Further Research on the M/G/1 Retrial Queueing Model with Server Breakdowns

Abstract

We study spectral properties of the operator which corresponds to the M/G/1 retrial queueing model with server breakdowns and obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and 0 is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model is probably strongly asymptotically stable.

1. Introduction

There has been considerable interest in retrial queueing systems, see Atencia et al. [1, 2], Choi et al. [3], Djellab [4], Gupur [5, 6], Kasim and Gupur [7], Li et al. [8], Li and Wang [9], Wang et al. [10, 11], and Yang and Templeton [12]. Many researchers studied the M/G/1 retrial queueing systems with server breakdowns in the steady-state case, see Atencia et al. [1, 2], Choi et al. [3], Djellab [4], Li and Wang [9], and Wang et al. [10]. Only few researchers studied transient solutions of M/G/1 retrial queueing systems, see Wang et al. [11], Gupur [5, 6], Kasim and Gupur [7]. And Wang et al. [11] studied the transient solution of the M/G/1 retrial queueing system with server failure by using Laplace transform and obtained the expression of the probability-generating function. In other words, they studied the existence of the time-dependent solution of the model. Gupur [5, 6, 13, 14], Gupur et al. [15], and Kasim and Gupur [7] did dynamic analysis for several queueing models including retrial queueing models by using functional analysis and obtained the existence and uniqueness of the time-dependent solution of several queueing models and asymptotic behavior of their time-dependent solutions. In this paper, by using the idea in Gupur [6] and Gupur et al. [15], we study the asymptotic behavior of the time-dependent solution of the M/G/1 retrial queueing system with server breakdowns in which the failure states of the server are absorbing states. This queueing system was studied by Wang et al. [10] in 2001. By using the supplementary variable technique they established the corresponding queueing model and obtained explicit expressions of some reliability indices such as the availability, failure frequency for steady-state cases under the following hypothesis: “the time-dependent solution of the model converges to a steady-state solution." By reading their paper we find that the above hypothesis, in fact, implies the following two hypotheses.

Hypothesis 1. The model has a nonnegative time-dependent solution.

Hypothesis 2. The time-dependent solution of the model converges to a steady-state solution.

In 2010, Gupur [6] studied the above two hypotheses. Firstly, he converted the model into an abstract Cauchy problem by selecting a suitable Banach space and defining the underlying operator which corresponds to the model and its domain. Next, by using the Hille-Yosida theorem and the Phillips theorem he proved that the model has a unique nonnegative time-dependent solution and therefore obtained that Hypothesis 1 holds. Then, when the service completion rate is a constant, he studied the asymptotic behavior of its time-dependent solution. By studying resolvent set of of the adjoint operator of the operator which corresponds to the model (in this case, the M/G/1 retrial queueing model with server breakdowns is called the M/M/1 retrial queueing model with server breakdowns) he obtained the resolvent set of the operator on the imaginary axis: all points on the imaginary axis except zero belong to its resolvent set. And he proved that 0 is not an eigenvalue of the operator. Thus, he suggested that the time-dependent solution of the model probably strongly converges to zero. Until now, any other results about this model have not been found in the literature. In this paper, we try to study the asymptotic behavior of the time-dependent solution of the above model when the service completion rate is a function. Firstly, we convert the model into an abstract Cauchy problem by Gupur [6] and we determine the resolvent set of the adjoint operator of the operator corresponding to the model when the service completion rate satisfies a certain condition. Also, we prove that 0 is not an eigenvalue of the operator. If we can verify 0 is not in the residue spectrum of the operator, then from the above results we conclude that the time-dependent solution of the model strongly converges to zero. Naturally, the results obtained in [6] are now special cases of our results.

