Chapter 7. Oversubscription-Based Design of Shared Backup Path Protection for MPLS or ATM
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9.8 Sample Results: Studies with MTRS
MTRS can, nonetheless, be implemented and solved to optimality for suitably small problems. Although too complex for direct general use
in planning tools, MTRS provides us with a research tool which serves as kind of microscope with which to view the structure and
properties of strictly perfect topology solutions. This leads to some interesting results and understanding about how topology responds to
the relative costs of capacity, edges and the demand matrix. It also leads to highly effective heuristic solution techniques that are usable in
practical network planning tools.
9.8.1 Effect of Edge-to-Capacity Cost Ratio,
28
Figure 9-15 shows how the total cost of optimal 8-node MTRS solutions, with its universal set of 28 possible edges (and hence 2
possible topologies) responds as the cost of edge establishment increases relative to the cost of a unit capacity on the same edge,
.
In these results, edge and capacity costs are proportional to the Euclidean distance between the 8 nodes as positioned in the plane for the
test case, and shown in Figure 9-16. For each edge, the edge establishment cost is
times the cost of a unit capacity, per unit length.
The relative positions of the 8 nodes and the demand between them are based on the COST-239 network and data shown ahead in
Figure 10-13. The test case here is essentially COST-239 without Paris, London, and Copenhagen and where each 2.5 Gb/s of traffic is
converted into one demand unit, e.g., a lightpath requirement. None of the spans shown in Figure 10-13 are assumed to exist here,
however, and we consider the universal set of candidates on 8 nodes.
Figure 9-15. Total cost of optimal 8-node MTRS solutions as
Figure 9-16. Evolution of the Optimal Topology from MTRS as
is increased.
is increased [Ezem03].
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Figure 9-15 shows that total cost rises relatively uniformly as
is increased, but there are significant plateaus and jumps in the
underlying breakdown of total cost between edges and capacity. These dynamics correspond to sudden jumps in the optimal graph
topologies, samples of which are shown in Figure 9-16 for the ranges of
indicated on Figure 9-15.
As expected, MTRS always produces a closed graph. Note also in Figure 9-16 how the topology evolves from richly connected down to a
Hamiltonian cycle as the cost of edges becomes increasingly important. Once the Hamiltonian cycle of Figure 9-16(f) is reached, total cost
rises completely linearly with
because no choice of fewer edges is feasible from a survivability standpoint. If this was an FCR solution
pushed to this extreme, the solution would be a minimum cost spanning tree of N-1 edges, rather than a cycle of N edges. More generally
note that for each
, or certain ranges of
, there is a specific optimal topology. This is the topology that balances efficient working
routing and spare capacity assignment costs against the cost to establish itself. In this regard the optimal graph of
= 20 to 40 in Figure
9-16(d) seems of special interest because it is the graph on which roughly equal edge and capacity costs arise (from inspection of
Figure
9-15) and is a graph with
= 2.5 which is characteristic of North American long-haul networks.
Figure 9-17 accompanies these results showing how nodal the degree of the optimal solution responds to increasing
. This plot has
axes of cost versus nodal degree but it does not have the general shape of Figure 9-6 as may be expected because, here,
independent variable and we are plotting the properties (cost and degree) of the optimal designs with
is the
as a parameter. In a plot such
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as Figure 9-6,
is constant and nodal degree is varied in search of a single optimal solution.Figure 9-17 clearly shows how capacity
costs vary almost continuously but are fixed at each "stack" of points in the edge costs. Each of the stacks represents an unchanging
topology whose set of edges has increased cost as
is increased, but does not change until suddenly some new configuration of fewer
edges costs less than the associated increase in routing (and hence capacity) costs, and then a sudden jump to the new optimum topology
takes place.
Figure 9-17. Total cost and nodal degree of optimal eight-node MTRS solutions as
increases.
