1. Capacity Allocation Paradox Isaac Keslassy
Joint Work with Asaf Baron and Ran Ginosar
EE Department, Technion, Haifa, Israel

2. The Capacity Allocation Paradox
Node A
Router CA RA
Node C CR
Node B CB RB
Finite (small) buffers
Unlimited queues
Capacity Allocation Paradox:
Adding Capacity Can Destabilize the Network

3. Marakana Soccer Stadium
UnStable
Stable
Safety Check
Brazillian Line
Enter the stadium
Fast Security Check
Argentinian Line
Fast Swipe Ticket Entrance
Fast Security Check
Slow Security Check

4. Motivation Small buffer networks are widely used
When QoS not met: add capacity [Guz et al., ’06]
May destabilize the network
Network On-Chip
SpaceWire
Interconnection of
Computers

5. Previous Work: Selfish Routing
Braess’s Paradox (1968)
Difference: We assume fixed routing

6. Previous Work: Cyclic Dependency
Kumar & Seidman (1990)
Instability even though capacity > data rate
Dai, Hasenbein & Vande Vate (1998)
Adding capacity may destabilize a network
Differences:
No cycles in dependency graph
Single router
Each packet visits router only once
Several simple arbitration policies
Independent of initial conditions
New fundamental reason: Finite buffers

7. Finite (small) buffers
A General Phenomenon
Round Robin
Exhaustive Round Robin
Strict Priority
General Processor Sharing
Arrivals:
Periodic, Poisson…
Finite (small) buffers
Node A
Router CA RA
Node C CR
Node B CB RB
Unlimited queues
When buffer is full:
Blocking: Wormhole Routing
Dropping (with retransmission): Store And Forward

8. Intuition (a) CA=1 (b) CA=2 A1 B1 A2 B2 A3 B3 A1 A2 A3 B1 (1) B1 (2)
Assume A has priority:
Node A
Router
CA=2
CA=1
1 [pkt/T]
Node C
CR=2
Node B
CB=1
1 [pkt/T]
Buffer of 1 bit 2T
Share of CR 2 T 1 3T
(a) A1 B1 A2 B2 A3 B3
(a) CA=1 2T
Share of CR 2 T 1 3T
(b) A1 A2 A3
(b) CA=2
B1 (1)
B1 (2)
B2 (1)
T/2
3T/2
5T/2 8

9. What are the conditions for stability?
Necessary conditions:
Node A
Node C
Node B
CR is constant
RA = RB

10. Case #1: Buffers in the router hold no more than one data unit?
Queue A
Buffer A CA
Node C CR
Queue B
Buffer B CB
Necessary conditions are also sufficient.

11. Example 1: Analysis Stability Picture
CR = 273[Kf/s] (Constant) 2 4
RA = RB =
100[Kf/s] 3 2
CB [Kf/s] 1
CA [Kf/s]

12. Example #1 – Capacity Allocations2 4 3 2
Stable
UnStable
Stable
Case #3
Case #2
Case #1 1
Node A
RA = 100
CA=190
CA=300
CA=110
Node C
CR=273
Node B
RB = 100
CB=150
CB=110
1000 [flits/pckt]
Buffer Size: 16 Flits
Exhaustive Round Robin, Wormhole

13. Results – Simulation Stability Regions
CR = 273[Kf/s] (Constant) 2 4
RA = RB =
100[Kf/s] 3 2 1

14. Example #2 – Wormhole Routing
Exhaustive Round Robin
Round-Robin
GPS
1000 [flits/pckt], Buffer Size: 16 Flits, RA = 500kf/s, RB = 100kf/s

15. Example #3 – Store and forward
Strict Priority, CR = 2.1[Mbit/s]
Exhaustive RR, CR = 2.1[Mbit/s]
Poisson Arrivals with Parameters: lA = 100, lB = 100
Packet Length 10^4 bit
Buffer Size 3-4 packets

16. Example #3 – Store and forward
RR, CR = 6.1[Mbit/s]
Exhaustive RR, CR = 6.1[Mbit/s]
Poisson Arrivals:
lA = 500
lB = 100
Packet = 10^4 bit
Buffer 3 packets
All packets
need to arrive
sometime

17. Summary Adding capacity may destabilize even a simple network
The scheduling algorithm affects the stability of the network (even if work-conserving)
GPS arbitration: always stable

18. Thank you.