Contel.final

Performance Analysis of First-Fit Wavelength
Assignment Algorithm in Optical Networks
¡ ¡
Xuehong Sun , Yunhao Li , Ioannis Lambadaris and Yiqiang Q. Zhao

School of Mathematics and Statistics
Carleton University, 1125 Colonel By Drive
Ottawa, Ontario CANADA K1S 5B6
¡ Email: xsun@math.carleton.ca
Department of Systems and Computer
Carleton University, 1125 Colonel By Drive
Ottawa, Ontario CANADA K1S 5B6
Email: ioannis@sce.carleton.ca

Abstract— This paper proposes a new analytical technique the exact optimal solution at the expense of complexity. The
for the performance analysis of all optical networks which use other approach is to consider the Routing problem and Wave-
the first-fit algorithm for wavelength assignment. We analyze length Assignment problem separately. The solution obtained
the wavelength usage on the links to calculate the blocking
probability of a source destination pair, taking into account is suboptimal but practical to use. Several routing algorithms
wavelength correlation and load correlation between links. Our are proposed in the literature of which some representatives
model is accurate even in a system with large number of are listed below:
wavelengths. ¢ Fixed routing [5]: The path for each source-destination (
§¥£
¦ ¤ ) pair is calculated off-line using an algorithm, say,
I. I NTRODUCTION
Dijkstra algorithm.
Transmission systems using copper line are limited in band- ¢ Fixed alternate routing [6], [5]: Instead of calculating one
width and constrained by electronic processing speed. With path for each pair, fixed alternate routing calculates off-
the advance of optical technology, copper lines are gradually line several alternate paths for each pair.
replaced by optical fiber. Optical transmission systems provide ¢ Adaptive routing [7]: The paths are calculated on-line, de-
high bandwidth. Sonet, a system using optical fiber, can pending on the network state which reflects the resource
transmit data in several gigabytes per second per fiber. Since usage.
it uses only one channel for each strand of fiber to transfer
Wavelength Assignment algorithms, among possible others,
data, such optical systems are inefficient in using the band-
include
width. Wavelength Division Multiplexing (WDM) technology
[1] is proposed as an alternative method for carrying data.
¢ Random wavelength assignment [8]: A wavelength is
With WDM, one strand of fiber can accommodate about 100 selected randomly from the available wavelengths.
channels nowadays (2003) and more in the future. WDM
¢ First-fit assignment [9], [7]: All wavelengths are num-
systems are also transparent for data format and data rate (i.e. bered. The wavelength with the lowest number is selected
they can transfer data in any format with any rate), therefore, from the available wavelengths.
they promise to integrate data and voice into one telecom-
¢ Most-used assignment [7]: The most used wavelength
munication system. Sitting in the heart of WDM technology is the wavelength that has the highest number of links
is the Routing and Wavelength Assignment (RWA) problem in the network that use the wavelength. The most-used
[2]. Different from the routing problem in traditional circuit algorithm selects the most used wavelength from the
switched systems, RWA needs to find a path and assign the available wavelengths on the path.
same wavelength to every link along the path. This is known
¢ Distributed relative capacity loss (DRCL) [10].
as the wavelength continuity constraint. This means there are First-fit does not require global knowledge about the net-
cases in which connections can not be established even if work. No storage is needed to keep the network states and
every link along the path has wavelengths available. Thus, no communication overhead is needed. The computational
the shortest path obtained by executing Dijkstra or Bellman- overhead is small and the complexity is low. Moreover, the
Ford algorithm [3] is not necessary the best solution to the performance in terms of blocking probability and fairness is
RWA problem. In the literature, there are two methodologies among the best. Therefore, first-fit is preferred in practice
proposed to tackle the RWA problem. One is to consider [10], [11]. Although the performance analysis of optical WDM
the Routing problem and Wavelength Assignment problem as networks is studied extensively in the literature [8], [12], [13],
coupled RWA problem and model the RWA as mixed integer [14]and [15], there is less work for analyzing the blocking
linear programming problem [4]. This approach tries to obtain performance of first-fit scheme. The overflow model is used