The form of these results, undertaken for research, differ in two important ways from the way we encounter the topology problem in
practice. One is that we are not often able to solve MTRS directly to reveal the optimal topology. MTRS run times with the universal edge
set go directly from hours to solve the eight-node problems above, to weeks for nine-node problems. Also the real problem context is one
where
is given by the costs of equipment and rights-of-way and is not under our direct control. In fact
may usually be different for
each possible edge. Nonetheless, the eight-node results, which we know are perfect solutions let us appreciate what happens in the actual
problem we face: if the topology is too rich we have excess cost from edges—and if it is too sparse then capacity costs are excessive. At
the extremes where general
is very low, capacity costs dominate total cost and we can be guided in a topology choice by principles
such as efficient working path routing. At the opposite extreme what would matter most to the solution is a minimum cost biconnected
graph, ultimately a particular cycle at the highest of
values. It is in the midrange in fact that the problem is the most general and
hardest to solve. CPLEX solution times exhibit this behavior quite strongly: at high or low
the results of Figure 9-16, the highest run times were seen exactly in the region of
, the MTRS problem solves most quickly. In
~ 25 where edge and capacity costs are in balance.
9.8.2 Effect of Demand Intensity and Demand Pattern
The comparative results so far presented on effects of
and nodal degree employ a common demand pattern across alternatives. So
how does the optimal topology vary with the pattern and relative intensity of the demand? To discuss this, let us separate the notion of
demand intensity from demand pattern.
By intensity we mean any bulk common scaling on all demands of one case relative to another. Let the multiplicative factor involved be
called b. In considering the effect of any such bulk scaling of demand, intuition and the total cost function for MTRS Equation
(
9.16),
suggests that any increase in b is equivalent to a corresponding divisor on
. The intuitive reason is that if we just scale up all demand
quantities (under the same routing rules for each O-D pair), then this implies b times more capacity use on each edge. Equivalently the
effect from a topology standpoint is also indistinguishable from an increase in the capacity cost coefficients cij in Equation 9.16 by a factor b
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at the same demand intensity. In summary, what we are saying is that the total cost function for edge (i,j) under variable intensity or edge
to capacity ratio, based on Equation 9.16 can be rewritten as:
Equation 9.33
where Fij/cij is simply rewritten as
0
0
and (wij+sij) is the total capacity in edge (i,j) under some baseline demand matrix D pattern relative
0
to which the current demand is b · D . The summation of total costs on all spans inEquation 9.16 does not alter the logic. But under
minimization no constant factor makes a difference to the solution details (only its absolute cost), so that relative to an initial problem
0
based on D , any other problem with the same
/b ratio has the same optimal graph topology. In other words, scaling demand intensity
is equivalent to lowering the edge-to-capacity cost ratio in terms of the effect on optimal topology. This was also verified experimentally in
[Ezem03]. The only situation where this would not follow is where the working capacity routing and/or SCA process for spare capacity
placement were not themselves invariant under b. Modularity and/or existing edges with fixed amounts of already installed capacity could
both have this effect.
Figure 9-18 presents a series of experimental design results with optimal solutions to MTRS using test cases created on 6 nodes from the
original COST-239 topology and demand pattern. The plot is similar in nature to that of Figure 9-17 but shows only total cost and includes
families of curves where demand undergoes flat multiplication on all O-D pairs by the factors "x1" through to "x10" and at each demand
intensity,
is varied to generate a family of solution topologies for each
, b combination. From Figure 9-17 we understand that the
stacks of rising cost at fixed degree are the signature of a graph topology that is robust under increasing
reasoning above in that certain topologies reappear under equivalent
. Results confirm the
/b ratios. For example, the graph forb=1, has the same graph for
10 <
< 40 as appears for b =5 for 40 <
< 100. What the reasoning underEquation 9.33 doesn't tell us, but Figure 9-18 portrays, is
just how often a single graph turns up as being optimal for a wide range of conditions. For instance Figure 9-18 shows that there are three
graphs (associated with the marker arrows) that emerge quite often. One, at d~3.7 serves well for a range of demand and edge cost
factors centred roughly around
/b ~5. The next, quite dominant, graph atd~2.6 is well suited for a wide range of conditions centered on
/b ~10. The most sparse graph of all d~2.01 serves in the
interchangeable under Equation 9.33, the "basins" of
/b ~ 50 range. We also note from Figure 9-18 that although
, b combinations that map into a common best graph topology become smaller at
high demand. At the lowest demand curve in Figure 9-18 we see only five different topologies to cover the whole range of 1 <
and only two of these cover most of the
cover the range 1 <
and b are
< 100,
range. On the other hand at the highest demand (x10) seven discrete graphs are needed to
< 100, and it becomes much more important in total cost terms to be at the right topology for each smaller basin of
, b combinations before the jump to the next discrete topology.
Figure 9-18. Complete relationship between demand and edge-to-capacity cost ratio ( ) for
C239-6n in determining optimal topology, indicating three very dominant topology options.
