# Efficient Uplink Modeling for Dynamic System-Level Simulations of Cellular and Mobile Networks

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## Abstract

A novel theoretical framework for uplink simulations is proposed. It allows investigations which have to cover a very long (real-) time and which at the same time require a certain level of accuracy in terms of radio resource management, quality of service, and mobility. This is of particular importance for simulations of self-organizing networks. For this purpose, conventional system level simulators are not suitable due to slow simulation speeds far beyond real-time. Simpler, snapshot-based tools are lacking the aforementioned accuracy. The runtime improvements are achieved by deriving abstract theoretical models for the MAC layer behavior. The focus in this work is long term evolution, and the most important uplink effects such as fluctuating interference, power control, power limitation, adaptive transmission bandwidth, and control channel limitations are considered. Limitations of the abstract models will be discussed as well. Exemplary results are given at the end to demonstrate the capability of the derived framework.

### Keywords

Radio Resource Management Schedule Probability Cell Edge User Intercell Interference Uplink Power Control## 1. Introduction

The requirements for simulation tools are changing with the introduction of novel advanced methods. In particular, investigation of self-organizing networks (SONs) [1, 2, 3, 4, 5] have to cover extremely long time intervals; however, they require a sufficient level of accuracy in terms of radio resource management (RRM), quality of service (QoS), and mobility at the same time. For instance, self-optimization of the downtilt angle [6] is a process which may cover at least several days, since the network has to make sure that meaningful statistics on user locations and signal strengths have been collected. Furthermore, there are certainly interactions and collisions between SON and RRM, so that RRM cannot be entirely excluded from the simulations. For instance, if the downtilt angle is changed too fast, RRM measurements might get confused leading to an unstable system. Similar things hold for other SON use cases such as load balancing [7], mobility robustness optimization, and automatic neighbor relation [5].

Typical system-level simulations [8] have a very exact implementation of RRM and QoS by explicitly modeling all the fast decisions, typically on a millisecond time scale or even below, for example [9]. This ends up in a very large simulation runtime, far beyond real-time. Simulating several hours, days, or even more is impossible with this class of simulators. Those simulators are used to make accurate performance evaluations given a fixed parameter configuration according to specified reference scenarios.

Alternatively, the use of *light,* snapshot-based tools is quite popular [10, 11]. Those allow for a rapid collection of network statistics. However, accuracy of RRM and QoS is lost to a wide extent. In particular, handover effects such as hysteresis and time to trigger. can not be modeled without having a true time axis implemented. Furthermore, traffic characteristics are poorly reflected, for example, the fact that users at the cell edge require much more resources than close users in many cases. It is also more than critical to investigate convergence behavior of dynamic SON loops without a real-time axis and without real mobility. Those simulators are used for network planning or for coarse studies to understand the interrelations of new features, for example, heterogeneous networks [12].

- (i)
Every terminal has its own individual power budget.

- (ii)
The uplink typically has a power control (due to near/far problem).

- (iii)
The intercell interference is heavily fluctuating.

- (iv)
Control channel limitations are more critical.

- (v)
The access scheme might be different so that the scheduling strategies are different.

Those aspects will be addressed in this work based on the principles introduced in [13]. Although the focus of this work is on the introduction of the simulation framework, we will also give some calibration results as well as some first SON results. The derivations are based on the 3GPP standard long-term evolution (LTE) [14]. However the principles can be applied to other systems such as HSPA and WiMAX as well.

We will start with definitions of the LTE uplink, the uplink power control, and the uplink SINR. In Section 3 we will discuss the scheduling strategies. We will consider different resource fair strategies, throughput fair strategies and QoS strategies targeting a certain bit rate. All derivations are done under the assumption of an*adaptive transmission bandwidth* scheduler. Performance metrics are introduced in Section 4, in particular, dissatisfaction levels due to overload, power limitation, and control channel limitation. Results with the new framework are given in Section 5, and Section 6 concludes this work. In the appendices important and interesting properties of fairness in the uplink in comparison to downlink fairness are discussed.

## 2. Definitions

We will discuss the LTE uplink, which is a *Single Carrier FDMA* system. [14]. The whole system bandwidth is divided into Open image in new window subbands which are called physical resource blocks (PRBs). In every transmission time interval (TTI) a user can be assigned a subset of those Open image in new window PRBs which, however, have to be adjacent. The user will spread the symbols to transmit over this group of PRBs. Note that this so-called single carrier constraint is different to the OFDMA downlink.

Due to the single carrier constraint a frequency selective scheduler for the LTE uplink may have a packing problem ("Tetris" problem), that is, it might not be able to fill the entire bandwidth in some cases. The more multiuser diversity the scheduler aims to exploit, the larger will be the packing problem. In this work we neglect those cutaways, that is, we assume that the scheduler can fill the entire bandwidth. Note that it is very easy to construct such a scheduler, but the frequency-selective multi-user gain will be poor.

Random variables will be written in bold letters, for example, Open image in new window or Open image in new window . It is very important for this work to distinguish between random and deterministic variables. All variables refer to linear values, except the first equations (1) to (4) that make use of the dB domain. For the sake of better notation we are using the same symbols nevertheless.

### 2.1. General Definitions

We are assuming a network given by Open image in new window users Open image in new window located at the coordinates Open image in new window , and Open image in new window cells Open image in new window . All propagation effects (comprising pathloss, antenna patterns, and shadowing) between position Open image in new window and cell Open image in new window are summarized in the *propagation maps* Open image in new window . Details on the included propagation effects are found in [13]. Note that the propagation maps are deterministic for our investigations even if the shadowing has been generated randomly. Fast Fading is not considered in this work. Open image in new window is the thermal noise on a single PRB.

Open image in new window is the *downtilt* angle of cell Open image in new window . We assume that this is the only propagation parameter which can be dynamically influenced, all others are either given by the environment (e.g., pathloss exponent, shadowing) or are configured statically (e.g., antenna height, azimuth orientation) and are therefore omitted. Please note that downtilt optimization is an important SON use case, and hence we leave the downtilt angle in the equations although we do not present results on that.

Furthermore, every cell Open image in new window can adjust individual power control settings given by the parameters Open image in new window and Open image in new window according to [15]. We assume that user Open image in new window is served by cell Open image in new window , where Open image in new window is the*connection function,* and every user is connected exactly to a single cell. In this work, we assume that Open image in new window is given by the best serving cell on downlink, that is, every user is connected to the strongest cell. This is a typical case; however we could in principle also optimize the connection function with the equations given in this work.

The number of users in cell Open image in new window is abbreviated by Open image in new window , and the set of users connected to cell Open image in new window is abbreviated by Open image in new window .

### 2.2. Power Control

*power spectral density*) as

### 2.3. Signal-to-Noise and Interference Ratio

*scheduling probabilities*Open image in new window . We assume that the scheduling probabilities are identically distributed over time and frequency but not independently. Correlations and further details of the random variables Open image in new window will be discussed later on. As a consequence, the interference produced from cell Open image in new window to a target cell Open image in new window is also a random variable:

- (i)
The received power Open image in new window is not a random variable.

- (ii)
The last expectation of (7) does not depend on user Open image in new window , only on the cell Open image in new window , that is, it is the same for all other users connected to cell Open image in new window .

- (iii)
It is interesting to see that the more the interference Open image in new window fluctuates, the smaller gets the average

**SINR**. This is easily derived from Jensen's inequality ( Open image in new window is a convex function).

Note that the random variable Open image in new window is actually a deterministic function of the random variable Open image in new window (cf. (5)),that is, the interference is determined as soon as the scheduler has selected a user Open image in new window .

### 2.4. Evaluation of the Expectation

Please note that cell Open image in new window does not contribute to the interference on itself. However, for the sake of better illustration we have left the corresponding sum in the equation. Unfortunately, the nested sum can hardly be evaluated numerically. For instance, in a typical scenario [16] with 57 cells and 10 users per cell we would have Open image in new window addends. Unfortunately, due to the nonlinearity of the Open image in new window function, there is no way to separate the random variables and thereby the nested sums. Restricting the interference impact to only close neighbors (e.g., first and second ring around a cell) reduces the problem a bit; however it is still hardly feasible. Note that we have used the abbreviation Open image in new window which is the set of users connected to cell Open image in new window .

*Monte Carlo*integration. We generate a large number Open image in new window of random Open image in new window

*-*tuples Open image in new window

*with*Open image in new window containing samples of the random variables Open image in new window . As long as the number of samples Open image in new window is sufficiently large, we can get a good approximation of the expectation by

Our investigations have shown that Open image in new window gives stable results and is still feasible from a complexity point of view. Note that for the Monte Carlo approach the generation of the random Open image in new window -tuples certainly must follow the scheduling probabilities Open image in new window . Accuracy can be increased by combining the two approaches: the first ring of interfering cells can be exactly evaluated whereas the rest of the cells is considered by the Monte Carlo approach. In this paper we have only used the Monte-Carlo approach.

### 2.5. Rate Function

Note that the Shannon bounds inherently assume a perfect selection of modulation and coding schemes. However in the uplink, due to fluctuating interference, this selection can not be perfect by definition, even not in static channel conditions. Furthermore imperfect channel estimation will also degrade the performance. The consequence is a loss of some dBs. On the other hand, the base stations typically have 2 receive antennas, which is also not considered in the Shannon bounds which will lead to a gain in the range of 3 dB. Furthermore, frequency selective scheduling (e.g., though proportional fair scheduling) will lead to multi-user diversity gain [18, 19].

In this work we will assume that those effects will compensate each other such that the rate function used here (red solid curve) is rather close to the Shannon bound considering the overhead through cyclic prefix and reference signals. Later on in Section 5.2 we will see that this assumption leads to a good agreement with existing simulation results.

## 3. Scheduling Probabilities

### 3.1. General Expression

We observe that the scheduling probabilities depend purely on the average number of assigned PRBs Open image in new window 's. Hence, we will investigate those elaborately in the following sections. We will be looking at individual cells; we assume that cells in general behave independently, that is, the random variables Open image in new window 's are mutually independent, too.

### 3.2. Adaptive Transmission Bandwidth

*power headroom:*

An uplink scheduler should never assign a user more PRBs than this limit Open image in new window . Otherwise, looking at the original power control equation (1), we observe that the users would have to spread the same power over the assigned PRBs instead of increasing the power with every assigned PRB (the Open image in new window operator in the PC equation (1) expires). This results in an SINR loss which would eat up at least part of the bandwidth gain. Furthermore, other (non-power-limited) users can make much better use of the bandwidth. Finally, spreading the maximum power over several PRBs would increase the dynamic range problems. Note that for the PC equation per PRB (2) we have already inherently assumed that the scheduler does not exceed the aforementioned limit. This behavior is typically called *adaptive transmission bandwidth [ 20 ]*.

*average number of PRBs as well, since every user can be scheduled at maximum in every time slot, hence we have*

### 3.3. Strict Resource Fair

An important observation is that this solution is also throughput fair in the case of Open image in new window (with the exception that power limited users would have smaller throughput). Otherwise ( Open image in new window ) close users get higher throughput since the received power is higher and the interference is the same for all users in a cell.

### 3.4. Modified Resource Fair

The previous scheduler has the disadvantage that it may leave a lot of resources unused although close users would still be able to extend their bandwidth. Unfortunately, users at the cell edge with high propagation loss cannot make use of the spare bandwidth due to power limitation.

In another extreme solution we could try to always give every user Open image in new window its maximum allowed bandwidth Open image in new window . If this does not exceed the available resources, that is, Open image in new window , this is a viable approach. However, this will be relatively unlikely in reality since already a single close user could have enough transmit power to occupy more than Open image in new window PRBs.

In this case we need to scale down the number of PRBs. The simplest solution would scale down all Open image in new window 's in the same way. However this would leave too much unfairness in the system. Instead we prefer scaling down large Open image in new window 's and bring this new solution as close as possible to the resource fair case. We will call this solution *modified resource fair* although it is in general not resource fair. However, in annex A we will observe that this solution achieves the same fairness as the typical resource fair definition in the downlink.

- (1)
Initialize: Open image in new window ; Open image in new window

- (2)
Abbreviate Open image in new window

- (3)if Open image in new window
- (a)
- (b)
- (c)
- (d)
exit

- (a)
- (4)
Increment Open image in new window and go to step 2

In every iteration, we check whether the remaining resource budget Open image in new window equally shared among the remaining Open image in new window exceeds the PRB limit Open image in new window of the worst of the remaining users Open image in new window . If yes, the worst remaining user gets its maximum number of PRBs Open image in new window , and we assign the remaining budget in the next iteration. Otherwise the remaining budget is equally shared among the remaining users, and we exit the algorithm.

Note again that in this solution the worst user gets the least amount of resources, but the maximum it can afford. With a high number of users this case will converge against the previous "Resource Fair" case.

### 3.5. Throughput Fair

for two users Open image in new window and Open image in new window in the same cell. Note that throughput fairness is required per cell. Unfortunately the SINRs are not known so far; recall that the Open image in new window 's are needed to calculated scheduling probabilities and thereby the SINRs. Therefore we will give two different approximations in the following.

Both approximations have the very nice property that they only depend on the positions of the users within a cell and not on intercell interference or other cells in general. With those assumptions, we can formulate the throughput fair (approximated) solution in three steps.

### 3.6. Quality of Service

A drawback of the previous methods is that we cannot define a target QoS or a user satisfaction level. Inherently the methods were based on the best effort and full buffer assumption. The users always have data to transmit on one hand; on the other hand they do not have to meet a certain target, that is, they are satisfied with whatever resources Open image in new window they get.

For a variety of services a certain QoS target has to be met. For instance, users are only satisfied if they get a certain bit rate Open image in new window . If they get less, they are unsatisfied. On the other hand, they typically cannot transmit more than Open image in new window , so the system will assign only the resources Open image in new window such that the target rate is fulfilled, not more. Such a behavior is called *constant bit rate*(CBR) service.

where Open image in new window is the rate function introduced in Section 2.5. It is important to observe that a user cannot be satisfied if the Open image in new window operator expires, irrespective of the traffic situation in the own cell (even if the user were alone). The only way to improve those users is to decrease the intercell interference, which requires modifications in the neighboring cell such as decreasing the P0 [21]. Note that any of those modifications is likely to reduce the QoS level in the neighboring cell.

This scaling procedure would basically make every user unsatisfied. However note that the scheduling probabilities here are needed to calculate SINRs. Performance metrics will be discussed in Section 4. Alternatively, we could make use of admission control functionality here, which basically would select a subset Open image in new window (and drops the other users) such that Open image in new window is fulfilled.

We would like to emphasize again that we have assumed that the Open image in new window 's are already known. However, we actually need the scheduling probabilities to calculate the Open image in new window 's based on (7). So in contrast to the strict resource fair, modified resource fair and (approximated) throughput fair solutions of the previous sections, we unfortunately have not found a closed form solution for the QoS case. This problem is very similar to the downlink problem as described in [13].

### 3.7. Comparison with Real-World Schedulers

In the following we will discuss how real schedulers would map to the previously introduced strategies. The most popular scheduler is a *proportional fair*(PF) scheduler. The pure PF strategy is resource fair [18, 19]. However, unfortunately the PF definition in the uplink is not as straightforward as it is in the downlink due to power control and power limitation. Most of the uplink PF strategies in LTE will use adaptive transmission bandwidth and will be very close to the modified resource fair definition introduced in Section 3.4, when assuming full buffer/best effort traffic models (i.e., no further QoS constraints), compare, for example, [20]. Note that the scheduling gain, that is, the fact that the SINR conditioned on a user being scheduled gets better, goes into the throughput mapping discussed in Section 2.5 and not into the scheduling probabilities. Hence, PF and round robin strategies are equivalent from the perspective of scheduling probabilities (both are resource fair).

Furthermore, the PF strategies typically have to be extended with QoS constraints such as a target bit rate, minimum bit rate, or delay constraints. Those extended PF versions will come closer to the QoS scheduler described in Section 3.6. Once again, the reduced scheduling gain (through more QoS constraints) is considered in the throughput mapping, rather than in the scheduling probabilities.

### 3.8. Initialization of the SINRs

The advantage is that it might be easier to make a guess on the SINR since it is a relative number rather than a guess on the expectation which is an absolute number. In particular the SINR of the worst user in a cell is rather likely to be very small. So the second proposal is to set the SINR of the worst user in every cell to a predefined value Open image in new window (e.g., 0 dB), and the other user's SINR in the same cell are derived from that according to (26). This method has the advantage that it also works with so-called *snapshot-* like simulators which do not have a time axis. In a dynamic simulator, this approach is probably less accurate than the first one.

## 4. Performance Metrics

So far, we have an (almost) analytical expression Open image in new window for the average SINR of every user in an LTE uplink network. Furthermore, we have already discussed the average number Open image in new window of assigned PRBs for different scheduling strategies. Note that in the QoS case the Open image in new window 's actually depend on the SINRs which are not known when calculating the Open image in new window 's. Hence, before calculating performance metrics we should update the Open image in new window 's with the more accurate values of the SINRs.

From these Open image in new window 's and Open image in new window 's we now can start deriving several capacity metrics such as average cell throughput, throughput percentiles, or number of (un)satisfied users.

### 4.1. Throughput Metrics

From those rates we can calculate a total network throughput, throughputs per cell, or throughput percentiles. In principle we could also check whether users are satisfied by comparing their data rates with the rate requirements Open image in new window 's. However recall that in (24) we have scaled down the Open image in new window 's of all users in case of an overload. In this case, *all users* would fall below their Open image in new window 's although in reality it might be sufficient to drop very few users to make the rest satisfied again. Furthermore, it would be interesting to have a quantitative notion of how much overloaded a cell is and how many users are unsatisfied in fact. So for the QoS case, we will define more appropriate performance metric in the following.

### 4.2. Overload and Unsatisfied Users

*virtual cell load*

## 5. Results

A dynamic system level simulator has been implemented based on the derivations in the previous chapters. In this section we will present some results with standard assumptions (such as full buffer traffic, proportional fair scheduler), and we will show that those are very close to other simulation results which have been agreed for by several companies in [9, 22]. Furthermore, we will present results with CBR traffic, and we will also look at an irregular network with SON adaptation of the power control parameters. Finally we will elaborate on the huge runtime performance.

### 5.1. Simulation Assumptions

We will use standard assumptions as proposed in [16], comprising a network of 19 LTE base stations with an intersite distance of 500 m, serving 57 hexagonal cells (sectors). Pathloss law, shadowing model, and horizontal beam pattern are also taken from [16], a vertical pattern is not used. The users are moving with a speed of 3 km/h, and they are handover to another cell if the received signal strength (measured on downlink reference signals) with respect to the new cell is 3 dB better than that with respect to the serving cell (handover hysteresis). One simulation step is 100 ms, that is, the network performance is evaluated 10 times a second.

*P0*values of Open image in new window or Open image in new window and a homogeneous Open image in new window value of Open image in new window . The resulting distribution of transmit power per PRB is shown in Figure 3. Note that this distribution does not depend on the scheduling mechanism or traffic model since we record one power value for every user per simulation step.

It is obvious that the larger P0 setting of Open image in new window 52 dBm leads to higher transmit powers. In this case we can also identify the maximum transmit power of 23 dBm.

### 5.2. Full Buffer Traffic

As expected we observe slightly higher user throughputs with the larger *P0* value. However, the difference between the curves is smaller in the lower part of the plot, since the power limitation is more critical with the smaller *P0* value. The 5% percentiles (which is typically referred to as *cell edge throughput)* are*420 * kbps and*503 * kbps whereas the average cell throughputs are 7.3 Mbps and 8.5 Mbps, respectively. This is in very good agreement with the simulations in [9, 22]. The results of different companies are compared in [22] for the reference case which we have used as well. The cell throughput results are in the range between 6.3 Mbps and 1.01 Mbps, with an average of 8.6 Mbps (which is also the result of [9]). The cell edge results span from 100 kbps to 460 kbps with an average of 260 kbps. Obviously our results are a bit too optimistic in terms of cell edge throughput which could be a consequence of the neglected fast fading, and, even more important, of handover gain, which is included in our simulations with full mobility.

### 5.3. Constant Bit Rate Traffic

- (i)
All curves reach a maximum and then do not grow any further. The reason is that the actual load is limited and cannot exceed 100%. So the interference will also not grow with the number of users, and the SINRs will not decrease.

- (ii)
The (power) dissatisfaction level is larger for higher data rates. This is quite obvious.

- (iii)
The (power) dissatisfaction level is larger for the larger Open image in new window . With smaller

*P0*, the users can afford more PRBs, compare (14), whereas the interference level goes down as well (note that the other cells will reduce P0 as well in our model). So the SINRs remain the same as long as we do not enter noise limited regimes. - (iv)
With 512 kbps and Open image in new window we even have a "dissatisfaction floor," that is, there will be power limited users even in an empty system. That is, high uplink data rates can only be supported with small Open image in new window values (or by relaxing the ATB power constraint (14)).

Certainly we can recognize the aforementioned dissatisfaction floor for 512 kbps and Open image in new window in this figure. Otherwise, the impact of the P0 value is almost negligible since adding users beyond 100% virtual load obviously means load-unsatisfied users hiding the aforementioned limit for the dissatisfaction level due to the power constraint. If we target a typical overall dissatisfaction level of 5%, the uplink can satisfy 10, 21, and 56 user with 512 kbps, 256 kbps, and 96 kbps, respectively. The cell throughput with the smaller rates is around 5.4 Mbps whereas the 512 kbps case is slightly worse with 5.4 Mbps due to the more critical power limitation.

As expected the CBR capacity is significantly below the best effort capacity. However, the difference is smaller than in the downlink, since the power control compensates for a part of the SINR loss of cell edge users.

### 5.4. Heterogeneous Scenario

*load adaptive power control*(LAPC) as proposed in [24] where the Open image in new window s are reduced in cells which only carry a small load. In the CBR model reducing Open image in new window blows up the resource consumption since the resulting SINR loss has to be compensated by bandwidth. We use a very similar approach to [24] and update the Open image in new window at time step Open image in new window depending on the previous value Open image in new window and the previous virtual load Open image in new window (note that this equation is in dB scale):

where Open image in new window is the virtual load which we are targeting. In theory we may want to target 100%; however, experience has shown that a margin should be left for handover users so that we will use Open image in new window . The rule means that we increase the current Open image in new window if the load is above target, and we decrease it if the load is below the target; however, we will not increase the initial Open image in new window which has been defined above. Note that this automatic adaptation of a cell parameter can already be considered as a SON mechanism.

*interference over thermal*(IoT) values. Those are based on the Open image in new window samples used for the Monte Carlo approach defined in (10); the exact definition of the (instantaneous) IoT is given by

The LAPC has degraded the SINR in the overloaded cell even though the Open image in new window has not been reduced there since the more fluctuating interference obviously offers more potential for the link adaptation (which is assumed to be ideal in our model). This is also visible in the virtual load of the overloaded cell no.11 in Figure 8 which has slightly been increased by LAPC (as a result of the decreased SINRs). This is a contrast to the results in [24] where LAPC helps to improve the system. Fluctuating interference has a negative impact on link adaptation (i.e., selection of modulation and coding schemes, scheduling, etc.). In other words, smoothening the interference through LAPC will improve link adaptation. Unfortunately this effect is not covered in our simplified model, where link adaptation is always assumed to be ideal. Hence, our model can exploit the aforementioned potential offered by fluctuating interference, which is not the case in reality.

Although we have gained important insights by this analysis, it also reveals a current limitation of the model. A remedy could be based on the principles of [25], where the rate function is elaborated by the introduction of a correlation between the SINR at the moment of choosing the modulation and coding scheme and the moment of applying it (where the interference might have changed). This correlation would increase through LAPC.

### 5.5. Simulation Runtime

## 6. Conclusions

We have presented a very efficient modeling approach for uplink investigations focussing on the LTE standard. QoS and radio resource management (which typically work on a millisecond time scale) are modeled in a very abstract but still accurate way such that the essential behavior is still included. By those means we can decrease simulation runtime far below real-time.

This is in particular helpful, even necessary, for simulations covering long time intervals. The most important applications are investigations on self-optimizing networks since SON mechanisms are typically very slow control loops and converge over hours or even days. Conventional system level simulators cannot serve this purpose. On the other hand, QoS and mobility issues are of utmost importance and can not be neglected when studying SON which makes static tools inappropriate as well. We are closing the gap between too slow but elaborate system level simulators with full mobility and QoS support on one side and rough static simulators which can lead to very fast results.

We have seen that uplink modeling is much more complicated than downlink modeling. The key differences are the uplink power control (including the associated individual UE power budgets) and the multiple-access structure of the uplink (leading to extremely fluctuating interference). We have derived an average uplink SINR which is equivalent to the downlink SINR which is typically intuitively used. We have observed that the uplink interference has to be averaged in a harmonic way.

Different traffic/scheduler assumptions have been discussed. Again in contrast to downlink, there is no unique definition of a resource fair scheduler in the full buffer case. We have given two solutions called strict and modified resource fair. Furthermore, throughput fair solutions as well as CBR solutions targeting a given bit rate have been defined. In order to evaluate the system performance we have discussed uplink satisfaction in the CBR case. In addition to load limitation, we have observed that satisfaction due to power limitation and due to control channel limitation is highly relevant in uplink, too.

Limitations of the abstract models have been addressed as well. In particular, RRM details such as the exact algorithms for MCS selection, multi-user diversity gains, and imperfect channel estimation are hidden behind the abstract models (although the essence of fairness issues is considered). Furthermore, the capability to consider more elaborate traffic models (different from pure CBR and pure best effort) is limited. We are giving an outlook on possible extensions in Appendix B.

We have given simulation results using the derived modeling approach. In the specified LTE test cases our results match very well the typical performance assumptions. We have also given results for the CBR case using different target bit rates and analyzed the impact of power limitation. Finally we were looking at a heterogeneous scenario with different cell sizes and non-uniform user placement. We have considered load-adaptive power control as an example for a SON mechanism. This scenario has revealed some limitations of our modeling approach which have to be improved in the future.

In this paper we have only looked at a small subset of the proposed SON use cases since the focus was on the introduction of the framework. Certainly the model can be used for all other SON use cases as well, such as load balancing, coverage and capacity optimization, or mobility robustness optimization.

## Appendices

### A. Discussion of Uplink Fairness

In this section we will compare the uplink fairness with the downlink fairness. We will show that the*modified resource fair* scheduler in the uplink achieves a comparable fairness as a typical resource fair scheduler in downlink.

In downlink, the SINRs on a PRB degrade towards the cell edge more severe than the pathloss law, since the interference grows in addition. With a resource fair scheduler, the throughputs behave accordingly. However, we can improve cell edge users arbitrarily by assigning them more PRBs (nonresource fair scheduling).

- (1)
The uplink interference is induced at the eNB antennas and therefore is the same for all users (as long as we do not assume intercell interference coordination).

- (2)
Assuming no power control Open image in new window (resulting in strict resource fairness, that is, one PRB per user), that is, each UE transmits with maximum power, the SINRs per PRB would degrade with the pathloss. Note that this is still "fairer" as in the downlink (where interference increases in addition).

- (3)
Assuming power control with full pathloss compensation Open image in new window (and adaptive transmission bandwidth), that is, all UEs are received with the same power per PRB, the SINRs per PRB would be the same for all UEs, but the assigned bandwidth degrades with the pathloss, copmared with (14) unless the Open image in new window s are rather small. So the throughputs degrade with the pathloss (which is a little steeper than the upper case due to concavity of the rate function).

- (4)
Assuming power control with fractional pathloss compensation Open image in new window (and adaptive transmission bandwidth), the SINRs per PRB would degrade less severely than the pathloss; however, the assigned bandwidth degrades with the "rest" of the pathloss law. So in total the throughputs again will degrade with pathloss, with the slope being in between the upper two cases.

So in either case the throughputs will degrade with the pathloss, either via SINR degradation (for Open image in new window ) and / or via the ATB scheduler. The degradation will be similar to the downlink. All cases refer to the definition of the *modified resource fair* scheduler.*The strict resource fair* solution will lead to more fairness, that is, less throughput degradation for Open image in new window . For the special case of Open image in new window strict resource fairness even leads to throughput fairness.

Despite similar fairness behavior, the uplink scheduler has much less degrees of freedom to trade throughput among the users (due to individual power budgets). The only way to extend the strict throughput limit of cell edge users is to reduce the intercell interference which unfortunately lies outside the responsibility of the serving cell. Even more severe, the neighbors can only decrease interference by degrading their own users. Hence, whereas in downlink we can trade throughputs between users, in uplink we need to trade throughputs between cells.

### B. Traffic Assumptions

We have already observed that the individual power limitation in the uplink results in a bandwidth limitation Open image in new window (the number of PRBs for edge users is limited) and thereby in a tight data rate limit. This is different from the downlink, where basically each PRB comes along with its own power budget, so that assigning more PRBs automatically means assigning more power.

As a consequence, in the uplink we can only guarantee small data rates to cell edge users. So if we want to assume a common CBR model for all users, it has to be very small such that the edge users have a fair chance to achieve it. With such a setting there are basically 3 different methods how capacity limit could be achieved.

( Open image in new window ) Common CBR

The straightforward solution is to consider a large number of users. This would increase simulation runtime, and large data rates would not occur at all.

( Open image in new window ) Common CBR Extended with Full Buffer

With a smaller number of users we could think about distributing the excess capacity (i.e., PRBs) among those users who still can afford more PRBs. Basically this would be a mixture of CBR and full buffer model, which can be referred to as *guaranteed bit rate*(GBR) model. The drawback is that we would need another metric (in addition to the satisfied users due to load and power) accounting for users with higher throughput (i.e., 95% percentile). Furthermore we have to set a rule how the excess capacity is distributed among the users.

( Open image in new window ) User-Specific CBR

- (a)
We define a minimum data rate Open image in new window which lower-bounds all rate requirements, that is, Open image in new window .

- (b)The rate requirement for the worst user Open image in new window (i.e., with largest pathloss) in every cell Open image in new window is set to the minimum rate, that is,
- (c)The rate requirements for the other users are upscaled according to the pathloss relation to the worst user Open image in new window
The

*fairness parameter*Open image in new window can be considered as "slope" of the rate requirements. Open image in new window obviously means no slope, that is, the same rate Open image in new window for all users which converges to the common CBR solution. There is an interesting relation to the power control parameter Open image in new window*;*setting Open image in new window will lead approximately to a resource fair behavior, since the rate increase would roughly be compensated by the increase in received power through fractional power control; this would be exactly true for a linear relation between SINR and throughput and if we neglect the power limitation. Open image in new window is the most aggressive setting which makes it approximately equally tough for every user to achieve its target.

- (i)
the resource fair behavior Open image in new window for Open image in new window ,

- (ii)
smaller resource consumption Open image in new window for close users towards the throughput fair solution Open image in new window ,

- (iii)
larger resource consumption Open image in new window for close users towards the aggressive solutions Open image in new window ;with this setting, we can initialize the SINR calculation for the user-specific CBR case depending only on a single parameter Open image in new window .

(4) Summary

- (i)
The first is a minimum rate requirement Open image in new window . We propose values between 64kbps and 96kbps. Higher values will already become critical for cell edge users.

- (ii)
The second is a slope Open image in new window for the rate requirements. Open image in new window means throughput fair behavior; smaller values increase the rate requirements for closer users depending on their pathloss relation to the worst user, thereby forcing more load in the system. We propose a setting between 0 and Open image in new window .

- (iii)
The third is a guess Open image in new window for SINR initialization. A first proposal is Open image in new window ; however, this requires further study.

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