Dynamic Modeling of Target Cell Location for Mobility Robustness Analysis in Cellular Networks: Technical Report

Dynamic Modeling of Target Cell Location for Mobility Robustness Analysis in Cellular Networks: Technical Report Kiichi Tokuyama

arXiv:2608.02467v1 [cs.NI] 3 Aug 2026

Graduate School of Engineering, Mie University, Japan tokuyama@info.mie-u.ac.jp Abstract—Mobility robustness optimization (MRO) requires an appropriate selection of handover (HO) parameters such as the time-to-trigger (TTT) and the offset margin to balance HO failures and ping-pong HOs. Existing stochastic-geometry-based analyses for MRO often characterize the target base station (BS) by assuming that its angular position is uniformly distributed over a feasible region. However, this assumption does not explicitly capture the spatial distribution of the target BS dynamically selected as a user equipment (UE) moves through the network. In this paper, we develop a stochastic-geometry-based analytical framework for MRO in sub-6 GHz cellular networks. We first derive the distribution of the HO triggering time and the spatial distribution of the dynamically selected target BS under straightline UE mobility. Based on these distributions, we formulate toolate HO and ping-pong HO events as mutually exclusive mobility events and analytically derive their probabilities. Numerical results validate the analytical expressions and demonstrate that explicitly characterizing the target BS distribution has a nonnegligible impact on the evaluated HO performance. Furthermore, the derived framework quantifies the tradeoff between toolate HO and ping-pong HO probabilities with respect to the TTT and enables the identification of a TTT value that minimizes their sum. Index Terms—Cellular networks, target cell modeling, handover failure, ping-pong handover, stochastic geometry

I. Introduction Mobile cellular communications have continuously evolved from 5G New Radio (NR) toward 6G, accompanied by the diversification of communication infrastructures. Meanwhile, terrestrial cellular networks, built upon 4G Long-Term Evolution (LTE), still play an important role. Ensuring seamless and stable cellular communications for user equipments (UEs) is still an active area of research. In recent cellular networks, Self-Organizing Networks have been introduced to autonomously optimize network parameters for the efficient operation of a large number of base stations (BSs). Among their functionalities, Mobility Robustness Optimization (MRO) has been extensively studied as a promising technique for maintaining communication quality by automatically and adaptively adjusting handover (HO) control parameters for mobile UE. In general, the timing of HO execution is controlled by HO parameters such as Time-toTrigger (TTT), Hysteresis, and A3 offset. Smaller values of the This work was supported in part by a research grant from the Okasan-Kato Foundation, Tsu, Japan. The author would like to thank Prof. Naoto Miyoshi for valuable discussions.

HO parameters reduce the likelihood of HO failures (HOFs) at the cost of increasing Ping-pong (PP) HOs, whereas larger values suppress PP HOs while increasing the probability of HOFs accompanied by communication interruptions. Therefore, there exists an inherent tradeoff between HOFs and PP HOs, highlighting the importance of appropriately optimizing HO parameters according to UE mobility conditions. To optimize HO parameters while addressing the tradeoff between HOFs and PP HOs, extensive studies on MRO have been conducted. In particular, machine learning-based optimization methods [1], [2] and theoretical approaches based on Markov chain [3] and Stochastic Petri Net [4] have been proposed. In addition, Geometry-based approaches that model cell boundaries and UE mobility using geometric arguments have been considered to analyze HOF and PP rates [5]–[7]. Recently, stochastic geometry based on Poisson point processes (PPPs) has been widely employed to evaluate HO, PP, and HOF rates in various network environments [8]– [14]. By capturing the spatial distribution of BSs and the mobility characteristics of UEs, stochastic geometry provides a powerful analytical framework for HO performance evaluation and offers valuable design insights for MRO. In [8], the HOF rate was analytically derived for UEs moving along straight trajectories in small-cell networks. In contrast, [9] derived the distribution of the sojourn time of the serving cell in multitier heterogeneous networks (HetNets) and evaluated the PP rate by combining it with the HO rate. In [10], both HOF and PP rates were analytically derived for two-tier HetNets. This analytical framework was subsequently extended to threetier HetNets with traffic hotspots [11] and two-tier HetNets with cellular-connected unmanned aerial vehicles (UAVs) [12]. However, [8] and [9] focused exclusively on either the HOF or PP rate, respectively. Moreover, the HOF analyses in [10]– [12] considered only the geometric relationship between the serving and target BSs. Consequently, Signal-to-Interferenceplus-Noise Ratio (SINR)-based evaluations of HOFs accounting for the aggregate interference from surrounding BSs were not considered. In contrast, recent studies have also investigated SINR-based HOF analyses that take into account the aggregate interference from surrounding BSs. In [13], both HOF and PP rates were analytically derived for multi-tier UAV networks by transforming the altitude information of UAVs into equivalent distances

on the ground plane, thereby establishing an analytical framework equivalent to that for terrestrial cellular networks. This framework was extended to UAV-assisted multi-tier HetNets, where the impact of channel fading was also evaluated [14]. However, these studies in [13] and [14] assume that, conditioned on the distance to the target BS, its angular position is uniformly distributed over the feasible region at the HO triggering location. Consequently, the spatial distribution of target BSs induced by the effect of moving UEs has not been considered. To the best of our knowledge, no analytical framework has been developed to model the target BS distribution induced by the UE dynamics and evaluate its impact on HO performance in the context of MRO. In this paper, we develop a stochastic geometry-based analytical framework for evaluating HO performance in Sub-6 GHz cellular networks. The main contributions of this paper are summarized as follows. • We analytically derive the spatial distribution of the target BSs induced by user mobility. The derived distribution provides the basis for analytically evaluating two HO performance metrics, the too-late HO and PP rates, which account for the target BS distribution. • We formulate the too-late HO and PP events as mutually exclusive mobility events in the HO process. Moreover, we analytically derive the too-late HO and PP rates based on the SINR while accounting for the aggregate interference from surrounding BSs. • We compare the derived target BS distribution with a baseline model that does not account for the derivation. Through this comparison, we numerically demonstrate that the modeling of the target BS distribution has a nonnegligible impact on the too-late HO and PP rates. • Using the derived analytical results, we numerically characterize the tradeoff between the too-late HO and PP rates with respect to TTT and investigate the TTT value that minimizes their sum. II. System Model A. Sub-6 GHz Cellular Networks We consider single-tier cellular networks where the allocation of BSs follows a homogeneous Poisson Point Process (PPP) Φ = {𝑋1 , 𝑋2 , · · · } on R2 with intensity 𝜆 > 0. Each BS located at 𝑋 𝑘 ∈ R2 has its corresponding cell, denoted by 𝑘 ∈ N. All BSs are assumed to have omnidirectional antenna patterns and operate on the same carrier frequency, transmitting at a constant power. The omnidirectional assumption has been widely adopted in the literature for sub-6 GHz communications, e.g., [7], [15], and [16]. Time is modeled as continuous, and the downlink channels are subject to power-law path loss and Rayleigh fading under a block-fading model [17], where the fading remains constant within each coherence interval and independent and identically distributed (i.i.d.) across different intervals. The shadowing effects are ignored. The received signal power at a UE located at 𝒖 𝑡 ∈ R2 at time 𝑡 ≥ 0 from cell 𝑘 ∈ N is given by 𝐻 𝑘,𝑛(𝑡 ) ∥ 𝑋 𝑘 − 𝒖 𝑡 ∥ −𝛽 , (1)

where 𝑛(𝑡) ∈ N denotes the index of the coherence interval containing 𝑡, {𝐻 𝑘,𝑛(𝑡 ) } are i.i.d. exponential random variables with unit mean representing the fading effects, and 𝛽 > 2 is the path-loss exponent. The constant transmit power is normalized to unity, and ∥ · ∥ denotes the Euclidean norm. For any time 𝑡 ≥ 0, we assume that each BS serves at least one UE, and that each UE is associated with exactly one BS, referred to as its serving cell. Then, from (1), the downlink signal-to-interference-plus-noise ratio (SINR) for the UE from the serving cell 𝑖 ∈ N at time 𝑡 is given by SINR𝑖 (𝑡) = Í

𝐻𝑖,𝑛(𝑡 ) ∥ 𝑋𝑖 − 𝒖 𝑡 ∥ −𝛽 , −𝛽 + 𝜎 2 𝑗 ∈N\{𝑖 } 𝐻 𝑗,𝑛(𝑡 ) ∥ 𝑋 𝑗 − 𝒖 𝑡 ∥

(2)

where 𝜎 2 ≥ 0 represents the constant noise power. B. RRC Handover Procedures We model the Radio Resource Control (RRC) protocol procedures relevant to HO control. These include the triggering condition of Event A3, RRC connection re-establishment, TTTbased mechanism for HO execution, the out-of-sync indication, and the radio link failure (RLF) detection. These procedures are essentially common across 4G LTE and 5G NR. 1) Triggering Condition of Event A3: The process of HO execution is started by measurement events defined by the 3rd Generation Partnership Project (3GPP). In particular, we focus on Event A3, which is based on a comparison of the measured signal strength between the serving cell and neighbouring cells. The Reference Signal Received Power (RSRP) of cell 𝑘 ∈ N measured by the UE located at 𝒖 𝑡 at time 𝑡 ≥ 0 is defined as RSRP 𝑘 (𝑡) = 10 log10 ∥ 𝑋 𝑘 − 𝒖 𝑡 ∥ −𝛽 ,

