Stochastic Geometry Analysis of Sojourn Time in Multi-Tier Cellular Networks

The next generations of cellular wireless networks are expected to support communications for highly mobile users and devices [1] with applications in new vertical sectors such as railway, unmanned aerial vehicle (UAV), and autonomous car. Therefore, addressing the mobility related challenges is necessary for the development of the next generation cellular networks. Impact of user/device mobility on its performance in cellular networks can be measured through mobility-aware performance metrics such as handoff rate, handoff probability, and sojourn time [1]. Sojourn time (or dwell time), time duration that a mobile user stays within a cell, is a key network parameter which allows studying other important network parameters such as channel occupancy time, new call and handoff call dropping probabilities [2]. Therefore, it is imperative to incorporate the sojourn time distribution in resource allocation and mobility management for improving the network performance. In general, modeling and analysis of mobility-related parameters and performances is however challenging in multi-tier (or heterogeneous) cellular networks (e.g. a two-tier macrocell-small cell network) since it needs to consider different aspects such as how to model the distributions of base stations (BSs) at the different tiers, how to model the user mobility and traffic at the different tiers, and how to model the radio access network performance at the different tiers [3].

In this above context, [4] derived the sojourn time distribution for the hexagonal (deterministic) cellular networks and Poisson (random) cellular networks, where the BSs are distributed according to a homogeneous Poisson point process (PPP). In [5], mean sojourn time of two-tier cellular networks was approximately derived, where the coverage areas of macro cells and small cells have regular shapes (circles) and within each macro cell multiple small cells are irregularly deployed. [6, 7] derived the mean sojourn time in small cells of two-tier cellular networks. The BSs of each tier are distributed following an independent homogeneous PPP. [8] also derived the mean sojourn time in two-tier cellular networks. However, it was assumed that a handoff occurs only when the mobile user crosses the boundary of a macro cell. Therefore, the mean sojourn time in [8] is similar to that in a single-tier network as in [4].

Moreover, the handoff rate, i.e. the expected number of handoffs in unit time, was derived in [4] for single-tier Poisson cellular networks and in [9] for multi-tier Poisson cellular networks. The handoff probability, i.e. the probability that the mobile user handoffs to a new BS at the end of a movement period, was also studied in [10] and [11] for single-tier and multi-tier Poisson networks, respectively. To derive the mean sojourn time (or distribution of the sojourn time), [4, 7, 8] used the chord length distribution (or linear contact distribution function) of Poisson Voronoi cells. However, in multi-tier networks with different transmission power and bias factor for each tier, we need the chord length distribution (or linear contact distribution function) of weighted Poisson Voronoi cells which is not available in the literature. For single-tier networks, [1] used the handoff probability to derive the distribution of sojourn time in the cell where connection is initiated. In single-tier networks, since the Voronoi cells are convex, we can directly use the handoff probability to derive the distribution of the sojourn time. However, in multi-tier networks, cells may not be convex. Therefore, the analytical method in [1] cannot be used for multi-tier networks.

A handoff is considered to be unnecessary when the dwell time of the mobile user in the new cell after the handoff is less than a predefined threshold. In [12, 13], handoff skipping schemes are employed to avoid unnecessary handoffs. Moreover, important system parameters such as channel occupancy time, new call and handoff call dropping probabilities depend on the sojourn time [2]. Therefore, sojourn time is fundamental for analysis and design of the mobile cellular networks. In [12, 2, 14, 15, 16], different distributions such as exponential, Erlang, gamma, Pareto, and Weibull were used for modeling the sojourn time distribution. Due to the principal role of the sojourn time in mobility management and resource allocation, in this paper, we derive the sojourn time distribution in multi-tier cellular networks.

To analyze the sojourn time distribution in multi-tier scenarios with PPP distributed BSs, the existing works either assume that the mobile user is always associated to only one of the tiers, or only focus on the small tier (in two-tier scenarios). For both the cases, the results are no different from the single-tier scenarios. In single-tier networks with maximum averaged received power association (nearest BS association), (Voronoi) cells are convex; however, in multi-tier networks with maximum biased averaged received power association, cells may not be convex depending on the transmission power and bias factor of each tier. Therefore, analysis of sojourn time of multi-tier cellular networks is more complicated compared to the single-tier networks. In this regard, the contributions of this paper can be summarized as follows:

• •

  We derive the distribution and mean of the sojourn time for multi-tier cellular networks. We show that the mean sojourn time is inversely proportional to velocity. We also study the impact of network parameters on the sojourn time.

• •

  To obtain the analytical results, we derive the linear contact distribution and chord length distribution of each tier.

• •

  We show that the mean sojourn time is inversely proportional to handoff rate. Also, using handoff rate and sojourn time, we calculate the ping-pong rate (i.e. rate of unnecessary handoffs) for each tier.

