Extension to Multiple Excess Load Sources

Up till now, the operational scenarios consisted of a single overloaded Grid site distributing its excess load to a collection of remote sites. In this section, we shed some light on the implications caused by considering Grid operational scenarios featuring multiple overloaded sites.

Assume that, in a Grid with N sites, each site’s computational resource has a constant processing capacity of C. Furthermore, assume k(1 ≤ k ≤ N − 1) overloaded sites are present and the total aggregate load on these sites is fixed at T ≤ N C. The amount of excess load generated at each of these k source sites is

T−kC

k . Assuming that the excess load is distributed uniformly among the N− k remote sites, let λk denote the total number of demanded wavelengths from all overloaded sites to each receiving remote site.

Using parameters D and B as in section 5.3.7, this quantity can be expressed as

λk= k Lk

k



(5.26)

where

Lk= D T− kC

(N − k)B (5.27)

Expressing Lkin terms of L1yields

Lk = L1+D(k − 1)(T − N C)

B(N − 1)(N − k) (5.28)

= L1− |∆k| (5.29)

Because of

|∆k+1|

|∆_k| = k(N − k)

(N − k − 1)(k − 1) (5.30)

≥ 1 (5.31)

we have that L1 ≥ L2 ≥ . . . ≥ LN−1. From equation 5.26, it follows that λ1= ⌈L1⌉ ≥ L1. We can now provide an upper bound on λk:

λk = k Lk

k



(5.32)

≤ L_k+ k (5.33)

= L1+ k − |∆k| (5.34)

≤ λ1+ k − |∆k| (5.35)

Since T ≤ N C we can distinguish between two cases:

• Case 1: T = N C. From the definition of |∆k| we can conclude that in this case|∆_k| = 0 and λ1= λ2= . . . = λN−1.

• Case 2: T < N C. In this case, we have |∆_k| > 0.

Because k and λkare both integers, we can state that in both cases the follow-ing upper bound is valid for2 ≤ k ≤ N − 1:

λ_k ≤ λ1+ k − 1 (5.36)

The above results are valid if excess load from every source is distributed to all remaining remote sites. An alternative set of multi-source scenarios can be envisioned, using the concept of partitioning as explained in chapter 4. This way, the collection of remote sites is partitioned into k (1 ≤ k ≤ ^N₂) subsets. Each subset is dedicated to the absorption of excess load from one source only. If all remote sites are to be engaged in each scenario then in general there will be k− 1 subsets containing_N−k

k  remote sites and 1 subset with ^N_k^−k remote sites in it.

Assuming a Grid setup where ^N−k_k is integer, and denoting the total number of wavelengths destined for a single remote site λ^′_k(obviously, λ^′₁= λ1), we have

λ^′_k=  L^′_k k



(5.37) where

L^′_k = Dk(T − kC)

(N − k)B (5.38)

= kLk (5.39)

Because⌈Lk⌉ ≤ k_Lk

k , it follows that in this case

λ^′_k ≤ λ_k (5.40)

Until now, the total aggregate load T on the collection of sources was assumed constant. Another set of multi-source scenarios consist of those scenarios where

the total excess load E generated by the source nodes is constant. For every k-source scenario, the total excess load is given by E= T −kC. Since schedulability requires that T ≤ N C, this kind of scenario is only realistic for values of k ≤

N C−E C .

When excess load is distributed to all of the N − k remote sites, we now have that

If we again consider the approach where the Grid is partitioned into k subgrids, we can deduce the following bound:

λ^′_k ≤DC

B + 1 (5.45)

The above formulas bound the change in wavelength path demand when adding additional sources to the base scenario. The resulting lambda Grid dimensioning cost, however, also depends on the number of fibers required to support all elemen-tary scenarios (see e.g. equation 5.11). To incorporate this factor some knowledge concerning the optical network’s topology must be present.

For regular topologies with k= 2 sources and constant aggregate excess load, the exact effects on the cost components can be derived analytically. In a bidirec-tional ring OTN topology (one of the regular topologies discussed in section 5.3.8), for example, the base scenarios require a number of fibers on each of the2N di-rected links equal to as there exists a link in each scenario that needs to support traffic to⌊^N₂⌋ remote sites. For the two-source partitioning scenarios, this figure is reduced to

& _N

2 − 1 λ^′₂ W

'

(5.47)

which is determined by observing the most loaded link in the ring in case the two source nodes are neighbors.

To calculate the average wavelength path cost, we continue as follows. Be-cause of the topology’s symmetry, we can limit our analysis to those scenarios where one source node is the node labeled0. If the second source node is node S and shortest path routing is enforced, we can decide for each node k (different from0 and S) which source node it should process excess load from. This is, be-cause of shortest path routing, the closest source to node k i.e. either node0 or node S, depending on Assigning the closest excess load source to each site, the N − 2 remote sites in the ring can be partitioned into4 sets. Set S₁^S contains the nodes1, 2, . . . ,_S analysis, we only need the size of these sets, given by

|S₁^S| =

Using these numbers, the average wavelength path cost over all ^N(N₂⁻¹⁾ sce-narios in the bidirectional ring dual excess load source case can be written as

1

equalling In contrast, the average number of wavelength paths carried on the links in the base scenarios (where the links directly connected to the source node carry at worst traffic for⌊^N₂⌋ nodes) is given by Comparing the costs of the dual source partitioning scenario to the base sce-nario on the bidirectional ring thus yields (for C= 1)

λ^′2

For our reference parameter choices (C = 1, W = 4, N = 13, wavelength granularity2.5Gbps and excess load generated as in section 5.4.4) this fraction equals0.9.

If we consider a full mesh topology instead, the average wavelength path cost changes from

2(N − 1)λ1 (5.59)

in the base scenario to

2(N − 2)λ^′₂ (5.60)

when two-source partitioning scenarios are being considered.

Each link now requires fibers required per link in the base scenario.

The change in cost incurred by the two-source partitioning scenario on a full mesh topology is thus (for C = 1)

which yields1.03 for the reference parameter settings we used.

In figure 5.17, we have plotted the dimensioning cost for the random network dual excess load source scenario problem and compared it to the dimensioning cost in the base scenario. The excess load is kept constant and has been generated following the recipe described in section 5.4.4. Again, for each value of p, 10 networks have been taken into account. Note how the cost differences shown in this figure are in line with the values predicted by equations 5.58 and 5.63, roughly corresponding to the cases p= 0.1 and p = 1, respectively. As p increases, more routing opportunities are available in the networks, decreasing the possibility of badly (i.e. resulting in high network costs) situated sources, as in the bidirectional ring scenario featuring two neighboring sources. Thus, for high values of p there is not much to be gained when half of the excess load is generated at a second

“well-placed” source. On the contrary, as the same excess load is now processed by|R| − 2 remote sites, the cost for the dual source scenario may surpass the cost of the base scenarios for high p values, as indicated by the numerical evaluation of expression 5.63.

3500 4000 4500 5000 5500 6000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Cost

p

Base Scenario Dual Source Scenario

Figure 5.17: Dual Source Scenario Cost for Random Networks