Network Working Group S. Yasukawa
Internet-Draft NTT
Intended Status: Informational A. Farrel
Created: December 14, 2008 Old Dog Consulting
Expires: June 14, 2009 O. Komolafe
Cisco Systems
An Analysis of Scaling Issues in MPLS-TE Core Networks
draft-ietf-mpls-te-scaling-analysis-05.txt
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Abstract
Traffic engineered Multiprotocol Label Switching (MPLS-TE) is
deployed in providers' core networks. As providers plan to grow these
networks, they need to understand whether existing protocols and
implementations can support the network sizes that they are planning.
This document presents an analysis of some of the scaling concerns
for the number of Label Switching Paths (LSPs) in MPLS-TE core
networks, and examines the value of two techniques (LSP hierarchies,
and multipoint-to-point LSPs) for improving scaling. The intention is
to motivate the development of appropriate deployment techniques and
protocol extensions to enable the application of MPLS-TE in large
networks.
This document only considers the question of achieving scalability
for the support of point-to-point MPLS-TE LSPs. Point-to-multipoint
MPLS-TE LSPs are for future study.
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Contents
1. Introduction .................................................. 3
1.1 Overview ..................................................... 3
1.2 Glossary of Notation ......................................... 4
2. Issues of Concern for Scaling ................................. 5
2.1 LSP State ................................................... 5
2.2 Processing Overhead .......................................... 5
2.3 RSVP-TE Implications ......................................... 6
2.4 Management ................................................... 7
3. Network Topologies ............................................ 7
3.1 The Snowflake Network Topology ............................... 8
3.2 The Ladder Network Topology .................................. 10
3.3 Commercial Drivers for Selected Configurations ............... 12
3.4 Other Network Topologies ..................................... 13
4. Required Network Sizes ........................................ 14
4.1 Practical Numbers ............................................ 14
5. Scaling in Flat Networks ...................................... 14
5.1 Snowflake Networks ........................................... 15
5.2 Ladder Networks .............................................. 16
6. Scaling Snowflake Networks with Forwarding Adjacencies ........ 19
6.1 Two Layer Hierarchy .......................................... 20
6.1.1 Tuning the Network Topology to Suit the Two Layer Hierarchy 21
6.2 Alternative Two Layer Hierarchy .............................. 21
6.3 Three Layer Hierarchy ........................................ 22
6.4 Issues with Hierarchical LSPs ............................... 23
7. Scaling Ladder Networks with Forwarding Adjacencies............ 24
7.1 Two Layer Hierarchy .......................................... 24
7.2 Three Layer Hierarchy ........................................ 25
7.3 Issues with Hierarchical LSPs ................................ 26
8. Scaling Improvements Through Multipoint-to-Point LSPs ......... 26
8.1 Overview of MP2P LSPs ........................................ 27
8.2 LSP State: A Better Measure of Scalability ................... 27
8.3 Scaling Improvements for Snowflake Networks .................. 28
8.3.1 Comparison with Other Scenarios ............................ 29
8.4 Scaling Improvements for Ladder Networks ..................... 30
8.4.1 Comparison with Other Scenarios ............................ 31
8.4.2 LSP State Compared with LSP Numbers ........................ 33
8.5 Issues with MP2P LSPs ........................................ 33
9. Combined Models ............................................... 34
10. An Alternate Solution ........................................ 35
10.1 Pros and Cons of the Alternate Solution ..................... 36
11. Management Considerations .................................... 37
12. Security Considerations ...................................... 37
13. Recommendations .............................................. 38
14. IANA Considerations .......................................... 38
15. Acknowledgements ............................................. 38
16. Intellectual Property Consideration .......................... 39
17. Normative References ......................................... 40
18. Informative References ....................................... 40
19. Authors' Addresses ........................................... 41
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1. Introduction
Network operators and service providers are examining scaling issues
as they look to deploy ever-larger traffic engineered Multiprotocol
Label Switching (MPLS-TE) networks. Concerns have been raised about
the number of Label Switched Paths (LSPs) that need to be supported
at the edge and at the core of the network. The impact on control
plane and management plane resources threatens to outweigh the
benefits and popularity of MPLS-TE, while the physical limitations of
the routers may constrain the deployment options.
Historically, it has been assumed that all MPLS-TE scaling issues
can be addressed using hierarchical LSP [RFC4206]. However, analysis
shows that the improvement gained by LSP hierarchies is not as
significant in all topologies and at all points in the network as
might have been presumed. Further, additional management issues are
introduced to determine the end-points of the hierarchical LSPs and
to operate them. Although this does not invalidate the benefits of
LSP hierarchies, it does indicate that additional techniques may be
desirable in order to fully scale MPLS-TE networks.
This document examines the scaling properties of two generic MPLS-TE
network topologies, and investigates the benefits of two scaling
techniques.
1.1 Overview
Physical topology scaling concerns are addressed by building networks
that are not fully meshed. Network topologies tend to be meshed in
the core, but tree-shaped at the edges giving rise to a snow-flake
design. Alternatively, the core may be more of a ladder shape with
tree-shaped edges.
MPLS-TE, however, establishes a logical full mesh between all edge
points in the network, and this is where the scaling problems arise
since the structure of the network tends to focus a large number of
LSPs within the core of the network.
This document presents two generic network topologies: the snow-flake
and the ladder, and attempts to parameterize the networks by making
some generalities. It introduces terminology for the different
scaling parameters and examines how many LSPs might be required to be
carried within the core of a network.
Two techniques (hierarchical LSPs, and multipoint-to-point LSPs) are
introduced and an examination is made of the scaling benefits that
they offer as well as of some of the concerns with using these
techniques.
Of necessity, this document makes many generalizations. Not least
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among these are a set of assumptions about the symmetry and
connectivity of the physical network. It is hoped that these
generalizations will not impinge on the usefulness of the overview of
the scaling properties that this document attempts to give. Indeed,
the symmetry of the example topologies tends to highlight the
scaling issues of the different solution models, and this may be
useful in exposing the worst case scenarios.
Although protection mechanisms like Fast Re-route (FRR) [RFC4090] are
briefly discussed, the main body of this document considers stable
network cases. It should be noted that make-before-break
re-optimisation after link failure may result in a significant number
of 'duplicate' LSPs. This issue is not addressed in this document.
It should also be understood that certain deployment models where
separate traffic engineered LSPs are used to provide different
services such as layer 3 VPNs [RFC4110] or pseudowires [RFC3985], or
different classes of service [RFC3270], may result in 'duplicate' or
'parallel' LSPs running between any pair of provider edge nodes
(PEs). This scaling factor is also not considered in this document,
but may be easily applied as a linear factor by the reader.
The operation of security mechanisms in MPLS-TE networks [MPLS-SEC]
may have an impact on the ability of the network to scale. For
example, they may increase both the size and number of control plane
messages. Additionally, they may increase the processing overhead as
control plane messages are subject to processing algorithms (such as
encryption), and security keys need to be managed. Deployers will
need to consider the trade-offs between scaling objectives and
security objectives in thier networks and should resist the
temptation to respond to a degradation of scaling performance by
turning off security techniques that have previously been deemed as
necessary. Further analysis of the effects of security measures on
scalability are not considered further in this document.
This document is designed to help service providers discover whether
existing protocols and implementations can support the network sizes
that they are planning. To do this, it presents an analysis of some
of the scaling concerns for MPLS-TE core networks, and examines the
value of two techniques for improving scaling. This should motivate
the development of appropriate deployment techniques and protocol
extensions to enable the application of MPLS-TE in large networks.
This document only considers the question of achieving scalability
for the support of point-to-point MPLS-TE LSPs. Point-to-multipoint
MPLS-TE LSPs are for future study.
1.2 Glossary of Notation
This document applies consistent notation to define various
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parameters of the networks that are analyzed. These terms are defined
as they are introduced though-out the document, but are grouped
together here for quick reference. Refer to the full definitions in
the text for detailed explanations.
n A network level. n = 1 is the core of the network.
See section 3, for more details of the definition of a level.
P(n) A node at level n in the network.
S(n) The number of nodes at level n. That is, the number of P(n)
nodes.
L(n) The number of LSPs seen by a P(n) node.
X(n) The number of LSP segment states held by a P(n) node.
M(n) The number of P(n+1) nodes subtended to a P(n) node.
R The number of rungs in a ladder network.
E The number of edge nodes (PEs) subtended below (directly or
indirectly) a spar node in a ladder network.
K The cost-effectiveness of the network expressed in terms of
the ratio of the number of PEs to the number of network nodes.
2. Issues of Concern for Scaling
This section presents some of the issues associated with the support
of LSPs at a Label Switching Router (LSR) or within the network.
These issues may mean that there is a limit to the number LSPs that
can be supported.
2.1 LSP State
LSP state is the data (information) that must be stored at an LSR in
order to maintain an LSP. Here we refer to the information that is
necessary to maintain forwarding plane state and the additional
information required when LSPs are established through control
plane protocols. While the size of the LSP state is implementation-
dependent, it is clear that any implementation will require some data
in order to maintain LSP state.
Thus LSP state becomes a scaling concern because, as the number of
LSPs at an LSR increases, so the amount of memory required to
maintain the LSPs increases in direct proportion. Since the memory
capacity of an LSR is limited, there is a related limit placed on the
number LSPs that can be supported.
Note that techniques to reduce the memory requirements (such as data
compression) may serve to increase the number of LSPs that can be
supported, but this will only achieve a moderate multiplier and may
significantly decrease the ability to process the state rapidly.
