# Call Blocking Probabilities under a Probabilistic Bandwidth Reservation Policy in Mobile Hotspots
## 2 The Probabilistic BR Loss Model  
### The Analytical Model  
Consider a vehicle with an AP installed on it and let the capacity of the AP's WLAN be equal to \(C\) b.u. The CAC proposed in [7] considers two different phases regarding the vehicle: a stop phase and a moving phase. During the stop phase, the vehicle remains for an exponentially distributed time in a location (e.g., a bus stop or a bus station), with mean \(\theta_{s}^{-1}\), and can accommodate Poisson arriving calls. The latter requests a single b.u. in order to be accepted in the system. If this b.u. is not available then an arriving call is blocked and lost without further affecting the system. Two types of users are considered during the stop phase: (1) new users who initiate a call after riding on the vehicle and (2) handover users who already have an ongoing call and should perform a handover from a NB to the vehicle's AP. To facilitate handover users, the probabilistic BR policy is considered whereby \(t_{new}\) b.u. are reserved for them. This means that a prioritization is applied to handover calls since they are 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. \(j=C-t_{new},\ldots,C-1\) and (ii) with probability 1 when the occupied b.u. \(j=C\). Note that in the case of the \"strict\" BR policy, the value of \(p_{s,new}=0\) when \(j=C-t_{new},\ldots,C-1\). The reason behind such a probabilistic BR policy is the following: when the \"strict\" BR policy is applied to prioritize a certain type of call, it can lead to a substantial CBP increase of the other types of calls that share the same link. In our case, the introduction of a probabilistic BR policy and the proper selection of \(p_{s,new}\) can protect new calls from such a substantial CBP increase. As a last comment, note that if \(t_{new}=0\) then \(p_{s,new}=1\), i.e., new calls are allowed to have access in all \(C\) b.u. During the moving phase, the vehicle moves from one location to another. We assume that the duration of the moving phase is also exponentially distributed with mean \(\theta_{m}^{-1}\). In the moving phase, handover calls (from the NB to the vehicle's AP) cannot be generated and therefore only new calls should be considered. Consequently, there is no reason to adopt the BR policy in that phase. More specifically, during a moving phase, the CS policy is adopted. 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 as a 2-D Markov chain, whose state space diagram is depicted in Figure 1, assuming for illustration purposes that the reservation space starts in state (0,2). Based on this chain, let \(P(i,n)\) be the steady-state probability that the vehicle is in phase \(i\) (\(i=0\) refers to the stop phase and \(i=1\) refers to the moving phase) and there exist \(n\) users in-service, where \(0\leq n\leq C\). Figure 1: The 2-D state transition diagram in the proposed probabilistic BR loss model. We initially consider the global balance equations for the boundary states (0,0) and (1,0):  
\[ \begin{array}{ll}\text{state (0,0)}:&(\lambda_{new}+\lambda_{h}+\theta_{s})P(0,0)- \mu_{s}P(0,1)-\theta_{m}P(1,0)=0\quad(for\,\,\,i=0,n=0)\\ \text{state (1,0)}:&(\lambda_{new}+\theta_{m})P(1,0)-\mu_{m}P(1,1)-\theta_{s}P(0,0)=0 \quad(fori=1,n=0)\end{array} \]  
or in a more compact way:  
\[(\lambda_{i}(0)+\theta_{i})P(i,0)-\mu_{i}P(i,1)-\theta_{j}P(j,0)=0 \tag{1}\]  
\[