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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1


Volume 13 (2015)

Functional analysis method for the M/G/1 queueing model with single working vacation

Ehmet Kasim / Geni Gupur
Published Online: 2018-07-26 | DOI: https://doi.org/10.1515/math-2018-0074


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.

Keywords: M/G/1 queueing model with single working vacation; C0−semigroup; Eigenvalue; Resolvent set; Dirichlet operator

MSC 2010: 47D03; 47A10; 60K25

1 Introduction

Queueing system with server working vacations have arisen many researchers’ attention because the working vacation policy is more appropriate to model the real system in which the server has additional task during a vacation[1, 2, 3]. Unlike a classical vacation policy, the working vacation policy requires the server working at a lower rate rather than completely stopping service during a vacation. Therefore, compared with the classical vacation model, there are also customers who leave the system due to the completion of the services during working vacation. In this way, the number of customers in the system may be reduced. For example, an agent in a call center is required to do additional work after speaking with a customer. The agent may provide service to the next customer at a lower rate while performing additional tasks. In 2002, Servi and Finn [1] first introduced the M/M/1 queueing system with multiple working vacation. Since then, many researchers have extended their work to various type of queueing system (see Chandrasekaran et al.[4]). Kim et al.[5] and Wu and Takagi [6] extended Servi and Finn’s [1] M/M/1 queueing system to an M/G/1 queueing system. Xu et al. [7] and Baba [8] studied a batch arrival MX/M/1 queueing with working vacation. Gao and Yao [9] generalized it to an MX/G/1 queueing system. Baba [10] introduced the general input GI/M/1 queueing model with working vacation. Du [11] and Arivudainambi et al. [12] developed retrial queueing model with the concept of working vacation, etc. In 2012, Zhang and Hou [13] established the mathematical model of the M/G/1 queueing system with single working vacation by using the supplementary variable technique and studied the queueing length distribution and service status at the arbitrary epoch in the steady-state case under the following hypothesis:


According to Zhang and Hou [13], the M/G/1 queueing model with single working vacation can be described by the following partial differential equations:


with the integral boundary conditions


If we assume the system states when there are no customers in the system and the server is in vacation, i.e.,


where (x, t) ∈ [0, ∞) × [0, ∞); p0,v(t) represents the probability that there is no customer in the system and the server is in a working vacation period at time t; pn,v(x, t)dx (n ≥ 1) is the probability that at time t the server is in a working vacation period and there are n customers in the system with elapsed service time of the customer undergoing service lying in (x, x + dx]; p0,b(t) represents the probability that there is no customer in the system and the server is in a regular busy period at time t; pn,b(x, t)dx (n ≥ 1) is the probability that at time t the server is in a regular busy period and there are n customers in the system with elapsed service time of the customer undergoing service lying in (x, x + dx]; λ is the mean arrival rate of customers; θ is the vacation duration rate of the server; μv(x) is the service rate of the server while the server is in a working vacation period and satisfies


μb(x) is the service rate of the server while the server is in a regular busy period and satisfying


In fact, the above hypothesis (H) implies the following two hypotheses in view of partial differential equations:

  • Hypothesis 1

    The model has a unique time-dependent solution.

  • Hypothesis 2

    The time-dependent solution converges to its steady-state solution.

In 2016, Kasim and Gupur [14] did the dynamic analysis for the above model and gave the detailed proof of the hypothesis 1. Moreover, when the service rates in a working vacation period and in a regular busy period are constant, by using the C0− semigroup theory they obtained that the hypothesis 2 also hold. In the general case, the service rates are function, the hypothesis 2 does not always hold, see Gupur [15] and Kasim and Gupur [16], and it is necessary to study the asymptotic behavior of the time-dependent solution of the model. This paper is an effort on this subject.