In this document we define X(n) as "The number of LSP segment states
held by a P(n) node." This definition observes that an LSR at the end
of an LSP only has to maintain state in one direction (i.e., into the
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network), while a transit LSR must maintain state in both directions
(i.e., toward both ends of the LSP). Furthermore, in multipoint-to-
point (MP2P) LSPs (see Section 8) a transit LSR may need to maintain
LSP state for one downstream segment (toward the destination) and
multiple upstream segments (from multiple sources). That is, we
define LSP segment state as the state necessary to maintain an LSP in
one direction to one adjacent node.
2.2 Processing Overhead
Depending largely on implementation issues, the number of LSPs
passing through an LSR may impact the processing speed for each LSP.
For example, control block search times can increase with the number
of control blocks to be searched, and even excellent implementations
cannot completely mitigate this fact. Thus, since CPU power is
constrained in any LSR, there may be a practical limit to the number
of LSPs that can be supported.
Further processing overhead considerations depend on issues specific
to the control plane protocols, and are discussed in the next
section.
2.3 RSVP-TE Implications
Like many connection-oriented signaling protocols, RSVP-TE requires
that state is held within the network in order to maintain LSPs. The
impact of this is described in Section 2.1. Note that RSVP-TE
requires that separate information is maintained for upstream and
downstream relationships, but does not require any specific
implementation of that state.
RSVP-TE is a soft-state protocol which means that protocol messages
(refresh messages) must be regularly exchanged between signaling
neighbors in order to maintain the state for each LSP that runs
between the neighbors. A common period for the transmission (and
receipt) of refresh messages is 30 seconds meaning that each LSR must
send and receive one message in each direction (upstream and
downstream) for every LSP it supports every 30 seconds. This has the
potential to be a significant constraint on the scaling of the
network, but various improvements [RFC2961] mean that this refresh
processing can be significantly reduced allowing an implementation to
be optimized to nearly remove all concerns about soft state scaling
in a stable network.
Observations of existing implementations indicate that there may be a
threshold of around 50,000 LSPs above which an LSR struggles to
achieve sufficient processing to maintain LSP state. Although refresh
reduction [RFC2961] may substantially improve this situation,
it has also been observed that under these circumstances the size of
the Srefresh may become very large and the processing required may
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still cause significant disruption to an LSR.
Another approach is to increase the refresh time. There is a
correlation between the percentage increase in refresh time and the
improvement in performance for the LSR. However, it should be noted
that RSVP-TE's soft-state nature depends on regular refresh messages,
thus a degree of functionality is lost by increasing the refresh
time. This loss may be partially mitigated by the use of the RSVP-TE
Hello message, and can also be reduced by the use of various GMPLS
extensions [RFC3473] such as the use of [RFC2961] message
acknowledgements on all messages.
RSVP-TE also requires that signaling adjacencies are maintained
through the use of Hello message exchanges. Although [RFC3209]
suggests that Hello messages should be retransmitted every 5ms, in
practice values of around 3 seconds are more common. Nevertheless,
the support of Hello messages can represent a scaling limitation on
an RSVP-TE implementation since one message must be sent and received
to/from each signaling adjacency every time period. This can impose
limits on the number of neighbors (physical or logical) that an LSR
supports, but does not impact the number of LSPs that the LSR can
handle.
2.4 Management
Another practical concern for the scalability of large MPLS-TE
networks is the ability to manage the network. This may be
constrained by the available tools, the practicality of managing
large numbers of LSPs, and the management protocols in use.
Management tools are software implementations. Although such
implementations should not constrain the control plane protocols, it
is realistic to appreciate that network deployments will be limited
by the scalability of the available tools. In practice, most existing
tools have a limit to the number of LSPs that they can support. While
a Network Management System (NMS) may be able to support a large
number of LSPs, the number that can be supported by anElement
Management System (EMS) (or the number supported by an NMS per-LSR)
is more likely to be limited.
Similarly, practical constraints may be imposed by the operation of
management protocols. For example, an LSR may be swamped by
management protocol requests to read information about the LSPs that
it supports, and this might impact its ability to sustain those LSPs
in the control plane. OAM, alarms, and notifications can further add
to the burden placed on an LSR and limit the number of LSPs it can
support.
All of these consideration encourage a reduction in the number of
LSPs supported within the network and at any particular LSR.
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3. Network Topologies
In order to provide some generic analysis of the potential scaling
issues for MPLS-TE networks, this document explores two network
topology models. These topologies are selected partly because of
their symmetry which makes them more tractable to a formulaic
approach, and partly because they represent generalizations of
real deployment models. Section 3.3 provides a discussion of the
commercial drivers for deployed topologies and gives more analysis
of why it is reasonable to consider these two topologies.
The first topology is the snowflake model. In this type of network,
only the very core of the network is meshed. The edges of the network
are formed as trees rooted in the core.
The second network topology considered is the ladder model. In this
type of network the core of the network is shaped and meshed in the
form of a ladder and trees are attached rooted to the edge of the
ladder.
The sections that follow examine these topologies in detail in order
to parameterize them.
3.1 The Snowflake Network Topology
The snowflake topologies considered in this document are based on a
hierarchy of connectivity within the core network. PE nodes have
connectivity to P-nodes as shown in Figure 1. There is no direct
connectivity between the PEs. Dual homing of PEs to multiple P
nodes is not considered in this document although it may be a
valuable addition to a network configuration.
P
/|\
/ | \
/ | \
/ | \
PE PE PE
Figure 1 : PE to P Node Connectivity
The relationship between P-nodes is also structured in a hierarchical
way. Thus, as shown in Figure 2, multiple P-nodes at one level are
connected to a P-node at a higher level. We number the levels such
that level 1 is the top level (top in our figure, and nearest to the
core of the network) and level (n) is immediately above level (n+1),
and we denote a P-node at level n as a P(n).
As with PEs, there is no direct connectivity between P(n+1) nodes.
Again, dual homing of P(n+1) nodes to multiple P(n) nodes is not
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considered in this document although it may be a valuable addition
to a network configuration.
P(n)
/|\
/ | \
/ | \
/ | \
P(n+1) P(n+1) P(n+1)
Figure 2 : Relationship Between P-Nodes
At the top level, P(1) nodes are connected in a full mesh. In
reality, the level 1 part of the network may be slightly less well
connected than this, but assuming a full mesh provides for
generality. Thus, the snowflake topology comprises a clique with
topologically equivalent trees subtended from each node in the
clique.
The key multipliers for scalability are the number of P(1) nodes, and
the multiplier relationship between P(n) and P(n+1) at each level
including down to PEs.
We define the multiplier M(n) as the number of P(n+1) nodes at level
(n+1) attached to any one P(n). Assume that M(n) is constant for all
nodes at level n. Since nodes at the same level are not
interconnected (except at the top level), and since each P(n+1) node
is connected to precisely one P(n) node, M(n) is one less than the
degree of the node at level n (that is, the P(n) node is attached to
M(n) nodes at level (n+1) and 1 node at level (n-1)).
We define S(n) as the number of nodes at level (n).
Thus: S(n) = S(1)*M(1)*M(2)*...*M(n-1)
So the number of PEs can be expressed as:
S(PE) = S(1)*M(1)*M(2)*...*M(n)
where the network has (n) layers of P-nodes.
Thus we may depict an example snowflake network as shown in Figure 3.
In this case:
S(1) = 3
M(1) = 3
S(2) = S(1)*M(1) = 9
M(2) = 2
S(PE) = S(1)*M(1)*M(2) = 18
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PE PE PE PE PE PE
\ \/ \/ /
PE--P(2) P(2) P(2) P(2)--PE
\ | | /
\| |/
PE--P(2)---P(1)------P(1)---P(2)--PE
/ \ / \
PE \ / PE
\/
P(1)
/|\
/ | \
/ | \
PE--P(2) P(2) P(2)--PE
/ /\ \
PE PE PE PE
Figure 3 : An Example Snowflake Network
3.2 The Ladder Network Topology
The ladder networks considered in this network are based on an
arrangement of routers in the core network that resembles a ladder.
Ladder networks typically have long and thin cores that are arranged
as conventional ladders. That is, they have one or more spars
connected by rungs. Each node on a spar may have:
- connection to one or more other spars
- connection to a tree of other core nodes
- connection to customer nodes.
Figure 4 shows a simplified example of a ladder network. A core of
twelve nodes make up two spars connected by six rungs.
PE PE PE PE
PE PE PE | PE | PE PE PE | PE | PE
\| \|/ |/ | \| \|/
PE-P-----P-----P-----P------P-----P--PE
| | | | | |\
| | | | | | PE
| | | | | |
PE-P-----P-----P-----P------P-----P
/| /|\ |\ |\ |\ \
PE PE PE | PE | PE | PE | PE PE
PE PE PE PE
Figure 4 : A Simplified Ladder Network.
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In practice, not all nodes on a spar (call them Spar-Nodes) need to
have subtended PEs. That is, they can exist simply to give
connectivity along the spar to other spar nodes, or across a rung to
another spar. Similarly, the connectivity between spars can be more
complex with multiple connections from one spar-node to another spar.
Lastly, the network may be complicated by the inclusion of more than
two spars (or simplified by reduction to a single spar).
These variables make the ladder network non-trivial to model. For the
sake of simplicity we will make the following restrictions:
- There are precisely two spars in the core network.
- Every spar-node connects to precisely one spar-node on the other
spar. That is, each spar-node is attached to precisely one rung.
- Each spar-node connects to either one (end-spar) or two (core-spar)
other spar-nodes on the same spar.
