n ≥0
is called an uncertain random renewal process. Theorem 3: Let η1 , η2 , . . . be a sequence of i.i.d. random variables with a common probability distribution Ψ, τ1 , τ2 , . . . be a sequence of i.i.d. uncertain variables, and f (x, y) be a positive and strictly monotone function. Assume that Nt is an uncertain random renewal process with interarrival times f (ξ1 , η1 ), f (ξ2 , η2 ), . . . Then +∞
−1 Nt → f (x, τ1 ) dΨ(x) t −∞ in the sense of convergence in distribution as t → ∞. III. ALTERNATING RENEWAL PROCESS
Then Sn → n
+∞
−∞
f (x, τ1 )dΨ(x)
in the sense of convergence in distribution as n → ∞. Definition 9 (see [26]): Let ξ be an uncertain random variable. Then, its expected value is defined by +∞ 0 E[ξ] = Ch{ξ ≥ r}dr − Ch{ξ ≤ r}dr −∞
0
provided that at least one of the two integrals is finite. Assume ξ has a chance distribution Φ. Liu [27] proved that if E[ξ] exists, then +∞ 0 E[ξ] = (1 − Φ(x))dx − Φ(x)dx. 0
−∞
For a random variable η and an uncertain variable τ , Liu [27] proved that E[η + τ ] = E[η] + E[τ ] and E[η × τ ] = E[η] × E[τ ]. Definition 10 (see [29]): Let T be a totally ordered set, and (Γ × Ω, L × A, M × Pr) be a chance space. An uncertain random process is a function Xt (γ, ω) from T × (Γ × Ω, L × A, M × Pr) to the set of real numbers such that for any Borel set B of real numbers, the set {Xt ∈ B} = {(γ, ω) ∈ Γ × Ω | Xt (γ, ω) ∈ B}
Definition 13: Let η1 , η2 , . . . be a sequence of i.i.d. positive random variables, and τ1 , τ2 , . . . be a sequence of i.i.d. positive uncertain variables. Define ⎧ Nt ⎪ ⎪ t − τi , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt Nt ⎪ ⎪ ⎪ if (ηi + τi ) ≤ t < (ηi + τi ) + ηN t +1 ⎪ ⎨ i=1 i=1 Rt = N ⎪ t +1 ⎪ ⎪ ⎪ ηi , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt N t +1 ⎪ ⎪ ⎩ (ηi + τi ) + ηN t +1 < t < (ηi + τi ) if i=1
and
Ut =
i=1
⎧ Nt ⎪ ⎪ τi , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt Nt ⎪ ⎪ ⎪ if (ηi + τi ) ≤ t < (ηi + τi ) + ηN t +1 ⎪ ⎨ i=1
i=1
N ⎪ t +1 ⎪ ⎪ ⎪ ηi , ⎪t − ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt N t +1 ⎪ ⎪ ⎩ if (ηi + τi ) + ηN t +1 < t < (ηi + τi ) i=1
i=1
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 23, NO. 5, OCTOBER 2015
where Nt is an uncertain random renewal process with interarrival times {ηi + τi }. Then, the tuple (Rt , Ut ) is called an uncertain random alternating renewal process. Remark 1: Let {ηi } denote the on-times of a system, and {τi } denote the off-times of a system. Then, Rt is the total ontime, Ut is the total off-time of a system before some given time t, and Rt + Ut = t. Remark 2: Since the renewal process Nt is an uncertain random process, the total on-time Rt is not a random process, and Ut is not an uncertain process. They both are uncertain random processes with not only random factors but also uncertain factors. Remark 3: Let η1 , η2 , . . . be a sequence of i.i.d. random ontimes, and τ1 , τ2 , . . . be a sequence of i.i.d. uncertain off-times. Then, the uncertain random alternating renewal process (Rt , Ut ) satisfies Nt
ηi ≤ Rt ≤
N t +1
i=1
uncertain off-times {τi }. Assume τ1 has a regular uncertainty distributions Φ. Then N
t x lim Ch τi /t ≤ x ≤ Φ E[η] · . t→∞ 1−x i=1 Proof: It follows from the definition of chance measure that N t τi /t ≤ x Ch i=1
i=1
N t +1
lim Ch
τi
N t
1
=
This section proves the uncertain random alternating renewal theorem, i.e.,
1
t→∞
Pr M
0
≥ r dr,
τi ≤ tx
i=1
lim Pr M
i=1
=
and
1
N t
≥ r dr
τi ≤ tx
≥ r dr.
i=1
lim M
Pr
N t
t→∞
0
Hence, we first consider
in the sense of convergence in distribution as t → ∞. If Rt denotes the total on-time of the system before sometime t, then Rt /t is just the interval availability of the system. Therefore, the alternating renewal theorem provides a limitation of the interval availability of an uncertain random alternating renewal system. Lemma 1 (see [24]): Let A, B be two uncertain events in uncertain space (Γ, L, M). Then
By the continuity property of probability measure, we have N t τi /t ≤ x lim Ch
E[η1 ] Rt → t E[η1 ] + τ1 Ut τ1 → t E[η1 ] + τ1
τi ≤ tx
N t
t→∞
0
τi /t ≤ x
i=1
t→∞
where Nt is an uncertain random renewal process with interarrival times {ηi + τi }.
τi /t ≤ x ≥ r dr
= lim
i=1
i=1
(2)
IV. ALTERNATING RENEWAL THEOREM
Pr M
N t
0
(1)
i=1
1
=
i=1
τi ≤ Ut ≤
N t
0
t→∞ Nt
Pr M
=
then ηi
1
lim M
t→∞
τi ≤ tx
≥ r dr.
i=1
N t
τi ≤ tx .
i=1
For any given ε > 0, there exists a positive number m such that ∞ M (τi ≤ m) ≥ 1 − ε i=1
by the independence of {τi }. For convenience, write
M{A} ≤ M{A ∩ B} + M{B c }. Proof: By the subadditivity and monotonicity of uncertain measure, we have M{A} = M{A ∩ (B ∪ B c )}
B=
∞
{τi ≤ m}.
i=1
Then M{B} ≥ 1 − ε,
= M{(A ∩ B) ∪ (A ∩ B )} c
≤ M{A ∩ B} + M{A ∩ B c } ≤ M{A ∩ B} + M{B }. c
Theorem 4: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and
M{B c } = 1 − M{B} ≤ ε
by the duality of uncertain measure. Since the renewal process Nt can only take integer values, we have ∞ k N t τi ≤ tx = M τi ≤ tx ∩ (Nt = k) . M i=1
k =0
i=1
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
Let Mt denote a stochastic renewal process with random interarrival times η1 , η2 , . . ., Then k +1 ηi > t − tx − m = {k ≥ Mt−tx−m } .
Noting the uncertain random event k
{Nt = k} =
(ηi + τi ) ≤ t
t ,
i=1
we have M
N t
∞
k
k =0
i=1
≤M
i=1
τi ≤ tx
i=1
= Φ τi ≤ tx
∩
k +1
i=1
τi ≤ tx
i=1
⊂
k
∩
k +1
τi ≤ tx
∩
and
τk +1 +
i=1
k +1
lim Φ
M
ηi > t − tx
i=1
τi ≤ tx
t→∞
∞
k
k =0
i=1
≤M
∞
k
k =0
i=1
≤M
τi ≤ tx
∩
τk +1 +
k +1
ηi > t − tx
i=1
τi ≤ tx ∩ τk +1 +
k +1
ηi > t − tx ∩B
i=1
+M{B c } by Lemma 1. Noting that τk +1 (γ) ≤ m for any γ ∈ B and any nonnegative integer k, we have M
N t
≤M
k
k =0
i=1
t→∞
τi ≤ tx
∩
k +1
ηi > t − tx − m
∩B
i=1
+M{B c } ∞ k k +1 ≤M τi ≤ tx ∩ ηi > t − tx − m
+ε
+ε
a.s.
t − tx − m tx · = lim Φ t→∞ Mt−tx−m t − tx − m
x , a.s. = Φ E[η1 ] · 1−x
i=1
x 1−x
x 1−x
+ε
,
a.s.
