This paper looks at time-dependent pricing schemes. A day is split into 48 half hour periods indexed by an integer. The system is known as TUBE (Time-dependent Usage-based Broadband-price Engineering). They use a control loop to adapt the prices ISPs charge users in response to changing behaviour.
A “waiting function” describes a users willingness to wait an amount of time given reward . The problem is first set as a convex optimisation problem given the target for the ISP which is to minimise the cost paid for links plus the cost paid in “rewards” to users. The cost function for links is assumed to be piecewise linear with bounded slope. Essentially the modelling occurs as if all ISP traffic went through a single link and a price were paid on this in every period where is the cost function, is the usage in period and is the available capacity in period .
Following this, a dynamic version of the model is developed in online and offline settings. The offline version assumes Poisson arrivals within a time period with a rate and exponentially distributed file sizes (the latter assumption is questionable as most studies show larger than exponential tails in file size distributions). The offline version chooses the rewards to maximise ISP profit. It is shown that this is equivalent to the static model but with “leftover” downloads which do not complete in one period added on to the start of the next. An online version is developed which solves the optimisation as a dynamic programming problem and calculates the optimal reward for steps ahead based on the current situation, existing rewards for steps ahead and predictions about arrivals based on past behaviour. The assumption remains that the system is constrained by a single bottleneck link.
The authors then describe a method which calculates the “waiting function” (a measure of the willingness of users to defer their downloads for given rewards). This is done by estimating the difference in demand between time independent and time dependent pricing. This is done by estimating a (potentially very large) number of parameters for each application type.
Simulation is performed using an input which is aggregate traffic from a large ISP. The authors show that the reward system can be used to smooth traffic throughout the day and hence to increase ISP profit.
More details and an expanded paper is C. Wong, S. Ha and M. Chiang, Time-Dependent Broadband Pricing: Feasibility and Benefits, ICDCS 2011. http://www.princeton.edu/~chiangm/timedependentpricing.pdf. This also contains a list of time dependent pricing papers in various fields.
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