On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation


This article considers a single server, infinite buffer, bulk service Poisson queue with single and multiple vacation. The customers are served in batches following ‘general bulk service’ (GBS) rule. The customers are arriving according to the Poisson process, and the service time of the batches follows an exponential distribution. Using bivariate probability generating function (PGF) method the steady-state joint distributions of the queue content and server content (when server is busy), and joint distribution of the queue content and type of the vacation taken by the server (when server is in vacation) have been obtained. Here by the ‘type of the vacation’ we mean the queue length at vacation initiation epoch. The information about these joint distributions may help in increasing the system performance. Finally, several numerical examples are carried out using MAPLE software to verify the analytical results.

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  1. 1.

    Altman, E., Nain, P.: Optimality of a threshold policy in the ${M}/{M}/1$ queue with repeated vacations. Math. Methods Oper. Res. 44(1), 75–96 (1996)

    Google Scholar 

  2. 2.

    Baba, Y.: Analysis of a ${GI}/{M}/1$ queue with multiple working vacations. Oper. Res. Lett. 33(2), 201–209 (2005)

    Google Scholar 

  3. 3.

    Banerjee, A., Gupta, U.C.: Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service. Perform. Eval. 69(1), 53–70 (2012)

    Google Scholar 

  4. 4.

    Banerjee, A., Gupta, U.C., Chakravarthy, S.R.: Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service. Comput. Oper. Res. 60, 138–149 (2015)

    Google Scholar 

  5. 5.

    Barbhuiya, F.P., Gupta, U.C.: A discrete-time ${GI}^{X}/{Geo}/1$ queue with multiple working vacations under late and early arrival system. Methodol. Comput. Appl. Probab. 1–26 (2019)

  6. 6.

    Chang, S.H., Choi, D.W.: Performance analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations. Comput. Oper. Res. 32(9), 2213–2234 (2005)

    Google Scholar 

  7. 7.

    Chaudhry, M.L., Templeton, J.G.C.: A First Course in Bulk Queues. A Wiley-interscience Publication. Wiley, New York (1983)

    Google Scholar 

  8. 8.

    Choi, B.D., Han, D.H.: ${G}/{M}^{(a, b)}/1$ queues with server vacations. J. Oper. Res. Soc. Jpn. 37(3), 171–181 (1994)

    Google Scholar 

  9. 9.

    Cong, T.D.: Application of the method of collective marks to some ${M}/{G}/1$ vacation models with exhaustive service. Queueing Syst. 16(1–2), 67–81 (1994)

    Google Scholar 

  10. 10.

    Doshi, B.T.: Queueing systems with vacationsa survey. Queueing Syst. 1(1), 29–66 (1986)

    Google Scholar 

  11. 11.

    Frey, A., Takahashi, Y.: An explicit solution for an ${M}/{GI}/1/{N}$ queue with vacation time and exhaustive service discipline. J. Oper. Res. Soc. Jpn. 41(3), 430–441 (1998)

    Google Scholar 

  12. 12.

    Gupta, G.K., Banerjee, A., Gupta, U.C.: On finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation. Qual. Technol. Quant. Manag. 1–27 (2019)

  13. 13.

    Gupta, U.C., Banik, A.D., Pathak, S.S.: Complete analysis of ${M}{A}{P}/{G}/{1}/{N}$ queue with single (multiple) vacation (s) under limited service discipline. Int. J. Stoch. Anal. 2005(3), 353–373 (2005)

    Google Scholar 

  14. 14.

    Gupta, U.C., Pradhan, S.: Queue length and server content distribution in an infinite-buffer batch-service queue with batch-size-dependent service. Adv. Oper. Res. 2015 (2015)

  15. 15.

    Gupta, U.C., Samanta, S.K., Sharma, R.K., Chaudhry, M.L.: Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations. Perform. Eval. 64(1), 1–19 (2007)

    Google Scholar 

  16. 16.

    Gupta, U.C., Sikdar, K.: The finite-buffer ${M}/{G}/1$ queue with general bulk-service rule and single vacation. Perform. Eval. 57(2), 199–219 (2004)

    Google Scholar 

  17. 17.

