A Theoretical Study of Optimization Techniques Used in Registration Area Based Location Management: Models and Online Al

1. A Theoretical Study of Optimization Techniques Used in Registration Area Based Location Management: Models and Online Algorithms Sandeep K. S. Gupta Goran KonjevodGeorgios Varsamopoulos Arizona State University

2. Location Management • Part of Mobile Communication System • Location Tracking (update, registration) • Call Delivery (search) • Components • Cells (base stations) • Registration Areas, Location Registers • Home Location Register (HLR) • Visiting Location Register (VLR) • Mobile Units (subscribers)

3. HLR HomeRegistrationArea Backbone Network Cells VLR 3 VLR1 VLR 2 Registration Area 1 Registration Area 2 Registration Area 3 Conceptual Configuration andrelated work (improvements) • Update • Subscriber moves to new RA • New VLR, HLR updated • Search • HLR is queried • Search cost improvements • Location Caching • Profile Replication • Prediction • Update cost improvements • Forwarding Pointers • Look-ahead Registration • Multi-layered Configuration • RA Overlapping

4. Previous Work • Dynamic Overlapping of Registration Areas • Find optimal size of a Registration Area by including and excluding cells from RAs • Optimal Registration Sequence • Minimize the number of registrations (updates) over a given user path in the service area • Online Algorithms and Competitiveness in Location Management

5. Overlapping RA1 RA2 RA1 RA2 • Overlapping • Eliminates updates due to subscriber oscillations at borders • Increases coverage of a Registration Area without increasing the number of users • Dynamic Overlapping • Reduces registration area planning time • Adapts to changes of call and mobility • Has higher requirements at component logic

6. Optimal Registration Sequence • Registration Areas (statically) overlap • Offline version • Mobile follows a predetermined path • Overlapping gives multiple choices on selection of Registration Area at each part of the mobile’s path • Find a sequence of registrations (updates) of minimal count • Greedy approach finds optimal solution

7. Offline and Online Computation • Offline problem • All input is given a-priori • Complete solution is given in “one time” • Online problem • Input is given one element at a time • Decision/output must be made upon arrival of an element • Sequence of output is the partial solution up to that point • Competitiveness • An online algorithm may not be able to find optimal solution • Competitive ratio : the worst possible “performance” or “size” ratio of an algorithm’s solution over the respective optimal solution for any input

8. Online ORS problem • Path is not known – a stochastic mobility model is known. • At each intersection decide if the mobile should register with another Registration Area • Competitiveness • No online ORS algorithm is inherently competitive

9. This paper • More on competitiveness • Modeling of Location Management techniques as Metrical Task Systems (MTS) • Known algorithms • Known bounds • Unified way of comparing LM schemes? • MTS lower bounds may not be good enough • Bounds depend on number of states • Number of states can be very large • We can get better bounds under restricted models

10. Metrical Task Systems (cont’d) Tc Tb Tc Tb • A Formal Definition • Μ=(Σ,Γ,c) metrical task system • Σ={S1, S2,…, Sn} set of states • Γ={T1, T2,…, Tm} set of tasks • c : Σ× (Σ Γ) → R cost function • triangular inequality on metric space (cost function) • MTS Problem • s=(Ti,Tj,…) sequence of tasks • Find sequence of states and executions that minimizes total cost for a given sequence of tasks S2 S1 S4 S3

11. Metrical Task Systems (cont’d) • Offline version • Has a simple solution • Can be mapped to a shortest path problem • Online version • Best known algorithm achieves polylogarithmic competitiveness ratio to the number of states • There is lower bound to competitiveness ratio of (logn) ( n is the number of states)

12. Example of LM problem as MTS:Registration Optimization • System Formulation • A state is a pair of a registration and a location • Incoming tasks are relocations • Problem definition • Given a sequence of relocations find a sequence of registrations • Performance • The number of states is polynomial to number of RAs • Example • Initial state S1 (location a) • Input relocations: b c b c b • Result execution: S1 S2 b S4 c S3 b c S2 b RA1a RA2b c 1,b 0 1 1,a 2,c 2,b 1 0

13. Bounds under restricted models:Competitiveness of ORS • A run is maximal constant subsequence of offline optimal sequence • There are as many runs as registrations made by the offline optimal sequence • RESTRICTION: Throughout a run there can be up to k different available RAs • At each run any algorithm cannot make more than (k-1) bad choices • Competitive ratio cannot be worse than k

14. Also in this paper • MTS formulations for • Pointer Forwarding • Multiple (replicated) registrations • Pre-emptive look-ahead registration • Bounds under restricted models for • Location Caching using sliding window • Dynamic Update using stochastic process

15. Conclusions • There are many optimization problems in Location Management • Many performance enhancements to LM can also be expressed as online decision/optimization problems • LM schemes can be modeled as Metrical Task Systems • Known bounds to Metrical Task Systems are not good enough • Under restricting yet reasonable assumptions, better bounds can be found.