Stability Analysis of MNCM Class of
Algorithms
and two more problems !
EE384Y Project Presentation
June 4, 2003
Nima Asgharbeygi
1
Outline
 MNCM
Class of Algorithms
 Fluid Analysis of LPF
 iSLIP Random
2
Introduction

Definition of MNCM : (Tabatabaee et. al. Infocom 2003)
A maximal size matching algorithm m belongs to MNCM
class iff m contains all nodes with maximum weight.

Node weights:
Bk ( n) 

Qij ( n)
( i , j ):( i , j )  k


MNCM includes LPF, MNM and MFM algorithms.
A port-based fluid model proof was represented.
3
Counter Examples

Deterministic arrivals,
 Example

due Da Chuang
IID Bernoulli arrivals,
shows instability for   0.8 uniform traffic.
0
.5(1   ) 
 0
 Counter example:
 Simulation

 0
 .5(1   )

0
.5(1   )
.5(1   )





Algorithm: Serve q33 only if q31 (n)  q32 (n)  q13 (n)  q23 (n)  0 ; otherwise
serve some other non-empty VOQ’s to maximize weight of the
matching.
 Rate of service to q33   , for some  and  .
4
What’s wrong with the proof?


Lyapunov function:
The issue:




f ( B(t ))  max{B1 (t ),..., B2 N (t )}
“Due to continuity properties of B(t), for every t0  0 there exists
some   0 such that for all t [t0 , t0   [ there is always one
common index q(t0 , t0   ) that f ( B(t ))  Bq (t0 ,t0  ) (t ) .”
This is wrong!
An interval of length   0 in continuous time, corresponds to an
interval of arbitrarily large length ( r as r   ) in discrete
time domain.
This is not guaranteed by MNCM (easy to see by a periodic
pattern counter example).
5
Important to Remember

To have a valid stability proof, we must
ensure that both fluid model policy and the
discrete policy always make the same
decision; i.e. equivalency of departure
processes.
6
Outline
 MNCM
Class of Algorithms
 Fluid Analysis of LPF
 iSLIP Random
7
Problem Statement

LPFf algorithm definition:
 Apply
MWM algorithm on these edge weights:


W (n)  f (Qij (n)).  Qik (n)   Qkj (n) 
k
 k

D
ij
Where f (Qij (n))  0 if Qij (n)  0.
 This is our famous LPF if f (Qij )  1{Q 0}.
ij
 Not straight forward to use fluid model on
original LPF, because of discontinuity of f .
8
Stability of Fluid Policy

Fluid model weights:


W (t )  g (Zij (t )).  Zik (t )   Z kj (t ) 
k
 k

F
ij


Theorem: This fluid model is weakly stable
under MWM policy if g ( z )  A and z.g '( z )  B, z  0
for some constants A, B  0.
Proof: Use L(t )   Z (t ),W F (t )  and show that:
L(t )  (1  A  B)  Z (t ),W F (t )   0
9
Equivalency of Fluid and Discrete Models

How g should relate f to ensure equivalency?

( i , j ) *
W 
D
ij
 W
( i , j )
D
ij


W 
F
ij
( i , j ) *
 W
( i , j )
F
ij
Qij  Qik
Qkj 
). 

 Recall that W  lim g (

r 
r  k r
r 
k
F
ij
Qij
)  f (Qij )

Enough to have lim g (

Reasonable to choose g ( z )  g r ( z )
r 
r
f (rz )
10
Example

Let f (Q)  1  e
 aQ
(a  0)
 arz
g
(
z
)

1

e
 Then
r
 Fluid model is based on
g ( z)  lim(1  e arz )
gr ( z)
1
z
g ( z )
1
r 
z
z.g ' ( z )  0
to see lim
z 
 So LPFf is efficient under general traffic.
 LPF is the limiting case of LPFf as a .
Uniformity of convergence proves efficiency of
LPF under general traffic.
 Easy
11
Outline
 MNCM
Class of Algorithms
 Fluid Analysis of LPF
 iSLIP Random
12
Problem Statement
iSLIP Random scheduling algorithm
 Wish to find results on stability and
convergence of iSLIP-R.

Input degree
Probability of being empty
n1  2
1 2 3 1 1
   
2 3 4 2 8
n2  3
2 3 1
 
3 4 2
n3  4
3
4
1 2 3 1 1
   
2 3 4 2 8
n4  2
1 iteration
13
Approach

The problem is to find
E[# of non-empty output bins]
 ( N )  min
size of maximal matching
all possible N N
graphs

Let
1
O j {1  | input i is connected to output j}
ni
 (O j )  p
pO j

Assume that size of maximal match=N, and
initially input i connected to output i (for all i).
14
Approach (continued)

Greedy algorithm:
an available input i with smallest ni and connect
it to a possible output with smallest  (O j ) ,
1
(add 1 
to O j ). Repeat until no available input
n
remains. i
 Pick

Theorem: Given (n1 , n2 ,..., nN ) and initially input i
connected to output i (for all i), the greedy
algorithm maximizes E[# of empty output bins].
15
Outline of Proof


The proof is based on the following lemma.
Lemma: If for given (n1 , n2 ,..., nN ) the sets O1 , O2 ,..., ON
N
maximize   (O j ) , then for any j and k:
j 1
S j  O j  Ok
 ( S j )   ( Sk )   ( S )   ( S ), 
S k  Ok  O j
c
j
c
k
16
Results
Need to search for best (n1 , n2 ,..., nN ) to
maximize E[# of empty output bins].
 I guess it is (1, 2,3,..., N ) but yet no proof!

 This
N 1
 (N ) 
2N
gives
 Therefore, iSLIP-R with only one iteration
would be stable by speedup 4 for large N.
 E[max
# of iterations needed]  log 2 N .
17
Thank You!
18