Semigroup Methods for the M / G / 1 Queueing Model with Working Vacation and Vacation Interruption

Abstract By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time-dependent solution which satisfies probability conditions. When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.


Introduction
According to (Zhang & Hou, 2010), the M/G/1 queueing system with working vacation and vacation interruption can be described by the following system of partial differential equations: p 1,0 (0, t) = λp 0,0 (t), p n,0 (0, t) = 0, ∀n ≥ 2, and initial condition: p 0,0 (0) = 1, p 0,1 (0) = 0, p m,0 (x, 0) = p m,1 (x, 0) = 0, ∀m ≥ 1. (1.3) Where, (x, t) ∈ [0, ∞) × [0, ∞); p 0,0 (t) represents the probability that there is no customer in the system and the server is in a working vacation period at time t; p n,0 (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]; p n,1 (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; µ 0 (x) is the service rate of the server while the server is in a working vacation period and satisfies µ 1 (x) is the service rate of the server while the server is in a regular busy period and satisfying Queuing situations in which the idle server may take vacations encounter in computer, communication and manufacturing systems, etc.In a classical vacation queue, a server may completely stop the service or do some additional work during a vacation.Proposing various vacation policies provides more flexibility for optimal design and operation control of the system.Therefore, many researchers studied such queueing system, see (Doshi, 1986;Takagi, 1990;Madan, 1992;Gupur, 2002;Gupur, 2010;Gupur & Guo, 2002;Lu & Gupur, 2010), for instance.
In many real life congestion situations, the server can be utilized for ancillary work and a different rate during the vacation period.Such a queueing situation is called queue with working vacation.(Servi & Finn, 2002) first studied the M/M/1 queueing system with multiple working vacation and obtained the transform formulae for the distribution of the number of customers in the system and the sojourn time in a steady state.Since then, Queueing models with working vacation have been studied by several researchers , see (Jain & Agrawal, 2007;Kim, Choi & Chae, 2003;Li, Tian, & Ma, 2007).Moreover, (Wu & Takagi, 2006) extended Servi and Finn's (Servi & Finn, 2002) M/M/1 queueing system to an M/G/1 queueing system with multiple working vacation, where the service times during regular service period and working vacation period.(Zhang & Hou, 2010) considered the M/G/1 queueing system with working vacation and vacation interruption where the server enters into vacations when there are no customers and it can take service at a lower rate during the vacation period.If there are customers in the system at the instant of a service completion during the vacation period, the server will come back to the normal working level no matter whether or not the vacation has ended.otherwise, it continues the vacation.Firstly, by using supplementary variable technique they established the above model and gave the Laplace-Stieltjes transform of the stationary waiting time.Then, they obtain the queue length distribution and service status at an arbitrary epoch in steady state condition under following hypothesis: In addition, they also perform some numerical examples to study the effect of various parameters on the system's characteristics.By reading the paper we have found that the above hypothesis implies the following two hypotheses: Hypothesis 1 The model has a unique time-dependent solution.
Hypothesis 2 The time-dependent solution converges to its steady-state solution.
So far, any results about this model have not been found in the literature.In this paper, we do dynamic analysis for the queueing model by using the idea of (Gupur, Li, & Zhu, 2001), that is, we investigate above two hypotheses.First of all, we convert the model into an abstract Cauchy problem by choosing a suitable Banach space as a state space and introducing an operator corresponding to the model and its domain.By using the Hille-Yosida theorem and Phillips theorem as well as Fattorini theorem we prove that the model has a unique positive time-dependent solutions and therefore we obtain that the Hypothesis 1 is hold.Next, when the both service rate in a working vacation period and service rate in a regular busy period are constant, we study the asymptotic behavior of its time-dependent solution, i.e., we study the Hypothesis 2. Firstly, we determine the expression of the adjoint operator of the operator corresponding to the model and deduce the resolvent set of the operator.Then, we verify that 0 is eigenvalue of the underlying operator and its adjoint operator with geometric multiplicity one.Thus, by using Theorem 14 in (Gupur, Li, & Zhu, 2001) we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.
For simplicity, we introduce some notations as follows: Take a state space as follows: It is obvious that X is a Banach space.Now we define operators and their domain as follows.), p n,0 (x) and p n,1 (x) (n ≥ 1) are absolutely continuous and p 0 (0) = Γ 1 p 0 ; here Then the above system of equations (1.1)-(1.3)can be written as an abstract Cauchy problem in Banach space X.
Well-posedness of the System (1.4) In this section, in order to obtain well-posedness of the system (1.4), we first need to prove that A + U + E generates a positive contraction C 0 − semigroup T (t) on X.
Theorem 2.1 If µ 0 (x) and µ 1 (x) are measurable functions and satisfy A detailed proof of the Theorem 2.1 can be found in the Appendix.It is not difficult to verify that X * , the dual space of X, is as follows.
It is obvious that X * is a Banach space.If we take a set S in X as Then S is cone in X. for (p 0 , p 1 ) ∈ D(A) ∩ S , we take For such (p * 0 , p * 1 ) ∈ X * , by using the boundary condition we have which shows that A + U + E is a conservative operator.Since the initial value (p 0 , p 1 )(0)) ∈ D(A 2 ) ∩ S , by using the Fattorini theorem (Fattorini, 1983) we obtain the following result.see (Gupur, Li, & Zhu, 2001) for detail proof.
Theorem 2.2 T (t) is isometric for the initial value of the system (1.4), that is, From Theorem 2.1 and Theorem 2.2 we obtain well-posedness of the system (1.4).
3. Asymptotic Behavior of the Time-Dependent Solution of the System (1.4) when µ 0 (x) = µ 0 and µ 1 (x) = µ 1 Lemma 3.1 (A + U + E) * , the adjoint operator of A + U + E, is as follows. where The proof is easy computaion, and we omit the detail proof.
belongs to the resolvent set of (A + U + E) * , Especially, all points on the imaginary axis except for zero belong to the resolvent set of (A + U + E) * and A + U + E.
From which together with (3.21) it follows that sup Substituting (3.20) into (3.17)we conclude similarly that We can easy to check that the following inequality holds.
(3.40)By (3.40) and the L'Hospital rule we calculate By combining (3.36), (3.37), (3.38) and (3.41), we have This shows that 0 is an eigenvalue of A + U + E.Moreover, by solving (3.29) -(3.31) we have From the above, we know that the eigenvectors corresponding to zero span one dimensional linear space, that is to say, the geometric multiplicity of 0 is one.2 From Theorem 2.3 and Lemma 3.2 we know that 0 is an eigenvalue of (A+U + E) * .Furthermore, we deduce the following result.
Theorem 1 If λ < µ 1 , then the time-dependent solution of the system (1.4) strongly converges to its steady-state solution, that is, here (p 0 , p 1 )(x) is the eigenvector in Lemma 3.3 and β is decided by the eigenvector in Lemma 3.4 and initial value (p 0 , p 1 )(0).

