Capacity Allocation Paradox Isaac Keslassy Joint Work with

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

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

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

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

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

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

A General Phenomenon Finite (small) buffers Arrivals: Periodic, Poisson… Node A CA RA Node B RB Router 1. 2. 3. 4. Round Robin Exhaustive Round Robin Strict Priority General Processor Sharing CR Node C CB Unlimited queues When buffer is full: 1. Blocking: Wormhole Routing 2. Dropping (with retransmission): Store And Forward 7

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

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

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

Example 1: Analysis Stability Picture CR = 273[Kf/s] (Constant) 2 4 CB [Kf/s] 0 RA = R B = 100[Kf/s] 2 3 1 0 CA [Kf/s] 11

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

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

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

Example #3 – Store and forward Strict Priority, CR = 2. 1[Mbit/s] Exhaustive RR, CR = 2. 1[Mbit/s] Poisson Arrivals with Parameters: l. A = 100, l. B = 100 Packet Length 10^4 bit Buffer Size 3 -4 packets 15

Example #3 – Store and forward Poisson Arrivals: l. A = 500 l. B = 100 Packet = 10^4 bit Buffer 3 packets RR, CR = 6. 1[Mbit/s] Exhaustive RR, CR = 6. 1[Mbit/s] All packets need to arrive sometime 16

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

Thank you. 18