5
to study the first-fit scheme [7], [16]. Reference [7] simply 1

uses Erlang-B formula to approximate the overflow traffic from
individual wavelength. However, the overflow traffic is bursty,
therefore, modeling it as Poisson process underestimates the 1 2

1
link blocking probability. Reference [16] applies the equivalent 2

random method [5] to approximately calculate the blocking
2 3 4
probability of the overflow traffic, but the accuracy of the
model decreases as the number of wavelength increases [16].
Fig. 1. A simple case.
In this paper, we provide a new method to analyze the first-
fit scheme which remains reasonably accurate with a large
number of wavelengths. n
The paper is organized as follows. Section 2 describes nC
1 2 3
the framework for the performance analysis. Numerical and
simulation results are provided in section 3. We conclude in
section 4.
Fig. 2. Link with interference.
II. A NALYTICAL M ODEL FOR F IRST- FIT P ERFORMANCE
A. Network and Traffic Assumptions
We use the fixed routing algorithm to calculate the path although only one wavelength is used in link and link . © ¨ ©
between a source-destination ( ¦ ¤ £
) pair. Therefore, we If there were no paths carrying traffic to ( ) and ( )
¨

assume that the path between the ¦ ¤ £
pair is calculated off- pairs, the traffic between nodes and would not have been ¨
line and remains fixed, therefore, the path uniquely identifies blocked. With respect to path between nodes and , we call ¨
the ¦ ¤ £ pair. When a call comes, a connection will be the ( ) and (

© ) pairs new coming paths and (
©
) !¥
¨
established on the path if the same wavelength is free on all the and ( ) pairs interference paths to (
#
) pair and ( ) %$
© %
©
links of the path. If there is no such a wavelength free, the call pair respectively. From this example, we can see that not only
is blocked and lost. The ratio of the calls lost to the total calls the new coming paths have influence on blocking probability,
is the blocking probability for calls transmitted on the path of but also that the interference paths to new coming paths have
the ¦ ¤ £ pair. The goal of our analysis is to calculate this a direct influence too.
blocking probability. In order to do the analysis, we further In order to calculate the blocking probability on the path,
make the following general assumptions: we first calculate the probabilities that wavelengths are used in
¢ The network is connected in an arbitrary topology. Each the new coming paths. This is done in the following section.
link of the network has the same number of
¡ wave-
C. Wavelengths Usage Probabilities
lengths.
¢ For a ¦ ¤ £pair with path in the network, the call
¢ When paths share a link, they contend for use of the
requests arrive according to Poisson Process with rate resources (the wavelengths) on that link. The wavelengths on
¤ ¥£
. The duration (holding time) of each call spending the link are assigned according to first-fit algorithm. Under
on the path is exponentially distributed with unit mean. the first-fit algorithm, different wavelengths will have different
Call requests are called the offered traffic to the path . ¢ probabilities of being occupied. The probability that a wave-
¢ The first-fit algorithm [9], [7] is used to assign the length is used is defined as the usage probability of that
wavelengths on the path. wavelength. In the following, we will calculate the wavelength
¢ We also assume the signaling time for establishing the usage probabilities in certain cases.
connection between a ¦ ¤ £
pair using first-fit algorithm is 1) Wavelength Usage Probabilities on Single Link: Let us
negligible compared to the traffic interarrival and service consider a link . We assume the paths that share are
, ( )'
times. , ,
1' 432 1'
5 22 0 . The call requests rates to these paths are , @86
97
, ,
D6 432 B¥6
C7 22 A7 respectively. We assume these call requests
B. A Simple Case
We will first discuss a simple case to show the traffic inter-
ference for the first-fit algorithm and then give a description
of general approach to calculate the blocking probability of a n
¦ ¤ £ pair in the network. Detailed results will be presented c