The rest of this paper is organized as follows. In Section 2 we convert the model into an abstract Cauchy problem. In Section 3, by investigating the spectral properties of the underlying operator we give the main results of this paper. Firstly, we prove that 0 is an eigenvalue of the underlying operator with geometric multiplicity one by using the probability generating function. Next, to obtain the resolvent set of the underlying operator we apply the boundary perturbation method. We obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator. Last, we determine the adjoint operator and verify that 0 is an eigenvalue of the adjoint operator with geometric multiplicity one. Finally, based on these results we present the desired result in this paper: the time-dependent solution of the model strongly converges to its steady-state solution. In addition, the asymptotic behavior of the queueing system’s indices are discussed. A conclusion is given in Section 4. Section 5 provides a detail proof of some lemmas.

2 Abstract Setting for the system

In this section, we reformulate the equation (1.1)-(1.3) as an abstract Cauchy problem. We start by introducing the state space as follows.


It is obvious that X × Y is a Banach space. Define an operator and its domain.



ϕvf=0μv(x)f(x)dx,ϕbf=0μb(x)f(x)dx,fL1[0,),ψvg=dg(x)dx(θ+λ+μv(x))g(x),gW1,1[0,),ψbg=dg(x)dx(λ+μb(x))g(x),gW1,1[0,).D(Am)={(pv,pb)X×Y|dpn,vdxL1[0,),dpn,bdxL1[0,),pn,v(x)andpn,b(x)(n1)are absolutelycontinuous andn=1dpn,vdxL1[0,)<,n=1dpn,bdxL1[0,)<}.

We choose the boundary space of X × Y


and define two boundary operators as


where Fg=0 g(x) dx,   gL1[0, ∞).

Now we define operator A and its domain as


Then the above equations (1.1)-(1.3) can be written as an abstract Cauchy problem:


Kasim and Gupur [14] have obtained the following results.

Theorem 2.1

If μv(x) and μb(x) are measurable functions and satisfy μv¯=supx[0,)μv(x) < ∞ and μb = supx[0,) μb(x) < ∞, then A generates a positive contraction C0-semigroup T(t). The system (2.1) has a unique positive time-dependent solution (pv, pb)(x, t) = T(t)(pv, pb)(0) satisfying


3 Main results

In this section, firstly we prove that 0 is an eigenvalue of A with geometric multiplicity one, next we study the resolvent set of operator A by using the Greiner’s idea [17] and obtain that all points on the imaginary axis except zero belong to the resolvent set of A. Thirdly, we determine the expression of A*, the adjoint operator of A, and verify that 0 is an eigenvalue of the adjoint operator A* with geometric multiplicity one. Thus, we conclude that the time-dependent solution of the system (2.1) strongly converges to its steady-state solution.

Lemma 3.1

If 0λxμb(x)e0xμb(τ)dτdx<1, then 0 is an eigenvalue of A with geometric multiplicity one.


We consider the equation A(pv, pb) = 0, which is equivalent to


with the boundary conditions


By solving (3.1) we obtain





The probability generating functions of these sequences are given by, for |z| < 1


From (3.1)-(3.4) we have


Let q0 = (p0,v,p0,b), qi = (pi,v(0), pi,b(0)), i ≥ 1, and define


Then (3.5) can be written as





Now, we introduce the row-vector generating functions


Hence, from (3.7)-(3.9) we deduce


An easy computation shows that




This together with (3.10) yields


Thus, we have


where e = (1, 1)T.

In the following, by using the Rouche’s theorem we conclude that zC(z) has a unique zero point inside unit circle |z| = 1. Let this root be denoted by γ, this must be root of the numerator of the equation (3.12) too. So, substituting z = γ into (3.12) we get


(3.6) and (3.13) give


By using the L’Hospital rule and (3.12)-(3.14), we determine



where e1 = (1, 0)T, e2 = (0,1)T.

