Call Blocking Probabilities under a Probabilistic Bandwidth Reservation Policy in Mobile Hotspots

In this paper we study a mobility-aware call admission control algorithm in a mobile hotspot. To this end, a vehicle is considered which has an access point with a fixed capacity. The vehicle alternates between stop and moving phases. When the vehicle is in the stop phase, it services new and handover calls by prioritizing them via a probabilistic bandwidth reservation (BR) policy. Based on this policy, new handover calls may enter the reservation space with a predefined probability. When the vehicle is in the moving phase, it services new calls only. In that phase, two different policies are considered: (a) the classical complete sharing (CS) policy, where new calls are accepted in the system whenever there exists available bandwidth, and (b) the probabilistic BR policy. Depending on the selected policy in the moving phase, we propose the probabilistic BR loss model (if the CS policy is selected) and the generalized probabilistic BR loss model (if the probabilistic BR policy is selected). In both stop and moving phases, where the call arrival process is Poisson, calls require a single bandwidth unit in order to be accepted in the system, while the service time is exponentially distributed. To analytically determine call blocking probabilities and the system’s utilization, we propose efficient iterative algorithms based on two-dimensional Markov chains. The accuracy of the proposed algorithms is verified via simulation.


Introduction
Call admission control (CAC) is a significant quality of service (QoS) mechanism in contemporary networks, not only because it controls the access to the bandwidth units (b.u.) required by new mobile users but also because it provides fairness among different mobile applications and at the same time affects the QoS of in-service calls [1][2][3][4].
Considering call-level traffic in wireless networks, such a QoS mechanism is a resource sharing policy that affects call blocking probabilities (CBP) of both new and handover calls. The term CBP refers to the blocking of new or handover calls due to the unavailability of b.u. To reduce CBP (especially of handover calls), various bandwidth sharing policies exist in the literature, with the most common being the guard channel or bandwidth reservation (BR) policy [5][6][7][8][9][10][11][12]. The main characteristic of the BR policy is that it introduces a priority to handover calls by reserving b.u. in favor of them. On the other hand, such a reservation usually results in a significant CBP increase of new calls [13][14][15][16].
We focus on a CAC mechanism proposed in [7] that is based on the BR policy and is applied on a mobile hotspot. The work of [7] considers a vehicle that has an access point (AP) installed on it with a wireless local area network (WLAN) capacity of C b.u. The vehicle can be in two different phases: (a) in the stop phase and (b) in the moving phase. During the stop phase, a vehicle remains in a location (e.g., a station area) and can accommodate two different types of users: new and handover users. New users may initiate a call when they enter the vehicle. On the other hand, handover users have an

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 θ −1 s , 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 , . . . , 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 , . . . , 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 θ −1 m . 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 λ new and λ 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 µ −1 s in the stop phase and µ −1 m 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 ≤ n ≤ C.
vehicle: a stop phase and a moving phase.
During the stop phase, the vehicle remains for an exponentially distributed time location (e.g., a bus stop or a bus station), with means θ 1 , and can accommodate Pois arriving calls. The latter requests a single b.u. in order to be accepted in the system. If b.u. is not available then an arriving call is blocked and lost without further affecting system. Two types of users are considered during the stop phase: (1) new users who tiate a call after riding on the vehicle and (2) handover users who already have an ongo call and should perform a handover from a NB to the vehicle's AP. To facilitate hando users, the probabilistic BR policy is considered whereby tnew b.u. are reserved for th This means that a prioritization is applied to handover calls since they are blocked onl all C b.u. are occupied while a new call is blocked: (i) with a predefined probability (1−ps when the occupied b.u. j = tnew, …, C − 1 and (ii) with probability 1 when the occupied j = C. Note that in the case of the "strict" BR policy, the value of ps,new = 0 when j = tnew, C − 1. The reason behind such a probabilistic BR policy is the following: when the "str BR policy is applied to prioritize a certain type of call, it can lead to a substantial C increase of the other types of calls that share the same link. In our case, the introduct of a probabilistic BR policy and the proper selection of ps,new can protect new calls fr such a substantial CBP increase. As a last comment, note that if tnew = 0 then ps,new = 1, 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 sume that the duration of the moving phase is also exponentially distributed with m m θ 1 . In the moving phase, handover calls (from the NB to the vehicle's AP) cannot be g erated and therefore only new calls should be considered. Consequently, there is no r son to adopt the BR policy in that phase. More specifically, during a moving phase, CS policy is adopted.
Let λnew and λh be the arrival rate for Poisson arriving new and handover calls, spectively. Accepted calls remain in the system for an exponentially distributed serv time, with means μ 1 in the stop phase andm μ 1 in the moving phase. The proposed analytical model can be described as a 2-D Markov chain, whose s space diagram is depicted in Figure 1, assuming for illustration purposes that tnew = C Based on this chain, let P(i,n) be the steady-state probability that the vehicle is in pha (i = 0 refers to the stop phase and i = 1 refers to the moving phase) and there exist n us in-service, where 0 n C   .    We initially consider the global balance equations for the boundary states (0,0) and (1,0): or in a more compact way: 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: Finally, we consider the boundary states (0,C) and (1,C). For these states, we have the following global balance equation: To determine the values of P(i,n) based on (1)- (3), an iterative algorithm is proposed in the following subsection.

