Transcript Capacity Assignment in Bluetooth Scatternets – Analytical

The Power of Tuning: A Novel Approach for
the Efficient Design of Survivable Networks
Ron Banner and Ariel Orda
Department of Electrical Engineering
Technion- Israel Institute of Technology
Introduction


Transmission rates have increased to 10 Gbit/s and
beyond.
Any failure may lead to a vast amount of data loss.
 Survivability
= the capability of the network to
maintain service continuity in the presence of
failures.
Current Survivability Schemes

Based on securing an independent resource for each
network element.

This is translated into the establishment of pairs of
disjoint paths.

Given a pair of disjoint paths, either 1+1 or 1:1 protection
can be employed
 With 1+1 protection, identical traffic is transmitted over a pair of
disjoint paths.
 With 1:1 protection, traffic is sent only on one (active) path.
 The other (backup) path is activated only upon a failure on the active
path.
Pros and cons of current survivability schemes
 Pro:
pairs of disjoint paths provide full (100%)
protection to single network failures.
 Cons:
in practice, too restrictive and requires
excessive redundancy.
 Often, very limiting (G. Maier, A. Pattavina, S. De Patre and M.
Martinelli, 2002).
 Sometimes, even infeasible (N. Taft-Plotkin, B. Bellur and R.
Ogier, 1999).
Tunable Survivability

Since full survivability is too restrictive, we propose
tunable survivability.

Tunable survivability allows any desired degree of
survivability in the range 0% to 100%.
 Provides a quantitative measure to specify the desired level of
survivability.
 Can substantially increase the space of feasible solutions.
 Enables to consider & quantify valuable tradeoffs (e.g.,
survivability vs. delay, survivability vs. jitter, survivability vs.
bandwidth…)
Survivable connections

We adopt the widely used single link failure model.
 Has been the focus of most studies on survivability.

Tunable survivability enables the establishment of psurvivable connections.
a
s
p=0.001
b
d
p=0.001
c
(0.999)23-survivable connection
(0.999) (0.999)
-survivable
connection
0.999survivable
connection
2 ≈ 0.998
(0.999)3 ≈ 0.997
p=0.001
t
Survivable connections (cont.)
 Connections
with tunable survivability can also
employ either 1+1 or 1:1 protection architecture.
 However,
the maximal traffic rate (bandwidth)
of a survivable connection with 1+1 protection
may be different than that with 1:1 protection.
 Indeed, for any given survivable connection, the flow
configuration induced by 1+1 protection is different than that
induced by 1:1 protection.
How much is gained by employing tunable survivability?

Through comprehensive simulations on random Internet networks
we demonstrate the major power of Tunable survivability.
 By
just slightly alleviating the requirement of
full survivability, major increase in bandwidth as
well as in feasibility is accomplished.
 Details
Analytical results

Motivated by the simulation results we investigated the
tunable survivability concept.

Established several fundamental properties of
survivable connections.

Designed polynomial (optimal) algorithmic schemes for
the establishment of survivable connections for 1:1 and
1+1 protection.

Derived for the tunable survivability approach a new
“hybrid” protection architecture that has several
advantages over both 1:1 and 1+1 protection.
Property: two paths are ENOUGH!
Claim:
Given a survivable connection
that admits more than two paths, it
is possible to obtain the same level of
survivability with only two paths.
Two paths are ENOUGH! (cont.)

Proof (sketch):
common (critical) links

Under the single link failure model, a failure in a link that is NOT
common to all paths can never fail a survivable connection.
 Hence, the probability to survive a single failure is equal to the probability
that all common links are operational.

It is possible to construct a pair of paths that intersect only
on the common links.
Types of survivable connections

Most Survivable connection= A connection that has the
maximum probability to survive a single failure.

Most survivable connection with a bandwidth of at least B=
A survivable connection that among all connections with a
bandwidth of at least B, has the maximum probability to
survive a single failure.

Widest p-survivable connection= A p-survivable connection
that has the maximum bandwidth.
Polynomial optimal algorithms for survivable connection



For each type of survivable connection, we designed a
polynomial optimal algorithm (both for 1+1 and 1:1
protection).
Most-survivable-connection-with-a-bandwidth-of-at-least-B
is established by a novel reduction to the min-cost flow
problem.
This reduction constitutes an algorithmic building block for
the establishment of the most survivable connection and the
widest p-survivable connection.
 Indeed, most survivable connection with a bandwidth of at least B=0
is a most survivable connection per se.
Polynomial optimal algorithms (cont.)


How to establish a widest p-survivable connection?
Idea: search for the largest B such that the most-survivableconnection-with-a-bandwidth-of-at-least-B is a p-survivable
connection.

We show that it is sufficient to perform a binary search over
the set
 ce

 e  E , k  1,2 .
k


Therefore, the widest p-survivable connection is established
within O(logN) executions of any min-cost flow algorithm.
 Indeed, the above set contains 2·M elements. Therefore, a binary search over this
set enables to consider O(log2·M)=O(logN) candidates.
Hybrid Protection
Up to now, only focused on 1:1 and 1+1 protection architectures.
 Tunable survivability gives rise to a third protection architecture
that combines 1:1 and 1+1 protection.

p1
e1
s
e4
u
e2
e3
t
v
e5
p2

Advantages
 Propagates data over minimum-latency paths.
 Produces better congestion level (over the common links) than 1+1
protection.
 Has better recovery time from a failure than 1:1 protection
 For 1:1, signaling is required to perform the switch-over operation.
Hybrid Protection (cont.)
e1
s
u
e2

e4
e3
t
v
e5
Disadvantage
 Requires additional nodal capabilities.

