Application of feasible vertex labeling for dynamic capacity allocation of a multicarrier multimode PON

Application of feasible vertex labeling for

dynamic capacity allocation of a multicarrier

multimode PON

R.O. Taniman1, A.C. van Bochove, P.T. de Boer, B. Sikkes

Chair for Design and Analysis of Communication Systems, University of Twente The Netherlands

Tel: +31-53-4893682 E-mail: r.o.taniman@ewi.utwente.nl

We propose a semi-heuristic dynamic capacity allocation algorithm. Benchmarked by simulation against the Binary Linear Programming solution, our proposed algorithm, with polynomial complexity order, resulted in a satisfactory performance in terms of achievable capacity, and also in terms of packet mean sojourn time.

1. Introduction

Multimode PON is considered as a potentially cost-effective solution for a Full Service Access Network. To avoid the limited bandwidth-distance product of multimode fiber when using baseband transmission, passband transmission which utilizes high-order transmission lobes in GHz region of the electrical-domain link frequency responses can be used instead [1]. However, due to the intermodal dispersion, the link frequency responses in general are frequency-selective for broadband serial transmission in the order of 100 Mbps (i.e., when only one subcarrier is used for transmission between the OLT and each ONU). Here, the impact of the chromatic dispersion is considered to be negligible for the distances of up to a few kilometers envisioned for the PON under consideration by using appropriate optical wavelengths. Therefore, we assume the use of multicarrier transmission which can mitigate the aforementioned impairment. Recent researches have considered the use of an optical OFDM scheme (e.g., [2,3]), which is a specific case of multicarrier transmission through an optical fiber, for similar reasons. An important aspect of this kind of system is that different subcarriers generally experience different subchannel normalized signal-to-noise ratios (nSNRs), i.e., the ratio of the subchannel gain to the subchannel signal-to-noise power. Moreover, these subchannel nSNRs are ONU-dependent and time-dependent. For this system, it is compelling to employ a dynamic capacity allocation which can exploit those time-dependent subchannel nSNRs in order to increase the overall system capacity such that the system can accommodate more traffic load.

In this paper, we propose a dynamic capacity allocation strategy which strives to maximize the system capacity under some constraints which then indirectly may result in low mean packet delay. The proposed dynamic capacity allocation employs three sequential stages. The first stage is the subcarrier allocation where the number of subcarriers to be assigned to each ONU is determined based on each ONU’s backlog and subchannel nSNRs. The second stage is the subcarrier assignment where we use a feasible vertex labelling method to find an optimal ONU-subcarrier matching with respect to the subchannel nSNRs. The third stage is the adaptive multilevel bitloading to

determine the modulation levels (M-QAM) of each subcarrier where generally the better the subchannel nSNR, the higher the modulation level used. We perform simulations to benchmark our proposed algorithm against a Binary Linear Programming solution of the dynamic capacity allocation problem. In our previous work [4], we used a fixed number of allocated subcarriers for each ONU. Compared to this work, the newly proposed dynamic capacity allocation strategy achieves better results especially in terms on packet mean sojourn time.

2. System description

The PON system considered in this paper consists of the OLT with MM ONUs. We focus on a case with tree topology employing ideal splitters. The distances between the OLT and each ONUs can be arbitrary, but nevertheless we assume a maximum value of a few kilometers (e.g., 3km as used in the simulation). Moreover, this maximum value is chosen such that for each subchannel, a distortionless transmission can be assumed.

As stated in the Introduction, we assume the use of multicarrier transmission. Like in an ordinary subcarrier-modulated system, the expression for the coupled optical power of the modulated light source is essentially as below

P (t) = P0 Ã 1 + N X i=1 misi(t) ! P (t) = P0 Ã 1 + N X i=1 misi(t) ! (1) where PP00 is the coupled CW optical power of the light source, NN is the number of subcarriers involved, ssii(t)(t) and mmii are the modulating signal and the optical modulation

index of subcarrier ii, respectively.

It should be noted that for each ii, ssii(t)(t) can have different mathematical expression

because of the use of multilevel modulation where the modulation levels can be different for each subcarrier (and also from time to time) depending on the subchannel nSNR γγi,ji,j (see Figure 1). Associated with this fact, we allow mmii to take different values

for each subcarrier too as long as no clipping happens. The no-clipping requirement is not mandatory, indeed, it can be relaxed in order to attain a better performace in physical layer; but this is not the focus of this paper so we just assume no clipping should happen. This point will be clearer in the description of the bit loading mechanism later.

