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

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)

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 Conceptual Configuration and related work (improvements) Home Registration Area Registration Area 3 Registration Area 2 Registration Area 1 Backbone Network Backbone Network HLR VLR1 VLR 2 VLR 3 Cells

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

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 RA 1 RA 2 RA 1 RA 2

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

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

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

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

Metrical Task Systems (cont’d) A Formal Definition Μ=(Σ,Γ,c) metrical task system Σ={S 1, S 2,…, S n } set of states Γ={T 1, T 2,…, T m } set of tasks c : Σ × (Σ  Γ) → R cost function triangular inequality on metric space (cost function) MTS Problem s=(T i,T j,…) sequence of tasks Find sequence of states and executions that minimizes total cost for a given sequence of tasks S2 S1 S3 S4 TcTbTcTbTcTbTcTb

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)

1,b 1,a 2,b 2,c 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 S 1 (location a) Input relocations: b c b c b Result execution: S 1 S 2 b S 4 c S 3 b c S 2 b RA 1 a RA 2 b c

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

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

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.