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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 ; −∞
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