Asymptotic behavior of the time-dependent solution of an M/G/1 queueing model

  • Geni Gupur1Email author and

    Affiliated with

    • Rena Ehmet2

      Affiliated with

      Boundary Value Problems20132013:17

      DOI: 10.1186/1687-2770-2013-17

      Received: 6 October 2012

      Accepted: 22 January 2013

      Published: 11 February 2013

      Abstract

      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.

      Keywords

      M/G/1 queueing model with exceptional service time for the first customer in each busy period C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq1_HTML.gif-semigroup eigenvalue resolvent set

      1 Introduction

      According to Takagi [1], the M/G/1 queueing system with exceptional service time for the first customer in each busy period can be described by the following partial differential equations with integral boundary conditions:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ1_HTML.gif
      (1.1)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ2_HTML.gif
      (1.2)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ3_HTML.gif
      (1.3)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ4_HTML.gif
      (1.4)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ5_HTML.gif
      (1.5)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ6_HTML.gif
      (1.6)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ7_HTML.gif
      (1.7)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ8_HTML.gif
      (1.8)

      where ( x , t ) [ 0 , ) × [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq2_HTML.gif; p 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq3_HTML.gif represents the probability that there is no customer in the system and the server is idle at time t; p n ( x , t ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq4_HTML.gif ( n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq5_HTML.gif) represents the probability that at time t there are n customers in the system and the server is busy with remaining service time lying between in [ x , x + d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq6_HTML.gif; Q n ( x , t ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq7_HTML.gif ( n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq5_HTML.gif) represents the probability that at time t there are n customers in the system and the server is busy with the elapsed service time of the first service lying between x and x + d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq8_HTML.gif; λ represents the arrival rate of customers; b ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq9_HTML.gif is the service rate at x; b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq10_HTML.gif is the exceptional service rate at x.

      Many papers have been published about queueing systems with server vacations. But most works on vacation models have been limited to the analysis of steady-states. There are few treatments of transient behavior, see Welch [2], Minh [3], Takagi [1], Gupur [4, 5] for instance. In 1990, Takagi [1] first established the mathematical model of the M/G/1 queueing system with exceptional service time for the first customer in each busy period by using the supplementary variable technique, then studied the time-dependent solution of the model by using probability generating functions and got the Laplace transform of the probability generating function. Roughly speaking, he obtained the existence of a time-dependent solution of the model. In 2002, by using C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq1_HTML.gif-semigroup theory in functional analysis, Gupur [6] proved that the model has a unique positive time-dependent solution which satisfies the probability condition. In 2003, Gupur [4] considered the asymptotic behavior of the time-dependent solution of the model when b ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq9_HTML.gif and b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq10_HTML.gif are constants. Firstly, he determined the resolvent set of the adjoint operator of the operator corresponding to the model; next he proved that 0 is an eigenvalue of the operator and its adjoint operator with geometric multiplicity one. Thus, by using Theorem 14 in Gupur, Li and Zhu [7] obtained that the time-dependent solution of the model converges strongly to its steady-state solution. In 2009, Zhang and Gupur [8] found that the operator has one eigenvalue on the left complex half-plane. In 2011, Lin and Gupur [9] proved that the operator has infinitely many eigenvalues on the left complex half-plane which converges to zero and therefore showed that the convergence of the time-dependent solution of the model obtained in Gupur [4] is the best result on the convergence, that is to say, it is impossible that the time-dependent solution exponentially (uniformly) converges to its steady-state solution. In the case that b ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq9_HTML.gif and b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq10_HTML.gif are functions, any literature about asymptotic behavior of the above model has not been found. This paper is an effort on this subject.

