Cost of searching : probabilistic analysis of the self-organizing move-to-front and move -to-root sorting rules

Abstract: Consider two (or more) users who request records in adatabase independently and with different frequencies. On arequest, the user's probability of calling a certain record iseither constant or changes deterministically in time. One userhas priority over the other(s) and rearranges the recordsaccording to seff-organizing rules.Extending the model to include not just constant but alsotime-dependent request probabilities allows us to treat a widerclass of situations naturally arising in applications. Theintroduction of several users provides means to model morecomplex systems such as e.g. cach-hierarchies.Assuming a finite number of records, the low-priority user'stransient and stationary/long-term search cost distribution(including mean, variance and generating function) is derivedwhen the self-organizing rule ismove-to-frontresp.move-to-root.For move-to-front with constant request probabilities, ratesof convergence to stationarity are considered and some examplesfor standard distributions are given.As the number of records tends to infinity, the transientresp. long-term search cost as a fraction of the total numberof records (under some regularity assumptions) converges indistribution. The asymptotic distribution function is derivedfor move-to-front assuming deterministically time-dependentrequest probabilities. Also, some stochastic ordering resultsallowing us to decide which low-priority user requestprobability distribution yields the better asymptotic searchcost fraction are given. Taking the request probabilitydistributions to be constant in time, the results stillapply.All results for the low-priority user reduce to analogousones for the high-priority one when all users are taken to beidentical. In particular, we recapture the probability functionfor move-to-root with uniform request probabilities.Numerically, we find that the 'worst' set of requestprobabilities, maximizing the expected search cost, closelyresembles the uniform which justifies the interest in thisspecial case.Keywords and phrases:Poisson embedding, self-organizingsearch, move-to-front, move-to-root, coupling, search cost,caching.