Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process

ISSN 1746-7659, ngland, K Journal of Information and Computer Science Vol 1, No 4, 6, pp 7-4 Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process Shunqin i, Ruiqing Zhao, Wansheng Tang Institute of Systems ngineering, Tianjin niversity, Tianjin 37, China Received Dec 5, 5, Accepted March 6, 6) Abstract By dealing with interarrival times as exponentially distributed fuzzy random variables, a fuzzy random homogeneous Poisson process and a fuzzy random compound Poisson process are respectively defined Several theorems on the two processes are provided, respectively Keywords Fuzzy Variables; Fuzzy Random Variables; Poisson Process; Fuzzy Random Homogeneous Poisson Process; Fuzzy Random Compound Poisson Process 1 Introduction The imprecise data are very common in renewal processes, and fuzzy sets theory developed by Zadeh 19] is able to effectively deal with the imprecise information Recently, fuzzy sets theory was used to renewal theory by a few authors Based on the expected value operator of a fuzzy variable defined by iu and iu 11], Zhao and iu 1] discussed a renewal process with fuzzy interarrival times and rewards, and established the fuzzy styles of elementary renewal theorem and renewal reward theorem i et al 6] defined a delayed renewal process with fuzzy interarrival times, and further discussed some properties on the average of the fuzzy renewal variable In practice, a hybrid uncertain process with randomness and fuzziness exists generally Thus, randomness and fuzziness should be considered simultaneously Fuzzy random theory is one of good tools to deal with such an uncertainty, where fuzzy random variables were introduced by Kwakernaak 4], Puri and Ralescu 17] to model a process which quantified fuzzily the outcomes of a random experiment Hwang 3] investigated a renewal process in which the interarrival times were expressed as independent and identically distributed iid) fuzzy random variables, and provided a theorem for the fuzzy rate of the fuzzy random renewal process Dozzi et al ] defined a fuzzy-set-indexed renewal counting process, and gave the elementary renewal theorem Popova and Wu 16] considered a renewal reward process with random interarrival times and fuzzy rewards, and presented a theorem on the asymptotic average fuzzy reward per unit time Homogeneous Poisson process is a special one of Poisson processes, where interarrival times and the rate of the process are two important quantities Conventionally, interarrival times are expressed as iid exponentially distributed random variables and the rate of the process is assumed to be a constant Mixed Poisson process, a special case of doubly stochastic Poisson process introduced by Cox 1] is an extension of homogeneous Poisson process, where the rate related to the process is assumed as a random variable Some conclusions on mixed Poisson process can be found in Mcfadden 14] However, an interval estimate or a point estimate of the rate is provided by the experiment data often with quite small sample sizes Sometimes, it is more realistic and appropriate to characterize the rate as a fuzzy variable rather than a crisp number or a random variable In this paper, by adopting the definition of fuzzy random variables defined by iu and iu 9] for a homogeneous -mail address: zhao@tjueducn Published by World Academic Press, World Academic nion

8 S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process Poisson process, the concept of a fuzzy random homogeneous Poisson process is defined, where interarrival times are assumed as iid exponentially distributed fuzzy random variables and the rate of the process is depicted as a fuzzy variable As an extension of the compound Poisson process, fuzzy random compound Poisson process is further defined Fuzzy Variables et Θ be a universe, P Θ) the power set of Θ, Pos a possibility measure see Zadeh ]), and Θ, P Θ), Pos) a possibility space Fuzzy variable is an important concept in fuzzy sets theory Kaufmann 5] first used it as a generalization of the concept of a Boolean variable, and Nahmias 15] defined it as a function from a pattern space to the set of real numbers Definition 1 8] A fuzzy variable ξ is 1) continuous if Posξ = r} is a continuous function of r; ) positive if Posξ } = ; 3) nonnegative if Posξ < } = ; 4) discrete if there exists a countable sequence x 1, x, } such that Posξ x 1, ξ x, } = Definition 1] et ξ be a fuzzy variable on the possibility space Θ, P Θ), Pos), and α, 1] Then ξ α = inf r Posξ r} α } and ξ α = sup r Posξ r} α } are called the α-pessimistic value and the α-optimistic value of ξ, respectively Definition 3 8, 15] The fuzzy variables ξ 1, ξ, are independent if and only if for any sets B 1, B, of R Posξ i B i, i = 1,, } = min i 1 Posξ i B i } Definition 4 8] The fuzzy variables ξ and η are identically distributed if and only if for any set B of R Posξ B} = Posη B} Proposition 1 et ξ and η be two independent fuzzy variables Then for any α, 1], ξ + η) α = ξ α + η α, ξ + η) α = ξ α + η α Proposition If ξ and η are two independently nonnegative fuzzy variables, for any α, 1], ξ η) α = ξ α η α, ξ η) α = ξ α η α Proposition 3 If ξ is a nonnegative fuzzy variable, for any α, 1] and positive integer k, ξ k) = ξ ) k α, ξ k) = ξ ) k α α α

