**5.2 Auction based access**

For the auction-based access we consider again the three scenarios defined in previous section. Additionally we define three classes of secondary users (SU), characterized by the price that they offer per minute of channel occupation. The bid offers per class are: class 1: 0.01 \$/m, class 2: 0.02 \$/m and class 3: 0.03 \$/m. Additionally, we define the probability of an SU incoming call being of each class. The SU class probability distribution is: class 1 probability: 0.5, class 2 probability: 0.3 and class 3 probability: 0.2. We summarize SU class definition in Table 2.


Table 2. Classification of SU in terms their bid offers and their probabilities.

Note that both the offered prices and their probability distributions are static, *i.e.* they do not change over time and are independent of the system occupation. It is not completely unrealistic taking into account typical tariff policies of wireless operators. In this environment the class structure and the probability distribution may be seen as types of contracts for secondary users and market penetration of each type of contract respectively. However, for a more dynamical auction process, where bidders are able to change their bid offers adaptively, the model should be revised. One possibility would be to define one probability distribution for each state. More detailed modeling strategies would increase the complexity of the MDP solving algorithm or even make them intractable. This is a classic problem of MDP formulation, known as the *curse of dimensionality* and is typically addressed by means of the heuristic approach of approximate dynamic programming.

Fig. 4. Pareto fronts obtained for the auction-based access in scenario 1 (a), scenario 2 (b) and scenario 3 (c)

Figure 4 shows the Pareto fronts for the auction-based system in the three scenarios. As in previous subsection, the MDP and the CMDP approaches provided similar results. It can be observed that, for the same traffic intensity (the three scenarios receive 40 calls per unit of time) when the traffic share of the secondary users is higher (scenarios with higher number) the Pareto front moves away from the y-axis, *i.e.* the income obtained from secondary users increases and it also approaches the x-axis, *i.e.* the blocking probability of the licensed users diminishes. It is interesting to check that, especially in scenarios 2 and 3, a very small increment of the blocking probability of licensed users can multiply the benefit obtained from spectrum leasing by a factor of 2 or 3. On the other hand, these figures also indicate that once the income surpasses certain threshold, Pareto-optimal policies can only produce small increments of the income by dramatically rising the blocking probability.
