Skip to main content
SpringerLink
Log in

A batch arrival multi phase queueing system with random feedback in service and single vacation policy

  • Original Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

We study a batch arrival Poisson input, k phases heterogeneous services with randomly feedback in services. The first phase of service is essential for all customers, but with probability γ 1 a tagged customer chose second phase, with probability α 1 feedback to tail of original queue or with probability β 1 = 1 − γ 1 − α 1 leave the system. Also, after completion of the second phase, with probability γ 2 the customer leads to the third service, or with probability α 2 feedback to tail of original queue, or leaves the system with probability β 2 = 1 − γ 2 − α 2. Like that, in kth phase feedback with probability α k or leaves the system with probability 1 − α k . After completion of each phases, the server either goes for a vacation with probability 𝜃(0 ≤ 𝜃 ≤ 1), or may continue to serve the next unit with probability 1 − 𝜃, if any. Otherwise, it remains in the system until a customer arrives, which is single vacation policy. We assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. In this paper we derive the steady- state equations, PGF’s of the system, and measures of sysytem.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Alnowibet, K., Tadj, L.: A quarum queueing system with Bernoulli vacation schedule and restricted admissibility. Adv. Model. Optim. 9(1), 171–180 (2007)

    Google Scholar 

  2. Artalejo, J.R., Choudhury, G.: Steady state analysis of an M/G/1 queue with repeated attempts and two-phase service. Qual. Technol. Quant. Manag. 1(2), 189–199 (2004)

    Google Scholar 

  3. Badamchizadeh, A., Shahkar, G.H.: A two phases queue system with Bernoulli feedback and Bernoulli schedule server vacation. Int. J. Inf. Manag. Sci. 19(2), 329–338 (2008)

    Google Scholar 

  4. Badamchizadeh, A.: An M x/(G 1, G 2)/1/G(B S)/V S with optional second service and admissibility restricted. Int. J. Inf. Manag. Sci. 20 (2), 305–316 (2009)

    Google Scholar 

  5. Chaudhry, M.L.: The queueing system M x/G/1 and its ramification. Naval Res. Logist. Quart. 26, 667–674 (1974)

    Article  Google Scholar 

  6. Choudhury, G.: A note on the M x/G/1 queue with a random set-up time under a restricted admissibility policy with a Bernoulli vacation schedule. Statistical Methodology, (to appear) (2008)

  7. Choudhury, G., Paul, M.: A batch arrival queue with a second optional service channel under N-policy. Stoch. Anal. Appl. 24, 1–22 (2006)

    Article  Google Scholar 

  8. Choudhury, G.: A bath arrival queueing system with an additional service channel. Inf. Manag. Sci. 14(2), 17–30 (2003)

    Google Scholar 

  9. Ke, J.C.: An M x/G/1 system with startup server and J additional options for service. Appl. Math. Model. 32, 443–458 (2006)

    Article  Google Scholar 

  10. Madan, K.C.: On a single server queue with two stage heterogeneous service and deterministic server vacation. Int. J. Syst. Sci. 32, 837–844 (2001)

    Article  Google Scholar 

  11. Madan, K.C., Abu-Dayyeh, W.: Restricted admissibility of batches into an M x/G/1 type bulk queue with modified Bernoulli schedule server vacations. ESSAIMP: Probab. Stat. 6, 113–125 (2002)

    Google Scholar 

  12. Madan, K.C., Abu-Dayyeh, W.: Steady state analysis of asingle server bulk queue with general vacation time and restricted admissibility of arriving batches. Revista Investigation Operacional 24(2), 113–123 (2002)

    Google Scholar 

  13. Madan, K.C., Choudhury, G.: An M x/G/1 queue with Bernoulli vacation schedule under restricted admissibility policy. Sankhaya 66(1), 175–193 (2004)

    Google Scholar 

  14. Madan, K.C, Choudhury, G.: A single server queue with two phases of heterogenous service under Bernoulli schedule and a general vcation time. Inf. Manag. Sci. 16(2), 1–16 (2005)

    Google Scholar 

  15. Madan, K.C., Al-Rawi, Z.R., Al-Naser, A.D.: On \(M^{X}/\left (\begin {array}{c}G_{1}\\G_{2} \end {array}\right ) /1/G(BS)/V_{s}\) vacation queue with two types of general heterogeneous service. J. Appl. Math. Decis. Sci., 123–135 (2005)

  16. Salehirad, M.R., Badamchizadeh, A.: On the multi-phase M/G/1 queueing system with random feedback. CEJOR 17, 131–139 (2009)

    Article  Google Scholar 

  17. Shahkar, G.H., Badamchizadeh, A.: On M/(G 1, G 2,…,G k )/V/1(B S). Far East J. Theor. Stat. 20(2), 151–162 (2006)

    Google Scholar 

  18. Wang, J.: An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47, 713–1723 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Allameh Tabataba’i University, Tehran, Iran

    A. Badamchi Zadeh

Authors
  1. A. Badamchi Zadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Badamchi Zadeh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zadeh, A.B. A batch arrival multi phase queueing system with random feedback in service and single vacation policy. OPSEARCH 52, 617–630 (2015). https://doi.org/10.1007/s12597-015-0206-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-015-0206-9

Keywords

Search

Navigation