2000 Mathematics Subject Classification: 60K05The spent life time and the residual life time are well investigated characteristics of an ordinary renewal process. In the present paper a generalization of these lifetime processes associated with an alternating renewal process is considered. Limiting distributions are presented in the case of in finite mean renewal periods.The paper is supported by NFSI-Bulgaria, Grant No. MM-1101/2001

Mitov, Kosto V.

Yanev, Nikolay M.

Bulgarian Digital Mathematics Library at IMI-BAS

Pliska Stud. Math. Bulgar. 16 (2004), 137-145STUDIA MATHEMATICABULGARICALIMITING DISTRIBUTIONS FOR LIFETIMES INALTERNATING RENEWAL PROCESSESKosto V. Mitov, Nikolay M. Yanev1The spent life time and the residual life time are well investigated character-istics of an ordinary renewal process. In the present paper a generalizationof these lifetime processes associated with an alternating renewal process isconsidered. Limiting distributions are presented in the case of infinite meanrenewal periods.1. Introduction and definitionsTwo important characteristics of renewal processes are the so called ”spent lifetime” and ”residual life time”. For the ordinary renewal processes these char-acteristics are widely studied in the literature (see e.g. [1], [3], [4], [9]). In therecent papers [7], [8] some processes of this kind were defined and limit theoremswere proved in the more general case of alternating renewal processes. The aimof the present paper is to represent in a systematic way the main results obtainedup to now in this area.On the probability space (Ω,A,P), letX = {Xi : i = 1, 2, . . . } and T = {Ti : i = 1, 2, . . . }1The paper is supported by NFSI-Bulgaria, Grant No. MM-1101/2001.2000 Mathematics Subject Classification: 60K05Key words: Alternating Renewal Processes, Spent Working Time, Spent Waiting Time,Residual Working Time, Limiting Distributions, Infinite Mean Renewal Periods138 K. Mitov, N. Yanevbe two independent sets of independent, non-negative random variables (r.v.)with distribution functions (d.f.)A(t) = P{Xi ≤ t} and F (t) = P{Ti ≤ t}respectively.Denote by Yi = Xi+Ti, i = 1, 2, . . . , the sequence of independent, identicallydistributed, non-negative r.v. with d.f.G(t) = P{Yi ≤ t} = (A ∗ F )(t) ≡∫ t0A(t− u)F (du).Define the ordinary renewal processS0 = 0, Sn+1 = Sn + Yn+1, n = 0, 1, 2, . . . ,and the corresponding counting processN(t) = max{n : Sn ≤ t}.The moments Sn are usually interpreted as times when an element in a systemis replaced instantly by another one of the same type. Two well known processesrelated to this sequence are the spent life time (t− SN(t), t ≥ 0) and the residuallife time (SN(t)+1− t, t ≥ 0). These processes are widely studied in the literature.