\(M/M/1\) queueing systems with inventory. (English) Zbl 1137.90429
Summary: We derive stationary distributions of joint queue length and inventory processes in explicit product form for various \(M/M/1\)-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical \(M/M/1/\infty\)-system.
MSC:
| 90B22 | Queues and service in operations research |
| 90B05 | Inventory, storage, reservoirs |
| 60K25 | Queueing theory (aspects of probability theory) |
Keywords:
queueing systems; inventory systems; performance analysis; inventory policy; queue lengths; service level; inventory level; stationary distribution; convex orderingReferences:
| [1] | F. Baccelli and P. Brémaud, Elements of queueing theory. Springer, New York, 2003. |
| [2] | O. Berman and E. Kim, Stochastic models for inventory management at service facilities, Stochastic Models 15(4) (1999) 695–718. · Zbl 0934.90002 · doi:10.1080/15326349908807558 |
| [3] | O. Berman and E. Kim, Dynamic order replenishment policy in internet-based supply chains. Mathematical Models of Operations Research 53 (2001) 371–390. · Zbl 1036.90017 |
| [4] | O. Berman and K.P. Sapna, Inventory management at service facilities for systems with arbitrarily distributed service times, Stochastic Models 16 (3, 4) (2000) 343–360. · Zbl 0959.90001 · doi:10.1080/15326340008807592 |
| [5] | O. Berman and K.P. Sapna, Optimal control of service for facilities holding inventory, Computers and Operations Research 28 (2001) 429–441. · Zbl 1080.90501 · doi:10.1016/S0305-0548(99)00128-8 |
| [6] | O. Berman and K.P. Sapna, Optimal service rates of a service facility with perishable inventory items, Naval Research Logistics, 49 (2002) 464–482. · Zbl 1013.90004 · doi:10.1002/nav.10021 |
| [7] | P. Brémaud, van Kannurpatti, and R. Mazumdar, Event and time averages: A review, Advances in Applied Probability 24 (1992) 377–411. · Zbl 0761.60045 · doi:10.2307/1427697 |
| [8] | D.J. Buchanan and R.F. Love, A (Q,R) Inventory model with lost sales and Erlang-Distributed Lead Times, Naval Research Logistics Quarterly 32 (1985) 605–611. · Zbl 0595.90012 · doi:10.1002/nav.3800320407 |
| [9] | F.M. Cheng and S.P. Sethi, Optimality of state-dependent (s,S) Policies in inventory models with markov-Modulated Demand and Lost Sales, Production and Operations Management 8 (1999) 183–192. · doi:10.1111/j.1937-5956.1999.tb00369.x |
| [10] | D. Gross and C.M. Harris, Fundamentals of Queueing Theory. 2nd edition. John Wiley and Sons, New York, 1985. · Zbl 0658.60122 |
| [11] | G. Hadley and T.M. Whitin, Analysis of Inventory Systems. Prentice Hall, Englewood Cliffs, N.J., 1963. · Zbl 0133.42901 |
| [12] | A.C. Hax and D. Candea, Production and Inventory Management. Prentice Hall, Englewood Cliffs, N.J., 1984. |
| [13] | F. P. Kelly, Reversibility and Stochastic Networks. John Wiley and Sons, Chichester–New York–Brisbane–Toronto, 1979. |
| [14] | E. Mohebbi and M.J.M. Posner, A continuous-review inventory system with lost sales and variable lead time, Naval Research Logistics, 45 (1998) 259–278. · Zbl 0941.90001 · doi:10.1002/(SICI)1520-6750(199804)45:3<259::AID-NAV2>3.0.CO;2-6 |
| [15] | H. Schneider, Effect of service-levels on order-points or order-levels in inventory models, International Journal of Production Research 19 (1981) 615–631. · doi:10.1080/00207548108956694 |
| [16] | K.C. Sevcik and I. Mitrani, The distribution of queueing networks states at input and output instants, Journal of the Association for Computing Machinery 28 (1981) 358–371. · Zbl 0456.68035 |
| [17] | K. Sigman and D. Simchi-Levi, Light traffic heuristic for an M/G/1 Queue with Limited Inventory, Annals of Operations Research 40 (1992) 371–380. · Zbl 0798.90068 · doi:10.1007/BF02060488 |
| [18] | E.A. Silver and R. Peterson, Decision Systems for Inventory Management and Production Planning. John Wiley and Sons, Inc., Chichester–New York–Brisbane–Toronto–Singapore, 1985. |
| [19] | R. Szekli, Stochastic Ordering and Dependence in Applied Probability. Springer-Verlag, New York–Berlin–Heidelberg, 1995. · Zbl 0815.60017 |
| [20] | H. Tempelmeier, Inventory control using a service constraint on the expected customer order waiting time. European Journal of Operational Research 19 (1985) 313–323. · Zbl 0573.90029 · doi:10.1016/0377-2217(85)90127-4 |
| [21] | H. Tempelmeier, Inventory service-levels in the customer supply chain, Operations Research Spektrum 22 (2000) 361–380. · Zbl 0973.90006 |
| [22] | R.J. Tersine, Principles of Inventory and Materials Management. 4th Ed. PTR Prentice Hall, Englewood Cliffs, N.J., 1994. |
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