(3)

where the received signal power (1) is converted to the decibelscale, with the fading gains are averaged out by assuming the measurement filtering to be applied perfectly1 . Let 𝑖 ∈ N denote the serving cell of the UE and 𝑗 ∈ N\{𝑖} a neighboring cell. According to 3GPP TS 36.331 [18], the triggering condition of Event A3 (hereafter, the A3 condition) at time 𝑡 ≥ 0 can be written as RSRP 𝑗 (𝑡) + CIO 𝑗 − Hys ≥ RSRP𝑖 (𝑡) + CIO𝑖 + OffA3 , (4) where CIO𝑖 and CIO 𝑗 are the cell-individual offsets, Hys is the hysteresis margin, and OffA3 is the A3 offset. The frequencyspecific offsets are removed since we assume the same carrier frequency. In 3GPP, Hys and OffA3 take values in [0, 15] dB and [−15, 15] dB, respectively, with a step size of 0.5 dB. We assume that CIO 𝑘 is identical for all 𝑘 ∈ N, and define the effective offset as Off = Hys + OffA3 ,

(5)

which is assumed to be a negative constant for analytical tractability. Substituting (3) and (5) into (4), and applying CIO𝑖 = CIO 𝑗 , the A3 condition at time 𝑡 ≥ 0 is written as 𝑎∥ ˆ 𝑋𝑖 − 𝒖 𝑡 ∥ ≥ ∥ 𝑋 𝑗 − 𝒖 𝑡 ∥,

(6)

1 The assumption of the perfect filtering in the Physical and Network Layers is widely seen in the literature, e.g., [10] and [11].

where, under the assumption Off < 0, Off

𝑎ˆ = 10− 10𝛽 > 1.

(7)

2) RRC Connection Re-establishment: A UE performs this process when it cannot maintain a reliable connection with the network (e.g., upon RLF detection). In this process, the UE releases its connection to the serving cell 𝑖 ∈ N and selects a new serving cell based on the cell selection criteria. Here, we assume that the cell with the largest RSRP is selected. The signaling delay is ignored, so that the UE immediately establishes a new association. We focus on a UE initially undergoing RRC connection reestablishment. The initial serving cell for the UE at time 𝑡 = 0, denoted by 𝐼, is given by 𝐼 = arg max RSRP 𝑘 (0),

(8)

𝑘 ∈N

where RSRP 𝑘 (0) is given in (3). We further define the initial non-triggering event for the UE as  N0 = 𝑎∥ ˆ 𝑋𝐼 − 𝒖 0 ∥ < ∥ 𝑋 𝑗 − 𝒖 0 ∥ ∀ 𝑗 ∈ N\{𝐼} , (9) under which the A3 condition in (6) does not hold at time 0. 3) TTT-based HO execution: When the A3 condition is satisfied for the UE, among the serving cell 𝑖 ∈ N and a neighbouring cell at time 𝑡 ≥ 0, the TTT timer starts. We assume that, among neighbouring cells, the one for which the A3 condition remains satisfied until the TTT timer expires is selected as the target cell. The HO is completed instantaneously, and the target cell becomes the new serving cell without any delay. We consider a common TTT duration for all cells, denoted by ΔT > 0. For the initial serving cell 𝐼 in (8) and neighbouring cell 𝑗 ∈ N\{𝐼}, let 𝑇˜𝐼, 𝑗 denote the first triggering time of the A3 condition for the UE, and 𝑇𝐼, 𝑗 the triggering time at which the TTT timer expires after ΔT . They are defined as  𝑇˜𝐼, 𝑗 = inf 𝜏 ≥ 0 : 𝑎∥ ˆ 𝑋𝐼 − 𝒖 𝜏 ∥ ≥ ∥ 𝑋 𝑗 − 𝒖 𝜏 ∥ , (10)  𝑇𝐼, 𝑗 = inf 𝜏 ≥ 0 : 𝑎∥ ˆ 𝑋𝐼 − 𝒖 𝑠 ∥ ≥ ∥ 𝑋 𝑗 − 𝒖 𝑠 ∥ ∀𝑠 ∈ [𝜏, 𝜏 + ΔT ) , (11) where 𝑇˜𝐼, 𝑗 and 𝑇𝐼, 𝑗 are set to infinity if the corresponding sets are empty. The first neighbouring cell and the target cell for the UE from 𝐼, denoted by f(𝐼) and t(𝐼), respectively, satisfy f(𝐼) ∈ arg min 𝑇˜𝐼, 𝑗 ,

(12)

𝑗 ∈N\{ 𝐼 }

t(𝐼) ∈ arg min 𝑇𝐼, 𝑗 .

(13)

quality for the UE at time 𝑡 falls below a threshold 𝑄 F ∈ R, corresponding to 𝑄 out in 3GPP TS 36.133 [19]. We define the duration of the RLF timer as ΔF > 0, which represents the combined effect of the parameters N310 and T310, and assume that an RLF is detected when an out-ofsync indication is continuously generated over the duration. Let 𝜌(𝑡 | 𝑋𝐼 , 𝑋t(𝐼 ) ) denote the downlink radio link quality for the UE in the initial serving cell 𝐼 conditioned on 𝑋𝐼 and 𝑋t(𝐼 ) , which we define2 as 𝜌(𝑡 | 𝑋𝐼 , 𝑋t(𝐼 ) ) = E[SINR𝐼 (𝑡) | N0 , 𝑋𝐼 , 𝑋t(𝐼 ) ],

(14)

where SINR𝐼 (𝑡) is given in (2). Then, the RLF detection time in the initial serving cell 𝐼, denoted by 𝐹𝐼 , is given by  𝐹𝐼 = inf 𝜏 ≥ ΔF : 𝜌(𝑡 | 𝑋𝐼 , 𝑋t(𝐼 ) ) < 𝑄 F ∀𝑠 ∈ [𝜏 − ΔF , 𝜏) , (15) where 𝐹𝐼 = ∞ if the set is empty. The initial in-sync event for the UE is further defined as  S0 = 𝜌(0 | 𝑋𝐼 , 𝑋t(𝐼 ) ) ≥ 𝑄 F , (16) under which no out-of-sync indication is generated at time 0. C. UE Mobility We assume that each UE moves in a straight line with a constant velocity on R2 . By the spatial stationarity and isotropy of the network model, we place a UE that performs RRC connection re-establishment at 𝑡 = 0 at the origin 𝒐 ∈ R2 and assume that it moves along the horizontal axis without loss of generality. We refer to this UE as the typical UE. Then, we have 𝒖 𝑡 = (𝑡𝑣, 0), where 𝑣 > 0 denotes the speed of the typical UE. D. HO Events with Initial Conditions We consider two types of undesirable HO events, namely too-late HOs and ping-pong HOs, following [7]. 1) Too-late HO Event: We regard that a too-late HO occurs if the HO execution from the serving cell 𝐼 to the target cell t(𝐼) is delayed compared to the detection of an RLF. The corresponding event L is represented by  L = N0 ∩ S0 ∩ 𝑇𝐼,t(𝐼 ) + ΔT ≥ 𝐹𝐼 . (17) 2) Ping-pong HO Event: We regard that a ping-pong HO occurs if, immediately after the HO execution from 𝐼 to t(𝐼), the HO triggering condition toward 𝐼 is satisfied at t(𝐼) and continues to hold until the expiration of the TTT timer with duration ΔT . The corresponding event P is represented by  P = N0 ∩ S0 ∩ 𝑇t(𝐼 ),𝐼 = 𝑇𝐼,t(𝐼 ) + ΔT < 𝐹𝐼 , (18)

𝑗 ∈N\{ 𝐼 }

The BS location of the target cell for the UE from 𝐼 is referred to as 𝑋t(𝐼 ) . 4) Out-of-sync Indication and RLF Detection: A main mechanism for RLF detection is governed by the parameters N310 and T310 [18]. The timer T310 is started when the UE receives N310 consecutive out-of-sync indications, and an RLF is detected upon its expiry. An out-of-sync indication in the serving cell 𝑖 ∈ N is generated when the downlink radio link

where we define 𝑇t(𝐼 ),𝐼 as  𝑇t(𝐼 ),𝐼 = inf 𝜏 ≥ 𝑇𝐼,t(𝐼 ) + ΔT : 𝑎∥ ˆ 𝑋t(𝐼 ) − 𝒖 𝑠 ∥ ≥ ∥ 𝑋𝐼 − 𝒖 𝑠 ∥ ∀𝑠 ∈ [𝜏, 𝜏 + ΔT ) ,

(19)

which denotes the triggering time after the HO from 𝐼 to t(𝐼) that leads to the TTT expiration after ΔT . 2 A similar use of the conditional expectation of the SINR for characterizing the radio link quality appears in the analytical derivation of [13].

