OSNR model to consider physical layer impairments in transparent optical networks

Noname manuscript No. (will be inserted by the editor)

OSNR Model to Consider Physical Layer Impairments in Transparent Optical Networks Helder A. Pereira · Daniel A. R. Chaves · Carmelo J. A. Bastos-Filho · Joaquim F. Martins-Filho

Received: date / Accepted: date

Abstract We propose a model that considers several physical impairments in all-optical networks based on the optical signal-to-noise degradation. Our model considers gain saturation effect and amplified spontaneous emission depletion in optical amplifiers, coherent crosstalk in optical switches, and four wave mixing in the transmission fibers. We apply our model to investigate the impact of different physical impairments in the performance of all-optical networks. The simulation results show the impact of each impairment on network performance in terms of blocking probability as a function of device parameters. We also apply the model as a metric for impairment-constraint routing in all-optical networks. We show that our proposed routing and wavelength assignment algorithm outperforms two common approaches.

Keywords Four Wave Mixing · Physical Impairments · Optical Communications · Optical Noise · Optical Signal-to-Noise Ratio · Routing and Wavelength Assignment · Wavelength Division Multiplexing

Helder A. Pereira University of Pernambuco, Department of Electrical Engineering, 52720-001, Recife-PE, Brazil. E-mail: [email protected] Daniel A. R. Chaves · Joaquim F. Martins-Filho Federal University of Pernambuco, Department of Electronics and Systems, 50740-530, Recife-PE, Brazil. E-mail: [email protected] Carmelo J. A. Bastos-Filho University of Pernambuco, Department of Computing and Systems, 52720-001, Recife-PE, Brazil. E-mail: [email protected]

1 Introduction All-optical networks have been considered as the most reliable and economic solution to achieve high transmission capacities with relative low cost. In these networks, the signal remains in the optical domain between the edge nodes, i.e., the signal propagates along the core of the optical network without any opticalelectrical-optical conversion. There are two main challenges to manage these networks providing quality of service (QoS): design an appropriate routing and wavelength assignment algorithm (RWA) and obtain an acceptable optical signal-to-noise ratio (OSNR) for every optical signal at the destination node [1; 2]. Linear and non-linear effects in fibers, and noise added by the network elements along transmission can lead to OSNR degradation, which have impact on the QoS. The main physical impairments that impact the optical network performance are the amplifier gain saturation, amplified spontaneous emission (ASE) and its saturation process, coherent crosstalk in switches, chromatic dispersion, polarization mode dispersion (PMD) and nonlinear effects in fibers [1; 3–11]. Furthermore, the routing process has a significant impact on the network performance. Some routing algorithms use heuristics based on pre-defined metrics. Some examples are: the shortest path (SP) [12], minor delay [12], load balance [12], hops count (HW), available wavelength (AW), hop count and available wavelengths (HAW), total wavelengths and available wavelengths (TAW), hop count and total wavelengths and available wavelengths (HTAW) [13], and least resistance weight (LRW) [14]. The main aim of these approaches is to achieve an improved load distribution, or to optimize network resources such as wavelength usage, to obtain lower blocking probabilities.

2

Many efforts have been made to develop RWA algorithms that consider physical impairments. In the optical network constricted by impairments, most reported studies concerning the solution of the RWA problem can be classified into three major categories. In the first category the RWA algorithm is treated in two steps: first a lightpath computation in a network layer module is provided, followed by a lightpath verification performed by the physical layer module. Different routing schemes have been proposed using this approach. In [4], Ramamurthy et al modeled their impairment-aware RWA algorithm taking into account the ASE noise generated in EDFA and cross talk added by the optical switch and compared the estimated bit error rate (BER) against a determined threshold. In [3], Huang et al modeled their impairment-aware RWA algorithm taking into account the PMD and OSNR performance parameters separately and compared them against two threshold levels at the end of the route. In the second category, the RWA algorithm is treated in three steps: first a lightpath computation in a network layer module is provided resulting in one (or none) feasible lightpath for each wavelength in the network. Then, for each feasible lightpath found, a verification is performed by the physical layer module. Among the lightpaths that passed in physical layer module verification the best one is chosen, considering some metric. Pointurier et al. [15] used this approach and developed a routing scheme based on Q-factor, which incorporates the effects of the compounded crosstalk in both physical layer module verification and choosing lightpath to set up the call. In [16], Anagnostopoulos et al developed a similar approach, nevertheless, considering the four wave mixing, cross phase modulation and EDFA ASE noise effects. In the third category, RWA algorithm itself is aware of physical impairments and uses the impairments information for routing procedures. Martins-Filho et al proposed in [8] a dynamic routing algorithm that selects the route based on the lowest physical impairments, including ASE accumulation, amplifier gain saturation and wavelength dependent gain along the path, and then calculate BER to check for the required signal quality. In [17], Cardillo et al proposed to use the OSNR model considered in [3] with some enhancements to consider non-linear penalties as well as the linear effects that occur along lightpath transmission. In [18], Kulkarni et al utilized the Q-factor as a performance parameter, and integrates the interplay of linear impairments (chromatic dispersion, PMD, ASE noise, cross-talk and filter concatenation). Although impairment-constraint routing schemes outperform the most common impairment unaware ap-

proaches, the use of these algorithms implies in higher computational complexity. Therefore, the development of models that use simple analytical equations is of paramount importance. Some models consider the SNR degradation or Q factor [4; 16; 19–22]. Another approach is to consider the degradation of the OSNR [3; 23–28]. In this paper we report on the development of an analytical model based on OSNR degradation to take into account the effect of the gain saturation and ASE noise depletion in amplifiers, coherent crosstalk in optical switches, FWM and PMD in the optical fibers. Differently from the previously reported work, our model considers these effects all together and it uses simple analytical equations obtained from well known fundamental or experimental behavior of network devices. We apply our model to investigate the impact of each physical impairment in the network performance [29], for two different network topologies. Network performance is evaluated in terms of blocking probability as a function of device parameters and network load. We also apply our model for impairment-constraint routing. We present a RWA algorithm that finds the route based on the OSNR using our proposed model (OSNR-R), and we compare its performance with other RWA algorithms. In Section II we describe in details our model that considers physical impairments in all-optical networks. In Section III we present the general characteristics and parameters used in our simulations considering two different optical network topologies. In Section IV we show the simulation results for the analysis of the impact of the physical impairments on the network performance using our proposed model. In Section V we present the proposed OSNR-R algorithm and we show the simulation results for network performance for three different RWA algorithms, including the OSNR-R. In Section VI we give our conclusions. 2 Physical impairments modeling Our formulation quantifies the OSNR degradation along the optical signal propagation in the all-optical networks. The impact of physical layer impairments is taken into account by considering the signal power and the noise power at the destination node, both affected by gains and losses along the lightpath. Moreover, some network elements add noise components or have a nonlinear response. The optical amplifiers add ASE noise power and are also affected by gain saturation and ASE depletion as the total input signal power increases. The optical switches add noise due to non-ideal isolation between ports. And the transmission fibers add noise due to FWM when the signal wavelength is close to

3

zero dispersion wavelength (λ0 ) and also induce pulse broadening due to PMD. In this paper, we neglect the effect of chromatic dispersion assuming that it is totally compensated in the network links. The consideration of residual dispersion in under development and will be the subject of future publication. Fig. 1 shows the configuration of the network wavelength routing node. It is similar to the one assumed in [4].

Switch 1

1

1

Switch 2

2

. .

. .

2

. .

. .

. . Switch W

M Optical fiber

Optical amplifier

M

DEMUX

1

MUX

2

..

W W

TX

..

2

Optical amplifier

Optical fiber

1

RX

WAVELENGTH ROUTING NODE

Fig. 1 Network wavelength routing node configuration, for M fibers with W wavelengths.

Fig. 2 shows the network devices considered in our model in each link. The links have the following elements: transmitter, optical switch, multiplexer, booster amplifier, optical fiber, pre-amplifier, demultiplexer, optical switch and receiver. The points a to h in Fig. 2 are evaluation points where the signal and noise can be determined in the optical domain. In point a, we have the input optical signal power (Pin ) and the input optical noise power (Nin ). The ratio between Pin and Nin defines the OSNR of the transmitter (OSNRin ). For the lightpath with i links, the elements between b and h are repeated i − 1 times before the signal reaches the receiver in the destination node.

TX

a

X

b

c

d

e

f

g

X

h

RX

generated by each optical switch in one wavelength is given by [30]:

NSwitch = ε

n X

PSwj (λ),

(1)

j=1

where PSwj (λ) is the optical power from the jth optical fiber in the same wavelength of the reference optical signal, ε is the switch isolation factor and n is the number of signals in the same wavelength at the switch input ports. For the node configuration used here, shown in Fig. 1, n = M . Pointurier et al [31], considered that ε has a dynamic effect depending on the network status, whereas in [32] the authors considered that fiber nonlinearities enhance the detuned crosstalk when signals are transmitted over long distances. For the sake of simplicity, we considered here that ε is the same for every wavelength, i. e. it is not channel-dependent. At points c and g we consider the contribution of the multiplexer and demultiplexer elements. BrandtPearce et al [33] considered a source of crosstalk due to imperfect WDM demultiplexing. They found that it is significant for high density of channels in the network. However, in this paper we take into account the MUX and DEMUX losses only. At points d and f in Fig. 2, we take into account the noise induced by the optical amplifiers, as well as the gain saturation effect. Considering the signal-spontaneous beating as the main noise source, this noise can be quantified by [34]:

Namp = PASE =

hf Bo Gamp Famp , 2

(2)

where h is the Planck constant, f is the optical signal frequency, Bo is the optical filter bandwidth, Gamp is the linear dynamic amplifier gain and Famp is the amplifier noise factor. The amplifier gain saturation effect is taken into account by using the following expression [8; 9]: G0 , Pout 1+ Psat

Fig. 2 The link configuration for adjacent nodes, with optical devices considered in our analytical model.

Gamp =

At points b and h in Fig. 2, we consider the noise induced by coherent crosstalk in the optical switches assuming that incoherent crosstalk can be filtered out on the switch output. This effect occurs basically because of energy leakage from other co-propagating signals to the reference optical signal, in the same wavelength, due to non-ideal optical switches. The optical noise power

where G0 is the non-saturated amplifier gain, Pout is the optical power at the amplifier output and Psat is the amplifier output saturation power. One can note in (3) that Gamp depends on the optical power at amplifier input. Since Famp also depends on the input optical power, we empirically developed the following expression to

(3)

4

obtain the real behavior of Famp as a function of the input optical power:    Famp = F0  1 + A1 −

A1  , Pin  1+ A2

(4)

5

10

15

30

10

25

9

20

8

15

7

10

6

5

5

0

4 -30 -25 -20 -15 -10 -5

0

5

10

η = D2 γ 2 Pi Pj Pk e−αd 9

"

1 − e−αd α2


2 #

,

(5)

′

where η is the FWM efficiency, D is the degeneracy factor which is equal to three or six for degenerate and nondegenerate FWM, γ is the nonlinear coefficient, Pi , Pj and Pk are the input powers for the signals at frequencies fi , fj and fk , respectively, α is the fiber attenuation coefficient and d is the fiber length. The frequencies generated by FWM process can be determined by [36]:

fijk = fi + fj − fk ,

(6)

where indexes i and j are different from k. Considering every optical power component generated by FWM in the reference signal wavelength, we have: Amplifier noise figure (dB)

Amplifier gain (dB)

0

PF W M (λ) = Pijk (λ) ′

where F0 is the amplifier noise factor for low input optical powers, A1 and A2 are function parameters. These parameters can be obtained by fitting experimental results. Fig. 3 shows the amplifier gain and noise figure as a function of input optical power per channel from an Erbium doped fiber amplifier (EDFA) built in our laboratories. The experimental results are typical from EDFAs and are represented by symbols and the model results are represented by solid curves. The function parameters of equations (3) and (4) that fit the experimental results of Fig. 3 are: G0 = 1000 (30 dB), F0 = 3 (4.77 dB), Psat = 15 dBm, A1 = 500 and A2 = 2 W. -30 -25 -20 -15 -10 -5

power can be evaluated using the following equation proposed by Song et al [37]:

15

Input optical power per channel (dBm) Fig. 3 Amplifier gain and amplifier noise figure as a function of input optical power per channel obtained from experimental results (symbols) and fitting model (solid curves).

In real optical amplifiers, both the ASE and the amplifier gain decrease as the input optical power increases [35]. In Fig. 3 one can observe that the noise figure increases with the input optical power. However, the amplifier gain decays faster than the increase in noise figure. As a consequence, using (2), the ASE power decreases when the optical power increases. At point e in Fig. 2, we consider the noise generated by FWM effect [36]. This nonlinear effect depends on the channel spacing, optical signal power per channel, number of wavelengths propagating in the optical fiber, fiber dispersion coefficient and the zero dispersion wavelength of the transmission fiber. Each FWM generated

NF W M =

m X

PF W Mj (λ),

(7)

j=1

where NF W M is the noise power due to FWM and PF W Mj (λ) is one of the m optical power components generated by the FWM effect that falls into the same propagating signal wavelength. Finally, at the point h in Fig. 2, one can evaluate the output optical signal power (Pout ) and the output optical noise power (Nout ). Pout is evaluated according to the gains and losses along the signal propagation and it is given by:

Pout =

Gamp1 e−αd Gamp2 Pin , L2Switch LMux LDemux

(8)

where Gamp1 and Gamp2 are the dynamic linear gains of the booster and pre-amplifier, α is the fiber loss coefficient, d is the fiber length, LSwitch , LMux and LDemux are the optical switch, multiplexer and demultiplexer losses. Nout is evaluated from the source node to the destination node, including the noise components in the respective points of evaluation (a to h in Fig. 2) along the lightpath and it is given by:

5

Nout =

where B is the transmission bit rate, DP MD (j) is the PMD coefficient, and d (j) is the length of the jth link belonging to the lightpath. The ∆t should be lower than a pre-determined maximum pulse broadening (δ).

Gamp1 e−αd Gamp2 Nin + LMux LDemux L2Switch n X Gamp1 e−αd Gamp2 ε PSw1,j (λ)+ + LMux LDemux LSwitch j=1

Gamp1 e−αd Gamp2 + LDemux LSwitch   hν (λ) Bo Famp2 + Famp1 + −αd 2 e Gamp1 m X Gamp2 PF W Mj (λ)+ + LDemux LSwitch j=1 +ε

s X

3 Simulation characteristics (9)

Fig. 4 shows the flowchart of our simulation algorithm using the shortest path as the routing metrics. This algorithm is used in Section 4 for physical impairments analysis purposes.

Call request

PSw2,j (λ),

j=1

where Nin is the noise power at the transmitter output. Dividing Pout by Nout , one can obtain the OSNR at destination node (OSNRout ). A threshold OSNR can be established to guarantee the QoS (OSNRQoS ) for call requests on the network. Considering a route with i links, we have:

Pouti =

Gampi,1 e−αdi Gampi,2 LMux LDemux LSwitch

!

(10)

Pouti−1

Nouti =

Gamp1,i e−αdi Gamp2,i Nouti−1 + LMux LDemux LSwitch Gamp1,i e−αdi Gamp2,i hν (λ) Bo + LDemux LSwitch 2   Famp2,i Famp1,i + −αdi + e Gamp1,i m X Gamp2,i PF W Mi,j (λ)+ + LDemux LSwitch j=1 +ε

s X

where Pout0

Yes

Acceptable OSNR

(11)

Pin and Nout0 LSwitch

=

Nin + LSwitch

PSw1,j (λ).

Futhermore, we also consider the pulse broadening effect caused by PMD in a route using the following expression [38]:

P MD

j=1

(j) d (j),

No

Yes

Fig. 4 Flowchart of the routing and wavelength assignment algorithm employed in the network simulations presented in Section 4.

j=1

v u i uX 2 D ∆t = B t

No

Establish call request

PSwi+1,j (λ),

=

Block call request

Yes

j=1

n X

No Wavelength available

Acceptable pulse broadening

and

ε

Route selection

(12)

For each network simulation, a set of at least 105 calls are generated choosing randomly the source-destination pair. The call request process is characterized as a Poisson process and the time duration for each established call is characterized as a exponential process. Our simulation algorithm works as follows: upon a call request it determines a route using a pre-defined metric. Then, it selects an available wavelength from a list using the first fit algorithm. The lightpath OSNR is evaluated. If it is above the pre-determined level the call is established (OSNRQoS ). Our algorithm blocks

6

a call if the pulse broadening (∆t) is above the maximum level (δ), if there is no available wavelength or if the OSNR for the respective wavelength is below the OSNRQoS . The blocked calls are lost. The blocking probability is obtained from the ratio of the number of blocked calls and the number of requested calls. We assume circuit-switched bidirectional connections in two fibers and no wavelength conversion capabilities. We used two different networks in our simulations to avoid any particularity that may raise from a specific network topology. Fig. 5(a) shows a regular network topology. Whereas the network presented in Fig. 5(b) has a irregular topology and it is similar to the NSFnet, but with different distances. The amplifier gains are initially set to compensate for the total link losses and the default parameters used in our simulations are shown in Table 1. The network parameters (mainly the number of wavelengths) were chosen such that the call blockings were mainly due to OSNR degradation, instead of lack of available wavelenghts. 120 km

1

2 60

km

km

60 70 km

70 km 70 km

8 60

km

km

60

3

(a)

11 76 km

km

m 24 k

22 km

35 km

7

5

6

9

8

72 km

55 km

12

45 km 65 km

3

4

km

35 km

21 km

22 km

25 km

26 km 27 42 km

1

30 km

km

25

km 48

2

100

Value

Psat

16 dBm

OSNRin

30 dB

OSNRQoS

23 dB

B Bo W

40 Gbps 100 GHz 36

∆f λi

100 GHz 1550.12 nm

λ0 α LM ux LDemux LSwitch F0 (NF)

1510 nm 0.2 dB/km 3 dB 3 dB 3 dB 3.162 (5 dB)

A1

100

A2

4W

ǫ δ DP M D Load

−40 dB 10% √ 0.05 ps/ km 60 Erlangs

Definition Amplifier output saturation power. Input optical signal-to-noise ratio. Optical signal-to-noise ratio for QoS criterion. Transmission bit rate. Optical filter bandwidth. Number of wavelengths in an optical link. Channel spacing. The lower wavelength of the grid. Zero dispersion wavelength. Fiber loss coefficient. Multiplexer loss. Demultiplexer loss. Optical switch loss. Amplifier noise factor (Noise figure). Noise factor model parameter. Noise factor model parameter. Switch isolation factor. Maximum pulse broadening. PMD coefficient. Network load.

on the network performance. We evaluate the network performance in terms of blocking probabilities using the shortest path algorithm to determine routes between the source and destination nodes, as described in Section 3. Otherwise stated, the simulation parameters are given in Table 1.

7

120 km

4

Parameter

120 km

6

70 km

120 km

5

Table 1 Default simulation parameters.

m 13 28 k

14 10

(b) Fig. 5 The optical networks used in our simulations: (a) regular topology and (b) american topology. Node distances are shown.

4 Physical impairments analysis In this section we apply our proposed model for the evaluation of the impact of each physical impairment

Fig. 6 shows the blocking probability as a function of input optical power per channel for different network loads, considering two different optical network topologies, and for a amplifier output saturation power equal to 19 dBm. Note that for each network topology the minimum blocking probability is obtained for a different input optical power per channel, which is −1 dBm for Fig. 6(a), and −3 dBm for Fig. 6(b). For higher laser powers, the coherent crosstalk, FWM effect and gain saturation effect are responsible for the increase in the blocking probability, since they increase with the increase in the signal power, as seen in Eq. (1), (3), (5) and (7). For lower laser powers, the ASE becomes more relevant, since this noise source does not decrease with the decrease in the signal power, as seen in Eq. (2). The blocking probability of requested calls increases when the network load increases. In fact, as the network traffic becomes higher, the physical impairments become more relevant, causing more OSNR degradation

7

to the signals. However, we verified in our simulations that for network loads of 80 Erlangs or higher the call blockings due to lack of available wavelengths is no longer negligible. 1

Blocking probability

0.1

0.01 Regular topology 40 Erlangs

in Fig. 7 (60 Erlangs). Fig. 7 also shows that the amplifier saturation power has a big impact on the network performance when the blocking probability is limited by the QoS constraint (i. e. for more than 30 wavelengths available). A 3 dB increase in its value (from 16 dBm to 19 dBm) leads to a drop in the blocking probability by one order of magnitude for the regular topology and by two orders of magnitude for the american topology. However, when the network blockings are due to lack of available wavelengths (i. e. for less than 20 wavelengths in the networks studied here), the physical impairments have little impact.

60 Erlangs

1E-3

80 Erlangs

1

100 Erlangs 120 Erlangs

1E-4 -15

-10

-5

0

5

Input optical power per channel (dBm)

(a) 1

0.1

Blocking probability

0.1

10 Blocking probability

-20

0.01

0.01

Regular topology

1E-3

Psat = 16dBm Psat = 19dBm American topology

1E-4

Psat = 16dBm Psat = 19dBm

1E-3 1E-5 1E-4

0

American topology

60 Erlangs

1E-5

20

30

40

50

60

Fig. 7 Blocking probability as a function of number of wavelengths in a link, considering two different optical network topologies for different amplifier output saturation powers.

80 Erlangs 100 Erlangs

1E-6

10

Number of wavelengths in a link

40 Erlangs

120 Erlangs

1E-7 -20

-15

-10

-5

0

5

10

Input optical power per channel (dBm)

(b) Fig. 6 Blocking probability as a function of input optical power per channel for different network loads, considering two different optical network topologies: (a) regular topology and (b) american topology, and for amplifier output saturation power equal to 19 dBm.

Fig. 7 shows the blocking probability as a function of the number of wavelengths in a link, considering two different optical network topologies, for different amplifier output saturation powers. The network load is 60 Erlangs. There is a threshold in terms of number of wavelengths in each network, above which the blocking probability is due to the physical impairments (OSNR degradation). For less than this threshold, the blocking probability is caused basically by the lack of available wavelengths. One can verify that 36 wavelengths in a link are necessary to obtain the minimum blocking probability for both amplifier output saturation powers in both network topologies considered. Adding more wavelengths would not result in any improvement of the network performance, for the network load considered

Fig. 8 shows the blocking probability as a function of the switch isolation factor, considering two different optical network topologies for different amplifier output saturation powers. We note that for ε below −40 dB the effect of coherent crosstalk can be neglected, and the network blockings are limited by other physical impairments. When ε is increased beyond −40 dB, we observe that the impact of coherent crosstalk increases sharply. Fig. 8 shows that the switch isolation factor is a critical device parameter for the performance of all-optical networks. Fig. 9 shows the blocking probability as a function of input optical power per channel, considering two different optical network topologies for different noise figures and amplifier output saturation powers. We observe that only 2 dB difference in the amplifier noise figure has a considerable impact on network performance. Also, one can note that for higher noise figures, the input optical power that achieves lowest blocking probability shifts toward high powers. This is because the ASE noise increases for higher noise figures. Fig. 10 shows the blocking probability as a function of input optical power per channel, considering two

8 1

Blocking probability

0.1

0.01

1E-3 Regular topology Psat = 16dBm

1E-4

Psat = 19dBm American topology

1E-5

Psat = 16dBm Psat = 19dBm

1E-6 -60

-50

-40

-30

-20

-10

Switch isolation factor (dB)

Fig. 8 Blocking probability as a function of switch isolation factor, considering two different optical network topologies for different amplifier output saturation powers.

dispersion shifted fiber (DSF) is used, λ0 = 1550 nm, we observe that the network blocking probabilities are higher than in the NZ-DSF case. In this case the use of amplifiers with higher output saturation power, such as Psat = 19 dBm, does not bring much improvement to the network performance. It occurs basically because the FWM effect becomes more relevant for this type of optical fiber. Fig. 10 also shows that for DSF fibers it is necessary to use lower input optical powers for achieving lower blocking probabilities. However, for the regular topology the blocking probability is very high and in this case one should use larger channel spacing to reduce the FWM effect. A FWM aware wavelength assignment algorithm, rather than first fit, should also improve network performance.

1

0.1 Blocking probability

Blocking probability

1

Regular topology Psat = 16dBm NF = 5dB

0.01

NF = 7dB Psat = 19dBm NF = 5dB

0.1 Regular topology Psat = 16dBm 0

0.01

0

0

0

1E-3 -15

-10

-5

0

5

10

= 1550nm

Psat = 19dBm

NF = 7dB

-20

= 1510nm

= 1510nm = 1550nm

1E-3

Input optical power per channel (dBm)

-20

(a)

-15

-10

-5

0

5

10

Input optical power per channel (dBm)

(a)

1 1

0.1 Blocking probability

Blocking probability

0.1

0.01 American topology Psat = 16dBm

1E-3

NF = 5dB NF = 7dB

1E-4

Psat= 19dBm NF = 5dB NF = 7dB

0.01

American topology Psat = 16dBm 0

1E-3 0

1E-4

0

0

-15

-10

-5

0

5

10

Input optical power per channel (dBm)

(b)

= 1550nm

Psat = 19dBm

1E-5 -20

= 1510nm

= 1510nm = 1550nm

1E-5 -20

-15

-10

-5

0

5

10

Input optical power per channel (dBm)

(b) Fig. 9 Blocking probability as a function of input optical power per channel, considering two different optical network topologies: (a) regular topology and (b) american topology, for different noise figures and amplifier output saturation powers.

different optical network topologies for optical fibers with different zero dispersion wavelengths and amplifier output saturation powers. When the non-zero dispersion shifted fiber (NZ-DSF) is used, λ0 = 1510 nm, we obtain the minimum blocking probabilities. When the

Fig. 10 Blocking probability as a function of input optical power per channel, considering two different optical network topologies: (a) regular topology and (b) american topology, for optical fibers with different zero dispersion wavelengths and amplifier output saturation powers.

Fig. 11 shows the blocking probability as a function of the PMD coefficient, considering two different optical network topologies for different amplifier out-

9

put saturation powers, for a bit rate of 40 Gbps. We note that there is a threshold behavior for the PMD coefficient (DPT hMD ). For DP MD below the threshold, the PMD can be neglected and the call blockings occur by the OSNR degradation. For DP MD above DPT hMD , the PMD effect becomes more relevant and it is the cause of most of the call blockings. And in this case, the blocking probability can not be improved by increasing the amplifier saturation power because the PMD affects the signals in the time domain.√ For the regular topology the amerwe found DPT hMD = 0.16 ps/ km, and for √ Th ican topology we found DP MD = 0.19 ps/ km. The regular topology is more sensitive to PMD because its average route length is longer (124.02 km) than for the american topology (84.99 km). 1

Blocking probability

0.1

OSNRQoS the connection is blocked. The wavelength assignment problem is solved by using the first fit algorithm [39]. It must be highlighted that for all wavelengths the route found by SP weight function is the same. This is not the case for LRW and OSNR-R algorithms. For the OSNR-R algorithm the different wavelengths in the same route may show different noise accumulation. The OSNR-R algorithm works as follows: upon a call request it selects a wavelength from a list using first fit algorithm. The route is determined by using the OSNR as the cost function in the Dijkstra algorithm. If the OSNR of the lightpath is above the pre-determined level (OSNRQoS ) the call is established. The algorithm blocks a call if the pulse broadening (∆t) is above the maximum level (δ), if there is no wavelength available or if the OSNR for the respective wavelength is below the OSNRQoS . Fig 4 shows the flowchart of SP and LRW algorithms and Fig 12 shows the flowchart of the OSNR-R algorithm employed in our network simulations.

0.01

Regular topology

1E-3

Psat = 16dBm Psat = 19dBm American topology

1E-4

Psat = 16dBm

Call request

Psat = 19dBm

1E-5 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Polarization mode dispersion coefficient (ps/km

1/2

)

Fig. 11 Blocking probability as a function of PMD coefficient, considering two different optical network topologies for different amplifier output saturation powers, for a bit rate of 40 Gbps.

No Wavelength available

Block call request

Yes Route selection

5 RWA analysis In this section we apply our model as a metric for a impairment-constraint routing and wavelength assignment algorithm. We present a RWA algorithm that finds the routes based on the minimum OSNR for a given available wavelength. Our OSNR based routing algorithm (OSNR-R) is compared to the shortest path (SP) [13] and to the least resistance weight (LRW) [14] algorithms in terms of blocking probability and computation costs. Each of these algorithms have their own metrics or weight functions. These weight functions are used as a cost function in Dijkstra’s shortest cost algorithm to solve the routing problem considering physical impairments along the lightpath. For the sake of fairness in the comparison among algorithms we use our proposed model to calculate the OSNR of the lightpaths determined by each routing algorithm. If the route found by the routing algorithm has an OSNR below the

Acceptable pulse broadening

No

Yes

Acceptable OSNR

No

Yes Establish call request

Fig. 12 Flowchart of the OSNR based routing and wavelength assignment algorithm employed in our network simulations.

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5.1 Shortest Path

Then, the route found by OSNR-R in λ can be expressed by

The link weight function wi,j is computed on a per-link basis for a given link between nodes i and j. The weight function is given by [13] wi,j = di,j ,

(13)

where di,j is the physical link length between nodes i and j.

5.2 Least Resistance Weight The link weight function wi,j is computed on a per-link basis for a given link between nodes i and j. The LRW weight function is given by [14]

wi,j

 T   Cmax A Ci,j =  ∞

A 6= 0, if Ci,j

(14)

A if Ci,j = 0,

A where Ci,j denotes the current number of available waT velengths on the link, Ci,j denotes the total number of T wavelengths on the link, and Cmax represents the maxT imum number of wavelengths in the link; i.e., Cmax = T max(Ci,j ).

5.3 OSNR based routing The idea behind this weight function is finding a route with minor OSNR degradation for a given wavelength. One can note from the noise formulation (Eq. (11) in Section 2) that the output OSNR in a chain of i links is dependent on the previous links in the chain. Therefore, the relative magnitude of the noise penalty induced by the ith link is dependent on noise accumulation in previous links in the chain. Thus, for the OSNR-R algorithm it is not correct to model a weight function for a given link independent of the chosen route as considered in the other two approaches (SP and LRW). However, OSNR-R can be easily implemented with Dijkstra’s minimum cost algorithm. It can recalculate total accumulated OSNR from source node to the currently visited node. It must be highlighted that a higher OSNR in lightpath means a better signal quality. Therefore, Dijkstra’s algorithm must be set to find the maximum value for OSNR instead of the common use of this algorithm that tries to minimize some metrics. Mathematically, if π(s, d) represents all possible routes between nodes s and d, and fOSN R [π(s, d), λ] represents the output OSNR for these routes in the wavelength λ.

λ = max {fOSN R [π (s, d) , λ]} . Rs,d

(15)

Fig. 13 shows the blocking probability as a function of network load, considering two different optical network topologies, for the three different routing algorithms. One can note that for both network topologies the OSNR-R algorithm presents lower blocking probabilities for all network loads. This is because the OSNRR algorithm finds the route with minor OSNR degradation. Thus, it provides load balance to the network, since if some links become busier, the OSNR of the signal passing these links will degrade and, therefore, the OSNR-R algorithm will tend to avoid such links, finding alternative routes. As a result, the blocking probability due to low QoS (unacceptable OSNR) is lower for the OSNR-R than for the other algorithms that do not take into account the physical impairments in the routing process. Fig. 13 shows that the OSNR-R algorithm outperforms the SP and LRW algorithms in terms of blocking probability. However, we must also compare the time spent by these approaches to solve the RWA problem. We used a Pentium Core 2 Duo with 2.13 GHz and 3 GB of RAM to perform this comparison. The results for the average time spent by the OSNR-R algorithm to solve the RWA per call are shown in Fig. 14, for the regular and American topologies, as a function of network load. The average time is obtained from 5 simulations of 10, 000 calls each. Fig. 14(a) shows that the OSNR-R algorithm can take up to 63 ms and 134 ms to find a route in the regular and American topologies, respectively. And also that the average route computation time has a linear dependence with network load, with a slope of 0.4 ms/Erlang for the regular and 1.0 ms/Erlang for the American topologies. Whereas the LRW algorithm solves the RWA problem in about 0.7 ms for the regular topology and about 1.2 ms for the American topology, and it presents no dependence with network load. It should be noted that the LRW algorithm considered here performs a QoS verification after route definition by calculating the signal OSNR at the end of the chosen route and checking it against the acceptable value (OSNRQoS ). We found that this OSNR evaluation is the main contribution to the LRW algorithm computation times. The results for the LRW are not shown in Fig. 14. Therefore, the OSNR-R algorithm solves the RWA problem up to 112 times slower than LRW. This is because of the time consuming calculations to evaluate the physical impairments performed by the OSNR-R.

