The Two Headed Disk: Stochastic Dominance of the Greedy Policy

Abstract

In his paper "Should the two-headed disk be greedy? - Yes, it should" Hofri defined a "greedy policy" as follows. Assuming that the range of disk addresses is [0,1], a request at location x is served by the closest arm while the other arm jockeys to a new position, z, where z = (1/3)x or z = 2/3 +x/3 depending on whether x is larger or smaller than 1/2. Hofri proved that this policy minimizes the expected seek distance for uniform request probabilities and conjectured that it stochastically dominates every other policy. Stochastic dominance is of practical importance in this context as it guarantees that a policy that optimizes expected seek distance also guarantees optimal seek time. The main result of this paper is a proof of Hofri's conjecture. The paper contains two proofs, the first establishes the conjecture, and the second shows that if the seek distance is stochastically minimized under a repositioning policy, then the policy must be Hofri's greedy policy and the request distribution must be uniform.