Skip to main content
SpringerLink
Log in

An ‘(s, S)’ inventory in a queueing system with batch service facility

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

This paper considers a single-server queueing model in which the customers are served in batches of varying size depending on predetermined thresholds as well as available inventory. There is a finite buffer for the inventory and the service of every customer requires an inventory item. An (s, S) -type inventory system is used for the models considered in this paper. Initially, the model is studied in detail using the matrix-analytic method by assuming all the underlying random variables to be exponentially distributed. Thereafter, an outline of the model in a more general set up is also presented. Due to complexity of the model when more general assumptions are made on the underlying random variables, simulation is opted after a satisfactory validation with the analytic counterpart of the exponential model. Finally, some illustrative numerical examples are also presented to accomplish our analysis.

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
Fig. 4

Similar content being viewed by others

References

  • Abolnikov, L., & Dukhovny, A. (2003). Optimization in HIV screening problems. International Journal of Stochastic Analysis, 16(4), 361–374.

    Article  Google Scholar 

  • Adan, I., & Resing, J. (2000). Multi-server batch-service systems. Statistica Neerlandica, 54(2), 202–220.

    Article  Google Scholar 

  • Arumuganathan, R., & Jeyakumar, S. (2005). Steady state analysis of a bulk queue with multiple vacations, setup times with n-policy and closedown times. Applied Mathematical Modelling, 29(10), 972–986.

    Article  Google Scholar 

  • Baba, Y. (1996). A bulk service \( GI/M/1\) queue with service rates depending on service batch size. Journal of the Operations Research Society of Japan, 39(1), 25–35.

    Article  Google Scholar 

  • Bailey, N. T. J. (1954). On queueing processes with bulk service. Journal of the Royal Statistical Society Series B (Methodological), 16(1), 80–87.

    Google Scholar 

  • Banik, A., Chaudhry, M., & Gupta, U. (2008). On the finite buffer queue with renewal input and batch markovian service process: \(GI/BMSP/1/N\). Methodology and Computing in Applied Probability, 10(4), 559–575.

    Article  Google Scholar 

  • Banik, A., Gupta, U., & Chaudhry, M. (2009). Finite-buffer bulk service queue under markovian service process: \(GI/MSP^ {(a, b)}/1/N\). Stochastic Analysis and Applications, 27(3), 500–522.

    Article  Google Scholar 

  • Bar-Lev, S. K., Parlar, M., Perry, D., Stadje, W., & Van der Duyn Schouten, F. A. (2007). Applications of bulk queues to group testing models with incomplete identification. European Journal of Operational Research, 183(1), 226–237.

    Article  Google Scholar 

  • Baron, O., Berman, O., & Perry, D. (2010). Continuous review inventory models for perishable items ordered in batches. Mathematical Methods of Operations Research, 72(2), 217–247.

    Article  Google Scholar 

  • Baron, O., Berman, O., & Perry, D. (2011). Shelf space management when demand depends on the inventory level. Production and Operations Management, 20(5), 714–726.

    Article  Google Scholar 

  • Berman, O., Kaplan, E. H., & Shevishak, D. G. (1993). Deterministic approximations for inventory management at service facilities. IIE Transactions, 25(5), 98–104.

    Article  Google Scholar 

  • Berman, O., & Kim, E. (1999). Stochastic models for inventory management at service facilities. Stochastic Models, 15(4), 695–718.

    Article  Google Scholar 

  • Berman, O., & Sapna, K. (2000). Inventory management at service facilities for systems with arbitrarily distributed service times. Stochastic Models, 16(3–4), 343–360.

    Article  Google Scholar 

  • Berman, O., & Sapna, K. (2001). Optimal control of service for facilities holding inventory. Computers & Operations Research, 28(5), 429–441.

    Article  Google Scholar 

  • Bradley, E. L. (1969). Queues with balking and their application to an inventory problem. In Technical Report, DTIC Document.