By combining (3.15) and (3.16) with (3.3) and (3.4), we estimate



(3.17) and (3.18) imply


Thus, 0 is an eigenvalue of A. Moreover, from (3.5) it easy to see that the eigenvectors corresponding to zero span one dimensional linear space, i.e., the geometric multiplicity of 0 is one. □

In order to obtain the asymptotic behavior of the time-dependent solution of the system (2.1) we need to know the spectrum of A on the imaginary axis (see Theorem 14 in Gupur et al.[18]). For that purpose we use boundary perturbation method, which is developed by the Greiner [17], through which the spectrum of the operator can be deduced by discussing the boundary operator. It is related to the resolvent set of operator A0 and spectrum of Φ Dγ, where Dγ is inverse of L in ker(γIAm). Hence, we first consider the operator


and discuss its inverse. For any given (y, z) ∈ X × Y, we consider the equation (γIA0)(pv, pb) = (y, z), i.e.,


By solving (3.19) we have







If we introduce the following two operators as


then (3.21), (3.22), (3.24) and (3.25) imply



Similarly, we have



By inserting (3.26) and (3.28) into (3.20), we obtain


(3.26)-(3.30) give the expression of (γIA0)−1 as follows if (γIA0)−1 exists.


Therefore, we obtain the following two lemmas and their proof given in the appendix.

Lemma 3.2





Lemma 3.3



If γρ(A0), then


Using the results in Greiner [17], observe that the operator L is surjective. So,


is invertible if γρ(A0). Its inverse will play an important role in the characterization of the spectrum of A on the imaginary axis and we denote its inverse by


and call it the Dirichlet operator. Furthermore, Lemma 3.3 gives the explicit formula of Dγ for all γρ(A0),




From the expression of Dγ and the definition of Φ, it is easy to determine the explicit form of Φ Dγ as follows.


Haji and Radl [19] gave the following result, which indicates the relations between the spectrum of A and spectrum of Φ Dγ.

Lemma 3.4

If γρ(A0) and there exists γ0 ∈ ℂ such that 1 ∉ σ(Φ Dγ0), then


From Lemma 3.4 and Nagel [20], we obtain the resolvent set of A on the imaginary axis.

Lemma 3.5



then all points on the imaginary axis except zero belong to the resolvent set of A.


Let γ = iβ, β ∈ ℝ ∖ {0}. The Riemann-Lebesgue lemma


implies that there exists a positive constant 𝓚 > 0 such that ∀ |β| > 𝓚,


In this formula, by replacing f(x) with μv(x)e0x(θ+λ+μv(ξ))dξ,μb(x)e0x(λ+μb(ξ))dξ, and


and using the fact 0μb(x)e0xμb(ξ)dξdx=1,0(θ+μv(x))e0x(θ+μv(ξ))dξdx=1, we estimate for a⃗v = (a1,v, a2,v, a3,v,⋯) ∈ l1 and a⃗b = (a1,b, a2,b, a3,b,⋯) ∈ l1,



(3.32) shows that 1 ∉ σ(Φ Dγ) when |β | > 𝓚. This together with Lemma 3.4 give


Theorem 2.1 and Lemma 3.1 ensures that T(t) is a positive contraction C0−semigroup and its spectral bound is zero. By Nagel [20] we know that σ(A) is imaginary additively cyclic (see also Thorem 1.88 in [21]) which states that

iβσ(A)iβhσ(A),all positive integerh.

From which together with (3.33) and Lemma 3.1 it follows that iℝ ∩ σ(A) = {0}. □

A trivial verification shows that X* × Y*, the dual space of X × Y, is as follows.




It is evident that X* × Y* is a Banach space.

Lemma 3.6

A*, the adjoint operator of A, is as follows.


here α in D(A*) is a constant which is independent of n.


By using integration by parts and the boundary conditions on (pv, pb) ∈ D(A), we have, for (qv,qb)D(A*)


From this together with the definition of adjoint operator the assertion follows. □

From Theorem 2.1, Lemma 3.1 and Arendt et al. [22], we know that 0 is an eigenvalue of A*. Furthermore, we deduce the following result.

