I am trying to find expectation of following expression: \begin{align*} V_n(c):=\sum_{j=1}^n p_j \mathbb{E}\left\{\min \left\{D_j,(c-D[1, j-1])^{+}\right\}\right\} \end{align*}
Here D_j has poison distribution and D[1,j] means sum of random variables from 1 to j. They are independent. I try to use partial moments however I could not. Could you help me to proceed. It is important to know how this is calculated since I will code this later. Thank you.