Random Access Game. Medium Access Control Design for Wireless Networks 1. Sandip Chakraborty. Department of Computer Science and Engineering,

Transcription

1 Random Access Game Medium Access Control Design for Wireless Networks 1 Sandip Chakraborty Department of Computer Science and Engineering, INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR October 22, Chen et al., Random Access Game and Medium Access Control Design, IEEE/ACM Transactions on Networking, Vol 18, No 4, August 2010, pp Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

2 Preface The choice of contention measure and contention resolution algorithm is key to the performance of medium access methods. In high load scenario, DCF results in excessive collisions Every node restart transmission from the base contention window regardless of the level of contention low throughput short term unfairness due to oscillation in the contention window Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

3 Preface The choice of contention measure and contention resolution algorithm is key to the performance of medium access methods. In high load scenario, DCF results in excessive collisions Every node restart transmission from the base contention window regardless of the level of contention low throughput short term unfairness due to oscillation in the contention window Requirements for achieving high efficiency (high throughput and low collision) and better fairness: Stabilize the network into a steady state that sustains an appropriate channel access probability (or, contention window size as per Bianchi s model) for each node. Properly estimate and implement the contention measure DCF adapts to packet collision, although can not distinguish between contention and corrupted frames. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

4 Random Access Games Capture the interaction among wireless nodes in wireless networks with contention-based medium access Basic idea: Regard the process of contention control as carrying out a distributed strategy update algorithm to achieve the equilibrium Game Model: Strategy: Decide the channel access probability Payoff: Utility gain from channel access - cost from packet collision Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

5 Random Access Games Consider a set N of wireless nodes in a wireless LAN with contention-based medium access Associated with each wireless node i N, Its channel access probability p i (t) at time t, Certain contention measure q i (t) 0 at time t. Node i can observe its own access probability p i (t) and contention measure q i (t), but not that of others. It adjusts its channel access probability p i (t) based only on p i (t) and q i (t), p i (t + 1) = F i (p i (t), q i (t)) The contention measure q i (t) depends on the channel access probability p(t) = {p i (t), i N}, chosen by wireless nodes, q i (t) = G i (p(t)) Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

6 Random Access Games p i (t + 1) = F i (p i (t), q i (t)) (1) q i (t) = G i (p(t)) (2) Assume that above two equations have an equilibrium (p, q). The fixed point of (1) defines an implicit relation between the equilibrium channel access probability p i and contention measure q i, p i = F i (p i, q i ) Assume F i is continuously differentiable and δf i /δq i 0 in [0, 1] Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

7 Random Access Games Implicit Function Theorem If G(x, y) = C, where G(x, y) is a continuous function and C is a constant, and δg/δy 0 at some point P, then y may be expressed as a function of x in some domain about P; i.e., there exists a function over that domain such that y = g(x). By implicit function theorem, there exists a unique continuously differentiable function F i such that q i = F i (p i ) Define the utility function for each node i as, U i (p i ) = F i (p i )dp i Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

8 Random Access Game U i (p i ) = F i (p i )dp i (3) Being an integral, and since F i (p i ) = q i 0, U i (.) is a continuous and non-decreasing function. The larger the contention, the smaller the channel access probability F i (.) is a decreasing function. U i (.) is strictly concave. Random Access Game A random access game G is defined as G := {N, (S i ) i N, (u i ) i N } where N is a set of players (wireless nodes), player i N strategy S i := {p i p i [ν i, ω i } with 0 ν i < ω i < 1, and payoff function u i (p) := U i (p i ) p i q i (p) with utility function U i (p i ) and contention measure q i (p). Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

9 Random Access Game Random Access Game A random access game G is defined as G := {N, (S i ) i N, (u i ) i N } where, N is a set of players (wireless nodes), Player i N strategy S i := {p i p i [ν i, ω i } with 0 ν i < ω i < 1, Payoff function u i (p) := U i (p i ) p i q i (p) Why the strategy p i S i is constrained to be strictly less than 1? Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

10 Random Access Game Random Access Game A random access game G is defined as G := {N, (S i ) i N, (u i ) i N } where, N is a set of players (wireless nodes), Player i N strategy S i := {p i p i [ν i, ω i } with 0 ν i < ω i < 1, Payoff function u i (p) := U i (p i ) p i q i (p) Why the strategy p i S i is constrained to be strictly less than 1? In order to prevent a node from exclusively occupying the wireless channel Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

11 Random Access Game Random Access Game A random access game G is defined as G := {N, (S i ) i N, (u i ) i N } where, N is a set of players (wireless nodes), Player i N strategy S i := {p i p i [ν i, ω i } with 0 ν i < ω i < 1, Payoff function u i (p) := U i (p i ) p i q i (p) Why the strategy p i S i is constrained to be strictly less than 1? In order to prevent a node from exclusively occupying the wireless channel Throughput of node i is proportional to p i if there is no collision q i can be seen as the contention price for node i. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