(See e.g. [1], [2], [4], [5], [6], [10]).In our setting the sequence{Sn, S′n+1}∞n=0whereS′n+1 = Sn + Xn+1, n = 0, 1, 2, . . . ,is called alternating renewal process.In the terms of reliability theory one can interpret Xi as the time for theinstallation or the repairing time of the i−th element in a system and Ti as theworking time of the same element. So, in the sequence {Sn, S′n+1} there aretwo types of renewal events: Sn the beginning of the installation and S′n+1 =Sn + Xn+1 the beginning of the work of the n−th element.For the alternating renewal process {Sn, S′n+1}∞n=0 the following random vari-ables can be defined:Limiting distributions for renewal processes 139i) the spent working timeσ(t) = t− S ′N(t)+1 = t− SN(t) −XN(t)+1, t ≥ 0;ii) the residual working timeτ(t) = min{SN(t)+1 − t, TN(t)+1}, t ≥ 0;iii) the spent waiting timeω(t) = min{t− SN(t), XN(t)+1}, t ≥ 0.It is clear from the definitions that the following inequalities hold almostsurely:−∞ < σ(t) ≤ t, t ≥ 0,0 ≤ τ(t) < ∞, t ≥ 0,0 ≤ ω(t) ≤ t, t ≥ 0.Obviously, one obtains a classical renewal process if all Xi are identicallyequal to zero. In this case σ(t) = t − SN(t) ≥ 0 a.s. is the spent life time,τ(t) = SN(t)+1 − t a.s. is the residual life time, and ω(t) = 0 a.s., i.e. there is nowaiting time.For the alternating renewal processes, σ(t) can take both negative and positivevalues:σ(t) = σ+(t)− σ−(t),where σ+(t) = max{σ(t), 0} and σ+(t) = max{−σ(t), 0}.Note that σ+(t) can be interpreted as the spent lifetime and σ−(t)−as theresidual waiting period.The limiting behaviour of the spent and residual life times in an ordinaryrenewal process is well studied in both cases when the interarival times Ti havefinite and infinite mathematical expectations. (See e.g. [1], [2], [4], [6], [9], [10]).In the recent papers [7] and [8] we investigated the limiting behaviour of the lifetime processes defined for an alternating renewal process in the case when at leastone of the mathematical expectations EXi or ETi is infinite. In the present paperthe results of those papers together with new ones are given without proof. Therest of the paper is organized as follows. Section 2 contains the basic conditionsand equations for interrarival times. The results and comments are stated inSection 3.140 K. Mitov, N. Yanev2. Basic Conditions and EquationsFurther we will suppose some of the following basic conditions:A(0) = F (0) = 0, A(x) and F (x) are non-lattice d.f.;(1)mA = EXi < ∞;(2)orEXi = ∞, A¯(t) = 1−A(t) ∼ t−αLA(t), t →∞, α ∈ (0, 1),(3)where LA(.) is a s.v.f., and for each h > 0 fixed A(t)−A(t−h) = O(1/t), t > 0;mF = ETi < ∞;(4)orETi = ∞, F¯ (t) = 1− F (t) ∼ t−βLF (t), t →∞, β ∈ (0, 1),(5)where LF (.) is a s.v.f.Finally it is assumed thatlimt→∞A¯(t)F¯ (t)= c, 0 ≤ c ≤ ∞.(6)Note that a sufficient condition for A(t)−A(t−h) = O(1/t), t > 0, is that LA(t)to be non-decreasing in t ∈ [0,∞).In the papers [7] and [8] the following inequalities are required 1/2 < α, β ≤ 1.It appears that these constrains are not necessary in all the cases. So they willbe explicitly stated only in the theorems where they are needed.The following notations are needed:U(t) =∞∑n=0P{Sn ≤ t} =∞∑n=0G∗n(t)andV (t) =∞∑n=0P{S′n+1 ≤ t} =∞∑n=0P{Sn + Xn+1 ≤ t} =∫ t0A(t− u)U(du).The first theorem is auxiliary. It provides the functional equations for thedistributions of σ(t), τ(t) and ω(t), which are the basic tools for the investigation.Limiting distributions for renewal processes 141Theorem 1. i). The distribution function of σ(t) isP{σ(t) ≤ x}(7)=∫ t0A¯(max(t, t− x)− u)U(du)+∫ tmin(t−x,t)F¯ (t− u)V (du), −∞ < x ≤ t.ii). The survival function of τ(t) isP{τ(t) ≥ x} =∫ t0F¯ (t + x− u)V (du) + F¯ (x)P{σ(t) < 0}, x ≥ 0.