According to Wang et al. [10], the M/G/1 retrial queueing system with server breakdowns can be described by the following system of partial differential equations with integral boundary conditions: 𝑑𝑝_𝐼,0,0(𝑡)𝑑𝑡=−𝜆𝑝^𝐼,0,0𝑑𝑝_𝐼,𝑖,0(𝑡)+^∞0𝜇(𝑥)𝑝_𝑤,0,1(𝑥,𝑡)𝑑𝑥,(𝑡)𝑑𝑡=−(𝜆+𝑖𝜃)𝑝^𝐼,𝑖,0𝜕𝑝_𝑤,0,1(𝑡)+^∞0𝜇(𝑥)𝑝_𝑤,𝑖,1(𝑥,𝑡)𝑑𝑥,𝑖≥1,(𝑥,𝑡)𝜕𝑡+𝜕𝑝^𝑤,0,1(𝑥,𝑡)𝜕𝑥=−(𝜆+𝛼+𝜇(𝑥))𝑝^𝑤,0,1𝜕𝑝_𝑤,𝑖,1(𝑥,𝑡),(𝑥,𝑡)𝜕𝑡+𝜕𝑝^𝑤,𝑖,1(𝑥,𝑡)𝜕𝑥=−(𝜆+𝛼+𝜇(𝑥))𝑝^𝑤,𝑖,1+𝜆𝑝_{𝑤,𝑖−1,1}(𝑥,𝑡)𝑝(𝑥,𝑡),𝑖≥1,_𝑤,𝑖,1𝑝(0,𝑡)=𝜆𝑝_𝐼,𝑖,0(𝑡)+(𝑖+1)𝜃𝑝_{𝐼,𝑖+1,0}(𝑡),𝑖≥0,_𝐼,0,0(0)=1,𝑝_𝐼,𝑖,0(0)=0,𝑖≥1;𝑝_𝑤,𝑗,1(𝑥,0)=0,𝑗≥0.(1.1) Here (𝑥,𝑡)∈[0,∞)×[0,∞);  𝑝_𝐼,𝑖,0(𝑡)(𝑖≥0) represents the probability that the server is idle and there are 𝑖 customers in the retrial group at time 𝑡; 𝑝_𝑤,𝑖,1(𝑥,𝑡) represents the joint probability that at time 𝑡 there are 𝑖 customers in the retrial group and the server is up and customer is being served with elapsed service time 𝑥;  𝜆 is the arrival rate of customers; 𝛼 is the server failing rate; 𝜃 is the successive interretrial times of customers; 𝜇(𝑥) is the service completion rate at time 𝑥 satisfying 𝜇(𝑥)≥0,^∞0𝜇(𝑥)𝑑𝑥=∞.(1.2) In this paper, we use the notations in Gupur [6]: Γ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⋮𝜆000⋮𝜆𝜃002⋮𝜆00𝜃3⋮𝜆00𝜃4⋮000𝜃⋱⋯⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(1.3) Take a state space as follows: 𝑋=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑝_𝐼,𝑝_𝑤|||||||||||||𝑝^𝐼=𝑝_𝐼,0,0,𝑝_𝐼,1,0,𝑝_𝐼,2,0,…∈𝑙¹𝑝_𝑤∈=𝐿₁𝑝_𝑤,0,1,𝑝_𝑤,1,1,𝑝_𝑤,2,1,…‖‖𝑝[0,∞)×𝐿¹[0,∞)×𝐿¹[0,∞)×⋯_𝐼,𝑝_𝑤‖‖=∑^∞_𝑖=0||𝑝_𝐼,𝑖,0||+∑^∞_𝑖=0‖‖𝑝_𝑤,𝑖,1‖‖_𝐿1[0,∞)<∞⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭.(1.4) It is obvious that 𝑋 is a Banach space and also a Banach lattice. In the following we define operators and their domains: 𝐴𝑝_𝐼,𝑝_𝑤=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−⋮000𝜆−(𝜆⋮000+𝜃)−(𝜆⋮000+2𝜃)−(𝜆⋮000+3𝜃)⋱⋯⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑝^𝐼,0,0𝑝_𝐼,1,0𝑝_𝐼,2,0𝑝_𝐼,3,0⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝑑⋮⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,0𝑑𝑥−0𝑑00⋯00𝑑𝑥−0𝑑0⋯000𝑑𝑥−0𝑑⋯⋮⋮⋮⋮𝑑𝑥⋱⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑝^𝑤,0,1𝑝(𝑥)_𝑤,1,1𝑝(𝑥)_𝑤,2,1𝑝(𝑥)_𝑤,3,1𝐷(𝐴)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑝_𝐼⋮(𝑥)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,,𝑝_𝑤∈𝑋||||||||||𝑝^𝑤,𝑖,1fu(n𝑥c)ti(o𝑖n≥s0a)ndarseataisbfsyol𝑝u_𝑤telycontinuous(0)=Γ𝑝_𝐼,_∞𝑛∑₌₀‖‖‖𝑑𝑝_𝑤,𝑖,1𝑑𝑥‖‖‖_𝐿¹_[0,∞)𝑈𝑝<∞_𝐼⎫⎪⎪⎪⎬⎪⎪⎪⎭,,𝑝_𝑤=⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝⋮000⋮000⋮000⋱⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝𝑝^𝐼,0,0𝑝_𝐼,1,0𝑝_𝐼,2,0⎛⎜⎜⎜⎜⎜⎜⎝⋮𝔲𝜆0⋮𝔲𝜆0⋮𝔲00⋱⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝𝑝⋮⎞⎟⎟⎟⎟⎟⎟⎠_𝑤,0,1,𝑝(𝑥)_𝑤,1,1𝑝(𝑥)_𝑤,2,1⋮(𝑥)⎞⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠,𝐷(𝑈)=𝑋,(1.5) where 𝔲=−(𝜆+𝛼+𝜇(𝑥)). 𝐸𝑝_𝐼,𝑝_𝑤=⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝∫^∞0∫𝜇(𝑥)𝑝_𝑤,0,1(𝑥)𝑑𝑥_∞0∫𝜇(𝑥)𝑝_𝑤,1,1(𝑥)𝑑𝑥_∞0𝜇(𝑥)𝑝⋮_𝑤,2,1(𝑥)𝑑𝑥⎞⎟⎟⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎜⎜⎝⋮000⎞⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠,𝐷(𝐸)=𝑋.(1.6) Then the above system of (1.1) can be expressed as an abstract Cauchy problem: 𝑑𝑝_𝐼,𝑝_𝑤(𝑡)𝑑𝑡=(𝐴+𝑈+𝐸)𝑝^𝐼𝑝,𝑝_𝑤(𝑡),∀𝑡∈(0,∞),_𝐼,𝑝_𝑤(0)=⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝⋮10⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝⋮00⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠.(1.7)

Gupur [6] have obtained the following results.

Theorem 1.1. If 𝜇=sup_{𝑥∈[0,∞)}𝜇(𝑥)<∞, then 𝐴+𝑈+𝐸 generates a positive contraction 𝐶₀-semi group  𝑇(𝑡). And the system (1.7) has a unique nonnegative time-dependent solution (𝑝_𝐼,𝑝_𝑤)(𝑥,𝑡)=𝑇(𝑡)(𝑝_𝐼,𝑝_𝑤)(0) satisfying ‖(𝑝_𝐼,𝑝_𝑤)(⋅,𝑡)‖≤1,∀𝑡∈[0,∞).

2. Asymptotic Behavior of the Time-Dependent Solution of the System (1.7)

In this section, firstly we determine the expression of (𝐴+𝑈+𝐸)^∗, the adjoint operator of 𝐴+𝑈+𝐸, next we study the resolvent set of (𝐴+𝑈+𝐸)^∗, through which we deduce the resolvent set of 𝐴+𝑈+𝐸 on the imaginary axis. Thirdly, we prove that 0 is not an eigenvalue of 𝐴+𝑈+𝐸. Thus, we state our main results in this paper.

It is easy to see that 𝑋^∗, the dual space of 𝑋, is as follows (see Gupur [6]): 𝑋^∗=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑞^∗𝐼,𝑞^∗𝑤,|||||||||||||𝑞^∗𝐼𝑞_∗𝑤=𝑞^∗𝐼,0,0,𝑞^∗𝐼,1,0,𝑞^∗𝐼,2,0,…∈𝑙^∞,=∈𝐿𝑞^∗𝑤,0,1_∞,𝑞^∗𝑤,1,1,𝑞^∗𝑤,2,1,…||‖‖𝑞[0,∞)×𝐿^∞[0,∞)×𝐿^∞[0,∞)×⋯_∗𝐼,𝑄^∗𝑤‖‖||=supsup_𝑖≥0||𝑞^{∗𝐼,𝑖,0}||,sup_𝑖≥0‖‖𝑞^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)}<∞⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭.(2.1) It is not difficult to verify that 𝑋^∗ is a Banach space.