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Much less can be said in general about how the relative pattern of demand between nodes affects the optimal topology. By and large this
is where we must resort to actual solution attempts using methods that are the main topic of this chapter. Nonetheless one general insight
is that significant variations in the pattern of demand can be "absorbed" without triggering or requiring a shift in the topology to remain
optimal. Each variation in pattern affects wij + sij costs in detail, but does not often trigger a shift in topology. This tendency to a stable (or
"pattern-robust" topology) seems to be strongest when both capacity and edge costs are significant in the solution. In these cases a
moderately rich topology tends to be selected, within which the routing and SCA subproblems have considerable latitude to absorb
detailed differences in pattern without requiring an edge addition or deletion to remain optimal. Conversely, we observe that for a given
total amount of demand, the optimum topology is more sensitive to the exact demand pattern at high
limiting cycle has been adopted).
(unless
is so high that a
These effects can be seen in some tests of different demand patterns conducted on a further subset of 6 of the 8 nodes used above.
Three different demand patterns, denoted C239(x2), Random, and GR50 were tested at three values of
. The resulting optimal
topologies are portrayed in Figure 9-19. The C239(x2) demand matrix uses the original published demand for the 6 nodes selected, scaled
by two to bring its range (largest to smallest demand difference) to match the other two patterns at 20. In "Random" each O-D pair is
assigned a demand quantity from a uniform random distribution of integers in the range [5,...,30] regardless of distance or nodal degree.
The total demand in Random is 2.2 times that in C239(x2). "GR50" is an inverse distance-weighted gravity model with demands generated
with a mutual attraction product term weighted by the inverse of the distance between the nodes. The importance factors for the mutual
attraction effect were taken as the degree of the corresponding nodes in the actual COST-239 network. GR50 has 1.2 times the total
demand of C239(x2). The important thing is that differences between solution topologies were analyzed with detailed reference to the
actual demand matrices, with the following main findings.
Figure 9-19. How demand pattern alters the topology (6 node optimal solutions of MTRS).
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The moderately connected 8-edge graph (given the dark shaded background in Figure 9-19) emerges as the optimal topology in five of the
nine trials. Even though demand pattern and working and spare capacity differ in each design, this one graph is optimal for all patterns at
the mid-range
=50 and for C239(x2) at lower
as well as Random at the higher
. In addition at
=20, both other patterns
contain this hub and spoke structure but add the same two edges to it in a way that inspection shows correlated with greater direct
demand between the extra node pairs connected in those cases. What seems to be so robust about this graph is that it uses only two
relatively short edges, above the minimum required for biconnectivity and it provides for shortest routes through the center of the network
which are nearly as low as the Euclidean distance between those node pairs. Note in particular that the evidently central node in this
topology is "star" connected to four of the other five nodes. Thus, out of 15 O-D pairs in total on 6 nodes the central node and its four
rds
immediately attached neighbors (2/3
of all the demand pairs) see relatively direct routing for their working demands. The general points
we draw from this are that to find an optimum topology, and moreover a topology that is preferably stable over a wide range of pattern
difference:
1.
Efficiency for the routing of working flows remains of first order important. Although by itself working flow routing efficiency
would never result in a biconnected graph, elements of the topology are clearly justified by their efficacy with respect to working
paths.
2.
While we may still be justified in asserting a priori limits on the overall network average degree, we need to permit the formation
of higher than average degree "hubs" which can play a significant role in stabilizing the optimum topology under pattern
variations.
3.
Optimum topology is more sensitive to demand pattern at high
the total cost.
4.
The more specific topology changes seen at
=100 are again primarily attributable to differences in the working demand
pattern, not primarily due to shifts in the strategy for placing spare capacity to protect such flows.
, where fewer edges are being chosen and edges dominate
The main conclusion seems to be that working considerations are of more direct importance to determining the MTRS topology. As long as
the topology is ultimately biconnected, protection arrangements seem to be of secondary dependence on the exact topology. Because
spare capacity is shared, and because any set of restoration routes under the hop limit for each failure scenario suffices, achievement of
an efficient spare capacity design is generally much less dependent on the topology. In contrast each working route consumes dedicated
capacity and shortest routes are far preferred over indirect routes for working capacity.
The insights obtained about MTRS from these experiments inspire a simple hypothesis: that optimal graphs for MTRS may be
approximated by first finding a good graph for the working-only problem and then just making that graph biconnected in an efficient way.