(8)iii). The survival function of ω(t) isP{ω(t) > x}(9)=∫ tx[∫ t−v0F¯ (t− v − u)U(du)]A(dv) +∫ t−x0A¯(t− u)U(du), 0 ≤ x ≤ t.iv). The joint distribution of (σ(t), τ(t)) has the following representation:P{σ(t) ≤ x, τ(t) > y}(10)= F¯ (y)∫ t0A¯(max(t, t− x)− u)U(du)+∫ tmin(t,t−x)F¯ (t + y − u)V (du), −∞ < x ≤ t, y > 0.3. Statement of the limiting resultsThe limit theorems stated below depends on the limit (6). So, provided thatc = ∞, the time for installation Xi has a heavier tail than the time for work Ti,and the condition 0 ≤ c < ∞ gives the opposite.The next theorem establishes proper limiting distributions under appropriatenormalization for σ(t) in the case c = ∞. The first assertion improves Theorem2.1 in [7] which is proved under the restriction α, β > 1/2. The assertion iii) isthe result of Theorem 2.2(ii) in [7]. The limiting distribution (11) is new.Theorem 2. Assume (1), (3), either (4) or (5), and (6) with c = ∞.i) There exist the limitslimt→∞P{σ(t) ≥ 0} = 0, limt→∞P{σ(t) < 0} = 1.142 K. Mitov, N. Yanevii) For −∞ < x < ∞,limt→∞P{σ(t)t≤ x} =sinpiαpi∫ 1/ max{1,1−x}0(1− u)−αuα−1du.(11)iii) If (5) with 1/2 < β < 1 holds thenlimt→∞P{σ(t)t≤ x|σ(t) ≥ 0} =1B(α, β)∫ x0uα−1(1− u)−βdu.The limiting distributions given in the following two theorems are new. Theydescribe the behavior of τ(t), ω(t), and that of the vector (σ(t), τ(t)).Theorem 3. Assume (1), (3), either (4) or (5), and (6) with c = ∞.i) For x > 0,limt→∞P{τ(t) ≤ x} = F (x).(12)ii) For −∞ < x < ∞, y > 0,limt→∞P{σ(t)t≤ x, τ(t) > y} = F¯ (y)sinpiαpi∫ 1/ max{1,1−x}0(1− u)−αuα−1du.(13)Theorem 4. Assume (1), (3) with α ∈ (1/2, 1), and (6) with c = ∞. For0 < x < 1,limt→∞P{ω(t)t≤ x} =sinpiαpi∫ x0u−α(1 − u)α−1du.(14)It is evident that in the case when the installation time Xi is ”longer” thenthe time for work Ti the limiting distributions depend on the tail parameter αof Xi. Only the conditional distribution in Theorem 2 depends on both tails. Itturns out that the restriction α, β > 1/2 is not necessary for most of the resultsin the above theorems. From the first part of Theorem 3 it becomes clear thatσ(t) and τ(t) are asymptotically independent. The joint limit distribution in (13)is the product of the limit distributions of σ(t) and τ(t) given by (11) and (12)respectively.The following theorems give the limiting distributions of σ(t), τ(t) and ω(t)for the case 0 ≤ c < ∞.Limiting distributions for renewal processes 143Theorem 5. Assume (1), (5), and (6) with 0 ≤ c < ∞.i) The following limits existlimt→∞P{σ(t) < 0} =c1 + c,limt→∞P{σ(t) ≥ 0} =11 + c.(15)ii) If β ∈ (1/2, 1) then for ∞ < x < ∞limt→∞P{σ(t)t≤ x} =c sinpiβpi(1 + c)∫ 1/ max{1,1−x}0(1− u)−βuβ−1du+sinpiβpi(1 + c)∫ g(x)0(1− u)β−1u−βdu,(16)whereg(x) =0, x < 0,x, 0 ≤ x ≤ 1,1, x > 1.iii) If β ∈ (1/2, 1) then for 0 ≤ x ≤ 1limt→∞P{σ(t)t≤ x} =c1 + c+c sin piβpi(1 + c)∫ x0(1− u)−βuβ−1du(17)andlimt→∞P{σ(t)t≤ x|σ(t) ≥ 0} =sinpiβpi∫ x0(1− u)−βuβ−1du(18)The first assertion of Theorem 5 improves Theorem 2.3 by removing thecondition α, β > 1/2. The third assertion is the