Lemma 2.1. (𝐴+𝑈+𝐸)^∗, the adjoint operator of 𝐴+𝑈+𝐸, is as follows: (𝐴+𝑈+𝐸)^∗𝑞^∗𝐼,𝑞^∗𝑤=(ℒ+𝒢+ℋ)𝑞^∗𝐼,𝑞^∗𝑤,∀𝑞^∗𝐼,𝑞^∗𝑤∈𝐷(ℒ),(2.2) where ℒ𝑞^∗𝐼,𝑞^∗𝑤=⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝−⋮00𝜆−(𝜆⋮00+𝜃)−(𝜆⋮00+2𝜃)⋱⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝𝑞^∗𝐼,0,0𝑞^∗𝐼,1,0𝑞^∗𝐼,2,0⎛⎜⎜⎜⎜⎜⎝𝑑⋮⎞⎟⎟⎟⎟⎟⎟⎠,𝑑𝑥−(𝜆+0𝛼+𝜇(𝑥))𝑑0⋯⋮𝑑𝑥−(𝜆+⋮𝛼+𝜇(𝑥))⋱⋯⎞⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝑞^∗𝑤,0,1𝑞(𝑥)_∗𝑤,1,1𝒢𝑞⋮(𝑥)_∗𝐼⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎠,,𝑞^∗𝑤=⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝⋮000⎞⎟⎟⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎜⎜⎝𝜇⋮00(𝑥)𝜇⋮00(𝑥)𝜇⋮00(𝑥)⋱⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝𝑞^∗𝐼,0,0𝑞^∗𝐼,1,0𝑞^∗𝐼,2,0ℋ𝑞_∗𝐼⋮⎞⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠,,𝑞^∗𝑤=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⋮𝜆𝜃002⋮𝜆00𝜃3⋮𝜆00𝜃⋮𝜆000⋱⋯⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑞^∗𝑤,0,1𝑞(0)_∗𝑤,1,1𝑞(0)_∗𝑤,2,1𝑞(0)_∗𝑤,3,1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⋮0000⋮𝜆000⋮𝜆000⋮𝜆000⋮𝜆000⋱⋯⋯⋯⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑞⋮(0_∗𝑤,0,1)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑞(𝑥)_∗𝑤,1,1𝑞(𝑥)_∗𝑤,2,1𝑞(𝑥)_∗𝑤,3,1𝐷(ℒ)=⎧⎪⎨⎪⎩𝑞_∗𝐼⋮(𝑥)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,,𝑞^∗𝑤∈𝑋^∗||||||𝑞_{∗𝑤,𝑖,1}and(s𝑥a)tisafrye𝑞abso_{∗𝑤,𝑖,1}lutelycontinuous𝐷(𝒢)=𝐷(ℋ)=𝑋_∗(∞)=𝜂,𝑖≥0⎫⎪⎬⎪⎭,.(2.3) Here, 𝜂 in 𝐷(ℒ) is a nonzero constant which is irrelevant to 𝑖.