      According to Theorem 14 in Gupur, Li and Zhu [7], to obtain the asymptotic behavior of the time-dependent solution of the above model, we need to know the spectrum of the underlying operator on the imaginary axis. By investigating the above model and comparing with Gupur [10], one may find that the main difficult points of the above equations (1.1)-(1.8) are that there are infinitely many equations and boundary conditions. When studying the population equation, Greiner [11] put forward an idea to perturb the boundary condition which states ‘one can introduce the maximal operator without the boundary condition and define a boundary operator, and by studying the spectrum of the boundary operator and the maximal operator can discuss the spectrum of the underlying operator which corresponds to the population equation.’ In 2007, Haji and Radl [12] successfully applied Greiner’s idea to the M / M B / 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq11_HTML.gif queueing model, in which both the service rate and arrival rate are constants, and studied the asymptotic behavior of its time-dependent solution. Gupur [5, 13] obtained the asymptotic behavior of the time-dependent solutions of two queueing models by using Greiner’s idea. In this paper, firstly, by using probability generating functions, we prove that 0 is an eigenvalue of the underlying operator; next, by using the idea in Gupur [5, 13], the result in Haji and Radl [12] and Corollary 2.3 in Nagel [14], we deduce the resolvent set of the underlying operator; thirdly, we show that 0 is an eigenvalue of the adjoint operator of the underlying operator, and therefore, by using Theorem 14 in Gupur, Li and Zhu [7], we obtain that the time-dependent solution of the above model converges strongly to its steady-state solution. Finally, by Lin and Gupur [9] we show that our result on convergence is optimal, that is to say, it is impossible that the time-dependent solution of the model converges exponentially to its steady-state solution. Although the idea and method in Gupur [4] are quite different, the main result is a special case of our result.

      In this paper, we use the notations in Gupur [5, 6, 13]. Take the state space as follows:
      X = { ( p , Q ) | p R × L 1 [ 0 , ) × L 1 [ 0 , ) × L 1 [ 0 , ) × , Q L 1 [ 0 , ) × L 1 [ 0 , ) × L 1 [ 0 , ) × , ( p , Q ) = | p 0 | + n = 1 p n L 1 [ 0 , ) + n = 1 Q n L 1 [ 0 , ) < } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equa_HTML.gif
      It is obvious that X is a Banach space. In addition, X is also a Banach lattice under the following order relation:
      ( p , Q ) ( y , z ) p 0 y 0 , p n ( x ) y n ( x ) , Q n ( x ) z n ( x ) , n 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equb_HTML.gif
      For convenience, we introduce
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equc_HTML.gif
      We define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equd_HTML.gif
      We choose a boundary space as
      X = l 1 × l 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Eque_HTML.gif
      and define the boundary operators
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equf_HTML.gif
      Now we introduce the underlying operator ( A , D ( A ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq12_HTML.gif by
      A p = A m p , D ( A ) = { p D ( A m ) L p = Φ p } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equg_HTML.gif
      Then the system of the above equations (1.1)-(1.8) can be written as an abstract Cauchy problem in the Banach space X, which is just the form given in Gupur [6]
      { d ( p , Q ) ( t ) d t = A ( p , Q ) ( t ) , t ( 0 , ) , ( p , Q ) ( 0 ) = ( ( 1 0 ) , ( 0 0 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ9_HTML.gif
      (1.9)

      Gupur [6] has proved the following result for the system (1.9).

      Theorem 1.1 The operator ( A , D ( A ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq13_HTML.gifgenerates a positive contraction C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq1_HTML.gif-semigroup T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq14_HTML.gifand the system (1.9) has a unique positive time-dependent solution ( p , Q ) ( x , t ) = T ( t ) ( p , Q ) ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq15_HTML.gifwhich satisfies
      ( p , Q ) ( , t ) = p 0 ( t ) + n = 1 0 p n ( x , t ) d x + n = 1 0 Q n ( x , t ) d x = 1 , t [ 0 , ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equh_HTML.gif

      2 Main results

      Lemma 2.1 If 0 λ x b ( x ) e 0 x b ( ξ ) d ξ d x < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq16_HTML.gif, then 0 is an eigenvalue of A with geometric multiplicity one.

