Portfolio optimization in secondary spectrum markets

In this paper, we address the spectrum portfolio optimization (SPO) question in the context of secondary spectrum markets, where bandwidth (spectrum access rights) can be bought in the form of primary and secondary contracts. While a primary contract on a channel provides guaranteed access to the channel bandwidth (possibly at a higher per-unit price), the bandwidth available to use from a secondary contract (possibly at a discounted price) is typically uncertain/stochastic. The key problem for the buyer (service provider) in this market is to determine the amount of primary and secondary contract units needed to satisfy uncertain user demand. We initially consider a single-region problem in which the spectrum contracts are valid only in the single-region in which the buyer wishes to provide service. We formulate the problem as one of minimizing the cost of the spectrum portfolio subject to constraints on bandwidth shortage. Two different forms of bandwidth shortage constraints are considered, namely, the demand satisfaction rate constraint, and the demand satisfaction probability constraint. While the SPO problem under demand satisfaction rate constraint is shown to be convex for all density functions, the SPO problem under demand satisfaction probability constraint is not convex in general. We derive some sufficient conditions for convexity for this case. The SPO problems can therefore be solved efficiently using standard convex optimization techniques. Later, we extend the problem formulation and the convexity results to the multiple-region setting, where the buyer's portfolio is intended to serve a set of disjoint geographical locations, each having its own customer demand. Finally, we perform a thorough simulation-based study of the single-region and the multiple-region problems for different choices of the problem parameters, and provide key insights regarding the portfolio composition and demonstrate the convexity of the efficient frontier. We provide several insights about the scaling behavior of the unit prices of the secondary contracts, as the stochastic characterization of the bandwidth available from secondary contracts change.