1. Dynamic Pricing of Internet Bandwidth via Chance Constrained Programming Xin Guo IBM Watson Research Center xinguo@us.ibm.com John Tomlin IBM Almaden Research Center tomlin@almaden.ibm.com

2. The Internet Reseller The Conceptual Model

3. Assumptions 1.The Reseller buys bandwidth in bulk and resells it to customers, grouped in classes. 2.A customer class i is defined by the mean l i and standard deviation r i 2 of the (assumed) Normal usage distribution for the class. 3.Customer arrival is described by a Poisson process with parameter k = k (t)

4. Simple case Consider a single customer class where the j th customer’s usage is described by a normally distributed random variable X j with mean l and standard deviation r 2. For any fixed time t, let Y t = X 1 + X 2 + ………+ X N(t) represent the usage of N(t) signed-on customers, where P(N(t) = k) = ( k t) k e – k t k!

5. Let b t be the bandwidth available at time t. We require that The customers collectively do not exceed this with some (high) probability d t, i.e. they satisfy the chance constraint: P( Y t > b t ) [ d t How do express this in a computationally tractable way? Consider the generating function: y r (Y t ) = E[exp(r Y t )] Where A > r > 0 such that this converges

6. First Attempt After much tedious algebra we can show that: E[exp(r Y t )] = exp(- k t) exp( k t exp(r l + r 2 r 2 /2)) and P( Y t > b t ) = P(exp(rY t ) > exp(rb t )) [ E[exp(r Y t )] / exp(r b t ) = exp( k t(exp( k t exp(r l + r 2 r 2 /2) - 1) – r b t ) (*) for all 0 < r < A The bad news is that the minimum of (*) has no analytic expression.

7. Second Attempt We note that by definition: d 2 y r (Y t ) | = E[ Y t 2 ] dr 2 | r=0 Thus (again after tedious algebra): E[ Y t 2 ] = k t r 2 + k t l 2 + ( k t) 2 l 2 And applying the Chebyshev bound: P( Y t > b t ) [ E[ Y t 2 ] = k t r 2 + k t l 2 + ( k t) 2 l 2 b t 2 b t 2

8. Expected Number of Customers Note that the expected number of customers k can be expressed as some function, say k (q), assuming a Poisson distribution with k (t) expected arrivals on [0,t]. In particular we assume that k is a function k (q) of q, the price charged, e.g. q k

9. Very simple model Let b 0 be initial bandwidth available, and a the amount bought at (wholesale) cost C. Let q, k, l, r be defined as above, Then, for some fixed t, we wish to: Maximize q k (q) - C a Subject to: b t = b 0 + a k t r 2 + k t l 2 + ( k t) 2 l 2 [ d t b t 2 k = k (q) a, q, k m 0

10. Model Bloat 1.Introduce multiple customer classes 2.Introduce multiple discrete time periods and contract durations 3.Allow for existing customers at start of model horizon, whose contracts expire during the model horizon and may or may not be renewed. The more general model then is of the form: Indices: i = 1,…, I customer classes defined by distributions N( l i, r i 2 ) t = 1, …,T time periods, each of length D

11. Data: d t tolerance on capacity violation in period t C t cost/unit of buying new capacity in period t d i duration of contract for customer class i D i actual duration (d i D ) of contract for customer class i n i t number of existing contracts of type i still active at start of period t Variables: b t bandwidth available in period t a t bandwidth purchased in period t q i t price charged to customers of type i in period t User supplied demand functions: k i = k i (q i t ) The expected number of new customers of type i arriving in a period if the price is set at q i t

12. Constraints: b t = b t -1 - a t ( t = 1,…,T)  [( n i t + k i tD ) r i 2 + k i tD l i 2 + (n i t + k i tD ) 2 l i 2 ] + S [ k i D i ( l i 2 + r i 2 ) + ( k i D i l i ) 2 ] – d t b t 2 [ 0 ( t = 1,…,T) i| t < d i i| t m d i Objective: Maximize S q i t k i (q i t ) - S C t a t All variables non-negative.

13. A Tiny Example Consider a single time period [0,1] and single customer class With l = 2, r 2 = 1. Let there be 10 units of initial bandwidth, the cost of purchase be 1 / unit and d = 0.99 Let k = 70 - 20 q Then the problem is to: Maximize q(70 –q) – a Subject to: b = 10 + a 5 k + 4 k 2 – 0.99 b [ 0 a, b, q, k m 0 The solution is a = 21.15, k = 14.88 with obj = 19.87

14. Conclusions 1.The chance constraints essential to this model can be approximated analytically in terms of the parameters of the distributions. 2.Supply/demand data can be explicitly incorporated in these constraints and in the rest of the model. 3.The model is highly nonlinear and needs larger scale testing with NLP solvers. 4.The solutions need to tested via simulation to check the satisfaction of the chance constraints

15. Reference Xin Guo and John Tomlin, “Dynamic pricing of internet bandwidth via chance constrained programming”, IBM Research Report RJ 10196 (95070), Nov., 2000.