The recently introduced $\{k\}$-packing function problem is considered in this paper. Special relation between a case when $k=1$, $k\ge 2$ and linear programming relaxation is introduced with sufficient conditions for optimality. For arbitrary simple connected graph $G$ there is construction procedure for finding values of $k$ for which $L_{\{k\}}(G)$ can be determined in the polynomial time. Additionally, relationship between $\{1\}$-packing function and independent set number is established. Optimal values for some special classes of graphs and general upper and lower bounds are introduced.