- Every spar-node has the same number of PEs subtended. This does not
mean that there are no P-nodes subtended to the spar-nodes, but
does mean that the edge tree subtended to each spar-node is
identical.
From these restrictions we are able to quantify a ladder network as
follows:
R - The number of rungs. That is, the number of spar-nodes on
each spar.
S(1) - The number of spar-nodes in the network. S(1)=2*R.
E - The number of subtended edge nodes (PEs) to each spar-node.
The number of rungs may vary considerably. A number less than 3 is
unlikely (since that would not be a significantly connected network),
and a number greater than 100 seems improbable (because that would
represent a very long, thin network).
E can be treated as for the snowflake network. That is, we can
consider a number of levels of attachment from P(1) nodes which are
the spar-nodes, through P(i) down to P(n) which are the PEs.
Practically, we need to only consider n=2 (PEs attached direct to the
spar-nodes) and n=3 (one level of P-nodes between the PEs and the
spar-nodes).
Let M(i) be the ratio of P(i) nodes to P(i-1) nodes, i.e. the
connectivity between levels of P-node as defined for the snowflake
topology. Hence, the number of nodes at any level (n) is:
S(n) = S(1)*M(1)*M(2)*...*M(n-1)
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So number of PEs subtended to a spar node is:
E = M(1)*M(2)*...*M(n)
And the number of PEs can be expressed as:
S(PE) = S(1)*M(1)*M(2)*...*M(n)
= S(1)*E
Thus we may depict an example ladder network as shown in Figure 5.
In this case:
R = 5
S(1) = 10
M(1) = 2
S(2) = S(1)*M(1) = 20
M(2) = 2
E = M(1)*M(2) = 4
S(PE) = S(1)*E = 40
PE PE PE PE PE PE PE PE PE PE PE PE PE PE PE PE
\| \| \| \| |/ |/ |/ |/
P(2) P(2) P(2) P(2) P(2) P(2) P(2) P(2)
\ \ | \ / | / /
PE \ \ | \ / | / / PE
\ \ \| \/ |/ / /
PE-P(2)---P(1)---P(1)---P(1)---P(1)---P(1)---P(2)-PE
| | | | |
| | | | |
| | | | |
PE-P(2)---P(1)---P(1)---P(1)---P(1)---P(1)---P(2)-PE
/ / / | /\ |\ \ \
PE / / | / \ | \ \ PE
/ / | / \ | \ \
P(2) P(2) P(2) P(2) P(2) P(2) P(2) P(2)
/| /| /| /| |\ |\ |\ |\
PE PE PE PE PE PE PE PE PE PE PE PE PE PE PE PE
Figure 5 : An Example Ladder Network
3.3 Commercial Drivers for Selected Configurations
It is reasonable to ask why these two particular network topologies
have been chosen.
The most important consideration is physical scalability. Each node
(label switching router - LSR) is only able to support a limited
number of physical interfaces. This necessarily reduces the ability
to fully mesh a network and leads to the tree-like structure of the
network toward the PEs.
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A realistic commercial consideration for an operator is the fact that
the only revenue-generating nodes in the network are the PEs. Other
nodes are needed only to support connectivity and scalability.
Therefore, there is a desire to maximize S(PE) while minimizing the
sum of S(n) for all values of (n). This could be achieved by
minimizing the number of levels, and by maximizing the connectivity
at each layer, M(n). Ultimately, however, this would produce a
network of just interconnected PEs, which is clearly in conflict with
the physical scaling situation.
Therefore, the solution calls for a "few" levels with "relatively
large" connectivity at each level. We might say that the
cost-effectiveness of the network can be stated as:
K = S(PE)/(S(1)+S(2) + ... + S(n)) where n is the level above the PEs
We should observe, however, that this equation may be naive in that
the cost of a network is not actually a function of the number of
routers (since a router chasis is often free or low cost), but is
really a function of the cost of the line cards which is, itself, a
product of the capacity of the line cards. Thus, the relatively high
connectivity decreases the cost-effectiveness, while a topology that
tends to channel data through a network core tends to demand higher
capacity (so more expensive) line cards.
A further consideration is the availability of connectivity (usually
fibers) between LSR sites. Although it is always possible to lay new
fiber, this may not be cost-effective or timely. As much of a problem
is likely to be the physical shape and topography of the country in
which the network is laid. If the country is 'long and thin' then a
ladder network is likely to be used.
This document examines the implications for control plane and data
plane scalability of this type of network when MPLS-TE LSPs are used
to provide full connectivity between all PEs.
3.4 Other Network Topologies
As explained in Section 1, this document is using two symmetrical
and generalized network topologies for simplicity of modelling. In
practice there are two other topological considerations.
a. Multihoming
It is relatively common for a node at level (n) to be attached to
more than one node at level (n-1). This is particularly common at
PEs that may be connected to more than one P(n).
b. Meshing within a level
A level in the network will often include links between P-nodes at
the same level including the possibility of links between PEs.
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This may result in a network that looks like a series of
concentric circles with spokes.
Both of these features are likely to have some impact on the scaling
of the networks. However, for the purposes of establishing the
ground-rules for scaling, this document restricts itself to the
consideration of the symmetrical networks described in Sections 2.1
and 2.2. Discussion of other network formats is for future study.
4. Required Network Sizes
An important question for this evaluation and analysis is the size of
the network that operators require. How many PEs are required? What
ratio of P to PE is acceptable. How many ports do devices have for
physical connectivity? What type of MPLS-TE connectivity between PEs
is required?
Although presentation of figures for desired network sizes must be
treated with caution because history shows that networks grow beyond
all projections, it is useful to set some acceptable lower bounds.
That is, we can state that we are interested in networks of at least
a certain size.
The most important features are:
- The network should have at least 1000 PEs.
- Each pair of PEs should be connected by at least one LSP in each
direction.
4.1 Practical Numbers
In practice, reasonable target numbers are as follows.
S(PE) >= 1000
Number of levels is 3. That is: 1, 2 and PE.
M(2) <= 20
M(1) <= 20
S(1) <= 100
5. Scaling in Flat Networks
Before proceeding to examine potential scaling improvements, we need
to examine how well the flat networks described in the previous
sections scale.
Consider the requirement for a full mesh of LSPs linking all PEs.
That is, each PE has an LSP to and from each other LSP. Thus, if
there are S(PE) PEs in the network, there are S(PE)*(S(PE) - 1) LSPs.
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Define L(n) as the number of LSPs handled by a level (n) LSR.
L(PE) = 2*(S(PE) - 1)
5.1 Snowflake Networks
There are a total of S(PE) PEs in the network and, since each PE
establishes an LSP with every other PE, it would be expected that
there are S(PE) - 1 LSPs incoming to each PE and the same number of
LSPs outgoing from the same PE, giving a total of 2(S(PE) - 1) on
the incident link. Hence, in a snowflake topology (see Figure 3),
since there are M(2) PEs attached to each P(2) node, it may tempting
to think that L(2), the number of LSPs traversing each P(2) node, is
simply 2*(S(PE) - 1)*M(2). However, it should be noted that of the
S(PE) - 1 LSPs incoming to each PE, M(2) - 1 originated from nodes
attached to the same P(2) node and so this value would count the LSPs
between the between the M(2) PEs attached to each P(2) node twice;
once when outgoing from the M(2) - 1 other nodes and once when
incoming into a particular PE.
There are a total of M(2)*(M(2) - 1) LSPs between these M(2) PEs and
since this value is erroneously included twice in 2*(S(PE) - 1)*M(2),
the correct value is:
L(2) = 2*M(2)*(S(PE) - 1) - M(2)*(M(2) - 1)
= M(2)*(2*S(PE) - M(2) - 1)
An alternative way of looking at this, that proves extensible for the
calculation of L(1), is to observe that each PE subtended to a P(2)
node has an LSP in each direction to all S(PE) - M(2) PEs in the rest
of the system, and there are M(2) such locally subtended PEs; thus
2*M(2)*(S(PE) - M(2)). Additionally there are M(2)*(M(2) - 1) LSPs
between the locally subtended PEs. So:
L(2) = 2*M(2)*(S(PE) - M(2)) + M(2)*(M(2) - 1)
= M(2)*(2*S(PE) - M(2) - 1)
L(1) can be computed in the same way as this second evaluation of
L(2). Each PE subtended below a P(1) node has an LSP in each
direction to all PEs not below the P(1) node. There are M(1)*M(2) PEs
below the P(1) node, so this accounts for
2*M(1)*M(2)*(S(PE) - M(1)*M(2)) LSPs. To this, we need to add the
number of LSPs that pass through the P(1) node and that run between
the PEs subtended below the P(1). Consider each P(2): it has M(2) PEs
that each has an LSP going to all of the PEs subtended to the other
P(2) nodes subtended to the P(1). There are M(1) - 1 such other P(2)
nodes, and so M(2)*(M(1) - 1) other such PEs. So the number of LSPs
from the PEs below a P(2) node is M(2)*M(2)*(M(1) - 1). And there
are M(1) P(2) nodes below the P(1), giving rise to a total of
M(2)*M(2)*M(1)*(M(1) - 1) LSPs. Thus:
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L(1) = 2*M(1)*M(2)*(S(PE) - M(1)*M(2)) + M(2)*M(2)*M(1)*(M(1) - 1)
= M(1)*M(2)*(2*S(PE) - M(2)*(M(1) + 1))
So, for example, with S(1) = 5, M(1) = 10, and M(2) = 20, we see:
S(PE) = 1000
L(PE) = 1998
L(2) = 39580
L(1) = 356000
Alternatively, with S(1) = 10, M(1) = 10, and M(2) = 20, we see:
S(PE) = 2000
L(PE) = 3998
L(2) = 79580
L(1) = 756000
In both examples, the number of LSPs at the core (P(1)) nodes is
probably unacceptably large even though there are only a relatively
modest number of PEs. In fact, L(2) may even be too large in the
second example.