As with 1+1 and 1:1 protection, designed for hybrid protection
optimal algorithms that establish survivable connections in a
polynomial running time.
Extensions

Additive QoS Extensions
 In many cases it is important to consider additive metrics as quality
criteria for survivable connections.
 Additive metrics: delay, jitter, cost…
 Fortunately, all the algorithms can be modified to consider additive
metrics while still admitting a polynomial running time.

Beyond the single link failure model
 Establishment of p-survivable connections is an NP-hard problem.
 Yet, we introduce an alternative survivability criterion for multiple
failures that admits optimal (polynomial) solutions.
 Approximation algorithms – good direction for future research.
Thank you!
How much is gained by employing tunable survivability?

Experiment: Generated random networks that include 10,000
Waxman topologies & 10,000 Power-law topologies.
Bandwidth Ratio r(p)= the ratio between the maximum bandwidth
of a p-survivable connection and the maximum bandwidth of a 1survivable connection.
2.4
Bandwidth Ratio r(p)

2.2
2
1.8
1.6
Waxman networks
1.4
1.2
1
0.8
99.2
99.4
99.6
99.8
100
level of survivability p
How much is gained? (cont.)
Bandwidth Ratio
r(p)
1+1 Protection Architecture
1.6
Power-law networks
1.4
Waxman networks
1.2
1
0.8
99.2
99.4
99.6
99.8
level of survivability p
100
How much is gained? (cont.)
Feasibility Ratio f(p)= the ratio between the number of networks
that have at least one p-survivable connections and the number of
networks that have at least one connection with full survivability.
f(p)
3
Feasibility Ratio

2.8
2.6
2.4
2.2
2
1.8
1.6
Waxman networks
1.4
1.2
1
99.5
99.6
99.7
99.8
99.9
level of survivability p
100
Establishing the widest p-survivable connection

c
Why is it enough to perform the search over the set  e e  E , k  1,2 ?
k


If one path admits a link e then the bandwidth of the connection is at
most ce.
 If both paths admit a link e then the bandwidth of the connection is
at most ce .
2
 Hence, by definition, there exists at least one tight link eE such
c
2
that the bandwidth of the connection is either cee or

.
Why O(logN) executions of a min cost flow algorithm ?
 The set contains 2·M elements.
 A binary search over the set enables to consider O(log2·M)=O(logN) values.
Waxman and Power-law topologies

10,000 Waxman networks:
 Source and destination are located at the diagonally opposite corner
of a square area of unit dimension.
 198 nodes are uniformly spread over the square.
 A link between two nodes u,v exists with the following probability,
which depends on the distance between them δ(u,v):
  ( u, v ) 
p ( u, v )    exp 



2


where α=1.8, β=0.05.

10,000 Power-law networks:
 We assigned a number of out-degree credits to each node, using the
power-law distribution β∙x-α where α=0.75 and β=0.05.
 Then, we connected the nodes so that every node obtained the
assigned out-degree.
Property: Only the Common Links Count

Under the single link failure model, only the links that are common
to all paths can affect a survivable connection.
common link

Therefore, the probability that a survivable connection remains
operational upon a failure is equal to the probability that all its
common links are operational upon that failure.

Hence, (p1,p2) is a most survivable connection if it maximizes
 ( 1  p ).
e p1  p1
e
Most Survivable Connections with a Bandwidth of at Least B

Established by reduction to the min cost flow problem.
Links in the transformed network
Discard the link
A link in the original network
ce,pe
B≤Ce<2∙B
ce=B, we=0
ce=B, we=0
ce=B, we=-ln(1-pe)
 The flow demand is set to 2∙B flow units.
 Since both the flow demand and the capacities are B-integral, the
resulting flow is B-integral.
 Hence, the flow decomposition algorithm can construct a pair of paths
each with a bandwidth B.
Most Survivability with a Bandwidth of at Least B (cont.)
Links in the transformed network
Discard the link
A link in the original network
ce,pe
B≤Ce<2∙B
ce=B, we=0
ce=B, we=0
ce=B, we=-ln(1-pe)

A min cost flow maximizes the success probability of the common
links.
 Only the common links incurs a non-zero cost of -B∙ln(1-pe).
 Hence, a min cost flow minimizes 
 hence, it maximizes
 (1  p ).
e p1  p1
e

e p1  p1
B  ln (1  pe )   B  ln
 (1  p ).
e p1  p1
e
Establishing Survivable Connections for 1:1 protection

The only difference in the reduction lies for the links that have
capacities in the range [B,2B].

For 1:1 protection, only one of the paths carries B flow units.

Hence, all links that have a capacity in the range [B,2B] can be
employed by both paths concurrently.
Links in the transformed network
A link in the original network
Discard the link
ce,pe
ce=B, we=0
ce=B, we=-ln(1-pe)
Go to 1+1
reduction