A Mobility Model for Studying Wireless Communication

1. A Mobility Model for Studying Wireless Communication Raymond Greenlaw Armstrong Atlantic State University Savannah, GA, USA Sanpawat Kantabutra Chiang Mai University Chiang Mai, Thailand

2. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

3. Introduction • Wireless networking is becoming prevalent because of low-cost, ease of installation, scalability, and convenience to users. • We consider a model of a mobile network; a wireless network in which the access points themselves may be moving. • Such networks are of great importance in supporting relief efforts for natural disasters or for military field exercises.

4. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

5. The Mobility Model • Goal is to model actual mobile networks. • Model needs to be sophisticated enough to model complex real-life situations. • Key features need to be abstracted out so the model is feasible to study and apply. • After defining the model, a communication protocol is defined to interpret how the model works.

6. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

7. Definition of the Model • Model operates on a 2-dimensional grid. • Model is an 8-tuple (S, D, U, L, R, V, C, O), where • Set S = {s1, s2, …, sm} is a finite collection of sources, where mN, m is the number of sources. Corresponding to each source si, for 1 ≤i ≤m, an initial location (xi, yi) is specified where xi, yiN. • Set D = {000, 001, 010, 101, 110} is called the directions and correspond to no movement, east, west, south, and north, respectively.

8. Definition of the Model • Model Definition (continued) • Set U = {u1, u2,…, up} is a finite collection of mobile devices, where pN. The set U is called the set of users. The value p is called the number of users. Corresponding to each user ui, for 1 ≤i ≤p, an initial location (xi, yi) is specified where xi, yiN.

9. Definition of the Model • Model Definition (continued) • Let tN. Set L = {l1, l2,…, lp} is a finite collection of “bit strings,” where liDt for 1 ≤i ≤t. Each group of three bits in li beginning with the first three defines a step in the given direction for the user ui’s movement or no movement at all if the string is 000. The value t is called the duration of the model.

10. Definition of the Model • Model Definition (continued) • Let t(i)N for 1 ≤i ≤m. The set R = {r1, r2, …, rm} is a finite collection of “bit strings,” where riDt(i) for 1 ≤i ≤m. Each group of three bits in ri beginning with the first three defines a step in a given direction for the source si’s movement or no movement at all if the string is 000. The set R is called the random walks of the mobility model.

11. Definition of the Model • Model Definition (continued) • Set V = {v1, v2, …, vm} is a finite collection of numbers, where viN.The valuevi is the corresponding number of steps from ri per unit time that si will take. This set is called the velocities. • Set C = {c1, c2, …, cm} is a finite collection of numbers, where ciN.The valueci is the corresponding diameter of the circular coverage of source si. This set is called the coverages.

12. Definition of the Model • Model Definition (continued) • The set O = {(x1, y1, x2, y2) | x1, y1, x2, y2N, x2 > x1, and y2 > y1} is a finite collection of rectangles in the plane. The set is called the obstacles.

13. Definition of the Model • Remarks • Sources in S correspond to wireless access points and are broadcasting and receiving signals. • Set D represents the four possible directions for movement in the grid, plus no movement. • Set U represents users with mobile devices. • Set L contains random walks used to model the movement of users. • Set R contains random walks to model the movement of sources.

14. Definition of the Model • Remarks (continued) • To accommodate for different source velocities, the walks in R have different lengths. • Relative speeds of sources are represented by natural numbers contained in set V. • Different sources will broadcast at different signal strengths depending on a variety of factors, available power being the main one.

15. Definition of the Model • Remarks (continued) • Various signal strengths are represented by specifying the diameter of a circle ci for each source indicating where its signal can be received. • This region is its coverage area. • Since buildings and other obstacles may interfere with signal transmission, the model incorporates a set of obstacles O. • For simplicity, only rectangular obstacles are permitted.

16. Definition of the Model • Communication Protocol • Illustrates how the model is interpreted. • Needed so that the model works as intended. • Source are always on. • Users with mobile devices are moving in and out of the range of each other and various sources. • Devices would like to communicate (send and receive messages) with one another.

17. Definition of the Model • Communication Protocol (continued) • Let k > 2 and kN. • Any two sources with overlapping-coverage areas may communicate with each other in full-duplex fashion as long as the intersection of their overlapping-coverage area is not completely contained inside obstacles. Those two sources are currently in range. • A series s1, s2, …, sk of sources are currently in range if si and si+1 are currently in range for 1 ≤i≤k-1.

18. Definition of the Model • Communication Protocol (continued) • Two mobile devices cannot communicate directly with one another. • A mobile device D1 always communicates with another mobile device D2 through a source or series of sources as defined on the following slides.