Remark 1. If ΔT < ΔF , the formulations of the events L and P correspond to those in [7]. The only difference is that the events in (9) and (16) are additionally imposed. These events imply that the signal from the serving cell is sufficiently strong compared to those from other cells right after RRC connection re-establishment, which is reasonable in network implementation. The condition ΔT < ΔF is also expected, since the TTT duration is often set shorter than the time to RLF detection, which depends on the parameters N310 and T310, in practical configurations. III. Distribition of the Cells In this section, we investigate the statistical distributions of the cells surrounding the typical UE. These results are used to analyze the events L and P defined in the preceding section. The goal of this section is to characterize 𝑋t(𝐼 ) ∈ Φ, i.e., the BS location of the target cell t(𝐼), under the initial nontriggering event N0 . As part of the analysis, the distribution of the triggering time 𝑇𝐼,t(𝐼 ) is also derived. Hereafter, we adopt polar coordinates to represent 𝒙 ∈ R2 and write 𝒙 = (𝑥, 𝜙), where 𝑥 ≥ 0 and |𝜙| ≤ 𝜋. A. Initial Serving Cell and Non-triggering Event We discuss the distribution of 𝑋𝐼 ∈ Φ and the probability of N0 . By the definition 𝐼 given in (8), together with (3) and 𝒖 0 = 𝒐, the joint probability density function (PDF) of 𝑋𝐼 , denoted by 𝑓 𝑋𝐼 (𝑟, 𝜃) for 𝑟 ≥ 0 and |𝜃| ≤ 𝜋, is well known as (see, e.g., [20, Eq. (2.17)]) 2 𝑓 𝑋𝐼 (𝑟, 𝜃) = 𝜆𝑟e− 𝜋𝜆𝑟 . (20) Given 𝑋𝐼 = (𝑟, 𝜃), we regard 𝑟 > 0 and |𝜃| < 𝜋. By the definition of N0 in (9), N0 occurs if and only if no points of Φ lie on the region 𝑏 𝒐 ( 𝑎∥ ˆ 𝑋𝐼 ∥)\𝑏 𝒐 (∥ 𝑋𝐼 ∥). Therefore,
2 2 P N0 |𝑋𝐼 = (𝑟, 𝜃) = e− 𝜋𝜆𝑟 ( 𝑎ˆ −1) . (21) B. Triggering Time Distribution Recall that 𝑇𝐼,t(𝐼 ) denotes the triggering time at which the TTT timer expires after ΔT , for the initial serving cell 𝐼 and its target cell t(𝐼) given in (13).

Proposition 1. Under the initial non-triggering event N0 and given 𝑋𝐼 = (𝑟, 𝜃), the conditional PDF of 𝑇𝐼,t(𝐼 ) for 𝑡 > 0 is 𝑓𝑇𝐼,t(𝐼) | N0 (𝑡 | 𝑟, 𝜃) (24) 
d   𝜆 𝜉 (𝑡, 𝑟, 𝜃) exp −𝜆𝜉 (𝑡, 𝑟, 𝜃) ,   d𝑡      when | 𝑎𝑟 ˆ − 𝑡𝑣| ≤ 𝑟ˆ (𝑡) ≤ 𝑎𝑟 ˆ + 𝑡𝑣,     = 2𝜋𝜆 𝑎ˆ 2 𝑣(𝑡𝑣 − 𝑟 cos 𝜃) exp −𝜋𝜆 𝑎ˆ 2 𝑡𝑣(𝑡𝑣 − 2𝑟 cos 𝜃)
,      when 𝑎𝑟  ˆ + 𝑡𝑣 ≤ 𝑟ˆ (𝑡),       0, otherwise,  where 

 𝑎𝑟 ˆ cos 𝜃 𝑡𝑣 + − 𝐴 𝑤𝑟,𝑡𝑣, 𝜃 𝑤𝑟,𝑡𝑣, 𝜃 √︂   𝑡𝑣  𝑡𝑣  2 − 𝑎ˆ 2 𝑟 2 cos−1 𝑎ˆ cos 𝜃 − 𝐴− + 𝑎𝑟𝑡𝑣 ˆ 1 − 𝑎ˆ cos 𝜃 − 𝐴− , 𝑟 𝑟

𝜉 (𝑡, 𝑟, 𝜃) = 𝑎ˆ 2 𝑤𝑟2,𝑡𝑣, 𝜃 cos−1

with 𝐴+ = ( 𝑎ˆ 2 + 1)/(2𝑎), ˆ and 𝐴− = ( 𝑎ˆ 2 − 1)/(2𝑎), ˆ and √︁ 𝑟ˆ (𝑡) = 𝑎𝑤 ˆ 𝑟 ,𝑡𝑣, 𝜃 = 𝑎ˆ 𝑟 2 + 𝑡 2 𝑣2 − 2𝑟𝑡𝑣 cos 𝜃. (25) Remark 2. Although 𝜉 (𝑡, 𝑟, 𝜃) admits a closed-form expression, its derivative with respect to 𝑡 becomes algebraically involved. For numerical evaluation of the PDF, we therefore compute d𝑡d 𝜉 (𝑡, 𝑟, 𝜃) numerically. The expression in (24) does not apply to the case Off > 0, in which Lemma 2 given below does not hold. Lemma 2. Let ℓ > 0, 𝑎 ≥ 1, and 𝒃 ∈ R2 . Define the disk region 𝑑 (𝑢) for 𝒖 = (𝑢, 0), 0 ≤ 𝑢 ≤ ℓ, as
𝑑 (𝑢) = 𝑏𝒖 𝑎∥𝒃 − 𝒖∥ , where 𝑏 𝒙 (𝑟) denotes an open ball in R2 with center 𝒙 and radius 𝑟 > 0. Then Ø 𝑑 (𝑢) = 𝑑 (0) ∪ 𝑑 (ℓ). (26) 0≤𝑢≤ℓ

Lemma 1. Under the initial non-triggering event N0 , 𝑇𝐼,t(𝐼 ) = 𝑇˜𝐼,f(𝐼 ) > 0,

(22)

where 𝑇˜𝐼,f(𝐼 ) is the first triggering time, with f(𝐼) given in (12). Proof. For 𝑗 ∈ N\{𝐼}, define 𝐷 𝐼, 𝑗 as 𝐷 𝐼, 𝑗 = {𝒛 ∈ R2 : 𝑎∥ ˆ 𝑋𝐼 − 𝒛∥ < ∥ 𝑋 𝑗 − 𝒛∥},

(23)

where 𝑎ˆ > 1 is given in (7). 𝐷 𝐼, 𝑗 represents the region in R2 where the A3 condition in (6) is not satisfied for 𝐼 and 𝑗. Under the event in (9), 𝒖 0 = 𝒐 ∈ 𝐷 𝐼, 𝑗 , which is an open disk since 𝑎ˆ > 1. Thus, 𝑇˜𝐼, 𝑗 > 0 follows by (10). Since 𝒖 𝑡 = (𝑡𝑣, 0), if 𝒖 𝑡 ′ ∉ 𝐷 𝐼, 𝑗 for some 𝑡 ′ , then 𝒖 𝑡 ∉ 𝐷 𝐼, 𝑗 for all 𝑡 ≥ 𝑡 ′ , which implies that the A3 condition, once satisfied, remains satisfied thereafter. Therefore, from (10) and (11), we have 𝑇˜𝐼, 𝑗 = 𝑇𝐼, 𝑗 . The claim follows from (12) and (13). □

Proof. For fixed 𝒙 ∈ R2 , define the function 𝑔(𝑢) = 𝑎 2 ∥𝒃−𝒖∥ 2 − ∥𝒙 − 𝒖∥ 2 . Note that 𝒙 ∈ 𝑑 (𝑢) if and only if 𝑔(𝑢) > 0. Since 𝑎 ≥ 1, 𝑔 is a linear function or a convex quadratic function of 𝑢. Hence, 𝑔(𝑢) attains its maximum over the compact interval [0, ℓ] when 𝑢 ∈ {0, ℓ}. This implies that 𝒙 ∈ 𝑑 (𝑢) for some 𝑢 ∈ [0, ℓ] if and only if 𝒙 ∈ 𝑑 (0) ∪ 𝑑 (ℓ). □ Proof of Proposition 1. By (10) and (12), 𝑇˜𝐼,f(𝐼 ) ≤ 𝑡 holds if and only if ∃ (𝑡 ′ , 𝑗) ∈ [0, 𝑡]×(N\{𝐼}) s.t. 𝑎∥ ˆ 𝑋𝐼 −𝒖 𝑡 ′ ∥ ≥ ∥ 𝑋 𝑗 −𝒖 𝑡 ′ ∥. (27) Let |𝐵| denote the area of 𝐵 ⊂ R2 and define the conditional non-empty probability of the homogeneous PPP Φ on 𝐵 as
V̄ (𝐵 | 𝑟, 𝜃) := P Φ(𝐵) ≠ 0 | N0 , 𝑋𝐼 = (𝑟, 𝜃) . (28)

Then, we have Probability density

P 𝑇𝐼,t(𝐼 ) ≤ 𝑡 | N0 , 𝑋𝐼 = (𝑟, 𝜃)  Ø 
= V̄ 𝑏𝒖 𝑡 ′ 𝑎∥ ˆ 𝑋𝐼 − 𝒖 𝑡 ′ ∥ 𝑟, 𝜃 𝑡 ′ ∈ [0,𝑡 ]





= V̄ 𝑏 𝒐 𝑎∥ ˆ 𝑋𝐼 ∥ ∪ 𝑏𝒖 𝑡 𝑎∥ ˆ 𝑋𝐼 − 𝒖 𝑡 ∥ 𝑟, 𝜃   ˆ 𝑟 ,𝑡𝑣, 𝜃 )\𝑏 𝒐 ( 𝑎𝑟) ˆ , = 1 − exp −𝜆 𝑏 (𝑡𝑣,0) ( 𝑎𝑤

(29)

where the first equality follows from (22) and (27), (26) is applied in the second equality, and the last equality follows from the void probability √ of Φ conditioned on 𝑋𝐼 = (𝑟, 𝜃) and N0 in (9). Here, 𝑤𝑟 ,𝑡𝑣, 𝜃 = 𝑟 2 + 𝑡 2 𝑣2 − 2𝑟𝑡𝑣 cos 𝜃. By the well-known expression for the area of intersection between two circles, |𝑏 (ℓ,0) (𝑞)\𝑏 𝒐 ( 𝑝)| (30)  2 2 2  2 2 2  𝑝 −𝑞 +ℓ 𝑝 −𝑞 −ℓ  2  − 𝑝 2 cos−1 𝑞 cos−1   2𝑞ℓ 2𝑝ℓ   √︄   2 2 2 2      + 𝑝ℓ 1 − 𝑝 −𝑞 +ℓ , when | 𝑝 − ℓ| ≤ 𝑞 ≤ 𝑝 + ℓ,  = 2𝑝ℓ   2 − 𝑝 2 ), when 𝑝 + ℓ ≤ 𝑞,  𝜋(𝑞      0, when 𝑝 − ℓ ≥ 𝑞,     𝜋𝑞 2 , when ℓ − 𝑝 ≥ 𝑞,  for 𝑝, 𝑞, ℓ > 0. Since 𝑡𝑣 − 𝑎𝑟 ˆ ≥ 𝑎𝑤 ˆ 𝑟 ,𝑡𝑣, 𝜃 never occurs for 𝑎ˆ > 1, substituting (30) into (29) and differentiating both sides of (29) with respect to 𝑡 lead to the result. □ C. Target Cell Distribution We represent the BS location of the target cell t(𝐼) relative to the location 𝒖𝑇𝐼,t(𝐼) = (𝑇𝐼,t(𝐼 ) 𝑣, 0) as 𝑋t(𝐼 ) − 𝒖𝑇𝐼,t(𝐼) = (𝑅t(𝐼 ) , Θt(𝐼 ) ).

