Filtered Gibbs Sampler fo r Estimating Filtered Gibbs Sampler fo r Estimating Blo cking Probabilities in WDM Optical Net w o rks Blo cking Probabilities in WDM Optical Net w o rks Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew D´ epartement d’informatique et recherche opr´ erationnelle Universit´ e de Montr´ eal, Qu´ ebec, CANADA couriel: vazquez@IRO.UMontreal.CA Department of Electronic and Electrical Engineering The University of Melbourne email: { fva,lha } @ee.mu.oz.au European Simulation Multiconference, 25 May 2000.
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 1 Outline of Presentation 1. Motivation • WDM optical networks 2. Clique packing • Stationary measure • Blocking probability 3. Monte Carlo simulation • Accept/reject Monte Carlo • Markov chain Monte Carlo 4. The Gibbs sampler • Periodic Gibbs • Filtered sequential Gibbs sampler 5. Future work
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 2 1.1 Motivation: WDM Optical Networks • Optical bandwidth >> electronic bandwidth. • Wavelength division multiplexing (WDM): – Λ independent wavelengths per fibre – Each wavelength modulated separately • Crossconnects: at nodes act as space switches, they can also switch wavelengths.
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 3 1.1 Motivation: WDM Optical Networks • Optical bandwidth >> electronic bandwidth. • Wavelength division multiplexing (WDM): – Λ independent wavelengths per fibre – Each wavelength modulated separately • Crossconnects: at nodes act as space switches, they can also switch wavelengths. • Optical carriers within fibres are wavelengths . Hop • Calls are connected using optical 2 carriers along the links on their paths: lightpath . Hop 1 Hop 3 • Connected calls use the bandwidth of each carrier wavelength along the lightpath. Lightpaths are shown in different shades of colour.
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 4 1.1 Motivation: Crossconnects Full wavelength conversion ⇒ standard circuit switched loss network λ 1 λ 2 λ 3 (a) (b) Space Switch Wavelength Converter M input and output fibres with W wavelengths on each, requirements: • wavelength continuous crossconnect: W different M × M space switches, • wavelength conversion crossconnect: a single MW × MW space switch. V ERY E XPENSIVE !!!!!!
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 5 1.1 Motivation: Resource Allocation Demand Model: Call arrivals to lightpath i follow a Point process N i ( t ) with intensity λ i (e.g. Poisson). Call durations: i.i.d holding times with mean 1 /µ . Resources: No (or partial) wavelength conversion : wavelength continuity constraints . Calls compete for bandwidth. • Dynamic allocation of lightpaths – Several methods available to assign LPs to incoming calls – Problem: Analysis and evaluation difficult (unless full conversion)
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 6 1.1 Motivation: Maximum and Clique Packing Dynamic lightpath allocation: Wavelength continuity constraint (no conversion). How to assign lightpaths to incoming calls at route i ? • Call arrives, search available wavelength (say First Fit assignment). • No wavelength available on path .... reject??? Fast tuning devices: Optical carriers can (in principle) change wavelength of on-going connections without affecting QoS.
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 7 1.1 Motivation: Maximum and Clique Packing Dynamic lightpath allocation: Wavelength continuity constraint (no conversion). How to assign lightpaths to incoming calls at route i ? • Call arrives, search available wavelength (say First Fit assignment). • No wavelength available on path .... reject??? Fast tuning devices: Optical carriers can (in principle) change wavelength of on-going connections without affecting QoS. Maximum packing: Fast tuning devices ⇒ rearrangement of wavelengths. Calls on route i connected if, upon rearrangement, there is a wavelength available.
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 8 1.1 Motivation: Maximum and Clique Packing Dynamic lightpath allocation: Wavelength continuity constraint (no conversion). How to assign lightpaths to incoming calls at route i ? • Call arrives, search available wavelength (say First Fit assignment). • No wavelength available on path .... reject??? Fast tuning devices: Optical carriers can (in principle) change wavelength of on-going connections without affecting QoS. Maximum packing: Fast tuning devices ⇒ rearrangement of wavelengths. Calls on route i connected if, upon rearrangement, there is a wavelength available. State description: occupancy, complex coupling equations. Analysis: Complex model for analytical results, state space too large. Simplified model: clique packing yields simple linear constraints
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 9 2.0 Clique Packing in WDM Optical Networks • R = number of routes in network (number of O/D pairs if fixed routing) • n i = number of calls currently using route i 1 Cliques Graph G = ( V, E ) • V : vertices = routes 2 • E : edge if routes share a link 3 • Clique: fully connected subgraph of G . Maximum packing Fast tuning devices: Allocate incoming calls whenever possible, allowing rearrangement ⇒ ( n -colouring of G ) Clique packing assumes that incoming calls can be connected iff � n j < Λ for all l with j ∈ C l j ∈C l Simplified Model: Occupancy vector n i ( t ) follows stochastic process: independent Poisson arrivals and i.i.d. holding times (not M/G/ ∞ server... boundaries!)
