Skip to main content
SpringerLink
Log in

Marked Markovian Arrivals in a Tandem G-Network with Blocking

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

Queueing networks with blocking have broad applications in computer modelling and manufacturing. The present paper focusses on the MMAP[2]/M/1/ ∞ →·/M/1/K + 1 G-queue with blocking. This network consists of a sequence of two single-server stations with an infinite queue allowed before the first server and an intermediate queue of finite capacity K ≥ 0 allowed between servers. This restriction results in the blocking of the first server whenever a unit having completed its service in Station 1 cannot enter into Station 2 due to K + 1 units are accommodated into Station 2. There are two types of arrivals, called units and signals, which are modelled by a single Markovian arrival process with marked transitions. Each unit is served at Stations 1 and 2 in that order, and then it exits of the network. A signal induces the last unit in queue or in service, if there is one, to leave the network instantly, and it has no effect otherwise. Our purpose is to study the influence of the dependence between units and signals on the performance evaluation of the continuous-time Markov chain describing the state of the network at arbitrary times, which constitutes a quasi-birth-and-death process. We present tractable formulas for a variety of probabilistic descriptors, with special emphasis on the distribution of inter-departure times.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abate J, Whitt W (1995) Numerical inversion of Laplace transforms of probability distributions. ORSA J Comput 7:36–43

    MATH  Google Scholar 

  • Akar N, Sohraby K (1997) Finite and infinite QBD chains: a simple and unifying algorithmic approach. In: Proceedings IEEE Infocom’97. IEEE Computer Society, Silver Spring, pp 1105–1113

    Google Scholar 

  • Albores F, Bocharov P, Ilhuicatzi R, Lyubin D (2003) Analysis of a two-node exponential queueing network with blocking. In: Queues, flows, systems, networks. Proceedings of the international conference ‘modern mathematical methods of analysis and optimization of telecommunication networks’, Gomel, 23–25 September 2003, pp 11–17

  • Artalejo JR (2000) G-networks: a versatile approach for work removal in queueing networks. Eur J Oper Res 126:233–249

    Article  MATH  MathSciNet  Google Scholar 

  • Asmussen S, Koole G (1993) Marked point processes as limits of Markovian arrival streams. J Appl Probab 30:365–372

    Article  MATH  MathSciNet  Google Scholar 

  • Balsamo S (1994) Properties and analysis of queueing network models with finite capacities. In: Donatiello L, Nelson R (eds) Performance evaluation of computer and communication systems. Lecture notes in computer science, vol 729. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Balsamo S, de Nitto Personé V, Onvural R (2001) Analysis of queueing networks with blocking. Kluwer, Boston

    MATH  Google Scholar 

  • Balsamo S, de Nitto Personé V, Inverardi P (2003) A review on queueing network models with finite capacity queues for software architectures performance prediction. Perform Eval 51:269–288

    Article  Google Scholar 

  • Bini DA, Meini B (1995) On cyclic reduction applied to a class of Toeplitz-like matrices arising in queueing models. In: Stewart WJ (ed) Computations with Markov chains. Kluwer, Boston, pp 21–38

    Google Scholar 

  • Bini DA, Latouche G, Meini B (2005) Numerical methods for structured Markov chains. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  • Bocharov PP, Vishnevskii VM (2003) G-networks: development of the theory of multiplicative networks. Autom Remote Control 64:714–739

    Article  MATH  MathSciNet  Google Scholar 

  • Bocharov PP, Viskova EV, Lyubin DYu (2005) The two-stage queueing system with negative arrivals in discrete time. In: Ermakov SM, Melas VB, Pepelyshev AN (eds) Proceedings of the fifth St. Petersburg workshop on simulation. NII Chemistry Saint Petersburg University Publishers, St. Petersburg, pp 155–161

    Google Scholar 

  • Disney RL, Kiessler PC (1987) Traffic processes in queueing networks: a Markov renewal approach. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Gelenbe E (1989) Random neural networks with negative and positive signals and product form solution. Neural Comput 1:502–510

    Article  Google Scholar 

  • Gelenbe E (1991) Product-form queueing networks with negative and positive customers. J Appl Probab 28:656–663

    Article  MATH  MathSciNet  Google Scholar 

  • Gelenbe E (1994) G-networks: a unifying model for neural and queueing networks. Ann Oper Res 48:433–461

    Article  MATH  MathSciNet  Google Scholar 

  • Golub GH, Van Loan CF (1996) Matrix computations. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Gómez-Corral A (2002a) On a tandem G-network with blocking. Adv Appl Probab 34:626–661

    Article  MATH  Google Scholar 

  • Gómez-Corral A (2002b) A tandem queue with blocking and Markovian arrival process. Queueing Syst 41:343–370

    Article  MATH  MathSciNet  Google Scholar 

  • Gómez-Corral A, Martos ME (2006) Performance of two-stage tandem queues with blocking: the impact of several flows of signals. Perform Eval 63:910–938

    Article  Google Scholar 

  • Grassmann WK, Drekic S (2000) An analytical solution for a tandem queue with blocking. Queueing Syst 36:221–235

    Article  MATH  MathSciNet  Google Scholar 

  • Green D (2000) Lag correlations of approximating departure processes for MAP/PH/1 queues. In: Latouche G, Taylor PG (eds) Advances in algorithmic methods for stochastic models. Notable Publications, Neshanic Station, pp 135–151

    Google Scholar 

  • Hall NG, Sriskandarajah C (1996) A survey of machine scheduling problems with blocking and no-wait in process. Oper Res 44:510–525

