Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions

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-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.


Introduction
Retrial queueing systems are widely used in teletraffic theory, computers networks, communication networks, and so on. So, retrial queueing systems received considerable attention over recent years; see, for instance, Artalejo et al. [1], Phung-Duc et al. [2], Aissani et al. [3], and Gomez-Corral [4]. This paper studies the M/G/1 retrial queueing system with general retrial times defined as follows: customers arrive according to a Poisson stream of rate λ > 0; upon arrival, the service of the arriving customer commences immediately; otherwise, the customer leaves the service area and enters a group of blocked customers called "orbit" in accordance with an FCFS (First Come, First Served) discipline; only the customer at the head of the orbit queue is allowed for access to the server; when a service is completed, the access from the orbit to the server is governed by an arbitrary law with common probability distribution functionÃ(x) (Ã(0) = 0), the density functionã(x), and the Laplace-Stieltjes transformα(θ ); the service times are independent with common probability distribution function B(x) (B(0) = 0), the density function b(x), the Laplace-Stieltjes transformβ(θ ), and the first momentsβ k = (-1) kβ k (0), k = 1, 2, 3; interarrival times, retrial times, and service times are mutually independent. In 1999, Gomez-Corral [4] studied the M/G/1 retrial queueing system with general retrial times and obtained the following results: (1) a necessary and sufficient condition for the system to be stable by establishing the ergodicity of the embedded Markov chain at the departure points; (2) the steady-state distribution of the server state and the orbit length by using the supplementary variable technique; (3) the Laplace-Stieltjes transform of the waiting time distribution of a primary customer who arrives at the system at time t; (4) the Laplace-Stieltjes transform of the joint distribution of busy periods and idle times; (5) the Laplace-Stieltjes transform of the server state and the orbit length. To get the results mentioned, Gomez-Corral [4] firstly established a mathematical model of the M/G/1 retrial queueing system with general retrial times by using a supplementary variable technique and studied its steady-state solution under the following hypotheses: p n (t) = lim t→∞ p n (x, t), n ≥ 1; Q n (t) = lim t→∞ Q n (x, t), n ≥ 0, which imply the following two hypotheses in view of partial differential equations:

Hypothesis 2 The time-dependent solution converges to the steady-state solution.
Hypothesis 2 does not hold for some queueing models; see, for instance, Zheng and Gupur [5], Kasim and Gupur [6], and Abla and Gupur [7]. On the other hand, three types of convergence are of interest in view of functional analysis: weak convergence, strong convergence, and uniform convergence, and Hypothesis 2 does not indicate which one holds for this queueing model. Hence, we need to study Hypotheses 1 and 2.
In 2005, by using the C 0 -semigroup theory Gupur [8] has proved that the model has a unique positive time-dependent solution that satisfies the probability condition, that is, he showed that Hypothesis 1 holds under certain conditions. So far, no results on Hypothesis 2 have been found in the literature.
When s(x) and r(x) are constants, the M/G/1 retrial queueing model with general retrial times is called the M/M/1 retrial queueing model with special retrial times. In 2005, by studying the resolvent set of the adjoint operator of the operator corresponding to the M/M/1 retrial queueing model with special retrial times Zhang and Gupur [9] obtained that all points on the imaginary axis except 0 belong to the resolvent set of the operator. In 2006, Jiang and Gupur [10] proved that 0 is an eigenvalue of the operator with algebraic multiplicity one and 0 is an eigenvalue of its adjoint operator. By combining these results with Theorem 14 in Gupur et al. [11] (Theorem 1.96 in Gupur [12]) we deduce that the time-dependent solution of the M/M/1 retrial queueing model with special retrial times converges strongly to its steady-state solution. In 2009, Lv and Gupur [13] found that the operator has one eigenvalue on the left real line. After that, Ismayil and Gupur [14] and Gupur [15] proved that all points in an interval in the left real line belong to its point spectrum under different conditions. Until now, any other results on the M/M/1 retrial queueing model with special retrial times have not been found in the literature.
In this paper, using Greiner's idea [16] and Gupur et al. [17], we prove that all points on the imaginary axis except 0 belong to the resolvent set of the underlying operator corresponding to the M/G/1 retrial queueing model with general retrial times, 0 is its eigenvalue with geometric multiplicity one,and 0 is an eigenvalue of its adjoint operator. Thus, using Theorem 14 in Gupur et al. [11] or Theorem 1.96 in Gupur [12], we deduce that the time-dependent solution of the M/G/1 retrial queueing model with general retrial times strongly converges to its steady-state solution. Hence, we answer that Hypothesis 2 holds in the sense of "strong convergence" under certain conditions. Next, we describe the point spectrum of the operator corresponding to the M/M/1 retrial queueing model with special retrial times and verify that an interval in the left real line that includes 0 belongs to the point spectrum of the operator. Moreover, we show that our result implies the main results obtained by Lv and Gupur [13], Ismayil and Gupur [14], and Gupur [15]. Lastly, by combining these results with the spectral mapping theorem we prove that the C 0 -semigroup generated by the underlying operator is not compact, not eventually compact, and even not quasi-compact, and our result on the convergence of the time-dependent solution of the M/G/1 retrial queueing model with general retrial times is optimal, that is, it is impossible that the time-dependent solution exponentially converges to its steady-state solution, which means that Hypothesis 2 holds at most in the sense of strong convergence.
According to Gomez-Corral [4], the M/G/1 retrial queueing model with general retrial times can be described by the following system of equations: ; p 0 (t) represents the probability that at time t there is no customer in the system and the server is idle; p n (x, t) ( n ≥ 1) represents the probability that at time t the server is idle and there are n customers in the system with elapsed retrial time x; Q n (x, t) (n ≥ 0) represents the probability that at time t the server is busy and there are n customers in the system with elapsed service time x of the customer who is undergoing service; λ is the arrival rate of customers; r(x) is the conditional completion rate for repeated attempt at x satisfying r(x) ≥ 0 and ∞ 0 r(x) dx = ∞; s(x) is the conditional completion rate for service at x satisfying s(x) ≥ 0 and ∞ 0 s(x) dx = ∞.
In this paper, we use the notations in Gupur [8] and choose the state space It is obvious that X is a Banach space. In addition, X is also a Banach lattice under the following order relation for almost all x ∈ [0, ∞): For convenience, we introduce We define , p n (x) and Q n (x) are absolutely continuous functions and ∞ n=1 dp n We choose the boundary space ∂X = l 1 × l 1 and define the boundary operators If we introduce the underlying operator (A, D(A)) by then Eqs. (1.1)-(1.8) can be written as an abstract Cauchy problem in the Banach space X, which is of the form given by Gupur [8]: (1.9) Gupur [8] has proved the following result for system (1.9).
2 Asymptotic behavior of the time-dependent solution of system (1.9)

