# Call Blocking Probabilities under a Probabilistic Bandwidth Reservation Policy in Mobile Hotspots
## 3 The Generalized Probabilistic BR Loss Model  
### The Analytical Model  
In the generalized probabilistic BR loss model, the probabilistic BR policy is considered in both stop and new phases. More specifically, the CAC mechanism, during the stop phase, is as follows: a handover call is blocked only if all \(C\) b.u. are occupied while a new call is blocked: (i) with a predefined probability (\(1-p_{s,new}\)) when the occupied b.u. \(j=C-t_{new}\),..., \(C-1\) and (ii) with probability 1 when the occupied b.u. \(j=C\). On the same hand, during the moving phase, a new call is blocked: (i) with a predefined probability (\(1-p_{m,new}\)) when the occupied b.u. \(j=C-t_{new}\),..., \(C-1\) and (ii) with probability 1 when the occupied b.u. \(j=C\). Note that if \(t_{new}=0\), then \(p_{s,new}=p_{m,new}=1\), i.e., new calls are allowed to have access in all \(C\) b.u. in both phases.  
Let \(\lambda_{new}\) and \(\lambda_{h}\) be the arrival rate for Poisson arriving new and handover calls, respectively. Accepted calls remain in the system for an exponentially distributed service time, with mean \(\mu_{s}^{-1}\) in the stop phase and \(\mu_{m}^{-1}\) in the moving phase.  
The proposed analytical model can be described via the state space diagram of Figure 2, assuming that the reservation space starts in state (0,2). Based on this 2-D Markov chain, let \(P_{g}(i,n)\) be the steady-state probability of the generalized loss model which expresses the fact that the vehicle is in phase \(i\) and there exist \(n\) users in-service, where \(0\leq n\leq C\).  
We initially consider the global balance equations for the boundary states (0,0) and (1,0):  
\[\begin{array}{ll}\text{state }\left(0,0\right):&\left(\lambda_{new}+\lambda_{ h}+\theta_{s}\right)P_{g}(0,0)-\mu_{s}P_{g}(0,1)-\theta_{m}P_{g}(1,0)=0\quad(for\quad i=0,n=0)\\ \text{state }\left(1,0\right):&\left(\lambda_{new}+\theta_{m}\right)P_{g}(1,0)- \mu_{m}P_{g}(1,1)-\theta_{s}P_{g}(0,0)=0\quad(for\quad i=1,n=0)\end{array}\]  
Figure 2: The 2-D state transition diagram in the proposed generalized probabilistic BR loss model.  
or in a more compact way:  
\[(\lambda_{i}(0)+\theta_{i})P_{g}(i,0)-\mu_{i}P_{g}(i,1)-\theta_{j}P_{g}(j,0)=0 \tag{18}\]  
\[\text{where }\lambda_{i}(0)=\left\{\begin{array}{ll}\lambda_{new}+\lambda_{h},&i=0 \\ \lambda_{new}&,\quad i=1\end{array}\right.,\quad\theta_{i}(0)=\left\{\begin{array} []{ll}\theta_{s}\,&i=0\\ \theta_{m},&i=1\end{array}\right.,\quad\mu_{i}=\left\{\begin{array}{ll}\mu_{ s}\,&i=0\\ \mu_{m},&i=1\end{array}\right.,\] \(j=1-i\).  
We consider now the states below the system's capacity, i.e., the states where \(0<n<C\). For these states, the following global balance equation can be written:  
\[(\lambda_{i}(n)+n\mu_{i}+\theta_{i})P_{g}(i,n)-\lambda_{i}(n-1)P_{g}(i,n-1)- \theta_{j}P_{g}(j,n)-(n+1)\mu_{i}P_{g}(i,n+1)=0 \tag{19}\]  
\[\text{where }\lambda_{i}(n)=\left\{\begin{array}{ll}\lambda_{new}+\lambda_{ h},&i=0,\quad 0<n\leq C-t_{new}-1\\ p_{s,new}\lambda_{new}+\lambda_{h},&i=0,\quad C-t_{new}\leq n<C\\ \lambda_{new},&i=1,\quad 0<n<C-t_{new}-1\\ p_{m,new}\lambda_{new},&i=1,\quad C-t_{new}\leq n<C\end{array}\right.,\ j=1-i.\]  
Finally, we consider the boundary states (0,C) and (1,C). For these states, we have the following global balance equation:  
\[(C\mu_{i}+\theta_{i})P_{g}(i,C)-\lambda_{i}(C-1)P_{g}(i,C-1)-\theta_{j}P_{g}(j,C)=0 \tag{20}\]  
To determine the values of \(P_{g}(i,n)\) based on (18)-(20), the iterative algorithm of Section 2.2 can be adopted.