  • Bretthauer, K. M., Shetty, B., Syam, S., & Vokurka, R. J. (2006). Production and inventory management under multiple resource constraints. Mathematical and Computer Modelling, 44(1–2), 85–95.

    Article  Google Scholar 

  • Chakravarthy, S. (1992). A finite capacity \(GI/PH/1\) queue with group services. Naval Research Logistics (NRL), 39(3), 345–357.

    Article  Google Scholar 

  • Chandra, C., & Grabis, J. (2008). Inventory management with variable lead-time dependent procurement cost. Omega, 36(5), 877–887.

    Article  Google Scholar 

  • Chaudhry, M. L., & Gupta, U. C. (1999). Modelling and analysis of \( M/G^{{(}a, b)}/1/N\) queue—A simple alternative approach. Queueing Systems, 31(1–2), 95–100.

    Article  Google Scholar 

  • Chaudhry, M., & Templeton, J. G. (1983). A first course in bulk queues. New York: Wiley.

    Google Scholar 

  • Claeys, D., Walraevens, J., Laevens, K., & Bruneel, H. (2010). A queueing model for general group screening policies and dynamic item arrivals. European Journal of Operational Research, 207(2), 827–835.

    Article  Google Scholar 

  • De Vries, J. (2013). The influence of power and interest on designing inventory management systems. International Journal of Production Economics, 143(2), 233–241.

    Article  Google Scholar 

  • Deepak, T., Krishnamoorthy, A., Narayanan, V., Vineetha, K., Deepak, T., Krishnamoorthy, A., et al. (2008). Inventory with service time and transfer of customers and inventory. Annals of Operations Research, 160(1), 191–213.

    Article  Google Scholar 

  • Fabens, A. J. (1961). The solution of queueing and inventory models by semi-markov processes. Journal of the Royal Statistical Society Series B (Methodological), 23(1), 113–127.

    Google Scholar 

  • Gallego, G., Katircioglu, K., & Ramachandran, B. (2007). Inventory management under highly uncertain demand. Operations Research Letters, 35(3), 281–289.

    Article  Google Scholar 

  • Germs, R., & van Foreest, N. (2013). Analysis of finite-buffer state-dependent bulk queues. OR Spectrum, 35(3), 563–583.

    Article  Google Scholar 

  • Gold, H., & Tran-Gia, P. (1993). Performance analysis of a batch service queue arising out of manufacturing system modelling. Queueing Systems, 14(3–4), 413–426.

    Article  Google Scholar 

  • Graham, A. (1981). Kronecker products and matrix calculus: With applications (Vol. 108). Chichester: Horwood.

    Google Scholar 

  • Graves, S. C. (1982). The application of queueing theory to continuous perishable inventory systems. Management Science, 28(4), 400–406.

    Article  Google Scholar 

  • Guan, R., & Zhao, X. (2011). Pricing and inventory management in a system with multiple competing retailers under \((r, Q)\) policies. Computers & Operations Research, 38(9), 1294–1304.

    Article  Google Scholar 

  • Hébuterne, G., & Rosenberg, C. (1999). Arrival and departure state distributions in the general bulk-service queue. Naval Research Logistics (NRL), 46(1), 107–118.

    Article  Google Scholar 

  • He, Q. M., Jewkes, E., & Buzacott, J. (2002). The value of information used in inventory control of a make-to-order inventory-production system. IIE Transactions, 34(11), 999–1013.

    Google Scholar 

  • Kelton, W. D., Sadowski, R. P., & Sadowski, D. A. (2002). Simulation with ARENA (Vol. 3). New York: McGraw-Hill.

    Google Scholar 

  • Kim, E. (2005). Optimal inventory replenishment policy for a queueing system with finite waiting room capacity. European Journal of Operational Research, 161(1), 256–274.