This leads to a 3-part heuristic based on this approach to approximating MTRS.
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.
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9.9 A Three-Part Heuristic for MTRS
In this section, we describe a heuristic for MTRS based on the hypothesis just stated and a decomposition into three smaller MIP
[5]
problems. The 3-step method is based on viewing MTRS as containing FCR-like aspects for working flows as well as mesh SCA- or
JCA -like aspects (for restoration). The basic idea is to first identify a topology that is good from an FCR standpoint, and another set of
edges that are meritorious from a mesh spare capacity viewpoint, and then solve the full MTRS model within the union of those specific
edge sets, rather than the universal edge set. The three steps, each to be described further, are:
[5]
Material in this section is adapted from prior publication on this heuristic with J.DoucetteGrDo01].
[
W1
Solve a (working-only) fixed charge plus routing problem (FCR). Edges identified at this stage are collectively sufficient for routing and our
philosophy is that they are of special merit for consideration in a complete design by virtue of their special role in serving working
demand flows. Any preexisting edges are represented as edges that have zero fixed charge for their establishment.
S2
Solve an artificial problem for the minimum cost of additional edges and capacity to ensure restorability of the working flows in W1. We
call this the reserve network fixed charge plus spare capacity problem (RN-FCS). Additional edges identified by this step are sufficient to
enable restoration by closing the graph and capacitating the needed spare capacity at minimum incremental cost. The philosophy is that
such edges are therefore of merit for further consideration in a complete design by virtue of their special efficiency from a restoration
standpoint. Any preexisting edges and new edges from W1 are asserted as inputs and the objective function excludes any fixed costs for
the latter edges.
J3
Solve a restricted instance of MTRS where the set of candidate edges is the union set from W1 and S2, not the universal set of all
possible edges. The idea is that since MTRS is exponential in |A| there is high run-time leverage on reducing the number of candidate
edges. But solution quality may not suffer greatly if the reduced edge set consists only of edges that are of special merit either from the
standpoint of routing (W1) or restoration (S2). The final restricted MTRS problem instance selects a set of edges that are collectively
sufficient and efficient for both routing and restoration.
The central hypothesis is that within the union of the edge sets arising from W1 and S2 lies a subgraph on which a high quality
approximation to MTRS can be found. The labels W1, S2, J3 are meant to suggest: "step 1 forWorking only, step 2 for Sparing only, step 3
for Joint reduced problem. It is important in this regard to realize that J3 and MTRS (or "full MTRS") have exactly the same problem
structure. The only difference is that a "J3 instance" of MTRS has a restricted edge set.
The computational advantage of this approach is significant because the most onerous component step (W1) can be a partially relaxed
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and/or time-limited FCR problem instance. FCR is itself still a difficult problem, but its role in the heuristic is only to help promote a
candidate edge set that is of interest from a working standpoint, so we do not necessarily need to solve it to optimality. In comparison, S2
generally solves quickly and J3 is much faster than unrestricted MTRS because we typically have reduced the candidate edge space
from Y=N(N-1)/2 to ~ 2N or 3N (edge sets corresponding to networks of maximum degree of six, say). Although the result is approximate
in the sense of global optimality, J3 is itself an exact instance of MTRS so its output is a fully feasible and constructible design in terms of
all edges, routing and restoration details. In other words, there are no functional details that are approximate as a result this heuristic.
The next sections define and discuss each step in more detail.
9.9.1 Step W1: Working-only Fixed Charge Plus Routing
W1 is an instance of FCR without regard to any survivability considerations, within the universe of all possible edges for the problem. As
such, the formulation for this stage is unchanged from that for classical FCR in Section 9.2. What we want going forward from this step
is, however, only the topology selection outcome and the objective function value for later bounding use in J3. There is nothing in the
FCR formulation that assures that a restorable topology emerges. In fact trees are very likely at this stage. Since the detailed routing
associated with the FCR solution is not retained, the working flow variables are candidates for relaxation to speed up this step. The idea
is only to produce a first topology that by itself is nearly optimal if the goal was only to serve the working demand flows. Nonetheless W1
remains the most complex stage of the 3-step method. This step may therefore also be time limited. An option discussed later for W1 is
to use an artificially low
to identify not just the edges that would strictly be part of the FCR solution at the true
edges that may have been close to this qualification.
