A Queueing Model With Randomized Depletion of Inventory

Abstract

Abstract. In this paper we study an M/M/1 queue, where the server continues to work during idle periods and builds up inventory. This inventory is used for new arriving service requirements, but it is completely emptied at random epochs of a Poisson process, whose rate depends on the current level of the acquired inventory. For several shapes of depletion rates, we derive differential equations for the stationary density of the workload and the inventory level and solve them explicitly. Finally numerical illustrations are given for some particular examples, and the effects of this depletion mechanism are discussed.

A queueing/inventory and an insurance risk model

“…In this section we use the analysis developed in [5] to state some explicit results when ω(x) = ax in the exponential service requirements case. In [5], the authors studied directly the functions v + and v − without considering their Laplace transforms.…”

“…In this section we use the analysis developed in [5] to state some explicit results when ω(x) = ax in the exponential service requirements case. In [5], the authors studied directly the functions v + and v − without considering their Laplace transforms. Indeed, differentiating (1) and (2), one can show that the functions v − and v + satisfy some well known differential equations.…”

“…In particular, if ν is odd, i.e. ν = 2n + 1, where n ∈ N, from [5], we have v − (x) = 2 −ν/2 Ke −(a/2)x 2 −λx H ν √ ax + λ + μ √ a for all x ≥ 0,…”

“…See Figure 2. In [5] this two sided queueing/inventory model has been analyzed for the case of exponentially distributed service times. Our queueing/inventory model is related to classical M/G/1 and inventory models (see [5] and the references therein).…”

“…In [5] this two sided queueing/inventory model has been analyzed for the case of exponentially distributed service times. Our queueing/inventory model is related to classical M/G/1 and inventory models (see [5] and the references therein). An important inventory model is the basestock model, in which a server produces products until the inventory has reached a certain basestock level, with requests for products arriving according to a Poisson process.…”