• •

  We show that the complement of the handoff probability provides an upper bound for the complementary cumulative distribution function (CCDF) of the sojourn time. We also discuss the scenarios where the CCDF of the sojourn time is equal to the complement of the handoff probability.

The rest of this paper is organized as follows: In Section II, the system model is presented. In Section III, we state the methodology for deriving the analytical results. In Sections IV and V, we obtain the main results related to the distribution and mean of the sojourn time and also discuss the effects of network parameters. Numerical and simulation results are provided in Section VI. Finally, in Section VII, we conclude the paper.

First we focus on the distribution of the sojourn time in the cell where connection is initiated $\tilde{S}$. Specifically, we derive the CCDF of $\tilde{S}$, i.e.

In single-tier cellular networks, Voronoi cells are convex [19] (as shown in Fig. 1(a)). A set $C$ is convex if the line segment between any two points in $C$ lies in $C$ [20]. Thus, in single-tier cellular networks, when the mobile user is connected to the same BS at time 0 and $T$, i.e. when $BS(T)=BS(0)$, the serving BS at any time between 0 and $T$ is also $BS(0)$, i.e. $BS(t)=BS(0)$, $\forall t\in(0,T)$. Hence, for single-tier cellular networks, (2) can be simplified as

However, for multi-tier cellular networks, Voronoi cells may not be convex depending on the values of $\beta_{kj}$, $k,j\in\mathcal{K}$ (as shown in Fig. 1(b)). Therefore, even when $BS(T)=BS(0)$, there may exist a time $t$ between 0 and $T$ for which $BS(t)\neq BS(0)$. To derive the CCDF of $\tilde{S}$ for multi-tier cellular networks, we must use (2), which makes the analysis of sojourn time in multi-tier cellular networks more complicated compared to the single-tier networks. Actually, single-tier scenario can be considered as a special case of multi-tier scenarios. Moreover, note that, (3) provides an upper bound for (2).

Given that the mobile user is initially connected to a tier-$k$ BS, the CCDF of $\tilde{S}$ can be obtained by

where $\theta$ is the angle between the serving BS at time 0 and direction of the movement (as shown in Fig. 2). $\theta$ is uniformly distributed in $[0,\pi]$. $f_{R}(r_{0}\mid\text{tier}=k)$ is the probability density function (PDF) of the serving link distance at time 0, given that $BS(0)$ belongs to tier-$k$. According to [21],

where $\mathbb{P}(\text{tier}=k)$ is the probability that $BS(0)$ belongs to tier-$k$ which is given by [21]:

Using the association strategy (1), we get

where $\mathcal{B}\left(0,\frac{r_{0}}{\beta_{kj}}\right)$ is excluded since we know there is no tier $j$ BS closer than $\frac{r_{0}}{\beta_{kj}}$ to the typical mobile user at time 0. Let us define

Using $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$, we can write

where $|A|$ denotes the area of $A$, (a) follows from the independence of different tiers’ point processes, and (b) is obtained by using the void probability of PPP. In Fig. 3, $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$ is illustrated for three different cases: a) $\beta_{kj}<1$, b) $\beta_{kj}=1$, and c) $\beta_{kj}>1$. To derive the distribution of $\tilde{S}$, we need to calculate the area of $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$ for all three cases. Further discussion about $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$ is provided in the next section.

Given that, at time 0, the mobile user is associated to a tier-$k$ BS, the origin is almost surely contained in the interior of a tier-$k$ Voronoi cell. In this paper, we define linear contact distribution function as the probability that a line segment $\ell$ containing the origin with length $r$ and random orientation crosses the cell boundaries. Therefore, given origin $o$ is inside a tier-$k$ cell, linear contact distribution function $H_{\ell}(z\mid\text{tier}=k)$ is equal to the probability that intersection of user’s trajectory with length $z$ and the cell boundaries is nonempty. Using the conditional CCDF of $\tilde{S}$, we can derive the linear contact distribution function as

So far, we have considered the sojourn time in the cell where connection is initiated ($\tilde{S}$). Distribution of the sojourn time ($S$), for tier-$k$, can be obtained using the chord length distribution. Due to the stationarity of our model, chord length distribution for tier-$k$, denoted by $F_{L}(z\mid\text{tier}=k)$, can be computed as follows [22]:

where $\mathbb{E}[L\mid\text{tier}=k]$ is the mean length of the chords lying in tier-$k$ cells, and is obtained by [23]

Finally, we can characterize the sojourn time distribution for tier-$k$ using the results from previous step. In particular, the mean and CCDF of the sojourn time in tier-$k$ are

The following proposition helps us to derive the area of $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$ for this case.

