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

Rajat Bhattacharjee

Ashish GoelStanford

UniversityInstability 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).

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 > 1

Overworked network

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

Sample task

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

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

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

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

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?

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

Basic Gadget: Topology

Edges: input, load, output edges

A Special Flow

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

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

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

Analysis of the flow

r1 = 1/T, ri = ri-1/T = 1/Ti

Rate of Escape R = krk k/(1+r)k

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

Network

Columns and connectors are formed by concatenationof gadgets

Chain Traversing Packets

Induction: Phases

Beginning of a phase

s packetsin each input

queue

End of phases’>s packets

in each input queue

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.

Next si time steps

In the next si time stepsrsi internal gadget

packetson each load edge andrsi/k chain traversing

packetson each input edge

are introduced

At the end of the phase

At the end of the phasethere are chain traversing packets

in each of the connectorswhich wish to traverse column B

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

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

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?