Fundamental Principles of Packet Switch Design

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Chapter 3

• Fundamental Principles ofPacket Switch Design

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Fig. 3.1. Packet arrivals in a 4x4 packet switch
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Fig. 3.2. Input packet processor
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Simplify VCI assignment algorithm Reduce
blocking due to shortage of valid VCI
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• Waiting System
• Contention Resolution mechanism to select
  packets
• to be switched
• Losing packets buffered at inputs or
  internally
• Output buffers needed if group size is greater
  than 1
• Throughput can be made arbitrarily close to 100

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Fig. 3.3. (a) Speeding up switch operation by N
times
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Fig. 3.3. (b) Dropping packets that cannot be
switched
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Fig. 3.3. (c) Queueing packets that cannot be
switched
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• Interconnection Networks
• Originally intended for multiprocessor computer
• interconnect
• distributed, self-routing algorithms
• regular topological interconnection pattern
• Rearrangeable nonblocking in circuit switching is
  the
• same as internally nonblocking in packet
  switching
• Speed is the practical difference !

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• Loss System
• No input or internal buffers. Packets may need
  to queue
• at outputs if group size is greater than 1
• Packets may be dropped where contention
  arises.
• Loss probability can be made arbitrarily small

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Fig. 3.4. (a) shuffle-exchange (omega) network
(b) reverse shuffle- exchange
network (c) banyan network (d) baseline network
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Fig. 3.5. Routing in the banyan network
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Fig. 3.6. Internal and external conflicts when
routing packets in a banyan network
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• Loss probability in a Banyan Network

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Fig. 3.7. Loss probability of the Banyan network
operating as a loss system
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• Combinatoric Properties of Banyan Networks

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• 3.2.4 Nonblocking Conditions for the Banyan
  Networks
• Banyan network is nonblocking if active inputs
• x1, xm, (xi, gt xj, if j gt 1) and their
  targeted outputs
• y1, ym satisfy
• 1.) Distinct and monotonic outputs
• y1 lt y2 lt lt ym or ym gt gt y2 gt y1
• 2.) Concentrated inputs

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• Fig. 3.8. (a) An example showing the banyan
  network
• is nonblocking for sorted
  inputs

• Fig. 3.8. (b) Nonblocking sort-banyan network

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Fig. 3.9. (a) Labeling of nodes in the banyan
network
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Fig. 3.9. (b) Sequence of nodes traversed by a
packet from input an a1 to
output bn b1
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• Proof Two packets
• 1st packet x ana1 y
  b1bn
• 2nd packet x ana1 y b1bn
• collide in stage k

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• But if conditions are satisfied
• 1.) There are x x 1 active inputs
• between x and x
• They must have distinct outputs
• 2.) y y 1 gt number of distinct
  outputs
• x x 1
• i.e. y y gt x x

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Fig. 3.10. An example of unsorted packets having
no conflict in the banyan network
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Fig. 3.11. Sorted packets (in P.19) remains
unblocked after their inputs are
shifted (mod 8) by 6
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• Concentration

Routing bits are used starting from L.S.B to
M.S.B.
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Fig. 3.12. (a) Sorting network switches correctly
when all inputs are active and have no
common outputs
Fig. 3.12. (b) Sorting network switches
incorrectly when some inputs are inactive
Fig. 3.12. (c) Sorting network switches
incorrectly when some inputs have common outputs
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• Fig 3.13. An example showing that dummy packets
  with
• nonconflicting destinations may be
  introduced
• to make the sorting network switch
  correctly
• when not all inputs are active,
  this requires
• knowledge of the destinations of
  active inputs

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Fig. 3.14. (a) A comparator
Fig. 3.14. (b) A compact way of representing a
comparator
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Fig. 3.15. (a) A 4x4 sorting network -- Compact
representation
Fig. 3.15. (b) A 4x4 sorting network -- Full
representation
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• Order-preserving Property
• Suppose a comparison network maps input sequence
• a lt a1, , aN gt to output sequence b lt b1, ,
  bN gt,
• Then for any monotonically increasing function
  f(.),
• it maps f(a) lt f(a1), , f(aN) gt to f(b) lt
  f(b1), , f(bN) gt
• ( Basic idea large numbers remain larger (no
  smaller) than
• small numbers after mapping
• ? Comparator states do not
  change )

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• Fig. 3.16. Illustration that a comparator has the
• order-preserving property

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• Fig. 3.17. (a) The inputs and outputs of a
  comparator at stage
• d when input sequence is a

• Fig. 3.17. (b) The inputs and outputs of the same
  comparator
• when input sequence is f(a)

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Fig. 3.18. Illustration of the proof of the
zero-one principle
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Fig. 3.19. Sorting based on merging, successive
shorter sorted sequences are
merged into longer sorted sequences
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• Example of sorting by merger

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Fig. 3.20. Bitonic Sorters
or
The two input sequences do not have to be the
same length The two input sequences are of
opposing directions
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Fig. 3.21. A half-cleaner
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Fig. 3.22. Operations performed by a half-cleaner
for different cases
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Fig. 3.22. Operations performed by a half-cleaner
for different cases
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• Examples of Bitonic Sequence (Circularly Bitonic)
  

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• Physical picture of half-cleaner action on
  arbitrary-number
• bitonic sequence

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Fig. 3.23. Recursive construction of a k-bitonic
sorter (merger)
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• Fig. 3.24. (a) A sorting network based on merging
  using
• bitonic sorters

