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In this study, we examined the existence and asymptotic behavior of the time-dependent solution of a system with two identical components whose lifetimes follow a bivariate exponential distribution. First, by using the strong continuous semigroup theory, we proved the existence and uniqueness of the nonnegative time-dependent solution of the system model. Next, by studying the spectral properties of the operator corresponding to the system model, we determined that zero is an eigenvalue of the operator and its adjoint operator with geometric multiplicity one, and all points on the imaginary axis except zero belong to the resolvent set of the operator. From the above results, we deduced that the time-dependent solution of the system model converges strongly to its steady-state solution.

By using the C0-semigroup theory, we study the asymptotic behavior of the time-dependent solution and the time-dependent indices of the M[X]/G/1 queuing model with feedback and optional server vacations based on a single vacation policy. This queuing model is described by infinitely many partial differential equations with integral boundary conditions in an unbounded interval. Under certain conditions, by studying spectrum of the underlying operator of this queuing model on the imaginary axis, we prove that the time-dependent solution of this queuing model strongly converges to its steady-state solution. Next, we prove that the time-dependent queuing length of this queuing system converges to its steady-state queuing length and the time-dependent waiting time of this queuing system converges to its steady-state waiting time as time tends to infinity. Our results extend the steady-state results of this queuing system.

In this paper, we study the asymptotic property of underlying operator corresponding to the M/G/1 queueing model with single working vacation, where both service times in a regular busy period and in a working vacation period are function. We obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of both the operator and its adjoint operator with geometric multiplicity one. Therefore, we deduce that the time-dependent solution of the queueing model strongly converges to its steady-state solution. We also study the asymptotic behavior of the time-dependent queueing system’s indices for the model.

By using the Hille–Yosida theorem, Phillips theorem and Fattorini theorem we prove that the M/G/1 queueing model with vacations and multiple phases of operation, which is described by infinitely many partial differential equations with integral boundary conditions, has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying the spectrum of the operator, which corresponds to the model, on the imaginary axis we prove that the time-dependent solution of the model strongly converges to its steady-state solution.

By studying the spectrum on the imaginary axis of the underlying operator, which corresponds to the M/G/1 retrial queueing model with general retrial times described by infinitely many partial differential equations with integral boundary conditions, we prove that the time-dependent solution of the model strongly converges to its steady-state solution. Next, when the conditional completion rates for repeated attempts and service are constants, we describe the point spectrum of the underlying operator and verify that all points in an interval in the left real line including 0 are eigenvalues of the underlying operator. Lastly, by using these results and the spectral mapping theorem we prove that the C0\\(C_{0}\\)-semigroup generated by the underlying operator is not compact, but not eventually compact and even not quasi-compact, and it is impossible that the time-dependent solution exponentially converges to its steady-state solution. In other words, our result on convergence is optimal.

This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treatsLp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics. This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to proveLp estimates for heat, Schrödinger, and wave type equations. A significant part of the results have been proved during the last decade. The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications. It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.

By studying the spectral properties of the underlying operator corresponding to the M / G / 1 queueing model with second optional service, we obtain that the time-dependent solution of the model strongly converges to its steady-state solution. We also show that the time-dependent queueing size at the departure point converges to the corresponding steady-state queueing size at the departure point.

We study the spectrum on the imaginary axis of the underlying operator which corresponds to the M/G/1 queueing model with exceptional service time for the first customer in each busy period that was described by infinitely many partial differential equations with integral boundary conditions and obtain that all points on the imaginary axis except 0 belong to the resolvent set of the operator and 0 is an eigenvalue of the operator and its adjoint operator. Thus, by combining these results with our previous results, we deduce that the time-dependent solution of the model converges strongly to its steady-state solution. Moreover, we show that our result on convergence is optimal. MSC: 47A10, 47D99.

The Baire category theorem for operators, concerning the denseness of the intersection of the ranges of certain sequences of continuous operators in Banach spaces, is generalized to multivalued linear operators. As an application we obtain a result which gives the denseness of the domains and ranges of certain sequences of paracomplete linear relations in Banach spaces. Results of Lennard and of Burlando about operators in Banach spaces are recovered.

By using the Hille–Yosida theorem and the Phillips theorem in functional analysis we prove that the M/G/1 retrial queueing model with server breakdowns has a unique nonnegative time-dependent solution. Next, when the service completion rate is a constant, by studying spectral properties of the operator corresponding to the model we study asymptotic behavior of its time-dependent solution. First of all, through considering the resolvent set of the adjoint operator of the operator we obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator. In addition, we prove that zero is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model strongly converges to zero.