Hence, we have N t τi /t ≤ x lim Ch t→∞
i=1
1
Pr
Mt−tx−m
holds for any ε > 0. As a result N t τi ≤ tx ≤ Φ E[η1 ] · lim M
0
τi ≤ tx
∞
⎭
tx
i=1
=
i=1
τi ≤ tx
⎫ ⎬
i=1
k =M t −t x −m
Then, the inequality N t τi ≤ tx ≤ Φ E[η1 ] · lim M
i=1
tx Mt−tx−m
t→∞
for every nonnegative integer k, we have N t
(ηi + τi ) > t
k
t − tx − m = E[η1 ], Mt−tx−m
lim
t→∞
i=1
∞
which is a random variable. By the strong law of large numbers for random variables, we have
(ηi + τi ) > t
by the monotonicity of uncertain measure. Since k
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1
≤
lim M
N t
t→∞
x = Φ E[η1 ] · 1−x
τi ≤ tx
i=1
Pr Φ E[η1 ] ·
0
x 1−x
≥ r dr
≥ r dr
.
The theorem is verified. Theorem 5: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and uncertain off-times {τi }. Assume τ1 has a regular uncertainty i=1 i=1 k =0 c distribution Φ. Then ∧M{B} + M{B } N +1
∞ k k +1 t x τi /t ≤ x ≥ Φ E[η] · lim Ch . ≤M τi ≤ tx ∩ ηi > t − tx − m + ε. t→∞ 1−x i=1 k =0
i=1
i=1
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 23, NO. 5, OCTOBER 2015
Proof: It follows from the definition of chance measure that N +1 t Ch τi /t > x i=1
1
=
Pr M
N +1 t
0
1
=
Pr M
N +1 t
lim Ch
1
1
=
Pr M
lim Pr M
N +1 t
t→∞
0
τi > tx
i=1
i=1
=
1
lim M
Pr
N +1 t
t→∞
0
Hence, we first consider
≥r
τi > tx
dr
lim M
t→∞
≥r
τi > tx
⊂
dr.
≥r
dr.
τi > tx .
For any given ε > 0, there exists a positive number m such that ∞ M (τi ≤ m) ≥ 1 − ε
i=1
≤M
≤M
(ηi + τi ) ≤ t
i=1
i=1
τi > tx
∩
k
{τi ≤ m}.
ηi ≤ t − tx + τk +1
i=1
∞
k +1
k =0
i=1
∞
k +1
k =0
i=1
τi > tx ∩
k
ηi ≤ t − tx + τk +1
i=1
k τi > tx ∩ ηi ≤ t − tx + τk +1 ∩ B i=1
+M{B } c
by Lemma 1. Noting that τk +1 (γ) ≤ m for any γ ∈ B and any nonnegative integer k, we have N +1 t M τi > tx i=1
by the independence of {τi }. For convenience, write
≤M
∞
k +1
k =0
i=1
τi > tx
∩
k
ηi ≤ t − tx + m ∩ B
i=1
+M{B c } ∞ k +1 k ≤M τi > tx ∩ ηi ≤ t − tx + m
i=1
Then M{B} ≥ 1 − ε,
∩
τi > tx
k
for every nonnegative integer k, we have N +1 t M τi > tx
i=1
B=
k +1
i=1
∞
i=1
i=1
i=1
N +1 t
k =0
i=1
i=1
k +1
≤M
By the continuity property of probability measure, we have N +1 t lim Ch τi /t > x t→∞
τi > tx
by the monotonicity of uncertain measure. Since k +1 k τi > tx ∩ (ηi + τi ) ≤ t
N +1 t
0
∞
dr,
τi /t > x
= lim
t→∞
≥r
i=1
M
N +1 t
τi > tx
(ηi + τi ) ≤ t
i=1
i=1
i=1
N +1 t
t→∞
we have
≥ r dr
τi /t > x
i=1
0
then
⊂
k
M{B c } = 1 − M{B} ≤ ε
k =0
i=1
i=1
∧M{B} + M{B } ∞ k +1 k ≤M τi > tx ∩ ηi ≤ t − tx + m + ε. c
by the duality of uncertain measure. Since the renewal process Nt can only take integer values, we have ∞ k +1 N +1 t τi > tx = M τi > tx ∩ (Nt = k) . M i=1
k =0
i=1
Noting the uncertain random event k k +1 {Nt = k} = (ηi + τi ) ≤ t < (ηi + τi ) i=1
i=1
k =0
i=1
i=1
Let Mt denote a stochastic renewal process with random interarrival times η1 , η2 , . . .. Then k ηi ≤ t − tx + m = {k ≤ Mt−tx+m } . i=1
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
Thus ⎧ ⎫ +1 N +1 t ⎨M t −t ⎬ x + m k τi > tx ≤ M τi > x +ε M ⎩ ⎭ i=1
i=1
k =0
= 1−Φ
tx Mt−tx+m + 1
+ε
which is a random variable. By the strong law of large numbers for random variables, we have Mt−tx−m 1 = , t − tx + m E[η1 ] and
lim Φ
i=1
⎞
tx ⎟ ⎠ Mt−tx−m (t − tx + m) + 1 t − tx + m
x = Φ E[η1 ] · , a.s. 1−x
t→∞
t→∞
Then, the inequality N +1 t τi > tx ≤ 1 − Φ E[η1 ] · lim M i=1
holds for any ε > 0. As a result N +1 t τi > tx ≤ 1 − Φ E[η1 ] · lim M t→∞
i=1
x 1−x
x 1−x
i=1
1
Pr
= 0
≤ 0
1
lim M
t→∞
t→∞
+ε
i=1
x 1−x
.
On the other hand, we have τ1 x ≤ x = M τ1 ≤ E[η1 ] · M E[η1 ] + τ1 1−x
x . = Φ E[η1 ] · 1−x
,
t→∞
by Theorems 4 and 5, we have lim Υt (x) = Φ E[η1 ] ·
a.s.
τ1 Ut → t E[η1 ] + τ1
N +1 t
Pr 1 − Φ E[η1 ] ·
≥r
τi > tx
i=1
x 1−x
in the sense of convergence in distribution as t → ∞. The uncertain random process Rt is the total on-times of the system before some given time t; therefore, Rt /t is just the interval availability of the system. Theorem 7: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and uncertain off-times {τi }. Then
dr
≥ r dr
Rt E[η1 ] → t E[η1 ] + τ1
.
By the duality of chance measure, we have N +1 N +1 t t τi /t ≤ x = 1 − lim Ch τi /t > x lim Ch i=1
i=1
Hence
x = 1 − Φ E[η1 ] · 1−x
t→∞
i=1
x = Φ E[η1 ] · 1−x
Hence, we have N +1 t τi /t > x lim Ch t→∞
in the sense of convergence in distribution as t → ∞. Proof: On the one hand, it follows from the inequality (2) that N +1 Nt t Ut ≤x ⊂ τi /t ≤ x ⊂ τi /t ≤ x t i=1 i=1
by the monotonicity of chance measure. Assume τ1 has an uncertainty distributions Φ. Since N N +1 t t τi /t ≤ x = lim Ch τi /t ≤ x lim Ch
⎜ = lim Φ ⎝
t→∞
τ1 Ut → t E[η1 ] + τ1
so Ut /t has a chance distribution Υt satisfying N +1 N t t Ch ξi /t ≤ x ≤ Υt (x) ≤ Ch ξi /t ≤ x
a.s.
Mt−tx+m + 1 ⎛
t→∞
uncertain off-times {τi }. Then
tx
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t→∞
i=1
x ≥ Φ E[η1 ] · 1−x
.
The theorem is verified. Theorem 6: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and
in the sense of convergence in distribution as t → ∞. Proof: Noting that Ut Rt =1− , t t
E[η1 ] τ1 =1− E[η1 ] + τ1 E[η1 ] + τ1
we immediately have Rt E[η1 ] → t E[η1 ] + τ1 in the sense of convergence in distribution as t → ∞ by Theorem 6.
