Game-Theoretic Analysis of Mobile Network Coverage David K.Y. Yau

Outline  Introduction  Mobility models  Cats’ strategies  Mouse’s strategies  Experimental results  Conclusion  Introduction  Mobility models  Cats’ strategies  Mouse’s strategies  Experimental results  Conclusion

Motivation – Why Mobile?  The mouse  Evade detection  Nature of “mission”  The cat  Improved coverage with fewer sensors  Robustness against contingencies  The mouse  Evade detection  Nature of “mission”  The cat  Improved coverage with fewer sensors  Robustness against contingencies

Problem Formulation  Two player game—cats and mouse  Closed rectangular area  Cats try to shorten detection time  Mouse tries to lengthen detection time  Both move at constant speed  Both have finite sensing range  Ends when mouse is within cat’s sensing range  Two player game—cats and mouse  Closed rectangular area  Cats try to shorten detection time  Mouse tries to lengthen detection time  Both move at constant speed  Both have finite sensing range  Ends when mouse is within cat’s sensing range

Mobility Model  Four-tuple  N: network area  M: accessibility constraints -- the “map”  T: trip selection  R: route selection  Random waypoint model is a special case  Null accessibility constraints  Uniform random trip selection  Cartesian straight line route selection  Four-tuple  N: network area  M: accessibility constraints -- the “map”  T: trip selection  R: route selection  Random waypoint model is a special case  Null accessibility constraints  Uniform random trip selection  Cartesian straight line route selection

The Sensing Range  Cats’ sensing range  R c  Mouse’s sensing range  R m  Blind mouse  R m < R c  Caught before evasion  Seeing mouse  R m > R c  Active evasion possible  Cats’ sensing range  R c  Mouse’s sensing range  R m  Blind mouse  R m < R c  Caught before evasion  Seeing mouse  R m > R c  Active evasion possible

Cats’ Strategies  Uniform scan  Bouncing  Random waypoint model  Uniform scan  Bouncing  Random waypoint model

Mouse Strategy  Blind  Hide at safe haven  Assume cats’ presence statistics is known  Seeing  Cats presence  Run  Maximize the minimum distance to all cats  Cats absence  Bouncing  Random waypoint model  Static  Don’t move  Blind  Hide at safe haven  Assume cats’ presence statistics is known  Seeing  Cats presence  Run  Maximize the minimum distance to all cats  Cats absence  Bouncing  Random waypoint model  Static  Don’t move

The Presence Matrix – ∏  Probability of a cat presence at an area  Divide network area into m × n cells  ∏ i,j = Probability of one or more cats present in cell (i, j)  Probability of a cat presence at an area  Divide network area into m × n cells  ∏ i,j = Probability of one or more cats present in cell (i, j)

Best Blind Mouse Play  Find an optimal path to move to a safest cell  Detection time is maximized along the path  ∏ i,j is lowest at the safest cell (usually)  Dynamic programming  Greedy does not always work  Find an optimal path to move to a safest cell  Detection time is maximized along the path  ∏ i,j is lowest at the safest cell (usually)  Dynamic programming  Greedy does not always work

Comparison with Local Greedy Strategy Greedy Dynamic Programming

Optimal Escape Path Formulation  Using ∏, computes  E j [T stay ] = Expected detection time if staying at cell i  E j [T move(k) ] = Expected detection time if moving to cell k  Cell k is a neighboring cell of i  Make decision—stay or move  Maximize expected detection time  Optimal escape path = sequence of movement until stay is chosen  How to compute the expected detection time?  Using ∏, computes  E j [T stay ] = Expected detection time if staying at cell i  E j [T move(k) ] = Expected detection time if moving to cell k  Cell k is a neighboring cell of i  Make decision—stay or move  Maximize expected detection time  Optimal escape path = sequence of movement until stay is chosen  How to compute the expected detection time?

