All Title Author
Keywords Abstract


Semigroup Method on a /G/1 Queueing Model

DOI: 10.1155/2013/893254

Full-Text   Cite this paper   Add to My Lib

Abstract:

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. 1. Introduction The queueing system when the server become idle is not new. Miller [1] 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 [2] 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 [6], Lee and Srinivasan [7], Lee et al. [8, 9], Borthakur and Choudhury [10], 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 [13] and Medhi [14]. In 2002, Choudhury [15] studied the /G/1 queueing model with vacation times. By using the supplementary variable technique [16] 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 [15], 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

References

[1]  L. W. Miller, Alternating priorities in multi-class queue [Ph.D. thesis], Cornel University, Ithaca, New York, USA, 1964.
[2]  Y. Levy and U. Yechiali, “Utilization of idle time in an M/G/1 queueing system,” Management Science, vol. 22, no. 2, pp. 202–211, 1975.
[3]  B. T. Doshi, “Queueing systems with vacations—a survey,” Queueing Systems, vol. 1, no. 1, pp. 29–66, 1986.
[4]  B. Doshi, “Generalizations of the stochastic decomposition results for single server queues with vacations,” Stochastic Models, vol. 6, no. 2, pp. 307–333, 1990.
[5]  H. Takagi, Vacation and Priority Systems, Part 1, vol. 1 of Queueing Analysis : A Foundation of Performance Evaluation, North-Holland, Amsterdam, The Netherland, 1991.
[6]  Y. Baba, “On the /G/1 queue with vacation time,” Operations Research Letters, vol. 5, no. 2, pp. 93–98, 1986.
[7]  H.-S. Lee and M. M. Srinivasan, “Control policies for the /G/1 queueing system,” Management Science, vol. 35, no. 6, pp. 708–721, 1989.
[8]  H. W. Lee, S. S. Lee, J. O. Park, and K. C. Chae, “Analysis of the /G/1 queue with -policy and multiple vacations,” Journal of Applied Probability, vol. 31, no. 2, pp. 476–496, 1994.
[9]  S. S. Lee, H. W. Lee, S. H. Yoon, and K. C. Chae, “Batch arrival queue with N-policy and single vacation,” Computers and Operations Research, vol. 22, no. 2, pp. 173–189, 1995.
[10]  A. Borthakur and G. Choudhury, “On a batch arrival Poisson queue with generalized vacation,” Sankhyā B, vol. 59, no. 3, pp. 369–383, 1997.
[11]  G. Choudhury, “On a batch arrival Poisson queue with a random setup time and vacation period,” Computers & Operations Research, vol. 25, no. 12, pp. 1013–1026, 1998.
[12]  G. Choudhury, “An /G/1 queueing system with a setup period and a vacation period,” Queueing Systems, vol. 36, no. 1–3, pp. 23–38, 2000.
[13]  K. C. Chae and H. W. Lee, “ /G/1 vacation models with N-policy: heuristic interpretation of the mean waiting time,” Journal of the Operational Research Society, vol. 46, no. 2, pp. 258–264, 1995.
[14]  J. Medhi, “Single server queueing system with Poisson input: a review of some recent developments,” in Advances in Combinatorical Method and Applications in Probability and Statistics, N. Balakrishnan, Ed., pp. 317–338, Birkh?user, Boston, Mass, USA, 1997.
[15]  G. Choudhury, “Analysis of the /G/1 queueing system with vacation times,” Sankhyā B, vol. 64, no. 1, pp. 37–49, 2002.
[16]  D. R. Cox, “The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables,” vol. 51, pp. 433–441, 1955.
[17]  R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975.
[18]  G. Gupur, X. Z. Li, and G. T. Zhu, Functional Analysis Method in Queueing Theory, Research Information, Herdfortshire, UK, 2001.
[19]  H. O. Fattorini, The Cauchy Problem, vol. 18 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Mass, USA, 1983.

Full-Text

comments powered by Disqus

Contact Us

service@oalib.com

QQ:3279437679

微信:OALib Journal