Note that, depending on radii of the two circles, $\frac{r_{0}}{\beta_{kj}}$ and $\frac{r_{0}(T)}{\beta_{kj}}$, and the distance between their centres, i.e. $vT$, three different situations can happen when $\beta_{kj}<1$:

Situation 1: When $\frac{r_{0}(T)}{\beta_{kj}}\geq\frac{r_{0}}{\beta_{kj}}+vT$, we have $\mathcal{B}\left(\textup{x}(0),\frac{r_{0}}{\beta_{kj}}\right)\subset\mathcal{B}\left(\textup{x}(T),\frac{r_{0}(T)}{\beta_{kj}}\right)$, which yields $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})=\mathcal{B}\left(\textup{x}(T),\frac{r_{0}(T)}{\beta_{kj}}\right)$.

Situation 2: When $\frac{r_{0}}{\beta_{kj}}\geq\frac{r_{0}(T)}{\beta_{kj}}+vT$, we have $\mathcal{B}\left(\textup{x}(T),\frac{r_{0}(T)}{\beta_{kj}}\right)\subset\mathcal{B}\left(\textup{x}(0),\frac{r_{0}}{\beta_{kj}}\right)$, which yields $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})=\mathcal{B}\left(\textup{x}(0),\frac{r_{0}}{\beta_{kj}}\right)$.

Situation 3: When $\frac{r_{0}(T)}{\beta_{kj}}<\frac{r_{0}}{\beta_{kj}}+vT$ and $\frac{r_{0}}{\beta_{kj}}<\frac{r_{0}(T)}{\beta_{kj}}+vT$, $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})=\mathcal{B}\left(\textup{x}(0),\frac{r_{0}}{\beta_{kj}}\right)\cup\mathcal{B}\left(\textup{x}(T),\frac{r_{0}(T)}{\beta_{kj}}\right)$. An example of which is illustrated in Fig. 3(a).

Using this information, now we can compute $|\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})|$ when $\beta_{kj}<1$.

where $V\left(\frac{r_{0}}{\beta_{kj}},\frac{r_{0}(T)}{\beta_{kj}},vT\right)$ is the area of intersection of two circles with radii $\frac{r_{0}}{\beta_{kj}}$ and $\frac{r_{0}(T)}{\beta_{kj}}$ whose centers are separated by $vT$, i.e., $V\left(\frac{r_{0}}{\beta_{kj}},\frac{r_{0}(T)}{\beta_{kj}},vT\right)=$

For this case, to derive the area of $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$, first we study the intersection of $\mathcal{B}\left(\textup{x}(t),\frac{r_{0}(t)}{\beta_{kj}}\right)$ and $\mathcal{B}\left(\textup{x}(t+{\rm d}t),\frac{r_{0}(t+{\rm d}t)}{\beta_{kj}}\right)$ as ${\rm d}t\to 0$ (Fig. 4(a)). From triangle equations, we have

Since $|vt-r_{0}\cos\theta|\leq r_{0}(t)$,

Dividing $r_{0}(t)-v{\rm d}t\leq r_{0}(t+{\rm d}t)\leq r_{0}(t)+v{\rm d}t$ by $\beta_{kj}$ yields

where (a) and (b) are obtained using $\beta_{kj}>1$. Therefore, as ${\rm d}t\to 0$, $\mathcal{B}\left(\textup{x}(t),\frac{r_{0}(t)}{\beta_{kj}}\right)$ and $\mathcal{B}\left(\textup{x}(t+{\rm d}t),\frac{r_{0}(t+{\rm d}t)}{\beta_{kj}}\right)$ partially overlap (boundaries of $\mathcal{B}\left(\textup{x}(t),\frac{r_{0}(t)}{\beta_{kj}}\right)$ and $\mathcal{B}\left(\textup{x}(t+{\rm d}t),\frac{r_{0}(t+{\rm d}t)}{\beta_{kj}}\right)$ intersect at two points), and we have

where the last result is proved in Appendix B.

Moreover, we can derive the intersection points of these two circles from their equations in Cartesian coordinate system, i.e.,

Combining these equations and solving for $x$ results in

which indicates that, for $\beta_{kj}>1$, the boundaries’ intersection points move along the positive $x$-axis as $t$ increases (Fig. 4(b)). Using this result, we can write

Let the time interval $[0,T]$ be partitioned by points $t_{i}=i{\rm d}t$, $i=0,...,\frac{T}{{\rm d}t}$. We can calculate the area of $\mathcal{A}_{kj}(r_{0},\theta,v,T,\beta_{kj})$ by setting $t=0$ in (20), i.e.,

where (a) is obtained by using (LABEL:eq:infinitesimal) and (b) follows from the Riemann integral.

For this case, similar to Appendix A, we can prove $\mathcal{A}_{kj}(r_{0},\theta,v,T,1)=\mathcal{B}\left(\textup{x}(0),r_{0}\right)\cup\mathcal{B}\left(\textup{x}(T),r_{0}(T)\right)$. Since $|r_{0}-vT|\leq r_{0}(T)\leq r_{0}+vT$, $\mathcal{B}\left(\textup{x}(0),r_{0}\right)$ and $\mathcal{B}\left(\textup{x}(T),r_{0}(T)\right)$ partially overlap. Therefore,