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

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

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

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

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

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

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

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

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

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

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

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

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

Performance 2005 Juan-Les-Pins Wouter Rogiest 14 Numerical example (1) BLP as function of D (E[B]=50.0  s, 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  s, N=20) l behaviour is similar to synchronous systems deterministic burst size distribution

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