The Generalized Stochastic Knapsack

In this chapter we again consider a stochastic system consisting of C resource units and K classes of objects. And we again suppose that class-k objects have size b_κ and arrive and depart at random times. But we now permit the arrival and service rates to depend on the current knapsack state. In particular, with n and S defined as in the previous chapter, the time until the next class-k arrival is exponentially distributed with parameter λ _κ (n) when the knapsack is in state n. Analogously, the time until the next class-k departure is exponentially distributed with parameter μ _κ (n) when the knapsack is in state n. Clearly, μ _κ (n) must satisfy μ _κ (n) = 0 whenever n _κ = 0. Note that the generalized stochastic knapsack becomes the stochastic knapsack, as studied in the previous chapter, if we set λ _κ (n) = λ _κ and μ _κ (n) = n _κ μ _κ for all n ∊ S.