5.2 Ladder Networks
In ladder networks L(PE) remains the same at 2*(S(PE) - 1).
L(2) can be computed using the same mechanism as for the snowflake
topology because the subtended tree is the same format. Hence,
L(2) = 2*M(2)*(S(PE) - 1) - M(2)*(M(2) - 1)
But L(1) requires a different computation because each P(1) not only
sees LSPs for the subtended PEs, but is also a transit node for some
of the LSPs that cross the core (the core is not fully meshed).
Each P(1) sees:
o all of the LSPs between locally attached PEs
o less the those LSPs between locally attached PEs that can be
served exclusively by the attached P(2) nodes
o all LSPs between locally attached PEs and remote PEs
o LSPs in transit that pass through the P(1).
The first three numbers are easily determined and match what we have
seen from the snowflake network. They are:
o E*(E-1)
o M(1)*M(2)*(M(2)-1) = E*(M(2) - 1)
o 2*E*E*(S(1) - 1)
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The number of LSPs in transit is more complicated to compute. It is
simplified by not considering the ends of the ladders, but examining
an arbitrary segment of the middle of the ladder such as shown in
Figure 6. We look to compute and generalize the number of LSPs
traversing each core link (labeled a and b in Figure 6) and so
determine the number of transit LSPs seen by each P(1).
: : : : : :
: : : : : :
P(2) P(2) P(2) P(2) P(2) P(2)
\ | \ / | /
\ | \ / | /
\| \/ |/
......P(1)---P(1)---P(1)......
| a | |
| |b |
| | |
......P(1)---P(1)---P(1)......
/| /\ |\
/ | / \ | \
/ | / \ | \
P(2) P(2) P(2) P(2) P(2) P(2)
: : : : : :
: : : : : :
Figure 6 : An Arbitrary Section of a Ladder Network
Of course, the number of LSPs carried on links a and b in the Figure
depends on how LSPs are routed through the core network. But if we
assume a symmetrical routing policy and an even distribution of LSPs
across all shortest paths, the result is the same.
Now we can see that each P(1) sees half of 2a+b LSPs (since each LSP
would otherwise be counted twice as it passed through the P(1))
except that some of the LSPs are locally terminated so are only
included once in the sum 2a+b.
So L(1) = a + b/2 -
(locally terminated transit LSPs)/2 +
(locally contained LSPs)
Thus:
L(1) = a + b/2 -
2*E*E*(S(1) - 1)/2 +
E*(E-1) - E*(M(2) - 1)
= a + b/2 +
E*E*(2 - S(1)) - E*M(2)
So all we have to do is work out a and b.
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Recall that the ladder length R = S(1)/2, and define X = E*E.
Consider the contribution made by all of the LSPs that make n hops on
the ladder to the totals of each of a and b. If the ladder was
unbounded then we could say that in the case of a, there are n*2X
LSPs along the spar only, and n(n-1)*2X/n = 2X(n-1) LSPs use a rung
and the spar. Thus, the LSPs that make n hops on the ladder
contribute (4n-2)X LSPs to a. Note that the edge cases are special
because LSPs that make only one hop on the ladder cannot transit a
P(1) but only start or end there.
So with a ladder of length R = S(1)/2 we could say:
R
a = SUM[(4i-2)*X] + 2RX
i=2
= 2*X*R*(R+1)
And similarly, considering b in an unbounded ladder, the LSPs that
only travel one hop on the LSP are a special case, contributing 2X
LSPs, and each other LSP that traverses n hops on the ladder
contributes 2n*2X/n = 4X LSPs. So:
R+1
b = 2X + SUM[4X]
i=2
= 2*X + 4*X*R
In fact the ladders are bounded and so the number of LSPs is reduced
because of the effect of the ends of the ladders. The links that see
the most LSPs are in the middle of the ladder. Consider a ladder of
length R; a node in the middle of the ladder is R/2 hops away from
the end of the ladder. So we see that the formula for the
contribution to the count of spar-only LSPs for a is only valid up to
n=R/2, and for spar-and-rung LSPs for n=1+R/2. Above these limits the
contribution made by spar-only LSPs decays as (n-R/2)*2X. However,
for a first order approximation, we will use the values of a and b as
computed above. This gives us an upper bound of the number of LSPs
without using a more complex formula for the reduction made by the
effect of the ends of the ladder.
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From this:
L(1) = a + b/2 +
E*E*(2 - S(1)) - E*M(2)
= 2*X*R*(R+1) +
X + 2*X*R +
E*E*(2 - S(1)) - E*M(2)
= E*E*S(1)*(1 + S(1)/2) +
E*E + E*E*S(1) +
2*E+E - E*E*S(1) - E*M(2)
= E*E*S(1)*(1 + S(1)/2) + 3*E+E - E*M(2)
= E*E*S(1)*S(1)/2 + E*E*S(1) + 3*E*E - E*M(2)
So, for example, with S(1) = 6, M(1) = 10, and M(2) = 17, we see:
E = 170
S(PE) = 1020
L(PE) = 2038
L(2) = 34374
L(1) = 777410
Alternatively, with S(1) = 10, M(1) = 10, and M(2) = 20, we see:
E = 200
S(PE) = 2000
L(PE) = 3998
L(2) = 79580
L(1) = 2516000
In both examples, the number of LSPs at the core (P(1)) nodes is
probably unacceptably large even though there are only a relatively
modest number of PEs. In fact, L(2) may even be too large in the
second example.
Compare the L(1) values with the total number of LSPs in the sytem
S(PE)*(S(PE) - 1) which is 1039380 and 3998000 respectively.
6. Scaling Snowflake Networks with Forwarding Adjacencies
One of the purposes of LSP hierarchies [RFC4206] is to improve the
scaling properties of MPLS-TE networks. LSP tunnels (sometimes known
as Forwarding Adjacencies (FAs)) may be established to provide
connectivity over the core of the network and multiple edge-to-edge
LSPs may be tunneled down a single FA LSP.
In our network we consider a mesh of FA LSPs between all core nodes
at the same level. We consider two possibilities here. In the first
all P(2) nodes are connected to all other P(2) nodes by LSP tunnels,
and the PE-to-PE LSPs are tunneled across the core of the network. In
the second, an extra layer of LSP hierarchy is introduced by
connecting all P(1) nodes in an LSP mesh and tunneling the P(2)-to-
P(2) tunnels through these.
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6.1 Two Layer Hierarchy
In this hierarchy model, the P(2) nodes are connected by a mesh of
tunnels. This means that the P(1) nodes do not see the PE-to-PE LSPs.
It remains the case that:
L(PE) = 2*(S(PE) - 1)
L(2) is slightly increased. It can be computed as the sum of all LSPs
for all attached PEs including the LSPs between the attached PE (this
figure is unchanged from Section 5.1: i.e. M(2)*(2*S(PE) - M(2) - 1))
plus the number of FA LSPs providing a mesh to the other P(2) nodes.
Since the number of P(2) nodes is S(2), each P(2) node sees
2*(S(2) - 1) FA LSPs. Thus:
L(2) = M(2)*(2*S(PE) - M(2) - 1) + 2*(S(2) - 1)
L(1), however, is significantly reduced and can be computed as the
sum of the number of FA LSPs to and from each attached P(2) to each
other P(2) in the network, including (but counting only once) the FA
LSPs between attached P(2) nodes. In fact, the problem is identical
to the L(2) computation in Section 5.1. So:
L(1) = M(1)*(2*S(2) - M(1) - 1)
So, for example, with S(1) = 5, M(1) = 10, and M(2) = 20, we see:
S(PE) = 1000
S(2) = 50
L(PE) = 1998
L(2) = 39678
L(1) = 890
Alternatively, with S(1) = 10, M(1) = 10, and M(2) = 20, we see:
S(PE) = 2000
S(2) = 100
L(PE) = 3998
L(2) = 79778
L(1) = 1890
So, in both examples, potential problems at the core (P(1)) nodes
caused by an excessive number of LSPs can be avoided, but any problem
with L(2) is made slightly worse as can be seen from the table below.
Example| Count | Unmodified | 2-Layer
| | (Section 5.1) | Hierarchy
-------+-------+---------------+----------
A | L(2) | 39580 | 39678
| L(1) | 356000 | 890
-------+-------+---------------+----------
B | L(2) | 79580 | 79778
| L(1) | 756000 | 1890
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6.1.1 Tuning the Network Topology to Suit the Two Layer Hierarchy
Clearly we can reduce L(2) by selecting appropriate values of S(1),
M(1), and M(2). We can do this without negative consequences, since
no change will affect L(PE), and since a large percentage increase in
L(1) is sustainable because L(1) is now so small.
Observe that:
L(2) = M(2)*(2*S(PE) - M(2) - 1) + 2*(S(2) - 1)
where S(PE) = S(1)*M(1)*M(2) and S(2) = S(1)*M(1). So L(2) scales
with M(2)^2 and we can have the most impact by reducing M(2) while
keeping S(PE) constant.
