SMM: Scalable Analysis of Power Delivery Networks by Stochastic Moment Matching Andrew B. Kahng, Bao Liu, Sheldon X.-D. Tan* UC San Diego, *UC Riverside

Outline Background Problem Formulation Random Walk Moment Computation in an RLC Tree SMM Theory Experiments Conclusion

P/G Supply Voltage Integrity Analysis Increasing Power/Ground supply voltage degradation in latest technologies  IR drop (DC/AC)  L dI/dt drop Effects:  Malfunction  Performance degradation P/G supply networks are special interconnects  Complex topology, numerous nodes, IOs Scalability improvement schemes  Top-down: multigrid-like, hierarchical, partition  Bottom-up: random walk

Random Walk A stochastic process which gives voltage of a specific P/G node Advantages:  Localization  Parallelism Limitations:  DC analysis  Transient analysis Our contribution:  Frequency domain analysis

Outline Background Problem Formulation Random Walk Moment Computation in an RLC Tree SMM Theory Experiments Conclusion

Problem Formulation Given  an RLC P/G supply network  power pads  supply current sources Find  P/G node voltages Challenges  Scalability  Accuracy

Kirchoff’s current law: A random wanderer pays for lodging every night, and has a probability to go to a neighboring location, until he reaches home A Monte Carlo method to a boundary value problem of partial differential equations Random Walk IqIq

Input: resistive network N, nodes B with known voltages Output: voltage of node s Start walking from a node s While (not reaching a node b  B) Pay A(q) at node q Walk to an adjacent node p with Pr(p, q) Gain V b the voltage of the boundary node b  B V s = net gain of the walk Random Walk in a Resistive Network

Moment Computation in an RLC Tree Current through Rpq charges all downstream capacitors Expanding the voltages in moments p q R pq

Input: RLC tree T, input nodes voltage moments Output: Output node voltage moments For each moment order j Depth-first traversal of the tree T In pre-order, compute m i-1 (p) for each node p In post-order, compute S k  Tp C k m i-1 (k) for each T p Moment Computation in an RLC Tree

Expanding moment computation in a tree to a general structure network Stochastic Moment Matching (SMM) IqIq CqCq q

A random walk process  Pr(p, q) transition probability  A(q) lodging cost Stochastic Moment Matching (SMM)

Input: RLC P/G network N, nodes B with known voltages, current sources S Output: P/G node voltages 1.For each current source s  S 2. Walk from s to a power pad with Pr(p, q) 3. For each node q in the path 4. For each moment order j 5. Compute m j (q) 6.Collect node moments 7.Compute poles and residues by moment matching 8.Output time domain waveforms and voltage drops SMM Algorithm

Numerical Stabilities Compute moments of all orders of a node based on the same random walk process  See algorithm Reduce number of random walks by reducing the number of node voltage moments needed  MMM vs. SMM Filtering out numerically instable solutions  Unvisited nodes, positive poles, etc. Take average

Runtime Number of moments M Average path length P (dominant)  = average distance from the node to a power pad  Independent to P/G network size Number of poles/residues for moment matching Time domain binary search for delay

Outline Background Problem Formulation Random Walk Moment Computation in an RLC Tree SMM Theory Experiments Conclusion

Convergence I.Solid curve: Random walk I II.Dashed curve: Random walk II III.Dotted curve: Liebmann’s method

Accuracy Randomly generated 100x100 power mesh of R=100W~1KW, C=0.1pF~1.0pF, L=0.1pH~1.0pH, Tr=0.5ns~2.5ns, Ip=0.5mA~2.0mA 1000 random walks vs. SPICE

Scalability Power mesh of R=1KW, C=1pF, Tr=1ns, Ip=1mA N/G1234 CPUVdopCPUVdropCPUVdropCPUVdrop

SMM vs. Transient Random Walk I.SMM: 100 random walks II.TRW: 100 random walks for each time step, each of 5ps ICPU Vdrop IICPU Vdrop

Summary We extend random walk to frequency domain analysis by computing moments for RLC P/G networks Much better efficiency/accuracy than transient analysis random walk Advantages of random walk: locality, runtime which depends on average distance to a power pad, parallelism More stable moment computation in a bunch of stochastic processes

Thank you !