Queueing Systems

, Volume 91, Issue 3–4, pp 347–365 | Cite as

Discrete-time queue with batch renewal input and random serving capacity rule: \(GI^X/ Geo^Y/1\)

  • F. P. BarbhuiyaEmail author
  • U. C. Gupta


In this paper, we provide a complete analysis of a discrete-time infinite buffer queue in which customers arrive in batches of random size such that the inter-arrival times are arbitrarily distributed. The customers are served in batches by a single server according to the random serving capacity rule, and the service times are geometrically distributed. We model the system via the supplementary variable technique and further use the displacement operator method to solve the non-homogeneous difference equation. The analysis done using these methods results in an explicit expression for the steady-state queue-length distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the underlying characteristic equation. Our approach enables one to estimate the asymptotic distribution at a pre-arrival epoch by a unique largest root of the characteristic equation lying inside the unit circle. With the help of few numerical results, we demonstrate that the methodology developed throughout the work is computationally tractable and is suitable for light-tailed inter-arrival distributions and can also be extended to heavy-tailed inter-arrival distributions. The model considered in this paper generalizes the previous work done in the literature in many ways.


Batch arrival Difference equation Discrete-time Random service capacity Renewal process Supplementary variable 

Mathematics Subject Classification




The author F. P. Barbhuiya is grateful to Indian Institute of Technology Kharagpur, India, for the financial support. The authors would like to thank the anonymous referee for their valuable remarks and suggestions which led to the paper in the current form.


  1. 1.
    Abolnikov, L., Dukhovny, A.: Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications. Int. J. Stoch. Anal. 4(4), 333–355 (1991)Google Scholar
  2. 2.
    Akar, N., Arikan, E.: A numerically efficient method for the \(MAP/D/1/K\) queue via rational approximations. Queueing Syst. 22(1), 97–120 (1996)CrossRefGoogle Scholar
  3. 3.
    Artalejo, J.R., Hernández-Lerma, Onésimo: Performance analysis and optimal control of the \(Geo/Geo/c\) queue. Perform. Eval. 52(1), 15–39 (2003)CrossRefGoogle Scholar
  4. 4.
    Bruneel, H., Kim, B.G.: Discrete-Time Models for Communication Systems Including \({ATM}\). Kluwer Acadmic, Boston (1993)CrossRefGoogle Scholar
  5. 5.
    Cardellini, V., Colajanni, M., Yu, P.S.: Dynamic load balancing on web-server systems. IEEE Internet Comput. 3(3), 28–39 (1999)CrossRefGoogle Scholar
  6. 6.
    Chang, S Ho, Choi, D .W.: Modeling and performance analysis of a finite-buffer queue with batch arrivals, batch services, and setup times: the \(M^X/G^Y/1/K+ B\) queue with setup times. INFORMS J. Comput. 18(2), 218–228 (2006)CrossRefGoogle Scholar
  7. 7.
    Chaudhry, M.L., Gupta, U.C.: Queue-length and waiting-time distributions of discrete-time \(GI^X/Geom/1\) queueing systems with early and late arrivals. Queueing Syst. 25(1–4), 307–324 (1997)CrossRefGoogle Scholar
  8. 8.
    Chaudhry, M.L., Kim, J.J.: Analytically simple and computationally efficient solution to \(GI^X/Geom/1\) queues involving heavy-tailed distributions. Oper. Res. Lett. 44(5), 655–657 (2016)CrossRefGoogle Scholar
  9. 9.
    Chaudhry, M.L., Gupta, U.C., Templeton, James GC: On the relations among the distributions at different epochs for discrete-time \(GI/Geom/1\) queues. Oper. Res. Lett. 18(5), 247–255 (1996)CrossRefGoogle Scholar
  10. 10.
    Claeys, D., Steyaert, B., Walraevens, J., Laevens, K., Bruneel, H.: Analysis of a versatile batch-service queueing model with correlation in the arrival process. Perform. Eval. 70(4), 300–316 (2013)CrossRefGoogle Scholar
  11. 11.
    Claeys, D., Steyaert, B., Walraevens, J., Laevens, K., Bruneel, H.: Tail probabilities of the delay in a batch-service queueing model with batch-size dependent service times and a timer mechanism. Comput. Oper. Res. 40(5), 1497–1505 (2013)CrossRefGoogle Scholar
  12. 12.
    Cordeau, J.L., Chaudhry, M.L.: A simple and complete solution to the stationary queue-length probabilities of a bulk-arrival bulk-service queue. Infor. 47, 283–288 (2009)Google Scholar
  13. 13.
    Economou, A., Fakinos, D.: On the stationary distribution of the \(GI^X/M^Y/1\) queueing system. Stoch. Anal. Appl. 21, 559–565 (2003)CrossRefGoogle Scholar
  14. 14.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. 1. Wiley, New York (1968)Google Scholar
  15. 15.
    Gravey, A., Hebuterne, G.: Simultaneity in discrete-time single server queues with bernoulli inputs. Perform. Eval. 14(2), 123–131 (1992)CrossRefGoogle Scholar
  16. 16.
    Harris, C.M., Brill, P.H., Fischer, M.J.: Internet-type queues with power-tailed interarrival times and computational methods for their analysis. INFORMS J. Comput. 12(4), 261–271 (2000)CrossRefGoogle Scholar
  17. 17.
    Hochbaum, D.S., Landy, D.: Scheduling semiconductor burn-in operations to minimize total flowtime. Oper. Res. 45(6), 874–885 (1997)CrossRefGoogle Scholar
  18. 18.
    Hunter, J .J.: Mathematical Techniques of Applied Probability: Discrete Time Models, Techniques and Applications. Academic Press, Cambridge (1983)Google Scholar
  19. 19.
    Kim, B., Choi, B.D.: Asymptotic analysis and simple approximation of the loss probability of the \(GI^X/M/c/K\) queue. Perform. Eval. 54(4), 331–356 (2003)CrossRefGoogle Scholar
  20. 20.
    Li, L., Li, S., Zhao, S., Indus: QoS-aware scheduling of services-oriented internet of things. IEEE Trans. Ind. Inform. 10(2), 1497–1505 (2014)CrossRefGoogle Scholar
  21. 21.
    Pacheco, A., Samanta, S.K., Chaudhry, M.L.: A short note on the \(GI/Geo/1\) queueing system. Stat. Probab. Lett. 82(2), 268–273 (2012)CrossRefGoogle Scholar
  22. 22.
    Singh, G., Gupta, U.C., Chaudhry, M.L.: Analysis of queueing-time distributions for \(MAP/D_N/1\) queue. Int. J. Comput. Math. 91, 1911–1930 (2014)CrossRefGoogle Scholar
  23. 23.
    Takagi, H.: Queuing Analysis: A Foundation of Performance Evaluation. Discrete Time Systems, vol. 3. North-Holland, Amsterdam (1993)Google Scholar
  24. 24.
    Vinck, B., Bruneel, H.: Analyzing the discrete-time \(G^{(G)}/Geo/1\) queue using complex contour integration. Queueing Syst. 18(1–2), 47–67 (1994)CrossRefGoogle Scholar
  25. 25.
    Woodward, M.E.: Communication and Computer Networks: Modelling with Discrete-Time Queues. Wiley-IEEE Computer Society Pr, Hoboken (1994)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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