Journal of Information and Computing Science, 16) 4, pp7-4 9 In order to define the expected value of a fuzzy variable ξ, iu and iu 11] introduced a set function Cr of a fuzzy event ξ r} as Crξ r} = 1 Posξ r} + 1 Posξ < r}) Based on the credibility measure, the expected value of a fuzzy variable ξ is defined as follows, ξ] = + Crξ r}dr Crξ r}dr 1) When the right side of 1) is of form, the expected value operator is not defined specially, if ξ is a nonnegative fuzzy variable, its expected value is defined as ξ] = + Crξ r}dr Proposition 4 1] et ξ be a fuzzy variable with finite expected value ξ], then we have 3 Fuzzy Random Variables ξ] = 1 ξ α + ξα ) dα et Ω, A, Pr) be a probability space and F a collection of fuzzy variables defined on the possibility space Θ, P Θ), Pos) Definition 31 9] A fuzzy random variable ξ is a function ξ : Ω F such that for any Borel set B of R, Posξω) B} is a measurable function of ω Definition 3 xponentially Distributed Fuzzy Random Variable) For each ω Ω, let ξαω) and ξα ω) be the α-pessimistic value and the α-optimistic value of ξω), respectively A fuzzy random variable ξ defined on the probability space Ω, A, Pr) is said to be exponential if ξαω) and ξα ω) are exponentially distributed random variables whose density functions are defined as f ξ α ω)x) = α e α x, x, x <, and f ξ α ω)x) = where is a fuzzy variable defined on the possibility space Θ, P Θ), Pos) with Pos = α} = min Pos ξω) = ξ α ω) }, ω Ω }, Pos = α } = min Pos ξω) = ξ α ω) }, ω Ω } α e α x, x, x <, An exponentially distributed fuzzy random variable is denoted by ξ XP ), and the fuzziness of fuzzy random variable ξ is said to be characterized by the fuzzy variable Definition 33 8] A fuzzy random variable ξ is nonnegative if and only if Posξω) < } = for any ω Ω Definition 34 8] Fuzzy Random Arithmetic On Different Spaces) et f : R n R be a Borel measurable function, and ξ 1, ξ,, ξ n fuzzy random variables on the probability spaces Ω i, A i, Pr i ) respectively Then ξ = fξ 1, ξ,, ξ n ) is a fuzzy random variable defined on the product probability space Ω 1 Ω Ω n, A 1 A A n, Pr 1 Pr Pr n ) as for any ω 1, ω,, ω n ) Ω 1 Ω Ω n ξω 1, ω,, ω n ) = fξ 1 ω 1 ), ξ ω ),, ξ n ω n )) )

1 S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process Definition 35 1] et ξ be a fuzzy random variable defined on Ω, A, Pr), then the average chance, denoted by Ch, of a fuzzy random event ξ r} is defined as Chξ r} = Remark 31 ] The equation 3) is equivalent to the following form Chξ r} = 1 Pr ω Ω Crξω) r} α } dα 3) Pr ω Ω ξ αω) r } + Pr ω Ω ξ α ω) r }) dα Proposition 31 1] The average chance Ch of a fuzzy random event ξ r} is self dual, ie, Chξ r} = 1 Chξ < r} Definition 36 9] et ξ be a fuzzy random variable defined on Ω, A, Pr) Then its expected value is defined by ξ] = Ω + ) Crξω) r}dr Crξω) r}dr Prdω) 4) provided that at least one of the two integrals is finite specially, if ξ is a nonnegative fuzzy random variable, then ξ] = + Ω Crξω) r}dr Prdω) Proposition 3 Based on Fubini theorem and Proposition 4, the equation 4) can be written as ξ] = 1 ξ α ω) ] + ξ α ω) ]) dα 5) Definition 37 11] The fuzzy random variables ξ 1, ξ,, ξ n are iid if and only if Posξ i ω) B 1 }, Posξ i ω) B },, Posξ i ω) B m }), i = 1,,, n are iid random vectors for any positive integer m and any Borel sets B i of R 4 Fuzzy Random Homogeneous Poisson Process with Fuzzy Rates et ξ i, known as interarrival times, be iid nonnegative fuzzy random variables defined on Ω i, A i, Pr i ) with ξ i XP i ) Furthermore, let Nt) denote the total number of events that have occurred by time t That is, Nt) = max n ξ 1 + ξ + + ξ n t } 6) n The counting process Nt), t } is called a fuzzy random homogeneous Poisson process with fuzzy rates i and Nt) is called a Poisson fuzzy random variable with fuzzy rates i Theorem 41 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then we have ] 1 ξ 1 ] = 1

Journal of Information and Computing Science, 16) 4, pp7-4 11 Proof Since ξ 1 XP 1 ), by Definition 3, we have It follows from Proposition 3 that The proof is finished ξ 1 ] = 1 = 1 ξ1,αω) ] = 1, ξ1,αω) ] = 1 1,α 7) 1,α = 1 1 = 1 ξ 1,α ω) ] + ξ 1,αω) ]) dα 1 + 1 1,α 1,α ) 1 ] 1 α ) 1 + 1 dα ) α ) dα by Proposition 4) Theorem 4 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then we have Nt)] = t 1 ] Proof For any ω, fuzzy variables ξ i ω) are defined on the possibility spaces Θ i, P Θ} i ), Pos i ) Assume } that Θ 1 = Θ = For any given α, 1], there exist points θ i, θ i Θ i with Pos θ i α, Pos θ i α such that ) ) ξi,αω) = ξ i ω) θ i, ξi,αω) = ξ i ω) θ i Taking θ 1 = θ =, θ 1 = θ =, we obtain Nt)ω) θ ) = max n ξ1,αω) + ξ,αω) + + ξn,αω) t }, 8) n where θ = Nt)ω) θ ) = max n ) ) θ 1, θ,, θ = θ 1, θ, For any θ i Θ i with Posθ i } α, by Definition, we have n ξ 1,αω) + ξ,αω) + + ξ n,αω) t }, 9) ξ i,αω) ξ i ω)θ i ) ξ i,αω) 1) Consequently, Nt)ω) θ ) Nt)ω)θ) Nt)ω) θ ), 11) where θ = θ 1, θ, ) et Nt) αω) and Nt) α ω) be the α-pessimistic value and the α-optimistic value of Nt)ω) for any ω Again by Definition, we have Nt) αω) = Nt)ω) θ ), Nt) α ω) = Nt)ω) θ )