      Proof We consider the equation A ( p , Q ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq17_HTML.gif, i.e.,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ10_HTML.gif
      (2.1)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ11_HTML.gif
      (2.2)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ12_HTML.gif
      (2.3)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ13_HTML.gif
      (2.4)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ14_HTML.gif
      (2.5)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ15_HTML.gif
      (2.6)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ16_HTML.gif
      (2.7)
      By solving (2.2)-(2.5), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ17_HTML.gif
      (2.8)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ18_HTML.gif
      (2.9)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ19_HTML.gif
      (2.10)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ20_HTML.gif
      (2.11)
      Through using (2.8)-(2.11) repeatedly, we deduce
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ21_HTML.gif
      (2.12)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ22_HTML.gif
      (2.13)
      By combining (2.10) and (2.11) with (2.7) and using (2.13), we deduce
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ23_HTML.gif
      (2.14)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ24_HTML.gif
      (2.15)
      It is difficult to determine directly all a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq18_HTML.gif and to verify k = 1 p k L 1 [ 0 , ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq19_HTML.gif. In the following, we use another method. We introduce the probability generating function P ( x , z ) = n = 1 p n ( x ) z n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq20_HTML.gif for all complex variables | z | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq21_HTML.gif. Theorem 1.1 ensures that P ( x , z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq22_HTML.gif is well defined. (2.2) and (2.3) give
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ25_HTML.gif
      (2.16)
      By applying (2.6), (2.16), (2.14), (2.1), 0 b ( x ) e 0 x b ( ξ ) d ξ d x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq23_HTML.gif, 0 b 0 ( x ) e 0 x b 0 ( ξ ) d ξ d x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq24_HTML.gif and the L’Hospital rule it follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ26_HTML.gif
      (2.17)
      (2.16) and (2.17) give
      n = 1 p n ( x ) = lim z 1 P ( x , z ) = 0 λ x b 0 ( x ) e 0 x b 0 ( ξ ) d ξ d x + 0 b 0 ( x ) e λ x 0 x b 0 ( ξ ) d ξ d x 1 0 λ x b ( x ) e 0 x b ( ξ ) d ξ d x λ | p 0 | × 0 e 0 x b ( ξ ) d ξ d x < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ27_HTML.gif
      (2.18)

      (2.18) and (2.15) show that 0 is an eigenvalue of A. Moreover, from (2.12), (2.14), (2.1) and (2.6), it is easy to see that the eigenvectors corresponding to zero span one dimensional linear space, that is, the geometric multiplicity of 0 is one. □

      According to Theorem 14 in Gupur, Li and Zhu [7], we know that in order to obtain the asymptotic behavior of the time-dependent solution of the system (1.9), we need the spectrum of A on the imaginary axis. Through investigating the system (1.9), we find that the infinite number of equations and the boundary conditions are the difficult points. Greiner [11] put forward an idea to study the spectrum of A by perturbing boundary conditions. And by using the Greiner idea, Haji and Radl [12] gave a result which was 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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq25_HTML.gif as
      A 0 p = A m p , D ( A 0 ) = { p D ( A m ) L p = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equi_HTML.gif
      and discuss the inverse of A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq26_HTML.gif. For any given ( y , z ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq27_HTML.gif, consider the equation ( γ I A 0 ) ( p , Q ) = ( y , z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq28_HTML.gif, that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ28_HTML.gif
      (2.19)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ29_HTML.gif
      (2.20)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ30_HTML.gif
      (2.21)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ31_HTML.gif
      (2.22)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ32_HTML.gif
      (2.23)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ33_HTML.gif
      (2.24)
      By (2.19)-(2.24) it is easy to calculate
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ34_HTML.gif
      (2.25)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ35_HTML.gif
      (2.26)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ36_HTML.gif
      (2.27)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ37_HTML.gif
      (2.28)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ38_HTML.gif
      (2.29)
      If we set
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equj_HTML.gif
      then the above equations (2.25)-(2.29) give, if the resolvent of A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq26_HTML.gif exists,
      ( γ I A 0 ) 1 ( y , z ) = ( ( 1 γ + λ 1 γ + λ ψ E 0 0 0 E 0 0 0 λ E 2 E 0 0 λ 2 E 3 λ E 2 E ) ( y 0 y 1 ( x ) y 2 ( x ) y 3 ( x ) ) + ( 1 γ + λ ϕ E 0 0 0 0 0 0 0 0 0 ) ( z 1 ( x ) z 2 ( x ) z 3 ( x ) ) , ( E 0 0 0 λ E 0 2 E 0 0 λ 2 E 0 3 λ E 0 2 E 0 ) ( z 1 ( x ) z 2 ( x ) z 3 ( x ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equk_HTML.gif

