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6 Stability of retrial queues with versatile retrial policy3. Stability of the retrial queue with versatile retrial policyWe consider a single server retrial queue at which primary customers arrive to the systemat times {t i , i = 1,2,...}. Letτ i = t i+1 − t i be the successive interarrival times, i = 1,2,....If the ith arriving customer finds the server free, he receives service and leaves the system.Otherwise, if the server is busy, the arriving customer moves immediately to an “orbit.”The probability of a repeated attempt from the orbit during the interval (t,t + Δt), giventhat j customers were in orbit at time t, is(θ(1 − δ 0j )+jμ)Δt + ◦(Δt). That is, after anexponential random time with rate θ (which we will call the orbit retrial time), independentof the arrival process, each customer in orbit generates a Poisson stream of repeatedattempts with parameter μ and behaves independently of other customers in orbit andof the external arrival process. This model, introduced by Artalejo and Gomez-Corral[4], incorporates simultaneously the classical linear retrial policy and the constant one.If μ = 0, we obtain the constant retrial policy with parameter θ; if the orbit retrial timeends before an external arrival, then one customer from the orbit (the customer at thehead of the queue or a randomly chosen one) occupies the server. The nth service durationis σ n , and we assume that 0 < Eσ n < ∞. We assume throughout this section that thesequence {σ n } is stationary and ergodic, the interarrival times {τ i } are i.i.d. exponentiallydistributed with parameter λ, the interarrival times, orbit retrial times, and retrial timesof each customer in orbit are mutually independent and independent of the sequence{σ n }.LetX(t) be the number of customers in orbit at time t. Defines n to be the instantwhen the (n − 1)st service time ends. We consider the process X(n) embedded immediatelyafter time s n , (i.e., X(n) = X(s + n)). After the end of the (n − 1)st service, a competitionbetween two independent (since the orbit retrial time and retrial times of eachcustomer in orbit are independent of the interarrival time) exponential laws with ratesλ and θ + X(n)μ determines the next customer that gains the server and the probabilitythat a retrial time expires earlier than the interarrival time is (θ + X(n)μ)/(λ + θ + X(n)μ).Let u 1 n and u 2 n be two i.i.d. sequences of random variables distributed uniformly on [0,1],mutually independent and independent of the sequence σ n . u 1 ={u 1 n } will generate thearrival process, and u 2 ={u 2 n} will generate the type of arrival (external or from the orbit)at the end of the successive service periods. Let ∏ : R + × [0,1] → N denote the inverse ofthe Poisson distribution∏(t,x) = inf{n ∈ N :n∑k=0t k e −tk!≥ x}, (3.1)so that ∏ (t,u 1 n) is a Poisson random variable with parameter t. We have the followingrepresentation of the process X(n):X(n +1)= ( X(n)+ξ n) +, (3.2)where x + = max[0,x]andξ n = ∏ ( λσn ,u 1 )n − I(u 2 n ≤ θ + X(n)μ )λ + θ + X(n)μ(3.3)is the governing sequence.

T. Kernane and A. Aïssani 7Theorem 3.1. The process X(n) is strong coupling convergent to a unique stationary ergodicregime if one of the following conditions is fulfilled:(1) θ>0, μ = 0,andλEσ 1 0,andλEσ 1 < 1.Proof. Consider the case θ>0andμ = 0, then the driving sequence (3.3)hastheformξ n = ∏ ( λσn ,u 1 )n − I(u 2 n ≤ θ ). (3.4)λ + θSince the sequence {u i n } is identically distributed for i = 1,2, and hence stationary, it canbe defined for all integers −∞

