1. Pricing Granularity for Congestion-Sensitive Pricing Murat Yüksel and Shivkumar Kalyanaraman Rensselaer Polytechnic Institute, Troy, NY {yuksem, shivkuma} @ecse.rpi.edu

2. Outline Congestion-sensitive pricing: “pricing interval” defined… Implementation issues Dynamics of congestion pricing Analytical models Discussion of analytical models Simulation experiments Summary

3. Congestion-Sensitive Pricing What is congestion pricing?: Increase the price when congestion, decrease when no congestion. A way of controlling user’s traffic demand and hence, a way of controlling network congestion Since congestion is dynamically variant, it is necessary to update the price at some time-scale, what we call “pricing interval”. So, length of pricing interval (i.e. pricing granularity) is a key implementation issue for congestion pricing..

4. Implementation Issues Users do not like price fluctuations: Users want “larger” pricing intervals for convenience… Congestion control is better with more frequent price updates: Providers need “smaller” pricing intervals for better control of congestion. Users want prior (or apriori) pricing: Provider has to undertake overhead of communicating the new price before it is applied. This overhead is more practical for “larger” pricing intervals. So, what is the best value for pricing intervals in order to keep it as large as possible while maintaining a reasonable control of congestion?

5. Dynamics of Congestion Pricing Provider employs pricing interval T. Provider observes the congestion level c, and advertises a price p. What is the steady-state dynamics? That is, how do p and c behave relative to each other?

6. Dynamics of Congestion Pricing (cont’d) Congestion measure c vs. congestion-sensitive price p T 1 > T 2   c 1 >  c 2,  p 1 >  p 2 There is an optimum price p* If p and c are more correlated, then apparently control is finer.. So, correlation between p and c should represent level of control over congestion!!

7. Analytical Models Let: T be the length of pricing intervals r be the # of observations in a pricing interval t be the length of observation intervals, i.e. T=rt c ij be the queue length at the end of j th observation interval of i th pricing interval. m ij and k ij be the # of arrivals and # of departures in j th observation interval of i th pricing interval.

8. Analytical Models (cont’d) Assume that: the customer has a fixed budget and generates traffic according to a continuous time stationary counting process, A(  ).  >= 0, with first and second moments of 1 and 2 respectively.. The packets leave the network according to a continuous time stationary counting process, B(  ),  >= 0, with first and second moments of  1 and  2 respectively.. packet drops in the network core are negligible. Congestion pricing algorithm calculates the price in i th pricing interval by p i = a(t,r) c (i-1)r

9. Analytical Models (cont’d) Conditioned correlation between p and c in n pricing intervals: We first write c ij and p i in terms of m ij and k ij. Then, we write each of the conditioned expectations above in terms of m ij and k ij. To get the final model, we need to relax the condition on m and k:

10. Analytical Models (cont’d) This relaxation is complicated!! Model-I: Assume B(  ) is independent from A(  ). Model-II: Consider the system as M/M/1, i.e. B(  ) and A(  ) are Poisson. We did find a closed form expression, but it is too long.. It is available as a Maple file.

11. Model Discussion The correlation degrades at most inversely proportional to an increase in T. Increase in traffic variances degrades the correlation. Increase in mean of incoming traffic degrades correlation.

12. Experiments We used Dynamic Capacity Contracting (DCC) as the congestion pricing scheme. It uses short-term contracts which corresponds to the pricing interval in our model. 5 customers using a 1Mb/s bottleneck link. We took the bottleneck queue as the congestion measure c, and the price advertised to the customers as p. The observation interval is 80ms, i.e. the bottleneck queue is sampled at every 80ms.

13. Experiments (cont’d) Control on congestion degrades less than linearly..

14. Experiments (cont’d) Control on congestion degrades when mean of incoming traffic increases!!

15. Experiments (cont’d) Control on congestion degrades when traffic variance increases..

16. Experiments (cont’d) Control on congestion vanishes at around 10r, i.e. 40RTTs in our experiment or 2-3secs in general..

17. Summary Length of pricing intervals is a key issue for deployment of congestion pricing. Correlation between p and c represents level of control achieved over congestion by pricing. We developed two approximate models for the correlation between p and c, and validated by simulation experiments. We observed that: the correlation is effected by various statistics about the incoming or outgoing traffic. the correlation vanishes when the pricing interval is about 40 RTT.

18. Questions, Ideas? THANK YOU!