    Article  MATH  MathSciNet  Google Scholar 

  • He QM (1996) Queues with marked customers. Adv Appl Probab 28:567–587

    Article  MATH  Google Scholar 

  • He QM (2000) Quasi-birth-and-death Markov processes with a tree structure and the MMAP[K]/PH[K]/N/LCFS non-preemptive queue. Eur J Oper Res 120:641–656

    Article  MATH  Google Scholar 

  • He QM (2001) The versatility of MMAP[K] and the MMAP[K]/G[K]/1 queue. Queueing Syst 38:397–418

    Article  MATH  MathSciNet  Google Scholar 

  • He QM (2004) Workload process, waiting times, and sojourn times in a discrete time MMAP[K]/SM[K]/1/FCFS queue. Stoch Models 20:415–437

    Article  MATH  MathSciNet  Google Scholar 

  • He QM, Li H (2003) Stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. Queueing Syst 44:137–160

    Article  MATH  MathSciNet  Google Scholar 

  • He QM, Neuts MF (1998) Markov chains with marked transitions. Stoch Process Their Appl 74:37–52

    Article  MATH  MathSciNet  Google Scholar 

  • Heindl A (2001) Decomposition of general tandem queueing networks with MMPP input. Perform Eval 44:5–23

    Article  MATH  Google Scholar 

  • Heindl A (2003) Decomposition of general queueing networks with MMPP inputs and customer losses. Perform Eval 51:117–136

    Article  Google Scholar 

  • Hunter JJ (1983) Mathematical techniques of applied probability, vol 1. Discrete time models: basic theory. Academic, New York

    Google Scholar 

  • Klimenok VI, Breuer L, Tsarenkov G, Dudin AN (2005) The \(BMAP/G/1/\bar{N}\to\cdot /PH/1/M\) tandem queue with losses. Perform Eval 61:17–40

    Article  Google Scholar 

  • Konheim AG, Reiser M (1976) A queueing model with finite waiting room and blocking. J Assoc Comput Mach 23:328–341

    MATH  MathSciNet  Google Scholar 

  • Konheim AG, Reiser M (1978) Finite capacity queuing systems with applications in computer modeling. SIAM J Comput 7:210–229

    Article  MATH  MathSciNet  Google Scholar 

  • Latouche G, Neuts MF (1980) Efficient algorithmic solutions to exponential tandem queues with blocking. SIAM J Algebr Discrete Methods 1:93–106

    Article  MATH  MathSciNet  Google Scholar 

  • Latouche G, Ramaswami V (1993) A logarithmic reduction algorithm for quasi birth and death processes. J Appl Probab 30:650–674

    Article  MATH  MathSciNet  Google Scholar 

  • Latouche G, Ramaswami V (1999) Introduction to matrix analytic methods in stochastic modeling. ASA-SIAM, Philadelphia

    MATH  Google Scholar 

  • Lucantoni DM, Meier-Hellstern KS, Neuts MF (1990) A single-server queue with server vacations and a class of non-renewal arrival processes. Adv Appl Probab 22:676–705

    Article  MATH  MathSciNet  Google Scholar 

  • Naoumov V, Krieger UR, Wagner D (1996) Analysis of a multiserver delay-loss system with a general Markovian arrival process. In: Chakravarthy SR, Alfa AS (eds) Matrix-analytic methods in stochastic models. Lecture notes in pure and applied mathematics, vol 183. Marcel Dekker, New York, pp 43–66

    Google Scholar 

  • Neuts MF (1994) Matrix-geometric solutions in stochastic models: an algorithmic approach, 2nd edn. Dover, New York

    Google Scholar 

  • Neuts MF, Li JM (2000) The input/output process of a queue. Appl Stoch Models Bus Ind 16:11–21

    Article  MATH  MathSciNet  Google Scholar 

  • Onvural RO (1990) Survey of closed queueing networks with blocking. ACM Comput Surv 22:83–121

    Article  Google Scholar 

  • Ost A (2001) Performance of communication systems. A model-based approach with matrix-geometric methods. Springer, Berlin

    Google Scholar 

  • Park D, Perros HG, Yamashita H (1994) Approximate analysis of discrete-time tandem queueing networks with bursty and correlated input traffic and customer loss. Oper Res Lett 15:95–104

    Article  MATH  MathSciNet  Google Scholar 

  • Perros HG (1989a) A bibliography of papers on queueing networks with finite capacity queues. Perform Eval 10:255–260

    Article  MathSciNet  Google Scholar 

  • Perros HG (1989b) Open queueing networks with blocking. In: Takagi H (ed) Stochastic analysis of computer and communication systems. North-Holland, Amsterdam

    Google Scholar 

  • Perros HG (1994) Queueing networks with blocking. Exact and approximate solutions. Oxford University Press, New York

    Google Scholar 

  • Shin YW, Choi BD (2003) A queue with positive and negative arrivals governed by a Markov chain. Probab Eng Inf Sci 17:487–501

    Article  MATH  MathSciNet  Google Scholar 

  • Shin YW (2004) Multi-server retrial queue with negative customers and disasters. In: Choi BD (ed) Proceedings of the fifth international workshop on retrial queues. TMRC - Korea University, Seoul, pp 53–60

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, Faculty of Mathematics, Complutense University of Madrid, 28040, Madrid, Spain

    A. Gómez-Corral & M. E. Martos

Authors
  1. A. Gómez-Corral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. E. Martos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Gómez-Corral.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gómez-Corral, A., Martos, M.E. Marked Markovian Arrivals in a Tandem G-Network with Blocking. Methodol Comput Appl Probab 11, 621–649 (2009). https://doi.org/10.1007/s11009-008-9080-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-008-9080-8

Keywords

AMS 2000 Subject Classification

Search

Navigation