Internal Fragmentation in a Class of Buddy Systems

A general class of buddy systems is defined and the overallocation of memory due to internal fragmentation is examined. It is shown that, for a uniform distribution of requests, the overallocation varies between $2x_{1}/(x_{1}+1)$ and $\frac{1}{2}(x_{1}+1)$, where $x_{1}>1$ is the largest real root of the characteristic equation of the particular buddy system. Bounds are found for the overallocation for request distributions characterized by a parameterization of Zipf’s law. The expected value of the overallocation is independent of the request distribution within wide values of the parameter, and is given by $(x_{1}-1)/\ln x_{1}$. For the binary buddy system $x_{1}= 2$; the overallocation varies between 1.33 and 1.50 and the expected value is $\approx 1.44$.