1 S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process Thus, equations 8) and 9) can be written as Nt) αω) = max n ξ1,αω) + ξ,αω) + + ξn,αω) t }, n 1) Nt) α ω) = max n ξ 1,α ω) + ξ,αω) + + ξn,αω) t } n 13) Since ξi,α ω) and ξ i,α ω) are iid random variables, respectively, then Nt) αω), t } defined by 1) is a homogeneous Poisson process with constant rate 1,α and Nt) α ω), t } defined by 13) is a homogeneous Poisson process with constant rate 1,α By the results of a homogeneous Poisson process see Ross 18]), It follows from Proposition 3 that The proof is finished Nt) αω) ] = t 1,α, Nt) α ω) ] = t 1,α 14) Nt)] = 1 = t = t 1 ] Nt) α ω) ] + Nt) α ω) ]) dα 1,α + ) 1,α dα Theorem 43 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then we have N t) ] = t 1 ] + t 1] Proof Note that, for any ω, the fuzzy variable Nt)ω) is defined on Θ, P Θ), Pos), where Θ = Θ 1 Θ, Posθ} = Posθ 1 } Posθ } for any θ = θ 1, θ, ) Θ For any θ Θ with Posθ} α, by Definition, we have Nt) αω) Nt)ω)θ) Nt) α ω) Consequently, Nt) α ω) ) N t)ω)θ) Nt) α ω) ) Again by Definition, N t) αω) = Nt) αω) ), N t) α ω) = Nt) α ω) ), where N t) αω) and N t) α ω) are the α-pessimistic value and the α-optimistic value of N t)ω) for each ω For the homogeneous Poisson processes defined by 1) and 13), by the results of a homogeneous Poisson process see Ross 18]), we obtain N t) αω) ] = Nt) α ω) ) ] = t 1,α + t 1,α) = t 1,α + t 1) α, N t) α ω) ] = Nt) α ω) ) ] = t 1,α + t 1,α) = t 1,α + t 1) α

Journal of Information and Computing Science, 16) 4, pp7-4 13 From Proposition 3, we have The proof is finished N t) ] = 1 = 1 = t N t) αω) ] + N t) α ω) ]) dα t 1,α + t 1) α + t 1,α + t ) 1) dα α 1,α + ) t 1,α dα + = t 1 ] + t 1] ) 1 α + ) 1) dα α Theorem 44 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then we have Nt) Nt + s)] = t 1 ] + t t + s) 1] Proof For any θ 1, θ Θ with Posθ 1 } α, Posθ } α, we have Nt) αω) Nt)ω)θ 1 ) Nt) α ω), 15) and consequently, Nt + s) αω) Nt + s)ω)θ ) Nt + s) α ω), 16) Nt) αω) Nt + s) αω) Nt)ω)θ 1 ) Nt + s)ω)θ ) Nt) α ω) Nt + s) α ω) By Definition, we have Nt) Nt + s)) αω) = Nt) αω) Nt + s) αω), Nt) Nt + s)) α ω) = Nt) α ω) Nt + s) α ω), where Nt) Nt + s)) αω) and Nt) Nt + s)) α ω) are the α-pessimistic value and the α-optimistic value of Nt) Nt + s))ω) for each ω For the homogeneous Poisson processes defined by 1) and 13), by the results of a homogeneous Poisson process see Ross 18]), we have Nt) Nt + s)) αω) ] = t 1,α + t t + s) 1,α) = t 1,α + t t + s) 1) α, Nt) Nt + s)) α ω) ] = t 1,α + t t + s) 1,α) = t 1,α + t t + s) 1) α From Proposition 3, we obtain Nt) t + s)] = 1 The proof is finished = 1 = t Nt) Nt + s)) α ω) ] + Nt) Nt + s)) α ω) ]) dα t 1,α + t t + s) 1) α + t 1,α + t t + s) ) 1) dα α 1,α + 1,α) dα + t t + s) = t 1 ] + t t + s) 1] ) 1 α + ) 1) dα α

