The M/M/1 queue with inventory, lost sale, and general lead times
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The M/M/1 queue with inventory, lost sale, and general lead times Mohammad Saffari · Søren Asmussen · Rasoul Haji
Received: 29 March 2012 / Revised: 25 September 2012 / Published online: 16 January 2013 © Springer Science+Business Media New York 2013
Abstract We consider an M/M/1 queueing system with inventory under the (r, Q) policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples. Keywords Queueing · Inventory · Stationary distribution · Lost sale · Regenerative process Mathematics Subject Classification
60K25 · 90B05 · 60G07 · 60G10
1 Introduction In classical inventory models, arriving demands are satisfied immediately if there is enough on-hand inventory. Most of these models consider optimization problems which choose the optimal policy or optimal value of decision variables without computing the stationary distribution of inventory states. Nevertheless, there are some studies that have derived stationary distributions to formulate long-run average cost M. Saffari (B) Department of Industrial Engineering, University of Tafresh, Tafresh, Iran e-mail: [email protected]; [email protected] S. Asmussen Department of Mathematical Sciences, Aarhus University, Aarhus, Denmark R. Haji Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran
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functions which are used for optimization. Sahin [6] considered (s, S) inventory system and general random demand process with fixed lead time and backordering. He derived time-dependent and stationary distribution to derive approximations for optimal control policy. Focusing on lost sale problems, let us take a brief look at some of these studies. Mohebbi and Posner [4] considered a continuous-review inventory system with compound Poisson demand, Erlang as well as hyper-exponentially distributed lead time and lost sales. They derived the stationary distribution of inventory level for the purpose of formulating long-run average cost functions with/without a service level constraint. Mohebbi and Hao [5] considered inventory system with compound Poisson demand, Erlang-distributed lead times, random supply interruptions and derived the stationary distribution of the inventory level under an (r, Q)-type control policy. Many recent studies deal with complex integrated production-inventory systems or service-inventory systems. In these models satisfying each demand needs on-hand inventory and involves a process or service that takes some time. Production-inventory or service-inventory systems can be discussed in connection with integrated supply ch
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