      From which together with the definition of the resolvent set we have the following result.

      Lemma 2.2 Let b ( x ) , b 0 ( x ) : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq29_HTML.gifbe measurable, 0 < inf x [ 0 , ) b ( x ) sup x [ 0 , ) b ( x ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq30_HTML.gifand 0 < inf x [ 0 , ) b 0 ( x ) sup x [ 0 , ) b 0 ( x ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq31_HTML.gif. Then
      { γ C | Re γ + λ > 0 , Re γ + inf x [ 0 , ) b ( x ) > 0 , Re γ + inf x [ 0 , ) b 0 ( x ) > 0 } ρ ( A 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equl_HTML.gif
      Proof For any f L 1 [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq32_HTML.gif, by using integration by parts, we estimate
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ39_HTML.gif
      (2.30)
      Similarly,
      E 0 1 Re γ + λ + inf x [ 0 , ) b 0 ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ40_HTML.gif
      (2.31)
      From (2.30), (2.31), ϕ sup x [ 0 , ) b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq33_HTML.gif and ψ sup x [ 0 , ) b ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq34_HTML.gif we deduce, for ( y , z ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq27_HTML.gif,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equm_HTML.gif

      which means that the result of this lemma is right. □

      Lemma 2.3 For γ ρ ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq35_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ41_HTML.gif
      (2.32)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ42_HTML.gif
      (2.33)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ43_HTML.gif
      (2.34)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ44_HTML.gif
      (2.35)
      Proof If ( p , Q ) ker ( γ I A m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq36_HTML.gif, then ( γ I A m ) ( p , Q ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq37_HTML.gif, which is equivalent to
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ45_HTML.gif
      (2.36)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ46_HTML.gif
      (2.37)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ47_HTML.gif
      (2.38)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ48_HTML.gif
      (2.39)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ49_HTML.gif
      (2.40)
      By solving (2.37)-(2.40) we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ50_HTML.gif
      (2.41)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ51_HTML.gif
      (2.42)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ52_HTML.gif
      (2.43)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ53_HTML.gif
      (2.44)
      Through inserting (2.41) and (2.43) into (2.36), it follows that
      p 0 = a 1 γ + λ 0 b ( x ) e ( γ + λ ) x 0 x b ( ξ ) d ξ d x + b 1 γ + λ 0 b 0 ( x ) e ( γ + λ ) x 0 x b 0 ( ξ ) d ξ d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ54_HTML.gif
      (2.45)
      By using (2.41), (2.42), (2.43) and (2.44) repeatedly, we deduce
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ55_HTML.gif
      (2.46)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ56_HTML.gif
      (2.47)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ57_HTML.gif
      (2.48)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ58_HTML.gif
      (2.49)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ59_HTML.gif
      (2.50)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ60_HTML.gif
      (2.51)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ61_HTML.gif
      (2.52)
      Since ( p , Q ) ker ( γ I A m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq38_HTML.gif, by the imbedding theorem in Adams [15],
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ62_HTML.gif
      (2.53)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ63_HTML.gif
      (2.54)

      (2.45)-(2.54) show that (2.32)-(2.35) are true.