Lemma 3.7

If 0λxμb(x)e0xμb(τ)dτdx<1, then 0 is an eigenvalue of A* with geometric multiplicity one.

Now, combining the Theorem 2.1, Lemma 3.1, Lemma 3.5 and Lemma 3.7 with Theorem 14 in Gupur et al. [18] we obtain the following main result.

Theorem 3.8



then the time-dependent solution of the system (2.1) strongly converges to its steady-state solution, i.e.,


here (pv, pb)(x) is the eigenvector in Lemma 3.1 and ω is decided by the eigenvector in Lemma 3.7 and the initial value (pv, pb)(0).

In the following, by applying the Theorem 3.8 we briefly discuss the queueing system’s indices. It is easily seen that the time-dependent queueing size at the departure point converges to a positive number, i.e.,


and the time-dependent queueing length L(t) converges to the steady-state queueing length L, that is,


From this we can obtain that other queuing indices Lq(t), W(t) and Wq(t) also converge to a positive number Lq, W and Wq respectively.

4 Conclusion

In this paper, we study an M/G/1 queueing model with single working vacation, in which the service time is generally distributed. The system is described by infinite number of partial differential equations with integral boundary conditions which we have converted into an abstract Cauchy problem in the Banach space. Then, by investigating the spectrum of the operator on the imaginary axis, which corresponds to the M/G/1 queueing model with single working vacation, we proved that the time-dependent solution of the model strongly converges to its steady-state solution. In other words, we verified that the hypothesis 2 holds in the sense of strong convergence.

In this paper and our previous paper, we only studied spectra of the operator on the right half complex plane and imaginary axis, which corresponds to the M/G/1 queueing model with single working vacation, so it is worth studying spectra of the operator on the left half complex plane.

5 Appendix

Proof of Lemma 3.2

For any fL1[0, ∞), by using integration by parts, we have



From (A.1) and (A.2) together with condition of this lemma and using ∥ϕv∥ ≤ μv, ∥ϕb∥ ≤ μb we deduce, for any (y, z) ∈ X × Y


This shows that the result of this lemma is right. □

Proof of Lemma 3.3

If (pv, pb) ∈ ker(γIAm), then (γIAm)(pv, pb) = 0, which is equivalent to







By solving (A.4), (A.5) and (A.7), (A.8), we have





By using (A.9) and (A.10) repeatedly, we obtain


Similarly, by applying (A.11) and (A.12) repeatedly, we deduce


Through inserting (A.9) and (A.11) into (A.3), we derive



Since (pv, pb) ∈ ker(γID(Am)), (pv, pb) ∈ D(Am) implies by the imbedding theorem in Adams [23],


from which together with (A.13)-(A.16) we know that (2.55) holds.

Conversely, if (2.55) is right, then by using 0xkeMxdx=k!Mk+1,k1,M>0, integration by parts and the Fubini theorem we estimate


Similarly, we get


(A.17) and (A.18) gives




It is immediately obtained



(A.17)–(A.20) show that (pv, pb) ∈ ker(γIAm). □

Proof of Lemma 3.7

We consider the equation A(qv,qb)=0, which is equivalent to


It is easy to see that


is a solution of (A.21). In addition, (A.21) is equivalent to


(A.22) show that we can determine each qn,v (x) and qn,b (x) for all n ≥ 1 if q1,v (x) and q1,b (x) are given. That is to say, geometric multiplicity of zero is one. □


The authors would like to express their sincere thanks to the anonymous referees and associated editor for his/her careful reading of the manuscript. The author’ research work was supported by the National Natural Science Foundation of China (No:11371303) and Natural Science Foundation of Xinjiang University(No:BS130104).


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About the article

Received: 2017-10-21

Accepted: 2018-05-24

Published Online: 2018-07-26

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 767–791, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0074.

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© 2018 Kasim and Gupur, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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