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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 893254, 10 pages
Semigroup Method on a /G/1 Queueing Model
Xinjiang Radio & TV University, Urumqi 830049, China
Received 24 December 2012; Revised 20 March 2013; Accepted 21 March 2013
Academic Editor: B. G. Konopelchenko
Copyright © 2013 Alim Mijit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem in functional analysis we prove that the /G/1 queueing model with vacation times has a unique nonnegative time-dependent solution.
The queueing system when the server become idle is not new. Miller  was the first to study such a model, where the server is unavailable during some random length of time for the M/G/1 queueing system. The M/G/1 queueing models of similar nature have also been reported by a number of authors, since Levy and Yechiali  included several types of generalizations of the classical M/G/1 queueing system. These generalizations are useful in model building in many real life situations such as digital communication, computer network, and production/inventory system [3–5].
At present, however, most studies are devoted to batch arrival queues with vacation because of its interdisciplinary character. Considerable efforts have been devoted to study these models by Baba , Lee and Srinivasan , Lee et al. [8, 9], Borthakur and Choudhury , and Choudhury [11, 12] among others. However, the recent progress of /G/1 type queueing models of this nature has been served by Chae and Lee  and Medhi .
In 2002, Choudhury  studied the /G/1 queueing model with vacation times. By using the supplementary variable technique  he established the corresponding queueing model and obtained the queue size distribution at a stationary (random) as well as a departure point of time under multiple vacation policy based on the following hypothesis. “The time-dependent solution of the model converges to a nonzero steady-state solution.” By reading the paper we find that the previous hypothesis, in fact, implies the following two hypothesis.
Hypothesis 1. The model has a nonnegative time-dependent solution.
Hypothesis 2. The time-dependent solution of the model converges to a nonzero steady-state solution.
In this paper we investigate Hypothesis 1. By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem we prove that the model has a unique nonnegative time-dependent solution, and therefore we obtain Hypothesis 1.
According to Choudhury , the /G/1 queueing system with vacation times can be described by the following system of equations: where ; represents the probability that there is no customer in the system and the server is idle at time ; represents the probability that at time there are customers in the system and the server is on a vacation with elapsed vacation time of the server lying in . represents the probability that at time there are customers in the system with elapsed service time of the customer undergoing service lying in . is batch arrival rate of customers. represents the probability that at every arrival epoch a batch of external customers arrives and satisfies . is the vacation rate of the server, which satisfies is the service rate of the server satisfying
2. Problem Formulation
We first formulate the system (1) as an abstract Cauchy problem on a suitable state space. For convenience we take some notations as follows: If we take state space then it is obvious that is a Banach space. In the following we define operators and their domains; where , , Then the previous system of equations (1) can be rewritten as an abstract Cauchy problem in the Banach space :
3. Well-Posedness of The System (8)
Theorem 1. If , , then generates a positive contraction -semigroup .
Proof. We split the proof of the theorem into four steps. Firstly, we prove that exists and is bounded for some . Secondly, we show that is dense in . Thirdly, we verify that and are bounded linear operators. Thus by using the Hille-Yosida theorem and the perturbation theorem of -semigroup we deduce that generates a -semigroup . Finally, we check that is dispersive, and therefore we obtain the desired result.
For any given , we consider the equation ; that is, Through solving (9)–(11), we have Combining (16) with (12) and (13), we obtain Substituting (19) into (16), it follows that By combining (15), (16), (17), and (20) with (14), we deduce If we set then (21) can be rewritten as follows:It is easy to calculateFrom which together with (24), we derive By using Fubini theorem we estimate (16) and (17) as follows (assume that : From (26) and Fubini theorem, we deduce By inserting (18), (19), and (28) into (27) and using inequality , we estimate Equation (29) shows that exists for and As far as the second step is concerned, from for it follows that, for any , there exists a positive integer such that , . Let then is dense in . If we set then by the relationship and in Adams , we know that is dense in . Hence in order to prove denseness of , it suffices to prove that is dense in . Take any , then there are a finite positive integer and positive numbers , ; such that which implies where . Define where where Then it is not difficult to verify that . Moreover, which shows that is dense in . Hence, is dense in . From the first step, the second step, and the Hille-Yosida theorem  we know that generates a -semigroup.
Next we will verify that and are bounded linear operators. From the definition of and and we have The previous two formulas show that and are bounded operators. It is easy to check that and are linear operators. Hence from the perturbation theorem of -semigroup , we obtain that generates a -semigroup .
Lastly, we will prove that is a dispersive operator. For we take as where If we define and for , then by a short argument we calculate Similar to (42), we get By using boundary conditions on , (42), (43), and for such , we derive Equation (44) shows that is a dispersive operator. From which together with the first step, the second step, and the Phillips theorem, we know that generates a positive contraction -semigroup . By the uniqueness of a -semigroup we conclude that this semigroup is just .
It is not difficult to see that , the dual space of , is
It is easy to check that is a Banach space. If we take a set in as then is a cone in . For , we take then we have that is For such , by using boundary conditions on and , we have Which shows that is a conservative operator. So we can use the Fattorini theorem  and state it as follows.
Theorem 2. is isometric for the initial value of the system (8); that is,
Proof. Since is conservative with respect to and , from the Taylor expansion of for we have
where as , uniformly in . Then
In view of (52) and the fact that is conservative with respect to , consider the set
Since , is nonempty, moreover is obviously a closed interval because of continuity of . If , then let be the right end point of and is so small that is bounded away from zero in . For any such we divide (53) by and get
where is positive and when , uniformly in .
Let now be are small positive number and such that for and . Let be partition of the interval such that . Then by (55), one has Since is arbitrary, it follows that , which contradicts the fact that is the right endpoint of . Hence . That is, for . The proof of the theorem is complete.
Theorem 3. If , , then the system (8) has a unique nonnegative time-dependent solution , which satisfies
Proof. Since , by Theorem 1 and Theorem 11 in Gupur et al. , we know that the system (8) has a unique nonnegative time-dependent solution which can be expressed as From which together with Theorem 2 (i.e., (51)) we have this just reflects the physical background of the problem.
4. Concluding Remarks
If we know the spectrum of on the imaginary axis, then by Theorem 1 and Theorem 14 in Gupur et al. , we obtain the asymptotic behavior of the time-dependent solution of the system (8), which describes Hypothesis 2. It is our next research work.
The author would like to thank the reviewer for his/her valuable suggestions and comments which helped to improve the quality and clarity of the paper. This work was supported by The Open University of China General Foundation (no: Q4301E-Y) and Xinjiang Radio & TV University Foundation (no: 2013xjddkt001).
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