      Conversely, if (2.32)-(2.35) hold, then by using the formulas
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equn_HTML.gif
      integration by parts and the Fubini theorem, we estimate
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ64_HTML.gif
      (2.55)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ65_HTML.gif
      (2.56)
      (2.33) and (2.34) give
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ66_HTML.gif
      (2.57)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ67_HTML.gif
      (2.58)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ68_HTML.gif
      (2.59)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ69_HTML.gif
      (2.60)
      By combining (2.57), (2.58), (2.59) and (2.60) with (2.55) and (2.56), we derive
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ70_HTML.gif
      (2.61)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ71_HTML.gif
      (2.62)

      (2.55)-(2.62) mean that ( p , Q ) D ( A m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq39_HTML.gif and ( γ I A m ) ( p , Q ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq37_HTML.gif. □

      It is not difficult to see that L is surjective. Moreover,
      L | ker ( γ I A m ) : ker ( γ I A m ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equo_HTML.gif
      is invertible for γ ρ ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq35_HTML.gif. For γ ρ ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq40_HTML.gif we define the Dirichlet operator as
      D γ : = ( L | ker ( γ I A m ) ) 1 : X ker ( γ I A m ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equp_HTML.gif
      Lemma 2.3 gives the explicit form of D γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq41_HTML.gif for γ ρ ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq35_HTML.gif
      D γ ( a , b ) = ( ( 1 γ + λ ψ ϵ 0 0 0 ϵ 0 0 0 ϵ 1 ϵ 0 0 ϵ 2 ϵ 1 ϵ 0 ) ( a 1 a 2 a 3 a 4 ) + ( 1 γ + λ ϕ δ 0 0 0 0 0 0 0 0 0 0 0 0 ) ( b 1 b 2 b 3 b 4 ) , ( δ 0 0 0 0 δ 1 δ 0 0 0 δ 2 δ 1 δ 0 0 δ 3 δ 2 δ 1 δ 0 ) ( b 1 b 2 b 3 b 4 ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ72_HTML.gif
      (2.63)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equq_HTML.gif
      From (2.63) and the definition of Φ, it is easy to determine the expression of Φ D γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq42_HTML.gif for γ ρ ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq35_HTML.gif.
      Φ D γ ( a , b ) = ( ( ψ ϵ 1 ψ ϵ 0 0 0 ψ ϵ 2 ψ ϵ 1 ψ ϵ 0 0 ψ ϵ 3 ψ ϵ 2 ψ ϵ 1 ψ ϵ 0 ) ( a 1 a 2 a 3 ) + ( ϕ δ 1 ϕ δ 0 0 0 ϕ δ 2 ϕ δ 1 ϕ δ 0 0 ϕ δ 3 ϕ δ 2 ϕ δ 1 ϕ δ 0 ) ( b 1 b 2 b 3 ) , ( λ γ + λ ψ ϵ 0 0 0 0 0 0 0 0 0 ) ( a 1 a 2 a 3 ) + ( λ γ + λ ϕ δ 0 0 0 0 0 0 0 0 0 ) ( b 1 b 2 b 3 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equr_HTML.gif

      Haji and Radl [12] gave the following result through which we deduce the resolvent set of A on the imaginary axis.

      Lemma 2.4 If γ ρ ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq35_HTML.gifand 1 σ ( Φ D γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq43_HTML.gif, then
      γ σ ( A ) 1 σ ( Φ D γ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equs_HTML.gif

      By using Lemma 2.4 and Nagel [14], page 297, we derive the following result.

      Lemma 2.5 Let b ( x ) , b 0 ( x ) : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq29_HTML.gifbe measurable, 0 < inf x [ 0 , ) b ( x ) sup x [ 0 , ) b ( x ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq30_HTML.gifand 0 < inf x [ 0 , ) b 0 ( x ) sup x [ 0 , ) b 0 ( x ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq31_HTML.gif. Then all points on the imaginary axis except zero belong to the resolvent set of A.

