Pricing Granularity for Congestion-Sensitive

Pricing

Murat Yüksel and Shivkumar KalyanaramanRensselaer Polytechnic Institute, Troy, NY

{yuksem, shivkuma} @ecse.rpi.edu

Outline

Congestion-sensitive pricing: “pricing interval” defined…

Implementation issues Dynamics of congestion pricing Analytical models Discussion of analytical models Simulation experiments Summary

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..

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?

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?

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!!

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 kij be the # of arrivals and # of departures

in jth observation interval of ith pricing interval.

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

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:

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.

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.

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.

Experiments (cont’d)

Control on congestion degrades less than linearly..

Experiments (cont’d)

Control on congestion degrades when mean of incoming traffic increases!!

Experiments (cont’d)

Control on congestion degrades when traffic variance increases..

Experiments (cont’d)

Control on congestion vanishes at around 10r, i.e. 40RTTs in our experiment or 2-3secs in general..

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.

Questions, Ideas?

THANK YOU!