later. Set i n
The readers should refer to Fig. 1. We have five nodes and
¦ ¤ £ 1 2 ... i i+1 ... L L+1
four pairs. The number of wavelength used by each
¦ ¤ £ l1 li lL
pair is also shown. We want to calculate the blocking
probability between nodes and . If we assume each link has
¦ § Fig. 3. Path E .
two wavelengths, the traffic between and will be blocked,
¦ §

undergo no interference before and after link . This implies We assume a path on link
b has probability of ¦ W
to
h C§ §
that the call requests to these paths are totally contributed to continue to link . Let
¦ be the usage probability of
the offered traffic of link without being thinned in other links.
wavelength on link¦ which can be calculated by (3). Let
If there is interference on these paths before and after link , i pW
be the usage probability of wavelength on link . Then b¦
the reduced traffic model [17], [18] is used to calculate the we approximate
i
by:
pW
offered traffic to a link. However, this interference is small
under light traffic, and we do not use it in order to reduce the W g f £ qW
i 0¡ ('$ ¦ CB£ `
) $%%% $ (4)
mathematical complexity. Since the call requests follow the f
Poisson Process, the offered traffic to link is also Poisson where can be calculated by (3).
W
Process with rate
¤ £ ©¨¦¤¢£
§ ¥ £ ¡
. With as input, we
¡ ¢£
Experiment study shows the approximation is accurate.
can calculate the wavelength usage probabilities on link . Let
0¡ ('$ ¦ #£ !¡
) $%%% $
be the usage probability of wavelength D. Wavelengths Correlation Model and Calculation of Path
on link . We define that is in state
if
0¡ 5''43£ 1 1
) $%%%$$ 2 Blocking
1 wavelengths are used on link . Let be the event that
¡7 6 Given a ¦ ¤ £
pair and its path , which consists of links ¢
is in state . According to Erlang-B Formula, we have
1 ) vt
u s
(Fig. 3), we will establish the model to V '$ W $
r $%%%
calculate the blocking probability on path .
¢ ©
7 ¡ Let ( s 5%''$ ¦ $C£ €
$ %% ) be the set of paths on link 'x1
yw
) ¡ A 6 ¢ ) 8£ £ ) ¡7 6 ¢ 0¡ ('$ ¦ CB£ 1
) $%%% $ £ „wx1 S‚ƒ©
that are new (not continuing from link ). Let
©
(1) y © Xr© …
@1 9 © , i.e. is the set of paths that are new along y„x1
w „x1 yw
TVSHQPHG¥ E £ ) ¡ A 6 ¢
U R © I F© D (2) the link
y 'w†1 . We assume wavelengths used in
ˆ7
‰‡ V %'%$ $5W '$ $ C#£
r $ % % % % $
( ) are independent. Let be the wavelength
¡ s ¦ €
)¡ ¡7 6 usage probability in . Let be the wavelength usage „x1 £ ©
y a
w
where '% X¢ 7 6 ¢ ¢
is the probability that
%% W occurs. Since we probability in ( s $5%'$ ¦ € „x1
). Then we have
%% $ yw
assumed , , ,
§ ¢ undergo no interference before and
after link and first-fit algorithm is used, we can argue that
% ) ¡ ¤ r ‘ ¤ B£ ‡ˆ7 ‰
(5)
when is in state , wavelengths to
1 are used. This is 1 H©
an approximation for the first-fit wavelength assignment and
implies that when is in state , wavelengths with indices
1 ¡
where can be approximated by (4). and
gXf f are defined
bigger than can be used. Therefore,
1 accordingly (the reader may refer to Fig.3).