14 S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process Theorem 45 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then for any a, we have Nt + a) Nt)] = Nt + a)] Nt)] = a 1 ] Proof It follows immediately from Theorem 4 that Nt + a)] Nt)] = a 1 ] In the following, we prove Nt + a) Nt)] = Nt + a)] Nt)] 17) For any θ 1, θ Θ with Posθ 1 } α, Posθ } α, from 15), 16) and using 1), Nt + a) αω) Nt) αω) Nt + a)ω)θ ) Nt)ω)θ 1 ) Nt + a) α ω) Nt) α ω) By Definition, Nt + a) Nt)) αω) = Nt + a) αω) Nt) αω), 18) Nt + a) Nt)) α ω) = Nt + a) α ω) Nt) α ω), 19) by taking expectations, Nt + a) Nt)) αω) ] = Nt + a) αω) Nt) αω) ] = Nt + a) αω) ] Nt) αω) ], Nt + a) Nt)) α ω) ] = Nt + a) α ω) Nt) α ω) ] = Nt + a) α ω) ] Nt) α ω) ] From Proposition 3, we obtain Nt + a) Nt)] = 1 = 1 = 1 Nt + a) Nt)) α ω) ] + Nt + a) Nt)) α ω) ]) dα Nt + a) α ω) ] Nt) αω) ] + Nt + a) α ω) ] Nt) α ω) ]) dα Nt + a) α ω) ] + Nt) α ω) ]) dα 1 Nt) α ω) ] + Nt) α ω) ]) dα = Nt + a)] Nt)] The proof is finished Theorem 46 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then for any a, we have Nt + a) Nt)] = Nt + a)] Nt)] = Na)] Proof It follows immediately from Theorem 4 Theorem 47 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then we have ] ChNt) = } = e t 1 )

Journal of Information and Computing Science, 16) 4, pp7-4 15 Proof Note that Nt) = } ξ 1 > t} Since ξ 1 XP 1 ), by Definition 3, Pr ξ 1,αω) > t } = e t 1,α, Pr ξ 1,αω) > t } = e t 1,α 1) Since e x t is a decreasing function of x for any fixed t >, then In addition, ] e t 1 = 1 Hence, by Remark 31 and 1), The proof is finished e 1 t ) α = e t 1,α, e 1 t ) ) e ) ) 1 t + e 1 t dα = 1 α α ChNt) = } = Chξ 1 > t} = 1 = 1 = α = e t 1,α ) e t 1,α + e t 1,α dα Pr ξ 1,α ω) > t } + Pr ξ 1,αω) > t }) dα ) e t 1,α + e t 1,α dα ] e t 1 Theorem 48 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then for any positive integer k, we have ChNt + s) Nt) k} = ChNs) k} = 1 k 1 e 1,α s 1,α s ) j + e 1,α s 1,α s ) j ) dα j= Proof For the homogeneous Poisson processes defined by 1) and 13), by the results of a homogeneous Poisson process see Ross 18]), we have Pr Ns) αω) k } = By Remark 31, we obtain k e 1,α s 1,α s ) j, Pr Ns) α ω) k } = j= ChNs) k} = 1 = 1 = 1 k j= k e 1,α s 1,α s ) j j= Pr Ns) α ω) k } + Pr Ns) α ω) k }) dα j= k e 1,α s 1,α s ) j k + e 1,α s 1,α s ) j dα j= e 1,α s 1,α s ) j + e 1,α s 1,α s ) j ) dα

16 S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process For any θ 1, θ Θ with Posθ 1 } α and Posθ } α, using 18), 19) and by the results of a homogeneous Poisson process see Ross 18]), Nt + s) Nt)) αω) = Ns) αω), Nt + s) Nt)) α ω) = Ns) α ω) Consequently, It follows from Remark 31 that The proof is finished Pr Nt + s) Nt)) αω) k } = Pr Ns) αω) k }, Pr Nt + s) Nt)) α ω) k } = Pr Ns) α ω) k } ChNt + s) Nt) k} = ChNs) k} ) Theorem 49 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then for any positive integer k, we have ChNt + s) Nt) k} = 1 1 k 1 1 e 1,α s 1,α s ) j + e 1,α s 1,α s ) j ) dα Proof It immediately follows from Proposition 31 and Theorem 48 The proof is finished j= Theorem 41 et Nt), t } be a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, iid nonnegative exponentially distributed fuzzy random interarrival times If 1 ] < +, then we have ChNt + h) Nt) 1} = ChNh) 1} = h 1 ] + oh), 3) where lim h oh) h = Proof For the homogeneous Poisson processes defined by 1) and 13), by the results of a homogeneous Poisson process see Ross 18]), we have Pr Nh) αω) 1 } = 1,α + oh), Pr Nh) α ω) 1 } = 1,α + oh) That is, Pr Nh) αω) 1 } 1,α Pr Nh) α ω) 1 } 1,α lim =, lim h h h h From Remark 31 and the dominated convergence theorem, we obtain ChNh) 1} h 1 ] lim h h 1 1 Pr Nh) = lim α ω) 1 } h h Pr Nh) α ω) 1 } = 1 = 1 =, lim h h 1,α + ) 1,α dα 1 ] = + Pr Nh) α ω) 1 } ) dα 1 ] h + Pr Nh) α ω) 1 } ) dα 1 ] h