An Iterative Algorithm for the Determination of P(i,n)
Initially, we express P(i,n) via the boundary states P(0,0) and P(1,0) as follows: At this point, all terms in (4) (including the coefficients S 0 i,n and S 1 i,n ) are unknown. A formula for the derivation of S 0 i,n and S 1 i,n can be obtained if we substitute (4) in (2), to have: where j = 1−i.
For a system of capacity C b.u., the determination of S 0 i,n and S 1 i,n requires 2C Equations of (6) and (7), respectively.
Up to this point, the coefficients S 0 i,n and S 1 i,n have been determined (via (6) and (7), respectively). The next step is to calculate the values of P(0,0) and P(1,0). To this end, a system of two equations is necessary. The first equation is based on (3), where the case of i = 0 is considered: Equation (8), via (4), can be written as: Equation (9) is the first equation to be used for the determination of P(0,0) and P(1,0). The second formula is based on the fact that: Equation (10), via (4), can be written as: Equation (11) is the second equation to be used for the calculation of P(0,0) and P(1,0). Having determined the values of P(0,0) and P(1,0) (via (9) and (11), respectively) and the values of S 0 i,n and S 1 i,n (via (6) and (7), respectively), we can calculate the values of P(i,n) via (4) and, consequently the CBP of new and handovers calls together with the link utilization, according to the next subsection.

CBP Determination of New and Handover Calls
The CBP of new calls in the stop phase, B s,new , can be determined via: In the case of the "strict" BR policy, the same CBP formula can be adopted, assuming that p s,new = 0 for j = C − t new , . . . , C − 1.
In the moving phase, call blocking of new calls occurs only when there are no available b.u. Consequently, the CBP determination of new calls in the moving phase, B m,new , can be based on the following formula: Based on (12) and (13), the total CBP of new calls, B new , can be determined via: The CBP of handover calls in the stop phase, B s,h , can be determined via: Telecom 2021, 2

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The CBP of handover calls given stop phase, B * s,h , can be determined via: Finally, the link utilization, U, can be computed via the following formula: A tutorial example at the end of this paper (see Appendix A) presents in detail all necessary calculations for the determination of B s,new ,B m,new ,B s,h and U.

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 Let λ new and λ 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 µ −1 s in the stop phase and µ −1 m 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 ≤ n ≤ C.
Finally, the link utilization, U, can be computed via the following formula:

The Analytical Model
In the generalized probabilistic BR loss model, the probabilistic BR policy ered in both stop and new phases. More specifically, the CAC mechanism, durin phase, is as follows: a handover call is blocked only if all C b.u. are occupied wh call is blocked: (i) with a predefined probability (1−ps,new) when the occupied b.
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: Finally, we consider the boundary states (0,C) and (1,C). For these states, we have the following global balance equation: To determine the values of P g (i,n) based on (18)-(20), the iterative algorithm of Section 2.2 can be adopted.

CBP Determination of New and Handover Calls
The CBP of new calls in the stop phase, B g,s,new , can be determined via: In the moving phase of the generalized model, call blocking of new calls occurs not only when there are no available b.u., but also probabilistically in states P g (1,n) where C − t new ≤ n < C. Consequently, the CBP computation of new calls in the moving phase,B g,m,new , is based on the following formula: Based on (21) and (22), the total CBP of new calls, B g,new , can be determined via: The CBP of handover calls in the stop phase, B g,s,h , can be determined via: The CBP of handover calls given stop phase, B * g,s,h , can be determined via: Telecom 2021, 2

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Finally, the link utilization, U g , can be computed via the formula:

