We consider a wireless provider who caters to two classes of customers, namely primary users (PUs) and secondary users (SUs). PUs have long term contracts while SUs are admitted and priced according to current availability of excess spectrum. The average rate at which SUs attempt to access the spectrum is a function on the currently advertised price, referred to as the demand function. We analyze the problem of maximizing the average profit gained by admissions of SUs, when the demand function is unknown. We introduce a new on-line algorithm, called Measurement-based Threshold Pricing (MTP), that requires the optimization of only two parameters, a price and a threshold, whereby SU calls are admitted and charged a fixed price when the channel occupancy is lower than the threshold and rejected otherwise. At each iteration, MTP measures the average arrival rate of SUs corresponding to a certain test price. We prove that these measurements of the secondary demand are sufficient for MTP to converge to a local optimal price and corresponding optimal threshold, within a number of measurements that is logarithmic in the total number of possible prices. We further provide an adaptive version of MTP that adjusts to time-varying demand and establish its convergence properties. We conduct numerical studies showing the convergence of MTP to near-optimal online profit and its superior performance over a traditional reinforcement learning approach.

We consider a wireless provider who caters to two classes of customers, namely primary and secondary users. Primary users have long term contracts while secondary users are admitted and priced according to current availability of excess spectrum. Secondary users accept an advertised price with a certain probability defined by an underlying demand function. We analyze the problem of maximizing profit gained by admission of secondary users. Previous studies in the field usually assume that the demand function is known and that the call length distribution is also known and exponentially distributed. In this paper, we analyze more realistic settings where both of these quantities are unknown. Our main contribution is to derive near-optimal pricing strategies under such settings. We focus on occupancy-based pricing policies, which depend only on the total number of ongoing calls in the system. We first show that such policies are insensitive to call length distribution except through the mean. Next, we introduce a new on-line, occupancy-based pricing algorithm, called Measurement-based Threshold Pricing (MTP) that operates by measuring the reaction of secondary users to a specific price and does not require the demand function to be known. MTP optimizes a profit function that depends on price only. We prove that while the profit function can be multimodal, MTP converges to one of the local optima as fast as if the function were unimodal. Lastly, we provide numerical studies demonstrating the near-optimal performance of occupancy-based policies for diverse sets of call length distributions and demand functions and the quick convergence of MTP to near-optimal on-line profit.

Recent deregulation initiatives enable cellular providers to sell excess spectrum for secondary usage. In this paper, we investigate the problem of optimal spot pricing of spectrum by a provider in the presence of both non-elastic primary users, with long-term commitments, and opportunistic, elastic secondary users. We first show that optimal pricing can be formulated as an infinite horizon average reward problem and solved using stochastic dynamic programming. Next, we investigate the design of efficient single pricing policies. We provide numerical and analytical evidences that static pricing policies do not perform well in such settings (in sharp contrast to settings where all the users are elastic). On the other hand, we prove that deterministic threshold pricing achieves optimal profit amongst all single-price policies and performs close to global optimal pricing. We characterize the profit regions of static and threshold pricing, as a function of the arrival rate of primary users. Under certain reasonable assumptions on the demand function, we show that the profit region of threshold pricing can be far larger than that of static pricing. Moreover, we also show that these profit regions critically depend on the support of the demand function rather than specific form of it. We prove that the profit function of threshold pricing is unimodal in price and determine a restricted interval in which the optimal threshold lies. These two properties enable very efficient computation of the optimal threshold policy that is far faster than that of the global optimal policy.