Abstract

We describe the point spectrum of the generator of a -semigroup associated with the M/M/1 queueing model that is governed by an infinite system of partial differential equations with integral boundary conditions. Our results imply that the essential growth bound of the -semigroup is 0 and, therefore, that the semigroup is not quasi-compact. Moreover, our result also shows that it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.

1. Introduction

In 1955, by considering service time of customers, Cox [1] first established the M/G/1 queueing model which was described by an infinite system of partial differential equations with integral boundary conditions and studied the steady-state solution of the model under the following hypothesis: the time-dependent solution of the model converges to its steady-state solution. In 2001, Gupur et al. [2] have proved that the underlying operator, which corresponds to the M/G/1 queueing model, generates a positive contraction -semigroup that is isometric for the initial value. Hence, they deduced that the model has a unique nonnegative time-dependent solution which satisfies the probability condition (i.e., its norm is 1). In 2011, by studying spectral properties of the underlying operator on the imaginary axis, Gupur [3] obtained that all points on the imaginary axis except belong to the resolvent set of the underlying operator and is an eigenvalue of the underlying operator and its adjoint operator with geometric multiplicity one. Thus, by using Theorem 14 in Gupur et al. [2] (Theorem 1.96 in Gupur [4]) it follows that the time-dependent solution of the model strongly converges to its steady-state solution; that is, Cox’s hypothesis holds in the sense of strong convergence. When the service rate is a constant, the M/G/1 queueing model is called M/M/1 queueing model. Well-posedness of the M/M/1 queueing model and asymptotic behavior of its time-dependent solution can be found in Gupur et al. [2] (see also Radl [5]). In 2008, Zhang and Gupur [6] have found that the underlying operator, which corresponds to the M/M/1 queueing model, has one negative real eigenvalue. In 2011, Kasim and Gupur [7] discovered that the underlying operator has uncountable negative real eigenvalues and therefore suggested that it is impossible that the time-dependent solution of the model exponentially converges to its steady-state solution. So far, no other results concerning this model can be found in the literature.

In this paper, we study eigenvalues of the underlying operator associated with the M/M/1 queueing model and obtain that if the mean arrival rate of customers and the mean service rate of the server satisfy , then all points in the set are eigenvalues of the underlying operator with geometric multiplicity one. In particular, the interval belongs to its point spectrum. These results together with the spectral mapping theorem for the point spectrum ([8], p. 277) imply that the -semigroup generated by the underlying operator has uncountable eigenvalues and therefore it is not compact, even not eventually compact. Moreover, by combining the result in this paper and the results in Gupur et al. [2] with Corollary 2.11 in Engel and Nagel [8], p. 258, we deduce that the essential growth bound of the -semigroup is and therefore it is not quasi-compact ([8], p. 332). Hence, queueing models are essentially different from population equations (see [9, 10]) and the reliability models that are described by finite partial differential equations with integral boundary conditions (see [4, 11]). In addition, we show that the essential spectral radius of the -semigroup is 1 and it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.

If we do not consider service time of customers, then the M/M/1 queueing model becomes an infinite system of ordinary differential equations. Its research can be found in Gupur et al. [2] and Zhao et al. [12].

2. The M/M/1 Queueing Model and Related Results

According to Cox [1] the M/M/1 queueing model can be described by the following system of partial differential equations with integral boundary conditions: Here is the mean arrival rate of customers; is the mean service rate of the server; is the probability that the system is empty at time is the probability that at time there are customers in the system and the service time of the customer undergoing service is .

In this paper, we use the notations in [2, 6, 7]. Select a state space as follows: It is obvious that is a Banach space. Moreover, is a Banach lattice under the following order relation: For simplicity, we introduce In the following we define operators and their domains: Then (2) can be rewritten as a Cauchy problem in : where is called M/M/1 operator.

The following results can be found in Gupur et al. [2].

Theorem 1. generates a positive contraction -semigroup .   is isometric for . Hence, the system (7) has a unique positive time-dependent solution for satisfying The set belongs to the resolvent set of , the adjoint operator of . In particular, all points on the imaginary axis except belong to the resolvent set of . When is an eigenvalue of and with geometric multiplicity 1. Therefore, the time-dependent solution of the system (7) strongly converges to its steady-state solution: here is the eigenvector with respect to .

In 2008, Zhang and Gupur [6] obtained the following result.

Theorem 2. If , then is an eigenvalue of with geometric multiplicity 1.

In 2011, Kasim and Gupur [7] proved the following result.

Theorem 3. If , then all points in are eigenvalues of with geometric multiplicity 1.

3. Main Results

Theorem 4. If , then all points in the set are eigenvalues of with geometric multiplicity 1. In particular, the interval belongs to the point spectrum of .

Remark 5. The condition in Theorem 4, which was used by Cox [1], means that the service rate of the server is larger than the arrival rate of customers, thus preventing long queues of customers and therefore ensuring that the queueing system exists. In other words, the condition is a necessary condition for the existence of the M/M/1 queueing system.

Proof. We consider the equation ; that is, By solving (13) and (14) we have By using (17) and (18) repeatedly we deduce Equation (20) and for imply Combining (15), (16), and (21) gives Multiplying (23) by and subtracting it from (24) yield If we assume then this together with (25) yields From (27) we determine Equation (26) implies From (29) we know By using (30) we derive, for , By inserting (17) into (12) and noting we have By combining (22) with (32) and it follows that Combining (23), (32), and (33) gives From (21), for , and the Cauchy product of series we estimate, when For simplicity, we introduce It is easy to see that Together with (35) and (31) to (34), this yields That is, all are eigenvalues of . Moreover, from (20) and (31) to (34) it is easy to see that the eigenvectors corresponding to each span 1-dimensional linear space; that is, their geometric multiplicity is one.
In the following, we discuss the case that is a real number and obtain explicit results.
Since Theorem 1 implies that all belong to the resolvent set of includes the following three cases.
(1) If , then by noting , This together with (38) implies that all points in are eigenvalues of , which is the main result in Kasim and Gupur [7] (see Theorem 3).
By applying the condition we have Since , (33) and (34) are simplified as By combining (40) and (41) with (31) and using (35) we have This means that is an eigenvalue of , which is the result in Gupur et al. [2] (see Theorem 1).
(2) If , then the condition implies Hence, is an eigenvalue of , which is the main result in Zhang and Gupur [6] (see Theorem 2).
(3) If , then the condition gives This yields that all points in are eigenvalues of .
By summarizing the above discussion we conclude that all points in are eigenvalues of with geometric multiplicity one. In particular, belongs to the point spectrum of .