      Proof Take γ = i m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq44_HTML.gif, m R { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq45_HTML.gif, a = ( a 1 , a 2 , ) l 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq46_HTML.gif and b = ( b 1 , b 2 , ) l 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq47_HTML.gif. Then by the Riemann-Lebesgue lemma,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equt_HTML.gif
      we know there exists M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq48_HTML.gif such that | m | > M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq49_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ73_HTML.gif
      (2.64)
      By replacing f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq50_HTML.gif in (2.64) with f ( x ) = e λ x 0 x b ( ξ ) d ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq51_HTML.gif, f ( x ) = e λ x 0 x b 0 ( ξ ) d ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq52_HTML.gif and using the fact
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equu_HTML.gif
      we derive, for | m | > M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq49_HTML.gif,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ74_HTML.gif
      (2.65)
      (2.65) means that when | m | > M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq49_HTML.gif, the spectral radius r ( Φ D γ ) Φ D γ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq53_HTML.gif, which implies 1 σ ( Φ D γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq54_HTML.gif for | m | > M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq49_HTML.gif, and therefore by Lemma 2.4, we know γ = i m σ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq55_HTML.gif for | m | > M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq49_HTML.gif, that is,
      { i m | m | > M } ρ ( A ) , { i m | m | M } σ ( A ) i R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equ75_HTML.gif
      (2.66)

      On the other hand, since T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq14_HTML.gif is positive uniformly bounded by Theorem 1.1, by Corollary 2.3 in Nagel [14], page 297, we know that σ ( A ) i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq56_HTML.gif is imaginary additively cyclic, which states that i m σ ( A ) i R i m k σ ( A ) i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq57_HTML.gif for all integer k, from which together with (2.66) and Lemma 2.1 we conclude σ ( A ) i R = { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq58_HTML.gif. □

      It is not difficult to prove X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq59_HTML.gif, dual space of X, is as follows:
      X = { ( p , Q ) | p R × L [ 0 , ) × L [ 0 , ) × , Q L [ 0 , ) × L [ 0 , ) × L [ 0 , ) × , ( p , Q ) = max { sup { | p 0 | , sup n 1 p n L [ 0 , ) } , sup n 1 Q n L [ 0 , ) } < } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equv_HTML.gif
      It is obvious that X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq59_HTML.gif is a Banach space. Gupur [4] gave the expression of A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq60_HTML.gif, the adjoint operator of A as follows:
      A ( p , Q ) = ( G + F + ) ( p , Q ) , ( p , Q ) D ( G ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equw_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equx_HTML.gif

      Since T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq14_HTML.gif is uniformly bounded, by Arendt and Batty [16] and Lemma 2.1, we know that 0 is an eigenvalue of A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq60_HTML.gif. Furthermore, by replacing μ and η in Lemma 3 in Gupur [4] with b ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq9_HTML.gif and b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq10_HTML.gif, respectively, we deduce the following result.

      Lemma 2.6 If 0 λ x b ( x ) e 0 x b ( ξ ) d ξ d x < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq16_HTML.gif, then 0 is an eigenvalue of A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq60_HTML.gifwith geometric multiplicity one.

      Since Theorem 1.1, Lemma 2.1, Lemma 2.5 and Lemma 2.6 satisfy the conditions of Theorem 14 in Gupur, Li and Zhu [7], the following conclusion is the direct result of Theorem 14 in Gupur, Li and Zhu [7].

      Theorem 2.7 Let b ( x ) , b 0 ( x ) : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq29_HTML.gifbe measurable, 0 < inf x [ 0 , ) b ( x ) sup x [ 0 , ) b ( x ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq30_HTML.gifand 0 < inf x [ 0 , ) b 0 ( x ) sup x [ 0 , ) b 0 ( x ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq31_HTML.gif. If 0 λ x b ( x ) × e 0 x b ( ξ ) d ξ d x < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq61_HTML.gif, then the time-dependent solution of the system (1.9) converges strongly to its steady-state solution, that is,
      lim t ( p , Q ) ( , t ) ( p , Q ) , ( p ( 0 ) , Q ( 0 ) ) ( p , Q ) ( ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_Equy_HTML.gif

      where ( p , Q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq62_HTML.gifand ( p , Q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq63_HTML.gifare the eigenvectors in Lemma 2.6 and Lemma 2.1, respectively.

