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A performance model for an asynchronous optical buffer

Performance 2005 Juan-Les-Pins, France. A performance model for an asynchronous optical buffer. W. Rogiest • K. Laevens • D. Fiems • H. Bruneel SMACS Research Group Ghent University. DWDM channels. edge nodes. (legacy) access networks. core nodes (possibly co-located

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A performance model for an asynchronous optical buffer

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  1. Performance 2005 Juan-Les-Pins, France A performance model for an asynchronous optical buffer W. Rogiest • K. Laevens • D. Fiems • H. Bruneel SMACS Research Group Ghent University

  2. DWDM channels edge nodes (legacy) access networks core nodes (possibly co-located with the edge nodes) Motivation opticalchannels vs. electrical nodes Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  3. optical switching (OBS/OPS) all-optical: new transport paradigm still need for contention resolution a solution: optical buffering (for now) light cannot be stored, only delayed → fibers aim: analyze model of an asynchronous equidistant fiber delay line (FDL) buffer set of fibers (N+1 in number) with equidistant fiber lengths → delays 0*D,1*D, ... N*D N is the size, D the granularity, N*D the capacity example for N=2 Aim Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  4. Overview Model Approach Analysis Numerical Results Conclusion Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  5. for FDL buffers system of infinite size (N=∞) only delays nD can be realized gives rise to “voids” scheduling horizon ≠ unfinished work (due to voids) as seen by arrivals queueing effect [x]+ (max{0,x}) FDL effect x (ceil(x)) valid for both slotted and unslotted systems burst size Bk Hk D Hk /D Model•system equation "work" being done at rate 1 void Hk+1 (k+1)st arrival kth arrival interarrival timeTk Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  6. Approach (1)•assumptions • unslotted model for an FDL buffer • single wavelength • uncorrelated arrivals • iid burst sizes • conventions slotted = synchronous = discrete time (DT) unslotted = asynchronous = continuous time (CT) (N = ∞) : infinite size buffer = infinite system (N < ∞) : finite size buffer = finite system • strategy • three mathematical domains • several steps involved Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  7. DT , N=∞ CT, N=∞ CT, N<∞ Approach (2)•domains mathematical approach z-domain • probability generating functions Laplace domain • Laplace transforms probability domain • probabilities resulting performance measures • sustainable load • tail probabilities • moments of the waiting time • loss probabilities Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  8. CT, N=∞ CT, N<∞ Approach (3)•steps “scratch” z-domain Laplace domain probability domain DT , N=∞ queueing effect FDL effect direct approach limit procedure CT, N=∞ queueing effect FDL effect heuristic (1) : dom. pole approx. heuristic (2) : heuristic approx. Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  9. Analysis (1)•z-domain • analysis assuming equilibrium • solution of queueing effect • memoryless arrivals, well-known solution (see paper) • analysis of FDL effect in DT • "solve“ • yields where • D’ is DT granularity, an integer multiple of slots Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  10. Analysis (2)•to Laplace domain first way: limit procedure • starting from results for a slotted model • slot length D (e.g. in ms) • take limit D 0 • time-related quantities scale accordingly • counting-related quantities do not • identity involving comb function second way: direct approach Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  11. Analysis (3)•Laplace domain • both ways yield • D is the CT granularity, a real number z-domain finite sum D’ is integer Laplace transform domain infinite sum D is real Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  12. Analysis (4)•to probability domain • special cases for burst size distribution: closed-form formulas • exponential • deterministic • mix of deterministic • heuristic, two parts: • (1) dominant pole approximation, allows to obtain overflow possibilities for infinite system • (2) heuristic approximation, involving special expressions (see paper), allows to obtain burst loss probabilities (BLP) for finite system Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  13. BLP Laplace transform domain exact N = ∞ probability domain approximate N = ∞ probability domain approximate N < ∞ Analysis (5)•probability domain • yields • applying steps for each special case yields numerical results Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  14. Numerical example (1) • BLP as function of D (E[B]=50.0 ms, N=20) exponential burst size distribution Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  15. Numerical example (2) • BLP as function of D (E[B]=50.0 ms, N=20) • behaviour is similar to synchronous systems deterministic burst size distribution Performance 2005 • Juan-Les-Pins • Wouter Rogiest

  16. Conclusions • performance measures for finite asynchronous optical buffers • derived from infinite synchronous buffer model • asynchronous operation • behaviour is similar to synchronous systems • further research • comparison “synchronous vs. asynchronous” by studying batch arrivals • contact Wouter.Rogiest@UGent.be Performance 2005 • Juan-Les-Pins • Wouter Rogiest

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