Pricing Granularity for Congestion-Sensitive Pricing

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 • T1 > T2  c1 > c2,p1 > p2 • 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 • cij be the queue length at the end of jth observation interval of ith pricing interval. • mij and kijbe the # of arrivals and # of departures in jth observation interval of ith 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 ith pricing interval by pi = 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 cij and pi in terms of mij and kij. • Then, we write each of the conditioned expectations above in terms of mij and kij. • 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!