Instability of FIFO at Arbitrarily Low Rates in the Adversarial Queuing Model

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Instability of FIFO at Arbitrarily Low Rates in
the Adversarial Queuing Model

• Rajat Bhattacharjee
• Ashish Goel
• Stanford University

Instability of FIFO at Arbitrarily Low Rates in
the Adversarial Queueing Model . IEEE Foundations
of Computer Science (FOCS), 2003. SIAM Journal
on Computing 34(2) 318-332 (2004).
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Overworked server
Betty

• Server processes tasks at rate 1
• Tasks are generated for the server at rate r
• What is the value of r such that the input quue
  of the server would become unbounded (unstable)?
• Equivalently, what is the value of r s.t. there
  would be a task, which would never be processed
  (unstable)?
• r gt 1

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Overworked network

• Is the network of servers stable at rate r lt 1?
• Unstable at r gt 0.85!!! Andrews et al.

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Adversarial Queuing Model

• Borodin et al. 1996
• Packets injected by an adversary instead of a
  stochastic process
• Route given at the time of injection
• Each edge forwards at most one packet in one time
  step
• Contention resolved by a protocol like FIFO

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Adversarial Queuing Model

• Limitations on the adversary
• In any window of T time steps, a (w,r) adversary
  can inject at most wrT packets that need to
  traverse any edge in the network
• w burst size, r injection rate (rlt1)
• No identifiable hotspots in the system

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Stability of protocols

• Stability bounded queue size and delay
• r-stable stable against all (w,r) adversaries
• Universally stable r-stable for all rlt1
• Andrews et al. 1996
• Rings and DAGs are universally stable networks
• Longest-in-system (global FIFO) and
  Shortest-in-system are universally stable
  protocols
• FIFO is unstable at rategt0.85

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Related Work

• Tsaparas 1997 Nearest-To-Go unstable at
  arbitrarily low rates
• Gamarnik 1998 Equivalence of Fluid Model and
  Adversarial Queuing Model
• Andrews 2000 Session-oriented model
• Goel 2001, Gamarnik1999, Alvarez et al.
  2002 Characterized universally stable networks
  
• Bramson studied FIFO in stochastic models
• Kelly-type networks
• Job-shop scheduling model

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Stability of FIFO

• Andrews et al. 1996 unstable at rate gt 0.85
• Diaz et al. 2001 0.83
• Koukopoulos et al. 2001 0.749
• Lotker et al. 2002 0.5
• Is FIFO stable below some threshold, or, is it
  unstable at arbitrarily low rates?

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Our Result

• FIFO is unstable at arbitrarily low injection
  rates in Adversarial Queuing Model
• Size of the network is polynomial in 1/r
• Stability not possible even at rates which are
  some inverse polylograthmic function of the
  network size (1/logc n)
• Main idea Construct a gadget which acts as a
  break
• Use gadget to create a network and flow which is
  unstable at arbitrarily low rates

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Basic Gadget Topology

• Edges input, load, output edges

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A Special Flow

• Gadget traversing packets arrive at rate 1
• Internal gadget packets arrive at rate r

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Proportional Share Property of FIFO

• T ?j r(j) Tlt1 R(i) r(i)
• Tgt1 R(i) r(i)/T we will use this a lot

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Analysis of the flow

• r(i) the rate of arrival of packets which have
  traversed i of the k load edges
• T ? 1r at all times

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Analysis of the flow

• r1 1/T, ri ri-1/T 1/Ti
• Rate of Escape R krk ? k/(1r)k

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Concatenation of gadgets

• Output edges of the first gadget act as Input
  edges of the second
• A chain is a sequence of concatenated gadgets
• More than one gadget can be concatenated to a
  gadget

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Network
Columns and connectors are formed by
concatenation of gadgets
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Chain Traversing Packets
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Induction Phases
Beginning of a phase s packets in each input
queue
End of phase sgts packets in each input queue
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Subphases
At the beginning of subphase i, there are si
packets waiting in the input queue of gadget
i. These packets are chain traversing for the
rest of column A.
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Next si time steps
In the next si time steps rsi internal gadget
packets on each load edge and rsi/k chain
traversing packets on each input edge are
introduced
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At the end of the phase
At the end of the phase there are chain
traversing packets in each of the
connectors which wish to traverse column B
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Putting it all together

• Parameters of the network
• Size of the ring k
• Length of column ?
• Length of connector ?
• Choose parameters such that
• (1r)k gt 64 k3/r2, ? 4k/r, ? ?2

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Putting it all together

• The number of packets in the column at the
  beginning of subphase i, si gt s/2
• s is the number of initial packets in each input
  queue of column A
• Due to exponentially small leak from a gadget
• Number of packets which survive each connector
  per load edge is gt rs/4k
• The number of connectors 4k/r
• Hence, the number of packets in each of the input
  queue of column B gt s

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Conclusion

• Polynomial size of the network excludes
  possibility of FIFO being stable even at rate
  O(1/logc n)
• Subsequently, Lotker has tightened our
  construction to Õ(1/r)
• Are there meaningful restrictions which can be
  imposed on the adversary to achieve stability
  using FIFO?
• Session-oriented model?