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Remark 4: If the on-times {ηi } and the off-times {τi } are all random variables, then the interval availability E[η1 ] Rt → t E[η1 ] + E[τ1 ] in the sense of almost sure convergence. Remark 5: If the on-times {ηi } and the off-times {τi } are all uncertain variables, then the interval availability η1 Rt → t η1 + τ1 in the sense of convergence in distribution. Theorem 8 (Alternating Renewal Theorem): Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and uncertain off-times {τi }. Then ! E[η1 ] E[Rt ] lim = E , t→∞ t E[η1 ] + τ1 ! τ1 E[Ut ] = E . lim t→∞ t E[η1 ] + τ1 Proof: It follows from the definition of expected value that 1 Rt E[Rt ] = lim ≥ r dr Ch lim t→∞ t→∞ 0 t t 1 Rt = ≥ r dr. lim Ch t 0 t→∞ By Theorem 7, we have
E[η1 ] ≥ r dr E[η1 ] + τ1 0 ! E[η1 ] = E . E[η1 ] + τ1
E[Rt ] = t→∞ t lim
1
M
Since E[η1 ] = exp(μ + σ 2 /2), the interval availability exp(μ + σ 2 /2) Rt → t exp(μ + σ 2 /2) + τ1
bx + exp(μ + σ 2 /2)(x − 1) bx − ax
for exp(μ + σ 2 /2) exp(μ + σ 2 /2) ≤x≤ . 2 exp(μ + σ /2) + b exp(μ + σ 2 /2) + a In addition, the limit expected interval availability ! exp(μ1 + σ12 /2) E[Rt ] lim = E t→∞ t exp(μ1 + σ12 /2) + τ1 =
exp(μ + σ 2 /2) + b exp(μ + σ 2 /2) · ln . b−a exp(μ + σ 2 /2) + a
Example 2: Consider an uncertain random alternating renewal process (Rt , Ut ) with i.i.d. random on-times {ηi } and i.i.d. uncertain off-times {τi }. Assume η1 has an exponential probability distribution Ψ(x) = 1 − exp(−λx),
x>0
and τ1 has a lognormal uncertainty distribution
−1 π(μ − ln x) √ , x > 0. Φ(x) = 1 + exp 3σ
Rt 1 → t 1 + λτ1
The proof is completed. Example 1: Consider an uncertain random alternating renewal process (Rt , Ut ) with i.i.d. random on-times {ηi } and i.i.d. uncertain off-times {τi }. Assume η1 has a lognormal probability density function
(ln x − μ)2 1 exp − Ψ(x) = √ , x>0 2σ 2 2πσx and τ1 has a linear uncertainty distribution x−a , a ≤ x ≤ b. Φ(x) = b−a
=
Since E[η1 ] = 1/λ, the interval availability
Since Rt /t + Ut /t = 1, we have E[Ut ] E[Rt ] = 1 − lim lim t→∞ t→∞ t t ! E[η1 ] = 1−E E[η1 ] + τ1 ! τ1 = E . E[η1 ] + τ1
in the sense of convergence in distribution, and its limit chance distribution exp(μ + σ 2 /2) Rt ≤x = M ≤x lim Ch t→∞ t exp(μ + σ 2 /2) + τ1
in the sense of convergence in distribution, and its limit chance distribution Rt ≤x lim Ch t→∞ t 1 =M ≤x 1 + λτ1
−1 π(ln(1−x) − ln(λx)− μ) √ = 1+ exp , x > 0. 3σ In addition, the limit expected interval availability lim
t→∞
E[Rt ] t =E = 0
1 1 + λτ1 +∞
! √
(λ exp(μ))π /( 3σ ) 1 √ √ · dx. (λ exp(μ))π /( 3σ ) + xπ /( 3σ ) (1 + x)2
Especially, if λ exp(μ) = 1, then we have +∞ E[Rt ] 1 1 1 √ lim = · dx = 2 π /( 3σ ) t→∞ t (1 + x) 2 1+x 0
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
√ for any σ > 0. If σ = π/ 3, we have +∞ 1 E[Rt ] λ exp(μ) lim = · dx t→∞ t λ exp(μ) + x (1 + x)2 0 =
λ exp(μ)(λ exp(μ) − μ − ln λ − 1) . (λ exp(μ) − 1)2
However, the above integration has no explicit form in general; therefore, we have to employ numerical integration for given parameters λ, μ, and σ. For example, if λ = 1, μ = 1, and σ = √ 2π/ 3, then lim
t→∞
E[Rt ] = 0.7031. t
V. APPLICATION Consider a system that can be in one of two states at any time: working or under repair. Generally, the working times of a system have many samples, and their probability distribution can be obtained via statistics; therefore, we employ random variables to model the working times. The repair processes involve human uncertainty, and the possible repairing times are often predicated by the repairmen; therefore, we employ uncertain variables to model the repairing times. Then, the on-times and off-times of the system form an uncertain random alternating renewal process. Assume the system is working at the initial time. After a random time η1 , the system fails and undergoes repair for an uncertain time τ1 . When the repair is complete, the system is as good as new and returns to work immediately. After a random time η2 , the system fails and undergoes repair for an uncertain time τ2 . The process repeats infinitely. Now, we consider two important indexes of the system: failure rate and interval availability. Let Ψ denote the common probability distribution of the successive working times {ηi }, and Φ denote the common uncertainty distribution of the successive repairing times {τi }. Consider a cycle from the time that the system begins working to the time that the repair is completed. Such cycles form an uncertain random renewal process Nt with interarrival times {ηi + τi }. Note that Nt is just the total failure times of the system before a given time t, and Nt /t is the failure rate on the interval [0, t]. By the operational law of uncertain random variables, the sum ξi + ηi has a chance distribution +∞ Υ(x) = Ch{ηi + τi ≤ x} = M{y + τi ≤ x}dΨ(y) −∞
+∞
= −∞
Φ(x − y)dΨ(y)
and the expected failure rate E[Nt ]/t has a limitation ! 1 E[Nt ] =E . lim t→∞ t E[η1 ] + τ1 Let Rt denote the total on-time, and Ut denote the total off-time. Then, (Rt , Ut ) is just an uncertain random alternating renewal process with alternating interarrival times
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η1 , τ1 , η2 , τ2 , . . ., and Rt /t is the interval availability of the system. By Theorem 8, the expected interval availability E[Rt ]/t has a limitation ! E[η1 ] E[Rt ] =E lim . t→∞ t E[η1 ] + τ1 Example 3: Consider an uncertain random alternating renewal process (Rt , Ut ) with i.i.d. random on-times {ηi } and i.i.d. uncertain off-times {τi }. Assume the working times {ηi } of the system have an exponential distribution with a parameter λ, i.e., Ψ(x) = 1 − exp(−λx),
x≥0
and the repairing times {τi } of the system have a linear uncertainty distribution Φ(x) =
x−a , b−a
a≤x≤b
evaluated by some repairmen. Since E[η1 ] = 1/λ, the failure rate Nt λ → t λ + τ1 in the sense of convergence in distribution, and the limitation of expected failure rate ! λ 1 + λb E[Nt ] 1 =E · ln . lim = t→∞ t 1 + λτ1 b−a 1 + λa The interval availability Rt 1 → t 1 + λτ1 in the sense of convergence in distribution, and the limitation of expected interval availability ! 1 E[Rt ] 1 + λb 1 =E · ln . lim = t→∞ t 1 + λτ1 λ(b − a) 1 + λa VI. CONCLUSION This paper mainly presents a concept of alternating renewal process with i.i.d. random on-times and i.i.d. uncertain off-times as the alternating interarrival times. An uncertain random alternating renewal theorem is verified, which provides the chance distribution of the interval availability. An application to system reliability is given to illustrate the alternating renewal theorem. REFERENCES [1] R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. New York, NY, USA: Wiley, 1965. [2] T. Nakagawa, Maintenance Theory of Reliability. London, U.K.: SpringerVerlag, 2005. [3] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, Jun. 1965. [4] R. Zhao and B. Liu, “Renewal process with fuzzy interarrival times and costs,” Int. J. Uncertainity Fuzziness, Knowl.-Based Syst., vol. 11, no. 3, pp. 573–586, Jun. 2003. [5] D. Hong, “Renewal process with T -related fuzzy inter-arrival times and fuzzy rewards,” Inf. Sci., vol. 176, no. 16, pp. 2386–2395, Aug. 2006. [6] S. Li, “Some properties of fuzzy alternating renewal processes,” Math. Comput. Model., vol. 54, no. 9/10, pp. 1886–1896, Nov. 2011.