12 Random Access Game Random Access Game A random access game G is defined as G := {N, (S i ) i N, (u i ) i N } where, N is a set of players (wireless nodes), Player i N strategy S i := {p i p i [ν i, ω i } with 0 ν i < ω i < 1, Payoff function u i (p) := U i (p i ) p i q i (p) Why the strategy p i S i is constrained to be strictly less than 1? In order to prevent a node from exclusively occupying the wireless channel Throughput of node i is proportional to p i if there is no collision q i can be seen as the contention price for node i. Payoff Function: Net gain utility from channel access discounted by the contention cost. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

13 Sandip Chakraborty You are(iit free Kharagpur) to select the utility CS and the contention measure!! October 22, / 23 Random Access Game Random Access Game A random access game G is defined as G := {N, (S i ) i N, (u i ) i N } where, N is a set of players (wireless nodes), Player i N strategy S i := {p i p i [ν i, ω i } with 0 ν i < ω i < 1, Payoff function u i (p) := U i (p i ) p i q i (p) Why the strategy p i S i is constrained to be strictly less than 1? In order to prevent a node from exclusively occupying the wireless channel Throughput of node i is proportional to p i if there is no collision q i can be seen as the contention price for node i. Payoff Function: Net gain utility from channel access discounted by the contention cost.

14 Random Access Game - Contention Measure Let I j denote the set of nodes that interfere with the transmission of node i. Use conditional collision probability as a contention measure: q i (p) := 1 j I i (1 p j ), i N This has two nice properties; It is an accurate measure of contention in the network provides multibit information of the contention. Equation based control gives sophistic method to derive the steady state network with a targeted fairness! Wireless nodes can estimate conditional collision probabilities without explicit feedback decouple congestion control from failed transmissions. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

15 Analyzing the Game Model Initial Assumptions A0. The utility function U i (.) is continuously differentiable, strictly concave, and with finite curvatures that are bounded away from zero, i.e. there exist some constant µ and λ such that 1/µ 1/U i (p i) 1/λ > 0 Existence and optimality of Nash Equilibrium A1. Let γ(p) := (1 p i ), and denote the smallest eigenvalue of i N 2 γ(p) over p by ν min. Then µ ν min < 0. Uniqueness of Nash Equilibrium A2. Functions Γ i (p i ) := (1 p i )(1 U i (p i)), i N are all strictly increasing or all strictly decreasing Optimality of non-trivial Nash equilibrium Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

16 Optimality of Nash Equilibrium There exists a Nash equilibrium: The strategy space S i are compact convex sets Payoff functions u i are continuous and concave in p i Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

17 Optimality of Nash Equilibrium There exists a Nash equilibrium: The strategy space S i are compact convex sets Payoff functions u i are continuous and concave in p i The Nash equilibrium gives globally optimal solution. At the Nash equilibrium, pi satisfies, (U i (p i ) q i (p ))(p i p i ) 0 p i S i (4) Define function V (p) as, V (p) := i N (U i (p i ) p i ) i N(1 p i ) Equation (4) is an optimality condition for max V (p) Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

18 The Random Access Game G has a Unique Nash Equilibrium V (p) := i N (U i (p i ) p i ) i N(1 p i ) The Hessian of function V (p) is written as, ( ) 2 V (p) = diag U 1 (p 1 ),..., U N (p N ) 2 γ(p) Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

19 The Random Access Game G has a Unique Nash Equilibrium V (p) := i N (U i (p i ) p i ) i N(1 p i ) The Hessian of function V (p) is written as, ( ) 2 V (p) = diag U 1 (p 1 ),..., U N (p N ) 2 γ(p) Note that, ( ) diag U 1 (p 1 ),..., U N (p N ) µi Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

20 The Random Access Game G has a Unique Nash Equilibrium V (p) := i N (U i (p i ) p i ) i N(1 p i ) The Hessian of function V (p) is written as, ( ) 2 V (p) = diag U 1 (p 1 ),..., U N (p N ) 2 γ(p) Note that, ( ) diag U 1 (p 1 ),..., U N (p N ) µi Also, 2 γ(p) ν min I Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

21 The Random Access Game G has a Unique Nash Equilibrium V (p) := i N (U i (p i ) p i ) i N(1 p i ) The Hessian of function V (p) is written as, ( ) 2 V (p) = diag U 1 (p 1 ),..., U N (p N ) 2 γ(p) Note that, ( ) diag U 1 (p 1 ),..., U N (p N ) µi Also, 2 γ(p) ν min I Then, under assumption A1, 2 V (p) (µ + ν min )I 0 Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

22 The Random Access Game G has a Unique Nash Equilibrium V (p) is strictly concave function over the strategy space. Therefore, the optimization has a unique Nash equilibrium. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

23 Non-trivial Nash Equilibrium (U i (p i ) q i (p ))(p i p i ) 0 p i S i The equilibrium condition implies that, p either takes value at the boundaries Or, the strategy space S i satisfies, U i (p i ) = q i (p ) Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

24 Non-trivial Nash Equilibrium (U i (p i ) q i (p ))(p i p i ) 0 p i S i The equilibrium condition implies that, p either takes value at the boundaries Or, the strategy space S i satisfies, U i (p i ) = q i (p ) We call a Nash equilibrium p a nontrivial equilibrium if for all nodes i, p i satisfies the second condition. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