    Gupta, U.C., Sikdar, K.: Computing queue length distributions in ${MAP}/{G}/1/{N}$ queue under single and multiple vacation. Appl. Math. Comput. 174(2), 1498–1525 (2006)

    Google Scholar 

  18. 18.

    Haridass, M., Arumuganathan, R.: Analysis of a ${M}^{X}/{G (a, b)}/1$ queueing system with vacation interruption. RAIRO Oper. Res. 46(4), 305–334 (2012)

    Google Scholar 

  19. 19.

    Jain, M., Singh, P.: State dependent bulk service queue with delayed vacations. Eng. Sci. 16(1), 3–15 (2005)

    Google Scholar 

  20. 20.

    Jeyakumar, S., Senthilnathan, B.: Modelling and analysis of a bulk service queueing model with multiple working vacations and server breakdown. RAIRO Oper. Res. 51(2), 485–508 (2017)

    Google Scholar 

  21. 21.

    Kalidass, K., Gnanaraj, J., Gopinath, S., Kasturi, R.: Transient analysis of an ${M}/{M}/1$ queue with a repairable server and multiple vacations. Int. J. Math. Oper. Res. 6(2), 193–216 (2014)

    Google Scholar 

  22. 22.

    Karaesmen, F., Gupta, S.M.: The finite capacity ${GI}/{M}/1$ queue with server vacations. J. Oper. Res. Soc. 47(6), 817–828 (1996)

    Google Scholar 

  23. 23.

    Ke, J.C., Wu, C.H., Zhang, Z.G.: Recent developments in vacation queueing models: a short survey. Int. J. Oper. Res. 7(4), 3–8 (2010)

    Google Scholar 

  24. 24.

    Kempa, W.M.: Transient workload distribution in the ${M}/{G}/1$ finite-buffer queue with single and multiple vacations. Ann. Oper. Res. 239(2), 381–400 (2016)

    Google Scholar 

  25. 25.

    Kim, S.J., Kim, N.K., Park, H.M., Chae, K.C., Lim, D.E.: On the discrete-time ${Geo}^{X}/{G}/1$ queues under ${N}$-policy with single and multiple vacations. J. Appl. Math. 2013 (2013)

  26. 26.

    Laxmi, P.V., Rajesh, P.: Analysis of variant working vacations on batch arrival queues. OPSEARCH 53(2), 303–316 (2016)

    Google Scholar 

  27. 27.

    Lee, H.W., Lee, S.S., Chae, K.C.: A fixed-size batch service queue with vacations. Int. J. Stoch. Anal. 9(2), 205–219 (1996)

    Google Scholar 

  28. 28.

    Lee, H.W., Lee, S.S., Chae, K.C., Nadarajan, R.: On a batch service queue with single vacation. Appl. Math. Model. 16(1), 36–42 (1992)

    Google Scholar 

  29. 29.

    Levy, Y., Yechiali, U.: Utilization of idle time in an ${M}/{G}/1$ queueing system. Manag. Sci. 22(2), 202–211 (1975)

    Google Scholar 

  30. 30.

    Li, H., Zhu, Y.: Analysis of ${M}/{G}/1$ queues with delayed vacations and exhaustive service discipline. Eur. J. Oper. Res. 92(1), 125–134 (1996)

    Google Scholar 

  31. 31.

    Mao, B., Wang, F., Tian, N.: Fluid model driven by an ${M}/{M}/1$ queue with multiple vacations and ${N}$-policy. J. Appl. Math. Comput. 38(1–2), 119–131 (2012)

    Google Scholar 

  32. 32.

    Medhi, J.: Stochastic Models in Queueing Theory. Academic Press, Cambridge (2002)

    Google Scholar 

  33. 33.

    Nadarajan, R., Subramanian, A.: A general bulk service queue with server’s vacation. Oper. Res. Manag. Syst. 127–135 (1984)

  34. 34.

    Neuts, M.F.: A general class of bulk queues with poisson input. Ann. Math. Stat. 38(3), 759–770 (1967)

    Google Scholar 

  35. 35.

    Niranjan, S., Chandrasekaran, V., Indhira, K.: Performance characteristics of a batch service queueing system with functioning server failure and multiple vacations. J. Phys. Conf. Ser. 1000, 012112 (2018)

    Google Scholar 

  36. 36.

