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.
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References
Abolnikov, L., & Dukhovny, A. (2003). Optimization in HIV screening problems. International Journal of Stochastic Analysis, 16(4), 361–374.
Adan, I., & Resing, J. (2000). Multi-server batch-service systems. Statistica Neerlandica, 54(2), 202–220.
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.
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.
Bailey, N. T. J. (1954). On queueing processes with bulk service. Journal of the Royal Statistical Society Series B (Methodological), 16(1), 80–87.
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.
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.
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.
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.
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.
Berman, O., Kaplan, E. H., & Shevishak, D. G. (1993). Deterministic approximations for inventory management at service facilities. IIE Transactions, 25(5), 98–104.
Berman, O., & Kim, E. (1999). Stochastic models for inventory management at service facilities. Stochastic Models, 15(4), 695–718.
Berman, O., & Sapna, K. (2000). Inventory management at service facilities for systems with arbitrarily distributed service times. Stochastic Models, 16(3–4), 343–360.
Berman, O., & Sapna, K. (2001). Optimal control of service for facilities holding inventory. Computers & Operations Research, 28(5), 429–441.
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.
Chakravarthy, S. (1992). A finite capacity \(GI/PH/1\) queue with group services. Naval Research Logistics (NRL), 39(3), 345–357.
Chandra, C., & Grabis, J. (2008). Inventory management with variable lead-time dependent procurement cost. Omega, 36(5), 877–887.
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.
Chaudhry, M., & Templeton, J. G. (1983). A first course in bulk queues. New York: Wiley.
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.
De Vries, J. (2013). The influence of power and interest on designing inventory management systems. International Journal of Production Economics, 143(2), 233–241.
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.
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.
Gallego, G., Katircioglu, K., & Ramachandran, B. (2007). Inventory management under highly uncertain demand. Operations Research Letters, 35(3), 281–289.
Germs, R., & van Foreest, N. (2013). Analysis of finite-buffer state-dependent bulk queues. OR Spectrum, 35(3), 563–583.
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.
Graham, A. (1981). Kronecker products and matrix calculus: With applications (Vol. 108). Chichester: Horwood.
Graves, S. C. (1982). The application of queueing theory to continuous perishable inventory systems. Management Science, 28(4), 400–406.
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.
Hébuterne, G., & Rosenberg, C. (1999). Arrival and departure state distributions in the general bulk-service queue. Naval Research Logistics (NRL), 46(1), 107–118.
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.
Kelton, W. D., Sadowski, R. P., & Sadowski, D. A. (2002). Simulation with ARENA (Vol. 3). New York: McGraw-Hill.
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.
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.
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.
Marcus, M., & Minc, H. (1992). A survey of matrix theory and matrix inequalities (Vol. 14). Mineola: Courier Dover Publications.
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.
Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Mineola: Courier Dover Publications.
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.
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.
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.
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.
Schwarz, M., Sauer, C., Daduna, H., Kulik, R., & Szekli, R. (2006). \(M/M/1\) queueing systems with inventory. Queueing Systems, 54(1), 55–78.
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.
Sikdar, K., & Gupta, U. (2005). Analytic and numerical aspects of batch service queues with single vacation. Computers & Operations Research, 32(4), 943–966.
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.
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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
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DOI: https://doi.org/10.1007/s10479-015-2041-z