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 its uncertain user demand. We formulate single and multi-region spectrum portfolio optimization problems 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 in this case. We also discuss application of the Bernstein approximation technique to approximate a non-convex demand satisfaction probability constraint by a convex constraint. The SPO problems can therefore be solved efficiently using standard convex optimization techniques. We then consider a discrete version of the SPO problem, in which the primary and secondary contracts can bought/sold in discrete units. We study the NP-hardness submodularity property of the discrete SPO problem and discuss a branch-and-bound algorithm to obtain the optimal solution for this problem. Finally, we perform a thorough simulation-based study of the singleregion and the multiple-region problems for different choices of the problem parameters, and provide key insights regarding the portfolio composition, the efficiency of the Bernstein convex approximation technique, and the closeness of the optimal discrete spectrum portfolio solutions to their continuous approximations. 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.