# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 3 System Behavior Analysis Model  
The suggested model, system behavior analysis model (SAM), is a state transition model that uses a price vector \(\{P_{i}\}\) as an input and produces a system state transition graph by estimating the decisions of users. This graph, in which each state is identified by the associated job arrival rate vector \(\{\lambda_{i}\}\), depicts the evolution of the system behavior over time. The state transition is discrete and occurs only when new values of \(\{\lambda_{i}\}\) are broadcast, as shown in Figure 2. \(S^{t}\) denotes the system state at time t, and \(\{\lambda_{i}\}^{t}\) denotes the job arrival rate vector at \(S^{t}\). The transition of the system state is derived by modeling the decisions of individual users. Recall that users make decisions based on the knowledge that is currently available, \(\{\lambda_{i}\}\), which remains fixed between updates. The actual \(\{\lambda_{i}\}\) may change as a result of the users' decisions during the interval between broadcasts, but this new \(\{\lambda_{i}\}\) is not available for users to base their decisions on. Depending on whether a given \(\{P_{i}\}\) guarantees that the system converges on a steady state, SAM handles the two cases differently. When the system converges, the SAM produces a single \(\{\lambda_{i}\}\) of the convergent system state. On the other hand, when the system oscillates, the SAM computes a set of \(\{\lambda_{i}\}\)'s. System divergence (i.e., oscillation among an infinite number of states) is theoretically possible but does not occur because the resolution of \(\{\lambda_{i}\}\) broadcast values is finite. The simulation indicates that this set contains a small number of states in most cases. Each \(\{\lambda_{i}\}\) in this set represents a recurring system state in the oscillation loop and is associated with a probability that indicates how long the system will stay in that corresponding state. The SAM first estimates the most preferred service level for each job type in a certain system state. For all pairs of \((i,j)\), it computes the Bernoulli variable, \(X^{t}_{i,j}\), which takes '\(1\)' if the level \(i\) is the most profitable choice for a type-\(j\) job for the given price vector \(\{P_{i}\}\) when the system is at \(S^{t}\) and takes '\(0\)' otherwise. That is,  
\[X^{t}_{i,j} = \left\{ \begin{array}{ll}1,&\text{if }U_{j}\Big{(}T^{t}_{ij}\Big{)}- P_{i}C_{j}\geq 0,&\text{and }U_{j}\Big{(}T^{t}_{ij}\Big{)}-P_{i}C_{j}\geq U_{j}\Big{(}T^{t}_{ij}\Big{)}-P_ {k}C_{j}\text{ for }\forall k\in I,\ k\
eq i\\ 0,&\text{otherwise}\end{array} \right.\]  
where \(T^{t}_{ij}\) is the expected completion time of a type-\(j\) job at level \(i\) at state \(S^{t}\). Note that the user's decision between time \(t\) and \(t+1\) is based on \(\{\lambda_{i}\}^{t}\). Second, using \(X^{t}_{ij}\), the SAM computes the probability that level \(i\) is selected at \(S^{t}\), \(\mathit{Prob}\big{(}i,S^{t}\big{)}\) by the following equation:  
\[\mathit{Prob}\big{(}i,S^{t}\big{)}\ =\ \sum_{j}\mathit{Prob}(j)\cdot X^{t}_{i,j} \tag{3}\]  
where \(\mathit{Prob}(j)\) is the probability that a job is of type \(j\). Finally, using \(\mathit{Prob}\big{(}i,S^{t}\big{)}\), the SAM computes the job arrival rate vector at \(S^{t-1}\), \(\{\lambda_{i}\}^{t+1}\), as follows:  
\[\lambda^{t+1}_{i}\ =\ \lambda\ \cdot\ \mathit{Prob}\big{(}i,\ S^{t}\big{)} \text{ for }\forall i\in I. \tag{4}\]  
In this way, the SAM constructs a state transition graph where a node corresponds to a system state visited, and a directional arc indicates a transition from one system state to another. The transition of \(\{\lambda_{i}\}\) may end up as a single state, as shown in Figure 2(a), or may  
Figure 2: \(\{\lambda_{i}\}\) Transition over time. oscillate, as shown in Figure 3b. In the latter case, the stationary probability at each state \(S\), \(W(S)\), in the oscillation loop is computed by counting the visit frequency.