Figure 1: An illustration of a multimode PON using multicarrier transmission.

f1 fN OLT ONU 1 ONU 2 ONU 3 ONU M f4f7f8 f1f9f12 f2f5f6 f3f17f28 0 2 4 8 10 −16 −14 −12 −10 −8 −6 −4 −2 0 frequency (GHz) re sp on se (d B) 6 f1f2f3 fN f1f2f3 fN 0 2 4 6 8 10 −16 −14 −12 −10 −8 −6 −4 −2 0 frequency (GHz) re sp ons e ( dB )

3. The capacity allocation problem

As implied before, an important aspect of the system to be considered is the time-varying nature of the link frequency responses. This time-dependence comes from the environmental disturbances. We anticipate that the link frequency responses can be considered stable or static for a duration of few hundreds milliseconds. Accordingly, we define a time epoch of 100ms wherein the OLT determines the capacity allocation so as to maximize the system capacity under relevant constraints. Considering also the bursty characteristic of traffic loads, for each time epoch the OLT determines the capacity allocation adaptively based on both the current subchannel nSNR γγi,ji,j and the current

ONU backlogs. Here, it is assumed that γγi,ji,j and the current ONU backlogs are known to

the OLT every time epoch.

3.1 The Binary Linear Programming formulation for the capacity allocation problem For each time epoch, it is natural to consider the capacity allocation problem as an optimization problem. In terms of the subcarriers and their adaptive modulation levels, the capacity allocation can be formulated into a Binary Linear Programming (BLP) problem. In the following BLP problem formulation, xxi,j,ki,j,k is the decision

variable where its value is either 0 or 1. If it is 1 then subcarrier jj is assigned to ONU ii and loaded with kk bits per symbol (i.e., 22kk-QAM modulation format is used,

with no modulation for k = 0k = 0, BPSK for k = 1k = 1 and QPSK for k = 2k = 2). If kk is odd, we assume the use of symmetrical constellations [5].

max M X i=1 N X j=1 K X k=0 k_{· x}i,j,k max M X i=1 N X j=1 K X k=0 k_{· x}i,j,k (2) subject to: - constraint 1: _XN j=1 K X k=0

xi,j,k = pi, ∀i where M X i=1 pi = N N X j=1 K X k=0

xi,j,k = pi, ∀i where M X i=1 pi = N (3) - constraint 2: _XM i=1 K X k=0 xi,j,k = 1,∀j M X i=1 K X k=0 xi,j,k = 1,∀j (4) - constraint 3: _XM i=1 N X j=1 K X k=0 ci,j,kxi,j,k ≤ 1 M X i=1 N X j=1 K X k=0 ci,j,kxi,j,k ≤ 1 (5)

where constraint 1, 2 and 3 represent exhaustive allocation (all subcarriers should be allocated), exclusive assignment (one subcarrier is assigned to only one ONU) and no-clipping bitloading, respectively. MM is the number of ONUs, NN is the number of subcarriers and KK is the maximum allowable bits per symbol. ppii is the

predetermined number of allocated subcarriers to ONU ii (details on this are described later). cci,j,ki,j,k is the bitloading cost which is affected by system parameters,

among others: link frequency response (evaluated at subcarrier frequencies), optical transmit power, required probability of bit error, modulation format and symbol time. For the bitloading cost, we use a continuous-time multicarrier transmission model (as expressed in equation 1). It should be noted that constraint 3 above applies only for downlink transmission. For uplink transmission, the

constraint should be modified such that the summation over the ONUs is dropped but then the modified constraint applies for each ONU. Also, if clipping is relaxed, the right-hand-side of the constraint 3 can be replaced with a number larger than 1, but cci,j,ki,j,k should also be modified accordingly. In this paper, we don’t pursue this

issue further.

The above BLP problem is in fact NP-hard, i.e., implying exponential complexity. This poses a problem if it is to be implemented in real-time which is the case for the system in this paper. Therefore, we develop a semi-heuristic algorithm to solve the capacity allocation problem with relatively low complexity. The BLP formulation above nevertheless will be used to benchmark the semi-heuristic algorithm by simulation.

3.2 Semi-heuristic algorithm employing feasible vertex labelling

A common way to solve a complicated problem is to divide it into subproblems. Here, we also use this approach. From the BLP problem formulation above, actually one can see that the number of allocated subcarries ppii should be

determined first. This then naturally comes as the first subproblem to be solved. The rest of the problem is then divided into two subproblems, i.e., a subcarrier assignment problem viewed as a maximum-weighted perfect matching problem and a bitloading problem. In other words, we divide the dynamic capacity allocation problem into 3 subproblems (see Figure 3b). Details on each subproblem are given in the following.

