- Research Article
- Open Access
A Fluid Model for Performance Analysis in Cellular Networks
© Jean-Marc Kelif et al. 2010
Received: 30 October 2009
Accepted: 8 June 2010
Published: 28 August 2010
We propose a new framework to study the performance of cellular networks using a fluid model and we derive from this model analytical formulas for interference, outage probability, and spatial outage probability. The key idea of the fluid model is to consider the discrete base station (BS) entities as a continuum of transmitters that are spatially distributed in the network. This model allows us to obtain simple analytical expressions to reveal main characteristics of the network. In this paper, we focus on the downlink other-cell interference factor (OCIF), which is defined for a given user as the ratio of its outer cell received power to its inner cell received power. A closed-form formula of the OCIF is provided in this paper. From this formula, we are able to obtain the global outage probability as well as the spatial outage probability, which depends on the location of a mobile station (MS) initiating a new call. Our analytical results are compared to Monte Carlo simulations performed in a traditional hexagonal network. Furthermore, we demonstrate an application of the outage probability related to cell breathing and densification of cellular networks.
Estimation of cellular networks capacity is one of the key points before deployment and mainly depends on the characterization of interference. As downlink is often the limited link w.r.t. capacity, we focus on this direction throughout this paper, although the proposed framework can easily be extended to the uplink. An important system parameter for this characterization is the other-cell interference factor (OCIF). It represents the "weight" of the network on a given cell.
OCIF is traditionally defined as the ratio of other-cell interference to inner-cell interference. In this paper, we rather consider an OCIF, which is defined as the ratio of total other-cell received power to the total inner-cell received power. Although very close, this new definition is interesting for three reasons. Firstly, total-received power is the metric mobile stations (MS) are really able to measure on the field. Secondly, the power ratio is now a characteristic of the network and does not depend on the considered MS or service. At last, the definition of OCIF is still valid if we consider cellular systems without inner-cell interference; in this case, the denominator of the ratio is reduced to the useful power. The precise knowledge of the OCIF allows the derivation of performance parameters, such as outage probabilities, capacity, as well as the definition of Call Admission Control mechanisms, in CDMA (Code Division Multiple Access) and OFDMA (Orthogonal Frequency Division Multiple Access) systems.
Pioneering works on the subject  mainly focused on the uplink. Working on this link,  derived the distribution function of a ratio of path-losses, which is essential for evaluating the external interference. For this purpose, authors approximated the hexagonal cell with a disk of same area. Based on this result, Liu and Everitt proposed in  an iterative algorithm for the computation of the OCIF for the uplink.
On the downlink, [4, 5] aimed at computing an averaged OCIF over the cell by numerical integration in hexagonal networks. In , Gilhousen et al. provided Monte Carlo simulations and obtained a histogram of the OCIF. In , other-cell interference was given as a function of the distance to the base station (BS) using Monte-Carlo simulations. Chan and Hanly  precisely approximated the distribution of the other-cell interference. They, however, provided formulas that are difficult to handle in practice. Baccelli et al.  studied spatial blocking probabilities in random networks. Focusing on random networks, power ratios were nevertheless not their main concern and authors relied on approximated formulas, which are not validated by simulations.
In contrast to previous works in the field, the modeling key of our approach is to consider the discrete BS entities of a cellular network as a continuum. Recently, the authors of  described a network in terms of macroscopic quantities such as node density. The same idea was used in  for ad hoc networks. They however assumed a very high density of nodes in both papers and infinite networks. We show hereafter that our model is accurate even when the density of BS is very low and the network size is limited (see Section 3.2).
Central idea of this paper has been originally proposed in conference papers [12–15], which provide a simple closed-form formula for the OCIF on the downlink as a function of the distance to the BS, the path-loss exponent, the distance between two BS, and the network size. We validate here the formulas by Monte Carlo simulations and show that it is possible to get a simple outage probability approximation by integrating the OCIF over a circular cell. In addition, as this ratio is obtained as a function of the distance to the BS, it is possible to derive a spatial outage probability, which depends on the location of a newly initiated call. Outage probability formula allows us to analyze the phenomenon of cell breathing, which results in coverage holes when the traffic load increases. An answer to this issue is to increase the BS density (i.e., network densification). Thanks to the proposed formulas, we are able to quantify this increase.
We first introduce the interference model and the notations (Section 2), then present the fluid model and its validation (Section 3). In the fourth section, we derive outage probabilities. In the last section, we apply the theoretical results to the characterization of cell breathing and to network densification.
2. Interference Model and Notations
The following power quantities are considered:
is the useful power received at mobile from serving station (for traffic data). Since we do not consider soft-handover (SHO), serving station is well defined and the subscript can be omitted for the sake of readability.