Proof. By using integration by parts and the boundary conditions on (𝑝_𝐼,𝑝_𝑤)∈𝐷(𝐴), we have, for any (𝑞^∗𝐼,𝑞^∗𝑤)∈𝐷(ℒ), (𝐴+𝑈+𝐸)𝑝_𝐼,𝑝_𝑤,𝑞^∗𝐼,𝑞^∗𝑤=^∞_𝑖=0−(𝜆+𝑖𝜃)𝑝^𝐼,𝑖,0+^∞0𝜇(𝑥)𝑝_𝑤,𝑖,1(𝑥)𝑑𝑥𝑞^{∗𝐼,𝑖,0}+^∞0−𝑑𝑝^𝑤,0,1(𝑥)𝑑𝑥−(𝜆+𝛼+𝜇(𝑥))𝑝^𝑤,0,1+(𝑥)𝑞^∗𝑤,0,1(𝑥)𝑑𝑥_∞_𝑖=1^∞0−𝑑𝑝^𝑤,𝑖,1(𝑥)𝑑𝑥−(𝜆+𝛼+𝜇(𝑥))𝑝^𝑤,𝑖,1×𝑞_{∗𝑤,𝑖,1}(𝑥)+𝜆𝑝_{𝑤,𝑖−1,1}(𝑥)=(𝑥)𝑑𝑥_∞_𝑖=0−(𝜆+𝑖𝜃)𝑝_𝐼,𝑖,0𝑞^{∗𝐼,𝑖,0}+^∞_𝑖=0^∞0+𝜇(𝑥)𝑝_𝑤,𝑖,1(𝑥)𝑞^{∗𝐼,𝑖,0}𝑑𝑥_∞_𝑖=0^∞0−𝑑𝑝^𝑤,𝑖,1(𝑥)𝑑𝑥𝑞^{∗𝑤,𝑖,1}−(𝑥)𝑑𝑥_∞_𝑖=0^∞0+(𝜆+𝛼+𝜇(𝑥))𝑝_𝑤,𝑖,1(𝑥)𝑞^{∗𝑤,𝑖,1}(𝑥)𝑑𝑥_∞_𝑖=1^∞0=𝜆𝑝_{𝑤,𝑖−1,1}(𝑥)𝑞^{∗𝑤,𝑖,1}(𝑥)𝑑𝑥_∞_𝑖=0−(𝜆+𝑖𝜃)𝑝_𝐼,𝑖,0𝑞^{∗𝐼,𝑖,0}+^∞_𝑖=0^∞0+𝑝_𝑤,𝑖,1(𝑥)𝜇(𝑥)𝑞^{∗𝐼,𝑖,0}𝑑𝑥_∞_𝑖=0⎡⎢⎢⎢⎣−𝑝_𝑤,𝑖,1(𝑥)𝑞^{∗𝑤,𝑖,1}(𝑥)||||||𝑥𝑥==∞0+^∞0𝑝_𝑤,𝑖,1(𝑥)𝑑𝑞^{∗𝑤,𝑖,1}(𝑥)+𝑑𝑥𝑑𝑥_∞⎤⎥⎥⎥⎦_𝑖=0^∞0+𝑝_𝑤,𝑖,1(𝑥)−(𝜆+𝛼+𝜇(𝑥))𝑞^{∗𝑤,𝑖,1}(𝑥)𝑑𝑥_∞_𝑖=0^∞0=𝑝_𝑤,𝑖,1(𝑥)𝜆𝑞^{∗𝑤,𝑖+1,1}(𝑥)𝑑𝑥_∞_𝑖=0−(𝜆+𝑖𝜃)𝑝_𝐼,𝑖,0𝑞^{∗𝐼,𝑖,0}+^∞_𝑖=0^∞0+𝑝_𝑤,𝑖,1(𝑥)𝜇(𝑥)𝑞^{∗𝐼,𝑖,0}𝑑𝑥_∞_𝑖=0+𝑝_𝑤,𝑖,1(0)𝑞^{∗𝑤,𝑖,1}(0)_∞_𝑖=0^∞0𝑝_𝑤,𝑖,1(𝑥)𝑑𝑞^{∗𝑤,𝑖,1}(𝑥)𝑑𝑥−(𝜆+𝛼+𝜇(𝑥))𝑞^{∗𝑤,𝑖,1}+(𝑥)𝑑𝑥_∞_𝑖=0^∞0=𝑝_𝑤,𝑖,1(𝑥)𝜆𝑞^{∗𝑤,𝑖+1,1}(𝑥)𝑑𝑥_∞_𝑖=0−(𝜆+𝑖𝜃)𝑝_𝐼,𝑖,0𝑞^{∗𝐼,𝑖,0}+^∞_𝑖=0^∞0+𝑝_𝑤,𝑖,1(𝑥)𝜇(𝑥)𝑞^{∗𝐼,𝑖,0}𝑑𝑥_∞_𝑖=0𝜆𝑝_𝐼,𝑖,0++(𝑖+1)𝜃𝑝_{𝐼,𝑖+1,0}𝑞^{∗𝑤,𝑖,1}(0)_∞_𝑖=0^∞0𝑝_𝑤,𝑖,1(𝑥)𝑑𝑞^{∗𝑤,𝑖,1}(𝑥)𝑑𝑥−(𝜆+𝛼+𝜇(𝑥))𝑞^{∗𝑤,𝑖,1}+(𝑥)𝑑𝑥_∞_𝑖=0^∞0=𝑝_𝑤,𝑖,1(𝑥)𝜆𝑞^{∗𝑤,𝑖+1,1}(𝑥)𝑑𝑥_∞_𝑖=0−(𝜆+𝑖𝜃)𝑝_𝐼,𝑖,0𝑞^{∗𝐼,𝑖,0}+^∞_𝑖=0^∞0𝑝_𝑤,𝑖,1(𝑥)𝑑𝑞^{∗𝑤,𝑖,1}(𝑥)𝑑𝑥−(𝜆+𝛼+𝜇(𝑥))𝑞^{∗𝑤,𝑖,1}+𝜆𝑞_{∗𝑤,𝑖+1,1}(𝑥)(𝑥)+𝜇(𝑥)𝑞^{∗𝐼,𝑖,0}+𝑑𝑥_∞_𝑖=0𝑝_𝐼,𝑖,0𝜆𝑞^{∗𝑤,𝑖,1}(0)+^∞_𝑖=0=𝑝_𝐼𝑝_{𝐼,𝑖+1,0}(𝑖+1)𝜃𝑞^{∗𝑤,𝑖,1}(0),𝑝_𝑤,(ℒ+𝒢+ℋ)𝑞^∗𝐼,𝑞^∗𝑤.(2.4) From the definition of the adjoint operator and (2.4) we know that the assertion of this lemma is true.

Lemma 2.2. Assume that there exist two positive constant 𝜇,𝜇 such that 0<𝜇=inf_{[𝑥∈0,∞)}𝜇(𝑥)≤𝜇=sup_{[𝑥∈0,∞)}𝜇(𝑥)<∞.(2.5) If 𝜇<𝛼+𝜇, then ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝛾∈ℂ||||||||||||||sup⎧⎪⎨⎪⎩𝜆||𝛾+𝜆||,sup_𝑖≥1||𝛾+𝜆𝜆++𝑖𝜃𝑖𝜃||,𝜆𝜇||𝛾+𝜆||Re𝛾+𝜆+𝛼+𝜇+𝜆Re𝛾+𝜆+𝛼+𝜇sup_𝑖≥1,⎧⎪⎨⎪⎩(𝜆+𝑖𝜃)𝜇Re𝛾+𝜆+𝛼+𝜇||𝛾+𝜆+𝑖𝜃||+𝜆Re𝛾+𝜆+𝛼+𝜇Re𝛾+𝜆+𝛼+𝜇⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭<1,>0⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(2.6) belongs to the resolvent set of (𝐴+𝑈+𝐸)^∗. In particular, all points on the imaginary axis except zero belong to the resolvent set of (𝐴+𝑈+𝐸)^∗, which implies that all points on the imaginary axis except zero belong to the resolvent set of 𝐴+𝑈+𝐸.