8 Stability of retrial queues with versatile retrial policyhence there is a number n 0 = n 0 (a)suchthatP(A n ) > 0forn ≥ n 0 . If, on the other hand,the events B n ,thenumberm, and the function g : R m+1 → R are defined asm = n 0 , B n = T m A n , g ( y 0 ,..., y m)≡ y+m , (3.11)then the events B n ∈I ξ n+m are renovating for {X(n)} on the segment [n,n + m] foralln ≥ 0, so one can assume that n 0 = 0. Hence, using Theorem 2.7, the sequence {X(n)}is strong coupling convergent to a unique stationary sequence {X n ≡ U n X 0 },whereX 0is I σ,u -measurable, obeying the equation X n+1 = (X n + ξ n ) + , and the ergodicity followsfrom Remark 2.4 and the fact that X n is compatible with the shift U.Let us now consider the case θ ≥ 0andμ>0. The renovation events A n will be constructednow in two steps. We will first introduce a majorizing SRS X(n) ∗ on the sameprobability space, which will enable us to obtain simple stationary renovating events A ∗ nwith positive probability, and A n will be obtained as some subsets of A ∗ n .TheSRSX(n) ∗has the following form:X(0) ∗ = X(0),X(n +1) ∗ = max ( )C,X(n) ∗ + ξn∗ , (3.12)whereξn ∗ = ∏ ( λσn ,u 1 )n − I(u 2 n ≤ θ + Cμ ). (3.13)λ + θ + CμThe sequence {ξ ∗ n } is measurable with respect to Iσ,u and ξ ∗ n+1 = Uξ ∗ n . From this andRemarks 2.3 and 2.4 it follows that the sequence {ξ ∗ n } is stationary and ergodic. We choosethe constant C such that Eξ ∗ n < 0 if the condition λEσ 1 < 1holds,whereE ( )ξn∗ = λEσ1 − θ + Cμλ + θ + Cμ . (3.14)It follows that there exist renovation events A ∗ n = Tn A ∗ 0 , n ≥ n 0 ,whereA ∗ 0 is defined as(3.7) with the sequence {ξn ∗},suchthatX(n)∗ = C on the set A ∗ n for all n ≥ n 0 .Definethe sets{ ∏ (B 0 = λσ−k ,u 1 )−k = 0, u2−k ≤ θ + kμ}λ + θ + kμ , k = 1,...,C , B n = T n B 0 . (3.15)The sets A n = A ∗ n−C ∩ B n form a stationary renovating sequence for X(n), since for all n ≥n 0 + C,wehaveonA n ,thevaluesX(n − k) ≤ k, k = 0,1,...,C, and in particular, X(n) = 0.Moreover, P(A 0 ) = P(A ∗ −C) P(B 0 | A ∗ −C) > 0 (see Altman and Borovkov [2, pages 353-354]). The strong coupling convergence of the process X(n) to a stationary ergodic regimefollows from Theorem 2.7.□Remark 3.2. Although the conditions of Theorem 3.1 are sufficient for stability, we canassert that they are necessary, since for SRS of the form X(n +1)= (X(n)+ξ n ) + ,the

T. Kernane and A. Aïssani 9condition Eξ n > 0 implies that the process X(n) converges in distribution to an improperlimiting sequence, that is, X(n) →∞almost surely, and it can be extended under wideassumptions to the case Eξ n = 0 (see Borovkov [8, Theorem 1.7]).4. Stability of the retrial queue with two types of customersConsider now a retrial queue with two types of arriving customers, known as “impatient”and “persistent.” If an impatient customer finds the server busy, then he leaves the system.On the other hand, if a persistent customer arrives and finds the server busy, then he mayhave access to the orbit and waits to be served later according to the versatile retrial policydescribed above. Assume that the interarrival times {τn 1} of type 1 (impatient) and {τ2 n }of type 2 (persistent) are sequences of independent and identically distributed randomvariables with exponential distributions with parameters λ 1 > 0andλ 2 > 0, respectively.The service times {σn 1} (for type 1) and {σ2 n } (for type 2) are stationary and ergodic,independent of each other and of {τn 1}, {τ2 n }, orbit retrial times and retrial times of eachcustomer in orbit, and 0 < E(σn) j < ∞, j = 1,2. The sequences {τn}, 1 {τn}, 2 orbit retrialtimes and retrial times of each customer in orbit are independent of each other. Let X(t)bethenumberofcustomersinorbitattimet. Defines n to be the instant when the (n −1)st service time ends. We consider the process X(n) embedded immediately after times n , (i.e., X(n) = X(s + n)). Let u 1 n and u 2 n defined as in Section 3, and independent of thesequences {σn 1} and {σ2 n },exceptthatu2 = u 2 n will generate now the type of request ofservice: impatient customer, external persistent customer or persistent customer fromthe orbit at the end of the successive service periods. The representation of the processX(n)isX(n +1)= ( X(n)+ξ n) +, (4.1)where the driving sequence {ξ n } is now of the formξ n = ∏ )(λ2 σn,u 1 1 )n I(u 2 n ≤ λ 1λ 1 + λ 2 + θ + X(n)μ+ ∏ ()(λ2 σn,u 2 1 ) λn I 1λ 1 + λ 2 + θ + X(n)μ

