cell infocom apr 2003
TRANSCRIPT
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Input Queued Switches:
Cell Switching vs. Packet Switching
Abtin Keshavarzian
Joint work with
Yashar Ganjali, Devavrat Shah
Stanford University
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Background
Time is slotted
Data units of fixed size cells
Buffers at input ports (Input-Queued Switch)
To avoid HoL blocking , virtual output queues are
used
VOQ11
VOQ1N
VOQN1
VOQNN
Output 1
Output N
Input 1
Input N
Switching Fabric
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VOQ11
VOQ1N
VOQN1
VOQNN
Motivation
Packets have different lengths
Splitter module Combiner module (memory)
Packet delays are more important than Cell delays
Packet Based Scheduling algorithms
Switch
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Notation Arrival rate
: Number of cells arrived to VOQij up to time n
: Number of cells departed from VOQij up to time n : Number of cells queued at VOQij at time n
(SLLN) almost surely
)(nAij
)(nDij
ij
ij
n n
nAP!
gp
)(
lim
)(nZij
VOQ11
VOQ1N
VOQN1
VOQNN
Output 1
Output N
Input 1
Input N
Switching Fabric
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Admissibility and Rate Stability
The arrival rate matrix is admissible
iff
A switch under a matching algorithm is stable
(rate stable) if, almost surely,
][ ijP!0
! !ePN
i
ij Nj1
,...,11 ! !eP
N
j
ij Ni1
,...,11
ij
ij
n n
nP!
gp
)(lim
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MWM algorithm
A matching
MWM: At each time slot, select the matching
with maximum weight
maxarg nWnm
mm
1
!
!! ji ijij nmnnW , )()(,)( Z
NNijm v! ][m
!otherwise0
outputtoconnectedisinputif1 jimij
! !N
i
ij jm1
1 ! !N
j
ij i1
1
)()(max)( nWnWnW !!1
mm
m
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MWM Stability
McKeown et al showed that
MWM is stable under i.i.d. Bernoulli traffic
Dai and Prabhakar using Fluid model technique showed
MWM is stable for any admissible traffic
J. G. Dai and B. Prabhakar, The throughput of data switches with or without
speedup,INFOCOM 2000,pp. 556-564.
N. McKeown,V. Ananthram, and J. Walrand, Achieving 100% throughput in an
input-queued switch, INFOCOM 1996, pp. 296-302.
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Outline
Cell based algorithms review:
Stability concept
Maximum Weight Matching algorithm
Packet based algorithms Packet-Based Algorithms
PB-MWM and its stability
Packet Based Algorithms Classification
Work Conserving Waiting
Waiting Packet Based Algorithms
Conclusion
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Packet-Based Switching
Once the scheduler starts transmitting the
first cell of a packet, it continues until thewhole packet is received at output port
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Packet-Based Switching
Once the scheduler starts transmitting the
first cell of a packet, it continues until thewhole packet is received at output port
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Packet-Based Switching
Once the scheduler starts transmitting the
first cell of a packet, it continues until thewhole packet is received at output port.
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Stability of PB-MWM
PB-MWM is stable under regenerative
admissible traffic
Traffic is called regenerative if on average it requires a
finite time to reach the state where all ports are free if it
keeps using any fixed matching.
Bernoulli i.i.d. is a regenerative traffic.
M.A. Marsan, A. Bianco, P. Giaccone, E. Leonardi, and F. Nari, Packet Scheduling in Input-
Queued Cell-based switches, INFOCOM2001, pp. 1085-10
94
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Proof Outline
Matchingm(n) is k-imperfect if
For PB
-MWM:
Lemma: A scheduling algorithm is rate stable if
the average value of its weight is larger than
maximum weight matching minus a bounded
constant.
)) knn ! mm
_ a )(2)()(|)( * TNnWnZnW EE u
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Question
CB-MWM is stable under any admissible
traffic
PB-MWM is stable under any admissibleregenerative traffic.
Is the regenerative condition necessary?
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
22
11
12
21
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
22
11
12
21
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
22
11
12
21
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
22
A11
A12
A21
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
A22
A11
A12
A21
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
A22
A11
A12
A21
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Counter-example
1 2
2
1 3 3
1
1 2 2
3
3 4
4
me
A22
A11
A12
A21
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Classification of PB algorithms
WorkConserving(non-waiting):
No input is left unmatched when it has a packetfor an unmatched output.
Waiting:
Input ports may wait(do not start sending apacket) forinfinite numberof time slots.
No work-conserving algorithm can be ratestable for all admissible traffic.
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PB-wMWM
I1
I/L
Segment #1 Segment #2
I/L
L L
Switch runs at speedup
Maximum packet length: L
If use usual PB-MWM Else wait till all ports are free.
PB-wMWM is rate stable for any admissible
traffic with known max packet length
1(,1[L
Lk
Lkn
I
I
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Modified PB-wMWM
Segment #1 Segment #2
I! /)2()2( eLM
)2(eL
I! /)1()1( eLM
)1(eL
The packet length is not known but has boundedexpectation
: the maximum length of packets left when
waiting starts during lth segment
Modified PB-wMWM is rate stable for any
admissible traffic with bounded packet length
)(lLe
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Conclusion
PB-MWM is rate stable under any admissibleregenerative traffic.
Work-conserving packet based algorithms can notbe rate stable for all admissible traffics
Waitingis essential PB-wMWM and its modified version are stable
under any admissible traffic (with bounded meanpacket length)
Future work: Find simpler algorithms that are stable for any
admissible traffic.
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Fluid model
: number of time slots matchingmbeing usedup to time n
)(nTm
!
!
!
1
1"
nnT
nTnTnn
nnAnZ
ijZijijij
ijijij
m
m
m
mm
)(
)1()(1)1()(
)()()(
}0{
!
x
x!
x
x
P!
"
ttT
t
tTm
t
tD
tDttZ
ijZij
ij
ijijij
m
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m
m
)(~
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}0~
{
/
tn
tDnD
ijij
ijij
Pp
p
)(
)(~
)(
-
r
rtZtZ
ij
rij
)(lim)(
~
gp
!