(31)

The following result contributes to the characterization of 𝑋t(𝐼 ) . Proposition 2. Under the initial non-triggering event N0 , given 𝑋𝐼 = (𝑟, 𝜃) and 𝑇𝐼,t(𝐼 ) = 𝑡 > 0, it holds that 𝑅t(𝐼 ) = 𝑟ˆ (𝑡),

(32)

where 𝑟ˆ (𝑡) is given in (25), and the conditional PDF of Θt(𝐼 ) is approximately given by 𝑓Θt(𝐼) | N0 (𝜙 | 𝑟, 𝜃, 𝑡) cos 𝜙 − 𝑎ˆ cos 𝜑𝑟 ,𝑡𝑣, 𝜃 , ≈ 2 sin 𝛼𝑟 ,𝑡𝑣, 𝜃 − 2𝑎𝛼 ˆ 𝑟 ,𝑡𝑣, 𝜃 cos 𝜑𝑟 ,𝑡𝑣, 𝜃

|𝜙| < 𝛼𝑟 ,𝑡𝑣, 𝜃 , (33)

 𝑟 cos 𝜃 − 𝑡𝑣 𝜑𝑟 ,𝑡𝑣, 𝜃 = cos , 𝑤𝑟 ,𝑡𝑣, 𝜃 𝛼𝑟 ,𝑡𝑣, 𝜃 = cos−1 (−1 ∨ 𝑎ˆ cos 𝜑𝑟 ,𝑡𝑣, 𝜃 ).



(34)

exact as Off → 0. The minimizer in (13) is almost surely (a.s.) unique under N0 , since the probability of multiple points vanishes, as shown in Appendix A. Remark 4. Previous studies such as [13] and [14] model the conditional distribution of the target cell angle, conditioned on the target cell distance, as uniform over the feasible angular region. Under the notation of Proposition 2, this baseline model is represented as (𝜙 | 𝑟, 𝜃, 𝑡) = 𝑓Θbase t(𝐼) | N0

1 , 2𝛼𝑟 ,𝑡𝑣, 𝜃

|𝜙| < 𝛼𝑟 ,𝑡𝑣, 𝜃 ,

(36)

which does not rely on the target cell definition in (13). A comparison of the HO performance obtained with the baseline and proposed models is presented in Sec. V. Fig. 1 compares the approximate analytical results given by (33) and the results from Monte Carlo simulations without the approximation for 𝑟 = 0.5 km, 𝑡 = 50 sec, 𝑣 = 0.01 km/sec, 𝜆 = 1 km−2 , Off = −3 dB, and 𝛽 = 4. Although a noticeable discrepancy is observed for small values of 𝜃, the analytical approximation captures the conditional distribution of Θt(𝐼 ) with good accuracy overall. IV. Handover Event Probabilities

A. Assumption for Downlink Radio Link Quality Define the event A as  A = 𝑋𝐼 = (𝑟, 𝜃), 𝑇𝐼,t(𝐼 ) = 𝑡, 𝑋t(𝐼 ) − 𝒖 𝑡 = (𝑟ˆ (𝑡), 𝜙) . (37)

(35)

where 𝑟 > 0, 𝜃 ∈ (−𝜋, 𝜋), 𝑡 > 0, and 𝜙 ∈ (−𝛼𝑟,𝑡𝑣, 𝜃 , 𝛼𝑟,𝑡𝑣, 𝜃 ). By (31) and (32), the above definition is equivalent to

□

A = {𝑋𝐼 = (𝑟, 𝜃), 𝑇𝐼,t(𝐼 ) = 𝑡, Θt(𝐼 ) = 𝜙}.

with 𝑥 ∨ 𝑦 = max(𝑥, 𝑦). Proof. See Appendix A.

Fig. 1. Conditional PDF of the target cell angle Θt(𝐼) given 𝑋𝐼 = (𝑟 , 𝜃 ) and 𝑇𝐼,t(𝐼) = 𝑡. The analytical approximation in (33) is compared with exact Monte Carlo simulations for different values of 𝜃.

In this section, we analyze the probabilities of the too-late HO and the ping-pong HO events described in Sec. II-D.

where −1

Θ ( ) = 𝜙 (rad)

Remark 3. The introduced
approximation allows the point 𝑋t(𝐼 ) to lie inside 𝑏 𝒐 𝑎∥ ˆ 𝑋𝐼 ∥ under N0 , which never occurs if Off = 0, i.e., 𝑎ˆ = 1. Therefore, this approximation becomes

(37′ )

For 𝑠 ≥ 0, under A, the downlink radio link quality 𝜌(𝑠 | 𝑋𝐼 , 𝑋t(𝐼 ) ) in (14) is abbreviated as 𝜌(𝑠 | A).

Lemma 3. The downlink radio link quality under A satisfies 𝜌(𝑠 | A) −𝛽 ∫ ∞ 𝑤𝑟,𝑠𝑣, 𝜃 2 2/𝛽 = e− 𝜎 𝑧− 𝜋𝜆𝐾𝛽 𝑧 𝜇(𝑧, 𝑠, 𝑡)d𝑧, (38) −𝛽 0 1+𝑧𝑤 𝑟ˆ (𝑡 ) , (𝑠−𝑡 )𝑣, 𝜙 √ where 𝑤𝑟,𝑠𝑣, 𝜃 = 𝑟 2 + 𝑠2 𝑣2 − 2𝑟 𝑠𝑣 cos 𝜃, and 2𝜋 2𝜋 csc , 𝛽 𝛽 ∫ 𝑡𝑣+𝑟ˆ (𝑡∫) 𝐶 ( 𝑥,𝑟ˆ (𝑡 ) ,𝑡𝑣) 𝜇(𝑧, 𝑠, 𝑡) = exp 2𝜆𝑧 0

(39) !

𝑥 𝛽

𝑧 + 𝑤 𝑥,𝑠𝑣, 𝜓

d𝜓d𝑥 , (40)

with 𝐶 (𝑥, 𝑟ˆ (𝑡), 𝑡𝑣) ( 𝜋,   2 2 𝑣2 − 𝑟ˆ (𝑡 ) 2 = cos−1 −1 ∨ 𝑥 +𝑡 2𝑥𝑡𝑣 ,

√︃ 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 𝑑A = 2 𝑎ˆ 2 − 2𝑎ˆ cos(𝜙 − 𝜑𝑟,𝑡𝑣, 𝜃 ) + 1, (44) 𝑎ˆ − 1 √ with 𝑤𝑟,𝑡𝑣, 𝜃 = 𝑟 2 + 𝑡 2 𝑣2 − 2𝑟𝑡𝑣 cos 𝜃, and 𝜑𝑟,𝑡𝑣, 𝜃 is in (34). Corollary 1. The probability of the too-late HO event is ∫ ∞∫ 𝜋∫ ∞∫ 𝛼𝑟,𝑡𝑣, 𝜃 2 2 P(L) = 𝜆 𝑟e− 𝜋𝜆𝑎ˆ 𝑟 𝜂(𝜙, 𝑡, 𝑟, 𝜃) 0 0 0 − 𝛼𝑟,𝑡𝑣, 𝜃

× 𝐻 𝜁 A (0) − 𝑄 F 𝐻¯ 𝜁 A (𝑡 + ΔT − ΔF ) − 𝑄 F d𝜙 d𝑡 d𝜃 d𝑟. where 𝐻¯ (𝑥) = 1 − 𝐻 (𝑥), and 𝜂(𝜙, 𝑡, 𝑟, 𝜃) is approximated by the expression in (41).

0 ≤ 𝑥 ≤ 𝑎𝑟, ˆ

Proof. See Appendix B.

