February 2012, 19(1): 119–123 www.sciencedirect.com/science/journal/10058885 http://jcupt.xsw.bupt.cn

The Journal of China Universities of Posts and Telecommunications

Improved wavelength decomposition approach for computing blocking probabilities in WRONs

YANG Peng, ZHANG Jie (�), ZHAO Yong-li, GU Wan-yi

State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract

Wavelength decomposition approach has been proposed to compute blocking probability (BP) of fixed routing in wavelength-routed optical networks (WRONs) without wavelength conversions. By means of wavelength decomposition, a WRON can be regarded as a set of different layers (colors), in which blocked traffic in one layer is overflowed to another layer. A novel iterative scheme is put forward in case of BP matching used to characterize the overflow traffic from one layer to another in this paper. Furthermore, the analysis of BP based on the improved wavelength decomposition approach is derived and results show that it yields higher calculation accuracy.

Keywords WRONs, wavelength decomposition, blocking probabilities

1 Introduction �

It is expected that future optical networks will be exposed to not only increase traffic volumes, but also improve diversity of services and dynamically vary traffic patterns. WRON architectures can potentially simplify routing and processing functions in high-capacity wavelength division multiplexing (WDM) networks. Considering wavelength continuity constraint a lightpath may fail to be established due to lack of available network resources. So it is important to estimate the connection BP for the expected traffic demands in WRONs.

However, it is hard for the performance analysis of BP in WRONs under the condition of first-fit wavelength assignment (WA) [1]. According to first-fit (FF) WA, wavelengths are searched in a fixed order and traffic flows that are blocked from using a wavelength are offered to the next wavelength in line [2]. The layered graph model based on wavelength decomposition approach has been studied for computing BP [3–6]. Link independence assumption leads to an overestimation of BP in Ref. [3]. It

Received date: 26-04-2011 Corresponding author: ZHANG Jie, E-mail: lgr24@bupt.edu.cn DOI: 10.1016/S1005-8885(11)60236-7

has to make an object independence assumption, where the object is a free link or path in Refs. [4–5] derives an iterative model to calculate BP in WDM optical networks, which stratifies the network according to wavelength continuity requirement as shown in Fig. 1. The overflow traffic from one layer to another is characterized by a moment matching method. The overflow traffic is characterized as a Bernoulli-Poisson-Pascal (BPP) process in Refs. [6–7] extends the technique presented in Ref. [5] to compute quality of transmissions (QoT) BP in transmission impaired optical networks.

Fig. 1 Layered network model for WRON

In this paper, we review the wavelength decomposition approach in Ref. [5] and propose a new overflow criterion, i.e. BP matching, to characterize the overflow traffic in layered network model. Analysis results show that the

120 The Journal of China Universities of Posts and Telecommunications 2012

novel approach yields higher calculation accuracy than the traditional one based on moment matching method.

2 Improved analytical scheme for computing the BP

In this section, the improved analytical scheme that is used in our approach to compute the BP is proposed. Y is the set of all source-destination pairs in the network. In general, a route from source node s to destination node d is denoted by ( , )r s d or ( )r y for y Y� . The following notations and assumptions will be used later:

1) y� is the Poisson arrival rate for y.2) 1� � is the mean of the call durations

(exponentially distributed). 3) w

yA is the equivalent Poisson offered load to

wavelength w for ( )r y , clearly 1yyA �� .

4) ,wi ja is the total equivalent Poisson offered load to

wavelength w for ( , )r i j .5) w

yP is the path BP for ( )r y on wavelength w.

2.1 BP analysis model using wavelength decomposition approach

As shown in Fig. 1, a WRON with W wavelengths in each link was decomposed to W layer networks. Each layer network has the same topology but one wavelength capacity in each link. According to the first-fit WA, the offered network traffic from source to destination first arrives at wavelength layer one, of which is blocked overflows to wavelength layer two, and so on. The overflow traffic from each layer is characterized by a BP matching method, adjusted for its bursty nature. The overflow traffic at layer W means the overall network blocked traffic, and then BP can be computed. The wavelength continuity constraint forces a connection to remain on the same wavelength along the path. It is automatically enforced in this approach.

Assuming the offered load is a Poisson traffic, the path BP in a single wavelength can be derived from an exact approach [5]. The state of an n hop path for a wavelengthw at time t can be expressed by a time-reversible Markov process and the stationary probability � can be given. The normalization constant (1, )

wr kG is computed

recursively as [5]

1

(1, ) (1, 1) (1, ) ,1

kw w w wr k r k r i i k

iG G G a

�

��

�

(1)

Where the sum of all equivalent Poisson traffic from all (s,d) paths on segment r(i, j) at wavelength w is calculated as [8]

,

, ,,

( , ): ( , ) ( , ) , assigned uniquely to ( , )

(1 )1

ws d

w ws d s dw

i j ws d r i j r s d i j

A i j

A Pa

P�

��

� (2)

Thus, the path r(1,k) BP is given by

(1, )(1, )

11 (0,0,...,0) 1wr k w

r k

PG

�� � � �

(3)

2.2 New overflow criterion based on BP matching

The traffic blocked at wavelength w flows down to the next wavelength (layer). However, the overflow traffic is in general non-Poisson distribution [8]. Many studies have analyzed the overflow traffic in conventional circuit switching network [8]. However, in Sect. 2.1, we calculate the path BP under the assumption that the offered load is Poisson traffic. The author gives an moment matching method uses an equivalent single-link system

11

wyN

wavelengths to find an equivalent Poisson traffic

with mean 1wyA and 1 1w

yZ � , that matches the overflow

traffic with mean 1wyA and 1 1w

yZ � [5]. Therefore, 1 1

11 11 1Er ,

w ww y yw wyy y w w

y y

A NA P AZ Z

� � ��� �� �

(4)

The value of 1w

yN

is calculated from

� �11 1Er ,ww wyy yP A N

�

(5)

The characteristics of the overflow traffic had been extensively studied [8]. The mean 1w

yA , the variance 1w

yV

and the peakedness 1wyZ are of particular interest.