      When b ( x ) = μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq64_HTML.gif and b 0 ( x ) = η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq65_HTML.gif, Lin and Gupur [9] proved that if http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq66_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq67_HTML.gif are eigenvalues of A with geometric multiplicity one for all θ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq68_HTML.gif. Which means that the result in Theorem 2.7 is optimal, that is to say, it is impossible that the time-dependent solution of the system (1.9) exponentially converges to its steady-state solution.

      Declarations

      Acknowledgements

      This work was supported by the Natural Science Foundation of Xinjiang (No: 2012211A023).

      Authors’ Affiliations

      (1)
      College of Mathematics and Systems Science, Xinjiang University
      (2)
      School of Mathematical Sciences, Xinjiang Normal University

      References

      1. Takagi H: Time-dependent analysis of M/G/1 vacation models with exhaustive service. Queueing Syst. 1990, 6: 369-390. 10.1007/BF02411484View Article
      2. Welch PD: On a generalized M/G/1 queueing process in which the first customer of each period receives exceptional service. Oper. Res. 1964, 12: 736-752. 10.1287/opre.12.5.736MathSciNetView Article
      3. Minh DL: Transient solutions of some exhaustive-service M/G/1 queues with generalized independent vacations. Cent. Eur. J. Oper. Res. 1998, 36: 197-201.MathSciNetView Article
      4. Gupur G: Asymptotic property of the solution of M/M/1 queueing model with exceptional service time for the first customer in each busy period. Int. J. Differ. Equ. Appl. 2003, 8: 23-94.MathSciNet
      5. Gupur G: Time-dependent analysis for a queue modeled by an infinite system of partial differential equations. Sci. China Math. 2012, 55: 985-1004. 10.1007/s11425-011-4351-1MathSciNetView Article
      6. Gupur G: Semigroup method for M/G/1 queueing system with exceptional service time for the first customer in each busy period. Indian J. Math. 2002, 44: 125-146.MathSciNet
      7. Gupur G, Li XZ, Zhu GT: Functional Analysis Method in Queueing Theory. Research Information Ltd, Herdfortshire; 2001.
      8. Zhang MQ, Gupur G: Another eigenvalue of the M/M/1 queueing model with exceptional service times for the first customer in each busy period. Acta Anal. Funct. Appl. 2009, 11: 62-68.MathSciNet
      9. Lin XJ, Gupur G: Other eigenvalues of the M/M/1 queueing model with exceptional service times for the first customer in each busy period. Acta Anal. Funct. Appl. 2011, 13: 383-391.MathSciNet
      10. Gupur G: Asymptotic property of the solution of a repairable, standby, human and machine system. Int. J. Pure Appl. Math. 2006, 8: 35-54.MathSciNet
      11. Greiner G: Perturbing the boundary conditions of a generator. Houst. J. Math. 1987, 13: 213-229.MathSciNet
      12. Haji A, Radl A:Asymptotic stability of the solution of the M / M B / 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-17/MediaObjects/13661_2012_Article_272_IEq11_HTML.gif queueing model. Comput. Math. Appl. 2007, 53: 1411-1420. 10.1016/j.camwa.2006.12.005MathSciNetView Article
      13. Gupur G: Advances in queueing models’ research. Acta Anal. Funct. Appl. 2011, 13: 225-245.MathSciNet
      14. Nagel R (Ed): One-Parameter Semigroups of Positive Operators. Springer, Berlin; 1986. LNM 1184
      15. Adams R: Sobolev Spaces. Academic Press, New York; 1975.
      16. Arendt W, Batty CJK: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 1988, 13: 837-852.MathSciNetView Article

      Copyright

      © Gupur and Ehmet; licensee Springer. 2013

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.