Abstract
In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.
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S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, J. Appl. Probab. 30 (1993) 365–372.
H. Chen and H. Zhang, Stabiliy of multiclass queueing networks under priority service disciplines, Oper. Res. 48 (2000) 26–37.
J.W. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1982).
J. Dai and S. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Trans. Automat. Control 40 (1995) 1889–1904.
G. Fayolle, V.A Malyshev and M.V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains (Cambridge Univ. Press, Cambridge, 1995).
F.R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959).
K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory (Cambridge Univ. Press, Cambridge, 1990).
Q.-M. He, Classification ofMarkov processes of M / G / 1 type with a tree structure and its applications to queueing systems, Oper. Res. Lett. 26 (2000) 67–80.
Q.-M. He, Classification of Markov processes of matrix M / G / 1 type with a tree structure and its applications to the MMAP[K]/PH[K]/1 queue, Stochastic Models 16 (2000) 407–433.
Q.-M. He, A fixed point approach to the classification ofMarkov chains with a tree stucture, Stochastic Models 19(1) (2003) 76–114.
Q.-M. He and A.S. Alfa, The discrete timeMMAP[K]/PH[K]/1/LCFS-GPR queues and its variants, in: Advances in Algorithmic Methods for Stochastic Models – Proc. of the 3rd Internat. Conf. on Matrix Analytic Methods, eds. G. Latouche and P.G. Taylor (Notable Publications, New Jersey, 2000) pp. 167–190.
Q.-M. He and M.F. Neuts, Markov arrival processes with marked transitions, Stochastic Process. Appl. 74(1) (1998) 37–52.
P.R. Kumar and S.P. Meyn, Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies, IEEE Trans. Automat. Control 41(1) (1996) 4–17.
R. Loynes, The stability of a queue with non-independent inter-arrival and service times, Proc. Cambridge Philos. Soc. 58 (1962) 497–520.
S.P. Meyn and R. Tweedie, Markov Chains and Stochastic Stability (Springer, Berlin, 1993).
M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (Johns Hopkins Univ. Press, Baltimore, MD, 1981).
M.F. Neuts, Structured Stochastic Matrices of M / G / 1 Type and Their Applications (Marcel Dekker, New York, 1989).
T. Takine, B. Sengupta and R.W. Yeung, A generalization of the matrix M/ G / 1 paradigm for Markov chains with a tree structure, Stochastic Models 11 (1995) 411–421.
B. Van Houdt and C. Blondia, Stability and performance of stack algorithms for random access communication modeled as a tree structured QBD Markov chain, Stochastic Models 17(3) (2001) 1–28.
R.W. Yeung and B. Sengupta, Matrix product-form solutions for Markov chains with a tree structure, Adv. in Appl. Probab. 26(4) (1994) 965–987.
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He, QM., Li, H. Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue. Queueing Systems 44, 137–160 (2003). https://doi.org/10.1023/A:1024420505098
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DOI: https://doi.org/10.1023/A:1024420505098