The total amount of power experienced by a mobile station in a cellular system consists of three terms: useful signal power ( ), interference and noise power ( ). It is common to split the system power into two terms: , where is the internal (or own-cell) received power and is the external (or other-cell) interference. Note that we made the choice of including the useful signal in , and, as a consequence, it should not be confused with the commonly considered own-cell interference.
In this paper, we use the signal to interference plus noise ratio (SINR) as the criteria of radio quality: is the SINR target for the service requested by MS . This figure is a priori different from the SINR evaluated at mobile station . However, we assume perfect power control, so that for all users.
2.1. CDMA Network
On the downlink of CDMA networks, orthogonality between physical channels may be approached by Hadamard multiplexing if the delay spread is much smaller than the chip duration . As a consequence, a coefficient may be introduced to account for the lack of perfect orthogonality in the own cell.
In HSDPA (High Speed Downlink Packet Access), (2) and (8) are valid for each TTI (Transmission Time Interval). Parameter should be now interpreted as an experienced SINR and not any more as a target. Sums are done on the number of scheduled users per TTI. If a single user is scheduled per TTI (which is a common case), sums reduce to a single term. Even in this case, parameter has to be considered since common control channels and traffic channels are not perfectly orthogonal due to multipath.
Even if previous equations use the formalism of CDMA networks, it is worth noting that they are still valid for other multiplexing schemes. For cellular technologies without internal interference (TDMA, Time Division Multiple Access, and OFDMA), reduces to and is the cochannel interference in (2). The definition of is unchanged provided that sums in (4) and (5) are done over the set of cochannel interfering BS (to account for frequency reuses different from reuse one).
2.2. OFDMA Network
Consequently, appears to be an important parameter characterizing cellular networks. This is the reason why we focus on this factor in the next section and, with the purpose of proposing a closed form formula of , we develop a physical model of the network.
3. Fluid Model
3.1. OCIF Formula
The key modelling step of the model we propose consists in replacing a given fixed finite number of interfering BS by an equivalent continuum of transmitters, which are spatially distributed in the network. This means that the transmitting interference power is now considered as a continuum field all over the network. In this context, the network is characterized by a MS density and a cochannel base station density . We assume that MS and BS are uniformly distributed in the network, so that and are constant. As the network is homogeneous, all base stations have the same output power .
For the assumed omnidirectional BS network, we use a propagation model, where the path gain, , only depends on the distance between the BS and the MS . The power, , received by a mobile at distance can thus be written , where is a constant and is the path-loss exponent.
This closed-form formula will allow us to quickly compute performance parameters of a cellular network. However, before going ahead, we need to validate the different approximations we made in this model.
3.2. Validation of the Fluid Model
The fluid model and the traditional hexagonal model are both simplifications of the reality. None is a priori better than the other but the latter is widely used, especially for dimensioning purposes. That is the reason why a comparison is useful.
The validation is done by Monte Carlo simulations:
at the end of the simulation, all values of corresponding to a given distance are averaged and we plot the average value in Figure 4.
Figure 4 shows the simulated OCIF as a function of the distance to the base station. Simulation parameters are the following: Km; between and ; ; is chosen, such that the number of rings of interfering BS is 15; and the number of snapshots is 1000. Equation (13) is also plotted for comparison. In all cases, the fluid model matches very well the simulations on a hexagonal network for various figures of the path-loss exponent. Note that at the border of the cell (between and Km), the model is a little bit less accurate because hexagon corners are not well captured by the fluid model.
3.3. OCIF Formula for Hexagonal Networks
Two frameworks for the study of cellular networks are considered in this paper: the traditional hexagonal model and the fluid model. While the former is widely used, the latter is very simple and allows the derivation of an analytical formula for . The last section has shown that both models lead to comparable results for the OCIF as a function of the distance to the BS. If we want to go further in the comparison of both models, in particular with the computation of outage probabilities, we need however to be more accurate.
Such calculations require indeed the use of the function (see Section 4.3 and (23) and (24)), which is very sensitive to its arguments (mean and standard deviation). This point is rarely raised in literature: analysis and Monte Carlo simulations can lead to quite different outage probabilities even if analytical average and variance of the underlying Gaussian distribution are very close to simulated figures.
In this perspective, we provide an alternative formula for that better matches the simulated figures in a hexagonal network. Note that this result is not needed if network designers use the new framework proposed in this paper.
4. Outage Probabilities
In this section, we compute the global outage probability and the spatial outage probability with the Gaussian approximation. Closed-form formulas for the mean and standard deviation of over a cell are provided.
Quality of service in cellular networks can be characterized by two main parameters: the blocking probability and the outage probability. The former is evaluated at the steady state of a dynamical system considering call arrivals and departures. It is related to a call admission control (CAC) that accepts or rejects new calls. The outage probability is evaluated in a semistatic system , where the number of MS is fixed and their locations are random. This approach is often used (see, e.g., ) to model mobility in a simple way: MS jump from one location to another independently. For a given number of MS per cell, outage probability is thus the proportion of configurations, where the needed BS output power exceeds the maximum output power: .
4.1. Global Outage Probability
4.2. Spatial Outage Probability
For a given number of MS per cell, a spatial outage probability can also be defined. In this case, it is assumed that MS have already been accepted by the system, that is, the output power needed to serve them does not exceed the maximum allowed power. The spatial outage probability at location is the probability that maximum power is exceeded if a new MS is accepted in .
4.3. Gaussian Approximation
where is given by (14). This equation allows us to precisely compute the influence of an entering mobile station whatever its position in a cell and is thus the starting point for an efficient call admission control algorithm.
Note that for an accurate fitting of the analytical formulas, which are presented in this section, to the Monte Carlo simulations performed in a hexagonal network, should be multiplied by , by and (14) replaced by (16).
4.4. Simulation Methodology
Monte Carlo simulations have been performed in order to validate the analytical approach. A fixed number of MS are uniformly drawn on a given cell. All interferers are assumed to have the same transmitted power (homogeneous network). OCIF is computed according to (5). Power transmitted by the cell is then compared to for the calculation of the global outage probability.
For the spatial outage probability calculation, only snapshots without outage are considered. A new MS is added in the cell. The new transmitted power is again compared to and the result is recorded with the distance of the new MS.
Figure 11 shows the spatial outage probability as a function of the distance to the BS for and for various numbers of MS per cell. Given that there are already , these curves give the probability that a new user, initiating a new call at a given distance, implies an outage. As an example, a new user in a cell with already ongoing calls, will cause outage with probability at m from the BS and with probability at m from the BS.
Traditional admission control schemes are based on the number of active MS in the cell. With the result of this paper, an operator would be able to admit or reject new connections also according to the location of the entering MS.
5. Application to Network Densification
In this section, we show the application of previous results to network densification. During the dimensioning process, the cell radius is determined by taking into account a maximum value of outage probability. This value characterizes the quality of service in terms of coverage the network operator wants to achieve. The number of BS to cover a given zone is directly derived from the cell radius.
Considering a maximum value of the outage probability, we first characterize cell breathing; that is, the fact that cell coverage decreases when the cell load increases. We then analyze BS densification as an answer to cell breathing.
5.1. Cell Breathing Characterization
As is the mobile density, we can write , where is the maximum number of mobiles served by a BS for maximum outage probability , and is the area covered by the BS. Let be the cell area. When mobile density increases, decreases, so that .
Equation (28) shows the link between the mobile density and the covered area and is now used to characterize cell breathing.
The solid curve shows the mobile density as a function of the coverage area of base stations. On this curve, the BS density is supposed to be constant, Km and thus Km2. The coverage area however shrinks when the traffic (characterized here by the density of mobiles ) increases. For example, going from point with mobiles/Km2 to point with mobiles/Km2 reduces the covered area from Km2 to approximately Km2. As a consequence, the cell is not completely covered, and due to cell breathing, coverage holes appear.
5.2. Base Station Densification
A way of solving the issue of cell breathing is to densify the network. The dotted line in Figure 12 plots the mobile density as a function of the covered area assuming full coverage of the cell. Along this curve, and when is decreasing, the BS density is increasing. We are thus able to find, for a given mobile density, the BS density that will ensure continuous coverage in the network. For example, for mobiles/Km2, the cell area should be approximately Km2 in order to avoid coverage holes.
The sequence - - shows an example of scenario, where BS densification is needed. In point , the network has been dimensioned for mobiles/Km2. If the cellular operator is successful and so more subscribers are accessing the network, mobile density increases along the solid line of Figure 12. At point , coverage holes appear and the operator decides to densify the network. While adding new BS, he has to jump to point in order to ensure continuous coverage. At point , he needs approximately BS per Km2 ( Km2), while at point 3, he needs about BS per Km2 ( Km2), which corresponds to a increase.
In this paper, we have proposed and validated by Monte Carlo simulations a fluid model for the estimation of outage and spatial outage probabilities in cellular networks. This approach considers BS as a continuum of transmitters and provides a simple formula for the other-cell power ratio (OCIF) as a function of the distance to the BS, the path-loss exponent, the distance between BS and the network size. Simulations show that the obtained closed-form formula is a very good approximation, even for the traditional hexagonal network. The simplicity of the result allows a spatial integration of the OCIF leading to closed-form formula for the global outage probability and for the spatial outage probability. At last, this approach allows us to quantify cell breathing and network densification.
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