Journal of Information and Computing Science, 16) 4, pp7-4 17 which implies ChNh) 1} = h 1 ] + oh) Furthermore, it follows from ) that the proof is finished Theorem 411 If Nt), t } is a fuzzy random homogeneous Poisson process with fuzzy rates i, and ξ i, i = 1,, are iid nonnegative exponentially distributed fuzzy random interarrival times, then we have ChNt + h) Nt) } = ChNh) } = oh) Proof For the homogeneous Poisson processes defined by 1) and 13), from the results of a homogeneous Poisson process see Ross 18]), we have Pr Nh) αω) } = oh), Pr Nh) α ω) } = oh) That is, By Remark 31, we obtain ChNh) } lim h h Pr Nh) lim αω) } Pr Nh) =, lim α ω) } = h h h h 1 = lim h = 1 = Pr Nh) α ω) } + Pr Nh) α ω) } ) dα h h Pr Nh) αω) } Pr Nh) + lim α ω) } ) h h h lim h Furthermore, it follows from ) that the result holds dα 5 Fuzzy Random Compound Poisson Process et η 1, η, be iid nonnegative fuzzy random variables defined on the probability spaces Ω i, A i, Pr i ), independent of Nt), a Poisson fuzzy random variable with fuzzy rates i et Xt) = Nt) η i 4) Then Xt), t } is called a fuzzy random compound Poisson process with fuzzy rates i and Xt) is called a compound Poisson fuzzy random variable with fuzzy rates i Theorem 51 If Xt), t } is a fuzzy random compound Poisson process with fuzzy rates i, then Xt)] = t 1 η 1 ] Proof For any given ω Ω, fuzzy variables η i ω) are defined on Γ i, P Γ i ), Pos i ) Assume that Γ 1 = Γ = For any θ Θ, γ i Γ i with Posθ} α, Posγ i } α, since Nt) αω) Nt)ω)θ) Nt) α ω), η i,αω) η i ω)γ i ) η i,αω),

18 S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process then That is, sing Definition, we have Nt) α ω) Nt) αω) Xt) αω) = η i,αω) Nt)ω)θ) η i ω)γ i ) η i,αω) Xt)ω)θ, γ i ) Nt) α ω) η i,αω), Xt) α ω) = Nt) α ω) Nt) α ω) η i,αω) η i,αω) 5) Nt) α ω) η i,αω) 6) Since ηi,α ω) and η i,α ω), i = 1,, are iid random variables, respectively, Xt) αω), t } and Xt) α ω), t } are two compound Poisson processes Applying Wald s quation to 6), Xt) αω) ] Nt) α ω) = ηi,αω) = Nt) αω) ] η1,αω) ], In addition, by 14), Xt) α ω) ] = Nt) α ω) ηi,αω) = Nt) α ω) ] η1,αω) ] Xt) αω) ] = t 1,α η 1,αω) ], Xt) α ω) ] = t 1,α η 1,αω) ] Since, for any θ 1 Θ 1, γ 1 Γ 1 with Posθ 1 } α and Posγ 1 } α, It follows from Definition that Consequently, by taking expectations, and then, Hence, by Proposition 3, The proof is finished 1,α η 1,αω) 1 θ 1 ) η 1 ω)γ 1 ) 1,α η 1,αω) 1 η 1 ) αω) = 1,α η 1,αω), 1 η 1 ) α ω) = 1,α η 1,αω) 1 η 1 ) αω) ] = 1,α η 1,αω) ], 1 η 1 ) α ω) ] = 1,α η 1,αω) ], 1 η 1 ] = 1 = 1 Xt)] = 1 = t = t 1 η 1 ] η1 ) αω) ] + η 1 ) α ω) ]) dα 1,α η 1,αω) ] + 1,α η 1,αω) ]) dα Xt) α ω) ] + Xt) α ω) ]) dα 1,α η 1,αω) ] + 1,α η 1,αω) ]) dα