Conclusion
In this paper, we do dynamic analysis for the M/G/1 queueing model with single working vacation and vacation interruption by using the functional analysis method.we prove the existence of a unique nonnegative time-dependent solution of the model, and when the service completion rate are constant we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.These results confirmed all two hypotheses which stated in the introduction.
In addition, from Theorem 3.1 we can prove the time-dependent queueing size at the departure point converges a positive number, the time-dependent queueing length and the time-dependent waiting time also converges to the corresponding steady-state queuing length and steady-state waiting time.
Our studies in this field, see (Kasim & Gupur, 2011) and (Gupur, 2011;Gupur, 2014) for instance, suggest that there are infinitely many eigenvalues of A + U + E on the left half complex plane, that is to say, it is impossible that the timedependent solution of the system (1.4) exponentially converges to its steady-state solution.Of course, it needs to verify.that is our next research work.
f s N,1 (x) = p N,1 (x), It is easy to verify that ( f s 0 , f s 1 ) ∈ D(A).Moreover This means that D(A) is dense in X.From the above two steps and Hille-Yosida Theorem we conclude that A generates a C 0 − semigroup.see (Gupur, Li, & Zhu, 2001).Now, we verify that U and E are bounded linear operators.From the definition of U and E we have, for (p 0 , p 1 ) ∈ X