F Let be the set of paths on link
that are not in ’„x1
gyw
) ¡7 6 ¢ Y £ ¡ 0¡ 5'$ ¦ a£ `
) $%%% $ (3)
. To calculate the blocking probability on path , we can
¢ ¡ „x1
yw
7 argue as follows. If wavelengths are used in
g py„wx1 , the call ¤ ¡
will be blocked. If
p„wx1
gy wavelengths are used in and
The formula tends to overestimate the usage probabilities of one wavelength is used in , the call will also be blocked. „x1
yw
wavelengths with small indices and underestimate the usage Exhausting all the cases, we obtain the following formula for
probabilities of wavelengths with big indices. the path blocking probability.
2) Wavelength Usage Probabilities on Link with Interfer- •‚
ence: In reality, the interference between paths are com- ˆ7 ) ”h ¡7
‰7 ‡‚ Y F D ) ”h ¡ £ “
plicated such that it is impossible to account all of them. F ¢ 6 ¢ A 7 F 6 ¢ (6)
A plausible approach is to make assumption to simplify
the problem and yet catch the essence of the problem. For where

g ’y„wx1 is the event that
0¡ 5''4¨£ 1 ”h ¡7 6
) $%%%$$ 2
is in state
example, In Fig. 1, path ( ) and ( ¦ ¤
) interfere with b c¤
1 and is the probability that at least wavelengths are
ˆ7 7
‰‡
each other directly. Path ( ) and ( b ¨¤ ¦
) also interfere § ¤¤ ¦ used in .
) $ % % % $ $ 2 1 7 6 ¢ '†1 ¢
1 0¡ ('C4¨£ ) ”h ¡
is calculated by (1) or yw
with each other directly. Although path ( ) and ( ) do ¦ § ¤ ¦ (2).
not directly interfere with each other, they do interfere with ˆ7 7
‰‡
b d¤ In order to calculate , we need to analyze the correla- ¢
each other indirectly by both affecting path ( ). Even path tion between wavelengths. Since the first-fit algorithm is used
( ¦ ¤
) and ( § ¤e
) have interference with each other. So it is not to assign wavelengths, a wavelength with low index has higher
possible for any analysis to faithfully model all interference. probability to be used. This leads to the following assumption.
Therefore, we only consider the direct interference between We assume the wavelengths used in have the following „x1
yw
two paths. We first discuss the scenario in Fig. 2. The result relationship: For any wavelengths with indices
0¡ 5'$ ¦ a£ W – $ – W – P –
) $%%% $ — such W X– $ –
will be used to calculate the blocking probability for a large that ,
network which is decomposed into parts as in Fig. 2.
In Fig. 2, we have three nodes and link and link . We ¦ b¦ ” ™hge™6
˜ 6 f d˜ (7)
assume there are paths on link
b¦ f and paths continuing¦ g f
¦
from link to link . We will calculate the wavelength usage where and ” 86
are the events that wavelength
˜ C86
d˜ – and W –
probabilities on link . b¦ are used on respectively.
„x1
yw

1 2 3 −1

−1.5

Fig. 4. Case 1: 3 nodes. −2

−2.5

Blocking probability (in log scale)
−3
This relationship is due to the first-fit algorithm. It indicates
−3.5
if a wavelength is used, then any wavelengths with lower index
are also used. Under this assumption, we have −4 # of wavelength=5

simulation
−4.5 numerical

¢ £ ‰ e7 7 ¢
‡ '%
%% ‰ e7 7 £ q)
‡
¡ ) ¥˜ 6
” ˜ 6 d˜ 6 −5
” ” ¨¦¢ £ £ ¤ ¢
¥ £
§ §
˜ © ¨
© ˜ ¥ F −5.5

ˆ7 7
‰‡ (8) −6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
where is given by (5). Arrival rate for a s−d pair

We first give an example and then the proof of (8). Let
b £ 1$ e £ ¡ . We use indices to represent the events and Fig. 5. Blocking prob. of 3 node network for path of length 2.

remove in the example to simplify the notation.