    Article  Google Scholar 

  • Krishnamoorthy, A., Manikandan, R., Lakshmy, B. (2013). A revisit to queueing-inventory system with positive service time. Annals of Operations Research. doi:10.1007/s10479-013-1437-x.

  • Krishnamoorthy, A., Shajin, D., Lakshmy, B. (2015). On a queueing-inventory with reservation, cancellation, common life time and retrial. Annals of Operations Research. doi:10.1007/s10479-015-1849-x.

  • Krishnamoorthy, A., & Viswanath, N. C. (2013). Stochastic decomposition in production inventory with service time. European Journal of Operational Research, 228(2), 358–366.

    Article  Google Scholar 

  • Manuel, P., Sivakumar, B., & Arivarignan, G. (2007). A perishable inventory system with service facilities, \(MAP\) arrivals and \(PH\)–service times. Journal of Systems Science and Systems Engineering, 16(1), 62–73.

    Article  Google Scholar 

  • Marcus, M., & Minc, H. (1992). A survey of matrix theory and matrix inequalities (Vol. 14). Mineola: Courier Dover Publications.

    Google Scholar 

  • Nair, A., Jacob, M., Krishnamoorthy, A. (2013). The multi server m/m/(s,s) queueing inventory system. Annals of Operations Research. doi:10.1007/s10479-013-1405-5.

  • Neuts, M. F. (1967). A general class of bulk queues with poisson input. The Annals of Mathematical Statistics, 38(3), 759–770.

    Article  Google Scholar 

  • Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Mineola: Courier Dover Publications.

    Google Scholar 

  • Powell, W. B., & Humblet, P. (1986). The bulk service queue with a general control strategy: Theoretical analysis and a new computational procedure. Operations Research, 34(2), 267–275.

    Article  Google Scholar 

  • Radke, A. M., & Tseng, M. M. (2012). A risk management-based approach for inventory planning of engineering-to-order production. CIRP Annals-Manufacturing Technology, 61(1), 387–390.

    Article  Google Scholar 

  • Saffari, M., Asmussen, S., & Haji, R. (2013). The \(M/M/1\) queue with inventory, lost sale, and general lead times. Queueing Systems, 75(1), 65–77.

    Article  Google Scholar 

  • Schwarz, M., & Daduna, H. (2006). Queueing systems with inventory management with random lead times and with backordering. Mathematical Methods of Operations Research, 64(3), 383–414.

    Article  Google Scholar 

  • Schwarz, M., Sauer, C., Daduna, H., Kulik, R., & Szekli, R. (2006). \(M/M/1\) queueing systems with inventory. Queueing Systems, 54(1), 55–78.

    Article  Google Scholar 

  • Sigman, K., & Simchi-Levi, D. (1992). Light traffic heuristic for an \(M/G/1\) queue with limited inventory. Annals of Operations Research, 40(1), 371–380.

    Article  Google Scholar 

  • Sikdar, K., & Gupta, U. (2005). Analytic and numerical aspects of batch service queues with single vacation. Computers & Operations Research, 32(4), 943–966.

    Article  Google Scholar 

  • Zee, D. J. V. D., Harten, A. V., & Schuur, P. (2001). On-line scheduling of multi-server batch operations. IIE Transactions, 33(7), 569–586.

    Article  Google Scholar 

Download references

Acknowledgments

We are thankful to the guest editor and anonymous reviewers for their suggestions that improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Industrial and Manufacturing Engineering, Kettering University, Flint, MI, 48504, USA

    S. R. Chakravarthy

  2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Arunava Maity & U. C. Gupta

Authors
  1. S. R. Chakravarthy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arunava Maity
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. U. C. Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arunava Maity.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chakravarthy, S.R., Maity, A. & Gupta, U.C. An ‘(s, S)’ inventory in a queueing system with batch service facility . Ann Oper Res 258, 263–283 (2017). https://doi.org/10.1007/s10479-015-2041-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-015-2041-z

Keywords

Search

Navigation