Flows in networks understanding how to manage them even when their definition is vague

1
Flows in networks understanding how to manage
them even when their definition is vague

• Ron Addie and Oleksiy Yevdokimov

2
Key Contributions

• By looking at flow states, we have found that
  identifying and measuring every flow is not
  necessary for managing a network.
• A good definition for how big has been found
  for flows and flow aggregates, and shown to be a
  semi-norm.
• A simple connection with an important real-world
  problem guides this work throughout.

3
What is a flow?

• A flow has a path
• At each node on the path, we record the bytes
  transferred to this node up to time t
• Ff,n(t) denotes the cumulative bytes of flowf
  at time t atnode n.

4
Problems with flows

• Some user actions generate a collection of
  closely related flows. Are these all one flow?
• The start and end of flows can easily be missed.
• Networks containmany more smallflows than
  largeones do we need to attendto every one?

• We are going to bypass these problems rather than
  solve them.
• A topology defines proximity (nearness) of
  flow states.
• If measurements, performance and control actions
  are well-defined and continuous on the space of
  flow states, our bypass is successful.

5
Virtual Buffers and Token Buckets

• A virtual buffer is a software object which
  simulates a real buffer.
• A virtual buffer can have a different service
  rate and capacity from the actual link, and can
  treat traffic from selected paths.
• A token bucket has a drip rate, of fresh tokens,
  and drops, marks, or queues packets when the
  bucket is empty.
• Token buckets and virtual buffers are equivalent

• Routers contain thousands of token buckets and
  buffers.

6
Aside how mathematics solves real world problems

• In the great applications of mathematics, masses
  of complicated details are reduced to elegant
  concepts.
• Hilbert and Banach spaces have solved problems in
  signal processing and statistical estimation for
  decades.
• Here we use topological vector spaces for
  problems in the performance of networks.

• Ultimately, solutions are obtained using linear
  algebra.

7
Flow States

• A flow state, F, is a collection of flows.
• Although we expect flows to be positive, we
  include negative ones (where the cumulative bytes
  functions decrease), so the space is
  algebraically complete.
• The collection of flow states is a vector space.

8
Semi-norms on the vector space of flow states

• The level of a virtual buffer, or a token bucket,
  at a certain moment, is a positive valued
  function on the space of flow states
• But it is not a semi-norm.
• Also, how can we define the a token bucket level
  on negative flows?
• Answer we replace a flow by its absolute value
  before putting it through the buffer

• A semi-norm must be homogenous satisfy the
  triangle inequality

9
Semi-norms on the vector space of flow states

• We can define a semi-norm, AB() on the space of
  flow states associated with a virtual buffer, B,
  and a time, t, as follows
• AB(F) infagt0 F/a B doesnt overflow at t
• This is a standard technique from Locally Convex
  Topological Vector Spaces.

What scaling wouldcause this flowstate to just
fill thebuffer?
10
Properties of a semi-norm

• These properties need to be verified
• a v av for positive a
• This is obvious for the vb semi-norms.
• The triangle inequality
• a b lt a b .
• Not obvious requires proof.
• Proof the set of flow states which pass through
  a virtual buffer without overflow is convex.

• A semi-norm should measure how big elements
  are.

11
Topology of Flow States

• We can define a topology by means of a collection
  of semi-norms.
• The sub-basic open sets are
• F F-Yk lt a, agt0, Y a flow state, .k a
  semi-norm.
• The basic open sets are finite intersections of
  these.
• The open sets are arbitrary unions of these.

• This is standard procedure for topological
  spaces.

12
Uses of Topology

• We can now define
• The Borel sets on the space the smallest sigma
  algebra containing the open sets
• Measurable functions the functions which can be
  defined by virtual buffer type measurements.
  These are the practical functions.
• Continuity of measurements, performance, and of
  control actions can now be defined.
• Probability measures on flow state space!

• Here we benefit from well developed mathematics.

13
Variations different topologies for different
purposes

• Each potential buffer, sampled at a moment in
  time, gives rise to a semi-norm.
• The maximum, over all time, of all these sampled
  norms, is also a semi-norm.
• The maximum, over the last busy period.
• As a principle, its best to use the smallest,
  simplest topology the smallest collection of
  measurements.

14
Applications

• Routers are currently designed to manage 1000s of
  buffers, in order to manage QoS.
• Proposed strategies include Fair Queueing (FQ)
  and DiffServ each of which uses one buffer per
  flow.
• But buffers will be empty most of the time!
• The corresponding semi-norms, however, are just
  as easily measured and more meaningful.

15
Applications Continuity

• Continuity of a mapping fF-gtR is the property
  that for any neighbourhoodN in R, there is a
  neighbourhood of flow states which maps into N.
• Continuity of control strategies ? stability.
• E.g. Strategies, like FQ or DiffServ, which
  require distinguishing every single flow, are
  probably not continuous.

• Continuity lt-gt action defined by a finite set
  of measurements.

16
A continuous control method which provides QoS
appears to exist

• This strategy monitors a small number of large
  flows
• and all other flows, in aggregate
• Then discriminates against the large flows.
• Because large and small is defined by the virtual
  buffer semi-norm, VOIP-like services will get
  good performance.
• All services will rcv good flow completion time.

17
Probability measures on Flow States

• Once a topology is defined on the space of flow
  states, we can easily define Gaussian probability
  measures.
• A Gaussian measure is defined to by the property
  that any finite collection of continuous linear
  measurements has a Gaussian distribution (normal
  distribution).
• Schilders theorem allows us to deduce
  probabilities of almost any set.

18
Implications

• An ideal strategy will monitor a small number of
  large flows to take actions which manage the
  performance of all flows.
• Because we know that the 1 of largest flows
  contain 20 of all bytes, such a strategy is
  possible.
• Once the largest flows are managed, all other
  flows will receive perfect performance.