¢ ¡ ) i 5W – –
–
˜ dd ¢˜ ” © ¢ ” ˜
§
©
¢ § −1

−1.5
… § b ¦ … eˆ§ … e b … § b … ¦ … ¦ … b ¦ ¢ £ ˆ§ ¦ … e b ¦
% ) b ¢ £ ) e e … § … § b ¢ £ ) e§ b … e
e −2

e™6 ” ¢6
For proving (8), we use (7) and get:
d˜ ˜ £ ˜ '%
¥ ™6
%% Blocking probability (in log scale)
−2.5

. Therefore, ) 0¥ ˜ ( ( C˜ ” ˜ '¨$6
d ( % # ! −3

˜™6 % C™6 2” ™6
%% ¡ ) ’) −3.5
d˜ ˜ ¢ ¥
§ '¢ £ £ £ ” ¢
¥ § ¨ −4 # of wavelength=10
F ¥ ˜ © 1© ” ˜
simulation
−4.5 numerical
) F 6 ¡ '% ¡ 47 6 ¡ 7 6 ¢ £
%% 3
−5

Also from (7), we have −5.5

7 6 ¢ ¡ 47
3
¡ '%
%% ¡ %) 7 6 ¢ £ ) F 6 −6
6 0.5 1 1.5 2
Arrival rate for a s−d pair
2.5 3

Therefore, formula (8) is true. Fig. 6. Blocking prob. of 3 node network for path of length 2.
We have given the formulas to approximate the blocking
probability on path that has length greater than one. For
the special case of path comprising a single link, we can
simply use Erlang-B formula to approximate the blocking −1

probability. This is because on the single link path the traffic −1.5
will not be blocked as far as there is an available wavelength
−2
on the link. We may have used the reduced traffic model
[17], [18] to calculate the blocking probability. However since −2.5

the blocking probabilities are small, we can get reasonable −3

estimates without using the reduced traffic model.
−3.5

III. N UMERICAL AND S IMULATION R ESULTS −4 # of wavelength=20

In this section we present the comparison of our numerical −4.5
simulation
numerical

results with the simulation results.
−5
In all the simulation cases, we assume the arrival traffic
follows a Poisson process and the service time is exponential
−5.5

distributed with mean second. The rates in the simulation of −6
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
numerical results are expressed in requests/sec. Arrival rate for a s−d pair

The first case is for the simple network topology as shown
in Fig. 4. We have 3 nodes and 3 pairs: ( ), ( ) ¦ ¤ £ ¦ ¤ b ¤ ¦ Fig. 7. Blocking prob. of 3 node network for path of length 2.

and ( b d¤
). Each pair has the same arriving rate and the same

−1 −1

−1.5 −1.5

−2 −2

−2.5 −2.5

−3 −3

−3.5 −3.5

−4 # of wavelength=30 −4 # of wavelength=10

simulation simulation
numerical numerical
−4.5 −4.5

−5 −5

−5.5 −5.5

−6 −6
5 6 7 8 9 10 11 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Arrival rate for a s−d pair Arrival rate for a s−d pair

Fig. 8. Blocking prob. of 3 node network for path of length 2. Fig. 11. Blocking prob. of 9 node network for path of length 1.

−1

−1.5

−2

−2.5

−3

−3.5

−4 # of wavelength=20

simulation
−4.5 numerical

−5

−5.5

−6
1 1.5 2 2.5 3

Fig. 9. Case 2: 9 nodes mesh torus. Fig. 12. Blocking prob. of 9 node network for path of length 1.

−1 −1

−1.5 −1.5

−2 −2

−2.5 −2.5


−3 −3

−3.5 −3.5

numerical numerical
−4.5 −4.5

−5 −5

−5.5 −5.5

−6 −6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2 2.5 3 3.5 4 4.5


−1

−1.5

−2

−2.5

−3

−3.5

simulation
−4.5 numerical

−5

−5.5

−6
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Fig. 14. Case 3: 16 nodes mesh torus.
Fig. 15. Blocking prob. of 16 node network for path of length 3.

number of wavelengths. Traffic is one way from source to
destination. We calculate the blocking probabilities ( ). b c¤ −1