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[7] H. Kwakernaak, “Fuzzy random variables-I: Definitions and theorems,” Inf. Sci., vol. 15, no. 1, pp. 1–29, Jul. 1978. [8] H. Kwakernaak, “Fuzzy random variables-II: Algorithms and examples for the discrete case,” Inf. Sci., vol. 17, no. 3, pp. 253–278, Apr. 1979. [9] B. Liu, Theory and Practice of Uncertain Programming. Heidelberg, Germany: Physica-Verlag, 2002. [10] Q. Shen, R. Zhao, and W. Tang, “Random fuzzy alternating renewal processes,” Soft Comput., vol. 13, no. 2, pp. 139–147, Apr. 2009. [11] E. Popova and H. C. Wu, “Renewal reward processes with fuzzy rewards and their applications to T -age replacement policies,” Eur. J. Oper. Res., vol. 117, no. 3, pp. 606–617, Sep. 1999. [12] C. M. Hwang, “A theorem of renewal process for fuzzy random variables and its application,” Fuzzy Set. Syst., vol. 116, no. 2, pp. 237–244, Dec. 2000. [13] M. Dozzi, E. Merzbach, and V. Schmidt, “Limit theorems for sums of random fuzzy sets,” J. Math. Anal. Appl., vol. 259, no. 2, pp. 554–565, Jul. 2001. [14] S. Wang, Y. K. Liu, and J. Watada, “Fuzzy random renewal process with queueing applications,” Comput. Math. Appl., vol. 57, no. 7, pp. 1232–1248, Apr. 2009. [15] S. Wang and J. Watada, “Fuzzy random renewal reward process with its applications,” Inf. Sci., vol. 179, no. 23, pp. 4057–4069, Nov. 2009. [16] R. Zhao and W. Tang, “Some properties of fuzzy random renewal process,” IEEE Trans. Fuzzy Syst., vol. 14, no. 2, pp. 173–179, Apr. 2006. [17] A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” Ann. Math. Statist., vol. 38, no. 2, pp. 325–339, 1967. [18] G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ, USA: Princeton Univ. Press, 1976. [19] D. Kahneman and A. Tversky, “Prospect theory: An analysis of decision under risk,” Econometrica, vol. 47, no. 2, pp. 263–292, Mar. 1979. [20] Z. Pawlak, “Rough sets,” Int. J. Parallel Program., vol. 11, no. 5, pp. 341–356, Oct. 1982. [21] B. Liu, Uncertainty Theory, 2nd ed. Berlin, Germany: Springer-Verlag, 2007. [22] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin, Germany: Springer-Verlag, 2010. [23] B. Liu, “Fuzzy process, hybrid process and uncertain process,” J. Uncertain Syst., vol. 2, no. 1, pp. 3–16, Feb. 2008. [24] K. Yao and X. Li, “Uncertain alternating renewal process and its application,” IEEE Trans. Fuzzy Syst., vol. 20, no. 6, pp. 1154–1160, Dec. 2012. [25] X. F. Zhang, Y. F. Ning, and G. W. Meng, “Delayed renewal process with uncertain interarrival times,” Fuzzy Optim. Dec. Making, vol. 12, no. 1, pp. 79-87, Mar. 2013. [26] Y. H. Liu, “Uncertain random variables: A mixture of uncertainty and randomness,” Soft Comput., vol. 17, no. 4, pp. 625–634, Apr. 2013. [27] Y. H. Liu, “Uncertain random programming with applications,” Fuzzy Optim. Dec. Making, vol. 12, no. 2, pp. 153–169, Jun. 2013. [28] K. Yao and J. Gao (2014, Apr. 1). Law of large numbers for uncertain random variables. [Online]. Available: http://orsc.edu.cn/online/120401.pdf [29] J. Gao and K. Yao, “Some concepts and theorems of uncertain random process,” Int. J. Intell. Syst., DOI: 10.1002/int.21681, Sep. 2014.
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[30] B. Liu, “Some research problems in uncertainty theory,” J. Uncertain Syst., vol. 3, no. 1, pp. 3–10, Feb. 2009. [31] Y. C. Hou, “Subadditivity of chance measure,” J. Uncertain. Anal. Appl., vol. 2, art. 14, pp. 1–8, Jun. 2014.
Kai Yao received the B.S. degree from Nankai University, Tianjin, China, in 2009 and the Ph.D. degree from Tsinghua University, Beijing, China, in 2013. He is currently an Assistant Professor with the School of Management, University of Chinese Academy of Sciences, Beijing. He has authored or coauthored more than 20 articles in several journals, including the IEEE TRANSACTIONS ON FUZZY SYSTEMS, Knowledge-Based Systems, Applied Soft Computing, Applied Mathematical Modelling, Applied Mathematics and Computation, Fuzzy Optimization and Decision Making, and Soft Computing. His current research interests include uncertain renewal processes, uncertain systems, and uncertain differential equations and their applications.
Jinwu Gao received the B.S. degree in mathematics from Shaanxi Normal University, Xian, China, in 1996 and the M.S. and Ph.D. degrees in mathematics from Tsinghua University, Beijing, China, in 2005. He is currently an Associate Professor of mathematics with the Renmin University of China, Beijing. His current research interests include fuzzy systems, uncertain systems and their application in optimization, game theory, and finance. He has authored or coauthored more than 30 papers that have appeared in the Soft Computing, Fuzzy Optimization and Decision Making, the International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, the Iranian Journal of Fuzzy Systems, Computer and Mathematics with Applications, and other publications. Dr. Gao has been the Coeditor-in-Chief of the Journal of Uncertain Systems since 2011 and the Executive Editor-in-Chief of the Journal of Uncertainty Analysis and Applications since 2013. He has served as the Vice President of Intelligent Computing Chapter of the Operations Society of China since 2007 and as Vice President of the International Consortium for Uncertainty Theory since 2013. He served as Vice President of the International Association for Information and Management Science from 2010 to 2013 and as President of the International Consortium for Electronic Business from 2012 to 2013.
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Uncertain Random Alternating Renewal Process With Application to Interval Availability Kai Yao and Jinwu Gao
Abstract—As a mixture of a random variable and an uncertain variable, an uncertain random variable is a tool to deal with the indeterminacy quantities involving randomness and human uncertainty. This paper aims at proposing a concept of an uncertain random alternating renewal process to model a repairable system with random on-times and uncertain off-times. An alternating renewal theorem is proved, which gives the limit chance distribution of the interval availability of the uncertain random alternating renewal system. Index Terms—Alternating renewal process, renewal process, uncertain random variable, uncertainty theory.
I. INTRODUCTION HE alternating renewal process, as one of the most popular processes in renewal theory, is used to model systems ON and OFF alternately for some indeterminacy time. Traditionally, the process is assumed to behave randomly, and parameters such as interarrival times or system lifetimes are usually regarded as random variables. An important application of alternating renewal process is to model the interval availability of a repairable system; see [1] or [2] for details. In order to model a quantity with unclear bounds, Zadeh [3] proposed a concept of fuzzy set via membership function. Then fuzzy renewal process was proposed, of which the interarrival times and other parameters are fuzzy variables instead of random variables. Zhao and Liu [4] proved a fuzzy elementary renewal theorem for a type of fuzzy renewal process, Hong [5] discussed a fuzzy renewal reward process whose interarrival times and rewards were fuzzy variables with triangular norm, and Li [6] verified the alternating renewal theorem for a type of alternating renewal process with convex fuzzy variables. As a mixture of fuzzy variable and random variable, concepts of fuzzy random variable and random fuzzy variable were proposed by Kwakernaak [7], [8] and Liu [9], respectively. Then, parameters of a renewal process are characterized as fuzzy random variables or random fuzzy variables in some literature. For example, Shen et al. [10] proposed a type of alternating renewal process with random fuzzy variables. See [11]–[16].