For example, with S(1) = 10, M(1) = 10, and M(2) = 10, we see:
S(PE) = 1000
S(2) = 100
L(PE) = 1998
L(2) = 20088
L(1) = 1890
And similarly, with S(1) = 20, M(1) = 20, and M(2) = 5, we see:
S(PE) = 2000
S(2) = 400
L(PE) = 3998
L(2) = 20768
L(1) = 15580
These considerable scaling benefits must be offset against the cost-
effectiveness of the network. Recall from Section 3.3 that
K = S(PE)/(S(1)+S(2) ... + S(n))
where n is the level above the PEs, so that for our network:
K = S(PE) / (S(1) + S(2))
Thus, in the first example the cost-effectiveness has been halved
from 1000/55 to 1000/110. And in the second example it has been
reduced to roughly one quarter, changing from 2000/110 to 2000/420.
So, although the tuning changes may be necessary to reach the desired
network size, they come at a considerable cost to the operator.
6.2 Alternative Two Layer Hierarchy
An alternative to the two layer hierarchy presented in section 6.1 is
to provide a full mesh of FA LSPs between P(1) nodes. This technique
is only of benefit to any nodes in the core of the level 1 network.
It makes no difference to the PE and P(2) nodes since they continue
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to see only the PE-to-PE LSPs. Furthermore, this approach increases
the burden at the P(1) nodes since they have to support all of the
PE-to-PE LSPs as in the flat model, plus the additional 2*(S(1) - 1)
P(1)-to-P(1) FA LSPs. Thus, this approach should only be considered
where there is a mesh of P-nodes within the ring of P(1) nodes, and
it is not considered further in this document.
6.3 Three Layer Hierarchy
As demonstrated by section 6.2, introducing a mesh of FA LSPs at the
top level (P(1)) has no benefit, but if we introduce an additional
level in the network (P(3) between P(2) and PE) to make a four level
snowflake, we can introduce a new layer of FA LSPs so that we have a
full mesh of FA LSPs between all P(3) nodes to carry the PE-to-PE
LSPs, and a full mesh of FA LSPs between all P(2) nodes to carry the
P(3)-to-P(3) LSPs.
The number of PEs is S(PE) = S(1)*M(1)*M(2)*M(3), and the number of
PE-to-PE LSPs at a PE remains as L(PE) = 2*(S(PE) - 1).
The number of LSPs at a P(3) can be deduced from section 6.1. It is
the sum of all LSPs for all attached PEs including the LSPs between
the attached PE plus the number of FA LSPs providing a mesh to the
other P(3) nodes.
L(3) = M(3)*(2*S(PE) - M(3) - 1) + 2*(S(3) - 1)
The number of LSPs at P(2) can also be deduced from section 6.1 since
it is the sum of all LSPs for all attached P(3) nodes including the
LSPs between the attached PE plus the number of FA LSPs providing a
mesh to the other P(2) nodes.
L(2) = M(2)*(2*S(3) - M(2) - 1) + 2*(S(2) - 1)
Finally, L(1) can be copied straight from 6.1.
L(1) = M(1)*(2*S(2) - M(1) - 1)
For example, with S(1) = 5, M(1) = 5, M(2) = 5, and M(3) = 8 we see:
S(PE) = 1000
S(3) = 125
S(2) = 25
L(PE) = 1998
L(3) = 16176
L(2) = 1268
L(1) = 220
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Similarly, with S(1) = 5, M(1) = 5, M(2) = 8, and M(3) = 10 we see:
S(PE) = 2000
S(3) = 200
S(2) = 25
L(PE) = 3998
L(3) = 40038
L(2) = 3184
L(1) = 220
Clearly, there are considerable scaling improvements with this three-
layer hierarchy, and all of the numbers (even L(3) in the second
example) are manageable.
Of course, the extra level in the network tends to reduce the cost-
effectiveness of the networks with values of K = 1000/155 and
K = 2000/230 (from 1000/55 and 2000/110) for the examples above.
That is, a reduction by a factor of 3 in the first case and 2 in the
second case. Such a change in cost-effectiveness has to be weighed
against the the desire to deploy such a large network. If LSP
hierarchies are the only scaling tool available, and networks this
size are required, the cost-effectiveness may need to be sacrificed.
6.4 Issues with Hierarchical LSPs
A basic observation for hierarchical scaling techniques is that it is
hard to have any impact on the number of LSPs that must be supported
by the level of P(n) nodes adjacent to the PEs (for example, it is
hard to reduce L(3) in section 6.3). In fact, the only way we can
change the number of LSPs supported by these nodes is to change the
scaling ratio M(n) in the network, in other words to change the
number of PEs subtended to any P(n). But such a change has a direct
effect on the number of PEs in the network and so the
cost-effectiveness is impacted.
Another concern with the hierarchical approach is that it must be
configured and managed. This may not seem like a large burden, but it
must be recalled that the P(n) nodes are not at the edge of the
network - they are a set of nodes that must be identified so that the
FA LSPs can be configured and provisioned. Effectively, the operator
must plan and construct a layered network with a ring of P(n) nodes
giving access to the level (n) network. This design activity is open
to considerable risk as failing to close the ring (i.e. allowing a
node to be at both level (n+1) and at level (n)) may cause
operational confusion.
Protocol techniques (such as IGP auto-mesh [RFC4972]) have been
developed to reduce the configuration necessary to build this type of
multi-level network. In the case of auto-mesh, the routing protocol
is used to advertise the membership of a 'mesh group', and all
members of the mesh can discover each other and connect with LSP
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tunnels. Thus the P(n) nodes giving access to level (n) can advertise
their existence to each other, and it is not necessary to configure
each with information about all of the others. Although this process
can help to reduce the configuration overhead, it does not eliminate
it as each member of the mesh group must still be planned and
configured for membership.
An important consideration for the use of hierarchical LSPs is how
they can be protected using MPLS Fast Re-Route (FRR) [RFC4090]. FRR
may provide link protection either by protecting the tunnels as they
traverse a broken link, or by treating each level (n) tunnel LSP as a
link in level (n+1), and providing protection for the level (n+1)
LSPs (although in this model fault detection and propagation time may
be an issue). Node protection may be performed in a similar way, but
protection of the first and last nodes of a hierarchical LSP is
particularly difficult. Additionally, the whole notion of scaling
with regard to FRR gives rise to separate concerns that are outside
the scope of this document as currently formulated.
Finally, observe that we have been explaining these techniques using
conveniently symmetrical networks. Consider how we would arrange the
hierarchical LSPs in a network where some PEs are connected closer to
the center of the network than others.
7. Scaling Ladder Networks with Forwarding Adjacencies
7.1 Two Layer Hierarchy
In Section 6.2, we observed that there is no value to placing FA LSPs
between the P(1) nodes of our example snowflake topologies. This is
because those LSPs would be just one hop long and would, in fact,
only serve to increase the burden at the P(1) nodes. However, in the
ladder model, there is value to this approach. The P(1) nodes are the
spar nodes of the ladder, and they are not all mutually adjacent.
That is, the P(1)-to-P(1) hierarchical LSPs can create a full mesh of
P(1) nodes where one does not exist in the physical topology.
The number of LSPs seen by a P(1) node is then:
o all of the tunnels terminating at the P(1) node
o any transit tunnels
o all of the LSPs due to subtended PEs.
This is a substantial reduction because all of the transit LSPs are
reduced to just one per remote P(1) that causes any transit LSP. So:
L(1) = 2*(S(1) - 1) +
O(S(1)*S(1)/2) +
2*E*E*(S(1) - 1) + E*(E-1) - E*(M(2) - 1)
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where O(S(1)*S(1)/2) gives an upper bound order of magnitude. So:
L(1) = S(1)*S(1)/2 + 2*S(1) + 2*E*E*(S(1) - 1) - E*M(2) - 2
So, in our two examples:
With S(1) = 6, M(1) = 10, and M(2) = 17, we see:
E = 170
S(PE) = 1020
L(PE) = 2038
L(2) = 34374
L(1) = 286138
Alternatively, with S(1) = 10, M(1) = 10, and M(2) = 20, we see:
E = 200
S(PE) = 2000
L(PE) = 3998
L(2) = 79580
L(1) = 716060
Both of these show significant improvements over the previous L(1)
figures of 777410 and 2516000. But the numbers are still too large to
manage, and there is no improvement in the L(2) figures.
7.2 Three Layer Hierarchy
We can also apply the three layer hierarchy to the ladder model. In
this case the number of LSPs between P(1) nodes is not reduced, but
tunnels are also set up between all P(2) nodes. Thus the number of
LSPs seen by a P(1) node is:
o all of the tunnels terminating at the P(1) node
o any transit tunnels between P(1) nodes
o all of the LSPs due to subtended P(2) nodes.
No PE-to-PE LSPs are seen at the P(1) nodes.
L(1) = 2*(S(1) - 1) +
O(S(1)*S(1)/2) +
2*(S(1) - 1)*M(1)*M(1) + M(1)*(M(1) - 1)
where O(S(1)*S(1)/2) gives an upper bound order of magnitude. So:
L(1) = S(1)*S(1)/2 + 2*S(1) + 2*M(1)*M(1)*S(1) - M(1)(M(1) + 1) - 2
Unfortunately, there is a small increase in the number of LSPs seen
by the P(2) nodes. Each P(2) now sees all of the PE-to-PE LSPs that
it saw before, and it is also an end point for a set of P(2)-to-P(2)
tunnels. Thus L(2) increases to:
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L(2) = 2*M(2)*(S(PE) - 1) - M(2)*(M(2) - 1) + 2*(S(1)*M(1) - 1)
So, in our two examples:
With S(1) = 6, M(1) = 10, and M(2) = 17, we see:
E = 170
S(PE) = 1020
L(PE) = 2038
L(2) = 34492
L(1) = 1118
Alternatively, with S(1) = 10, M(1) = 10, and M(2) = 20, we see:
E = 200
S(PE) = 2000
L(PE) = 3998
L(2) = 79778
L(1) = 1958
This represents a very dramatic decrease in LSPs across the core.
