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

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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

- 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)

HLR

HomeRegistrationArea

Backbone

Network

Cells

VLR 3

VLR1

VLR 2

Registration

Area 1

Registration

Area 2

Registration

Area 3

- 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

- 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

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

- 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 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

- 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

- 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

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

- 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)

- 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

- 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

- 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

- 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.