(𝑥A , 𝑦A )   3 𝑎ˆ 3 sin 𝜙 − sin 𝜑𝑟,𝑡𝑣, 𝜃 𝑎ˆ cos 𝜙 − cos 𝜑𝑟,𝑡𝑣, 𝜃 𝑤 , 𝑤 = 𝑟,𝑡𝑣, 𝜃 𝑟,𝑡𝑣, 𝜃 , 𝑎ˆ 2 − 1 𝑎ˆ 2 − 1 (43) and

𝐾𝛽 =

0

with 𝑓𝑇𝐼,t(𝐼) | N0 (𝑡 | 𝑟, 𝜃) and 𝜌(𝑠 | A) given in (24) and (38), √︃ ± = 𝑥 ± 𝑑 2 − 𝑦2 , where respectively. Furthermore, 𝑞A A A A

𝑎𝑟 ˆ ≤ 𝑥 ≤ 𝑡𝑣 + 𝑟ˆ (𝑡). □

To facilitate our analysis, we introduce the following assumption, whose validity is supported numerically in Appendix C. Assumption 1. 𝜌(𝑠 | A) given in (38) is monotonically decreasing or unimodal with respect to 𝑠 ≥ 0. B. Results for HO Event Probabilities This subsection presents analytical results for the probabilities of the events L and P defined in (17) and (18). We first derive the expression for P(P), and then obtain the result for P(L) as its corollary. In this paper, the Heaviside step function 𝐻 (𝑥) is defined as  1, 𝑥 ≥ 0, 𝐻 (𝑥) = 0, 𝑥 < 0. Theorem 1. The probability of the ping-pong HO event is ∫ ∞∫ 𝜋∫ ∞∫ 𝛼𝑟,𝑡𝑣, 𝜃 2 2 P(P) = 𝜆 𝑟e− 𝜋𝜆𝑎ˆ 𝑟 𝜂(𝜙, 𝑡, 𝑟, 𝜃) 0 0 0 − 𝛼𝑟,𝑡𝑣, 𝜃 h
i + −
} − max{ΔT 𝑣, 𝑞A } 𝐻 𝑑A − 𝑦A × 1 − 𝐻 min{2ΔT 𝑣, 𝑞A
× 𝐻 min{𝜁 A (0), 𝜁 A (𝑡 + ΔT − ΔF )} − 𝑄 F d𝜙 d𝑡 d𝜃 d𝑟. where 𝜂(𝜙, 𝑡, 𝑟, 𝜃) = 2 𝑓𝑇𝐼,t(𝐼) | N0 (𝑡 | 𝑟, 𝜃) 𝑓Θt(𝐼) | N0 ,𝑋𝐼 ,𝑇𝐼,t(𝐼) (𝜙 | 𝑟, 𝜃, 𝑡) cos 𝜙 − 𝑎ˆ cos 𝜑𝑟,𝑡𝑣, 𝜃 ≈ 𝑓𝑇 | N (𝑡 | 𝑟, 𝜃), sin 𝛼𝑟,𝑡𝑣, 𝜃 − 𝑎𝛼 ˆ 𝑟,𝑡𝑣, 𝜃 cos 𝜑𝑟,𝑡𝑣, 𝜃 𝐼,t(𝐼) 0 (41) ( 𝜌(𝑠 | A), 𝑠 ≥ 0, 𝜁 A (𝑠) = (42) 𝑄F, 𝑠 < 0,

Proof of Theorem 1. By (20) and (21), and using the PDFs in (24) and (33), P(P) defined in (18) is expressed as ∫ ∞∫ 𝜋 ∫ ∞∫ 𝛼𝑟,𝑡𝑣, 𝜃 2 2 P(P) = 𝜆𝑟e− 𝜋𝜆𝑎ˆ 𝑟 0

−𝜋

0

− 𝛼𝑟,𝑡𝑣, 𝜃

× 𝑓𝑇𝐼,t(𝐼) | N0 ,𝑋𝐼 (𝑡 | 𝑟, 𝜃) 𝑓Θt(𝐼) | N0 ,𝑋𝐼 ,𝑇𝐼,t(𝐼) (𝜙 | 𝑟, 𝜃, 𝑡)
× P S0 ∩ {𝑇t(𝐼 ),𝐼 = 𝑡 + ΔT < 𝐹𝐼 } | N0 , A × d𝜙 d𝑡 d𝜃 d𝑟,

(45)

where A is given in (37). Since 𝑇t(𝐼 ),𝐼 defined in (19) is conditionally independent of 𝐹𝐼 and S0 under A, we have
P S0 ∩ {𝑇t(𝐼 ),𝐼 = 𝑡 + ΔT < 𝐹𝐼 } | N0 , A

= P 𝑇t(𝐼 ),𝐼 = 𝑡 + ΔT | A P S0 ∩ {𝑡 + ΔT < 𝐹𝐼 } | N0 , A , (46) where the conditioning on N0 is omitted in the first term, since 𝑇t(𝐼 ),𝐼 depends only on 𝑋𝐼 , 𝑋t(𝐼 ) , and 𝑇𝐼,t(𝐼 ) . By (19), under A, 𝑇t(𝐼 ),𝐼 = 𝑡 + ΔT holds if and only if 𝑎∥ ˆ 𝑋t(𝐼 ) − 𝒖 𝑠 ∥ ≥ ∥ 𝑋𝐼 − 𝒖 𝑠 ∥

∀𝑠 ∈ [𝑡 + ΔT , 𝑡 + 2ΔT ).

(47)

Let 𝒙˜ 𝐼 = 𝑋𝐼 − 𝒖 𝑡 , and 𝒙˜ t(𝐼 ) = 𝑋t(𝐼 ) − 𝒖 𝑡 . Under A, they are expressed as 𝒙˜ 𝐼 = (𝑟, 𝜃) − (𝑡𝑣, 0) = (𝑤𝑟,𝑡𝑣, 𝜃 , 𝜑𝑟,𝑡𝑣, 𝜃 ), 𝒙˜ t(𝐼 ) = ( 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 , 𝜙), where (25) is used in the second line, and 𝜑𝑟,𝑡𝑣, 𝜃 is given in (34). Substituting these expressions into (47) and using 𝒖 𝑠 = (𝑠𝑣, 0) yields 𝑎∥ ˆ 𝒙˜ t(𝐼 ) − (𝑠𝑣, 0) ∥ ≥ ∥ 𝒙˜ 𝐼 − (𝑠𝑣, 0) ∥

∀𝑠 ∈ [ΔT , 2ΔT ).

This implies that {𝑇t(𝐼 ),𝐼 = 𝑡 +ΔT | A} is equivalent to the event that the interval [ΔT 𝑣, 2ΔT 𝑣) does not intersect the following region  𝐷˜ t(𝐼 ),𝐼 = 𝒛 ∈ R2 : 𝑎∥ ˆ 𝒙˜ t(𝐼 ) − 𝒛∥ < ∥ 𝒙˜ 𝐼 − 𝒛∥ ,

which is the interior of a circle with radius 𝑑A given in (44) and centered at (𝑥A , 𝑦A ) in Cartesian √︃ coordinates, as defined in 3 ± 2 − 𝑦2 , (43) . Therefore, by 𝑞A = 𝑥A ± 𝑑A A
P 𝑇t(𝐼 ) ,𝐼 = 𝑡 + ΔT | A
+ −
= 1 − 𝐻 min{2ΔT 𝑣, 𝑞A } − max{ΔT 𝑣, 𝑞A } 𝐻 𝑑A − 𝑦A , (48) which follows from the fact that [𝑎, 𝑏] ∩ [𝑐, 𝑑] ≠ ∅ if and only if max{𝑎, 𝑐} ≤ min{𝑏, 𝑑}. By Assumption 1 and the definitions in (15) and (16), under N0 and A, S0 ∩ {𝑡 + ΔT < 𝐹𝐼 } occurs if and only if 𝜌(0 | A) ≥ 𝑄 F , and either 𝑡 + ΔT < ΔF ,
or 𝑡 + ΔT ≥ ΔF and 𝜌(𝑡 + ΔT − ΔF | A) ≥ 𝑄 F , which implies that P S0 ∩ {𝑡 + ΔT < 𝐹𝐼 } | N0 , A

= 𝐻 min{𝜁 A (0), 𝜁 A (𝑡 + ΔT − ΔF )} − 𝑄 F

(49)

where 𝜁 A (0) and 𝜁 A (𝑡 + ΔT − ΔF ) are given in (42). Consequently, substituting (46), (48), and (49) into (45), and combining with (33), the result follows by changing the variable as 𝜃 ′ = |𝜃|, since the integrand is even in 𝜃. □ Proof of Corollary 1. By the definition in (17), P(L) can be expressed similarly to (45) as ∫ ∞∫ 𝜋 ∫ ∞∫ 𝛼𝑟,𝑡𝑣, 𝜃 2 2 P(L) = 𝜆𝑟e− 𝜋𝜆𝑎ˆ 𝑟 0

−𝜋

0

− 𝛼𝑟,𝑡𝑣, 𝜃

× 𝑓𝑇𝐼,t(𝐼) | N0 ,𝑋𝐼 (𝑡 | 𝑟, 𝜃) 𝑓Θt(𝐼) | N0 ,𝑋𝐼 ,𝑇𝐼,t(𝐼) (𝜙 | 𝑟, 𝜃, 𝑡)
× P S0 ∩ {𝑡 + ΔT ≥ 𝐹𝐼 } | N0 , A d𝜙 d𝑡 d𝜃 d𝑟. By letting 𝜃 ′ = |𝜃|, the result follows by an argument analogous to (49), yielding
P S0 ∩ {𝑡 + ΔT ≥ 𝐹𝐼 } | N0 , A

 = 𝐻 𝜁 A (0) − 𝑄 F 1 − 𝐻 𝜁 A (𝑡 + ΔT − ΔF ) − 𝑄 F

= 𝐻 𝜁 A (0) − 𝑄 F 𝐻¯ 𝜁 A (𝑡 + ΔT − ΔF ) − 𝑄 F . □ V. Numerical Evaluation This section presents numerical results to validate the proposed analytical framework. The analytical results are compared with Monte Carlo simulations and a conventional target-cell model, and the tradeoff between the too-late HO and pingpong HO probabilities is investigated with respect to the TTT duration. The common parameters throughout the experiments are 𝑣 = 0.01 km/sec, 𝛽 = 4, 𝜎 2 = 0, ΔF = 3000 ms, and 𝑄 F = 0.5. The BS density is chosen from 𝜆 = 1, 3, and 10 km−2 , representing sparse, typical, and moderately dense deployment scenarios, respectively. Depending on the experiment, the TTT duration is varied over 0 < ΔT ≤ 5000 ms, which covers almost the entire range specified in 3GPP TS 36.331 [18]. The negative effective offset is varied over −3 ≤ Off < 0 dB, which lies within the HO-margin range commonly adopted in the literature [24]. 3 These results of the radius and the center can be derived using elementary geometry; the derivation is omitted for conciseness.