T
Manuscript received February 2, 2014; revised June 25, 2014; accepted August 15, 2014. Date of publication September 26, 2014; date of current version October 2, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61403360 and Grant 61374082. (Corresponding author: Jinwu Gao.) K. Yao is with the School of Management, University of Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]). J. Gao is with the School of Information, Renmin University of China, Beijing 100872, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2014.2360551
In our daily life, we sometimes meet with a situation where we have no historical data about the event when making decisions. In this case, we have to rely on the experts for their belief degree about the possible outcome. In order to deal with the belief degree, some theories have been founded such as Dempster–Shafer theory (see [17] and [18]), prospect theory (see [19]), and rough set theory (see [20]). In order to deal with the uncertainty arising from the belief degree, an uncertainty theory was founded by Liu [21] and refined by Liu [22] based on normality, duality, subadditivity, and product axioms. In order to model the evolution of uncertain phenomenon, Liu [23] proposed a concept of uncertain process. Meanwhile, Liu [23] designed an uncertain renewal process as an important example, of which interarrival times are uncertain variables instead of random variables or fuzzy variables. Furthermore, Liu [22] proposed an uncertain renewal reward process whose interarrival times and rewards are both uncertain variables. After that, Yao and Li [24] proposed an uncertain alternating renewal process, and Zhang et al. [25] proposed an uncertain delayed renewal process. Inspired by the concept of fuzzy random variable, Liu [26] proposed an uncertain random variable to deal with a system behaving randomly and uncertainly. It can be regarded as a mixture of an uncertain variable and a random variable. Concepts of chance distribution, expected value, and variance were also provided by Liu [26] to describe an uncertain random variable. Following that, Liu [27] gave an operational law of uncertain random variables, and Yao and Gao [28] proved the law of large numbers for a sequence of uncertain random variables. In addition, Gao and Yao [29] presented a concept of uncertain random renewal process to model the sudden jumps in an uncertain random system. In our everyday life, some uncertain random systems are usually in two states: ON and OFF. They are initially ON and remain ON for some time; then, they go OFF and remain OFF for some time alternately. In this paper, we propose an uncertain random alternating renewal process to model such a system of which the on-times and off-times are uncertain random variables. The emphasis of this paper is put on the limit chance distribution of the interval availability, i.e., the ratio of the total on-times to the total time. The rest of this paper is organized as follows. In Section II, we review some concepts and properties about uncertain variable and uncertain random variable. After that, a type of uncertain random alternating renewal process is presented in Section III, and its total on-times and off-times are bounded by some inequalities. Then, an alternating renewal theorem is proved in Section IV, which gives the limit chance distribution of the interval availability of a system. After that, an application
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of the alternating renewal theorem to system reliability is given in Section V. Finally, some remarks are made in Section VI. II. PRELIMINARY In this section, we introduce some basic concepts about uncertain variable and uncertain random variable, which are used throughout the paper. A. Uncertain Variable Uncertainty theory is a branch of axiomatic mathematics to deal with human uncertainty arising from the belief degree. First, a concept of uncertain measure was proposed to indicate the belief degree on the possible events. Definition 1 (see [21]): Let L be an σ-algebra on a nonempty set Γ. A set function M : L → [0, 1] is called an uncertain measure, if it satisfies the following axioms. Axiom 1 (Normality Axiom): M{Γ} = 1, for the universal set Γ. Axiom 2 (Duality Axiom): M{Λ} + M{Λc } = 1, for any event Λ. Axiom 3 (Subadditivity Axiom): For every countable sequence of events Λ1 , Λ2 , . . . , we have ∞ ∞ Λi ≤ M {Λi } . M i=1
i=1
Axiom 4 (Product Axiom): Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . . , n. Then, the product uncertain measure M on the product σ-algebra satisfies n n M Λk = Mk {Λk }. i=1
k =1
An uncertain variable is a measurable function from an uncertain space to real numbers, just as a random variable is on a probability space. Definition 2 (see [21]): An uncertain variable ξ is a measurable function from an uncertainty space (Γ, L, M) to the set of real numbers , i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ | ξ(γ) ∈ B} is an event. Definition 3 (see [21]): Let ξ be an uncertain variable. Then, its uncertainty distribution is defined by Φ(x) = M{ξ ≤ x} for any real number x. Definition 4 (see [30]): The uncertain variables {ξi } are said to be independent if ∞ ∞ M (ξi ∈ Bi ) = M{ξi ∈ Bi } i=1
i=1
for any Borel sets {Bi } of real numbers. The inverse function Φ−1 is called the inverse uncertainty distribution of ξ if it exists and is unique for each α ∈ (0, 1). For a sequence of independent and identically distributed (i.i.d.)
uncertain variables {ξi } with a common uncertainty distribution Φ, we have ∞ ∞ −1 (ξi ≤ Φ (α)) = M{ξi ≤ Φ−1 (α)} = α M i=1
i=1
for any α ∈ (0, 1). Inverse uncertainty distribution plays a crucial role in operations of independent uncertain variables. Theorem 1 (see [22]): Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with uncertainty distributions Φ1 , Φ2 , . . . , Φn , respectively. If f (x1 , x2 , . . . , xn ) is strictly increasing with respect to x1 , x2 , . . . , xm and strictly decreasing with respect to xm +1 , xm +2 , . . . , xn , then ξ = f (ξ1 , ξ2 , . . . , ξn ) is an uncertain variable with an inverse uncertainty distribution −1 Φ−1 (α) = f Φ−1 1 (α), . . . , Φm (α),
−1 Φ−1 m +1 (1 − α), . . . , Φn (1 − α) . Definition 5 (see [21]): Let ξ be an uncertain variable. Then, the expected value of ξ is defined by +∞ 0 E[ξ] = M{ξ ≥ r}dr − M{ξ ≤ r}dr −∞
0
provided that at least one of the two integrals is finite. For an uncertain variable ξ with a regular uncertainty distribution Φ, we have +∞ 0 1 E[ξ] = (1 − Φ(r))dr − Φ(r)dr = Φ−1 (α)dα. −∞
0
0
B. Uncertain Random Variable Let (Γ, L, M) be an uncertainty space, and (Ω, A, Pr) be a probability space. Then (Γ, L, M) × (Ω, A, Pr) = (Γ × Ω, L × A, M × Pr) is called a chance space. Definition 6 (see [26]): Let (Γ, L, M) × (Ω, A, Pr) be a chance space, and Θ ∈ L × A be an uncertain random event. Then, the chance measure Ch of Θ is defined by 1 Ch{Θ} = Pr{ω ∈ Ω | M{γ ∈ Γ | (γ, ω) ∈ Θ} ≥ r}dr. 0
Liu [26] verified that the chance measure Ch satisfies normality, duality, and monotonicity properties, that is, 1) Ch{Γ × Ω} = 1; 2) Ch{Θ} + Ch{Θc } = 1 for any event Θ; and 3) Ch{Θ1 } ≤ Ch{Θ2 } for any events Θ1 and Θ2 with Θ1 ⊂ Θ2 . Besides, Hou [31] proved the subadditivity of chance measure, that is, ∞ ∞ Θi ≤ Ch{Θi } Ch i=1
i=1
for a sequence of events {Θi }. Definition 7 (see [26]): An uncertain random variable ξ is a measurable function from a chance space (Γ, L, M) × (Ω, A, Pr) to the set of real numbers, i.e., {ξ ∈ B} = {(γ, ω) | ξ(γ, ω) ∈ B} is an uncertain random event for any Borel set B.
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
Random variables and uncertain variables can be regarded as special cases of uncertain random variables. Let η be a random variables, and τ be an uncertain variable. Then, η + τ and η × τ are both uncertain random variables. Definition 8 (see [26]): Let ξ be an uncertain random variable. Then, its chance distribution is defined by Φ(x) = Ch {ξ ≤ x}
∀x ∈ .
The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain variable is just its uncertainty distribution. It follows from Definition 6 that the chance distribution of an uncertain random variable ξ is 1 Pr{ω ∈ Ω | M{γ ∈ Γ|ξ(γ, ω) ≤ x} ≥ r}dr. Ch{ξ ≤ x} = 0
As to a random variable η with a probability distribution Ψ, an uncertain variable τ , and a measurable function f , Liu [27] proved that f (η, τ ) has a chance distribution +∞ Φ(y) = F (x, y)dΨ(x) −∞
where F (x, y) is the uncertainty distribution of f (x, τ ). Theorem 2 (see [24], Law of Large Numbers): Let η1 , η2 , . . . be a sequence of i.i.d. random variables with a common probability distribution Ψ, τ1 , τ2 , . . . be a sequence of i.i.d. uncertain variables, and f (x, y) be a strictly monotone function. Define S0 = 0, and Sn = f (η1 , τ1 ) + · · · + f (ηn , τn ) ∀n ≥ 1.