, but to also reveal
9.9.2 Step S2: Reserve Network Fixed Charge and Sparing (RN-FCS)
This step augments the topology from W1 to become at least two-connected while simultaneously minimizing the fixed costs for
additional edges and the spare capacity placed on all edges to achieve restorability. The result from W1 is an initial topology and set of
working capacity wij values that fully serve the demand matrix. In S2 the topology from W1 is asserted as "already existing" edges. In S2
only edges from W1 are considered as failure scenarios whereas from a restoration flow standpoint all existing or possible edges are
considered. Restoration flows are subject to the same fixed charges that apply in W1 for any edge that is added at this stage. Thus, new
edges are added to the topology at this stage if they are justified on their combined merits of closing the graph and providing restoration
capacity.
This step benefits computationally relative to MTRS in three ways: (1) The edge decision space is already reduced from Y possible
edges to no more than (Y-N+1) remaining edge choices (because at least N-1 edges were decided in W1). (2) All working capacity and
2
working flow variables and constraints are eliminated. (3) Not all Y (i.e., O(N )) possible span failure scenarios have to be considered,
only ~ O(N) corresponding to the edges in the FCR solution from W1.
RN-FCS:
Equation 9.34
Equation 9.35
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Equation 9.36
Equation 9.37
Equation 9.38
Equation 9.39
Equation 9.40
Equation 9.41
Equation 9.42
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For clarity the objective function is expressed in two parts. First is the edge and spare capacity costs for additional edges at this stage.
The second sum recognizes spare capacity that may be added at this stage to edges already selected in W1. The set of edge selections
already made (and "paid for") in W1 are passed into S2 in the set E1 where they are directly asserted as part of the S2 solution (in
Equation 9.40). In Equation 9.39 the edge space is correspondingly reduced to {A-E1}. Constraints Equation 9.35 - Equation 9.36 relate
the restoration flow variables
for each (i,j) failure scenario to the amount of working flow to be protected in the form of source-sink
and transshipment constraints for each failure scenario. Equation 9.38 dimensions the spare capacity variable on each new and existing
edge. The remaining constraints define the edge selection variables and restate the added valid knowledge constraints.
9.9.3 Step J3: Restricted MTRS for Final Topology Selection
The last step addresses the global coordination of working, spare, and topology considerations that are inherent in the full MTRS
problem but not present in the design at the end of S2. The augmented edge set at the end of S2 can only be retained or reduced by this
step. Note that J3 does not adopt the union of edges from W1 and S2 as a given topology on which to solve routing and spare capacity.
Rather the union of the prior edge sets is again viewed as a candidate edge space. Thus J3 can either keep all edges promoted to it
from prior steps, or further reduce the final set of edges employed. J3 can also make use of a bound obtainable from the result of W1,
described below. In addition, the J3 objective value can be fed into an unrestricted instance of MTRS as an upper bound when
attempting to solve a fully optimal MTRS problem. In summary, the three steps play the following roles:
W1 Finds a minimal topology and capacity as justified by working flows alone.
S2 Finds a min-cost topology augmentation as justified by restorability considerations.
J3 Revises the working flows of W1 to exploit the augmented topology of S2 and coordinates the assignment of spare
capacity and final selection of edges.
9.9.4 Useful Bounds from Steps W1 and J3
At two stages in the heuristic we can also obtain bounds to aid in solution of either restricted or unrestricted MTRS problem. Specifically,
if step W1 is individually solved to optimality it follows that the objective value of W1 is a lower bound on J3. Any feasible solution for
MTRS, or a J3 instance, has to cost more than an optimal solution for FCR alone. (It can only cost more to add protection.) Even more
specifically, we can lower-bound a subset of the variables in the MTRS objective function by applying the same line of reasoning to the
topology plus working capacity variables only within an MTRS problem. That is to say that:
Equation 9.43
because the component costs for the fixed charge and routing solution (alone) that are embodied within a full MTRS solution can only
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make compromises to accommodate the wider set of considerations in MTRS compared to the pure FCR solution from W1.
Similarly, J3 provides an upper bound on the cost of the full MTRS problem because it has considered only a reduced edge set. A full
MTRS problem, by considering all candidate edges can only possibly improve on a J3 solution. Thus we can write the additional
bounding relationship:
Equation 9.44
We make use of the first relationship to help solve the J3 problem in the 3-step heuristic and the second one to help solve instances of
unrestricted MTRS.
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