A Semigroup Approach to Queueing Systems

Spectral properties of the M [X]/G/1 operator and its application

“…And Greiner [7] promoted an idea through which the spectrum of the underlying operator can be deduced by discussing the boundary operator. By applying the Greiner's idea, Haji and Radl [12,13] derived a lemma which is useful for us. This lemma was related to the resolvent set of the operator A 0 and the spectrum of D γ , where D γ is the inverse of L in ker(γ I − A m ).…”

“…The M [X ] /G/1 retrial queue with server breakdowns and constant rate of repeated attempts was governed by infinite number of partial differential equations with integral boundary conditions and the service rates are functions. By using a lemma in Haji and Radl [12,13] and Nagel [18] we obtain that all points on the imaginary axis except zero belong to the resolvent set of the underlying operator. Last, we determine the adjoint operator of the underlying operator and verify that 0 is an eigenvalue of the adjoint operator with geometric multiplicity one.…”

“…In 2007, Haji and Radl [12] have used successfully the Greiner's idea to study the asymptotic behavior of the time-dependent solution of a reliability model which was described by finite number of partial differential equations with boundary conditions. At the same year, they have applied the Greiner's idea to research the asymptotic behavior of the time-dependent solution of the M/M B /1 queueing model which was modeled by infinite number of partial differential equations with integral boundary conditions, but the service rate is constant [13]. The M [X ] /G/1 retrial queue with server breakdowns and constant rate of repeated attempts was governed by infinite number of partial differential equations with integral boundary conditions and the service rates are functions.…”

Asymptotic Stability of a Series-Parallel Repairable System Consisting of Three-Unit with Multiple Vacations of a Repairman

“…Applying the same method as in [7] we can express the resolvent of A in terms of the resolvent of 0 A , the Dirichlet operator D γ and the boundary operator as follows. Lemma 2.5: If Using Lemma 2.2, Lemma 2.3, Theorem 2.6and the same method as in ( [7], Th.…”

“…Lemma 2.5: If Using Lemma 2.2, Lemma 2.3, Theorem 2.6and the same method as in ( [7], Th. 3.11) we obtain the following result.…”