Abstract

We propose a model based on stochastic geometry to assess downlink interference and signal over interference ratio (SIR) in LTE networks. The originality of this work lies in the proposition and combination of resource blocks assignment strategies, transmission power control, and realistic traffic patterns into a stochastic geometry model. For this model, we compute the first two moments of interference. They are used to parameterize its distribution from which we deduce the SIR distribution. Outage and transmission rates (modulation and coding rate) are then derived to evaluate the system performance. Simulations that cover a large set of scenarios show the accuracy of our proposal and allow us to compare these strategies with more complex ones that aim to minimize global interference. Numerical evaluations highlight the behavior of the LTE network for different traffic patterns/load, eNodeB density, and amount of resource blocks and offer insights about possible parameterization of LTE networks.

1. Introduction

The amount of mobile data that cellular networks must carry is continuously increasing. The capacity of the wireless systems must continuously increase in order to satisfy the growing demand of traffic from users and applications. Long-Term Evolution and Long-Term Evolution-Advanced [1] (LTE-A) have been recently standardized to improve the network capacity and support this traffic growth. One of the solutions brought by LTE is the enhancement of the radio spectrum reuse. The smallest radio resource that can be allocated to a user is a resource block (RB). An RB is a channel (an OFDMA channel composed of a set of OFDM subcarriers) for the duration of one time slot. Considering the number of RB is finite, they are reused in different cells generating potential intercell interference. The algorithms that assign RB to users located in different cells have thus an important role in the system performance. A static RB assignment where disjoint resources are distributed to each cell may lead to an inefficient resource usage as the unused RB in a cell cannot be reused in another one. Instead, algorithms that assign RB can be centralized in a scheduler/controller that controls a certain number of neighborhood cells and adapts the RB assignments to the cells load. Also, it may improve the spatial reuse while ensuring a low level of interference.

Several studies have proposed assignment strategies performed at the scheduler to minimize global interference [2, 3]. These strategies aim to minimize interference for a given configuration and are evaluated exclusively through simulations. However, the assignment strategies have to be evaluated for more general scenarios and at larger scale.

Stochastic geometry offers a powerful tool to analyze large scale networks through a few parameters and to understand the role of these key parameters on the whole system. The other benefit of stochastic geometry is to consider realistic Base stations (BS) or eNodeB (evolved Node B) locations. It uses random point processes rather than deterministic (grid or hexagonal patterns for instance) or predetermined locations of BS/eNodeB. For instance, the Poisson Point Process (PPP) has been shown to be suitable to model the spatial location of BS [4–6]. Nevertheless, interference as experienced by a user is not generated by all BS but only by the ones using the same radio resources. The resource allocation strategies have thus to be mapped to the point process modeling BS/eNodeB to determine which points/BS are interfering with a given communication. Consequently, the traffic demand must also be taken into account as it sets the number of resources used at a given time.

In this work, we propound a combination of several assignment strategies, realistic traffic demands, and transmission power control mechanisms into a stochastic geometry model. We begin by reviewing related works and our contributions.

1.1. Related Work

In a downlink LTE system, a resource block (RB) is the smallest radio resource unit that can be allocated to a user. The LTE system has to schedule and assign RB to users as a function of the link qualities, traffic demands, and potential quality of service requirements. In this paper, we focus on a system where a controller assigns RB for a set of eNodeB. We do not overview RB assignment techniques in LTE network as they aim to optimize RB assignments and modulation/coding rates for a given topology and a traffic demand. Instead, this paper deals with the macroscopic design of the network: the impact of eNodeB density, allocation scheme, and power allocation on the global performance of a downlink LTE system. Nevertheless, the reader can refer to [7, 8] for recent surveys. Also, interesting contributions on the optimization of the downlink system for a given configuration are described in [3, 9–12].

Stochastic geometry has emerged as an efficient tool to analyze the performance of cellular networks. It offers, through simple models, a way to study wireless architectures at a large scale. Recent surveys [13, 14] summarize the numerous wireless architectures and models for which stochastic geometry has been applied. One of the main difficulties in the analysis of large wireless systems is to characterize interference. This quantity does not depend only on BS location and radio environment (path loss, shadowing/fading, etc.), but also on the way that radio resources (time, frequency, and power) are allocated. The point process modeling interfering nodes is thus of crucial importance. The PPP offers an accurate model to describe BS location [4–6]. This process is tractable, and it is possible to derive closed formulas for some key performance metrics of the system: interference, coverage, outage, Signal over Interference plus Noise Ratio (SINR), etc. But the PPP models all BS/eNodeB and not the subset of interfering eNodeB for a given communication. The process has thus to be thinned to take into account interference coordination (IC) techniques and radio resources assignment, for example, leading to processes that are no more Poisson. In the next paragraph, we focus on recent contributions, and on studies where resources allocation and more generally IC techniques are taken into account.

IC refers to techniques that aim to mitigate interference at the receivers. Surveys on such techniques can be found in [2, 15]. A common IC approach consists in controlling the allocated radio resources (frequency/time/power) in order to alleviate the interference impact on communications. In [16], the authors consider a random resources allocation strategy where the BS are distributed as a PPP. This simple and tractable strategy allows model interfering BS as an independent thinning of a PPP and deriving closed formulas for the coverage probability. They also deduce the minimal reuse factor achieving a given coverage probability. The performance of strict FFR (Fractional Frequency Reuse) and SFR (Soft FFR) allocation strategies is evaluated using stochastic geometry in [17]. With these two techniques, different radio resources are allocated to users that are at the edge of a cell (Voronoï cells here) with regard to the ones close to the BS. The criteria distinguishing core and edge users is based on the SINR at each user computed from the underlying PPP modeling all BS. For strict FFR, the radio resources used at the edge and in the core are disjoint. Instead, the radio resources may be reused between the two regions for SFR. For these two strategies, the authors derive closed formulas for the coverage probability and discuss pros and cons of these approaches. A superior interference reduction is observed for FFR but SFR benefits from a greater resource efficiency. This work is generalized in [18, 19] to the context of K-tier and heterogeneous networks considering different point processes for each tier or network technology. It is also extended and studied in [20] with the dynamic strict FFR (DSFFR) where the edges of the cells are dynamically divided into sectors with the help of directional antennas. In [21], a coordinated beamforming is employed to ensure that a set of closed BS, “a cluster”, will use different resources. A user associated with a BS is then not subject to interference from BS belonging to the same cluster. The authors derive analytical expressions for the Signal over Interference Ratio (SIR) for this strategy and discuss the impact of the clusters cardinality. A similar approach is used in [22], where the set of coordinated BS corresponds to the most interfering ones. Interference level takes into account path loss and long-term shadowing. The interfering BS are outside this set. They are selected randomly and independently leading to a thinned PPP. For this model, the authors study the coverage probability for different scenarios. In [23], a user is served by its 1 or 2-closest BS according to the position of these BS with regard to the user. When the two BS are coordinated, the transmission power is split into the two transmissions. The total transmission power is thus the same with one or two coordinated BS. Interference is generated by the other BS without restriction which is assumed to be distributed through a PPP. The authors derive a closed-form expression for the SIR distribution and the network coverage probability and discuss the benefit of this approach. In [24], an IC technique is evaluated for a user at the edge of its cell. When the resource of this communication is used by neighboring cells, they may not transmit any signal for a certain period to mitigate interference at this user. This coordination technique is analytically evaluated assuming that interfering nodes are still distributed as a PPP.

Besides the modeling of IC, [22, 25–27] propose spatial and tractable models that take into account the traffic demand in the interference computation, but they do not consider concrete RB assignment algorithms. In particular, the authors in [26] study SIR coverage for a cellular network based on PPP. A queue is associated with each BS that determines the BS transmission activity as a function of the traffic. Considering the traffic at each BS is independent, interferers at a given time are then an independent thinning of the initial PPP and are still Poisson. This model differs with this paper as we do not take into account eNodeB activity as a function of the traffic but instead the resource allocation as a function of the number of associated users to each eNodeB. Also, stochastic geometry models can be specific to certain power control scheme [28] or radio technologies as in [29] where the authors consider a K-tier heterogeneous network with transmissions operating on the millimeter wave band.

1.2. Contributions

The primary contribution of this work is to offer an analytical model based on stochastic geometry to evaluate the performance of a downlink LTE system taking into account RB allocation strategies, power control, and traffic demands. All these mechanisms have never been combined into a single stochastic geometry model. The number of allocated resources for an eNodeB is assumed to follow the distribution of the number of clients in an M/M/C/C queue. It models the number of communications in progress when both the interarrival of the communications and their duration follow an exponential distribution. Such assumptions are pertinent in cellular networks as it has been recently shown in [30]. We associate to these traffic demands several resource allocation strategies. All these algorithms are combined with a power control mechanism that depends on the channels quality. Allocation strategies lead to non-PPP as correlation appears between the locations of the interfering nodes. It prevents the use of the convenient properties of the PPP to compute interference distribution. Nevertheless, we propose approximations that allow us to deal with these correlations and to obtain an analytical method that is shown very accurate with regard to simulations.

We compare our model to classical optimization approaches where, for a given configuration/sample, the allocation is optimized with regard to an objective function. To our knowledge, such comparison has never been done before. It shows that geometry stochastic based model may be relevant to offer tight approximations on wireless system performance.

Models are evaluated through a large set of simulations that highlights benefits of our approach to design some key parameters of the wireless system. Results show that the obtained values for SIR, coding, and modulation rates correspond to the reference values of the standards and technical LTE documents, empirically proving that our model is able to approximate performance of real systems. This work has been partially presented in [31].

1.3. Paper Organization

The remainder of this work is organized as follows. In the next section we present the system model. We expose the assignment strategies in Section 3. In Section 4, we derive the first and second moments of interference for each allocation strategy. SIR distribution is assessed in Section 5. Numerical results and simulations are presented and discussed in Section 6. We conclude the paper in Section 7.

2. System Model

2.1. eNodeB Location and Interference

eNodeB location is modeled by a point process distributed in with intensity . Its distribution is detailed in Section 2.2. The eNodeB are numbered with regard to their distance to the origin, eNodeB at being the closest one. We consider a downlink system between a typical user and its attached eNodeB. Without loss of generality, we assume that this user is located at the origin. The users are assumed to be associated with their closest eNodeB with regard to the Euclidean distance. The channels between eNodeB and the typical user are modeled through a sequence of i.i.d. random variables . The transmission power between an eNodeB and one associated user is given by the function . This function models the power control algorithm implemented by the eNodeB and depends on the distance between a user and its attached eNodeB. The scheduler/controller manages a set of resource blocks that are common to all eNodeB. They are thus shared between eNodeB. The RB are numbered from to . The scheduler assigns one RB for each user, but the model can be easily extended with a random number of RB for each demand. Traffic demands exceeding at an eNodeB are not served. The RB with index is allocated to the typical user. We show, in the appendix, that this choice does not impact the computations, and any other index could be chosen instead. An eNodeB interferes with the typical user if and only if it reuses this RB. Interference at the typical user can be expressed aswhere is the path loss function. The argument of interference is the distance between the typical user and its eNodeB (eNodeB ). expresses interference when this distance is random and depends on the r.v. . expresses interference when this distance is given and equal to . This notation is motivated by the fact that the r.v. and are correlated to . Moreover, the mean and the variance of interference will be computed for both a given value of and with regard to its distribution. The r.v. indicates whether eNodeB interferes with the typical user ( or ). is the random variable modeling the location of a user attached to the eNodeB (at ). We assume that is uniformly distributed in the Voronoï cell formed by the process and with nucleus . The main notations used throughout this paper are given in Table 1.

2.2. Point Process Modeling eNodeB

The point process modeling eNodeB is a modified homogeneous PPP. The process is Poisson in where is the ball centered at the origin and with radius . We choose a distribution for that makes the process different of a PPP. Indeed, with a PPP the typical user at the origin lies in a Voronoï cell that is greater in average than the other cells. Intuitively, “big cells” cover more space than “small cells” and consequently the cell covering the origin has a greater size in average. It is consistent with a modeling where users are homogeneously scattered in the plane but not with our assumption where the network has been dimensioned to have the same load in average (the same number of users) in each cell. It is more realistic as it has been shown in [30], where a homogeneous load is observed for the different cells independently of their sizes. In this case, the typical cell covering the typical user must have the same distribution as the other cells. Therefore, we consider the distribution of the distance under Palm measure. More precisely, this distribution corresponds to the distance between the nucleus of a typical cell under Palm measure and a point uniformly distributed in this cell. The distribution of this distance is not known, but we use the approximation presented in [32] (page ). We set the distribution of the distance between the typical user and its closest eNodeB at aswith . The angle between the lines and the abscissa is uniformly distributed in .

The distribution of the distance between a user in a given Voronoï cell and its nucleus () follows the same definition, and consequently the same distribution. In the next section, we present the different RB assignment strategies evaluated in this paper.

3. Assignment Strategies

We consider four different allocation strategies. We begin by two simple RB allocation schemes: independent and static allocations. Then, we develop a more realistic allocation strategy using the queue, named allocation hereafter. Also, a more global approach where the RB are assigned in order to minimize the sum of interference at each user is considered but for which we do not propose a mathematical resolution. All the allocation strategies are set in such a way that a given RB is reused every () eNodeB in average. The mean load in each cell and equivalently the mean number of RB used by an eNodeB are then . The reuse factor reflects the network load.

3.1. Independent Allocation: Thinning

With this strategy, each eNodeB selects its resources independently of the other eNodeB. Therefore, we assume that an eNodeB has a probability to reuse the RB with index . The point process describing the interfering eNodeB is then a thinned PPP in with intensity .

3.2. Static Allocation

We assign a constant proportion of the available resources to each eNodeB. They are allocated in their index order: eNodeB uses RB from to (it includes the typical user), eNodeB from to , etc. We take the integer part of these fractions when is not a multiple of . We loop when all resources have been used. Consequently, the eNodeB interfering with the typical user has an index with the form with .

3.3. Allocation

In an queue, customers arrive according to a Poisson process (in ) and the service times are exponentially distributed. It models a system with resources/servers and a capacity of the same size. A customer cannot enter in the system if all resources/servers are busy. We associate with each eNodeB an independent queue to model the number of RB in use. The servers model the RB. Upon the arrival of a request/user, an RB/server is used for a time exponentially distributed. If no RB is available, the request is rejected. In order to have a mean reuse factor of , the parameter of the queue (the load) denoted is set in such a way that the mean number of customers in the system, or equivalently the mean number of busy resource blocks, is equal to . The distribution of the number of busy resource blocks for a given eNodeB () denoted is then given bywhere and .

These RB are allocated in a cyclic order. If the last used by eNodeB has index , eNodeB uses indexed from to .

For eNodeB , we do not consider the total number of allocated RB (), but instead, a random variable . It describes the index of the last RB used by eNodeB . Indeed, for this particular eNodeB, the quantity used in practice to compute the next allocation (RB indexes used by eNodeB ) is rather than . A formal definition of is given in appendix (Appendix A). An example of allocation is given in Figure 1.

(a) RB allocation

(b) Impact of RB allocation on interferer

(a) RB allocation
(b) Impact of RB allocation on interferer

Figure 1 

In Figure 1(a) an example of RB allocation is given. eNodeB uses RB (the first one has index , and the other ones are indexed from to ). It uses the RB with index to communicate with the typical user. is the index of the last resource used by eNodeB . As it uses RB indexed from to , here. The number of used RB for the other eNodeB are , , etc. An eNodeB interferes if and only if it reuses the resource : in this example eNodeB and . In Figure 1(b), we plot a sample of the point process describing eNodeB location with the same RB assignment. The eNodeB interfering with the typical user is then a dependent thinning of the original point process where the thinning involves eNodes , , and (this last interfering eNodeB was not shown in Figure 1(a)). Their Voronoï cells are colored in green.

The distribution of is set according to the stationary distribution of a Markov chain. The transition probabilities of this Markov chain arewhere . The motivation for this particular construction is to keep the probability of using the resource between eNodeB homogeneous. More precisely, it is built in order to verify the property given in Proposition 1. Details about the Markov chain construction are given in appendix (Appendix A). It is worth noting that other distributions for the resource demands (given by (3) in our case) can be considered as well. As soon as the distribution of verifies Proposition 1, the method proposed in this paper holds.

3.4. Property of These Assignment Strategies

We define more precisely the sequence of r.v. . It indicates which eNodeB interferes with the typical user. It was already used in (1).

By convention, we set a.s. In the following, we shall thus assume that .

Proposition 1. For the three allocation strategies defined in Sections 3.1, 3.2, and 3.3, the following property holds:

The proofs for the first two strategies are straightforward. For the strategy, the distribution of has been set to verify this property (see Appendix A).

3.5. Heuristics

We compare these strategies to heuristics that aim to optimize interference or spatial reuse for a given configuration. It allows us to compare our RB assignment and performs in a cyclic manner around a typical user, to strategies where a controller in charge of a set of eNodeB will assign RB in order to optimize a certain objective function. For the heuristic minimizing the sum of interference, named “minimize interference” hereafter, the considered optimization problem is similar to the one developed in [3]. The problem has been shown NP-hard, so we use a greedy algorithm to find a solution. The number of users associated with each eNodeB follows the same distribution as the allocation. Then, we consider users in a random order and apply Algorithm 1 to associate an RB to a user. It chooses the resource block that minimizes the sum of interference. Obviously, we compute interference only for the users already assigned. The typical user is considered in last when the system has reached the targeted load. This algorithm mimics an assignment strategy where the RB are assigned at the arrival of the users request without changing the already assigned RB.

The second heuristic, “maximize reuse distance”, maximizes the distance at which the RB are reused. Each user is considered in a random order. Different RB are assigned to the first users. When assigning an RB to the other users, the controller chooses the RB for which the reuse distance is maximum. The typical user is considered in last. This second heuristic may correspond to a case where the controller does not have information on channel conditions and interference but knows the distances between eNodeB.

4. Interference Characterization

We derive the mean and the variance of interference for the three assignment strategies defined in the previous section.

The point process modeling interferers is a dependent thinning of the original PPP. Consequently, conditions for mean (respectively variance) to be finite with a PPP also hold for our point process: the path loss function must belong to (respectively, ).

4.1. Distribution of Distances between the Typical User and eNodeB ()

As a preamble, we give the PDF of the distance between the typical user at the origin and eNodeB. Both PDF of and joint distribution of are derived. These PDF are used in the computation of the mean and the variance of interference.

In the numerical evaluation, we shall condition interference by the distance . It allows us to study interference for a given distance between the typical user and its attached eNodeB. It is also motivated by the computation of the SIR where both interference and the typical user signal strength depend on the distance .

For our model, the PDF of with given is

The joint PDF of with given is

To obtain the PDF when is not set ( and ), it suffices to integrate the two conditional PDF with regard to the PDF of given in (2).

4.2. Mean of Interference

The mean is derived from (1):

In this equation, has been separated from the expectation as it is independent of the process (according to the defined strategies). We derive and in the two next sections.

4.2.1. Computation of

No Power Control. In absence of power control, i.e., when is constant or independent of the process , a closed formula may be expressed for (10). is then given by . Expectation of is obtained from the distribution of given in Section 4.1.

Power Control. When the transmission power depends on the distance between the receiver and its attached eNodeB (), the computations are more complex. cannot be approximated by as the size of the Voronoï cell with nucleus depends on its distance to the origin. The joint distribution of being unknown, we propose the following approximation:

with

The PDF of is the same as given by (2). Its parameter depends on : with and . The motivation and the computation details for this PDF are given in appendix (Appendix B). We obtain

Often, in real systems, the transmission power cannot be set arbitrarily and is limited to a set of predetermined values. The transmission power function can then be represented as a step function, where is the number of possible transmission powers, the transmission power value, and the distance interval between a user and its eNodeB at which this transmission power is used. An example of such setting is given in the numerical evaluation section. In this case, (14) becomes

When the computation is performed for a given distance , the PDF in (12) and (15) must be replaced by (given in Section 4.1).

4.2.2. Computation of

Finally, in order to compute (10), we need to express . First, note that .

Proposition 2. The probability for eNodeB to interfere with the typical user is given by (i)Independent allocation(ii)Static allocation(iii) allocation

The computation details for the allocation are given in appendix (Appendix C).

4.3. Variance of Interference

Variance of interference is defined as

For the second moment, we obtain

As for the mean, complexity lies in the correlation between and . The term is computed with the same method as the first moment.

Computation of . As (respectively, ) depends on (respectively ), we condition by the distribution of given in Section 4.1. Given and , we use the same PDF as in (B.1) assuming that and are independent. The considered joint distribution of becomes

When the distance is fixed, must be replaced by the PDF given in Section 4.1.

Computation of . It has been shown that the sequence verifies (7) for the three strategies. It allows us to express with as

Proposition 3. The joint probability for two eNodeB and () to interfere with the typical user is given by(i)Independent allocation(ii)Static allocation(iii) allocation where is given by (18).

5. Signal over Interference Ratio (SIR)

In our model, there is a strong correlation between the interfering eNodeB. It is generated by the allocation strategies and cannot be neglected. Also, a correlation exists between the location of an eNodeB and the size of its Voronoï cell. Consequently, classical approach based on PPP which uses Laplacian transform for instance cannot be applied here and a formal derivation of interference distribution seems intractable.

Nevertheless, the different simulations presented in the next section will show that the PDF of interference can be approximated by a log-normal distribution. The parameters of this distribution, mean and variance denoted by and , are directly derived from the previous analytical computations. The classical mapping between log-normal and normal parameters can be applied to derive parameters of the normal distribution when interferences are expressed in decibel. In the following, a variable is indexed by when it is expressed in decibel.

We get

Assuming that is normally distributed with mean and variance , we obtain

When is not constant, the expectation with regard to its distribution must be taken into account in (28). In (29), is assumed to be constant that equals to . In these two equations, is the error function.

6. Numerical Results

We consider an E-UTRA channel with a bandwidth of 5MHz with [33]. The path loss function is the same as [3]. It is expressed in dB: where is the distance (in km). is set in such a way to guarantee to each user a minimum receiving power. We set the transmission power function to ensure a signal power greater than or equal to at the reception as specified in [33]. For each 50 meters (from 50 to 500 meters), we compute the minimum transmitting power required to reach this threshold ( for each interval of 50 meters leading to 10 possible transmission powers). This step function models the case where eNodeB has a finite set of predetermined power. The process intensity modeling eNodeB is equal to 2.25 per km². It corresponds to the intensity of base stations in Paris (https://www.antennesmobiles.fr/). Random variables are supposed constant equal to . This assumption facilitates interpretation of the results but any distribution can be considered as well. It simply adds a factor in terms of variance (cf. (20)). We simulate the different strategies through a simulator coded in C available here (http://www.anthonybusson.fr/index.php/publications). In all simulations and numerical results we consider 50 eNodeB. The different sums in the equations (e.g., (10) and (20)) are then limited to 50. For each set of parameters, simulations have be run from 10,000 to several millions times depending on the evaluated quantities. The number of simulations/samples has been set in order to have negligible confidence intervals. They are consequently not shown in the different figures.

Mean and Variance of Interference. In Figure 2, we plot the mean and the standard deviation of interference obtained from simulations and computed from formulas (10), (19), and (20) when the distance varies. The theoretical evaluation closely matches empirical estimators obtained by simulations. As expected, the highest interference level is observed for the independent allocation, and the lowest level for the static allocation. The static and the allocations offer equivalent results with a multiplication factor varying from to for the mean interference. Instead, the independent allocation differs deeply. Mean and standard deviation are multiplied by a factor ranging from to for the mean and can reach up to for the standard deviation (for  km). The use of the thinned point process is thus questionable to approximate realistic assignment strategies. This strategy is no more considered in the following.

(a) Mean

(b) Standard deviation

(a) Mean
(b) Standard deviation

Figure 2 

Interference: mean and standard deviation as a function of the distance between the typical user and its eNodeB (). Simulation results are given by points and solid lines, and theoretical evaluations by the dotted lines.

Extrapolation of Interference Distribution. We compare empirical distributions obtained from simulations to known distributions for the different strategies and the two heuristics when , 6, and . Three of these empirical distributions are shown in Figure 3 (in dB). The distribution parameters have been set according to the maximum likelihood. The distributions that best fit simulations vary according to the allocation strategy and the reuse factor . For cases over , the distribution minimizing error is the normal distribution. As soon as an asymmetry is observed, other distributions are more accurate: log-normal, Weibull, and Inverse Gamma. We perform two hypothesis tests, T-test and Smirnov-Kolmogorov, for all these scenarios. The alternative hypothesis is systematically ruled out. For the strategy, best fits are given by the normal ( and ) and Weibull distributions ().

(a) Heuristic, = 3

(b) allocation, = 6

(c) Static allocation, = 9

(a) Heuristic, = 3
(b) allocation, = 6
(c) Static allocation, = 9

Figure 3 

Interference distribution: empirical (histogram) and extrapolation (curve). Among the classical distributions, the best extrapolations are Gamma (m=10.03, var=9.77), normal (m=28.23, var=22.16), and Weibull (a=25.14,b=5.75) for the heuristic, MMRB allocation and static allocation , respectively.