Implementation Issues

Before really applying the determined rearrangement, the AP can constantly perform the rearranging algorithms in the background and compare the derived result with that of the current configurations. If the current configuration results in a much worse condition of the distances between existing scheduled events when compared with the background result, e.g., the minimum distance of the system is less than half of the background one, the application of the rearrangement should be triggered.

Since the periodic beacon is important for issues such as timing synchronization, it is better to leave its scheduled transmission times unchanged. Therefore, the target beacon transmission times can be a reference for the new schedules to take effect. Similar to what described in Section 3.4.1, the installation of the rearranging results, i.e., the revised sched-ules, can be easily done by the AP according to the repetition pattern of the established periodic schedules and the current time since the notion of sequence can be extended to both directions of the time line as shown in the system model. According to this idea, the

longest delay experienced by certain TS for the starting of the rearranged service is upper bounded by the value of its SI, rather than the period of the total repetition pattern of different periodic service schedules. Moreover, the longest possible interval for certain TS between the original schedule and the renewed one is upper bounded by 2 × SI. Although it could violate the delay constraint for some TSs when the new schedule takes effect, it is only transient condition.

There could be two possible ways to notify the change of schedule to STAs. In addition to the established S-APSD service, STAs should regularly listen to the beacons for admin-istrating issues of the associated basic service set. Therefore, the renewed schedule can be attached in the beacons and broadcast to the STAs. However, if some STAs intentionally lengthen the listening interval of the beacons, the application of the new schedule cannot be read immediately. Moreo-ver, the STAs using this method should be able to understand the new arrangement and update its wake-up time according to the current time. An al-ternative method is to individually notify the STAs during the unicast communications. A more energy efficient approach is to inform the STA before the new schedule taking effect.

Therefore, given the target schedule renewal time, the AP should trace the last scheduled instants of every STA before the application of the new arrangement. Denote the time to apply the new arrangement as T_new which corresponds to the l^th scheduled instants of the reference Xr with period p_r and offset O_r with respect to X1 within the total repetition interval. Therefore, the time to notify the STA attaching the m^th TS with relative offset Om

to the reference can be found as follows. Assume that

O_m+ s_m× p_m < O_r+ l × p_r< O_m+ (s_m+ 1) × p_m. (4.17) Then s_m can be obtained as

sm =jOr+ l × p_r− O_m p_m

k

. (4.18)

Therefore, the time to notify the i^th TS is T_new− (O_r+ j × p_r− O_m− s_m× p_m). Then STAs can sleep till the revised time which could be either earlier or later than the original wake-up

time, depending on the repetition pattern of the new schedule and the current time. The sleeping interval for the transition is bounded between 0 and 2 × SI. After that, it returns to SI.

4.4 Performance Evaluations

The considered scenario is composed of K scheduled TSs which belong to 5 classes. The 5 classes of traffic, listed in Table 4.2, in the system are real-time voice, real time video, streaming audio, streaming video, and real-time gaming, and they are referred to Class 1 to 5 in the following discussion. The traffic characteristics are obtained from [36], [38], and [41]. The video traces are available online [49]. It is assumed that there are K STAs, each is configured with a scheduled TS belonging to one of the 5 classes. The number of STAs is increased in multiples of 5 STAs to keep the ratio of existing traffic classes. The system parameters are listed in Table 4.3 and conform to the IEEE 802.11a OFDM wireless LANs. The calculation for frame transmission time can be found in [14]. The precision used in our simulations is set to 100µs. We consider the scenario of increasing the number of TSs in each class, ni, from 1 to 10.

4.4.1 Comparison of Computational Complexity

In this numerical evaluation, we increase the number of existing TS in each class of applica-tion, and check the average online complexity of LCS-APSD and those of the proposed GMD greedy method and the Simply Sorted greedy method. Given the number of existing TSs, the average complexity for finding the SST is derived by averaging the number of necessary online operations when a new TS belonging to any class of application joins.