7.3 Issues with Hierarchical LSPs
The same issues exist for hierarchical LSPs as described in Section
6.4. Although dramatic improvements can be made to the scaling
numbers for the number of LSPs at core nodes, this can only be done
at the cost of configuring P(2) to P(2) tunnels. The mesh of P(1)
tunnels is not enough.
But the sheer number of P(2) to P(2) tunnels that must be configured
is a significant management burden that can only be eased by using a
technique like automesh [RFC4972].
It is significant, however, that the scaling problem at the P(2)
nodes cannot be improved by using tunnels, and the only solution to
ease this in the hierarchical approach would be to institute another
layer of hierarchy (that is, P(3) nodes) between the P(2) nodes and
the PEs. This is, of course, a significant expense.
8. Scaling Improvements Through Multipoint-to-Point LSPs
An alternative (or complementary) scaling technique has been proposed
using multipoint-to-point (MP2P) LSPs. The fundamental improvement in
this case is achieved by reducing the number of LSPs toward the
destination as LSPs toward the same destination are merged.
This section presents an overview of MP2P LSPs and describes their
applicability and scaling benefits.
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8.1 Overview of MP2P LSPs
Note that the MP2P LSPs discussed here are for MPLS-TE and are not
the same concept familiar in the Label Distribution Protocol (LDP)
described in [RFC5036].
Traffic flows generally converge toward their destination and this
can be utilized by MPLS in constructing an MP2P LSP. With such an
LSP, the LFIB mappings at each LSR are many-to-one so that multiple
pairs {incoming interface, incoming label} are mapped to a single
pair {outgoing interface, outgoing label}. Obviously, if per-platform
labels are used, this mapping may be optimized within an
implementation.
It is important to note that with MP2P MPLS-TE LSPs the traffic flows
are merged. That is, some additional form of identifier is required
if demerging is required. For example, if the payload is IP traffic
belonging to the same client network, no additional demerging
information is required since the IP packet contains sufficient data.
On the other hand, if the data comes, for example, from a variety of
VPN client networks, then the flows will need to be labeled in their
own right as point-to-point (P2P) flows, so that traffic can be
disambiguated at the egress of the MP2P LSPs.
Techniques for establishing MP2P MPLS-TE LSPs and for assigning the
correct bandwidth downstream of LSP merge points are out of the scope
of this document.
8.2 LSP State: A Better Measure of Scalability
Consider the network topology shown in figure 3. Suppose that we
establish MP2P LSP tunnels such that there is one tunnel terminating
at each PE, and that that tunnel has every other PE as an ingress.
Thus, a PE-to-PE MP2P LSP tunnel would have S(PE)-1 ingresses and one
egress, and there would be S(PE) such tunnels.
Note that there still remain 2*(S(PE) - 1) PE-to-PE P2P LSPs that are
carried through these tunnels.
Let's consider the number of LSPs handled at each node in the
network.
The PEs continue to handle the same number of PE-to-PE P2P LSPs, and
must also handle the MP2P LSPs. So:
L(PE) = 2*(S(PE) - 1) + S(PE)
But all P(n) nodes in the network only handle the MP2P LSP tunnels.
Nominally, this means that L(n) = S(PE) for all values of n. This
would appear to be a great success with the number of LSPs cut to
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completely manageable levels.
But the number of LSPs is not the only issue (although it may have
some impact for some of the scaling concerns listed in section 4). We
are more interested in the amount of LSP state that is maintained by
an LSR. This reflects the amount of storage required at the LSR, the
amount of protocol processing, and the amount of information that
needs to be managed.
In fact, we were also interested in this measure of scalability in
the earlier sections of this document, but in those cases we could
see a direct correlation between the number of LSPs and the amount of
LSP state since transit LSPs had two pieces of state information (one
on the incoming and one on the outgoing interface), and ingress or
egress LSPs had just one piece of state.
We can quantify the amount of LSP state according to the number of
LSP segments managed by an LSR. So (as above), in the case of a P2P
LSP, an ingress or egress has one segment to maintain, while a
transit has two segments. Similarly, for an MP2P LSP, an LSR must
maintain one set of state information for each upstream segment
(which, we can assume, is in a one-to-one relationship with the
number of upstream neighbors), and exactly one downstream segment -
ingresses obviously have no upstream neighbors, and egresses have no
downstream segments.
So we can start again on our examination of the scaling properties of
MP2P LSPs using X(n) to represent the amount of LSP state held at
each P(n) node.
8.3 Scaling Improvements for Snowflake Networks
At the PEs, there is only connectivity to one other network node, the
P(2) node. But note that if P2P LSPs need to be used to allow
disambiguation of data at the MP2P LSP egresses, then these P2P LSPs
are tunneled within the MP2P LSPs . So X(PE) is:
X(PE) = 2*(S(PE) - 1) if no disambiguation is required
and
X(PE) = 4*(S(PE) - 1) if disambiguation is required
Each P(2) node has M(2) downstream PEs. The P(2) sees a single MP2P
LSP targeted at each downstream PE with one downstream segment (to
that PE) and M(2) - 1 upstream segments from the other subtended PEs.
Additionally, each of these LSPs has an upstream segment from the one
upstream P(1). This gives a total of M(2)*(1 + M(2)) LSP segments.
There are also LSPs running from the subtended PEs to each other PE
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in the network. There are S(PE) - M(2) such PEs, and the P(2) sees
one upstream segment for each of these from each subtended PE. It
also has one downstream segment for each of these LSPs. This gives
(M(2) + 1)*(S(PE) - M(2)) LSP segments.
Thus:
X(2) = M(2)*(1 + M(2)) + (M(2) + 1)*(S(PE) - M(2))
= S(PE)*(M(2) + 1)
Similarly, at each P(1) node there are M(1) downstream P(2) nodes and
so a total of M(1)*M(2) downstream PEs. Each P(1) is connected in a
full mesh with the other P(1) nodes so has (S(1) - 1) neighbors.
The P(1) sees a single MP2P LSP targeted at each downstream PE. This
has one downstream segment (to the P(2) to which the PE is connected)
and M(1) - 1 upstream segments from the other subtended P(2) nodes.
Additionally, each of these LSPs has an upstream segment from each of
the P(1) neighbors. This gives a total number of LSP segments of
M(1)*M(2)*(M(1) + S(1) - 1).
There are also LSPs running from each of the subtended PEs to each
other PE in the network. There are S(PE) - M(1)M(2) such PEs, and the
P(1) sees one upstream segment for each of these from each subtended
P(2) (since the aggregation form the subtended PEs has already
happened at the P(2) nodes). It also has one downstream segment to
appropriate next hop P(1) neighbor for each of these LSPs. This gives
(M(1) + 1)*(S(PE) - M(1)*M(2)) LSP segments.
Thus:
X(1) = M(1)*M(2)*(M(1) + S(1) - 1) +
(M(1) + 1)*(S(PE) - M(1)*M(2))
= M(1)*M(2)*(S(1) - 2) + S(PE)*(M(1) + 1)
So, for example, with S(1) = 10, M(1) = 10, and M(2) = 10, we see:
S(PE) = 1000
S(2) = 100
X(PE) = 3996 (or 1998)
X(2) = 11000
X(1) = 11800
And similarly, with S(1) = 20, M(1) = 20, and M(2) = 5, we see:
S(PE) = 2000
S(2) = 400
X(PE) = 5996 (or 2998)
X(2) = 12000
X(1) = 39800
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8.3.1 Comparison with Other Scenarios
For comparison with the examples in sections 5 and 6, we need to
convert those LSP-based figures to our new measure of LSP state.
Observe that each LSP in sections 5 and 6 generates two state units
at a transit LSR and one at an ingress or egress. So we can provide
conversions as follows:
Section 5 (flat snowflake network)
L(PE) = 2*(S(PE) - 1)
L(2) = M(2)*(2*S(PE) - M(2) - 1)
L(1) = M(1)*M(2)*(2*S(PE) - M(2)*(M(1) + 1))
X(PE) = 2*(S(PE) - 1)
X(2) = 2*M(2)*(2*S(PE) - M(2) - 1)
X(1) = 2*M(1)*M(2)*(2*S(PE) - M(2)*(M(1) + 1))
For the example with S(1) = 10, M(1) = 10, and M(2) = 10, this
gives a comparison table as follows:
Count | Unmodified | MP2P
------+-------------+----------
X(PE) | 1998 | 3996
X(2) | 39780 | 11000
X(1) | 378000 | 11800
Clearly this technique is a significant improvement over the flat
network within the core of the network, although the PEs are more
heavily stressed if disambiguation is required.
Section 6.1 (Two-layer hierarchy snowflake network)
L(PE) = 2*(S(PE) - 1)
L(2) = M(2)*(2*S(PE) - M(2) - 1) + 2*(S(2) - 1)
L(1) = M(1)*(2*S(2) - M(1) - 1)
X(PE) = 2*(S(PE) - 1)
X(2) = 2*M(2)*(2*S(PE) - M(2) - 1) + 2*(S(2) - 1)
X(1) = 2*M(1)*(2*S(2) - M(1) - 1)
Note that in the computation of X(2) the hierarchical LSPs only add
one state at each P(2) node.
