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

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

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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

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

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

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

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

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

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.

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]

Example #1 – Capacity Allocations 2 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

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

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

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

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

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

Thank you.