result of Theorem 2.3(ii) in [7].The second one is new.The last three theorems state the limiting behavior of τ(t), ω(t) and the vector(σ(t), τ(t)) in the case when the working time Ti is ”longer” then the time forinstallation Xi. All the results are new.Theorem 6. Assume (1), (5), and (6) with 0 ≤ c < ∞. Then for y > 0limt→∞P{τ(t) ≤ y} =c1 + cF (y),144 K. Mitov, N. Yanevandlimt→∞P{τ(t)t≤ y} = 1−sinpiβpi(1 + c)∫ 1/(1+y)0(1− u)−βuβ−1du.Note that there are two singular limiting distributions for τ(t) under differentnormalization. The first one has an atom at infinity equal to 1/(1+c) over whichis the support of the second distribution with an atom at zero equal to c/(1 + c).Theorem 7. Assume (1), (5) with β ∈ (1/2, 1), and (6) with 0 ≤ c < ∞.Then for x ∈ (0, 1)limt→∞P{ω(t)t≤ x} =11 + c+c sin piβpi(1 + c)∫ x0u−β(1− u)β−1du.Theorem 8. Assume (1), (5) with β ∈ (1/2, 1), and (6) with 0 ≤ c < ∞.Then for y ≥ 0, x ∈ Rlimt→∞P{σ(t)t≤ x, τ(t) > y} = F¯ (y)c sin piβpi(1 + c)∫ 1/ max{1,1−x}0(1− u)−βuβ−1du+sinpiβpi(1 + c)∫ g(x)0(1− u)β−1u−βdu,andlimt→∞P{σ(t)t≤ x,τ(t)t> y} =sinpiβpi(1 + c)∫ 11+y1−g(x)1+y(1− u)−βuβ−1du.The only case which is not considered here is the boundary case when α = 1or β = 1 and the corresponding mathematical expectations are infinite. This caseis under consideration and the results will be published later.REFERE NCES[1] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Cam-bridge University Press, 1987.[2] E. B. Dynkin, Some limit theorems for sums of independent random vari-ables with infinite mathematical expectations, Izv. Acad. Nauk. SSSR Ser.Mat., 19 (1955), 247–266 (in Russian).Limiting distributions for renewal processes 145[3] W. Feller, An Introduction to Probability Theory and its Applications,John Wiley, New York, vol.2, 1nd edn., 1968.[4] W. Feller, An Introduction to Probability Theory and its Applications,John Wiley, New York, vol.2, 2nd edn., 1971.[5] K. B. Erickson, Strong renewal theorems with infinite mean, Trans. Amer.Math. Soc., 151 (1970), 263–291.[6] J. Lamperti, Some limit theorems for stochastic processes, J. Math. Mech.,7 (1958), 433–448.[7] K. V. Mitov, N. M. Yanev, Limit theorems for alternating renewal pro-cesses in the infinite mean case, Adv. Appl. Prob., 33 (2001), 896–911.[8] K. V. Mitov, N. M. Yanev, Regenerative processes in the infinite meancycle case, J. Appl. Prob., 38 (2001), 165–179.[9] K. Sigman, R. W. Wolff, A review of regenerative processes, SIAM Rev.,35 (1993), 269–288.[10] J. L. Teugels, Renewal theorems when the first or the second moment isinfinite, Ann. Math. Statist., 39 (1968), 1210–1219.Kosto V. MitovFaculty of AviationNational Military University ’V. Levski’5856 D. Mitropolia, Pleven, BULGARIAe-mail: kmitov@af-acad.bgNikolay M. YanevDepartment of Probability and StatisticsInstitute of Mathematics and InformaticsBulgarian Academy of Sciences1113 Sofia, BULGARIAe-mail: yanev@math.bas.bg

Limiting Distributions for Lifetimes in Alternating Renewal Processes

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Limiting Distributions for Lifetimes in Alternating Renewal Processes

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