Preliminary Result in Routing Games

1
Preliminary Result in Routing Games

• Joao P. Hespanha Stephan Bohacek
• Presented by Weiying Dai

2
Layout

• Motivation of the Routing Game
• Introduction of data transmission network
• Maximum Flow
• Routing game
• online game
• offline game
• Conclusion

3
Motivation

• What have been done before this paper?
• flow control, allocation of link capacities,
  server allocation, the trade-off between delay
  and throughput along virtual circuits,
  competitive routing.
• what is this paper doing?
• security routing policies where packets are
  under threat of being intercepted by an adversary.

4
Notation of data transmission network

• Nodes N 1,2,, n.
• Set of links L a link from node j to
  node i
  
• bl bandwidth of link l in packets per second.
• Tl the time it takes for a packet to traverse
  the link l.
  
• xl the number of packets per second along the
  link l.
  
• percentage of all the packets arriving at
  node i are routed through link ? L .
  
• probability of a packet arriving at
  node i is routed through link ? L .
  

5
Maximum Flow

• Objective maximum number of packets per second
  can flow in a network.
  
• Deterministic multi-path routing policies
  
  
• l ? L , where 1.

1
6
Maximum throughput

• Assume that ? packets per second are sent from
  node 1 to node n without any drops. By
  conservation law, we have
  

is the maximum throughput of
7
Maximum bandwidth

• Maximum throughput depends on the routing policy
  R, how about the maximum throughput obtained by
  any routing policy?
  

Thm1
8
Routing Games

• Who is playing the game?
• the network designer that specifies the
  routing algorithm.
  
• An adversary that attempts to intercept data
  in the network.
  
• what are their purpose?
• the designer wants to minimize the time it
  takes for a packet to be sent from node 1 to node
  n.
  
• the adversary wants to maximize the time. He
  will scan link l ? L when he tries to intercept
  the packet at that link.

9
Several version of game definition

• Offline Online
• Effectiveness
• Delay resend

10
Online Game

• Assumption intercepting a packet simply results
  in a fixed extra delay T.
  
• pl the probability of intercepting a packet
  traveling in link l ? L . L is the link being
  scanned by adversary.
  
• at ? L , the next link determined by the routing
  algorithm. bt ? L , the link scanned by the
  adversary. The transition probability function
  can be described as

11
Online Games(continued)

• The cost to be optimized is the average time it
  takes to send the package from node 1 to node n.

12
Online Games(continued)

• Vi the average time it takes to send a packet
  from node I to node n.
  

Suppose that a packet just arrived at node i ?
N . The Designer decides to route it through th
e link ? L , the adversary decides to scan l
? L . For the particular choice,
The average cost will be
13
MinMax Model

• The average cost is .
• a(i) ak ? L , the distribution of links
  with which the designer decides to route the
  packet out of node i.
  
• b(i)bl l ? L , the distribution of the
  links to be scanned.
  
• MiV is a matrix defined by

14
Solution of MinMax

• Define two operators Tminmax and Tmaxmin.

Theses two operators satisfy the regularity
assumption. MinMax Theorem proved that they are e
qual
Thm2 Tminmax (which is same as Tmaxmin) has a
fixed point V, and .
And the fixed point V is the value of V at
the saddle solution.
15
One more case for online game

• An intercepted packet is sent back to node 1
  after a delay T.
  

the cost for the designer choosing link ? L
and the adversary choosing to scan link l ? L is
Again, the average cost is
16
Offline Game

• The adversary selects which link to be scanned
  before routing starts, but the player responsible
  for designing the routing policy does not know
  which link selected.
• Two methods to solve the offline game.
• build a matrix M with one row for each
  possible path from node 1 to node n and one
  column for each possible link that adversary can
  scan.
• convert it into a Markov chain to memorize
  its state (qt , st), where qt denotes the node
  where the packet is before the tth hop and st is
  the link being scanned by the adversary.

17
Stochastic routing policy

• Define the cost of zero-sum game as
• where 1 if the packet is intercepted and 0
  otherwise.

When a packet arrives at node k ? N , is
the probability that it will be routed through t
he link
.
L
rl l? L . Where

For a stochastic routing policy R and a link l
? L to be scanned by adversary, the cost JRlPRl
( 1).
18
Offline game Model

• Assume that once the packet is caught it will not
  routed any more.
  
• Xl(t) denotes the probability that the packet
  will be sent to link l for the hop t ? 1,2,,.
  

pl denotes the probability that a packet is
caught in the tth hop if the packet is sent to l
ink l in the hop. Then the
Cost is
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Matrix form of offline game

• Write the probability that the packet will be
  sent to link l for hop t in a matrix form
  

Write the cost in matrix form.
Suppose that dl is the probability that
adversary scans Link l ? L . Then for fixed routi
ng policy R, and the distribution Ddll ? L ,
the cost is
20
Solution of Offline game

• Consider two operators Tminmax and Tmaxmin .

The saddle solution exists in case
Conjecture the saddle solution exists?
The security policy R is defined by
21
Cycle-free routing

• What is Cycle-free routing policy?
• there is no sequence of links S
• Rnocycle denotes set of cycle-free routing
  policies.
  
• Lemma1. For cycle-free routing policies

then the optimization will become
Unique solution to
Maximum flow problem(LP)
22
Bias Towards shortest Path

• Remember so far the cost is . The
  cost doesnt favor shorter path.
  
• Bias routing towards shorter path

More hops result in larger costs since
increases with t.
As it converges to minimum hop
routing. As it converges to maximum
flow routing.

23
Conclusion

• Determined routing policies for a data
  transmission network that are robust with respect
  to packet interception/eavesdropping.
  
• Formulated the problem as a zero-sum game between
  the router and an attacker.
  
• Considered the several versions of the game
• online game reduced to shortest path and
  solved through dynamic programming.
  
• offline game reduced to max-flow problem and
  solved through linear programming.
  
• Future Work
• investigate if policies found for the offline
  game are saddle solution.

study the interaction between stochastic
routing and
TCP congestion control.