For the same example with S(1) = 10, M(1) = 10, and M(2) = 10, this
gives a comparison table as follows:
Count | 2-Layer | MP2P
| Hierarchy |
------+-------------+----------
X(PE) | 1998 | 3996
X(2) | 39978 | 11000
X(1) | 3780 | 11800
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And we can observe that the MP2P model is better at P(2), but the
hierarchical model is better at P(1).
In fact, this comparison can be generalized to observe that the MP2P
model produces best effects toward the edge of the network, while the
hierarchical model makes most impression at the core. However, the
requirement for disambiguation of P2P LSPs tunneled within the MP2P
LSPs does cause a double burden at the PEs.
8.4 Scaling Improvements for Ladder Networks
MP2P LSPs applied just within the ladder will not make a significant
difference, but applying MP2P for all LSPs and at all nodes makes a
very big difference without requiring any further configuration.
LSP state at a spar node may be divided into those LSPs segments that
enter or leave the spar node due to subtended PEs (local LSP
segments), and those that enter or leave the spar node due to remote
PEs (remote segments).
The local segments may be counted as:
o E LSPs targeting local PEs
o (S(1)-1)*E*M(1) LSPs targeting remote PEs
The remote segments may be counted as:
o (S(1)-1)*E outgoing LSPs targeting remote PEs
o <= 3*S(1)*E incoming LSPs targeting any PE (there are precisely
P(1) nodes attached to any other P(1) node).
Hence, using X(1) as a measure of LSP state rather than a count of
LSPs, we get:
X(1) <= E + (S(1)-1)*E*M(1) + (S(1)-1)*E + 3*S(1)*E
<= (4 + M(1))*S(1)*E - M(1)*E
The number of LSPs at the P(2) nodes is also improved. We may also
count the LSP state in the same way so that there are:
o M(2) LSPs targeting local PEs
o M(2)*(S(1)*E) LSPs from local PEs to all other PEs
o S(1)*E - M(2) LSPs to remote PEs.
So using X(2) as a measure of LSP state and not a count of LSPs, we
have:
X(2) = M(2) + M(2)*(S(1)*E) + S(1)*E - M(2)
= (M(2) + 1)*S(1)*E
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Our examples from Section 5.2 give us the following numbers:
With S(1) = 6, M(1) = 10, and M(2) = 17, we see:
E = 170
S(PE) = 1020
X(PE) = 2038
X(2) = 18360
X(1) <= 12580
Alternatively, with S(1) = 10, M(1) = 10, and M(2) = 20, we see:
E = 200
S(PE) = 2000
X(PE) = 3998
X(2) = 42000
X(1) <= 26000
8.4.1 Comparison with Other Scenarios
The use of MP2P compares very favorably with all scaling scenarios.
It is the only technique able to reduce the value of X(2), and it
does this by a factor of almost two. The impact on X(1) is better
than everything except the three level hierarchy.
The following table provides a quick cross-reference for the figures
for the example ladder networks. Note that the previous figures are
modified to provide counts of LSP state rather than LSP numbers.
Again, each LSP contributes one state at its end points, and two
states at transit nodes.
Thus, for the all cases we have
X(PE) = 2*(S(PE) - 1) or
X(PE) = 4*(S(PE) - 1) if disambiguation is required.
In the unmodified (flat) case, we have:
X(2) = 2*(M(2)*(2*S(PE) - M(2) - 1))
X(1) = 2*(M(1)*M(2)*(2*S(PE) - M(2)*(M(1) + 1)))
In the 2-level hierarchy, we have:
X(2) = 2*(2*M(2)*(S(PE) - 1) - M(2)*(M(2) - 1))
X(1) = S(1)*S(1) + 2*S(1) + 4*E*E*(S(1) - 1) - 2*E*M(2) - 2
In the 3-level hierarchy, we have:
X(2) = 2*(2*M(2)*(S(PE) - 1) - M(2)*(M(2) - 1)) + 2*(S(1)*M(1) - 1)
X(1) = S(1)*S(1) + 2*S(1) + 4*M(1)*M(1)*S(1) - 2*M(1)(M(1) + 1) - 2
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Example A: S(1) = 6, M(1) = 10, and M(2) = 17
Example B: S(1) = 10, M(1) = 10, and M(2) = 20
Example| Count | Unmodified | 2-Level | 3-Level | MP2P
| | | Hierarchy | Hierarchy |
-------+-------+------------+------------+-------------+-------
A | X(2) | 68748 | 68748 | 68866 | 18360
| X(1) | 1554820 | 572266 | 2226 | 12580
-------+-------+------------+------------+-------------+-------
B | X(2) | 159160 | 159160 | 159358 | 42000
| X(1) | 5032000 | 1433998 | 3898 | 26000
8.4.2 LSP State Compared with LSP Numbers
Recall that in Section 8.3, the true benefit of MP2P was analyzed
with respect to the LSP segment state required, rather than the
actual number of LSPs. This proved to be a more accurate comparison
of the techniques because the MP2P LSPs require state on each branch
of the LSP so the saving is not linear with the reduced number of
LSPs.
A similar analysis could be performed here for the ladder network.
The net effect is that it increases the state by an order of two for
all transit LSPs in the P2P models, and by a multiplier equal to the
degree of a node in the MP2P model.
A rough estimate shows that, as with snowflake networks, MP2P
provides better scaling than the 1-level hierarchical model, and is
considerably better at the core. But MP2P compares less will with the
2-level hierarchy especially in the core.
8.5 Issues with MP2P LSPs
The biggest challenges for MP2P LSPs are the provision of support in
the control and data planes. To some extent support must also be
provided in the management plane.
Control plane support is just a matter of defining the protocols and
procedures [MP2P-RSVP], although it must be clearly understood that
this will introduce some complexity to the control plane.
Hardware issues may be a little more tricky. For example, the
capacity of the upstream segments must never (allowing for
statistical over-subscription) exceed the capacity of the downstream
segment. Similarly, data planes must be equipped with sufficient
buffers to handle incoming packet collisions.
The management plane will be impacted in several ways. Firstly the
management applications will need to handle LSPs with multiple
senders. This means that, although the applications need to process
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fewer LSPs, they will be more complicated and will, in fact, need to
process the same number of ingresses and egresses. Other issues like
diagnostics and OAM would also need to be enhanced to support MP2P,
but might be borrowed heavily from LDP networks.
Lastly, note that when the MP2P solution is used, the receiver (the
single egress PE of an MP2P tunnel) cannot use the incoming label as
an indicator of the source of the data. Contrast this with P2P LSPs.
Depending on deployment, this might not be an issue since the PE-PE
connectivity may in any case be a tunnel with inner labels to
discriminate the data flows.
In other deployments, it may be considered necessary to include
additional PE-PE P2P LSPs and tunnel these through the MP2P LSPs.
This would require the PEs to support twice as many LSPs. Since PEs
are not usually as fully specified as P-routers, this may cause some
concern although the use of penultimate hop popping on the MP2P LSPs
might help to reduce this issue.
In all cases, care must be taken not to confuse the reduction in the
number of LSPs with a reduction in the LSP state that is required. In
fact, the discussion in Section 8.3 is slightly optimistic since
LSP state toward the destination will probably need to include
sender information and so will increase depending on the number of
senders for the MP2P LSP. Section 8.4, on the other hand, counts LSP
state rather than LSPs. This issue is clearly dependent on the
protocol solution for MP2P RSVP-TE which is out of scope for this
document.
MPLS Fast Re-route (FRR) [RFC4090] is an attractive scheme for
providing rapid local protection from node or link failures. Such a
scheme has, however, not been designed for MP2P at the time of
writing, so it remains to be seen how practical it would be,
especially in the case of the failure of a merge node. Initial
examination of this case suggests that FRR would not be a problem
for MP2P given that each flow can be handled separately.
As a final note, observe that the MP2P scenario presented in this
document may be optimistic. MP2P LSP merging may be hard to achieve
between LSPs with significantly different traffic and QoS parameters.
Therefore, it may be necessary to increase the number of MP2P LSPs
arriving at an egress.
9. Combined Models
There is nothing to prevent the combination of hierarchical and MP2P
solutions within a network.
Note that if MP2P LSPs are tunneled through P2P FA LSPs across the
core, none of the benefit of LSP merging is seen for the hops during
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which the MP2P LSPs are tunneled.
On the other hand, it is possible to construct solutions where MP2P
FA LSPs are constructed within the network resulting in savings from
both modes of operation.
10. An Alternate Solution
A simple solution to reducing the number of LSP tunnels handled by
any node in the network has been proposed. In this solution it is
observed that part of the problem is caused purely by the total
number of LSP in the network, and that this is a function of the
number of PEs since a full mesh of PE-PE LSPs is required. The
conclusion of this observation is to move the tunnel end-points
further into the network so that, instead of having a full mesh of
PE-PE tunnels, we have only a full mesh of P(n)-P(n) tunnels.
Obviously, there is no change in the physical network topology, so
the PEs remain subtended to the P(n) nodes, and the consequence is
that there is no TE on the links between PEs and P(n) nodes.
In this case we have already done the hard work for computing the
number of LSP in the previous sections. The power of the analysis in
the earlier sections is demonstrated by its applicability to this new
model - all we need to do is make minor changes to the formulae. This
is most simply done by removing a layer from the network. We
introduce the term "tunnel end-point" (TEP), and replace the P(n)
nodes with TEPs. Thus, the example of a flat snowflake network in
Figure 3 becomes as shown in Figure 7. Corresponding changes can be
made to all of the sample topologies.