The heuristic subcarrier allocation

We use the following sub-algorithm to determine ppii:

Require: ri = 0∧ ˆpi = 0∧ pi = 0 {pre-conditions} Ensure: PM_i=1pi = N {post-condition}

1: for i = 1 to M do 2: ri ⇐ Pq¯i/¯γi i(¯qi/¯γi) 3: pˆi ⇐ bri(N − M)c + 1 4: pi ⇐ ˆpi 5: end for 6: δ_⇐³N₋PM_i=1pˆi ´ 7: while δ > 0 do 8: i∗ ⇐ arg maxiri 9: pi∗ ⇐ pi∗ + 1 10: ri∗ ⇐ −∞ 11: δ _{⇐ δ − 1} 12: end while Require: ri = 0∧ ˆpi = 0∧ pi = 0 {pre-conditions} Ensure: PM_i=1pi = N {post-condition}

1: for i = 1 to M do 2: ri ⇐ Pq¯i/¯γi i(¯qi/¯γi) 3: pˆi ⇐ bri(N − M)c + 1 4: pi ⇐ ˆpi 5: end for 6: δ_⇐³N₋PM_i=1pˆi ´ 7: while δ > 0 do 8: i∗ ⇐ arg maxiri 9: pi∗ ⇐ pi∗ + 1 10: ri∗ ⇐ −∞ 11: δ _{⇐ δ − 1} 12: end while

Figure 2: The sub-algorithm for subcarrier allocation.

In the above sub-algorithm, rrii is the weighting coefficient for ONU ii, where ¯qq¯ii is the average of the last hh backlog sample values corresponding to the last hh time epochs for ONU ii (we record each ONU backlog every time epoch but only keep

the last hh samples). This averaging is used to reduce overshoot/undershoot that may happen. In this paper, hh is fixed empirically to 3. ¯γ¯γii is the average of the γγi,ji,j over the subchannels corresponding to ONU ii. Assuming that all ONUs are active, the formulae for ppˆˆii makes sure that each ONU is at least allocated one subcarrier to

reduce the starvation possibility. In general, there can be leftover, i.e., δ > 0δ > 0 (some subcarriers are not yet allocated because of the rounding-down operation at line 3). The leftover subcarriers are distributed according to line 7-12. Concerning the correctness of the above sub-algorithm for subcarrier allocation, it can be shown that N − M ≤PM

i=1pˆi ≤ N

N _{− M ≤}PM_i=1pˆi ≤ N because rrii(N(N − M) − 1 ≤ br− M) − 1 ≤ brii(N(N − M)c ≤ r− M)c ≤ rii(N(N − M)− M) so

that after the sub-algorithm is done, PM

i=1pi = N

PM

i=1pi = N . Regarding the sub-algorithm

complexity, it can be seen that its complexity order is essentially O (MN)O (MN).

Figure 3: (a) An illustration of a complete bipartite graph with weighted edges. (b) The diagram of the semi-heuristic algorithm.

The feasible vertex labeling for the subcarrier assignment problem

The fact that ppii is the number of allocated subcarriers to ONU ii can be viewed as

follows: ONU ii is replicated such that the number of ONU ii becomes ppii for

subsequent sub-algorithms. For the subcarrier assignment problem, we use the so-called feasible vertex labeling method to solve it. After solving the first subproblem, we can construct a symmetrical complete bipartite graph GG (see Figure 3a). The left vertex subset XX represents the (replicated) ONUs, while the right vertex subset YY represents the subcarriers. Every edge connecting a certain ONU and a certain subcarrier is given a weight w(x, y)w(x, y) which in this case is the subcarrier’s subchannel nSNR with respect to the corresponding ONU. Now, the subcarrier assignment problem can be regarded as a maximum-weighted perfect matching problem. Perfect matching is a matching where each vertex of XX is matched to a vertex of YY and all vertices are matched.

A feasible vertex labeling ll is a mapping from the graph vertices to real numbers (hence, labeling) such that ∀x ∈ X, y ∈ Y∀x ∈ X, y ∈ Y

l(x) + l(y) _{≥ w(x, y)}

l(x) + l(y) _{≥ w(x, y)} (6)

For a feasible vertex labeling ll, the equality subgraph GGll is defined as the subgraph of GG which contains the edges where l(x) + l(y) = w(x, y)l(x) + l(y) = w(x, y) and the associated vertices. There is a theorem concerning feasible vertex labeling (presented without proof here) stating that if GGll contains a perfect matching MMll(x, y)(x, y) on GG then Ml(x, y) = M∗(x, y)

Ml(x, y) = M∗(x, y) where MM∗∗(x, y)(x, y) is a maximum-weighted perfect matching on GG. By virtue of this theorem, the problem of finding M_M∗∗_{(x, y)(x, y) is replaced by the problem}

to find a feasible vertex labeling ll such that MMll(x, y)(x, y) exists. A well-known w(x,y) X Y x1 x4 x3 x2 y1 y4 y3 y2 Subcarrier allocation (# subcarriers / ONU) ONU backlog Subchannel nSNR Capacity allocation Subcarrier assignment (subcarriers ONUs) Bitloading (# bits / symbol) (a) (b)

implementation of this method is the so-called Hungarian algorithm which is of O (N3₎

O (N3_{) complexity. We use this algorithm in this paper to solve the second}

subproblem. More details about the feasible vertex labeling and the Hungarian algorithm can be found, e.g., in [6]. Related to the w(x, y)w(x, y) as a subchannel nSNR, the Hungarian algorithm results in a perfect matching where a maximum sum of w(x, y)

w(x, y) translates into a maximum sum of subchannel nSNRs (at the corresponding time epoch). This way, it can be expected that the system capacity is maximized in terms of the total loaded bits per symbol.

