Multi-Stage PON ILP Model

Given the lack of existing literature on this topic, the option for the development of a new model followed some guidelines. The first was to approach the problem in the broadest possible sense, that is, imposing the least possible amount of restrictions on the PON configuration. This allows for more exotic configurations than the ones actual deployments might consider (like the distributed split). However, it enables the formulation to act as an uncompromised common ground that can be adapted to different settings. Furthermore, it is interesting to investigate how an “ideal” scenario compares in cost terms to a real one with more topological restrictions. The second main principle was the use of exact optimization algorithms (ILP) again. This was done on the one hand to maintain coherence with the previous work, and on the other because while potentially more complex and intractable (as it would be the case), the ILP model can be used as a benchmark to evaluate the performance of other algorithms.

The formulation of the problem borrowed its principles from the CLP, meaning the OLT site is geographically fixed, as are all the possible locations for splitters and ONTs. Also, the connection costs between network elements are known in advance. Because it is a problem with multiple levels of concentrators, one idea would be to pre-assign the level of each splitter in advance. This would limit the total possible number of connections and reduce the overall number of variables. However, it would constraint the PON’s flexibility which is against the principles enunciated above. The selected option enables every splitter to be connected to any other (in any hierarchical order), so the level of each splitter is only known when looking at the final solution.

Before introducing the ILP model developed for the multi-stage problem, it is important to clarify how the problem of appropriately setting splitter capacities described in 3.1 was addressed. Because in a PON the biggest constraint to capacity is ultimately optical power, the logical way to address the problem would be to create a variable that reflects a splitter’s capacity in terms of the optical power it disposes of. The solution found was to have a set of variables, , that denote the ratio of power splitting that can occur after splitter . Because in a PON optical power is always divided by a power of two, this will fit quite nicely in an ILP formulation. The reason for this is that we can use the base 2 logarithm to represent the number of splitting stages possible and since only splitting of 1:2, 1:4, 1:8, 1:16, 1:32 and 1:64 is possible, the base 2 logarithm converts these ratios into a integer consecutive sequence. Also, this way a division in power becomes a logarithmic subtraction which can also be handled by the ILP. Let us take the example of the left side of Figure 3.3. For GPON, the OLT has a capacity of 6 (it allows 6 splitting stages of 1:2). Splitter 1 divides the power by 4 output ports. It’s “capacity” is then:

'b_c\d V 'b_cd \ V W d (3.1)

Which makes sense, because any splitter connected to it, can subdivide its power by at most 2⁴=16 times. But the capacities in Figure 3.3 are incremented by one unit relative to what was previously explained. This is because a difference had to exist between a splitter that allows no further division (like splitter 5) and a splitter that simply isn’t used in the solution (splitter 6). By incrementing the capacity values, zero capacity univocally denotes a splitter that is not used, and the structure’s problem remains the same.

Figure 3.3 - Examples of splitter capacity variables.

The model can now be described as follows. Let us consider splitter locations and ONTs. The cost coefficients are:

e - Cost of connecting ONT to splitter ;

f - Cost of connecting splitter to the OLT and of OPEX associated with a PON ;

f - Cost of connecting splitter upwards to splitter . This means that if a connection exists between and , then is hierarchically below - (Recall that a splitter’s level is not known beforehand);

g[- Cost of a splitter of type . For GPON, \, referring to 1:2, 1:4, 1:8 etc...

The integrality conditions correspond to:

2 h/!%!ⁱ0 !" #!"$!! jh 0F'""!%k

. j"/!%$0!k 1 (3.2)

h/!%!ⁱ0 !" #!"$!! 0F'""!% 0F'""!%Djlhm .B

$/!%!0 !'!n!'#!'$k . j"/!%$0!k

1 (3.3)

I _[ 2 oF'""!%+5p,qr)sDs Wtd]\Bk

. j"/!%$0!k 1 (3.4)

Variables and are similar to the ones considered in the CLP, with extended to allow connections between splitters. Variables I refer to the splitter type, which, more than differentiating the cost of the splitters, are used to shape the constraints.

Additional parameters are defined as:

o – Base 2 logarithm of the maximum split ratio of the PON technology considered. Indicates how many splitter types there are;¹

u - Very large number to ensure constraint coherency;

The objective function is:

e ?

f ?

f ?

v

g_[? I _[

w [

(3.5)

Subject to:

k (3.6)

v

k (3.7)

W^[? I _[

w [

S

v

k (3.8)

u= V > V k k . k x (3.9) o ?

v

k (3.10)

Vu= V > ? I [ w [

V k k . k x (3.11)

1 1:64 and 1:32 maximum ratios considered for GPON and EPON respectively.

u= V > ? I [ w [

S V k k . k x (3.12)

v

u ?

v

k W (3.13)

v

k W (3.14)

I [ w [

k (3.15)

o (3.16)

The two first terms of the objective function come directly from the CLP. The third term reflects the costs of connecting splitters amongst themselves and the fourth term denotes the costs of each type of splitter used. Because the differences between costs of splitter types are very low compared to the other costs considered, the fourth term can be discarded with some confidence if running time is a problem.

The I variables are still needed for the constraints though, because it is the type of each splitter that indicates how many output ports it has.

The constraints act as following: (3.6) ensures every ONT is connected to one and only one splitter, (3.7) makes every splitter connect to at most one other splitter or the OLT, (3.8) states that the type of each splitter (its split ratio) is such that its output ports are enough for every ONT and splitter connected to it. (3.9) assures the capacity of a splitter is necessarily lower than that of a splitter if there is an upwards connection between and . u is a number sufficiently large so that the inequality still holds even if there is no connection between the splitters. Constraint (3.10) sets the splitter’s capacity to zero if it is not upwardly connected to a splitter or the OLT. It also sets its maximum capacity to o. Constraints (3.11) and (3.12) form an equality together, stating that the difference in capacities between two splitters that are connected is determined by the splitting ratio of the lower-level splitter. Constraint (3.13) states that if a splitter has ONTs or splitters connecting to it, then it must be connected to a splitter or the OLT. (3.14) ensures that a splitter with upwards connections has a capacity greater than zero. (3.15) implies that there can only be one type of splitter per location (at most) and (3.16) sets the OLT capacity. Of course (3.16) is not a constraint in the strictest sense because the technology used is known beforehand so

can be set appropriately but it would require modifying some constraints for the special case of connecting to the OLT while this allows a more clean and compact formulation. Figure 3.4 shows a two- level PON obtained from this model. The OLT feeds one splitter which in turn feeds another three that connect 10 ONTs. As expected, given the model’s size and complexity, medium and large networks can have exceedingly long running times.

Figure 3.4 - Multi-Level PON obtained from ILP.

The single-stage problem can be obtained from this formulation by considering only the two first terms of the multi-stage formulation. Constraint (3.6) remains unaltered relative to (2.7), while constraint (3.8) is equal to (2.8) if we remove the connections between splitters and on the left hand side of (3.8) consider a fixed capacity for every splitter.

Table 3.1 presents the running times for different sized networks:

Table 3.1 - Running times for multi-stage ILP model.

Splitter locations 5 5 10 10 20

ONTs 20 50 50 100 100

Running time [hh:mm:ss]

00:00:00 00:01:18 04:32:27 > 120 hours > 120 hours

Results show that the formulation is limited, for current everyday computational power, to small networks. Additionally, the increase in the running times is quite sudden, which makes even medium sized networks hard to obtain by this method in useful time. This led to the study of heuristic procedures described in the following section.