Proof. For any given (𝑦^∗𝐼,𝑦^∗𝑤)∈𝑋^∗, consider the equation (𝛾𝐼−ℒ−𝒢)(𝑞^∗𝐼,𝑞^∗𝑤)=ℋ(𝑦^∗𝐼,𝑦^∗𝑤), that is, (𝛾+𝜆+(𝜃𝛾)+𝑞𝜆)_∗𝐼,1,0𝑞^∗𝐼,0,0=𝜆𝑦^∗𝑤,0,1(0),(2.7)(𝛾+𝜆+𝑖𝜃)𝑞_{∗𝐼,𝑖,0}=𝜃𝑦^∗𝑤,0,1(0)+𝜆𝑦^∗𝑤,1,1(0),(2.8)𝑑𝑞_{∗𝑤,𝑖,1}=𝑖𝜃𝑦^{∗𝑤,𝑖−1,1}(0)+𝜆𝑦^{∗𝑤,𝑖,1}(0),𝑖≥1,(2.9)(𝑥)𝑑𝑥=(𝛾+𝜆+𝛼+𝜇(𝑥))𝑞^{∗𝑤,𝑖,1}𝑞(𝑥)−𝜇(𝑥)𝑞^{∗𝐼,𝑖,0}−𝜆𝑦^{∗𝑤,𝑖+1,1}(𝑥),𝑖≥0,(2.10)_{∗𝑤,𝑖,1}(∞)=𝜂,𝑖≥0.(2.11) By solving (2.7)–(2.10), we have 𝑞^∗𝐼,0,0=𝜆𝛾+𝜆𝑦^∗𝑤,0,1𝑞(0),(2.12)_{∗𝐼,𝑖,0}=𝑖𝜃𝛾+𝜆+𝑖𝜃𝑦^{∗𝑤,𝑖−1,1}(0)+𝜆𝛾+𝜆+𝑖𝜃𝑦^{∗𝑤,𝑖,1}𝑞(0),𝑖≥1,(2.13)_{∗𝑤,𝑖,1}(𝑥)=𝑎_𝑖𝑒^{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}+𝑒^{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}^𝑥0−𝜇(𝜏)𝑞^{∗𝐼,𝑖,0}=𝑎_𝑖−𝜆𝑦^{∗𝑤,𝑖+1,1}(𝜏)𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏𝑒^{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}−𝑒^{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}^𝑥0𝜇(𝜏)𝑞^{∗𝐼,𝑖,0}+𝜆𝑦^{∗𝑤,𝑖+1,1}(𝜏)𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏,𝑖≥0.(2.14) Through multiplying 𝑒^{−∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏} to two sides of (2.14), we obtain 𝑒^{−∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}𝑞^{∗𝑤,𝑖,1}(𝑥)=𝑎_𝑖−^𝑥0𝜇(𝜏)𝑞^{∗𝐼,𝑖,0}+𝜆𝑦^{∗𝑤,𝑖+1,1}(𝜏)𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏,𝑖≥0.(2.15) By combining (2.11) with (2.15), we deduce 𝑎_𝑖=^∞0𝜇(𝜏)𝑞^{∗𝐼,𝑖,0}+𝜆𝑦^{∗𝑤,𝑖+1,1}(𝜏)𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏,𝑖≥0.(2.16) By inserting (2.16) into (2.14), we get 𝑞^{∗𝑤,𝑖,1}(𝑥)=𝑒^{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}^∞𝑥𝜇(𝜏)𝑞^{∗𝐼,𝑖,0}=𝑒_{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}+𝜆𝑦^{∗𝑤,𝑖+1,1}(𝜏)𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏^∞𝑥+𝜆𝑒_{∫𝑥0(𝛾+𝜆+𝛼+𝜇(𝜏))𝑑𝜏}𝜇(𝜏)𝑞^{∗𝐼,𝑖,0}𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏^∞𝑥⟹‖‖𝑞𝑦^{∗𝑤,𝑖+1,1}(𝜏)𝑒^{−∫𝜏0(𝛾+𝜆+𝛼+𝜇(𝑠))𝑑𝑠}𝑑𝜏(2.17)_{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)}≤^∞0𝑒^{∫𝑥0(Re𝛾+𝜆+𝛼)𝑑𝜏}^∞𝑥+𝜆_∞0𝜇(𝜏)||𝑞^{∗𝐼,𝑖,0}||𝑒^{−∫𝜏0(Re𝛾+𝜆+𝛼)𝑑𝑠}𝑒^{−∫𝜏𝑥𝜇(𝑠)𝑑𝑠}𝑑𝜏𝑑𝑥𝑒^{∫𝑥0(Re𝛾+𝜆+𝛼)𝑑𝜏}^∞𝑥≤𝜇(𝜏)||𝑦^{∗𝑤,𝑖+1,1}(𝜏)||𝑒^{−∫𝜏0(Re𝛾+𝜆+𝛼)𝑑𝑠}𝑒^{−∫𝜏𝑥𝜇(𝑠)𝑑𝑠}𝑑𝜏𝑑𝑥𝜇Re𝛾+𝜆+𝛼+𝜇||𝑞^{∗𝐼,𝑖,0}≤+𝜆||Re𝛾+𝜆+𝛼+𝜇‖‖𝑦^{∗𝑤,𝑖+1,1}‖‖_{𝐿∞[0,∞)},𝑖≥0.(2.18) Equation (2.12) gives ||𝑞^∗𝐼,0,0||≤𝜆||𝛾+𝜆||||𝑦^∗𝑤,0,1(0)||≤𝜆||𝛾+𝜆||‖‖𝑦^∗𝑤,0,1‖‖_{𝐿∞[0,∞)}.(2.19) From (2.13) we estimate ||𝑞^{∗𝐼,𝑖,0}||≤𝑖𝜃||𝛾+𝜆+𝑖𝜃||||𝑦^{∗𝑤,𝑖−1,1}(0)||+𝜆||𝛾+𝜆+𝑖𝜃||||𝑦^{∗𝑤,𝑖,1}≤𝑖𝜃(0)||||𝛾+𝜆+𝑖𝜃||‖‖𝑦^{∗𝑤,𝑖−1,1}‖‖_{𝐿∞[0,∞)}+𝜆||𝛾+𝜆+𝑖𝜃||‖‖𝑦^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)}≤𝜆+𝑖𝜃||𝛾+𝜆+𝑖𝜃||sup_𝑖≥0‖‖𝑦^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)},𝑖≥1.