• Fig. 3.24. (b) The same network broken down into
  comparators

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Fig. 3.25. The operation of a comparator used in
a sorting network for packet
switching
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Fig. 3.25. The operation of a comparator used in
a sorting network for packet
switching
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Fig. 3.24(1) Recursion for Odd-even Sorting
Network
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Fig. 3.26. An 8x8 Batcher-banyan network
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• Number of comparators in a Batcher bitonic
  sorting network
• f(k) stages in a k-bitonic sorter
• log2k

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Fig. 3.27. (a) Three-phase scheme for sort-banyan
network ( Stage 1 probing
for conflict )
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Fig. 3.27. (b) Three-phase scheme for sort-banyan
network ( Stage 2
acknowledgement of winning packets )
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Fig. 3.27. (c) Three-phase scheme for sort-banyan
network ( Stage 3 routing
winning packets )
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• Packet switching with Clos networks
• ?The Clos network first studied by C. Clos in
  1953
• ?Features
• Rearrangeably non-blocking
• Requires centralized route assignment
• Self-routing is impossible in genernal 1
• 1 B.G. Douglass and A.Y. Oruc, On self-routing
  in Clos connection networks, IEEE Trans. On
  Commun., Vol. 41, No. 1, Jan 1993, pp.121-124

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• 3.4.1 Non-blocking Route Assignment
• Generalization of the sort-banyan principle
• The non-blocking and self-routing properties of
  Clos network
• Simple route assignment with an appropriate
  addressing scheme
• General Clos-type network from the cascade
  combination of a MIN and its reverse network

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• Address numbering scheme

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• Continued

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• A sufficient non-blocking condition
• Let p (s1, d1), , (sn, dn)
• An assignment f p? C is non-blocking if

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• A fundamental lemma
• Let x1, x2,, xn be a strictly monotonic
  sequence of integers and for all i, define
• g(xi) miq for all i
• where m and q are constant integers. Then

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• Proof
• Without loss of generality, assume that the
  sequence is increasing and let i lt j. We have

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• Rank-based assignment algorithm
• Route assignment based on the rank of each
  connetion request
• Let p (s1, d1),,(sn, dn) be monotonic. The
  assignment
• f (s1, d1) m iq
• where m is a constant integer and i is the rank
  of connection (si, di) in p, is non-blocking 2
• 2 K. Sezaki, Y. Tanaka and M. Akiyama, N1
  Connection Switching Networks Suited for Time
  Division Switching, Computer Networks and ISDN
  Systems, No. 20, 1990, pp. 383-389

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• Proof
• The sequences (s1,, sn) and (d1,, dn) are
  monotonic
• Let
• g1(si) f (si, p(si)) f (si, di) m
  iq
• g2(di) f (p-1(di), di) f (si, di) m
  iq
• Thus f (si, di) f (sj, dj) implies
• g1(si) g1(sj) and g2(di) g2(dj)

Hence the assignment is non-blocking
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• An example

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• Order preserving property

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• 3.4.2 Recursiveness
• ? Consider a 3-stage Clos network with parameter
  p0 and q0.
• Let the rank of a packet from si to di be r0(si)
• ?The middle-stage module assigned to the request
  (si, di) is r0(si)q0 which can be obtained by
  decomposing r0(si)
• r0(si) a1(q0) a0

? Suppose the subnetwork has parameters p1 and
q1. The route assignment in the subnetwork
is
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• If N q0q1qn-1 , number of stages 2n 1

r0(si) a1(q0) a0 (a2(q1) a1)q0
a0 a2q1q0 a1q0 a0
((a3(q2) a2)q1 a1)q0 a0 a3q2q1q0
a2q1q0 a1q0 a0
The routing tag is (a0,a1,a2,,an-2,ßn-1,ßn-2,
,ß1,ß0)
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• Example
• Benes network qi 2 for all i

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• General Clos-type Networks

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• The Omega Network

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• The Reverse Omega Network

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• Cascade combination

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• Cascading a MIN and its reverse network results
  in a
• general Clos-type network

?Unique path for each input-output pair in MIN
and its reverse network ?Total N/p alternate
paths in Clos-type network
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• Self-Routing Properties of Sort-Clos Network
  Problems

3 J.Y. Hui and E. Arthurs, A Broadband Switch
for Integrated Transport, IEEE JSAC, Vol. SAC-5,
No. 8, Oct. 1987, pp. 1264-1273
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• Multicast connections
• ?Extension to broadcast Clos network
• ?Let the set of active inputs be (s0,s1,,sn-1)
• Let their corresponding sets of outputs be
  (D0,D1,,Dn-1)
• The set of connection requests is monotonic if

?Non-blocking route assignment by the Rank-based
Assignment Algorithm
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• An example

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• Routing and replication

?Routing from input to middle-stage modules by
decomposing the rank
?Routing tag to middle-stage module
(a0,a1,,an-2) ?Replication and routing
controlled by the General Interval Splitting
Algorithm
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• General Interval Splitting Algorithm
• ?Each packet is assigned an address interval
  represented by minimum and maximum

min (i-1) mn-1 m2n-2 max (i-1) Mn-1 M2n-2
?Replication is controlled by the digits mi and Mi
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• General Interval Splitting Algorithm (continued)
  
• The following procedure is performed
• If mi Mi , then send the packet out on link mi
• If mi ? Mi , then (Mi - mi 1) copies are
  required Replicate the packet, modify the headers
  and send the packets out on link mi to Mi

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• Header modifications

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• An example

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• Decomposition and generalization

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• Generalized copy network architectures

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• Conclusions
• ?The principle governing the non-blocking and
  self-routing properties of a large class of
  interconnection networks
• ?Non-blocking and self-routing properties of Clos
  networks
• ?Construction of a general Clos-type network by
  cascade combination of any MIN and its reverse
  network
• ?Extension to multicast network based on
  broadcast Clos network

END