A. Comparison with Simulation and Previous Model Fig. 2 compares the analytical results of the proposed model with the Monte Carlo simulation results and the baseline model. Figs. 2(a) and (b) show the probabilities of the too-late HO and ping-pong HO events, respectively. The results of the proposed model are obtained from Corollary 1 in Fig. 2(a) and Theorem 1 in Fig. 2(b). The baseline model results are obtained by replacing 𝑓Θt(𝐼) | N0 ,𝑋𝐼 ,𝑇𝐼,t(𝐼) (𝜙 | 𝑟, 𝜃, 𝑡) in (42) with the baseline density (36) instead of (33). The Monte Carlo results for the events (17) and (18) are obtained by averaging over 1000 independent PPP realizations. For each realization, the conditional expectation in (14) is estimated from 10000 independent realizations of the fading and inner PPP. Here, the BS density is set to 𝜆 = 3 km−2 , and three TTT duration ΔT = 1000 ms, 2000 ms, and 3000 ms, are considered. For both HO events, the analytical results obtained with the proposed model closely match the simulation results over the considered range of Off and for all three TTT durations. In particular, the proposed analysis accurately captures the increase in the too-late HO probability as Off increases, as well as the corresponding decrease in the ping-pong HO probability. Despite the approximation errors observed in Fig. 1, their impact on the evaluated HO event probabilities is negligible for these parameter settings.. For ΔT = 1000 ms, however, the ping-pong HO probability exhibits a slight increase around −3 dB ≤ Off ≤ −2 dB. This is due to the initial non-triggering
event N0 in (18). As follows from (21), P N0 | 𝑋𝐼 = (𝑟, 𝜃) decreases as the effective offset becomes more negative, thereby suppressing the unconditional ping-pong HO probability. The baseline model also captures the overall trends of both HO events. However, noticeable differences are observed between the baseline and proposed models, particularly for the ping-pong HO probability. The discrepancy is particularly pronounced for moderately negative effective offsets (approximately −1.5 dB ≤ Off ≤ −0.5 dB), where the selected target cell defined by (13) exhibits a markedly non-uniform angular distribution that is not captured by the baseline model. B. Joint HO Events and Optimal TTT Timer In this subsection, we investigate the tradeoff between the too-late HO and ping-pong HO events with respect to the TTT duration ΔT > 0. Fig. 3 shows the probabilities of the too-late HO event and the ping-pong HO event obtained from Corollary 1 and Theorem 1, respectively, together with their sum, P(L) + P(P) = P(L ∪ P), as functions of ΔT . The equality follows since the events defined in (17) and (18) are mutually disjoint. Here, the BS density is varied over 𝜆 = 1, 3, and 10 km−2 , while the effective offset is fixed at Off = −1 dB. As the TTT duration ΔT increases, the probability of the toolate HO event monotonically increases, whereas the probability of the ping-pong HO event monotonically decreases. This behavior clearly illustrates the inherent tradeoff between the two HO events, since increasing the TTT duration suppresses unnecessary HOs while simultaneously delaying legitimate HOs.

HO event probabilities

Too-late HO event probability

Effective offset Off (dB)

(a) P(L).

Ping-pong HO event probability

Common TTT duration ΔT (ms) Fig. 3. Probabilities P( L ), P( P ), and P( L ) + P( P ) versus the common TTT duration ΔT for different BS densities 𝜆 (km −2 ). The results illustrate the tradeoff between the too-late HO and ping-pong HO events and identify the TTT duration that minimizes P( L ) + P( P ).

Effective offset Off (dB)

(b) P(P). Fig. 2. Probabilities of (a) the too-late HO event P( L ) and (b) the ping-pong HO event P( P ) versus the effective offset Off for different TTT durations ΔT . The proposed analytical model is compared with the baseline model and exact Monte Carlo simulations.

The total HO event probability P(L) + P(P) exhibits a distinct minimum for each BS density, indicating the existence of an optimal value of ΔT that balances the two competing HO events. For sparse BS deployments, the minimum is attained at a relatively large value of ΔT because both the increase in the too-late HO probability and the decrease in the ping-pong HO probability are gradual. As the BS density increases, the optimal TTT duration shifts toward smaller values because the too-late HO probability increases more rapidly, whereas the ping-pong HO probability decreases more rapidly. These observations demonstrate that the optimal TTT duration depends on the BS density and should therefore be adapted to the deployment environment. VI. Conclusion This paper presented an analytical framework for mobility robustness analysis in cellular networks by explicitly modeling the spatial distribution of the dynamically selected target cell. Unlike previous analytical studies based on a uniform angular distribution, the proposed framework derives the conditional

distribution of the target cell location by incorporating user mobility and the HO triggering condition. Based on this result, analytical expressions for the probabilities of the too-late and ping-pong HO events were derived under the SINR model with aggregate interference. Numerical results demonstrated that the proposed analysis closely matches exact Monte Carlo simulations and provides more accurate HO performance evaluation than the baseline model. Furthermore, the analytical framework quantitatively revealed the tradeoff between the too-late and ping-pong HO events and identified the TTT duration that minimizes the probability of their union. Future work includes extending the proposed framework in several directions. First, the proposed analytical framework can be extended to multi-tier heterogeneous, UAV-assisted, and non-terrestrial networks by incorporating the corresponding BS characteristics and SINR expressions. Another direction is to jointly optimize the TTT duration and the effective offset by extending the analysis to positive Off values. Finally, eliminating the approximation introduced in the derivation of (33) would provide an exact analytical characterization of the target cell distribution. Appendix A. Proof of Proposition 2 On the event 𝑇𝐼,t(𝐼 ) ∈ (𝑡, 𝑡 + 𝛿𝑡 ] for 𝑡 > 0, let 𝑅t(𝐼 ) (𝒖 𝑡 ) and Θt(𝐼 ) (𝒖 𝑡 ) denote the magnitude and polar angle of 𝑋t(𝐼 ) − 𝒖 𝑡 , respectively. We then consider their distributions, which converge as 𝛿𝑡 → 0 to those of 𝑅t(𝐼 ) and Θt(𝐼 ) , since the displacement between 𝒖 𝑡 and 𝒖𝑇𝐼,t(𝐼) vanishes as 𝛿𝑡 → 0. Define the conditional void probability of the homogeneous PPP Φ on 𝐵 ⊂ R2 as
V (𝐵 | 𝑟, 𝜃) = P Φ(𝐵) = 0 | N0 , 𝑋𝐼 = (𝑟, 𝜃) .

From the second equality of (29), we have
P 𝑇𝐼,t(𝐼 ) ∈ (𝑡, 𝑡 + 𝛿𝑡 ] | N0 , 𝑋𝐼 = (𝑟, 𝜃)  


= V 𝑏 𝒐 𝑎∥ ˆ 𝑋𝐼 ∥ ∪ 𝑏𝒖 𝑡 𝑎∥ ˆ 𝑋𝐼 − 𝒖 𝑡 ∥ 𝑟, 𝜃 V̄ 𝐷 (𝛿𝑡 ) 𝑟, 𝜃 , where V̄ (𝐵 | 𝑟, 𝜃) is defined in (28), and the region 𝐷 (𝛿𝑡 ) is


𝐷 (𝛿𝑡 ) = 𝑏𝒖 𝑡+ 𝛿𝑡 𝑎∥ ˆ 𝑋𝐼 − 𝒖 𝑡+ 𝛿𝑡 ∥ 𝑏 𝒐 𝑎∥ ˆ 𝑋𝐼 ∥ 𝑏𝒖 𝑡 𝑎∥ ˆ 𝑋𝐼 − 𝒖𝑡 ∥ . (50) Therefore, given N0 , 𝑋𝐼 = (𝑟, 𝜃), and 𝑇𝐼,t(𝐼 ) ∈ (𝑡, 𝑡 + 𝛿𝑡 ], the point 𝑋t(𝐼 ) ∈ Φ lies in 𝐷 (𝛿𝑡 ) ≈ 𝐷˜ (𝛿𝑡 ) = 𝑏 (𝑡𝑣+ 𝛿𝑡 𝑣,0) ( 𝑎𝑤 ˆ 𝑟,𝑡𝑣+ 𝛿𝑡 𝑣, 𝜃 )\𝑏 (𝑡𝑣,0) ( 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 ), (51) √ where 𝑤𝑟,𝑡𝑣, 𝜃 = 𝑟 2 + 𝑡 2 𝑣2 − 2𝑟𝑡𝑣 cos 𝜃. The approximate region 𝐷˜ (𝛿𝑡 ) is obtained by ignoring the subtraction of 𝑏 𝒐 ( 𝑎𝑟). ˆ Let 𝒛 ∈ 𝐷˜ (𝛿𝑡 ), and set ℓ = (𝑡𝑣, 0), and ℓ + = (𝑡𝑣 + 𝛿𝑡 𝑣, 0). Let 𝑙 𝒛 denote the half-line emanating from ℓ and passing through 𝒛, and write 𝒛 − ℓ = (𝑟 𝒛 , 𝜃 𝒛 ). Let 𝒛 0 and 𝒛+ denote the inner and outer intersection points of 𝑙 𝒛 with the boundary 𝜕 𝐷˜ (𝛿𝑡 ), respectively, and define ℎ 𝜃𝒛(𝛿𝑡 ) = ∥𝒛+ − 𝒛 0 ∥. Due to (51), ∥𝒛 + − ℓ + ∥ = 𝑎𝑤 ˆ 𝑟,𝑡𝑣+ 𝛿𝑡 𝑣, 𝜃 ,

∥𝒛0 − ℓ∥ = 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 .

