Optical Networks

Poompat Saengudomlert

Session 14

Traffic Grooming in WDM Networks (continued)

P. Saengudomlert (2018) Optical Networks Session 14 1 / 10

5.2.1 (Continued) Single-Hub Uniform Traffic

Grooming for a BLSR

Approach:

View working directed wavelengths of a BLSR as CW directedwavelengths for a UPSR

Modify the RWA solution of a UPSR to obtain a solution for a BLSR

Theorem:

For a BLSR-based feeder ring with the uniform single-hub traffic,

ABLSRmin = AUPSR

min = N +

⌈N

⌊g/r⌋

⌉

W BLSRmin = W UPSR

min = ⌈Nr/g⌉

P. Saengudomlert (2018) Optical Networks Session 14 2 / 10

Example:

N = 4, g = 7, r = 5,CW for working on λ1 and λ2, CCW for working on λ3 and λ4

⇒ W BLSRmin = ⌈4× 5/7⌉ = 3

λ4

λ1λ2λ3

(a) (b)Arrow labels are the traffic units.

EN5

5

2

EN5

2

1

5

55

5

AN 2

CCW: working CW: backup

λ4

λ1λ2λ3

(a) (b)Arrow labels are the traffic units.

EN5

5

2

EN5

2

1

5

55

5

AN 2

CCW: working CW: backup

NOTE: Only the RWA of traffic to/from AN2 is illustrated in the rightfigure.

P. Saengudomlert (2018) Optical Networks Session 14 3 / 10

Non-Uniform Single-Hub Traffic Grooming

Consider a UPSR (with results applicable for BLSR).

1 EN and N ANs

AN i transmits ri units to EN and receives ri units from EN

WA in order to minimize the number of ADMs

As before, traffic splitting does not help.⇒ focusing on WA with no traffic splitting

Example:

N = 4, g = 7, (r1, r2, r3, r4) = (3, 2, 5, 4): Greedy WA (on the left) not optimal

λ2λ3

λ1

(a) (b)

Arrow labels are the traffic units.

3

5

24

EN

AN 1

AN 2AN 3

AN 43

5

24

EN

AN 1

AN 2AN 3

AN 4λ2λ3

λ1

(a) (b)

Arrow labels are the traffic units.

3

5

24

EN

AN 1

AN 2AN 3

AN 43

5

24

EN

AN 1

AN 2AN 3

AN 4

P. Saengudomlert (2018) Optical Networks Session 14 4 / 10

ADM Allocation as Bin Packing Problem

Minimizing the number of ADMs viewed as bin packing problem

N objects of sizes r1, . . . , rN (with ri < g)

Each bin can contain objects up to total size g .

Use the minimum number of bins to hold all objects

Known to be NP-complete

Can be formulated as ILP problem

Given parameters

B: number of available bins (indexed from 1 to B)

N: number of objects (indexed from 1 to N)

g : size of bin

ri : size of object i

Variables

aij ∈ {0, 1}: equal to 1 iff object i is assigned to bin j

bj ∈ {0, 1}: equal to 1 iff bin j is used

P. Saengudomlert (2018) Optical Networks Session 14 5 / 10

Objective

Minimize the number of used bins

minimizeB∑

j=1

bj

Constraints

Assign each object to exactly one bin

∀i ∈ {1, . . . ,N},B∑

j=1

aij = 1

Bin usage and bin size limitation

∀j ∈ {1, . . . ,B},N∑

i=1

riaij ≤ gbj

Integer constraints

∀i ∈ {1, . . . ,N}, ∀j ∈ {1, . . . ,B}, aij ∈ {0, 1}∀j ∈ {1, . . . ,B}, bj ∈ {0, 1}

P. Saengudomlert (2018) Optical Networks Session 14 6 / 10

Additional comments on bin-packing

Can use the number of bins from first-fit heuristic as the value of B.

First-fit may not be optimal but uses no more than twice theminimum number of bins.

Example:

N = 5, g = 7, (r1, r2, r3, r4, r5) = (2, 4, 3, 2, 3).First-fit ⇒ 3 bins: {2, 4}, {3, 2}, {3}Optimization ⇒ 2 bins: {2, 3, 2} and {4, 3}

Theorem:

The number of bins used by the first-fit bin-packing heuristic is at mosttwice the minimum number of bins.

Proof: See notes.

P. Saengudomlert (2018) Optical Networks Session 14 7 / 10

5.2.2 Static Traffic Grooming with General Traffic

With general traffic, AUPSRmin and ABLSR

min may differ.Can use ILP to minimize the number of ADMs.Focus on using ILP for BLSR; see problem 4.5 for UPSR (optional)

Given information

W: set of wavelength channels in each fiber

g : capacity of each wavelength channel (e.g. in time slot)

N : set of nodes

L: set of links (one link equivalent to one fiber)

WDw ∈ {CW,CCW}: working direction of wavelength w

(w , t): time slot t on wavelength w in direction WDw1

S: set of s-d pairs with nonzero traffic

S(i ,·): set of s-d pairs whose source nodes are node i

S(·,j): set of s-d pairs whose destination nodes are node j1The term circle will be used to refer to such pair (w , t).P. Saengudomlert (2018) Optical Networks Session 14 8 / 10

Given information (continued)

ts : traffic demand (in time slot) for s-d pair s

psc : path for s-d pair s in ring direction c (CW or CCW)

Variables

f sw ,t ∈ {0, 1}: working traffic flow for s-d pair s on circle (w , t)

aiw ∈ {0, 1}: equal to 1 iff an ADM is used at node i for wavelength w

Objective

Minimize the number of ADMs used

minimize∑

i∈N

∑

w∈Waiw

P. Saengudomlert (2018) Optical Networks Session 14 9 / 10

Constraints

No collision on any link on any circle

∀l ∈ L, ∀w ∈ W , ∀t ∈ {1, . . . , g},∑

s: l∈psWDw

f sw ,t ≤ 1

Satisfaction of traffic demands

∀s ∈ S,∑

w∈W

∑

t∈{1,...,g}f sw ,t = ts

ADM termination capacity contraints (transmit and receive)

∀i ∈ N , ∀w ∈ W,∑

s∈S(i,·)

∑

t∈{1,...,g}f sw ,t ≤ gaiw

∀j ∈ N , ∀w ∈ W,∑

s∈S(·,j)

∑

t∈{1,...,g}f sw ,t ≤ gajw

Integer constraints

∀s ∈ S, ∀w ∈ W, ∀t ∈ {1, . . . , g}, f sw ,t ∈ {0, 1}∀i ∈ N , ∀w ∈ W, aiw ∈ {0, 1}

P. Saengudomlert (2018) Optical Networks Session 14 10 / 10