(2.20) By inserting (2.19) and (2.20) into (2.18), we have ‖‖𝑞^∗𝑤,0,1‖‖_{𝐿∞[0,∞)}≤𝜇Re𝛾+𝜆+𝛼+𝜇𝜆||𝛾+𝜆||‖‖𝑦^∗𝑤,0,1‖‖_{𝐿∞[0,∞)}+𝜆Re𝛾+𝜆+𝛼+𝜇‖‖𝑦^∗𝑤,1,1‖‖_{𝐿∞[0,∞)}≤⎧⎪⎨⎪⎩𝜆𝜇||𝛾+𝜆||Re𝛾+𝜆+𝛼+𝜇+𝜆Re𝛾+𝜆+𝛼+𝜇×sup_𝑖≥0⎫⎪⎬⎪⎭‖‖𝑞_{∗𝑤,𝑖,1}‖‖𝑦^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)},‖‖_{𝐿∞[0,∞)}≤𝜇Re𝛾+𝜆+𝛼+𝜇𝜆+𝑖𝜃||𝛾+𝜆+𝑖𝜃||sup_𝑖≥0‖‖𝑦^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)}+𝜆Re𝛾+𝜆+𝛼+𝜇sup_𝑖≥0‖‖𝑦^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)}=⎧⎪⎨⎪⎩(𝜆+𝑖𝜃)𝜇Re𝛾+𝜆+𝛼+𝜇||𝛾+𝜆+𝑖𝜃||+𝜆Re𝛾+𝜆+𝛼+𝜇×sup_𝑖≥0⎫⎪⎬⎪⎭‖‖𝑦^{∗𝑤,𝑖,1}‖‖_{𝐿∞[0,∞)},𝑖≥0.(2.21) By combining (2.19) and (2.20) with (2.21), we derive ||‖‖𝑞^∗𝐼,𝑞^∗𝑤‖‖||≤sup⎧⎪⎨⎪⎩𝜆||𝛾+𝜆||,sup_𝑖≥1𝜆||𝛾+𝜆𝜆++𝑖𝜃𝑖𝜃||,𝜇||𝛾+𝜆||Re𝛾+𝜆+𝛼+𝜇+𝜆Re𝛾+𝜆+𝛼+𝜇sup_𝑖≥1,⎧⎪⎨⎪⎩(𝜆+𝑖𝜃)𝜇Re𝛾+𝜆+𝛼+𝜇||𝛾+𝜆+𝑖𝜃||+𝜆Re𝛾+𝜆+𝛼+𝜇⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭.(2.22) This shows that [𝐼−(𝛾𝐼−ℒ−𝒢)⁻¹ℋ]⁻¹ exists and is bounded when 𝛾 belongs to the set ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝛾∈ℂ||||||||||||||sup⎧⎪⎨⎪⎩𝜆||𝛾+𝜆||,sup_𝑖≥1||𝛾+𝜆𝜆++𝑖𝜃𝑖𝜃||,𝜆𝜇||𝛾+𝜆||Re𝛾+𝜆+𝛼+𝜇+𝜆Re𝛾+𝜆+𝛼+𝜇sup_𝑖≥1,⎧⎪⎨⎪⎩(𝜆+𝑖𝜃)𝜇Re𝛾+𝜆+𝛼+𝜇||𝛾+𝜆+𝑖𝜃||+𝜆Re𝛾+𝜆+𝛼+𝜇Re𝛾+𝜆+𝛼+𝜇⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭<1,>0⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭.(2.23) Through discussing the solution of the equation (𝛾𝐼−ℒ−𝒢)(𝑞^∗𝐼,𝑞^∗𝑤)=(𝑦^∗𝐼,𝑦^∗𝑤) for any given (𝑦^∗𝐼,𝑦^∗𝑤)∈𝑋^∗, it is not difficult to verify (𝛾𝐼−ℒ−𝒢)⁻¹ exists and is bounded when 𝛾 satisfies (2.23). Therefore, by the resolvent equation [𝛾𝐼−(ℒ+𝒢+ℋ)]⁻¹=(𝛾𝐼−ℒ−𝒢)𝐼−(𝛾𝐼−ℒ−𝒢)⁻¹ℋ⁻¹=𝐼−(𝛾𝐼−ℒ−𝒢)⁻¹ℋ⁻¹(𝛾𝐼−ℒ−𝒢)⁻¹,(2.24) we know that (𝛾𝐼−ℒ−𝒢−ℋ)⁻¹ exists and is bounded when 𝛾 belongs to the set (2.23), which means that (2.23) belongs to 𝜌(ℒ+𝒢+ℋ).
In particular, if 𝛾=𝑖𝑎,𝑎∈ℝ⧵{0},𝑖²=−1, then all 𝛾’s belong to (2.23). In fact, by using the condition on this lemma, we have 𝜆√𝑎²+𝜆²sup_𝑘≥1<1,𝜆+𝑘𝜃√𝑎²+(𝜆+𝑘𝜃)²<1,𝜇<𝛼+𝜇⟹𝜆𝜇<𝛼+𝜇√𝑎²+𝜆²⟹𝜆𝜇+𝜆√𝑎²+𝜆²<𝛼+𝜇√𝑎²+𝜆²+𝜆√𝑎²+𝜆²⟹𝜆𝜇+𝜆√𝑎²+𝜆²<𝜆+𝛼+𝜇√𝑎²+𝜆²⟹𝜆𝜇√𝑎²+𝜆²𝜆+𝛼+𝜇+𝜆𝜆+𝛼+𝜇<1,𝜇<𝛼+𝜇⟹(𝜆+𝑘𝜃)𝜇<𝛼+𝜇𝑎²+(𝜆+𝑘𝜃)²⟹(𝜆+𝑘𝜃)𝜇+𝜆𝑎²+(𝜆+𝑘𝜃)²<𝛼+𝜇𝑎²+(𝜆+𝑘𝜃)²+𝜆𝑎²+(𝜆+𝑘𝜃)²⟹(𝜆+s𝑘u𝜃p_𝑘≥1)𝜇+𝜆𝑎²+(𝜆+𝑘𝜃)²<𝜆+𝛼+𝜇𝑎²+(𝜆+𝑘𝜃)²,⎧⎪⎨⎪⎩(𝜆+𝑘𝜃)𝜇√𝑎²+(𝜆+𝑘𝜃)²𝜆+𝛼+𝜇+𝜆+𝛼𝜆+𝜇⎫⎪⎬⎪⎭<1.(2.25) The above inequalities show that all points on the imaginary axis except zero belong to the resolvent set of (𝐴+𝑈+𝐸)^∗. From the relation between the spectrum of 𝐴+𝑈+𝐸 and spectrum of (𝐴+𝑈+𝐸)^∗ we know that all points on the imaginary axis except zero belong to the resolvent set of 𝐴+𝑈+𝐸.

