Network Modeling

We use G= (V,E)to denote the graph of an OBS network; the set of nodes is denoted as V , and the set of links is denoted as E. Link e∈E comprises Ce wave-lengths.P denotes the set of paths predefined between source s and destination t nodes, s,t∈V , and s=t. Each individual path p∈P is identified with a subset p⊆E. SubsetPst⊆P identifies all paths from source s to destination t; the sets Pst are disjoint in our model. SubsetPe⊆P identifies all paths that go through link e.

The reservation (holding) times on each link are independent and identically dis-tributed random variables with the mean equal to the mean burst duration h; for simplicity we assume h=1. We assume that the network is capable of full wave-length conversion, i.e., a burst can be transmitted on any available wavewave-length in each link. The demand traffic pattern is described by matrix[γst]s,t∈Vand bursts des-tined to given node t arrive at node s according to a Poisson process of (long-term) rateγst/h=γst.

Later we useρpandρeto denote the traffic offered to path p∈Pand the traffic offered to link e∈E, respectively.

In the following two subsections we deal with the modeling of the volume of burst traffic lost in the OBS network. The procedure consists, in the first step, of the calculation of burst loss probabilities Eeon individual links, and in the second step,

of the calculation of BLP in the entire network. Finally, we introduce a multi-path source routing model.

6.3.1 Link Loss Calculation

By assuming the network has full wavelength conversion capability, i.e., each wave-length can be selected whenever it is available, the blocking probability Eeon each link is given by the following Erlang loss formula (see [21]):

Ee=E(ρe,Ce) =ρe^Ce

Ce!

"

C_e i=0

∑

ρeⁱ

i!

#₋₁

, e∈E. (6.1)

In order to determine Ee, e∈E, we have to calculate the traffic loadρe offered to individual links; recall that Ce, e∈E is given. Below, we provide two models of such a calculation.

Reduced load (RL). A common loss model of an OBS network was proposed by Rosberg et al. [21] and it makes use of a reduced load calculation. This model is an extension of the model proposed by Kelly [9] for circuit-switching (CS) networks.

In the OBS network, it is assumed that the traffic offered to link e is obtained as a sum of the traffic offered to all the paths that cross this link reduced by the traffic lost in the preceding links along these paths.

This relation can be expressed as ρe=

∑

p∈Pe

ρpΛpe, e∈E, (6.2)

where Λpe=

∏

f∈rpe

$1−Ef

%, p∈P,e∈E, (6.3)

and subset rpe⊂p identifies all links that precede link e along path p.

The difference between this model and the CS network model is that in the latter the subset rpe contains all the links that succeed link e along path p, on top of all preceding links. This difference reflects the fact that a burst offered to path p in OBS uses a single wavelength from each link along the path until the first link where it is being blocked or until it exists in the network. In contrast, a connection in CS either occupies a channel in all the links along the path or is blocked.

The calculation of link loss probabilities Ee, e∈E, together with the calculation of offered burst trafficρe, given by the reduced load model (6.2), leads to a fixed-point equation with a solution known as the Erlang fixed fixed-point. The fixed fixed-point cannot be solved in a closed form but its approximation can be found through re-peated substitution of (6.1) in (6.2). It is known that the fixed point exists in both

vp

CS and OBS networks (see [9] and [21], respectively). Although the fixed point is unique in CS networks, its uniqueness has not been proved in OBS networks.

Although the traffic offered to a route is Poisson, it may still be thinned by blocking at consecutive links and thus no longer remain Poisson. Since there is no straightforward solution to this problem, we make a simplification that the burst arrival process to each link is Poisson.

Non-reduced load (NRL). Formulation (6.2) may bring some computational diffi-culty, especially with regard to the calculation of partial derivatives for optimization purposes. Therefore, we also consider a simplified non-reduced load model, where the traffic offered to link e is calculated as a sum of the traffic offered to all paths that cross this link:

ρe=

∑

p∈Pe

ρp, e∈E. (6.4)

The rationale behind this assumption is that under low link losses Ef, f ∈E, ob-served in a properly dimensioned network, model (6.2) can be approximated by (6.4).

Figure 6.1 presents illustrative examples of the reduced load calculation for both CS and OBS networks, as well as of the non-reduced load calculation.

6.3.2 Network Loss Calculation

Overall network loss (NL). The calculation of overall burst loss or blocking prob-ability in an OBS network is presented in [21], and it uses the same formulation as was proposed for CS networks [9]. In further discussion we name this model an overall network loss (NL) model.

The main modeling steps include the calculation of

1. burst loss probabilities Eeon links, given by (6.1), 2. loss probabilities Lpof bursts offered to paths

Lp=1−

∏

e∈p(1−Ee), p∈P,and (6.5)

3. the overall burst loss probability BNL, BNL=

∑

p∈P

ρpLp

"

p∈P

∑

ρp

#₋₁

. (6.6)

In order to calculate the path loss probability Lp, p∈P, we make an assumption that burst blocking events occur independently at the network links. Then formula (6.5) accounts for blocking probabilities in all links e that belong to path p.

The overall burst loss probability BNLis calculated simply as the volume of burst traffic lost in the network normalized to the volume of burst traffic offered to the network.

Overall link loss (LL). Another method for calculation of burst losses in the entire network is based on an overall link loss (LL) model [6]. In this method we sum up the volumes of traffic lost on individual network links.

The main modeling steps include the calculation of 1. burst loss probabilities Eeon links, given by (6.1), and

2. BLL, a sum of the burst traffic lost on individual links relative to the overall traffic offered to the network

LLL=

∑

e∈EρeEe

"

p∈P

∑

#₋₁

. (6.7)

LL overestimates actual burst losses given by (6.6) in NL because it counts twice the intersection of blocking events that occur on distinct links. In fact, BLL may be higher than 1, and thus it cannot be considered as the probability metric. Neverthe-less, for Ee→0, e∈E, the blocking events that occur simultaneously vanish rapidly, and model (6.7) converges to model (6.6).

6.3.3 Multi-path Source Routing

We assume that the network applies source-based routing, so that the source node determines the path of a burst that enters the network (see Figure 6.2). Moreover, the network uses multi-path routing where each subsetPstcomprises a (small) number of paths, and a burst can follow one of them. We assume that the selection of a route from setPstis random for each burst and is performed according to a given traffic splitting factor xp, such that

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