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Zero-automatic queues and product form

Part of: Computer system organization Special processes

Published online by Cambridge University Press:  01 July 2016

Thu-Ha Dao-Thi  and
Jean Mairesse
Thu-Ha Dao-Thi*
Affiliation:
LIAFA
Jean Mairesse*
Affiliation:
LIAFA
*
∗ Postal address: LIAFA, CNRS-Université Paris 7, case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France.
∗ Postal address: LIAFA, CNRS-Université Paris 7, case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France.
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Abstract

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We introduce and study a new model: zero-automatic queues. Roughly, zero-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable zero-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of zero-automatic queues, we recover the simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers.

Keywords

Queueing theoryM/M/1 queueG-queuequasi-reversibilityproduct formquasi-birth-and-death process

MSC classification

Primary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 429 - 461
Copyright
Copyright © Applied Probability Trust 2007 

References

Aliprantis, C. and Border, K. (1999). Infinite Dimensional Analysis: A Hitchhiker's Guide, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Baccelli, F. and Foss, S. (1995). On the saturation rule for the stability of queues. J. Appl. Prob. 32, 494–507.CrossRefGoogle Scholar
Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues (Texts Appl. Math. 31). Springer, New York.Google Scholar
Chao, X. and Miyazawa, M. (2000). Queueing networks with instantaneous movements: a unified approach by quasi-reversibility. Adv. Appl. Prob. 32, 284–313.Google Scholar
Chao, X., Miyazawa, M. and Pinedo, M. (1999). Queueing Networks. Customers, Signals, and Product Form Solutions. John Wiley, New York.Google Scholar
Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
Dao-Thi, T.-H. and Mairesse, J. (2005). Zero-automatic queues. In Formal Techniques for Computer Systems and Business Processes LNCS 3670, Springer, Berlin, pp. 64–78.Google Scholar
Dao-Thi, T.-H. and Mairesse, J. (2006). Zero-automatic networks. In Proc. VALUETOOLS (Pisa, Italy), ACM, New York.Google Scholar
Dayar, T. and Quessette, F. (2002). Quasi-birth-and-death processes with level-geometric distribution. SIAM J. Matrix Anal. Appl. 24, 281–291.Google Scholar
Dynkin, E. and Malyutov, M. (1961). Random walk on groups with a finite number of generators. Soviet Math. Dokl. 2, 399–402.Google Scholar
Epstein, D. et al. (1992). Word Processing in Groups. Jones and Bartlett, Boston, MA.Google Scholar
Fourneau, J.-M., Gelenbe, E. and Suros, R. (1996). G-networks with multiple classes of negative and positive customers. Theoret. Comput. Sci. 155, 141–156.Google Scholar
Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28, 656–663.Google Scholar
Gelenbe, E. and Pujolle, G. (1998). Introduction to Queueing Networks, 2nd edn. John Wiley, Chichester.Google Scholar
Guivarc'h, Y. (1980). Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. Astérisque 74, 47–98.Google Scholar
Kelly, F. (1979). Reversibility and Stochastic Networks. John Wiley, New York.Google Scholar
Kingman, J. (1973). Subadditive ergodic theory. Ann. Prob. 1, 883–909.Google Scholar
He, Q.-M. (2003). The classification of matrix GI/M/1-type Markov chains with a tree structure and its applications to queueing. J. Appl. Prob. 40, 1087–1102.Google Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia, PA.Google Scholar
Ledrappier, F. (2001). Some asymptotic properties of random walks on free groups. In Topics in Probability and Lie Groups: Boundary Theory (CRM Proc. Lecture Notes 28), American Mathematical Society, Providence, RI, pp. 117–152.Google Scholar
Mairesse, J. (2005). Random walks on groups and monoids with a Markovian harmonic measure. Electron. J. Prob. 10, 1417–1441.Google Scholar
Mairesse, J. and Mathéus, F. (2007). Random walks on free products of cyclic groups. J. London Math. Soc. 75, 47–66.Google Scholar
Neuts, M. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Sawyer, S. and Steger, T. (1987). The rate of escape for anisotropic random walks in a tree. Prob. Theory Relat. Fields 76, 207–230.Google Scholar
Serfozo, R. (1999). Introduction to Stochastic Networks. Springer, Berlin.CrossRefGoogle Scholar
Stallings, J. (1966). A remark about the description of free products of groups. Proc. Camb. Philos. Soc. 62, 129–134.Google Scholar
Yeung, R. and Sengupta, B. (1994). Matrix product-form solutions for Markov chains with a tree structure. Adv. Appl. Prob. 26, 965–987.Google Scholar