10 Stability of retrial queues with versatile retrial policyProof. Consider the first case θ>0andμ = 0. Then the sequence {ξ n } has the formξ n = ∏ )(λ2 σn,u 1 1 )n I(u 2 n ≤ λ 1λ 1 + λ 2 + θ+ ∏ ((λ2 σn,u 2 1 ) λn I 1λ 1 + λ 2 + θ

12 Stability of retrial queues with versatile retrial policyDefine the σ-algebras I σ,u,r n = σ(σ k ,r (k) ,u (k) ,u 1 k ,u2 k ,u3 k ; k ≤ n) and Iσ,u,r = σ(σ k ,r (k) ,u (k) ,u 1 k ,u2 k ,u3 k ; −∞

T. Kernane and A. Aïssani 13process of the regular customers, u 2 ={u 2 n } will generate the arrival process of the negativecustomers, and u 3 n will generate the type of arrival that gains the server (external orfrom the orbit) at the end of the successive service periods. Let X(n)bedefinedasaboveand it has now the following representation:X(n +1)= ( X(n)+ζ n − η n) +, (6.1)whereζ n = ∏ ( λσn ,u 1 )n − I(u 3 n ≤ θ + X(n)μ ), (6.2)λ + θ + X(n)μη n = ∏ ( δσn ,u 2 n) . (6.3)Theorem 6.1. The process X(n) is strong coupling convergent to a unique stationary andergodic regime if one of the following conditions is fulfilled:(1) θ>0, μ = 0,and((λ − δ)(λ + θ)/θ)Eσ 1 < 1;(2) θ ≥ 0, μ>0,and(λ − δ)Eσ 1 < 1.Proof. Consider the first case θ>0andμ = 0. Then the sequence (6.2)hastheformζ n = ∏ ( λσn ,u 1 )n − I(u 3 n ≤ θ ). (6.4)λ + θDefine the σ-algebras I σ,u n= σ(σ k ,u 1 k ,u2 k ,u3 k ; k ≤ n) andIσ,u = σ(σ k ,u 1 k ,u2 k ,u3 k ; −∞

14 Stability of retrial queues with versatile retrial policy7. Stability of the retrial queue with batch arrivalsConsider a single server retrial queue with the versatile retrial policy. Let us now considerthat at every arrival epoch t k , k = 1,2,...,arandombatchofa k customers enters the system.If the server is busy at the arrival epoch, then the whole batch of customers joins theorbit, whereas if the server is free, then one of the arriving customers starts his serviceand the others join the orbit. We assume that the input flow of customers occurs accordingto a Poisson process with rate λ, the sequence of batch sizes {a k } is independent andidentically distributed with general distribution and mean a, where0< a0andμ = 0, so the driving sequence (7.2) will have thefollowing form:ξ n =∏ (λσn,u 1 n)∑k=1a (n)k + ( a 1 − 1 ) (I u 2 n ≤λ ) ( )λ− Iλ + θ λ + θ

T. Kernane and A. Aïssani 15and if the condition λaEσ 1 < (λ(1 − a)+θ)/(λ + θ) holds,thenEξ n < 0. The conditionλ(a − 1) 0. The majorizing SRS has now theformX ∗ (0) = X(0),X ∗ (n +1)= max ( )C,X ∗ (n)+ξn∗ , (7.5)where∏ (λσn,u 1 n)ξn ∗ = ∑k=1a (n)k + ( a 1 − 1 ) (I u 2 n ≤) (λ− Iλ + θ + Cμ)λλ + θ + Cμ