PE PE PE PE PE PE
\ \/ \/ /
PE--TEP TEP TEP TEP--PE
\ | | /
\| |/
PE--TEP---P(1)------P(1)---TEP--PE
/ \ / \
PE \ / PE
\/
P(1)
/|\
/ | \
/ | \
PE--TEP TEP TEP--PE
/ /\ \
PE PE PE PE
Figure 7 : An Example Snowflake Network With Tunnel End-Points
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To perform the scaling calculations we need only replace the PE
counts in the formulae with TEP counts, and observe that there is
one fewer layer in the network. For example, in the flat snowflake
network shown in Figure 7 we can see that the number of LSPs seen
at a TEP is:
L(TEP) = 2*(S(TPE) - 1)
In our sample networks S(TPE) is typically of the order of 50 or 100
(the original values of S(2)), so L(TEP) is less than 200 which is
quite manageable.
Similarly, the number of LSPs handled by a P(1) node can be derived
from the original formula for the number of LSPs seen at a P(2) node
since all we have done is reduce n in P(n) from 2 to 1. So our new
formula is:
L(1) = M(1)*(2*S(TEP) - M(1) - 1)
With figures for M(1) = 10 and S(TEP) = 100, this gives us
L(1) = 1890. This is also very manageable.
10.1 Pros and Cons of the Alternate Solution
On the face of it, this alternate solution seems very attractive.
Simply by contracting the edges of the tunnels into the network, we
have shown a dramatic reduction in the number of tunnels needed, and
there is no requirement to apply any additional scaling techniques.
But what of the PE-P(n) links? In the earlier sections of this
document we have assumed that there was some requirement for PE-PE
LSPs with TE properties that extended to the PE-P(n) links at both
ends of each LSP. That means that there was a requirement to provide
reservation-based QoS on those links, to be able to discriminate
traffic flows for priority-based treatment, and to be able to
distinguish applications and sources that send data based on the LSPs
that carry the data.
It might be argued that, since the PE-P(n) links do not offer any
routing options (each such link provides the only access to the
network for a PE), most of the benefits of tunnels are lost on these
peripheral links. However, TE is not just about routing. Just as
important are the abilities to make resource reservations, to
prioritize traffic, and to discriminate between traffic from
different applications, customers, or VPNs.
Furthermore, in multihoming scenarios where each PE is connected to
more than one P(n) or where a PE has multiple links to a single P(n),
there may be a desire to preselect the link to be used and to direct
the traffic to that link using a PE-PE LSP. Note that multihoming
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has not been considered in this document.
Operationally, P(n)-P(n) LSPs offer the additional management
overhead that is seen for hierarchical LSPs described in Section 6.
That is, the LSPs have to be configured and established through
additional configuration or management operations that are not
carried out at the PEs. As described in Section 6, automesh [RFC4972]
could be used to ease this task. But it must be noted that, as
mentioned above, some of the key uses of tunnels require that traffic
is classified and placed in an appropriate tunnel according to its
traffic class, end-points, originating application, and customer
(such as client VPN). This information may not be readily available
for each packet at the P(n) nodes since it is PE-based information.
Of course, it is possible to conceive of techniques to make this
information available such as assigning a different label for each
class of traffic, but this gives rise to the original problems of
larger numbers of LSPs.
Our conclusion is, therefore, that this alternate technique may be
suitable for the general distribution of traffic based solely on the
destination, or on a combination of the destination and key fields
carried in the IP header. In this case it can provide a very
satisfactory answer to the scaling issues in an MPLS-TE network. But
if more sophisticated packet classification and discrimination is
required, this technique will make the desired function hard to
achieve, and the trade-off between scaling and feature-level will
swing too far towards solving the scaling issue at the expense of
delivery of function to the customer.
11. Management Considerations
The management issues of the models presented in this document
have been discussed in line. No one solution is without its
management overhead.
Note, however, that scalability of management tools is one of the
motivators for this work and that network scaling solutions that
reduce the active management of LSPs at the cost of additional effort
to manage the more static elements of the network represent a
benefit. That is, it is worth the additional effort to set up MP2P or
FA LSPs if it means that the network can be scaled to a larger size
without being constrained by the management tools.
The MP2P technique may prove harder to debug through OAM methods than
the FA LSP approach.
12. Security Considerations
The techniques described in this document use existing or
yet-to-be-defined signaling protocol extensions and are subject to
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draft-ietf-mpls-te-scaling-analysis-05.txt December 2008
the security provided by those extensions. Note that we are talking
about tunneling techniques used within the network and both
approaches are vulnerable to the creation of bogus tunnels that
deliver data to an egress or consume network resources.
The fact that the MP2P technique may prove harder to debug through
OAM methods than the FA LSP approach is a security concern since it
is important to be able to detect misconnections.
General issues of the realtionship between scaling and security are
covered in Section 1.1, but the details are beyond the scope of this
document. Readers are refered to [MPLS-SEC] for details of MPLS
security techniques.
13. Recommendations
The analysis in this document suggests that the ability to signal
MP2P MPLS-TE LSPs is a desirable addition to the operator's MPLS-TE
toolkit.
At this stage, no further recommendations are made, but it would be
valuable to consult more widely to discover:
- The concerns of other service providers with respect to network
scalability.
- More opinions on the realistic constraints to the network
parameters listed in Section 4.
- Desirable values for the cost-effectiveness of the network
(parameter K)
- The applicability, manageability and support for the two techniques
described
- The feasibility of combining the two techniques as discussed in
Section 9.
- The level of concern over the loss of functionality that would
occur if the alternate solution described in Section 10 was
adopted.
14. IANA Considerations
This document makes no requests for IANA action.
15. Acknowledgements
The authors are grateful to Jean-Louis Le Roux for discussions and
review input. Thanks to Ben Niven-Jenkins, JP Vasseur, Loa Andersson,
Yasukawa, Farrel, and Komolafe [Page 38]
draft-ietf-mpls-te-scaling-analysis-05.txt December 2008
Anders Gavler, and Tom Polk for their comments. Thanks to Dave Allen
for useful discussion of the math.
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For the avoidance of doubt, each Contributor to the IETF Standards
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Yasukawa, Farrel, and Komolafe [Page 39]
draft-ietf-mpls-te-scaling-analysis-05.txt December 2008
17. Normative References
[RFC4206] Kompella, K. and Y. Rekhter, "Label Switched Paths (LSP)
Hierarchy with Generalized Multi-Protocol Label
Switching (GMPLS) Traffic Engineering (TE)", RFC 4206,
October 2005.
18. Informative References
[RFC2961] Berger, L., Gan, D., Swallow, G., Pan, P., Tommasi, F.,
and S. Molendini, "RSVP Refresh Overhead Reduction
Extensions", RFC 2961, April 2001.
[RFC3209] Awduche, D., Berger, L., Gan, D., Li, T., Srinivasan, V.,
and G. Swallow, "RSVP-TE: Extensions to RSVP for LSP
Tunnels", RFC 3209, December 2001.
[RFC3270] Le Faucheur, F., et al., "Multi-Protocol Label Switching
(MPLS) Support of Differentiated Services", RFC 3270, May
2002.
[RFC3473] Berger, L., et al., Editor, "Generalized Multi-Protocol
Label Switching (GMPLS) Signaling Resource ReserVation
Protocol-Traffic Engineering (RSVP-TE) Extensions",
RFC 3473, January 2003.
[RFC3985] Bryant, S., and Pate, P., "Pseudo Wire Emulation Edge-to-
Edge (PWE3) Architecture", RFC 3985, March 2005.
[RFC4090] P. Pan, G. Swallow, A. Atlas (Editors), "Fast Reroute
Extensions to RSVP-TE for LSP Tunnels", RFC 4090, May
2005.
[RFC4110] Callon, R., and Suzuki, M., "A Framework for Layer 3
Provider-Provisioned Virtual Private Networks (PPVPNs)",
RFC 4110, July 2005.
[RFC4972] Vasseur, JP., and Le Roux, JL., "Routing Extensions for
Discovery of Multiprotocol (MPLS) Label Switch Router
(LSR) Traffic Engineering (TE) Mesh Membership",
RFC 4972, July 2007.
[RFC5036] Andersson, L., Doolan, P., Feldman, N., Fredette, A., and
B. Thomas, "LDP Specification", RFC 5036, October 2007.
[MP2P-RSVP] Yasukawa, Y., "Supporting Multipoint-to-Point Label
Switched Paths in Multiprotocol Label Switching Traffic
Engineering", draft-yasukawa-mpls-mp2p-rsvpte, work in
progress.
Yasukawa, Farrel, and Komolafe [Page 40]
draft-ietf-mpls-te-scaling-analysis-05.txt December 2008
[MPLS-SEC] Fang, L. (Ed.), "Security Framework for MPLS and GMPLS
Networks", draft-ietf-mpls-mpls-and-gmpls-security-
framework, work in progress.
19. Authors' Addresses
Seisho Yasukawa
NTT Corporation
9-11, Midori-Cho 3-Chome
Musashino-Shi, Tokyo 180-8585 Japan
Phone: +81 422 59 4769
EMail: s.yasukawa@hco.ntt.co.jp
Adrian Farrel
Old Dog Consulting
EMail: adrian@olddog.co.uk
Olufemi Komolafe
Cisco Systems
96 Commercial Street
Edinburgh
EH6 6LX
United Kingdom
EMail: femi@cisco.com
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Yasukawa, Farrel, and Komolafe [Page 41]