Average case analysis of disk scheduling, increasing subsequences and spacetime geometry

We consider the problem of estimating the average tour length of the asymmetric TSP arising from the disk scheduling problem with a linear seek function and a probability distribution on the location of I/O requests. We provide a law of large numbers expression, for the expected tour size and compute it explicitely in the most interesting cases. We also provide finer asymptotic estimates in the case of a uniform request distribution. Our methods relate disk scheduling to the problem of finding the longest increasing subsequence in a permutation. Our tour length estimates also answer questions on the asymptotics of the connect-the dots problem with Lipschitz functions which was raised in Adv. Appl. Probab. 37, 571-603 (2005). In the last section we show that all the above problems can be interpreted naturally, in terms of Lorentzian geometry. We generalize the main results to this setting.