Lemma 2.3. If 0<𝜇=inf_{[𝑥∈0,∞)}𝜇(𝑥)≤𝜇=sup_{[𝑥∈0,∞)}𝜇(𝑥)<∞,(2.26) then 0 is not an eigenvalue of 𝐴+𝑈+𝐸.

Proof. We consider the equation (𝐴+𝑈+𝐸)(𝑝_𝐼,𝑝_𝑤)=0, which is equivalent to (𝜆+𝑖𝜃)𝜆𝑝𝑝_{𝐼,0,0𝐼,𝑖,0}=^∞0𝜇(𝑥)𝑝_𝑤,0,1(𝑥)𝑑𝑥,(2.27)𝑑𝑝_𝑤,0,1=^∞0𝜇(𝑥)𝑝_𝑤,𝑖,1(𝑥)𝑑𝑥,𝑖≥1,(2.28)(𝑥)𝑑𝑥=−(𝜆+𝛼+𝜇(𝑥))𝑝^𝑤,0,1𝑑𝑝_𝑤,𝑖,1(𝑥),(2.29)(𝑥)𝑑𝑥=−(𝜆+𝛼+𝜇(𝑥))𝑝^𝑤,𝑖,1𝑝(𝑥)+𝜆𝑝_{𝑤,𝑖−1,1}(𝑥),𝑖≥1,(2.30)_𝑤,𝑖,1(0)=𝜆𝑝_𝐼,𝑖,0+(𝑖+1)𝜃𝑝_{𝐼,𝑖+1,0},𝑖≥0.(2.31) It is hard to determine the concrete expressions of all 𝑝_𝐼,i,0,𝑝_𝑤,𝑖,1(𝑥) and to prove (𝑝_𝐼,𝑝_𝑤)∈𝐷(𝐴). In the following we use another method. We introduce the probability-generating functions 𝑄_𝐼(𝑧)=∑^∞𝑖=0𝑝_𝐼,𝑖,0𝑧^𝑖,    𝑄_𝑤(𝑥,𝑧)=∑^∞𝑖=0𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖 for |𝑧|<1. Theorem 1.1 ensures that 𝑄_𝐼(𝑧),𝑄_𝑤(𝑥,𝑧) are well-defined. By applying the basic knowledge of power series, the Fubini theorem and (2.27)-(2.28), we have 𝜆𝑝_𝐼,0,0+^∞_𝑖=1(𝜆+𝑖𝜃)𝑝_𝐼,𝑖,0𝑧^𝑖=^∞0𝜇(𝑥)𝑝_𝑤,0,1(𝑥)𝑑𝑥+^∞_𝑖=1^∞0⟹𝜆𝑄_𝐼𝜇(𝑥)𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖𝑑𝑥(𝑧)+𝜃𝑧^∞_𝑖=0𝑝_𝐼,𝑖,0𝑧^𝑖^=^∞0𝜇(𝑥)^∞_𝑖=0⟹𝜆𝑄_𝐼𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖𝑑𝑥(𝑧)+𝜃𝑧𝑑𝑑𝑧𝑄^𝐼(𝑧)=^∞0𝜇(𝑥)𝑄_𝑤(𝑥,𝑧)𝑑𝑥.(2.32) By (2.29) and (2.30), we deduce 𝑑𝑝_𝑤,0,1(𝑥)𝑑𝑥+^∞_𝑖=1𝑑𝑝_𝑤,𝑖,1(𝑥)𝑑𝑥𝑧^𝑖=−−((𝜆𝜆++𝛼𝛼++𝜇𝜇((𝑥𝑥))))𝑝_𝑤,0,1∞(𝑥)_𝑖=1𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖+^∞_𝑖=1𝜆𝑝_{𝑤,𝑖−1,1}(𝑥)𝑧^𝑖𝜕⟹∑_∞𝑖=0𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖𝜕𝑥=−(𝜆+𝛼+𝜇(𝑥))^∞_𝑖=0𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖+𝜆^∞_𝑖=1𝑝_{𝑤,𝑖−1,1}(𝑥)𝑧^𝑖𝜕⟹𝑄_𝑤(𝑥,𝑧)𝜕𝑥=−(𝜆+𝛼+𝜇(𝑥))𝑄^𝑤(𝑥,𝑧)+𝜆𝑧^∞_𝑖=0𝑝_𝑤,𝑖,1(𝑥)𝑧^𝑖𝜕⟹𝑄_𝑤(𝑥,𝑧)𝜕𝑥=(𝜆𝑧−𝜆−𝛼−𝜇(𝑥))𝑄^𝑤𝑄⟹(𝑥,𝑧)_𝑤(𝑥,𝑧)=𝑄_𝑤(0,𝑧)𝑒^{∫𝑥0(𝜆𝑧−𝜆−𝛼−𝜇(𝜏))𝑑𝜏}.(2.33) Equation (2.31) gives 𝑄_𝑤(0,𝑧)=^∞_𝑖=0𝑝_𝑤,𝑖,1(0)𝑧^𝑖=𝜆^∞_𝑖=0𝑝_𝐼,𝑖,0𝑧^𝑖+^∞_𝑖=0(𝑖+1)𝜃𝑝_{𝐼,𝑖+1,0}𝑧^𝑖=𝜆𝑄_𝐼(𝑧)+𝜃𝑑𝑑𝑧𝑄^𝐼(𝑧).(2.34) By combining (2.34) with (2.33) and using (2.32), we calculate 𝜆𝑄_𝐼(𝑧)+𝜃𝑧𝑑𝑑𝑧𝑄^𝐼(𝑧)==^∞0𝜇(𝑥)𝑄_𝑤(𝑥,𝑧)𝑑𝑥_∞0=𝜆𝑄𝜇_𝐼(𝑥)𝑄_𝑤(0,𝑧)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥(𝑧)+𝜃𝑑𝑑𝑧𝑄^𝐼𝑑⟹𝑄_𝐼(𝑧)^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥(𝑧)𝑄_𝐼(𝑧)=𝜆𝜃∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥−1𝑄⟹𝑧−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑑𝑧_𝐼(𝑧)=𝑄_𝐼(0)exp𝜆𝜃^𝑧0∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥−1=𝑝_𝐼,0,0𝑠−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑑𝑠exp𝜆𝜃^𝑧0∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥−1𝑠−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑑𝑠.(2.35) This together with (2.34) gives 𝑄_𝑤(0,𝑧)=𝜆(𝑧−1)𝑧−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑄^𝐼(𝑧).(2.36) From (2.35), we derive ^∞_𝑖=0𝑝_𝐼,𝑖,0==lliimm_{𝑧→1𝑧→1}𝑄_𝐼(𝑧)𝑝_𝐼,0,0exp𝜆𝜃^𝑧0∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥−1=𝑝_𝐼,0,0𝑠−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑑𝑠exp𝜆𝜃¹⁰∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥−1𝑠−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑑𝑠.(2.37) In the following, by the Rouche theorem we know that 𝑠−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑠−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥 has a unique zero point inside the unit circle |𝑧|=1. And by the residue theorem we determine the above integral.
