Abstract
Queueing networks with blocking have proved useful in modelling of data communications and production lines. We study such a network consisting of a sequence of two service stations with an infinite queue allowed before the first station and no intermediate queue allowed between them. This restriction results in the blocking of the first station whenever a unit having completed its service in that station cannot enter into the second one due to the presence of another unit there. The input of units to the network is the MAP (Markovian Arrival Process). At the first station, service requirements are of phase type whereas service times at the second station are arbitrarily distributed. The focus is on the embedded process at departures. The essential tool in our analysis is the general theory on Markov renewal processes of M/G/1-type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, J. Appl. Probab. 30 (1993) 365-372.
B. Avi-Itzhak and S. Halfin, Servers in tandem with communication and manufacturing blocking, J. Appl. Probab. 30 (1993) 429-437.
B. Avi-Itzhak and M. Yadin, A sequence of two servers with no intermediate queue, Managm. Sci. 11 (1965) 553-564.
S. Balsamo, V. de Nitto Personé and R. Onvural, Analysis of Queueing Networks with Blocking (Kluwer Academic, Norwell, 2001).
R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).
E. Ç inlar, Introduction to Stochastic Processes (Prentice-Hall, Englewood Cliffs, NJ, 1975).
A.N. Dudin and V.I. Klimenok, Multi-dimensional quasi-Toeplitz Markov chains, J. Appl. Math. Stochastic Anal. 12 (1999) 393-415.
F.G. Foster and H.G. Perros, On the blocking process in queue networks, European J. Oper. Res. 5 (1980) 276-283.
H.R. Gail, S.L. Hantler and B.A. Taylor, Spectral analysis of M/G/1 and G/M/1 type Markov chains, Adv. in Appl. Probab. 28 (1996) 114-165.
H.R. Gail, S.L. Hantler and B.A. Taylor, Non-skip-free M/G/1 and G/M/1 type Markov chains, Adv. in Appl Probab. 29 (1997) 733-758.
H.R. Gail, S.L. Hantler and B.A. Taylor, Use of characteristic roots for solving infinite state Markov chains, in: Computational Probability, ed. W.K. Grassmann (Kluwer Academic, Boston, 2000) pp. 205-255.
F.R. Gantmacher, Applications of the Theory of Matrices (Interscience, New York, 1959).
A. Graham, Kronecker Products and Matrix Calculus with Applications (Ellis Horwood, Chichester, 1981).
W.K. Grassmann and D.A. Stanford, Matrix analytic methods, in: Computational Probability, ed. W.K. Grassmann (Kluwer Academic, Boston, 2000) pp. 153-203.
N.G. Hall and C. Sriskandarajah, A survey of machine scheduling problems with blocking and no-wait in process, Oper. Res. 44 (1996) 510-525.
J.G. Kemeny, J. Snell and A.W. Knapp, Denumerable Markov Chains (Van Nostrand, Princeton, 1966).
V.I. Klimenok, Sufficient conditions for existence of three-dimensional quasi-Toeplitz Markov chain stationary distribution, Queues: Flows Systems Networks 13 (1997) 142-145.
V.I. Klimenok, Application of the Rouché theorem for establishing the 2D distribution of the quasi-Toeplitz Markovian chain, Automat. Control Comput. Sci. 32 (1998) 23-29 (in Russian).
C. Langaris, The waiting-time process of a queueing system with gamma-type input and blocking, J. Appl. Probab. 23 (1986) 166-174.
C. Langaris and B. Conolly, Three stage tandem queue with blocking, European J. Oper. Res. 19 (1985) 222-232.
G. Latouche, Algorithms for infinite Markov chains with repeating columns, in: Linear Algebra, Markov Chains, and Queueing Models, eds. C.D. Meyer and R.J. Plemmons (Springer, Berlin, 1993) pp. 231-265.
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling (ASA-SIAM, Philadelphia, PA, 1999).
D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models 7 (1991) 1-46.
D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Adv. in Appl. Probab. 22 (1990) 676-705.
M.F. Neuts, A versatile Markovian point process, J. Appl. Probab. 16 (1979) 764-779.
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).
H.G. Perros, A bibliography of papers on queueing networks with finite capacity queues, Performance Evaluation 10 (1989) 255-260.
H.G. Perros, Queueing Networks with Blocking (Oxford Univ. Press, New York, 1994).
H.G. Perros and T. Altiok, Queueing networks with blocking: a bibliography, Performance Evaluation Rev., ACM, Sigmetrics 12 (1984) 8-12.
N.U. Prabhu, Transient behaviour of a tandem queue, Managm. Sci. 13 (1967) 631-639.
V. Ramaswami, A stable recursion for the steady state vector in Markov chains of M/G/1 type, Stochastic Models 4 (1988) 183-188.
T. Suzuki, On a tandem queue with blocking, J. Oper. Res. Soc. Japan 6 (1964) 137-157.
Y.Q. Zhao, Censoring technique in studying block-structured Markov chains, in: Advances in Algorithmic Methods for Stochastic Models, eds. G. Latouche and P. Taylor, Notable Publications (Chennai, 2000) pp. 417-433.
Y.Q. Zhao, W. Li and W.J. Braun, Censoring, factorizations, and spectral analysis for transition matrices with block-repeating entries (2001) submitted.
Y. Zhu, Tandem queue with group arrivals and no intermediate buffer, Queueing Systems 17 (1994) 403-412.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gómez-Corral, A. A Tandem Queue with Blocking and Markovian Arrival Process. Queueing Systems 41, 343–370 (2002). https://doi.org/10.1023/A:1016235415066
Issue Date:
DOI: https://doi.org/10.1023/A:1016235415066