Lemma 2.1 If r(x) and s(x) satisfy
then 0 is an eigenvalue of A with geometric multiplicity one.
Proof Consider the equation A(p, Q) = 0, which is equivalent to Using (2.9) and (2.10) repeatedly, we deduce It is hard to determine concrete expressions of all p n (x) and Q n (x) and to prove that (p, Q) ∈ D(A). We further use another method. We introduce the probability generating functions where the summation and differential are interchangeable because of the convergence of ∞ n=1 [λ + r(x)]p n (x)z n and the Lebesgue control theorem.
The convergence of ∞ n=0 [λ + s(x)]Q n (x)z n and ∞ n=1 Q n-1 (x)z n allows us to change the order of summation and differential, so (2.3) and (2.4) imply Applying (2.5), (2.13), and (2.1) and noting the convergence of (2.14) By combining (2.6) and (2.7) with (2.12) and (2.14) and noting the convergence of ∞ 0 ∞ n=1 p n (x)z n dx and the Lebesgue control theorem we have , the Lebegue control theorem, and the l'Hospital rule it follows that By using (2.12), (2.13), (2.16), (2.17), the condition of this lemma, Theorem 1.1, and the Lebesgue control theorem we derive Equations (2.18) and (2.19) show that 0 is an eigenvalue of A. Moreover, from (2.1), (2.5)-(2.8), and (2.11) we know that the eigenvectors corresponding to 0 span the following linear space: It is easy to see that a n and b n are decided by p 0 , and p n (x) and Q n (x) are decided by a n and b n , respectively. Therefore, p n (x) and Q n (x) are determined by p 0 . Since p 0 ∈ R and c ∈ R, Υ is a one-dimensional linear subspace of X, that is, the geometric multiplicity of 0 is one.
According to Theorem 14 in Gupur, Li, and Zhu [11] or Theorem 1.96 in Gupur [12], we know that to obtain the asymptotic behavior of the time-dependent solution of system (1.9), we need to know the spectrum of A on the imaginary axis. By comparing to Zhang and Gupur [9] we find that the main difficult point is the boundary conditions and that there are infinitely many equations. In 1987, Greiner [16] put forward an idea to study the spectrum of A by perturbing boundary conditions when he studied a population equation which was described by a partial differential equation with an integral boundary condition. Using Greiner's idea, Haji and Radl [18] obtained the resolvent set of the operator corresponding to the M/M B /1 queueing model where all parameters are constants and gave a result described by the Dirichlet operator. In the following, by applying the result, we deduce the resolvent set of A on the imaginary axis. To do this, define (A 0 , D(A 0 )) as and discuss the inverse of A 0 . For any given (y, z) ∈ X, consider the equation (γ I -A 0 )(p, Q) = (y, z), that is, By solving (2.20)-(2.24) we have , then Eqs. (2.25)-(2.28) give, provided that the resolvent of A 0 exists, From this, together with the definition of the resolvent set, we obtain the following result. Then Proof. For any f ∈ L 1 [0, ∞), using integration by parts, as in Gupur and Ehmet [17], we estimate .
Since an absolutely convergent number series converges to the original limit if the original orders of its terms are changed, (2.29 which means that the result of this lemma is right.

