The El Farol Bar Problem for next generation systems

More documents

Recommendations

Info

$Modelling Network Traffic Via a Multifractal Wavelet Model ...$

26 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM attendance attendance attendance 100 80 60 40 20 0 0 500 1000 1500 2000 2500 3000 time (iterations) 100 80 60 40 20 0 0 500 1000 1500 2000 2500 3000 time (iterations) 100 80 60 40 20 0 0 500 1000 1500 2000 2500 3000 time (iterations) probabities for each agent probabities for each agent probabities for each agent 1.0 0.8 0.6 0.4 0.2 0.0 0 500 1000 1500 2000 2500 3000 time (iterations) 1.0 0.8 0.6 0.4 0.2 0.0 0 500 1000 1500 2000 2500 3000 time (iterations) 1.0 0.8 0.6 0.4 0.2 0.0 0 500 1000 1500 2000 2500 3000 time (iterations) Figure 3.2: The overall attendance and the probabilities for each of the M agents for ’partial’ information tax, modified tax algorithms and the full information tax algorithm.
3.2. TAXING/PAYOFFS ALGORITHMS 27 In the following a set of equations that describe the stationary state behaviour of the probabilities will be derived pi(k). The number of agents attending at time k is N(k) = M i=1 xi(k) and xi(k) are independent Bernoulli trials given by: 1, pi(k) xi(k) = (3.2.5) 0, 1 − pi(k) In the stationary state we have: pi(k + 1) = pi(k) ∀i ∈ N Taking the expectation values results to: E(pi(k + 1)) = E(pi(k)) ⇒ E(pi(k + 1)) − E(pi(k)) = 0 ⇒ E(µtax(N(k) − N )) = 0 N(k) is the number of agents attending at time k is equal to M i=1 xi(k), where xi(k) are independent Bernoulli trials. The expectation value is calculated using the following formula: E(XY ) = fx,y(x, y) where X,Y random variables and fx,y(x, y) joined mass function. In this case Y = µtax and X = N(k) − N . By definition, Y = µtax is a function of N(k). So, Y = g(X) and fx,y(x, y) = fx(x, g(x)) = fx(x). Hence: E(µtax(N(k)) · (N(k) − N )) = (N(k) − N ) · µtax(N(k)) · fN(k)(N(k)) ⇒ cµ N(k)>N N(k)−N (N(k) − N ) · fN(k)(N(k)) + µ cµ N(k)>N µ N(k)≤N x N(k)≤N N(k) · fN(k)(N(k)) − cµN (N(k) − N ) · fN(k)(N(k)) ⇒ N(k)>N N(k) · fN(k)(N(k)) − µN N(k)≤N fN(k)(N(k))+ fN(k)(N(k)) (3.2.6) Kolmogorov-Smirnov test indicated that z = N(k) might follow Poisson distribution f(z) = λz z! e−λ , with λ = E(z). Hence: E(f(z)) = zf(z) = zf(z) + zf(z) = λ ⇒ z z≤N z>N
Page 1: The El Farol Bar Problem for next g
Page 4 and 5: List of Figures 1.1. Bar attendance
Page 6 and 7: Acknowledgements I would like to de
Page 8 and 9: 2 CHAPTER 1. CHAOS, COMPLEXITY AND
Page 16 and 17: 10 CHAPTER 2. PREVIOUS APPROACHES T
Page 30 and 31: 24 CHAPTER 3. ANALYSIS AND EXTENSIO
Page 44 and 45: 38 CHAPTER 4. CASE STUDY: MULTIPLE
Page 50 and 51: 44 CHAPTER 5. CONCLUSIONS Another n
Page 52 and 53: 46 CHAPTER 5. CONCLUSIONS
Page 54 and 55: 48 APPENDIX A. C CODE FOR THE EL FA
Page 66: 60 BIBLIOGRAPHY [14] Challet D. and

The El Farol Bar Problem for next generation systems

Create successful ePaper yourself

Delete template?

Save as template?