For LCS-APSD and K existing TSs belonging to C classes, we add a new TS as the (K + 1)^th TS of Class j. It is assumed that the contents of Ai,j have had been calculated and stored in advance. To construct the scheduling matrix, we need K × GL_K+1 comparisons to

obtain d(X, Y + k), 0 ≤ k ≤ GLK+1− 1. Then it takes GLK+1 comparisons to determine k^∗. As for the suggested implementation method in Section 3.3.1 for the Class-based system, i.e., given that the contents of B_i,j have had been renewed to the up to date situation, the complexity is reduced to C × GL_K+1comparisons after the scheduling matrix is constructed.

Finally, GL_K+1 comparisons to determine k^∗ is necessary.

1 2 3 4 5 6 7 8 9 10

0 0.5 1 1.5 2 2.5 3 3.5x 10⁶

Number of traffic streams in each class

Number of required operations

LCS−APSD CLCS−APSD GMD greedy Simply Sorted greedy

Figure 4.4: Comparison of implementation complexity.

Regarding the GMD greedy method, in addition to constructing necessary scheduling matrices to determine the offsets, firstly it needs to divide TSs into groups and then performs equal spacing within each group. For Class i, n_i is represented by

n_i = Q_i(1) × f_i(1) + Q_i(2) × f_i(2) + ... =X

t

Q_i(t) × f_i(t).

We assume that the number of TSs of Class i after the grouping becomes R_i. Therefore, R_i = P

tQ_i(t). In summary, we need R_i subtractions to obtain these R_i groups. To determine

fi(t) from Fi, the number of comparisons is upper bounded by Qi(t) × ui. Therefore, to find out fi(t)’s from Fi for all t, it is bounded by P

tQi(t) × ui operations. The overall complexities for grouping C classes are P_C

i=1R_i subtractions and P_C

i=1R_i× u_i comparisons.

When the grouping is done, there are R =P_C

i=1R_i ≤ (K + 1) period-revised TSs belonging to C⁰ classes, where C⁰ is the number of distinct revised periods. Then we sort the TSs by the size of the revised periods. If the Merge Sort [40] is used, a sorting of the elements requires (R log₂R) comparisons. Subsequently, we add the period-revised TSs one after another by gradually constructing the scheduling matrices. From Theorem 4.2, we can easily determine the relative distances by G⁰_i,j operations, where G⁰_i,j, 1 ≤ i, j ≤ C⁰, is the GCD of two revised periods. The complexity in preparing all necessary relative distances is thus P_C⁰

j=2

P_j−1

i=1G⁰_i,j operations. The complexity for R period-revised TSs to generate all necessary scheduling matrices to obtain the impacts from scheduled TSs isP_R

m=2(m−1)×GL⁰_mcomparisons where we have the period-revised TS m belongs to Class i and GL⁰_m = lcm{G⁰_1,i, G⁰_2,i, ..., G⁰_j,i, ...}.

To determine the offset for each TS, it needs P_R

m=2GL⁰_m comparisons.

For the Simply Sorted greedy method, because it does not divide any TS into the groups, there are K + 1 TSs to be scheduled. Firstly, it needs to sort the TSs according to the length of SIs and requires (K + 1)log₂(K + 1) comparisons if the Merge Sort is adopted. The complexity for these K + 1 TSs to generate all necessary scheduling matrices to obtain the impacts from scheduled TSs requires P_K+1

m=2(m − 1) × GL_m comparisons. To determine the offset for each TS, it needs P_K+1

m=2GLm comparisons.

The comparisons of computational complexity are shown in Fig.4.4. As can be seen, the complexity of the GMD greedy method is close to LCS-APSD. The complexity varies be-cause that the decomposition does not necessarily generate more period-revised TSs with the increasing number of TSs . Moreover, due to the fact that TSs are divided into a few groups in the GMD greedy method, the complexity in constructing scheduling matrices and compar-isons for determining the offsets is not that much. For the Simply Sorted greedy method, the TSs are not divided into groups. Therefore, the number and the size of scheduling matrices

would tend to grow with the increasing number of TSs to be scheduled.