Functional analysis method for the M/G/1 queueing model with single working vacation
-
Ehmet Kasim
and
Geni Gupur
Abstract
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.
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:
The model has a unique time-dependent solution.
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.
where
We choose the boundary space of X × Y
and define two boundary operators as
where
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
Ifμv(x) andμb(x) are measurable functions and satisfy
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
Proof
We consider the equation A(pv, pb) = 0, which is equivalent to
with the boundary conditions
By solving (3.1) we obtain
Define
The probability generating functions of these sequences are given by, for |z| < 1
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
and
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 z − C(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
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
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 ΦDγ, where Dγ is inverse of L in ker(γI − Am). Hence, we first consider the operator
and discuss its inverse. For any given (y, z) ∈ X × Y, we consider the equation (γI − A0)(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 (γI − A0)−1 as follows if (γI − A0)−1 exists.
Therefore, we obtain the following two lemmas and their proof given in the appendix.
Lemma 3.2
If
then
Lemma 3.3
Let
Ifγ ∈ ρ(A0), then
Using the results in Greiner [17], observe that the operator L is surjective. So,
is invertible if γ ∈ ρ(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 γ ∈ ρ(A0),
where
From the expression of Dγ and the definition of Φ, it is easy to determine the explicit form of ΦDγ as follows.
Haji and Radl [19] gave the following result, which indicates the relations between the spectrum of A and spectrum of ΦDγ.
Lemma 3.4
Ifγ ∈ ρ(A0) and there existsγ0 ∈ ℂ such that 1 ∉ σ(ΦDγ0), then
From Lemma 3.4 and Nagel [20], we obtain the resolvent set of A on the imaginary axis.
Lemma 3.5
If
then all points on the imaginary axis except zero belong to the resolvent set ofA.
Proof
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
and using the fact
(3.32) shows that 1 ∉ σ(ΦDγ) 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
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.
here
It is evident that X* × Y* is a Banach space.
Lemma 3.6
A*, the adjoint operator ofA, is as follows.
hereαinD(A*) is a constant which is independent ofn.
Proof
By using integration by parts and the boundary conditions on (pv, pb) ∈ D(A), we have, for
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
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
If
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 f ∈ L1[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(γI − Am), then (γI − Am)(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(γI − D(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
Similarly, we get
Since
It is immediately obtained
Proof of Lemma 3.7
We consider the equation
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
Acknowledgement
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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© 2018 Kasim and Gupur, published by De Gruyter
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- A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
- Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
- Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
- Introduction to disoriented knot theory
- Restricted triangulation on circulant graphs
- Boundedness control sets for linear systems on Lie groups
- Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
- Disjointed sum of products by a novel technique of orthogonalizing ORing
- A parametric linearizing approach for quadratically inequality constrained quadratic programs
- Generalizations of Steffensen’s inequality via the extension of Montgomery identity
- Vector fields satisfying the barycenter property
- On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
- Biderivations of the higher rank Witt algebra without anti-symmetric condition
- Some remarks on spectra of nuclear operators
- Recursive interpolating sequences
- Involutory biquandles and singular knots and links
- Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
- Topological entropy for positively weak measure expansive shadowable maps
- Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
- On 𝓠-regular semigroups
- One kind power mean of the hybrid Gauss sums
- A reduced space branch and bound algorithm for a class of sum of ratios problems
- Some recurrence formulas for the Hermite polynomials and their squares
- A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
- On f - prime radical in ordered semigroups
- Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
- Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
- A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
- Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
- The points and diameters of quantales
- Directed colimits of some flatness properties and purity of epimorphisms in S-posets
- Super (a, d)-H-antimagic labeling of subdivided graphs
- On the power sum problem of Lucas polynomials and its divisible property
- Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
- On generalized P-reducible Finsler manifolds
- On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
- On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
- On the different kinds of separability of the space of Borel functions
- Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
- Functional analysis method for the M/G/1 queueing model with single working vacation
- Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
- The existence of solutions to certain type of nonlinear difference-differential equations
- Domination in 4-regular Knödel graphs
- Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
- Algebras of right ample semigroups
- Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
- Nontrivial periodic solutions to delay difference equations via Morse theory
- A note on the three-way generalization of the Jordan canonical form
- On some varieties of ai-semirings satisfying xp+1 ≈ x
- Abstract-valued Orlicz spaces of range-varying type
- On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
- Arithmetic of generalized Dedekind sums and their modularity
- Multipreconditioned GMRES for simulating stochastic automata networks
- Regularization and error estimates for an inverse heat problem under the conformable derivative
- Transitivity of the εm-relation on (m-idempotent) hyperrings
- Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
- Simultaneous prediction in the generalized linear model
- Two asymptotic expansions for gamma function developed by Windschitl’s formula
- State maps on semihoops
- 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
- Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
- New topology in residuated lattices
- Optimality and duality in set-valued optimization utilizing limit sets
- An improved Schwarz Lemma at the boundary
- Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
- Toeplitz matrices whose elements are coefficients of Bazilevič functions
- Epi-mild normality
- Nonlinear elastic beam problems with the parameter near resonance
- Orlicz difference bodies
- The Picard group of Brauer-Severi varieties
- Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
- Weak group inverse
- Infinite growth of solutions of second order complex differential equation
- Semi-Hurewicz-Type properties in ditopological texture spaces
- Chaos and bifurcation in the controlled chaotic system
- Translatability and translatable semigroups
- Sharp bounds for partition dimension of generalized Möbius ladders
- Uniqueness theorems for L-functions in the extended Selberg class
- An effective algorithm for globally solving quadratic programs using parametric linearization technique
- Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of Szász-mirakjan operators of blending type
- Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