Using the law of cosines applied to the triangle with vertices 𝒛+ , ℓ, and ℓ + , with angle |𝜃 𝒛 |, and solving for ℎ 𝜃𝒛 (𝛿𝑡 ), we obtain ℎ 𝜃𝒛(𝛿𝑡 ) = −( 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 − 𝛿𝑡 𝑣 cos 𝜃 𝒛 ) +

√︃

P Θt(𝐼 ) (𝒖 𝑡 ) ∈ [𝜙, 𝜙 +d𝜙] N0 , 𝑋𝐼 = (𝑟, 𝜃), 𝑇𝐼,t(𝐼 ) ∈ (𝑡, 𝑡 +𝛿𝑡 ] {𝒛 ∈ 𝐷˜ (𝛿𝑡 ) : 𝜃 𝒛 ∈ [𝜙, 𝜙 + d𝜙]} + 𝑜(1)d𝜙 ≈ 𝐷˜ (𝛿𝑡 ) ∫  𝑟ˆ (𝑡 )+ℎ 𝜙( 𝛿𝑡 ) 𝑟 𝒛 d𝑟 𝒛 d𝜙 𝑟ˆ (𝑡 ) =∫𝛼 + 𝑜(1)d𝜙 +𝑜(d𝜙) ∫ 𝑟ˆ (𝑡 )+ℎ 𝜃𝒛 ( 𝛿𝑡 ) 𝑟,𝑡𝑣, 𝜃 𝑟 d𝑟 d𝜃 𝒛 𝒛 𝒛 − 𝛼𝑟,𝑡𝑣, 𝜃 𝑟ˆ (𝑡 )
cos 𝜙 − 𝑎ˆ cos 𝜑𝑟,𝑡𝑣, 𝜃 + 𝑜(1) d𝜙 = + 𝑜(1)d𝜙 +𝑜(d𝜙), 2 sin 𝛼𝑟,𝑡𝑣, 𝜃 − 2𝑎𝛼 ˆ 𝑟,𝑡𝑣, 𝜃 cos 𝜑𝑟,𝑡𝑣, 𝜃 + 𝑜(1) where the first step follows from P(Φ(𝐷 (𝛿𝑡 )) > 1 | N0 , 𝑋𝐼 = (𝑟, 𝜃), 𝑇𝐼,t(𝐼 ) ∈ (𝑡, 𝑡 + 𝛿𝑡 ]) = 𝑜(1) as 𝛿𝑡 → 0 and the approximation (51), the second step uses a first-order approximation of the numerator in d𝜙, and the last step follows from (52). Here 𝑜(1) (resp., 𝑜(d𝜙)) is with respect to 𝛿𝑡 → 0 (resp., d𝜙 → 0). Dividing both sides by d𝜙 and taking the limits d𝜙 → 0 followed by 𝛿𝑡 → 0, we obtain (33). B. Proof of Lemma 3 By the definition in (14) along with the event A in (37), 𝜌(𝑠 | A) = E[SINR𝐼 (𝑡) | N0 , A]. Thus, using (2), 𝜌(𝑠 | A)  ∫ ∞  𝐻 𝐼,𝑛(𝑠) ∥ (𝑟, 𝜃) − 𝒖 𝑠 ∥ −𝛽 ′ > 𝑧 N0 , A d𝑧 ′ = P 𝜎 2 + Iagg 0 ∫ ∞   h i −𝛽 𝑤𝑟,𝑠𝑣, 𝜃 exp −𝜎 2 𝑧 E e−𝑧 Iagg N0 , A d𝑧, (53) = 0

2 2 2 2 𝑎ˆ 2 𝑤𝑟,𝑡𝑣+ 𝛿𝑡 𝑣, 𝜃 − 𝛿 𝑡 𝑣 sin 𝜃 𝒛

= 𝛿𝑡 𝑣(cos 𝜃 𝒛 − 𝑎ˆ cos 𝜑𝑟,𝑡𝑣, 𝜃 ) + 𝑜(𝛿𝑡 ),

𝛿𝑡 → 0,

(52)

where we used 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 − 𝛿𝑡 𝑣 cos 𝜃 𝒛 > 0 for sufficiently small 𝛿𝑡 , and the first-order expansion

where Iagg is defined as ∑︁ Iagg = 𝐻 𝑗,𝑛(𝑠) ∥ 𝑋 𝑗 − 𝒖 𝑠 ∥ −𝛽 .

(54)

𝑗 ∈N\{𝐼 }

𝑤𝑟,𝑡𝑣+ 𝛿𝑡 𝑣, 𝜃 − 𝑤𝑟,𝑡𝑣, 𝜃 = −𝛿𝑡 𝑣 cos 𝜑𝑟,𝑡𝑣, 𝜃 + 𝑜(𝛿𝑡 ),

Besides, in the second equality, we apply [21, Eq. (5)] with 𝜇 𝑃𝑡 𝐴 = 1, along with the change of variables

with 𝜑𝑟,𝑡𝑣, 𝜃 in (34). Since 𝑡 lies in the support of 𝑓𝑇𝐼,t(𝐼) | N0 (𝑡 | 𝑟, 𝜃) in (24), we have | 𝑎𝑟 ˆ − 𝑡𝑣| ≤ 𝑟ˆ (𝑡). Using 𝑟ˆ (𝑡) = 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 from (25), this is equivalent to 𝑎𝑟 ˆ cos 𝜃 ≤ 𝑡𝑣𝐴 − + 𝑟, where 𝐴 − = ( 𝑎ˆ 2 − 1)/(2𝑎). ˆ Hence, using (34) and 𝐴− < 𝑎ˆ − 1 for 𝑎ˆ > 1, we have 𝑎ˆ cos 𝜑𝑟,𝑡𝑣, 𝜃 < 1.

𝑧 = 𝑧 ′ ∥ (𝑟, 𝜃) − 𝒖 𝑠 ∥ 𝛽 = 𝑧 ′ 𝑤𝑟,𝑠𝑣, 𝜃 , √ where 𝑤𝑟,𝑠𝑣, 𝜃 = 𝑟 2 + 𝑠2 𝑣2 − 2𝑟 𝑠𝑣 cos 𝜃. Under A, ∥ 𝑋t(𝐼 ) − 𝒖 𝑡 ∥ = 𝑟ˆ (𝑡), and the polar angle of 𝑋t(𝐼 ) − 𝒖 𝑡 is 𝜙. This means that √︁ ∥ 𝑋t(𝐼 ) − 𝒖 𝑠 ∥ = 𝑟ˆ (𝑡) 2 + (𝑠 − 𝑡) 2 𝑣2 − 2𝑟ˆ (𝑡) (𝑠 − 𝑡)𝑣 cos 𝜙

Since cos 𝜃 𝒛 > 𝑎ˆ cos 𝜑𝑟,𝑡𝑣, 𝜃 from (52), it follows that 𝜃 𝒛 ∈ (−𝛼𝑟,𝑡𝑣, 𝜃 , 𝛼𝑟,𝑡𝑣, 𝜃 ), where 𝛼𝑟,𝑡𝑣, 𝜃 is given in (35). Let | · | denote the two-dimensional Lebesgue measure. Since 𝑋t(𝐼 ) lies in 𝐷 (𝛿𝑡 ) ⊂ 𝐷˜ (𝛿𝑡 ), and ∥𝒛 0 − ℓ∥ = 𝑎𝑤 ˆ 𝑟,𝑡𝑣, 𝜃 = 𝑟ˆ (𝑡), it follows that 𝑅t(𝐼 ) (𝒖 𝑡 ) ∈ [𝑟ˆ (𝑡), 𝑟ˆ (𝑡) + ℎΘt(𝐼) (𝒖 𝑡 ) (𝛿𝑡 )), where ℎΘt(𝐼) (𝒖 𝑡 ) (𝛿𝑡 ) is given in (52). Taking the limit 𝛿𝑡 → 0 in the above relation, we obtain (32). As 𝛿𝑡 → 0, the probability of having more than one point in 𝐷 (𝛿𝑡 ) defined in (50) is 𝑜(𝛿𝑡 ) (see, e.g., Sec 2.3 of [20]). Hence, given N0 , 𝑋𝐼 = (𝑟, 𝜃), and 𝑇𝐼,t(𝐼 ) ∈ (𝑡, 𝑡 + 𝛿𝑡 ], 𝑋t(𝐼 ) is asymptotically uniformly distributed over 𝐷 (𝛿𝑡 ). Therefore, for 𝜙 ∈ (−𝛼𝑟,𝑡𝑣, 𝜃 , 𝛼𝑟,𝑡𝑣, 𝜃 ),

𝛽

= 𝑤𝑟ˆ (𝑡 ), (𝑠−𝑡 )𝑣, 𝜙 . Therefore, using (54) and the Laplace transform of 𝐻 𝑗,𝑛(𝑠) ∼ exp(1),  Ö  h i   1  E e−𝑧 Iagg N0 , A = E  N , A 0  −𝛽   𝑗 ∈N\{𝐼 } 1 + 𝑧∥ 𝑋 𝑗 − 𝒖 𝑠 ∥   1 = −𝛽 1 + 𝑧 𝑤𝑟ˆ (𝑡 ), (𝑠−𝑡 )𝑣, 𝜙  Ö    1  ×E N0 , A  . (55) −𝛽  𝑗 ∈N\{𝐼,t(𝐼 ) } 1 + 𝑧∥ 𝑋 𝑗 − 𝒖 𝑠 ∥   

