Paging Area Optimization Based on Interval Estimation in Wireless Personal Communication Networks

1. Paging Area Optimization Based on Interval Estimation in Wireless Personal Communication Networks By Z. Lei, C. U. Saraydar and N. B. Mandayam

2. Roadmap • Introduction / the problem • Background • Modeling • Optimization • Experimental results • Conclusion

3. Introduction: Definitions • Paging Area (PA): Region of the line/plane. Send paging signals from all base stations within the paging area • Want to minimize PA because cost is proportional to PA • At the same time, want to have a high probability of finding the mobile in the PA because a missed page is even more expensive • In other words, want to OPTIMIZE the PA.

4. Introduction: Motivation • Minimize transmissions, energy use • Similar techniques may be applicable with other cost structures • Keep track of user locations for other algorithms, such as location aided routing

5. Introduction: Problems • Optimization given user location probabilities - Given probabilities of user locations, what’s the least amount of effort required to find user (I.e. what’s the optimal PA)? • Optimization given user movement over time -Given a time-varying probability distribution, what are the optimal paging procedures? • Determining user motion patterns - How can these time-varying distributions be estimated based on measurements and models of user motion? • All three problems need to be solved.

6. Problem Definition: Cost Structure What are we trying to optimize? • Fix a probability of finding the user within the PA. • Subject to this probability, minimize the cost function: • Equivalent to minimizing , the area of the PA

7. Background: Location Distributions • The density function is related to the probability of being at a location. • Shaded area is probability of being in the interval • Higher density implies greater likelihood of presence at that point • This density is Unimodal and Symmetric, both are useful properties

8. Background: Confidence Intervals • Specify the probability of an interval • There are infinitely many intervals with the specified probability; here, they are and • Select the smallest one: for symmetric, unimodal densities this is easy – the region

9. Background: 2-D Densities • 2-D density is a function defined on the plane • Regions in the plane correspond to intervals on the line. Shown using contours here • Probability of being in a region equals volume under the density function over that region

10. Modeling: The Set-up

11. Modeling: Illustration

12. Modeling: Formal Mobility Model • Uses Brownian Motion with Drift as the mobility model • Start at time tn, at location xn. Let x(t) be location at time t, t > tn. Then: • E[x(t)] = xn + V(t – tn) • Var[x(t)] = D(t – tn) • V is the velocity of motion • D is the diffusion parameter – it represents location uncertainty/erraticity of motion • The primary result of the paper is an estimate for V

13. Optimization: Parameter Estimation • If D is known, it is easier to estimate V with fixed confidence G. • Can calculate the mean location from V (location is simply xn + V(t – tn)) based on Gaussian confidence intervals.

14. Optimization: Parameter Estimation Contd. • The size of the interval in which V lies turns out to be: • Increasing in the confidence parameter G • Proportional to sqrt(D) • Inversely proportional to square root of the interval over which observations were taken (I.e. tn – t0) • Roughly proportional to (t – tn) • The estimate itself is not dependent on the number of sample points, but the variance of the estimate decreases as the number of sample points increases.

15. Optimization: Parameter Estimation Contd. • If D is unknown, we must first estimate D. This can be done if the observation time increments are all equal. D is estimated as sample variance, denoted • The estimate for V is now based on a Student’s t distribution instead of a Gaussian distribution

16. Optimization: Parameter Estimation Contd. • The characteristics of the estimate obtained here are the same as those for the known-D case, except that its size is proportional to the square root of , the estimate for D, rather than the square root of D itself

17. Simulation Results: Known D

18. Simulation Results: PA Sizes

19. Simulation Results: Actual PA Sizes

20. Simulation Results: Effect of Sample Size

21. Conclusions • The results are analytically optimal under the assumptions made in the paper (can’t do better) • Growth of paging area is linear as time progresses, which is good • The parameter G, which determines probability of a correct page, is crucial – when G is very close to 1, PA increases drastically • V doesn’t affect paging area – this is expected • Can select an optimal sample size for a given problem as well

22. Observations/Reservations • The results in the paper have been well known for over 50 years • No results on whether the model chosen is representative of real user mobility • What happens if D and V are dependent on time, I.e. of the form D(t) and V(t)? • depends on several factors – signaling cost, pressure on MAC layers, etc. How easy is it to determine? • G depends the cost of a missed page. How easily can it be determined?

23. Problem Definition: Role of