However, the moment matching method only considers the mean moment of the overflow traffic, but ignores the other overflow traffic moments—variance and peakedness. And the overflow traffic has

1wyV

> 1w

yA (or 1wyZ >1), which

indicates that the overflow traffic is bursty. Hence, using only the mean value to characterize the overflow traffic will underestimates the BP.

In this paper, we propose a BP matching method to characterize the overflow traffic. With a single-link system of

11

wyN

wavelengths, we find an equivalent Poisson

Issue 1 YANG Peng, et al. / Improved wavelength decomposition approach for computing blocking probabilities in WRONs 121

load with mean 1wyA and 1 1w

yZ � , which makes the BP

matches the one when the offered load is a non-Poisson traffic with mean

1wyA

and variance

1wyV

. We can learn

from the Fredericks & Hayward’s approximation that a system with Non-Poisson traffic 1Z � has the same BP as a system with N/Z channels, offered traffic A/Z and peakedness value = 1 (Poisson). Therefore,

� �1 1

11 11 1Er , Er ,

w wwy yw wyy yw w

y y

A NP A NZ Z

� � �� �� �� �

(6)

The mean of the overflow traffic to the next layer is 1w w w

y y yA A P

�

(7)

The variance of the overflow traffic 1w

yV

is computed using Riordan’s formula as

1 1 1

111

w w w yy yy ww

yyy

V A AN A

�

�

� � � � �� � �� �

(8)

where wyN is the capacity of an equivalent single-link

system for layer 1 to w, which can be obtained from 1

Er( , )ww

y y yyN A� �

�

(9)

where Er( , )wy yN� is the generalized Erlang-B formula

for non-integral capacity. Then the peakedness is defined as the ratio between the

variance and the mean value, 1

1

1

ww yy w

y

VZA

� (10)

Above all, we can approximate the blocking probabilities and overflow traffics for each layer. Then the overall path BP is computed as

1W WWy yy

yy y

A P AP� �

� � (11)

The overall network BP is given by 1W

yy Y

yy Y

AP

�

�

�

�

(12)

3 Numerical examples and validation by simulations

In this section, the analytical technique described above is used to predict the BP for several all-optical networks; simulations are used to validate the results. We first compare the results of the moment matching to the BP matching given in Sect. 3 for the 7-node ring network shown in Fig. 2, and then do the same work for another

larger topology: the 14-nodes National Science Foundation network (NSFNET) network shown in Fig. 3. A uniform load is assumed for both analysis and simulation. Unless otherwise stated, simulation results are obtained by running more than 106 calls. In all analysis results,

210� �� . In Figs. 4, 5, 6, and 7, the conditions are shown as Table 1.

Fig. 2 Topology of 7-node ring network

Fig. 3 Topology of the NSFNET network with 14 nodes

Table 1 The conditions used in Figs. 4, 5, 6, and 7. Topology Nodes Offered load (Poisson) Link capacities

Ring 7 1 W = 7 Mesh 14 1 W = 14

We plot the overall BP against the link capacity (number of wavelengths W) as shown in Fig. 4 and Fig. 6. As the link capacity (wavelength) increases, the BP decreases.

Fig. 4 FF WA BP for the 7-node ring network with uniform load = 1

By means of wavelength decomposition, a WRON can be regarded as a set of different layers (colors). With more and more layers, the error of BP becomes larger, the original model in Ref. [5] loses accuracy and our improved

122 The Journal of China Universities of Posts and Telecommunications 2012

approach performs better. In Fig. 5 and Fig. 7, we plot the overall BP against the total network traffic load. As the traffic decreases, the BP decreases. Our improved approach also performs better.

Fig. 5 FF WA BP for the 7-node ring network with uniform W = 7

Fig. 6 FF WA BP for 14-node NSFNET network with uniform load = 1

Fig. 7 FF WA BP for 14-node NSFNET network with uniform load = 1

The results show that, our improved wavelength decomposition approach with BP matching method yields higher calculation accuracy than the traditional one [5] based on moment matching method in ring and mesh network. The moment matching method only considers the mean moment of the overflow traffic, but ignores the other

overflow traffic moments—variance and peakedness, which would underestimate the BP. By using BPs matching, the improved approach will fix the underestimating part. And with more and more complex network topology, the effect is more obvious.

4 Conclusions

We propose a BP matching method to characterize the overflow traffic from one layer to another, resulting in an improved model to calculate the blocking probabilities for fixed routing in WRON with arbitrary topologies and without conversions. As the moment matching method only considers the mean moment of the overflow traffic, but ignores the second overflow traffic moment--variance. Our approach is more accurate, which is showed by the final results.

Acknowledgements

This work was supported by the National Basic Research Program of China (2010CB328204), the Hi-Tech Research and Development Program of China (2008AA01A328, 2009AA01Z255), the National Natural Science Foundation of China (60932004), RFDP Project (20090005110013) and 111 Project (B07005) of China, and the Fundamental Research Funds for the Central Universities.

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