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is an uncertain random event, i.e., for each t ∈ T , the function Xt is an uncertain random variable. For each fixed γ ∗ ∈ Γ, the function Xt (γ ∗ , ·) is a stochastic process. For each fixed ω ∗ ∈ Γ, the function Xt (·, ω ∗ ) is an uncertain process. Definition 11 (see [29]): Assume Xt is an uncertain random process on a chance space (Γ, L, M) × (Ω, A, Pr). Then, for each fixed γ ∗ ∈ Γ and ω ∗ ∈ Ω, the function Xt (γ ∗ , ω ∗ ) is called a sample path of the uncertain random process Xt . Definition 12 (see [29]): Let η1 , η2 , . . . be a sequence of i.i.d. random variables, τ1 , τ2 , . . . be a sequence of i.i.d. uncertain variables, and f (x, y) be a positive measurable function. Define S0 = 0 and Sn = f (η1 , τ1 ) + f (η2 , τ2 ) + · · · + f (ηn , τn ) for n ≥ 1. Then Nt = max{n | Sn ≤ t} n ≥0
is called an uncertain random renewal process. Theorem 3: Let η1 , η2 , . . . be a sequence of i.i.d. random variables with a common probability distribution Ψ, τ1 , τ2 , . . . be a sequence of i.i.d. uncertain variables, and f (x, y) be a positive and strictly monotone function. Assume that Nt is an uncertain random renewal process with interarrival times f (ξ1 , η1 ), f (ξ2 , η2 ), . . . Then +∞
−1 Nt → f (x, τ1 ) dΨ(x) t −∞ in the sense of convergence in distribution as t → ∞. III. ALTERNATING RENEWAL PROCESS
Then Sn → n
+∞
−∞
f (x, τ1 )dΨ(x)
in the sense of convergence in distribution as n → ∞. Definition 9 (see [26]): Let ξ be an uncertain random variable. Then, its expected value is defined by +∞ 0 E[ξ] = Ch{ξ ≥ r}dr − Ch{ξ ≤ r}dr −∞
0
provided that at least one of the two integrals is finite. Assume ξ has a chance distribution Φ. Liu [27] proved that if E[ξ] exists, then +∞ 0 E[ξ] = (1 − Φ(x))dx − Φ(x)dx. 0
−∞
For a random variable η and an uncertain variable τ , Liu [27] proved that E[η + τ ] = E[η] + E[τ ] and E[η × τ ] = E[η] × E[τ ]. Definition 10 (see [29]): Let T be a totally ordered set, and (Γ × Ω, L × A, M × Pr) be a chance space. An uncertain random process is a function Xt (γ, ω) from T × (Γ × Ω, L × A, M × Pr) to the set of real numbers such that for any Borel set B of real numbers, the set {Xt ∈ B} = {(γ, ω) ∈ Γ × Ω | Xt (γ, ω) ∈ B}
Definition 13: Let η1 , η2 , . . . be a sequence of i.i.d. positive random variables, and τ1 , τ2 , . . . be a sequence of i.i.d. positive uncertain variables. Define ⎧ Nt ⎪ ⎪ t − τi , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt Nt ⎪ ⎪ ⎪ if (ηi + τi ) ≤ t < (ηi + τi ) + ηN t +1 ⎪ ⎨ i=1 i=1 Rt = N ⎪ t +1 ⎪ ⎪ ⎪ ηi , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt N t +1 ⎪ ⎪ ⎩ (ηi + τi ) + ηN t +1 < t < (ηi + τi ) if i=1
and
Ut =
i=1
⎧ Nt ⎪ ⎪ τi , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt Nt ⎪ ⎪ ⎪ if (ηi + τi ) ≤ t < (ηi + τi ) + ηN t +1 ⎪ ⎨ i=1
i=1
N ⎪ t +1 ⎪ ⎪ ⎪ ηi , ⎪t − ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ Nt N t +1 ⎪ ⎪ ⎩ if (ηi + τi ) + ηN t +1 < t < (ηi + τi ) i=1
i=1
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where Nt is an uncertain random renewal process with interarrival times {ηi + τi }. Then, the tuple (Rt , Ut ) is called an uncertain random alternating renewal process. Remark 1: Let {ηi } denote the on-times of a system, and {τi } denote the off-times of a system. Then, Rt is the total ontime, Ut is the total off-time of a system before some given time t, and Rt + Ut = t. Remark 2: Since the renewal process Nt is an uncertain random process, the total on-time Rt is not a random process, and Ut is not an uncertain process. They both are uncertain random processes with not only random factors but also uncertain factors. Remark 3: Let η1 , η2 , . . . be a sequence of i.i.d. random ontimes, and τ1 , τ2 , . . . be a sequence of i.i.d. uncertain off-times. Then, the uncertain random alternating renewal process (Rt , Ut ) satisfies Nt
ηi ≤ Rt ≤
N t +1
i=1
uncertain off-times {τi }. Assume τ1 has a regular uncertainty distributions Φ. Then N
t x lim Ch τi /t ≤ x ≤ Φ E[η] · . t→∞ 1−x i=1 Proof: It follows from the definition of chance measure that N t τi /t ≤ x Ch i=1
i=1
N t +1
lim Ch
τi
N t
1
=
This section proves the uncertain random alternating renewal theorem, i.e.,
1
t→∞
Pr M
0
≥ r dr,
τi ≤ tx
i=1
lim Pr M
i=1
=
and
1
N t
≥ r dr
τi ≤ tx
≥ r dr.
i=1
lim M
Pr
N t
t→∞
0
Hence, we first consider
in the sense of convergence in distribution as t → ∞. If Rt denotes the total on-time of the system before sometime t, then Rt /t is just the interval availability of the system. Therefore, the alternating renewal theorem provides a limitation of the interval availability of an uncertain random alternating renewal system. Lemma 1 (see [24]): Let A, B be two uncertain events in uncertain space (Γ, L, M). Then
By the continuity property of probability measure, we have N t τi /t ≤ x lim Ch
E[η1 ] Rt → t E[η1 ] + τ1 Ut τ1 → t E[η1 ] + τ1
τi ≤ tx
N t
t→∞
0
τi /t ≤ x
i=1
t→∞
where Nt is an uncertain random renewal process with interarrival times {ηi + τi }.
τi /t ≤ x ≥ r dr
= lim
i=1
i=1
(2)
IV. ALTERNATING RENEWAL THEOREM
Pr M
N t
0
(1)
i=1
1
=
i=1
τi ≤ Ut ≤
N t
0
t→∞ Nt
Pr M
=
then ηi
1
lim M
t→∞
τi ≤ tx
≥ r dr.
i=1
N t
τi ≤ tx .
i=1
For any given ε > 0, there exists a positive number m such that ∞ M (τi ≤ m) ≥ 1 − ε i=1
by the independence of {τi }. For convenience, write
M{A} ≤ M{A ∩ B} + M{B c }. Proof: By the subadditivity and monotonicity of uncertain measure, we have M{A} = M{A ∩ (B ∪ B c )}
B=
∞
{τi ≤ m}.
i=1
Then M{B} ≥ 1 − ε,
= M{(A ∩ B) ∪ (A ∩ B )} c
≤ M{A ∩ B} + M{A ∩ B c } ≤ M{A ∩ B} + M{B }. c
Theorem 4: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and
M{B c } = 1 − M{B} ≤ ε
by the duality of uncertain measure. Since the renewal process Nt can only take integer values, we have ∞ k N t τi ≤ tx = M τi ≤ tx ∩ (Nt = k) . M i=1
k =0
i=1
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
Let Mt denote a stochastic renewal process with random interarrival times η1 , η2 , . . ., Then k +1 ηi > t − tx − m = {k ≥ Mt−tx−m } .
Noting the uncertain random event k
{Nt = k} =
(ηi + τi ) ≤ t
t ,
i=1
we have M
N t
∞
k
k =0
i=1
≤M
i=1
τi ≤ tx
i=1
= Φ τi ≤ tx
∩
k +1
i=1
τi ≤ tx
i=1
⊂
k
∩
k +1
τi ≤ tx
∩
and
τk +1 +
i=1
k +1
lim Φ
M
ηi > t − tx
i=1
τi ≤ tx
t→∞
∞
k
k =0
i=1
≤M
∞
k
k =0
i=1
≤M
τi ≤ tx
∩
τk +1 +
k +1
ηi > t − tx
i=1
τi ≤ tx ∩ τk +1 +
k +1
ηi > t − tx ∩B
i=1
+M{B c } by Lemma 1. Noting that τk +1 (γ) ≤ m for any γ ∈ B and any nonnegative integer k, we have M
N t
≤M
k
k =0
i=1
t→∞
τi ≤ tx
∩
k +1
ηi > t − tx − m
∩B
i=1
+M{B c } ∞ k k +1 ≤M τi ≤ tx ∩ ηi > t − tx − m
+ε
+ε
a.s.
t − tx − m tx · = lim Φ t→∞ Mt−tx−m t − tx − m
x , a.s. = Φ E[η1 ] · 1−x
i=1
x 1−x
x 1−x
+ε
,
a.s.
Hence, we have N t τi /t ≤ x lim Ch t→∞
i=1
1
Pr
Mt−tx−m
holds for any ε > 0. As a result N t τi ≤ tx ≤ Φ E[η1 ] · lim M
0
τi ≤ tx
∞
⎭
tx
i=1
=
i=1
τi ≤ tx
⎫ ⎬
i=1
k =M t −t x −m
Then, the inequality N t τi ≤ tx ≤ Φ E[η1 ] · lim M
i=1
tx Mt−tx−m
t→∞
for every nonnegative integer k, we have N t
(ηi + τi ) > t
k
t − tx − m = E[η1 ], Mt−tx−m
lim
t→∞
i=1
∞
which is a random variable. By the strong law of large numbers for random variables, we have
(ηi + τi ) > t
by the monotonicity of uncertain measure. Since k
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1
≤
lim M
N t
t→∞
x = Φ E[η1 ] · 1−x
τi ≤ tx
i=1
Pr Φ E[η1 ] ·
0
x 1−x
≥ r dr
≥ r dr
.