19. Definition of the Model • Communication Protocol (continued) • The mobile devices D1 at location (x1, y1) and D2 at location (x2, y2) communicate through a single source s located at (x3, y3) if at a given instance in time the lines between points (x1, y1) and (x3, y3), and points (x2, y2) and (x3, y3) are within the area of coverage of s, and do not intersect with any obstacle from O.

20. Definition of the Model • Communication Protocol (continued) • The mobile devices D1 at location (x1, y1) and D2 at location (x2, y2) communicate through a series of sources s1 at location (a1, b1), s2 at location (a2, b2), …, and sk at location (ak, bk) that are currently in range if the line between points (x1, y1) and (a1, b1) is inside s1’s coverage area and does not intersect any obstacle from O and the line between points (x2, y2) and (ak, bk) is inside sk’s coverage area and does not intersect any obstacle from O.

21. Definition of the Model • Communication Protocol (continued) • Mobility of the sources and the users are built into this model. • Reflects the situation in a real mobile network where access points and users are moving around. • For simplicity, we implicitly assumed that all users are moving at the same rate of speed, whereas we explicitly modeled sources moving at different velocities.

22. Definition of the Model • Communication Protocol (continued) • Model can handle users moving at different rates of speed by having some users remain stationary while others are moving at each step.

23. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

24. A Sample Instance of the Model • Let S = {s1, s2, s3, s4} with initial locations (2, 5), (5, 5), (6, 4), and (5, 2) respectively. • Let D = {000, 001, 010, 101, 110}. • Let U = {u1, u2, u3} with initial location (3, 4), (2, 1), and (6, 2), respectively. • Let t = 3 and L = {l1, l2, l3}, where li = (000, 000, 000) for 1 ≤i≤ 3.

25. A Sample Instance of the Model • Let R = {r1, r2, r3, r4}. For clarity, the figure only shows r1 = (101, 001, 101) and omits other ri’s, which we assume are all (000, 000, 000) except r2 which is twice as long. • Let V = {1, 2, 1, 1}. • Let C = {2, 2, 2, 4}. • Let O = {(2, 1, 4, 2)}.

26. A Sample Instance of the Model • Three stationary users. • Four sources. • One obstacle. • s1-3 have coverage of 2, s4 has coverage of 4. • s1 moves south, east, and south with a velocity of v1 = 1, or one step per unit of time.

27. A Sample Instance of the Model • Initially, s2 and s3 are currently in range, s2, s3, and s4 are a series of sources currently in range, and sources s1 and s2 are not currently in range. • Initially, users u1 and u3 cannot communicate; after three steps, u1 can communicate with u3 through the series of sources s1 and s4.

28. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

29. Problem Definitions • User Communication Problem • Instance: Mobility model (S, D, U, L, R, V, C, O), two designated users ua and ub from U, and a time k. • Question: Can users ua and ub communicate at time k?

30. Problem Definitions • Sources Reachability Problem • Instance: Mobility model (S, D, U, L, R, V, C, O), two designated sources sa and sb from S, and a time k. • Question: Are sources sa and sb in range at time k?

31. Problem Definitions • Access Point Location Problem • Instance: Mobility model (S, D, U, L, R, V, C, O), two designated users ua and ub from U, an access point diameter d, and a natural number k. • Question: Can users ua and ub communicate if k or fewer access points of diameter d are placed appropriately in the grid?

32. Problem Definitions • Access Point Placement Problem • Instance: Two mobility models M = (S, D, U = {u1, u2}, L, R, V, C, O), and M’ = (S’, D, U = {u1, u2}, L, R’, V’, C’, O’). • Question: Can u1 and u2 communicate for more steps in model M than they can in model M’?

33. Problem Definitions • Obstacle Removal Problem • Instance: Mobility model (S, D, U, L, R, V, C, O), two designated users ua and ub from U, and a natural number k. • Question: Can ua and ub communicate throughout the duration of the model if k or fewer obstacles are removed?

34. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

35. Conclusion • Description of a mobility model and several interesting decision problems related to the model were presented. • It would be interesting to examine the complexity of other related problems. • Mobility model itself can be studied further, including extending the model to three dimensions.

36. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

37. Acknowledgments • Ray is very grateful to the Computer Science Department at Chiang Mai University for its generosity and hospitality during his stay there during the spring semester of 2006. • Ray’s research was supported by a Fulbright Lecturing/Research Fellowship. • Ray thanks the Fulbright Commissions of Thailand and the United States.

38. Outline • Introduction • The Mobility Model • Definition of the Model • A Sample Instance of the Model • Problem Definitions • Conclusion • Acknowledgments • References

39. References • Paul Goransson and Raymond Greenlaw, Secure Roaming in 802.11 Networks, Chapter 11, Elsevier, 2007.