Let 𝑓(𝑧)=𝑧 and 𝑔(𝑧)=∫^∞0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥. It is well known that 𝑓(𝑧) and 𝑔(𝑧) are differentiable inside and continuous on the contour |𝑧|=1. And |𝑓(𝑧)|=1 on |𝑧|=1. Since, for (𝜆+𝛼−𝜆Re𝑧)>0, ||𝑔(𝑧)||≤=||||−^∞0_∞0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥||||𝑒^{−(𝜆+𝛼−𝜆Rez)𝑥}𝑑𝑒^{−∫𝑥0𝜇(𝜏)𝑑𝜏}=−⎡⎢⎢⎢⎣𝑒^{−(𝜆+𝛼−𝜆Re𝑧)𝑥}𝑒^{−∫𝑥0𝜇(𝜏)𝑑𝜏}+(𝜆+𝛼−𝜆Rez)_∞0||||||𝑥𝑥==∞0=1−(𝜆+𝛼−𝜆Re𝑧)_∞0𝑒^{∫−(𝜆+𝛼−𝜆Re𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥⎤⎥⎥⎥⎦<1=||𝑓(𝑧)||,𝑒^{∫−(𝜆+𝛼−𝜆Re𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥(2.38) we have |𝑓(𝑧)|<|𝑔(𝑧)| on |𝑧|=1. Consequently, all conditions of Rouche’s theorem are satisfied. Obviously 𝑓(𝑧)−𝑔(𝑧) has only one zero inside the unit circle |𝑧|=1, since 𝑓(𝑧) has one. If we denote this zero by 𝑧₀, then it is a simple pole of ∫^∞0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥−1𝑧−∫^∞0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥.(2.39) By the residue theorem, we calculate ¹⁰∫^∞0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑠)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥−1=𝑠−lim∫^∞0_𝑧→𝑧0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑠)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥𝑑𝑠∫^∞0𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥−1=∫1−∫^∞0𝜆𝑥𝜇(𝑥)𝑒^{∫−(𝜆+𝛼−𝜆𝑧)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥_∞0𝜇(𝑥)𝑒^{−(𝜆+𝛼−𝜆𝑧0∫)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥−11−∫^∞0𝜆𝑥𝜇(𝑥)𝑒^{−(𝜆+𝛼−𝜆𝑧0∫)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥.(2.40) Thus, we obtain ^∞_𝑖=0𝑝_𝐼,𝑖,0=𝑝_𝐼,0,0exp𝜆𝜃∫^∞0𝜇(𝑥)𝑒^{−(𝜆+𝛼−𝜆𝑧0∫)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥−11−∫^∞0𝜆𝑥𝜇(𝑥)𝑒^{−(𝜆+𝛼−𝜆𝑧0∫)𝑥−𝑥0𝜇(𝜏)𝑑𝜏}𝑑𝑥<∞.(2.41) From (2.36), we derive lim_𝑧→1𝑄_𝑤(0,𝑧)=lim_𝑧→1𝜆(𝑧−1)𝑧−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑄^𝐼=lim_𝑧→1(𝑧)𝜆(𝑧−1)𝑧−∫^∞0𝜇(𝑥)𝑒^{∫𝑥0[𝜆𝑧−𝜆−𝛼−𝜇(𝜏)]𝑑𝜏}𝑑𝑥lim^𝑧→1=𝜆(1−1)𝑄_𝐼(𝑧)1−∫^∞0𝜇(𝑥)𝑒^{−∫𝑥0[𝛼+𝜇(𝜏)]𝑑𝜏}𝑑𝑥𝑄^𝐼=0.(1)(2.42) By combining (2.33) with (2.42), we estimate ^∞_𝑖=0𝑝_𝑤,𝑖,1(𝑥)=lim_𝑧→1𝑄_𝑤(𝑥,𝑧)=lim_𝑧→1𝑄_𝑤(0,𝑧)𝑒^{∫𝑥0(𝜆𝑧−𝜆−𝛼−𝜇(𝜏))𝑑𝜏}=lim_𝑧→1𝑄_𝑤(0,𝑧)𝑒^{−∫𝑥0(𝛼+𝜇(𝜏))𝑑𝜏}=0.(2.43) Since Theorem 1.1 shows that each 𝑝_𝑤,𝑖,1(𝑥) is nonnegative, (2.43) implies 𝑝_𝑤,𝑖,1(𝑥)=0 for all 𝑖≥0. This together with (2.27) and (2.28) gives 𝑝_𝐼,𝑖,0=0 for all 𝑖≥0. In other words, (𝐴+𝑈+𝐸)(𝑝_𝐼,𝑝_𝑤)=0 has only zero solution; that is, 0 is not an eigenvalue of 𝐴+𝑈+𝐸.

Lemma 2.3 shows that the system (1.7) does not have nonzero steady-state solution.

If we can prove that 0 is not in the residue spectrum of 𝐴+𝑈+𝐸, then by Theorem 1.1, Lemmas 2.2 and 2.3, and ABLV Theorem (see [16] or [17]), we deduce that the time-dependent solution of system (1.7) is strongly asymptotically stable. This result is quite different from other queueing models, see Gupur [13–15] and Kasim and Gupur [7].

Acknowledgments

The authors’ research work is supported by Science Foundation of Xinjiang University (no: XY110101) and the Natural Science Foundation of Xinjiang (no: 2012211A023).