25 Non-trivial Nash Equilibrium (U i (p i ) q i (p ))(p i p i ) 0 p i S i The equilibrium condition implies that, p either takes value at the boundaries Or, the strategy space S i satisfies, U i (p i ) = q i (p ) We call a Nash equilibrium p a nontrivial equilibrium if for all nodes i, p i satisfies the second condition. Suppose assumption A2 holds. If the random access game G has a nontrivial Nash equilibrium, it must be unique. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

26 Dynamics of Random Access Game Random Access Settings: Players (wireless nodes) can observe the outcome (e.g. packet collision or successful packet transmissions) of the actions of others, but do not have direct knowledge of other player actions or payoffs. Repeated play of random access game: Players repeatedly adjust strategies in response to observations of other player actions so as to achieve the Nash equilibrium. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

27 Strategy Update in Random Access Game Straightforward way: At each stage, every node chooses the best response to the actions of all other nodes in the previous round. At stage t + 1, node i N chooses a channel access probability, p i (t + 1) := G i (p(t)) := arg max p S i U i (p) pq i (p(t)) If the above dynamics reach a steady state, then it is the Nash equilibrium. There is no convergence result for this dynamics. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

28 Gradient Play based Strategy Update Better response strategy in contrast to best response strategy. Every player adjusts the current channel access probability gradually in a gradient direction suggested by the observations of other players actions. Every player i N updates its strategy according to, p i (t + 1) := [p i (t) + f i (p i (t))(u i (p i (t)) q i (p(t)))] S i f i (.) > 0 is the step size which is a function of the strategy of player i S i denotes the projection on user i s strategy space. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

29 Gradient Play based Strategy Update Better response strategy in contrast to best response strategy. Every player adjusts the current channel access probability gradually in a gradient direction suggested by the observations of other players actions. Every player i N updates its strategy according to, p i (t + 1) := [p i (t) + f i (p i (t))(u i (p i (t)) q i (p(t)))] S i f i (.) > 0 is the step size which is a function of the strategy of player i S i denotes the projection on user i s strategy space. This has an economical interpretation: If the marginal utility U i (p i(t)) is more than the contention price q i (p(t)), we increase the access probability Otherwise the access probability is decreased. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

30 Gradient Play based Strategy Update The equilibrium condition is: (U i (p i ) q i (p ))(p i p i ) 0 p i S i By the equilibrium condition, the Nash equilibrium of random access game G are the fixed points of the gradient play and vice versa. Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

31 Medium Access Control Design Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

32 Design of Utility Function: A Case Study Consider a single-cell wireless networks with L classes of users. Each class l L is associated with a weight φ l, we assume, φ 1 = φ max > φ 2 > φ 3 >... > φ l We want to achieve maximum throughput under weighted fairness constraint; T l = φ l ; 1 l, m L T m φ m where T l is the throughput of class-l node. Let ζ = p i i N Under the assumption of Poisson arrival process, the channel idle probability is approximately, (1 p i ) e ζ i N Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

33 Design of Utility Function: A Case Study The aggregate successful packet transmission probability is ζe ζ Let P is the packet payload, σ is the duration of an idle slot, T s and T c are the durations that the channel is sensed busy because of a successful transmission and during a collision, respectively. The throughput can be expressed as; T = ζe ζ P e ζ σ + ζe ζ T s + (1 e ζ ζe ζ )T c This attains maximum at ζ that solves; (1 ζ )e ζ = 1 σ T c ; 1 l, m L Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

34 Design of Utility Function: A Case Study Now consider how to achieve weighted fairness. When there is a large number of nodes accessing the channel, each user should sense approximately the same environment on average we can assume that each user has the same conditional collision probability. With equal packet payload sized, the throughput ratio between the users of different classes is approximately the ratio between their channel access probabilities. Therefore, p l p m = φ l φ m Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

35 Design of Utility Function: A Case Study The users of each class l are associated with the same utility U l (.) they are associate with same function, Γ l (p l ) := (1 p l )(1 U l (p l)) At non-trivial Nash equilibrium, Γ l (p l ) = γ(p) for all 0 l L. Therefore, the condition p l p m = φ l φ m can be achieved at a non-trivial Nash equilibrium if we design the utility functions such that, ( ) pl Γ l (p l ) = h, 1 l L φ l A non-trivial equilibrium p that achieves maximum throughput should satisfy, ( p ) h l = e ζ φ l Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

36 Design of Utility Function: A Case Study When the number of wireless nodes is large, pl should be very small. A convenient choice that approximately satisfies the condition is, ( ) ( pl h = e ζ 1 + p ) l φ l φ l From here, Chen et al. develop an utility function; ) ( U l (p l ) = (1 + e ζ p l + e ζ 1 + p ) l ln(1 p l ) φ l φ l Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23

37 Thank You Sandip Chakraborty (IIT Kharagpur) CS October 22, / 23