  The weighted $k$-server problem is a generalization of the $k$-server problem
in which the cost of moving a server of weight $\beta_i$ through a distance $d$
is $\beta_i\cdot d$. The weighted server problem on uniform spaces models
caching where caches have different write costs. We prove tight bounds on the
performance of randomized memoryless algorithms for this problem on uniform
metric spaces. We prove that there is an $\alpha_k$-competitive memoryless
algorithm for this problem, where $\alpha_k=\alpha_{k-1}^2+3\alpha_{k-1}+1$;
$\alpha_1=1$. On the other hand we also prove that no randomized memoryless
algorithm can have competitive ratio better than $\alpha_k$.
  To prove the upper bound of $\alpha_k$ we develop a framework to bound from
above the competitive ratio of any randomized memoryless algorithm for this
problem. The key technical contribution is a method for working with potential
functions defined implicitly as the solution of a linear system. The result is
robust in the sense that a small change in the probabilities used by the
algorithm results in a small change in the upper bound on the competitive
ratio. The above result has two important implications. Firstly this yields an
$\alpha_k$-competitive memoryless algorithm for the weighted $k$-server problem
on uniform spaces. This is the first competitive algorithm for $k>2$ which is
memoryless. Secondly, this helps us prove that the Harmonic algorithm, which
chooses probabilities in inverse proportion to weights, has a competitive ratio
of $k\alpha_k$.