31)
p n (x) = a n e -(γ +λ)x-x 0 r(ξ ) dξ , ∀n ≥ 1, (2.32) Since (p, Q) ∈ ker(γ I -A m ), by Theorem 4.12 in Adams [19], which implies that integration by parts, and the Cauchy product, we estimate From (2.53) and the definition of Φ it is easy to determine the expression of ΦD γ by Haji and Radl [18] gave the following result, through which we deduce the resolvent set of A on the imaginary axis.
By using Lemma 2.4 and p. 297 of Nagel [20] we derive the following result.

Then all points on the imaginary axis except zero belong to the resolvent set of A.
Proof. Let γ = im, m ∈ R\{0}, − → a = (a 1 , a 2 , . . .) ∈ l 1 , and implies that there exist M > 0 and θ 1 ∈ (0, 1) such that recalling that a convergent positive number series still converges to the original limit if the orders of its terms are changed we derive, for |m| > M and ( This means that when |m| > M, the spectral radius r(ΦD γ ) ≤ ΦD γ ≤ θ 2 < 1, which implies 1 / ∈ σ (ΦD γ ) for |m| > M, and therefore by Lemma 2.4 we know that γ = im / ∈ σ (A) for |m| > M, that is, On the other hand, since T(t) is positive uniformly bounded by Theorem 1.1, by Corollary 2.3 in Nagel [20], p. 297, we know that σ (A) ∩ iR is imaginary additively cyclic, which states that im ∈ σ (A) ∩ iR ⇒ imk ∈ σ (A) ∩ iR for all integers k, from which, together with (2.56) and Lemma 2.1, we conclude that σ (A) ∩ iR = {0}, that is, all points on the imaginary axis except zero belong to the resolvent set of A.
According to Zhang and Gupur [9] and Jiang and Gupur [10], X * , the dual space of X, is as follows: It is obvious that X * is a Banach space. Zhang and Gupur [9] gave the following expression of A * , the adjoint operator of A: Since T(t) is uniformly bounded, by Lemma 2.3 in Arendt and Batty [21] and Lemma 2.1 we know that 0 is an eigenvalue of A * . Furthermore, replacing α and β in [9] with r(x) and s(x), respectively, we deduce the following result.

Lemma 2.6 If
then 0 is an eigenvalue of A * with geometric multiplicity one.
Since by combining Lemmas 2.1 and 2.6 and Lemma 5 in Gupur [22] we know that 0 is an eigenvalue of A * with algebraic multiplicity one, Theorem 1.1, Lemma 2.1, Lemma 2.5, and Lemma 2.6 satisfy the conditions of Theorem 14 in Gupur, Li, and Zhu [11] (Theorem 1.96 in Gupur [12]). Thus, we obtain the desired result in this section, which is the direct result of Theorem 14 in Gupur, Li, and Zhu [11].
then the time-dependent solution of system (1.9) converges strongly to its steady-state solution, that is, where (p * , Q * ) and (p, Q) are the eigenvectors in Lemmas 2.6 and 2.1, respectively.
Remark 2.1 Gomez-Corral [4] obtained that the Markov process is ergodic if and only if λβ 1 <α(λ). This condition is quite different from the conditions in Theorem 2.1. Hence, it is worth studying the relation between the ergodicity of the Markov process and the conditions in Theorem 2.1. This is our future research topic.
Theorem 3.1 If α, β, λ > 0 and λ(λ+α) αβ < 1, then all points in the set are eigenvalues of A with geometric multiplicity 1. Especially, the interval belongs to the point spectrum of A.

Conclusion and discussion
Let σ p (T(t)) and σ p (A) be the point spectra of T(t) and A, respectively. From Theorem 1. we know that T(t) has uncountably many eigenvalues, and therefore it is not compact and even not eventually compact ( [23], p. 330). Corollary 2.11 in Engel and Nagel [23], p. 258, states that if T(t) is a C 0 -semigroup on the Banach space X with generator A, then I. ω 0 = max{ω ess ,s(A)}, where ω 0 is the growth bound of T(t), ω ess is the essential growth bound of T(t), ands(A) is the spectral bound of A. II. σ (A) ∩ {γ ∈ C | Re γ ≥ ω} is finite for each ω > ω ess . Here σ (A) is the spectrum of A. Theorem 1.1, Lemma 2.5, and Theorem 3.1 imply that ω 0 = 0 ands(A) = 0. These, together with items I and II above, yield ω ess = 0. From this and Proposition 3.5 in [23], p. 332, we conclude that T(t) is not quasi-compact. Hence, queueing models are essentially different from the population equation [24] and the reliability models that are described by a finite number of partial differential equations with integral boundary conditions [12].
Until now, we have not described the essential spectrum of A for r(x) = α and s(x) = β. We have not found an efficient way to describe the spectrum of A when r(x) and s(x) are nonconstant. All of them are our next research topic.