Dynamic Page Migration with Stochastic Requests

1. Dynamic Page Migration with Stochastic Requests Marcin Bieńkowski

2. Motivation Mobility: • mobile nodes on the (unbounded) plane • nodes move with constant speeds • wireless (radio) communication Parallel program: • A parallel application running on all the nodes • Application uses shared variables stored in the local memories of nodes We restrict our attention to one such variable called (memory) page

3. Motivation (movement) • Typically page is big (of size ) and indivisible • Nodes want to access (read or modify) just one unit of data from the page

4. Motivation (costs) • Cost incurred can be measured in terms of energy usage • In one step it is proportional to the distance to some power (propagation exponent of the medium) plus a constant overhead for communication • For any two nodes and • their distance: • cost of sending one unit of data Snapshot at

5. Motivation (costs cont.) • After serving a request the • page can be moved (migrated) • to a new node • Cost of moving the page from node to is equal to (in this example: ) Goal: minimize the total cost of communication

6. Dynamic Page Migration: Modelling An online problem • We assume discrete time steps • processors in a metric space • Input: configuration + request sequences In one step • Positions of the nodes are defined by (constant speed!) • Node issues a request • Algorithm pays for serving the request • Algorithm optionally moves the page Performance metric: We measure the efficiency of an algorithm by standard competitive analysis – competitive ratio

7. Previous results Results for Dynamic Page Migration for B., Byrka, Dynia, Korzeniowski, Meyer auf der Heide: Competitive ratios • For adaptive adversaries: • For oblivious adversaries: For the lower bounds are even higher - is replaced by

8. Possible relaxations of the model A) Replace the adversarial description of the mobility by the random walk of the nodes (considered in [BKM04,BK05] for a kind of Brownian motion of the nodes) B) Generate requests randomly (this paper) • In one step is drawn uniformly and independently according to the probability distribution • The mobility is still dictated by an adversary! Performance metric: algorithm is -competitive with prob. if for all configuration sequences and all holds

9. Our contribution • A deterministic algorithm MTFR, which achieves constant competitive ratio with high probability • High probability = for any we may achieve a probability if the input sequence is sufficiently long. • Using potential functions and amortized analysis we proved also that in the worst case (both and generated by an adversary) the competitive ratio of MFTR is • The expected value of the competitive ratio is • High probability = for any we may achieve a probability • if the input sequence is sufficiently long on sufficiently long sequences

10. Static placement is bad • Assume that our algorithm knows • Then the best possible static placement strategy is to choose a node with maximum • Strategy for the adversary: Then OPT places the page at exp. cost The cost of the algorithm is Thus, for large the competitive ratio is at least

11. A better solution • We may choose a node randomly according to constant competitive ratio in expectation, but the variance is high. • Better: we may choose a new position for the page randomly not once, but from time to time, once for steps • Our algorithm MTFR: • Divides time into phases of length • use actual outcomes of the random process: at the beginning of a phase moves to the node which issued a request • We prove that MTFR is -competitive

12. Average cost of communication Consider any positions of the nodes in time We define average cost of communication in step Let be the maximum cost of communication between two nodes in time . We relate costs of OPT and MTFR to and

13. Lower bound on OPT Lemma: Situation in step Proof by triangle inequality (on costs) Thus, Moreover, this lower bound on depends only on steps and .

14. Expected value -> High probability • We have a sequence of independent variables , s.t. • We have no global bound on ... • ... but we can relate in two consecutive steps: If , then two consecutive differ by at most Combinatorial Lemma:If we have more than such variables, then their sum differs by at most a constant factor from its expectation with probability w.h.p.

15. Cost of MTFR phase • The cost in time step depends only on and the node holding the page (equal to ) The expectation is also • Problem: more dependencies • Solution: Consider a cost in the whole phase without its first step – independent for two different phases • Bound cost in first steps separately (phases are quite long) Variant of Combinatorial Lemma implies that for input sequences longer than holds w.h.p.

16. Final remarks • Is there an algorithm which performs well both in the adversarial scenario and under stochastic requests? • What happens if we make different for each step? • In general – we cannot get a better bound than the fully adversarial scenario • What if depends only on the node which issued request in the last step?

17. Thank you for your attention.

18. Lower bound for adversarial scenario • For the deterministic case: time decision point