Queueing Systems

, 60:111 | Cite as

Analysis of a retrial queue with two-phase service and server vacations



A queueing system with a single server providing two stages of service in succession is considered. Every customer receives service in the first stage and in the sequel he decides whether to proceed to the second phase of service or to depart and join a retrial box from where he repeats the demand for a special second stage service after a random amount of time and independently of the other customers in the retrial box. When the server becomes idle, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service times are arbitrarily distributed. For such a system the stability conditions and the system state probabilities are investigated both in a transient and in a steady state. A stochastic decomposition result is also presented. Numerical results are finally obtained and used to investigate system performance.


Poisson arrivals Two-phase service Retrial queue General services Single vacation 

Mathematics Subject Classification (2000)

60K25 90B22 


  1. 1.
    Artalejo, J.R.: A classified bibliography of research on retrial queues: Progress in 1990–1999. Top 7(2), 187–211 (1999) CrossRefGoogle Scholar
  2. 2.
    Choi, D.I., Kim, T.: Analysis of a two-phase queueing system with vacations and Bernoulli feedback. Stoch. Anal. Appl. 21(5), 1009–1019 (2003) CrossRefGoogle Scholar
  3. 3.
    Choudhury, G.: Steady state analysis of a M/G/1 queue with linear retrial policy and two-phase service under Bernoulli vacation schedule. Appl. Math. Model. (2007). doi: 10.1016/j.apm.2007.09.020 Google Scholar
  4. 4.
    Choudhury, G., Madan, K.C.: A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Math. Comput. Model. 42, 71–85 (2005) CrossRefGoogle Scholar
  5. 5.
    Cinlar, E.: Introduction to Stochastic Processes. Prentice Hall, New York (1975) Google Scholar
  6. 6.
    Doshi, B.T.: Analysis of a two-phase queueing system with general service times. Oper. Res. Lett. 10, 265–272 (1991) CrossRefGoogle Scholar
  7. 7.
    Falin, G.I., Fricker, C.: On the virtual waiting time in an M/G/1 retrial queue. J. Appl. Probab. 28, 446–460 (1991) CrossRefGoogle Scholar
  8. 8.
    Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman and Hall, London (1997) Google Scholar
  9. 9.
    Falin, G.I., Artalejo, J.R., Martin, M.: On the single server retrial queue with priority customers. Queueing Syst. 14, 439–455 (1993) CrossRefGoogle Scholar
  10. 10.
    Katayama, T., Kobayashi, K.: Sojourn time analysis of a queueing system with two-phase service and server vacations. Nav. Res. Logist. 54(1), 59–65 (2006) CrossRefGoogle Scholar
  11. 11.
    Krishna, C.M., Lee, Y.H.: A study of a two-phase service. Oper. Res. Lett. 9, 91–97 (1990) CrossRefGoogle Scholar
  12. 12.
    Kulkarni, V.G., Liang, H.M.: Retrial queues revisited. In: Dshalalow, J.H. (ed.) Frontiers in Queueing, pp. 19–34. CRC Press, Boca Raton (1997) Google Scholar
  13. 13.
    Krishna Kumar, B., Vijayakumar, A., Arivudainambi, D.: An M/G/1 retrial queueing system with two-phase service and preemptive resume. Ann. Oper. Res. 113, 61–79 (2002) CrossRefGoogle Scholar
  14. 14.
    Krishna Kumar, B., Arivudainambi, D., Vijayakumar, A.: On the M (x)/G/1 retrial queue with Bernoulli schedule and general retrial times. Asia-Pac. J. Oper. Res. 19, 117–194 (2002) Google Scholar
  15. 15.
    Langaris, C., Katsaros, A.: Time dependent analysis of a queue with batch arrivals and N levels of non-preemptive priority. Queueing Syst. 19, 269–288 (1995) CrossRefGoogle Scholar
  16. 16.
    Madan, K.C.: On a single server queue with two-stage heterogeneous service and deterministic server vacations. Int. J. Syst. Sci. 32(7), 837–844 (2001) CrossRefGoogle Scholar
  17. 17.
    Moutzoukis, E., Langaris, C.: Two queues in tandem with retrial customers. Probab. Eng. Inf. Sci. 15, 311–325 (2001) CrossRefGoogle Scholar
  18. 18.
    Pakes, A.G.: Some conditions of ergodicity and recurrence of Markov chains. Oper. Res. 17, 1058–1061 (1969) CrossRefGoogle Scholar
  19. 19.
    Takacs, L.: Introduction to the Theory of Queues. Oxford Univ. Press, New York (1962) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IoanninaIoanninaGreece

Personalised recommendations