The theorem is verified. Theorem 5: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and uncertain off-times {τi }. Assume τ1 has a regular uncertainty i=1 i=1 k =0 c distribution Φ. Then ∧M{B} + M{B } N +1
∞ k k +1 t x τi /t ≤ x ≥ Φ E[η] · lim Ch . ≤M τi ≤ tx ∩ ηi > t − tx − m + ε. t→∞ 1−x i=1 k =0
i=1
i=1
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 23, NO. 5, OCTOBER 2015
Proof: It follows from the definition of chance measure that N +1 t Ch τi /t > x i=1
1
=
Pr M
N +1 t
0
1
=
Pr M
N +1 t
lim Ch
1
1
=
Pr M
lim Pr M
N +1 t
t→∞
0
τi > tx
i=1
i=1
=
1
lim M
Pr
N +1 t
t→∞
0
Hence, we first consider
≥r
τi > tx
dr
lim M
t→∞
≥r
τi > tx
⊂
dr.
≥r
dr.
τi > tx .
For any given ε > 0, there exists a positive number m such that ∞ M (τi ≤ m) ≥ 1 − ε
i=1
≤M
≤M
(ηi + τi ) ≤ t
i=1
i=1
τi > tx
∩
k
{τi ≤ m}.
ηi ≤ t − tx + τk +1
i=1
∞
k +1
k =0
i=1
∞
k +1
k =0
i=1
τi > tx ∩
k
ηi ≤ t − tx + τk +1
i=1
k τi > tx ∩ ηi ≤ t − tx + τk +1 ∩ B i=1
+M{B } c
by Lemma 1. Noting that τk +1 (γ) ≤ m for any γ ∈ B and any nonnegative integer k, we have N +1 t M τi > tx i=1
by the independence of {τi }. For convenience, write
≤M
∞
k +1
k =0
i=1
τi > tx
∩
k
ηi ≤ t − tx + m ∩ B
i=1
+M{B c } ∞ k +1 k ≤M τi > tx ∩ ηi ≤ t − tx + m
i=1
Then M{B} ≥ 1 − ε,
∩
τi > tx
k
for every nonnegative integer k, we have N +1 t M τi > tx
i=1
B=
k +1
i=1
∞
i=1
i=1
i=1
N +1 t
k =0
i=1
i=1
k +1
≤M
By the continuity property of probability measure, we have N +1 t lim Ch τi /t > x t→∞
τi > tx
by the monotonicity of uncertain measure. Since k +1 k τi > tx ∩ (ηi + τi ) ≤ t
N +1 t
0
∞
dr,
τi /t > x
= lim
t→∞
≥r
i=1
M
N +1 t
τi > tx
(ηi + τi ) ≤ t
i=1
i=1
i=1
N +1 t
t→∞
we have
≥ r dr
τi /t > x
i=1
0
then
⊂
k
M{B c } = 1 − M{B} ≤ ε
k =0
i=1
i=1
∧M{B} + M{B } ∞ k +1 k ≤M τi > tx ∩ ηi ≤ t − tx + m + ε. c
by the duality of uncertain measure. Since the renewal process Nt can only take integer values, we have ∞ k +1 N +1 t τi > tx = M τi > tx ∩ (Nt = k) . M i=1
k =0
i=1
Noting the uncertain random event k k +1 {Nt = k} = (ηi + τi ) ≤ t < (ηi + τi ) i=1
i=1
k =0
i=1
i=1
Let Mt denote a stochastic renewal process with random interarrival times η1 , η2 , . . .. Then k ηi ≤ t − tx + m = {k ≤ Mt−tx+m } . i=1
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
Thus ⎧ ⎫ +1 N +1 t ⎨M t −t ⎬ x + m k τi > tx ≤ M τi > x +ε M ⎩ ⎭ i=1
i=1
k =0
= 1−Φ
tx Mt−tx+m + 1
+ε
which is a random variable. By the strong law of large numbers for random variables, we have Mt−tx−m 1 = , t − tx + m E[η1 ] and
lim Φ
i=1
⎞
tx ⎟ ⎠ Mt−tx−m (t − tx + m) + 1 t − tx + m
x = Φ E[η1 ] · , a.s. 1−x
t→∞
t→∞
Then, the inequality N +1 t τi > tx ≤ 1 − Φ E[η1 ] · lim M i=1
holds for any ε > 0. As a result N +1 t τi > tx ≤ 1 − Φ E[η1 ] · lim M t→∞
i=1
x 1−x
x 1−x
i=1
1
Pr
= 0
≤ 0
1
lim M
t→∞
t→∞
+ε
i=1
x 1−x
.
On the other hand, we have τ1 x ≤ x = M τ1 ≤ E[η1 ] · M E[η1 ] + τ1 1−x
x . = Φ E[η1 ] · 1−x
,
t→∞
by Theorems 4 and 5, we have lim Υt (x) = Φ E[η1 ] ·
a.s.
τ1 Ut → t E[η1 ] + τ1
N +1 t
Pr 1 − Φ E[η1 ] ·
≥r
τi > tx
i=1
x 1−x
in the sense of convergence in distribution as t → ∞. The uncertain random process Rt is the total on-times of the system before some given time t; therefore, Rt /t is just the interval availability of the system. Theorem 7: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and uncertain off-times {τi }. Then
dr
≥ r dr
Rt E[η1 ] → t E[η1 ] + τ1
.
By the duality of chance measure, we have N +1 N +1 t t τi /t ≤ x = 1 − lim Ch τi /t > x lim Ch i=1
i=1
Hence
x = 1 − Φ E[η1 ] · 1−x
t→∞
i=1
x = Φ E[η1 ] · 1−x
Hence, we have N +1 t τi /t > x lim Ch t→∞
in the sense of convergence in distribution as t → ∞. Proof: On the one hand, it follows from the inequality (2) that N +1 Nt t Ut ≤x ⊂ τi /t ≤ x ⊂ τi /t ≤ x t i=1 i=1
by the monotonicity of chance measure. Assume τ1 has an uncertainty distributions Φ. Since N N +1 t t τi /t ≤ x = lim Ch τi /t ≤ x lim Ch
⎜ = lim Φ ⎝
t→∞
τ1 Ut → t E[η1 ] + τ1
so Ut /t has a chance distribution Υt satisfying N +1 N t t Ch ξi /t ≤ x ≤ Υt (x) ≤ Ch ξi /t ≤ x
a.s.
Mt−tx+m + 1 ⎛
t→∞
uncertain off-times {τi }. Then
tx
1339
t→∞
i=1
x ≥ Φ E[η1 ] · 1−x
.
The theorem is verified. Theorem 6: Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and
in the sense of convergence in distribution as t → ∞. Proof: Noting that Ut Rt =1− , t t
E[η1 ] τ1 =1− E[η1 ] + τ1 E[η1 ] + τ1
we immediately have Rt E[η1 ] → t E[η1 ] + τ1 in the sense of convergence in distribution as t → ∞ by Theorem 6.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 23, NO. 5, OCTOBER 2015
Remark 4: If the on-times {ηi } and the off-times {τi } are all random variables, then the interval availability E[η1 ] Rt → t E[η1 ] + E[τ1 ] in the sense of almost sure convergence. Remark 5: If the on-times {ηi } and the off-times {τi } are all uncertain variables, then the interval availability η1 Rt → t η1 + τ1 in the sense of convergence in distribution. Theorem 8 (Alternating Renewal Theorem): Let (Rt , Ut ) be an uncertain random alternating renewal process with alternating random on-times {ηi } and uncertain off-times {τi }. Then ! E[η1 ] E[Rt ] lim = E , t→∞ t E[η1 ] + τ1 ! τ1 E[Ut ] = E . lim t→∞ t E[η1 ] + τ1 Proof: It follows from the definition of expected value that 1 Rt E[Rt ] = lim ≥ r dr Ch lim t→∞ t→∞ 0 t t 1 Rt = ≥ r dr. lim Ch t 0 t→∞ By Theorem 7, we have
E[η1 ] ≥ r dr E[η1 ] + τ1 0 ! E[η1 ] = E . E[η1 ] + τ1
E[Rt ] = t→∞ t lim
1
M
Since E[η1 ] = exp(μ + σ 2 /2), the interval availability exp(μ + σ 2 /2) Rt → t exp(μ + σ 2 /2) + τ1
bx + exp(μ + σ 2 /2)(x − 1) bx − ax
for exp(μ + σ 2 /2) exp(μ + σ 2 /2) ≤x≤ . 2 exp(μ + σ /2) + b exp(μ + σ 2 /2) + a In addition, the limit expected interval availability ! exp(μ1 + σ12 /2) E[Rt ] lim = E t→∞ t exp(μ1 + σ12 /2) + τ1 =
exp(μ + σ 2 /2) + b exp(μ + σ 2 /2) · ln . b−a exp(μ + σ 2 /2) + a
Example 2: Consider an uncertain random alternating renewal process (Rt , Ut ) with i.i.d. random on-times {ηi } and i.i.d. uncertain off-times {τi }. Assume η1 has an exponential probability distribution Ψ(x) = 1 − exp(−λx),
x>0
and τ1 has a lognormal uncertainty distribution
−1 π(μ − ln x) √ , x > 0. Φ(x) = 1 + exp 3σ
Rt 1 → t 1 + λτ1
The proof is completed. Example 1: Consider an uncertain random alternating renewal process (Rt , Ut ) with i.i.d. random on-times {ηi } and i.i.d. uncertain off-times {τi }. Assume η1 has a lognormal probability density function
(ln x − μ)2 1 exp − Ψ(x) = √ , x>0 2σ 2 2πσx and τ1 has a linear uncertainty distribution x−a , a ≤ x ≤ b. Φ(x) = b−a
=
Since E[η1 ] = 1/λ, the interval availability
Since Rt /t + Ut /t = 1, we have E[Ut ] E[Rt ] = 1 − lim lim t→∞ t→∞ t t ! E[η1 ] = 1−E E[η1 ] + τ1 ! τ1 = E . E[η1 ] + τ1
in the sense of convergence in distribution, and its limit chance distribution exp(μ + σ 2 /2) Rt ≤x = M ≤x lim Ch t→∞ t exp(μ + σ 2 /2) + τ1
in the sense of convergence in distribution, and its limit chance distribution Rt ≤x lim Ch t→∞ t 1 =M ≤x 1 + λτ1
−1 π(ln(1−x) − ln(λx)− μ) √ = 1+ exp , x > 0. 3σ In addition, the limit expected interval availability lim
t→∞
E[Rt ] t =E = 0
1 1 + λτ1 +∞
! √
(λ exp(μ))π /( 3σ ) 1 √ √ · dx. (λ exp(μ))π /( 3σ ) + xπ /( 3σ ) (1 + x)2
Especially, if λ exp(μ) = 1, then we have +∞ E[Rt ] 1 1 1 √ lim = · dx = 2 π /( 3σ ) t→∞ t (1 + x) 2 1+x 0
YAO AND GAO: UNCERTAIN RANDOM ALTERNATING RENEWAL PROCESS WITH APPLICATION TO INTERVAL AVAILABILITY
√ for any σ > 0. If σ = π/ 3, we have +∞ 1 E[Rt ] λ exp(μ) lim = · dx t→∞ t λ exp(μ) + x (1 + x)2 0 =
λ exp(μ)(λ exp(μ) − μ − ln λ − 1) . (λ exp(μ) − 1)2
However, the above integration has no explicit form in general; therefore, we have to employ numerical integration for given parameters λ, μ, and σ. For example, if λ = 1, μ = 1, and σ = √ 2π/ 3, then lim
t→∞
E[Rt ] = 0.7031. t
V. APPLICATION Consider a system that can be in one of two states at any time: working or under repair. Generally, the working times of a system have many samples, and their probability distribution can be obtained via statistics; therefore, we employ random variables to model the working times. The repair processes involve human uncertainty, and the possible repairing times are often predicated by the repairmen; therefore, we employ uncertain variables to model the repairing times. Then, the on-times and off-times of the system form an uncertain random alternating renewal process. Assume the system is working at the initial time. After a random time η1 , the system fails and undergoes repair for an uncertain time τ1 . When the repair is complete, the system is as good as new and returns to work immediately. After a random time η2 , the system fails and undergoes repair for an uncertain time τ2 . The process repeats infinitely. Now, we consider two important indexes of the system: failure rate and interval availability. Let Ψ denote the common probability distribution of the successive working times {ηi }, and Φ denote the common uncertainty distribution of the successive repairing times {τi }. Consider a cycle from the time that the system begins working to the time that the repair is completed. Such cycles form an uncertain random renewal process Nt with interarrival times {ηi + τi }. Note that Nt is just the total failure times of the system before a given time t, and Nt /t is the failure rate on the interval [0, t]. By the operational law of uncertain random variables, the sum ξi + ηi has a chance distribution +∞ Υ(x) = Ch{ηi + τi ≤ x} = M{y + τi ≤ x}dΨ(y) −∞
+∞
= −∞
Φ(x − y)dΨ(y)
and the expected failure rate E[Nt ]/t has a limitation ! 1 E[Nt ] =E . lim t→∞ t E[η1 ] + τ1 Let Rt denote the total on-time, and Ut denote the total off-time. Then, (Rt , Ut ) is just an uncertain random alternating renewal process with alternating interarrival times
1341
η1 , τ1 , η2 , τ2 , . . ., and Rt /t is the interval availability of the system. By Theorem 8, the expected interval availability E[Rt ]/t has a limitation ! E[η1 ] E[Rt ] =E lim . t→∞ t E[η1 ] + τ1 Example 3: Consider an uncertain random alternating renewal process (Rt , Ut ) with i.i.d. random on-times {ηi } and i.i.d. uncertain off-times {τi }. Assume the working times {ηi } of the system have an exponential distribution with a parameter λ, i.e., Ψ(x) = 1 − exp(−λx),
x≥0
and the repairing times {τi } of the system have a linear uncertainty distribution Φ(x) =
x−a , b−a
a≤x≤b
evaluated by some repairmen. Since E[η1 ] = 1/λ, the failure rate Nt λ → t λ + τ1 in the sense of convergence in distribution, and the limitation of expected failure rate ! λ 1 + λb E[Nt ] 1 =E · ln . lim = t→∞ t 1 + λτ1 b−a 1 + λa The interval availability Rt 1 → t 1 + λτ1 in the sense of convergence in distribution, and the limitation of expected interval availability ! 1 E[Rt ] 1 + λb 1 =E · ln . lim = t→∞ t 1 + λτ1 λ(b − a) 1 + λa VI. CONCLUSION This paper mainly presents a concept of alternating renewal process with i.i.d. random on-times and i.i.d. uncertain off-times as the alternating interarrival times. An uncertain random alternating renewal theorem is verified, which provides the chance distribution of the interval availability. An application to system reliability is given to illustrate the alternating renewal theorem. REFERENCES [1] R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. New York, NY, USA: Wiley, 1965. [2] T. Nakagawa, Maintenance Theory of Reliability. London, U.K.: SpringerVerlag, 2005. [3] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, Jun. 1965. [4] R. Zhao and B. Liu, “Renewal process with fuzzy interarrival times and costs,” Int. J. Uncertainity Fuzziness, Knowl.-Based Syst., vol. 11, no. 3, pp. 573–586, Jun. 2003. [5] D. Hong, “Renewal process with T -related fuzzy inter-arrival times and fuzzy rewards,” Inf. Sci., vol. 176, no. 16, pp. 2386–2395, Aug. 2006. [6] S. Li, “Some properties of fuzzy alternating renewal processes,” Math. Comput. Model., vol. 54, no. 9/10, pp. 1886–1896, Nov. 2011.
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Kai Yao received the B.S. degree from Nankai University, Tianjin, China, in 2009 and the Ph.D. degree from Tsinghua University, Beijing, China, in 2013. He is currently an Assistant Professor with the School of Management, University of Chinese Academy of Sciences, Beijing. He has authored or coauthored more than 20 articles in several journals, including the IEEE TRANSACTIONS ON FUZZY SYSTEMS, Knowledge-Based Systems, Applied Soft Computing, Applied Mathematical Modelling, Applied Mathematics and Computation, Fuzzy Optimization and Decision Making, and Soft Computing. His current research interests include uncertain renewal processes, uncertain systems, and uncertain differential equations and their applications.
Jinwu Gao received the B.S. degree in mathematics from Shaanxi Normal University, Xian, China, in 1996 and the M.S. and Ph.D. degrees in mathematics from Tsinghua University, Beijing, China, in 2005. He is currently an Associate Professor of mathematics with the Renmin University of China, Beijing. His current research interests include fuzzy systems, uncertain systems and their application in optimization, game theory, and finance. He has authored or coauthored more than 30 papers that have appeared in the Soft Computing, Fuzzy Optimization and Decision Making, the International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, the Iranian Journal of Fuzzy Systems, Computer and Mathematics with Applications, and other publications. Dr. Gao has been the Coeditor-in-Chief of the Journal of Uncertain Systems since 2011 and the Executive Editor-in-Chief of the Journal of Uncertainty Analysis and Applications since 2013. He has served as the Vice President of Intelligent Computing Chapter of the Operations Society of China since 2007 and as Vice President of the International Consortium for Uncertainty Theory since 2013. He served as Vice President of the International Association for Information and Management Science from 2010 to 2013 and as President of the International Consortium for Electronic Business from 2012 to 2013.