We use 4 different numbers of wavelengths on each link i.e.
2 ¦ „ e
2 2b −1.5

, , and wavelengths. The results are shown in Fig.
−2
5 - Fig. 8.
−2.5
The second case is the nine node mesh torus network as
shown in Fig. 9. Each node has the same probability to send −3

packet to the rest of nodes. The routing algorithm for the nine −3.5

node mesh torus network is: 1) shortest path: If one node want
to send packet to other node inside the network, it will choose simulation
numerical
the shortest path. 2) column first: If there are two or more −4.5

shortest paths, choose the path which goes through the node −5

within the same column of the source node first, e.g., if node −5.5
0 wants to send packet to node 4, the packet will be sent to
node 3 first, then node 3 will send it to node 4. In this case, −6
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
we can reckon that each link has
¦ ¤ £
pairs to share its
resources with. Among which, are new coming paths.
§ Fig. 16. Blocking prob. of 16 node network for path of length 3.
We simulate the blocking probabilities between node 3 and
node 4 with 4 different numbers of wavelengths. The results
are shown in Fig. 10 - Fig 13.
−1
The third simulation experiment involves the sixteen node
mesh torus network as shown in Fig. 14. Each node has the −1.5

same probability to send packet to the rest of nodes. The −2

routing algorithm is same as for the nine node mesh torus −2.5

network.
In this case, we can see that each link has
pairs¦ ¤ £ −3

¦

to share its resource with. Among which, are new coming −3.5

pairs. We simulate the blocking probabilities between node 5 −4 # of wavelength=30

and node 7 and between node 4 and node 7 with 3 different −4.5
simulation
numerical

numbers of wavelengths. The results are shown in Fig. 15 -
−5
Fig 20.
In most cases, the numerical results are a bit higher than the −5.5

simulation results. A possible reason for this is that we do not −6
0.5 0.6 0.7 0.8 0.9 1 1.1
use the reduced traffic model [17], [18]. Finally, we see that Arrival rate for a s−d pair

the results are not sensitive to the number of wavelength and
Fig. 17. Blocking prob. of 16 node network for path of length 3.
our numerical approach remains accurate even with a relative
large number of wavelengths.

−1 −1

−1.5 −1.5

−2 −2

−2.5 −2.5

−3 −3

−3.5 −3.5


−4.5 numerical −4.5 numerical

−5 −5

−5.5 −5.5

−6 −6
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1


−1

−1.5 [3] D. Bertsekas, and R. Gallager, “Data Networks,” Upper Saddle River, NJ:
Prentice Hall, 1992.
−2 [4] R. Ramaswami and K. N. Sivarajan, “Routing and wavelength assignment
in all-optical networks,” IEEE/ACM Transactions on Networking, vol. 3,
−2.5

pp. 489-500, Oct 1995.
−3 [5] A. Girard, “Routing and dimensioning in circuit-switched networks,”
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−3.5 [6] H. Harai, M. Murata and H. Miyahara, “Performance of Alternate
Routing Methods in All-Optical Switching Networks,” Proceedings, IEEE
INFOCOM ’97, pp. 516-524.
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−4.5 numerical [7] Ahmed Mokhtar and Murat Azizoglu, “Adaptive Wavelength Routing in
All-Optical Networks,” IEEE/ACM Transactions on Networking, vol. 6,
−5 No. 2, pp. 197-206, April 1998.
[8] R. A. Barry and P .A. Humblet, “Models of Blocking Probability in
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Journal Of Selected Areas of Communication, volume 14, pp. 878-867,
−6
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 June 1996.
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[11] S. Seetharaman, A. Durresi, R. Jagannathan, R. Jain, N. Chandhokand
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estimate the blocking probabilities even in a network with with Sparse Wavelength Conversion,” IEEE/ACM Transactions on Net-
working, volume 4, pp. 544-557, August 1996.
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position Approach for Computing Blocking Probabilities in Wavelength
ACKNOWLEDGMENT Routing Networks,” IEEE/ACM Transactions on Networking, vol. 8, no.
The authors would like to acknowledge that this work 6, pp. 747-762, December 2000.
[16] Ezhan Karasan and Ender Ayanoglu, “Effects of wavelength routing
was supported from a grant from MITACS (Mathematics for and selection algorithms on wavelength conversion gain in WDM optical
Information Technology and Complex Systems). networks,” IEEE/ACM Transactions on Networking, pp. 186-196, June,
1998.
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