  Ö   1  N0 , A  E −𝛽 1 + 𝑧∥ 𝑋 − 𝒖 ∥ 𝑗 𝑠   𝑗 ∈N\{ 𝐼,t(𝐼 ) }   ! ∫ 1 = exp −𝜆𝑧 d𝒙 𝛽 R2 \𝑏𝒖 0 ( 𝑎𝑟 ˆ )\𝑏𝒖 𝑡 ( 𝑟ˆ (𝑡 ) ) 𝑧 + ∥𝒙 − 𝒖 𝑠 ∥ ! ∫ 1 − 𝜋𝜆𝐾𝛽 𝑧 2/𝛽 =e exp 𝜆𝑧 d𝒙 , 𝛽 𝑏𝒐 ( 𝑎𝑟 ˆ )∪𝑏 (𝑡𝑣,0) ( 𝑟ˆ (𝑡 ) ) 𝑧 + ∥𝒙 − 𝒖 𝑠 ∥ (56) where 𝐾 𝛽 is given in (39). The last line follows since     ∫ ∫ 1 1 d𝒙 = exp −𝜆𝑧 d𝒙 , exp −𝜆𝑧 𝛽 𝛽 R2 𝑧 + ∥𝒙∥ R2 𝑧 + ∥𝒙 − 𝒖 𝑠 ∥ to which we apply the last equality of [23, Eq. (26)]. Since 𝑡 lies in the support of the PDF in (24), we have either the case | 𝑎𝑟 ˆ − 𝑡𝑣| ≤ 𝑟ˆ (𝑡) ≤ 𝑎𝑟 ˆ + 𝑡𝑣, or the case 𝑎𝑟 ˆ + 𝑡𝑣 < 𝑟ˆ (𝑡), in which 𝑏 𝒐 ( 𝑎𝑟) ˆ ⊂ 𝑏 (𝑡𝑣,0) (𝑟ˆ (𝑡)). Therefore, similar techniques to the ones in [23, Eq. (32)] yield ∫ 1 d𝒙 𝑧 + ∥𝒙 − 𝒖𝑠 ∥ 𝛽 𝑏 (𝑡𝑣,0) ( 𝑟ˆ (𝑡 ) )\𝑏𝒐 ( 𝑎𝑟 ˆ ) 

∫ 𝑡𝑣+𝑟ˆ (𝑡∫) cos−1 −1∨ 𝑥2 +𝑡 2 𝑣2 −𝑟ˆ (𝑡 ) 2



𝑥

2𝑥𝑡𝑣

=2

𝛽

𝑎𝑟 ˆ

0

𝑧 + 𝑤 𝑥,𝑠𝑣, 𝜓

By the above equation together with ∫ ∫ 𝑎𝑟 ˆ∫ 𝜋 1 𝑥 d𝒙 = 2 𝛽 𝛽 𝑏𝒐 ( 𝑎𝑟 ˆ ) 𝑧 + ∥𝒙 − 𝒖 𝑠 ∥ 0 0 𝑧+𝑤

d𝜓d𝑥.

d𝜓d𝑥,

𝑥,𝑠𝑣, 𝜓

we have ! 1 exp 𝜆𝑧 d𝒙 = 𝜇(𝑧, 𝑠, 𝑡), 𝛽 𝑏𝒐 ( 𝑎𝑟 ˆ )∪𝑏 (𝑡𝑣,0) ( 𝑟ˆ (𝑡 ) ) 𝑧 + ∥𝒙 − 𝒖 𝑠 ∥ (57) where 𝜇(𝑧, 𝑠, 𝑡) is given in (40). Plugging (55), (56), and (57) into (53) leads to the result. ∫

C. Numerical Justification of Assumption 1 In this subsection, we numerically justify Assumption 1. We numerically evaluate 𝜌(𝑠 | A) using (38) and observe its evolution for 𝑠 ≥ 0. For the event A in (37), we consider 𝑟 = 0.5 km, 𝑡 = 50 sec, and 𝜙 = 𝜋/6, while 𝜃 is varied among 𝜋/6, 2𝜋/6, 3𝜋/6, 4𝜋/6, and 5𝜋/6. The remaining system parameters are set to (𝑣, 𝜆, 𝛽, 𝜎 2 , Off) = (0.01 km/sec, 1 km−2 , 4, 0, −3 dB). These parameter settings are presented as a representative example and are consistent with those used in Fig. 1, except that 𝜙 and 𝜎 2 are fixed at 𝜋/6 and 0, respectively. Fig. 4 shows the numerical results. The figure shows that Assumption 1 is satisfied for the considered parameter setting. When 𝜃 = 𝜋/6 or 2𝜋/6, the initial serving BS 𝑋𝐼 = (𝑟, 𝜃) lies in the direction of motion of the typical UE, and thus the distance ∥ 𝑋𝐼 − 𝒖 𝑠 ∥ initially decreases and then increases as 𝑠 grows

Conditional expectation of SINR

By the definitions of the events in (9) and (37), Φ\{𝑋𝐼 , 𝑋t(𝐼 ) } lies in R2 \𝑏𝒖0 ( 𝑎𝑟)\𝑏 ˆ 𝒖 𝑡 ( 𝑟ˆ (𝑡)), where 𝒖 𝑡 = (𝑡𝑣, 0). Thus, the generating functional of a PPP (e.g., [22, Example 9.4(c)]) gives

Elapsed time 𝑠 (sec) Fig. 4. Conditional expectation of 𝜌(𝑠 | A ) given in (38) as a function of elapsed time 𝑠 for different values of the serving cell angle 𝜃.

from zero. As a result, 𝜌(𝑠 | A) first increases and subsequently decreases for 𝑠 ≥ 0. In contrast, when 𝜃 = 4𝜋/6 or 5𝜋/6, 𝑋𝐼 does not lie in the direction of motion of the UE. Hence, ∥ 𝑋𝐼 − 𝒖 𝑠 ∥ increases monotonically with 𝑠. As a result, 𝜌(𝑠 | A) decreases monotonically for 𝑠 ≥ 0. References [1] M. -T. Nguyen and S. Kwon, “Machine Learning–Based Mobility Robustness Optimization Under Dynamic Cellular Networks,” IEEE Access, vol. 9, pp. 77830-77844, 2021. [2] R. Karmakar, G. Kaddoum and S. Chattopadhyay, “Mobility Management in 5G and Beyond: A Novel Smart Handover With Adaptive Time-toTrigger and Hysteresis Margin,” in IEEE Trans. Mobile Comput., vol. 22, no. 10, pp. 5995-6010, Oct. 2023. [3] F. Guidolin, I. Pappalardo, A. Zanella and M. Zorzi, “Context-Aware Handover Policies in HetNets,” in IEEE Trans. Wireless Commun., vol. 15, no. 3, pp. 1895-1906, March 2016. [4] Z. Zhu, J. Zheng, E. Takimoto, P. Finnerty, and C. Ohta, “Directional Handover Analysis with Stochastic Petri Net and Poisson Point Process in Heterogeneous Networks,” Mathematics, vol. 13, Art. no. 349, 2025. [5] D. Lopez-Perez, I. Guvenc and X. Chu, “Theoretical analysis of handover failure and ping-pong rates for heterogeneous networks,” in Proc. IEEE Int. Conf. Commun. (ICC), Ottawa, ON, Canada, June, 2012, pp. 6774-6779. [6] K. Vasudeva, M. Simsek, D. Lopez-Perez and I. Guvenc, “Analysis of Handover Failures in Heterogeneous Networks With Fading,” in IEEE Trans. Veh. Technol., vol. 66, no. 7, pp. 6060-6074, July 2017. [7] M. -T. Nguyen and S. Kwon, “Geometry-Based Analysis of Optimal Handover Parameters for Self-Organizing Networks,” IEEE Trans. Wireless Commun., vol. 19, no. 4, pp. 2670-2683, April 2020. [8] C. H. M. de Lima, M. Bennis and M. Latva-aho, “Modeling and analysis of handover failure probability in small cell networks,” 2014 IEEE Conf. Comput. Commun. Workshops (INFOCOM WKSHPS), Toronto, ON, Canada, April, 2014, pp. 736-741. [9] M. Salehi and E. Hossain, “Stochastic Geometry Analysis of Sojourn Time in Multi-Tier Cellular Networks,” in IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1816-1830, March 2021. [10] X. Xu, Z. Sun, X. Dai, T. Svensson and X. Tao, “Modeling and Analyzing the Cross-Tier Handover in Heterogeneous Networks,” IEEE Trans. Wireless Commun., vol. 16, no. 12, pp. 7859-7869, Dec. 2017. [11] H. Zhou, H. Zhou, J. Li, K. Yang, J. An and X. Shen, “Heterogeneous Ultradense Networks With Traffic Hotspots: A Unified Handover Analysis,” IEEE Internet of Things J., vol. 10, no. 10, pp. 8825-8838, May, 2023. [12] Y. Guo and H. Zhang, “3D Boundary Modeling and Handover Analysis of Aerial Users in Heterogeneous Networks,” in IEEE Trans. Veh. Technol., vol. 72, no. 10, pp. 13523-13529, Oct. 2023. [13] H. Wei and H. Zhang, “Equivalent Modeling and Analysis of Handover Process in K-Tier UAV Networks,” in IEEE Trans. Wireless Commun., vol. 22, no. 12, pp. 9658-9671, Dec. 2023.

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