Compute Expected Detection Time  Initialize E j [T detect ] as E j [T stay ]  Insert all the cells into max-heap  i := Extract-max  Update E j [T detect ] for each neighbor cell k of i  E k [T detect ] := max(E k [T detect ], E i [T move(k) ])  Heapify  Repeat until heap becomes empty  Initialize E j [T detect ] as E j [T stay ]  Insert all the cells into max-heap  i := Extract-max  Update E j [T detect ] for each neighbor cell k of i  E k [T detect ] := max(E k [T detect ], E i [T move(k) ])  Heapify  Repeat until heap becomes empty

Example Optimal Paths V m = 10 m/s V m = 15 m/s

Blind Mouse Strategies Compared Expected Detection Time Cat Strategies ScanBouncingRWP Mouse Strategies DP RWP Stay V c = 10 m/s, V m = 10 m/s, R c = 25 m, R m = 0 m

Other Options for Cats  Increase sensing range  Increase speed  Increase quantity  Increase sensing range  Increase speed  Increase quantity R c (m) T detect (s) V c (m/s) T detect (s) NcNcNcNc

Best Seeing Mouse Play  Find and move at the optimum direction  Minimum distance to all cats is maximized  Distance between cat and mouse  d(β, t) = ║C(t) – M(β, t)║  Minimum distance moving at direction β  d * (β) = min t ≥ 0 { d(β, t) }  Optimal escape direction  β * = argmax d * (β)  Find and move at the optimum direction  Minimum distance to all cats is maximized  Distance between cat and mouse  d(β, t) = ║C(t) – M(β, t)║  Minimum distance moving at direction β  d * (β) = min t ≥ 0 { d(β, t) }  Optimal escape direction  β * = argmax d * (β)

Strategies When Cat Absence  Bouncing  Centric  Random waypoint model  Static  Don’t move  Bouncing  Centric  Random waypoint model  Static  Don’t move

Seeing Mouse Strategies Compared Expected Detection Time Cat Strategies BouncingRWP Mouse Strategies Bouncing Centric Static Stay V c = 10 m/s, V m = 10 m/s, R c = 5 m, R m = 10 m

Result Explained  Why Bouncing is better for cats?  All area are equally likely to be visited (approx.)  Uniform presence matrix (approx.)  Safe haven eliminated  Why Centric is better for mouse?  More choices of direction  Why Bouncing is better for cats?  All area are equally likely to be visited (approx.)  Uniform presence matrix (approx.)  Safe haven eliminated  Why Centric is better for mouse?  More choices of direction

Presence Matrices Random Waypoint Model Bouncing

Where The Mouse Were Caught? Detection Time (s)

Other Options—Sensing Range

Other Options—Speed

Other Options—Quantity

Conclusions  Detect intelligent mobile target using mobile sensor  Mobile sensors increase robustness  Strategies to evade detection without full knowledge of cat movement  Movement model is important  Presence matrix determine the coverage performance  Bouncing movement is better than random waypoint model  Stochastic movement prevent movement prediction  Optimal escape direction helps seeing mouse  Dynamic programming algorithm helps blind mouse  Effects of sensing range, speed and number of cats are quantified  Detect intelligent mobile target using mobile sensor  Mobile sensors increase robustness  Strategies to evade detection without full knowledge of cat movement  Movement model is important  Presence matrix determine the coverage performance  Bouncing movement is better than random waypoint model  Stochastic movement prevent movement prediction  Optimal escape direction helps seeing mouse  Dynamic programming algorithm helps blind mouse  Effects of sensing range, speed and number of cats are quantified

Future Work  Radioactive, chemical plume detection  Explosion  Dispersion  Mobile target detection with presence of obstacle  Model for sensor reliability, interference, etc.  Quantification of sensing uncertainty  Radioactive, chemical plume detection  Explosion  Dispersion  Mobile target detection with presence of obstacle  Model for sensor reliability, interference, etc.  Quantification of sensing uncertainty

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