Performance Evaluation
In this section, we consider an application example of a vehicle with an AP installed on it and let the capacity of the AP's WLAN be equal to C = 58 b.u. During the stop phase, the vehicle remains for an exponentially distributed time in a location with mean θ −1 s =3. We also assume that the duration of the moving phase is exponentially distributed with mean θ −1 m = 6. The vehicle accommodates Poisson arriving calls with λ h = 24 (fixed value) and λ new = 2 (variable value). Accepted calls remain in the system for an exponentially  [7]). Note that the probabilistic BR loss model refers to the case of p m,new = 1.0 (the CS policy is applied to new calls and therefore no reservation occurs) and p s,new < 1.0. Since this model leads to similar conclusions with the generalized loss model, we present analytical and simulation results of the generalized probabilistic BR loss model only. The simulation results presented herein are based on Simscript III and are the mean values of eight runs [25]. In each run, the initial 5% of the generated calls (one million generated calls per run) is not taken into account in the CBP results, in order to obtain a warm-up period [26,27]. Finally, reliability ranges which are less than two orders of magnitude are not presented in the CBP graphs. Table 1 includes the list of simulation parameters used in this application example. In the x-axis of Figures 3-5, the values of λ new increase in steps of 2. Thus, point 1 (in the x-axis) refers to λ new = 2 and point 10 refers to λ new = 20. In the x-axis of Figures 6-8, we consider 10 different values of θ −1 s , assuming that λ h = 24 (fixed value), λ new = 16 (fixed value) and θ −1 m = 6 (fixed value). More precisely, we consider the values 0.2, 0.25, 0.3333, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0 and 6.0. To increase the readability of all graphs, simulation results are represented with different types of dots, while analytical results are represented with different types of lines. Besides, two different arrows have been included in each graph in order to identify the results obtained via [7]. An arrow identifies the case of p s,new = p m,new = 0.0 (the cutoff priority scheme of [7]) and another arrow identifies the case of p s,new = p m,new = 1.0 (the nonpriority scheme of [7]). Figure 3 presents the analytical together with the simulation results of the total CBP of new calls for the five different sets of p s,new and p m,new . Figure 4 presents the corresponding results of the CBP of handover calls (based on (15) and (24)). Finally, Figure 5 presents the analytical and simulation results of the system's utilization. All graphs show that the analytical results are similar to the corresponding simulation results. Regarding  [7] cannot capture the behavior of the proposed models, a fact that is expected since the models of [7] do not consider the case of the probabilistic BR policy. Regarding Figure 4, we observe that: (i) the increase of λ new increases the CBP of handover calls (in the stop phase); (ii) the choice of p s,new and p m,new affects the CBP of handover calls leading to high CBP values (when p s,new = p m,new = 1.0) and low CBP values (when p s,new = p m,new = 0.0) (i.e., the increase of p s,new and p m,new has the opposite impact on the CBP of new and handover calls); (iii) the CBP results obtained via [7] cannot capture the behavior of the proposed models. Finally, regarding Figure 5, we observe that: (i) the increase of λ new increases the system's utilization and (ii) a slightly higher utilization appears when p s,new = p m,new = 1.0 (i.e., when the BR policy is not applied in favor of handover calls).
In Figures 6-8, we present the total CBP of new calls (Figure 6), the CBP of handover calls in the stop phase (

Conclusions
We propose two loss models for a mobile hotspot that accommodates new and hand-

Conclusions
We propose two loss models for a mobile hotspot that accommodates new and handover users. To favor handover users but at the same time not substantially increase the CBP of new users, we propose the probabilistic BR policy where new calls may enter the reservation space with a certain, predefined, probability. In the first loss model, the probabilistic BR policy is applied only to new calls during the stop phase of a vehicle. In the second loss model, a generalization is considered, where the probabilistic BR policy can be applied not only during the stop phase but also during the moving phase. Both models are analytically described with the aid of two-dimensional Markov chains, while iterative algorithms are proposed for the determination of the steady-state probabilities and the various performance measures such as CBP and system's utilization.
As a future study, we will extend these models to include the case of quasi-random traffic, where a finite number of users may generate calls in the system not only during the stop but also during the moving phase. Additionally, we intend to study the case of multiservice traffic where users generate traffic (either random or quasi-random) and have different bandwidth-per-call requirements, i.e., a user may request more than a single b.u. in order to be accepted in the system. Such extensions have been considered in the literature in wired/wireless networks but not in the case of mobile hotspots [28][29][30][31][32][33][34][35][36].    In what follows, we consider the proposed algorithm of Section 2.2 and present the calculations of the coefficients S 0 i,n and S 1 i,n . According to (6) and since S 0 0,0 = 1, S 0 1,0 = 0 and S 0 0,y = 0 for y < 0, we have the following values of Having determined the values of P(0,0) and P(1,0), (via (9) and (11), respectively) and the values of S 0 i,n and S 1 i,n (via (6) and (7), respectively), we can calculate the values of P(i,n) via (4) and, consequently the CBP of new and handovers calls together with the link utilization.
For reference, we present the values of the various performance measures. More specifically, the CBP of new calls in the stop phase,B s,new , can be determined via (12) as follows: