Single Class Model

With this model all users use the same traffic model, hence no QoS differentiation is present. The users generate infinite elastic ON/OFF traffic represented by an ON period file size Xon and an OFF period of waiting time represented by To f f.

model explained earlier in section 7.1 with arrival rate λ (see equation 7.1) and a departure rate ofμ(n) (see equation 7.9).

μ(n) depends on the TD scheduling discipline since it is inherently dependent

onT BS(n). Looking back at equation 7.12, it can be seen that the TDS is reflected

in Xk,i. Where Xk,i is chosen to be 0 or 1 depending on the time domain scheduler.

Three different TDS schedulers are modeled in this thesis: BET, OSA and MaxT. As a generalization, equation 7.11 can be rewritten in a more generic form that can incorporate all of the three TD schedulers as:

T BS(n₀,...,nK) =

∑

i=[i₁,...,i_η]

T BSi(η) (7.13)

with

η = min[n − n0,ψ] (7.14)

where[i1,...,iη] represents the index of the chosen η users out of the n active users

considered for scheduling within a TTI, and ψ is the number of scheduled users by the FDS scheduler. Obviously the choice of the η users is determined by the TD scheduler.

Now, substituting equation 7.13 in equation 7.9, the generic departure rate μ(n) will be expressed as:

μ(n) = 1 Xon· TTI (n,...,n)

∑

_∑

∏

Once all departure rates have been calculated for every state then the Markov chain steady state probability can be calculated as well as the average file download time using the techniques explained earlier (by using equation 7.6 and 7.3).

In order to model the different TD schedulers, the procedure (scheduler depen- dent) on how to choose η users out of the n active users has to be explained. For simplicity, the MaxT scheduler procedure is explained first. According to the def- inition of the MaxT scheduler (section 6.2.1) only the users with the best channel conditions are served, i.e., in every TTI the n active users are prioritized based on their channel conditions and then only the highest η users are scheduled. Now relating this to the analytical model theη users in the highest MCS are considered in the scheduling and represent the i user indices ([i1,...,iη]). This can be seen in

MCS0 MCS1 MCS2 MCS3 n0 n1 n2 n3 Findη UEs MCS4 MCS5 MCS6 MCS7 n4 n5 n6 n7 M

Figure 7.2: MaxT scheduling of users (example with 7 MCSs)

The reason why the MaxT is explained first is the fact that it only depends on the instantaneous channel conditions (i.e., MCSs). This makes the choice of theη users rather straight forward by only choosing the ones with the highest MCSs. In contrast to other scheduler disciplines like BET or OSA, where the choice of theη users does not only depend on the channel conditions but also on the accumulated sent throughput in the past. This makes the choice rather difficult, as there is a memory aspect in the decision which contradicts one of the basic principles of the Markov chain, that is the memoryless property.

In order to overcome the above mentioned problem, i.e., memory dependency, some work around is required in the analytical model. This can be done by ap- proximating and emulating the behavior of the schedulers without actually dealing with the memory aspect. For example, the BET scheduler prioritizes the users based on their accumulated throughput from the past as described in equation 6.1. The main target of the BET scheduler is to equalize the users’ throughputs, i.e., all users will get the same average throughput at the end. This can be approximated by choosing theη users from the MCSs that correspond to the average throughput. This can be seen in Figure 7.3.

MCS0 MCS1 MCS2 MCS3 n0 n1 n2 n3 Find (η-1) UEs MCS4 MCS5 MCS6 MCS7 n4 n5 n6 n7 M M Find 1 UE

Figure 7.3: BET scheduling of users (example with 7 MCSs)

Figure 7.4 shows the MCSs static probability. This can be obtained for each specific scenario using simulations (with specific cell coverage, channel model and mobility model). The example figure was created using an eNodeB cell coverage

of 375 km, with the channel model explained earlier in section 4.3.5 and both mobility models (RandomWay Point (RWP), and Random Direction (RD)).

Now, since the BET scheduler tries to let the users achieve the same average throughput, the analytical BET model chooses the η users from the MCS that achieves the average throughput. In the case of the RWP, it is MCS5, whereas for

the RD mobility model it is MCS4. However, since the MCSs static probability

(seen in Figure 7.4) shows that with a large probability a user can be in MCS7

(50% for RWP, and above 30% for RD mobility model), one user out of the η chosen users is always chosen from MCS7. The average MCS is chosen by roughly

estimating the average throughput for each mobility model as follows:

AverageT hroughput =

K

∑

k=1

T BSk· Pk

where K represents the number of MCSs, TBSk is the transport block size that can

be transmitted when using MCSk, and Pk is the static probability of MCSk.

0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Modulation and Coding Scheme (MCS)

Probability of MCS (P

k

)

Random WayPoint Random Direction

Figure 7.4: MCSs static probability obtained from simulations

The OSA scheduler has the same memory dependency problem similar to the BET scheduler. Thus, a similar approximation is required in order to overcome this problem, so as to emulate the OSA behavior. The OSA scheduler gives higher probability to users with better channel conditions. This can be seen in equation 6.5, where a normalized average channel condition factor in considered in the denominator of the time domain priority factor of the OSA scheduler.

The emulation of the OSA scheduler, i.e. the choice of theη users is shown in Figure 7.5. The figure shows that a higher average MCS is chosen (compared to the BET scheduler) due to the fact that the OSA scheduler prefers users with better channel conditions. MCS0 MCS1 MCS2 MCS3 n0 n1 n2 n3 Find (η-1) UEs MCS4 MCS5 MCS6 MCS7 n4 n5 n6 n7 M Find 1 UE

Figure 7.5: OSA scheduling of users (example with 7 MCSs)