Journal of Information and Computing Science, 16) 4, pp7-4 19 Remark 51 If Nt) degenerates to a Poisson random variable with constant rate, then the result in Theorem 51 degenerates the following form Xt)] = t η 1 ] Remark 5 If η i degenerate to iid random variables, then the result in Theorem 51 degenerates the following form Xt)] = t 1 ] η 1 ] Theorem 5 If Xt), t } is a fuzzy random compound Poisson process with fuzzy rates i, then X t) ] = t 1 η1 ] t + 1,α η1,αω) ]) + 1,α η1,αω) ]) ) dα Proof sing 5), That is, From Definition, Nt) α ω) ηi,αω) Nt) α ω) X t) αω) = ηi,αω) Nt) α ω) Nt)ω)θ) η i ω)γ i ) X t)ω, γ i ) ηi,αω) Nt) α ω), X t) α ω) = Nt) α ω) ηi,αω) Nt) α ω) ηi,αω) ηi,αω) By the results of a compound Poisson process see Ross 18]), X t) αω) ] η = t ) ] 1,α 1 α ω) + t ) 1,α η1,αω) ] X t) α ω) ] η = t ) ] 1,α 1 α ω) + t 1,α) η1,αω) ] It is easy to prove that 1 η1 ) η α ω) = ) ] 1,α 1 α ω), and then 1 η1 ] 1 = sing Proposition 3, X t) ] = 1 The proof is finished = t + 1 1 η1 ) η α ω) = ) ] 1,α 1 α ω) η ) ] η 1,α 1 α ω) + ) ]) 1,α 1 α ω) dα X t) αω) ] + X t) α ω) ]) dα η ) ] η 1,α 1 α ω) + ) ]) 1,α 1 α ω) dα t 1,α η 1,αω) ]) + t 1,α η 1,αω) ]) ) dα = t 1 η1 ] t + 1,α η 1,αω) ]) + 1,α η 1,αω) ]) ) dα

S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process Remark 53 If Nt) degenerates to a Poisson random variable with constant rate, then the result in Theorem 5 degenerates the following form X t) ] = t η1 ] t + η 1,αω) ] + η 1,αω) ]) dα Remark 54 If η i degenerate to iid random variables, then the result in Theorem 5 degenerates the following form X t) ] = t 1 ] η 1] + t 1] η 1 ] Theorem 53 If Xt), t } is a fuzzy random compound Poisson process with fuzzy rates i, then we have X 3 t) ] = t 1 η1 3 ] + 3t + t3 ) 1,α η 1,α ω) ] η ) ] 1 α ω) + ) α η 1,α ω) ] η ) ]) 1 α ω) dα ) 3 1,α 3 η1,αω) ] + ) 3 α 3 η1,αω) ]) dα Proof Similarly, we have X 3 t) αω) = Nt) αω) ηi,αω) By the results of a compound Poisson process see Ross18]), X 3 t) αω) ] = t 1,α η 3 1 ) α ω) ] + 3 1,α) η 1,α ω) ] X 3 t) α ω) ] = t 1,α η 3 1 ) α ω) ] + 3 1,α) η 1,α ω) ] It is easy to prove that 3, X 3 t) α ω) = Nt) α ω) ηi,αω) η 1 ) α ω) ] + t 3 1,α) 3 3 η 1,α ω)], η 1 ) α ω) ] + t 3 1,α) 3 3 η 1,α ω)] 3 1 η1 3 ) η α ω) = ) ] 3 1,α 1 α ω), 1 η1 3 ) η α ω) = ) ] 3 1,α 1 α ω) and then 1 η1 3 ] 1 = η ) ] 3 η 1,α 1 α ω) + ) ]) 3 1,α 1 α ω) dα

