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

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.

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

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  1. 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).

  2. Overworked server • 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 > 1 Betty

  3. Sample task Overworked network • Is the network of servers stable at rate r < 1? • Unstable at r > 0.85!!! [Andrews et al.]

  4. 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

  5. Adversarial Queuing Model • Limitations on the adversary • In any window of T time steps, a (w,r) adversary can inject at most w+rT packets that need to traverse any edge in the network • w: burst size, r: injection rate (r<1) • No identifiable hotspots in the system

  6. Stability of protocols • Stability: bounded queue size and delay • r-stable: stable against all (w,r) adversaries • Universally stable: r-stable for all r<1 • 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 rate>0.85

  7. 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], Gamarnik[1999], Alvarez et al. [2002]: Characterized universally stable networks • Bramson studied FIFO in stochastic models: • Kelly-type networks • Job-shop scheduling model

  8. Stability of FIFO • Andrews et al. [1996]: unstable at rate > 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?

  9. 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

  10. Basic Gadget: Topology • Edges: input, load, output edges

  11. A Special Flow • Gadget traversing packets arrive at rate 1 • Internal gadget packets arrive at rate r

  12. Proportional Share Property of FIFO • T = j r(j) T<=1: R(i)= r(i) • T>1: R(i)= r(i)/T: we will use this a lot

  13. Analysis of the flow • r(i) – the rate of arrival of packets which have traversed i of the k load edges • T  1+r at all times

  14. Analysis of the flow • r1 = 1/T, ri = ri-1/T = 1/Ti • Rate of Escape R = krk k/(1+r)k

  15. 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

  16. Network Columns and connectors are formed by concatenation of gadgets

  17. Chain Traversing Packets

  18. Induction: Phases Beginning of a phase s packets in each input queue End of phase s’>s packets in each input queue

  19. 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.

  20. 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

  21. 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

  22. Putting it all together • Parameters of the network • Size of the ring: k • Length of column:  • Length of connector:  • Choose parameters such that: (1+r)k > 64 k3/r2,  = 4k/r,  = 2

  23. Putting it all together • The number of packets in the column at the beginning of subphase i, si > 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 > rs/4k • The number of connectors = 4k/r • Hence, the number of packets in each of the input queue of column B > s

  24. 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?

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