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 10 2.1 Analysis of clique packing: stationary measure Model Arrivals ∼ Poisson ( λ i ) , holding times ∼ exp ( µ ) , { n ( t ) } occupancy process: each dimension B&D with state dependent reflecting boundaries. λ i λ i λ i λ i Λ 1 n (n) 0 n -1 n +1 i i i µ µ (n +1) µ Λ µ n (n) i i i Result The limit occupancy distribution (stationary probabilites) are: R � ρ n i � π ( n ) = 1 � i , n ∈ S G n i ! i =1 n ∈ N R : � S = n j ≤ Λ; l = 1 , . . . , L j ∈C l R � ρ n i � � � i G = n i ! n ∈ S i =1 Result: This result may be generalised for other renewal arrival processes and holding time distribution.
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 11 2.2 Blocking probability R Y i ( t ) � B = lim Y i ( t ) = number of lost arrivals on route i at time t A ( t ) t →∞ A ( t ) = total number of arrivals at time t . i =1 Blocking states on route i : states n ∈ B i ⇒ incoming calls at i are lost: R � λ i � � � B = π ( B i ) B i = n ∈ S : max n j = Λ λ { l : i ∈C l } i =1 j ∈C l ...solved the problem?
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 12 2.2 Blocking probability R Y i ( t ) � B = lim Y i ( t ) = number of lost arrivals on route i at time t A ( t ) t →∞ A ( t ) = total number of arrivals at time t . i =1 Blocking states on route i : states n ∈ B i ⇒ incoming calls at i are lost: R � λ i � � � B = π ( B i ) B i = n ∈ S : max n j = Λ λ { l : i ∈C l } i =1 j ∈C l ...solved the problem? Realistic network sizes: > 20 nodes, 8–64 wavelengths, R = n 2 / 2 + o ( n 2 ) # states ≈ O (Λ R ) . For 10 nodes and 8 wavelengths, computation of G requires ≈ 8 45 ≈ 10 40 multiplications,
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 13 2.2 Blocking probability R Y i ( t ) � B = lim Y i ( t ) = number of lost arrivals on route i at time t A ( t ) t →∞ A ( t ) = total number of arrivals at time t . i =1 Blocking states on route i : states n ∈ B i ⇒ incoming calls at i are lost: R � λ i � � � B = π ( B i ) B i = n ∈ S : max n j = Λ λ { l : i ∈C l } i =1 j ∈C l ...solved the problem? Realistic network sizes: > 20 nodes, 8–64 wavelengths, R = n 2 / 2 + o ( n 2 ) # states ≈ O (Λ R ) . For 10 nodes and 8 wavelengths, computation of G requires ≈ 8 45 ≈ 10 40 multiplications, which takes 10 21 years of CPU time on a 1 TFlops computer...
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 14 3. Simulation methods: Monte Carlo Idea: Estimate B directly, rather than find G then B R � λ i � � B = E ( 1 { X ∈ B i } ) , X ∼ π λ i =1 Simulation: • Generate a sample { X 1 , . . . , X N } i.i.d., X i ∼ π • Use the sample average: R N Y ( N ) = 1 � λ i � ˆ � � 1 { X s ∈ B i } N λ i =1 s =1
Felisa J. V´ azquez-Abad and Lachlan L. H. Andrew 15 3. Simulation methods: Efficiency R N � λ i � Y ( N ) = 1 ˆ � � 1 { X s ∈ B i } N λ s =1 i =1 LLN and CLT ⇒ confidence intervals can be estimated to give � r ( ˆ approximate error ǫ = z 1 − α/ 2 V a Y ( N )) ⇒ � r ˆ V a Y ( N )) Relative error ≈ B 2 Definition: Relative efficiency of estimator ˆ Y ( N ) : B 2 E r ( ˆ Y ( N )) = CPU [ ˆ r [ ˆ Y ( N )] Y ( N )] V a Trade-off between relative error and CPU time .
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