The bitloading subproblem

The third subproblem, namely, the bitloading, is now easier, as by first fixing the matching described above, we can treat the multiuser multicarrier bitloading problem as a single-user multicarrier bitloading (which is essentially a greedy algorithm) which is of O (NK)O (NK) complexity. This has been described in our previous paper [4], along with the statistical link model we use for the simulation.

In summary, we use three sub-algorithms sequentially to solve the dynamic capacity allocation problem (see Figure 3b). Overall, because N À MN À M and also N À K

N À K , the complexity order of the semi-heuristic algorithm we propose is essentially O (N3₎

O (N3). This is a significant complexity reduction from the BLP approach which has exponential complexity order.

4. Simulation results

We performed Monte Carlo simulation using OPNET 12.0.A (PL3) package to benchmark the proposed semi-heuristic algorithm against the BLP solution, focusing on the downlink transmission. For each ONU, a 2-phase Markov-modulated Poisson process (MMPP) traffic generators (with each average load switches between 100Mbps and 200Mbps) was used to generate bursty traffic load. These parameter values put the simulated system in a heavy-load operation regime. The OLT-ONU lengths were fixed but uniformly distributed (see Table 1). Also, KK was fixed to 8. We incorporated the C implementation of the Hungarian algorithm developed by Cyrill Stachniss of University of Freiburg, Germany into our simulation code. To solve the BLP for benchmarking, we additionally used the lp_solve 5.5.0.10 code library.

Description Value Description Value

Number of ONUs 16 Optical wavelength 1310 nm

Number of subcarriers 128 Fiber attenuation factor 0.5 dB/km Time epoch 100 ms Photodiode responsivity 0.8 A/W Number of simulated time epochs 500 Coupled optical transmit power 10 mW

OLT-ONU distances 1-3 km Receiver load resistance 50 Ω Packet size 1538 bytes Receiver temperature 300 K Required bit error rate 10-9 _{Subcarrier frequency 1 5 GHz}

Symbol time 100 ns Subcarrier frequency interval 20 MHz Table 1: Parameters used in the simulation.

Figure 4: The simulation results by average values (note: 1 bits/symbol = 10Mbps)

Figure 4 shows that the Hungarian-algorithm-based (HAB) capacity allocation method could result in a very close performance in terms of allocated capacity relative to the BLP method, i.e., 98% on average. It should be noted that from the simulation results, each ONU was allocated capacity around 190Mbps on average. The result on allocated capacity is also reflected in terms of mean packet sojourn time, with the relative value (HAB/BLP) is 125% on average. The packet jitter also showed the same pattern with the relative value of 119%. The important point is both methods (BLP and HAB) resulted in low mean packet sojourn time (around 70 ms) though the model was quite heavily loaded. Certainly, the BLP results, especially in terms of mean packet sojourn time, are better, but it should be noted again that the HAB complexity is much lower than the BLP one. In fact, the BLP method may not be implementable for adaptive real-time systems. Nevertheless, this complexity reduction manifests here as small performance penalty which in most cases can be accepted. From Figure 4, it can also be seen that the heuristic subcarrier allocation (the first sub-algorithm) has largely mitigated the influence of different ONU distances. This means the fairness among ONUs could be maintained.

5. Conclusion

We have developed a low-complexity semi-heuristic dynamic capacity allocation algorithm for multicarrier multimode PON which has comparable performance with the Binary Linear Programming solution in terms of allocated capacity, mean packet sojourn time and packet jitter. The adaptive subcarrier allocation sub-algorithm helps ensure a good degree of fairness among ONUs despite diferrent distances to the OLT.

Mean packet sojourn time

0 0.02 0.04 0.06 0.08 0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ONU sec o n d s HAB BLP Allocated capacity 16 18 20 22 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ONU b its /s ym bol HAB BLP Packet jitter 0 0.02 0.04 0.06 0.08 0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ONU se co nd s HAB BLP

.

Acknowledgements

This work is supported by the Dutch Ministry of Economic Affairs in the IOP GenCom project "Full Service Access Networks using Multimode Fibre".

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