Journal of Information and Computing Science, 16) 4, pp7-4 1 sing Proposition 3, = t X 3 t) ] = 1 1,α + 1 + 1 = t 1 η1 3 ] + 3t + t3 The proof is finished X 3 t) αω) ] + X 3 t) α ω) ]) dα η ) ] 3 η 1 α ω) + ) ]) 3 1,α 1 α ω) dα 3 1,α) η 1,α ω) ] η ) ] 1 α ω) + 3 1,α) η 1,α ω) ] η ) ]) 1 α ω) dα t 3 1,α) 3 3 η 1,αω) ] + t 3 1,α) 3 3 η 1,αω) ]) dα ) 1,α η 1,α ω) ] η ) ] 1 α ω) + ) α η 1,α ω) ] η ) ]) 1 α ω) dα ) 3 1,α 3 η1,αω) ] + ) 3 α 3 η1,αω) ]) dα Remark 55 If Nt) degenerates to a Poisson random variable with constant rate, then the result in Theorem 53 degenerates the following form X 3 t) ] = t η1 3 ] + 3t + t3 3 η1,αω) ] η ) ] 1 α ω) + η1,αω) ] η ) ]) 1 α ω) dα 3 η 1,αγ) ] + 3 η 1,αγ) ]) dα Remark 56 If η i degenerate to iid random variables, then the result in Theorem 53 degenerates the following form X 3 t) ] = t 1 ] η 3 1] + 3t 1] η1 ] η 1] + t3 3 1] 3 η 1 ] Theorem 54 et Xt), t } be a fuzzy random compound Poisson process with fuzzy rates i If h is a strictly increasing nonnegative function, then Xt) hxt))] = 1 η 1 hxt) + η 1 )] Proof Since h is a strictly increasing nonnegative function, using 5), Nt) αω) h ηi,αω) h Xt)ω)θ, γ i )) h Nt) αω) ηi,αω) h Nt) αω) Nt) α ω) ηi,αω), ηi,αω) Xt)ω)θ, γ i ) hxt)ω)θ, γ i )),

S i, et al: Fuzzy Random Homogeneous Poisson Process and Compound Poisson Process From Definition, Xt)ω)θ, γ i ) hxt)ω)θ, γ i )) Xt) hxt)) αω) = Xt) hxt)) αω) = Nt) αω) Nt) α ω) Nt) α ω) By the results of a compound Poisson process see Ross 18]), It is easy to prove that and consequently, ηi,αω) h ηi,αω) h ηi,αω) h Nt) αω) Nt) α ω) Nt) α ω) ηi,αω), ηi,αω) ηi,αω) Xt) hxt)) αω) = 1,α η 1,αω) h Xt) αω) + η 1,αω) ), Xt) hxt)) α ω) = 1,α η 1,αω) h Xt) α ω) + η 1,αω) ) 1 η 1 hxt) + η 1 ) αω) ] = 1,α η 1,αω) h Xt) αω) + η 1,αω) )], 1 η 1 hxt) + η 1 ) α ω) ] = 1,α η 1,αω) h Xt) α ω) + η 1,αω) )], 1 η 1 hxt) + η 1 )] = 1 It follows from Proposition 3 that Xt) hxt))] = 1 = 1 1,α η 1,αω) h Xt) αω) + η 1,αω) )] + 1,α η 1,α ω) h Xt) α ω) + η 1,α ω))]) dα Xt) hxt)) α ω) ] + Xt) hxt)) α ω) ]) dα 1,α η 1,α h Xt) αω) + η 1,αω) )] + 1,α η 1,α h Xt) α ω) + η 1,αω) )]) dα = 1 η 1 hxt) + η 1 )] The proof is finished Remark 57 If Nt) degenerates to a Poisson random variable with rate constant, then the result in Theorem 54 degenerates the following form Xt) hxt))] = η 1 hxt) + η 1 )] Remark 58 If η i degenerate to random variables, then the result in Theorem 54 degenerates the following form Xt) hxt))] = 1 ] η 1 hxt) + η 1 )]

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