    Panda, G., Banik, A.D., Guha, D.: Stationary analysis and optimal control under multiple working vacation policy in a ${GI}/{M}^{(a, b)}/1$ queue. J. Syst. Sci. Complex. 31(4), 1003–1023 (2018)

    Google Scholar 

  37. 37.

    Pradhan, S., Gupta, U.C.: Analysis of an infinite-buffer batch-size-dependent service queue with markovian arrival process. Ann. Oper. Res. 1–36 (2017)

  38. 38.

    Pradhan, S., Gupta, U.C., Samanta, S.K.: Queue-length distribution of a batch service queue with random capacity and batch size dependent service: ${M}/{G^{Y}_{r}}/1$. OPSEARCH 53, 329–343 (2016)

    Google Scholar 

  39. 39.

    Samanta, S.K., Chaudhry, M.L., Gupta, U.C.: Discrete-time ${Geo}^{X}/{G}^{(a, b)}/1/{N}$ queues with single and multiple vacations. Math. Comput. Model. 45(1–2), 93–108 (2007)

    Google Scholar 

  40. 40.

    Scholl, M., Kleinrock, L.: On the ${M}/{G}/1$ queue with rest periods and certain service-independent queueing disciplines. Oper. Res. 31(4), 705–719 (1983)

    Google Scholar 

  41. 41.

    Servi, L.D., Finn, S.G.: ${M}/{M}/1$ queues with working vacations ${M}/{M}/1/{WV}$. Perform. Eval. 50(1), 41–52 (2002)

    Google Scholar 

  42. 42.

    Sikdar, K., Gupta, U.C.: Analytic and numerical aspects of batch service queues with single vacation. Comput. Oper. Res. 32(4), 943–966 (2005)

    Google Scholar 

  43. 43.

    Sikdar, K., Gupta, U.C.: On the batch arrival batch service queue with finite buffer under servers vacation: ${M}^{X}/{G}^{Y}/1/{N}$ queue. Comput. Math. Appl. 56(11), 2861–2873 (2008)

    Google Scholar 

  44. 44.

    Sikdar, K., Samanta, S.K.: Analysis of a finite buffer variable batch service queue with batch Markovian arrival process and servers vacation. OPSEARCH 53(3), 553–583 (2016)

    Google Scholar 

  45. 45.

    Takagi, H.: Time-dependent analysis of ${M}/{G}/1$ vacation models with exhaustive service. Queueing Syst. 6(1), 369–389 (1990)

    Google Scholar 

  46. 46.

    Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation. Vacation and Priority, vol. 1. North-Holland, New York (1991)

    Google Scholar 

  47. 47.

    Thangaraj, M., Rajendran, P.: Analysis of batch arrival queueing system with two types of service and two types of vacation. Int. J. Pure Appl. Math. 117(11), 263–272 (2017)

    Google Scholar 

  48. 48.

    Tian, N., Zhang, Z.G.: Vacation Queueing Models: Theory and Applications, vol. 93. Springer, Berlin (2006)

    Google Scholar 

  49. 49.

    van der Duyn Schouten, F.A.: An ${M}/{G}/1$ queueing model with vacation times. Zeitschrift für Oper. Res. 22(1), 95–105 (1978)

    Google Scholar 

  50. 50.

    Wu, D.A., Takagi, H.: ${M}/{G}/1$ queue with multiple working vacations. Perform. Eval. 63(7), 654–681 (2006)

    Google Scholar 

  51. 51.

    Wu, W., Tang, Y., Yu, M.: Analysis of an ${M}/{G}/1$ queue with ${N}$-policy, single vacation, unreliable service station and replaceable repair facility. OPSEARCH 52(4), 670–691 (2015)

    Google Scholar 

  52. 52.

    Yang, D., Ke, J.: Cost optimization of a repairable ${M}/{G}/1$ queue with a randomized policy and single vacation. Appl. Math. Model. 38(21–22), 5113–5125 (2014)

    Google Scholar 

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Tamrakar, G.K., Banerjee, A. On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation. OPSEARCH (2020). https://doi.org/10.1007/s12597-020-00446-9

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  • Bulk-service
  • Bivariate probability generating function
  • Infinite buffer queue
  • Single vacation
  • Multiple vacation