11 150

Blocking probability

0.1

0.01

1E-3

Regular topology

1E-4

LRW SP OSNR-R

1E-5 20

40

60

80

100

Average route computation time (ms)

1

120

90

60

30

All physical impairments Regular topology American topology

120

0

Network load (Erlangs)

30

(a)

60

90

120

Network load (Erlangs)

(a)

1 2.0

0.01

1E-3 American topology LRW

1E-4

SP OSNR-R

1E-5 40

60

80

100

120

140

160

Network load (Erlangs)

(b)

Average route computation time (ms)

Blocking probability

0.1

1.5

1.0

0.5

No FWM effect Regular topology American topology 0.0 30

60

90

120

Network load (Erlangs)

Fig. 13 Blocking probability as a function of network load, considering two different optical network topologies: (a) regular topology and (b) american topology, for three different routing algorithms.

These long route computation times presented by the OSNR-R algorithm, reaching a hundred milliseconds, can be regarded as prohibitive for practical applications. However, we found that the FWM effect is the main responsible for them. In Fig. 14(b) we present similar results to Fig. 14(a), but considering no FWM effect in the OSNR calculations. In this case one can see that the OSNR-R algorithm can take up to 0.6 ms and 1.5 ms to find a route in the regular and American topologies, respectively. The average route computation time still presents a linear dependence with network load, with a slope of 0.003 ms/Erlang for the regular and 0.010 ms/Erlang for the American topologies. Whereas the LRW algorithm solves the RWA problem in about 0.06 ms for the regular topology and about 0.13 ms for the American topology, and again it presents no dependence with network load. In this situation the OSNR-R algorithm is up to 11.5 times slower than the LRW algorithm. And more important, the OSNR-R route computation times lie within the range of sub-

(b) Fig. 14 Average time spent by the OSNR-R algorithm to solve the RWA per call, for the regular and American topologies, as a function of network load, considering: (a) all physical impairments and (b) no FWM effect.

milliseconds, reaching just over a millisecond, which can be considered reasonable for practical applications. Moreover, the average route computation times can be significantly reduced if one uses a more powerful computer for route calculations. Also the FWM effect evaluation algorithm should be further improved and simplified to reduce its computation complexity. We did not consider the SP algorithm for time analysis since it is not an adaptive routing algorithm.

6 Conclusions We presented a novel model to consider several physical impairments in all-optical networks. Our model is based on the degradation of OSNR along the lightpaths and it considers the effects of gain saturation and ASE noise in amplifiers, coherent crosstalk in optical switches, and FWM in the optical fibers.

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To our knowledge, we are the first to consider these effects all together in a simple model based on OSNR, using analytical equations obtained from well known fundamental or experimental behavior of network devices. Moreover, we are also the first to consider the dependence of gain, noise factor and overall amplifier noise power on the total signal power. We presented an application of our model for the evaluation of network performance in terms of blocking probability. Our results show the impact of each impairment on network performance as a function of device parameters. For low signal powers the blocking probability is mainly due to the amplifiers noise, whereas for high signal powers the main contribution to the blocking probability comes from the FWM effect, coherent crosstalk and gain saturation effect. We note that the optimum signal power depends on network topology and network device parameters. Our simulation results also show that network performance is highly dependent on device parameters, such as amplifier output saturation power, amplifier noise figure, switch isolation factor, and fiber type. These device parameters have considerable impact on device costs. To our knowledge, we are the first to present such type of evaluation, i. e. network performance in terms of device parameters. Therefore, the model, as well as the simulation results for the network cases presented in this paper may be of great interest to network designers. We also presented an application of our proposed model in impairment-constraint routing. The proposed RWA algorithm finds the route based on the OSNR of the lightpaths. The simulation results showed that our OSNR-R algorithm outperforms the shortest path (SP) and least resistance weight (LRW) algorithms in terms of blocking probability. However, the OSNR-R is much slower (about 100 times) than the LRW algorithm, reaching average route computation times in the order of a hundred milliseconds. We also found that the most time consuming part of the OSNR calculation is the FWM effect evaluation. If we neglect the FWM effect the average route computation time is reduced to about one millisecond. The FWM effect can be neglected if NZ-DSF or standard fibers are used. Therefore, we believe our model has applications in routing and wavelength assignment algorithms, and also in network planning to balance costs and performance. Moreover, since our model is based on OSNR, it is compatible with some signal monitoring techniques, that may be implemented in some strategically chosen points in the network to measure the actual OSNR and feed this information to the model, improving its accuracy and functionality.

Acknowledgments The authors acknowledge the financial support from FACEPE, CNPq and CAPES for scholarships and grant.

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