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A performance model for an asynchronous optical buffer

This paper presents a detailed performance model for asynchronous optical buffers, focusing on the analysis of fiber delay line (FDL) buffers. Conducted by Wouter Rogiest and colleagues at Ghent University, the study emphasizes the transition of optical switching paradigms and the critical role of contention resolution through optical buffering. The authors explore the system's behavior, leveraging mathematical domains such as z-domain and Laplace transforms. Key findings highlight the similarities between synchronous and asynchronous systems, urging further research into optimizing optical network performance.

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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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