Queueing Systems

, Volume 82, Issue 3–4, pp 353–380 | Cite as

Analysis of generalized QBD queues with matrix-geometrically distributed batch arrivals and services



In a quasi-birth–death (QBD) queue, the level forward and level backward transitions of a QBD-type Markov chain are interpreted as customer arrivals and services. In the generalized QBD queue considered in this paper, arrivals and services can occur in matrix-geometrically distributed batches. This paper presents the queue length and sojourn time analysis of generalized QBD queues. It is shown that, if the number of phases is N, the number of customers in the system is order-N matrix-geometrically distributed, and the sojourn time is order-\(N^2\) matrix-exponentially distributed, just like in the case of classical QBD queues without batches. Furthermore, phase-type representations are provided for both distributions. In the special case of the arrival and service processes being independent, further simplifications make it possible to obtain a more compact, order-N representation for the sojourn time distribution.


Matrix-analytic methods Age process Batch arrival Batch service Queue length analysis Sojourn time analysis 

Mathematics Subject Classification

60K25 68M20 


  1. 1.
    Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems, vol. 6. SIAM, Philadelphia (2005)CrossRefGoogle Scholar
  2. 2.
    Bini, D., Meini, B., Steffé, S., Van Houdt, B.: Structured Markov chains solver: software tools. In: Proceeding from the 2006 Workshop on Tools for Solving Structured Markov Chains, ACM, p. 14 (2006)Google Scholar
  3. 3.
    Chakka, R., Harrison, P.: The MMCPP/GE/c queue. Queueing Syst. 38(3), 307–326 (2001)CrossRefGoogle Scholar
  4. 4.
    Éltető, T., Telek, M.: Numerical analysis of M/G/1 type queueing systems with phase type transition structure. J. Comput. Appl. Math. 212(2), 331–340 (2008)CrossRefGoogle Scholar
  5. 5.
    Gail, H., Hantler, S., Taylor, B.: Non-skip-free M/G/1 and G/M/1 type Markov chains. Adv. Appl. Probab. 29, 733–758 (1997)CrossRefGoogle Scholar
  6. 6.
    Golub, G.H., Nash, S., Van Loan, C.: A Hessenberg–Schur method for the problem AX + XB = C. IEEE Trans. Autom. Control 24(6), 909–913 (1979)CrossRefGoogle Scholar
  7. 7.
    Harrison, P.G., Zatschler, H.: Sojourn time distributions in modulated G-queues with batch processing. In: Proceedings of First International Conference on the Quantitative Evaluation of Systems, 2004, pp. 90–99 (2004)Google Scholar
  8. 8.
    He, Q.: Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: age process, sojourn times, and queue lengths. J. Syst. Sci. Complex. 25(1), 133–155 (2012)CrossRefGoogle Scholar
  9. 9.
    Horváth, G., Van Houdt, B., Telek, M.: Commuting matrices in the queue length and sojourn time analysis of MAP/MAP/1 queues. Stoch. Models 30(4), 554–575 (2014)CrossRefGoogle Scholar
  10. 10.
    Jafari, R., Sohraby, K.: Combined M/G/1-G/M/1 type structured chains: a simple algorithmic solution and applications. In: INFOCOM 2001. Proceedings of Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies, vol 2, pp 1065–1074 (2001)Google Scholar
  11. 11.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  12. 12.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Corporation, Baltimore (1981)Google Scholar
  13. 13.
    Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Applications. Dekker, New York (1989)Google Scholar
  14. 14.
    Ozawa, T.: Sojourn time distributions in the queue defined by a general QBD process. Queueing Syst. 53(4), 203–211 (2006)CrossRefGoogle Scholar
  15. 15.
    Sengupta, B.: Phase-type representations for matrix-geometric solutions. Stoch. Models 6(1), 163–167 (1990a)CrossRefGoogle Scholar
  16. 16.
    Sengupta, B.: The semi-Markovian queue: theory and applications. Stoch. Models 6(3), 383–413 (1990b)CrossRefGoogle Scholar
  17. 17.
    Steeb, W.H.: Matrix Calculus and Kronecker Product with Applications and C++ Programs. World Scientific, Singapore (1997)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.MTA-BME Information Systems Research GroupBudapestHungary
  2. 2.Department of Networked Systems and ServicesBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations