Performance Analysis of a WDMA Protocol with a Multiple Tunable
Receivers Node Architecture for High-speed Optical Fiber Lans
Peristera A. Baziana
School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece
Keywords: Wavelength Division Multiple Access, Multiple Tunable Receivers, Receiver Collision, Rejection
Probability.
Abstract: A synchronous transmission WDMA protocol for high-speed optical fiber LANs of passive star topology is
studied in this paper. The packet rejection at destination, referred as receiver collision, is extensively
examined. A network interface with more than one tunable receivers per destination station is considered.
This means that each station is capable of receiving more than one data packets during a time frame. The
presented WDMA protocol expands previous studies about the receiver collision phenomenon that assume a
single tunable receiver per station, while it provides an analytical framework about its effect on the total
network performance. The average throughput and rejection probability are analytically derived, while the
bandwidth utilization improvement provided by the use of the multiple tunable receivers station interface is
estimated. The analysis considers Poisson arrivals and finite station population. Numerical results are
comparatively studied for various numbers of data channels and stations.
1 INTRODUCTION
Latest technology achievements concerning the
high-speed optical fiber networks deployment have
introduced a variety of communication techniques in
order to exploit the total fiber bandwidth provided.
Wavelength Division Multiplexing (WDM) (Zheng
and Mouftah, 2004) technique has been proven as
the most preferable and widely used technique to
divide the inefficient high fiber data rate into
multiple parallel channels of lower data rates, each
corresponding to a different optical wavelength.
Moreover, the WDM technique utilization in
conjunction with a variety of WDM access
(WDMA) strategies that have been proposed for
optical networks, have -without objection- given the
opportunity to increase the total throughput achieved
comparatively to the single channel system of the
same bandwidth.
Similar to any multi-channel network, there are
two main reasons for packet loss in WDM networks.
First, packets are destroyed if two or more stations
transmit a packet over the same WDM channel and
the transmissions are overlapped in time. This
phenomenon is referred as WDM channel collisions,
while it is distinguished in two main categories:
control channel collisions and data channel
collisions, depending on the type of packet
transmissions over each channel category (either
control or data packet). Second, additional packets
are aborted in case of the WDM receiver collisions
phenomenon (Pountourakis, 1998). Particularly, a
receiver collision occurs if a data packet that has
been successfully transmitted over a WDM data
channel cannot be picked up by the intended
destination station since its tunable receiver is
currently tuned to another WDM channel to receive
a packet from another source station.
In literature, the WDM channel collisions have
been extensively studied by means of analytical
methods or simulations in local and metropolitan
area scale (Zheng and Mouftah, 2004). On the other
hand, the receiver collisions are not extensively
studied in the majority of the WDMA protocols due
to complexity reasons. Nevertheless, some studies
take under consideration the receiver conflicts and
provide the performance measures estimation via
either analytical or simulation models. It is worth
mentioning that the receiver collisions phenomenon
can be evaluated in case that the destination station
is capable of receiving packets transmitted over
different channels, i.e. it is equipped with at least
one tunable receiver (TR) or more than one fixed
receivers (FR).
5
Baziana P..
Performance Analysis of a WDMA Protocol with a Multiple Tunable Receivers Node Architecture for High-speed Optical Fiber Lans.
DOI: 10.5220/0005517100050013
In Proceedings of the 6th International Conference on Optical Communication Systems (OPTICS-2015), pages 5-13
ISBN: 978-989-758-116-8
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
A quick research about studies for passive star
Local Area Networks (LANs) can show that there
have been introduced some WDMA protocols which
suffer from the receiver conflicts loss, while each
station uses a tunable receiver that can be tuned over
all WDM channels for reception. Especially, the
receiver collisions effect on both synchronous and
asynchronous transmission WDMA protocol cases
with Poisson aggregated traffic is analytically
examined by Pountourakis (1998), where a separate
control channel is introduced to exchange control
information in order to coordinate the data packets
successful transmissions. Also, the receiver
collisions impact on a synchronous transmission
WDMA protocol is analytically explored by Baziana
(2014) based on Poisson aggregated traffic scenario,
while the Multichannel Control Architecture (MCA)
is used in order to exchange the appropriate control
information over multiple parallel control channels
aiming to reduce the control loss probability. The
use of the MCA is also introduced by Baziana and
Pountourakis (2007 and 2012), where two
synchronous transmission WDMA protocols are
proposed assuming the receiver collisions effect for
different access strategies on the MCA. In these
studies, two different analytical Markovian models
are extensively adopted for the rigorous analytical
performance measures estimation.
On the other hand, in case of WDM Metropolitan
Area Networks (MANs) the receiver collisions are
considered in a slight different way. Thus, in order
to face them many WDMA protocols assume a
specific network and station configuration:
according to it, each station around the ring may
receive packets only from a dedicated channel
especially assigned to it for reception, while it is
equipped with a fixed receiver (FR) that is always
tuned to the dedicated reception channel (Bengi and
As, 2002), (Bengi, 2004), (Bregni et al., 2006),
(Herzog et al., 2004) and (Yang et al., 2004).
Although this assumption aims to face the packet
loss due to the receiver collisions, it provides
bandwidth under-utilization. This is because it
restricts the transmission of packets destined to a
specific destination station over its dedicated
reception channel, although there may exist other
available channels for transmission in case that it is
not currently free. In order to overcome the above
drawback and to efficiently exploit the available
fiber bandwidth, the use of a set of tunable
transmitter and receiver (TT-TR) per station is
proposed by Baziana and Pountourakis (2008 and
2010), by Turuk and Kumar (2004 and 2005) and by
Turuk et al. (2004), while all WDM channels can be
used for both transmission and reception. The
transceivers tunability benefits to significantly
reduce the dropping probability are given by
MacGregor et al. (2002).
The up to now investigations about the receiver
collisions effect in WDM networks performance
mainly consider that each station is equipped with a
single receiver, fixed or tunable. Since the recent
technology evolutions provide us with reliable
tunable receivers whose cost gradually decreases,
the utilization of more than one tunable receivers per
station appears as an interesting idea in order to
reduce the packet loss at destination and
consequently to increase the system performance. In
other words, the utilization of a multiple tunable
receivers station interface aims to provide gradual
reduction of packet rejection probability at
destination, improving the total throughput and
eliminating the system delay.
This paper introduces a synchronous
transmission WDMA protocol that takes under
consideration the receiver collisions effect in a
single-hop, passive star LAN that interconnects a
finite number of stations. The single-hop
architecture ensures that the communication between
the source and the destination station is realized over
the same channel without any wavelength
conversion. The proposed network configuration
uses a separate control WDM channel for the control
information exchange prior to the data
communication in order to coordinate the data
packets successful transmission without data channel
collisions. At each station a network interface is
assumed that contains a tunable transmitter and a
number x of tunable receivers (TT-TR
x
). In this way,
each station is capable of receiving at the end of
each time frame more than one (and up to x) data
packets that have been successfully transmitted over
the data channels and are destined to it. In this way,
the proposed protocol effectively faces the WDM
receiver collisions phenomenon providing essential
rejection probability reduction and total performance
improvement, as compared to the singe tunable
receiver per station case.
The present study expands previous studies, like
this of Pountourakis (1998), about the impact of
receiver collisions on the total network performance.
Especially, in this study we provide an analytical
model based on a Poisson arrival process in order to
derive in close mathematical formulas the average
throughput and the average rejection probability at
destination. Numerical results for diverse finite
numbers of stations and WDM channels are
OPTICS2015-InternationalConferenceonOpticalCommunicationSystems
6
comparatively studied, giving the total performance
improvement.
The proposed WDMA protocol performance
depends on a number of key factors which are taken
under consideration by the network configuration
and the analysis. Some of these are the number of:
station population, WDM data channels, and tunable
receivers.
This study is organized as follows. The network
model and the assumptions are described in Section
2. In Section 3 the protocol analysis is extensively
described and the performance measures are
analytically derived. Comparative numerical results
and comments are discussed in Section 4. Finally,
Section 5 outlines the concluding remarks. The
Appendix explores some mathematical formulation.
2 NETWORK MODEL AND
ASSUMPTIONS
We assume the passive star network presented in
Figure 1. The total fiber bandwidth is divided into
N+1 parallel WDM channels, each operating in a
different wavelength {λ
0
, λ
1
,.... λ
Ν
}. The channel λ
0
is called control channel and it transmits the control
packets, while the remaining channels {λ
1
, λ
2
,.... λ
Ν
}
are called data channels and they transmit the data
packets. The passive star coupler interconnects a
finite number M of stations (M > N). Each station
network interface is equipped with a tunable
transmitter and a set of x (1 x N) tunable
receivers that can be tuned to all channels {λ
0
, λ
1
,....
λ
Ν
}, as Figure 1 shows.
The control packet transmission time is defined
as time unit reference and is called control slot or
mini-slot. Thus, the data packet transmission time
normalized in time units is L and is called data slot
(L > 1). The control packet consists of the source
address, the destination address and the data channel
λ
k
that belongs to the set of {λ
1
, λ
2
,.... λ
Ν
} and has
been chosen for the corresponding data packet
transmission. Both control and data channels use the
same time reference which we call frame. We define
as frame the time interval that includes N time units
for the control packets transmissions plus the
normalized data packet transmission time L, as
Figure 2 depicts. Thus, the frame time duration F
d
is:
F
d
=N+L time units (1)
We assume a common clock to all stations. Time
axis is divided into contiguous frames of equal
length and stations are synchronized for
transmission over the control and data channels
Figure 1: Network model.
Figure 2: Frame duration. At the bottom: a receiver
collision case.
during a frame. Each frame consists of the control
and the data phase, as Figure 2 shows. The control
phase consists of N time units, while the control
packets transmissions occur. The data phase that
follows lasts for L time units, while the data packets
transmissions take place. At the beginning of a
frame data phase, each station is able to transmit at a
given wavelength λ
T
and simultaneously receive
from a set of wavelengths {λ
R1,
λ
R2,…
λ
Rx
}. Finally, in
our analysis we assume that the tunable transceivers
have negligible tuning time and very large tuning
range.
We assume that each station is equipped with a
buffer with capacity of one data packet. If the buffer
is empty the station is said to be free, otherwise it is
backlogged. Packets are collectively generated in a
Poisson stream. If a station is backlogged and
generates a new packet, the packet is lost. Finally,
the aggregated traffic from new generated and
retransmitted packets obeys Poisson statistics.
The successfully transmitted data packets are
uniformly distributed among the M stations, each
randomly selected with equal probability (for the
sake of generality we suppose that a station may
send to and receive from itself). Thus, if more than x
successfully transmitted over different data channels
PerformanceAnalysisofaWDMAProtocolwithaMultipleTunableReceiversNodeArchitectureforHigh-speedOptical
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packets are destined to the same destination, the
destination is able to receive only x packets of them
with its tunable receivers and rejects all the others.
This phenomenon is called receiver collision.
Especially at the beginning of each frame, each
station tunes one of its tunable receivers to the
control channel λ
0
in order to monitor the control
packets transmissions from all stations during the
control phase. Also at the beginning of each frame,
if it has to send a data packet to another, first it tunes
its tunable transmitter to the control channel λ
0
.
Then, it chooses randomly one of the data channels
over which the data packet will be transmitted, let’s
say data channel i. Then, it informs the other stations
about the i-th data channel selection, by transmitting
a control packet during the i-th control mini-slot of
the control phase with its tunable transmitter. The
control packets from all stations compete according
to the Slotted Aloha scheme to gain access over the
N control mini-slots. Since the station continuously
monitors the control channel with its tunable
receiver during the control phase, by the end of this
time period it is informed about the outcome of its
control packet transmission. This means that, grace
to the broadcast nature of the control channel, the
station is aware of the data channel claims for
transmission of all stations. Especially if one or
more other stations have selected the same i-th data
channel for transmission, the corresponding control
packets have collided during the i-th control mini-
slot and are all aborted, while all involved stations
become backlogged. In the contrary if the station
control packet has been successfully transmitted
over the i-th control mini-slot, the station gains
access to the i-th data channel for successful
transmission during the frame data phase. This fact
does not mean that the corresponding data packet
will be correctly received by the destination. This
fact depends on the number of the other data packets
that are successfully transmitted over other data
channels during the data phase and have the same
destination. In this case, the destination station may
receive up to x data packets with its tunable
receivers, while the corresponding source stations
become free. It is evident that the destination station
aborts all the others packets destined to it due to the
receiver collisions phenomenon, while the relative
stations become backlogged. We may consider
several arbitration rules for the selection of the data
packets that are finally correctly received by the
destination while the others are aborted, such as
priority etc.
At the end of the control phase, the station is
informed about the data packets that will be
successfully transmitted over the N data channels
and are destined to it. Based on this information and
the above arbitration rules, the station decides which
of these data packets it is going to receive, let’s say z
(z x) of them. Thus at the beginning of the frame
data phase, it tunes z of its tunable receivers to the
corresponding data channels while the data packets
reception immediately starts.
3 ANALYSIS
We denote as G the offered load, i.e. the average
number of transmitted control packets per time unit
on the control channel. According to Sudhakar et al.
(1991), the probability P
suc
of a successful data
packet transmission over the data channel j
(j=1,2,…, N) during a frame is given by:
G
suc
GeP
=
(2)
Let S
N
be a random variable representing the
number of successful data packet transmissions over
the N data channels during a frame, 0≤S
N
≤N.
The probability
]sSPr[
N
= of finding s
successfully transmitted data packets over the N data
channels during a frame conforms to the binomial
probability law and is given by Pountourakis (1998):
sN
suc
s
sucN
)P1(P
s
N
]sSPr[
==
(3)
Also, let A
N
(s) be the number of the correctly
received data packets at destination given that s
successful transmissions over the N data channels
occurred during a frame,
NN
S)s(A1 for
0S
N
> .
The probability
]r)s(APr[
N
= of finding r
correctly received data packets at destination from s
successful transmissions over the N data channels
during a frame is given by (see the Appendix):
==
==
sets all
s
1z
k
s
0i
i
s
N
z
)!z(!kM
!s!M
]r)s(APr[
(4)
where: the sets of integers
{}
M210
k,...,k,k,k
{}
M,...,2,1,0ki,
i
are defined in the Appendix.
Thus, the probability S
rc
(r) of finding r correctly
received data packets at destination during a frame
in steady state is given by:
OPTICS2015-InternationalConferenceonOpticalCommunicationSystems
8
=
===
)N,Mmin(
rs
NNrc
]r)s(APr[]sSPr[)r(S
(5)
It is obvious that:
)Mx,smin(r)x,smin(
(6)
We define the throughput S
rc
as the average
number of correctly received data packets at
destination during a frame in steady state. Thus:
=
=
)N,Mmin(
1r
rcrc
)r(rSS
(7)
Also, we define the average rejection probability
at destination P
rej
in steady state, as the ratio of the
average number of data packets rejected at
destination due to the receiver collisions
phenomenon to the average number of successfully
transmitted data packets over the N data channels,
during a frame. Thus, P
rej
is given by:
S
SS
P
rc
rej
=
(8)
where: S is the average number of successfully
transmitted data packets over the N data channels,
during a frame. In other words, S represents the
average throughput per frame without the receiver
collisions effect and it is given by Sudhakar
et al.
(1991):
suc
NPS =
(9)
while its maximum value S
max
is provided for
offered load G
max
=1 and is given by Pountourakis
(1998):
e
N
S
max
=
(10)
Finally, we define the normalized system
throughput during a frame S
nor
as:
rc
d
nor
S
F
L
S =
(11)
4 PERFORMANCE
EVALUATION
In this Section, we present the numerical solution of
the proposed protocol, for various numbers of
stations M, data channels N, and tunable receivers
per station x. In the following figures, we consider
that the data packet length is L=100 time units.
Figure 3 illustrates the normalized throughput
S
nor
versus the offered load G for M=50 stations,
N=3,8,10,13 data channels for x=2 tunable receivers
per station. The curves provided are compared with
the case of a single tunable receiver per station.
Let’s study in Figure 3 the S
nor
value for N=13 data
channels. As it is observed the network
configuration with x=2 tunable receivers per station,
as it is compared to the single tunable receiver case,
provides higher S
nor
value for a wide range of
offered load conditions. Especially for low offered
load (lower than G=0.2 control packets/control slot),
the S
nor
values provided in cases of x=1,2 are almost
equal. This is because, in low offered load
conditions the number of transmitted control packets
over the control slots is low, while the consequent
control channel collisions are not few. In this case,
the number of successfully transmitted data packets
over the N data channels is also low, introducing
low number of rejection events at destination. In
other words, for low offered load conditions the
impact of the receiver collisions phenomenon is not
significant, providing almost equal values of
throughput. On the contrary, as the offered load
increases up to almost G=2 control packets/control
slot for values around G
max
=1 control
packets/control slot, the S
nor
in case of the x=2 is
essentially higher than in the x=1 case, while the
maximum improvement is reached for G= G
max
. This
behavior is explained by the fact that for offered
load around the G
max
value, the system reaches
maximum number of successfully transmitted data
packets over the N data channels. Thus, under these
offered load conditions the number of data packets
that are distributed to the M destination stations is
maximum, providing higher number of rejection
events at destination. Thus, the utilization of x=2
instead of x=1 tunable receivers per station provides
maximum throughput improvement, as it is observed
in Figure 3. For high offered load conditions (higher
than G=2 control packets/control slot), the
throughput values for x=1,2 are almost equal. This is
because, for this offered load the number of control
channel collisions are getting higher, while the
probability of a successful data packet transmission
over the N data channels is getting lower. This is the
reason why the impact of the receiver collisions
phenomenon on the system throughput decreases
too, while the use of higher number of tunable
receivers per station (from x=1 to x=2) does not
seem to improve the throughput achieved.
PerformanceAnalysisofaWDMAProtocolwithaMultipleTunableReceiversNodeArchitectureforHigh-speedOptical
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Also, the above behavior is noticed for N=10 and
N=8. Thus, in Figure 3 it is shown that the S
nor
maximum
improvement provided by the utilization
of x=2 instead of x=1 tunable receivers per station
occurs for G=G
max
, while it is analogous to the value
of N. This means that as the number N of data
channels decreases, the S
nor
for G=G
max
decreases
too. This can be understood since, as the N value
decreases the number of successfully transmitted
data packets over the N data channels decreases too,
providing lower rejections at destination due to the
receiver collisions. Consequently, the exploitation of
more tunable receivers per station is not able to
provide higher throughput values, as the number N
decreases. This behavior can be representatively
noticed when N=3, where the probability of a
control channel collision is extremely high providing
almost zero probability of a receiver collision. This
is the reason why, the S
nor
values for x=1, 2 tunable
receivers per station are equal. The above remarks
are validated by studying the S
nor
improvement when
increasing the number of tunable receivers per
station from x=1 to x=2. For example for G=1.6
control packets/control slot, the S
nor
increases about:
0.65% for N=3, 2.25% for N=8, 2.89% for N=10,
and 3.84% for N=13.
Figure 3: S
nor
versus G, for M=50 stations, N=3,8,10,13
data channels and x=1,2 tunable receivers per station.
In Figure 4 the average rejection probability P
rej
versus the offered load G is shown, for M=50
stations, N=8,10,13 data channels for x=2 tunable
receivers per station, while the curves are compared
with the case of a single tunable receiver per station.
The previous results are validated in Figure 4.
Particularly, it is illustrated that the increase of the
number of tunable receivers per station from x=1 to
x=2 provides significant P
rej
reduction that reaches
almost 100% in a wide range of offered load values,
while it obtains its maximum value when G=G
max
.
Also, it is remarkable that the P
rej
reduction is a
decreasing function of N. This is understood since as
N increases for a given number of stations, the
probability of a destination conflict increases too, as
previously described. As a direct result, the
utilization of more tunable receivers per station
provides lower rejection probability. For example,
for G=1 control packets/control slot, the P
rej
reduction when increasing the number of tunable
receivers per station from x=1 to x=2, is: 98.5% for
N=8, 98% for N=10, and 97.3% for N=13.
The proposed protocol performance is studying
in Figure 5, when the station population varies.
Especially, Figure 5 depicts the average rejection
probability P
rej
versus the offered load G for N=13
data channels, M=50,100,150 stations for x=2
tunable receivers per station, while the curves are
compared with the case of a single tunable receiver
per station. As in Figure 4, the utilization of x=2
instead of x=1 tunable receivers per station provides
essential P
rej
reduction that becomes maximum when
G=G
max
, while it is almost 100% in the whole
offered load range. As Figure 5 illustrates, P
rej
reduction is an increasing function of M for finite
number of N. This is because, as M increases the
offered load to the control channel is getting higher.
This means that the probability of control channel
collisions increases, while consequently the number
of successfully transmitted packets that are
distributed to the destination stations is getting
lower. This is the reason why, the P
rej
reduction
provided by the high number of tunable receivers
utilization increases as the station population
increases. For example, for G=1 control
packets/control slot, the P
rej
reduction when
increasing the number of tunable receivers per
station from x=1 to x=2, is: 97.3% for M=50, 98.4%
for M=100, and 99.1% for M=150.
It is obvious that in each network
implementation, the determination of the number of
tunable receivers per station has to take under
consideration the desired performance level
achieved (in terms of P
rej
) in conjunction with the
implementation cost. Figure 6 and 7 illustrate the
rejection probability maximum value P
rej-max
for
various numbers of stations M and data channels N,
in the cases of number of tunable receivers per
station x=2, 3. As expected, the increase of x from 2
to 3 provides significant performance improvement.
For example for N=13, the P
rej-max
reduction when
increasing from x=2 to x=3 is: 98.2% for M=50,
98.9% for M=100, and 99.4% for M=150. In other
words, the P
rej-max
reduction is an increasing function
OPTICS2015-InternationalConferenceonOpticalCommunicationSystems
10
Figure 4: P
rej
versus G, for N=3,8,10,13 data channels,
M=50 stations and x=1,2 tunable receivers per station.
Figure 5: P
rej
versus G, for M=50,100,150 stations, N=13
data channels and x=1,2 tunable receivers per station.
Figure 6: P
rej-max
, for N=3,8,13 data channels,
M=50,100,150 stations, and x=1,2 tunable receivers per
station.
Figure 7: P
rej-max
, for M=50,100,150 stations, N=3,8,13
data channels and x=1,2 tunable receivers per station.
of M. This is because as M increases for fixed N, the
probability of a receiver collision decreases, fact that
becomes noticeable with the concurrent increase of
x. Similar, for M=50, the P
rej-max
reduction when
increasing from x=2 to x=3 is: 99.1% for N=8,
98.5% for N=10, and 98.2% for N=13. This
behavior is an immediate result of the above
discussion. It is obvious that P
rej
=0 when N=3 and
x=3, since there is no receiver collisions probability.
5 CONCLUSIONS
This paper proposes a synchronous transmission
WDMA protocol that examines the effect of receiver
collisions in high-speed optical fiber LANs. As the
cost of the optical tunable receivers gradually
decreases, we exploit the idea to introduce at each
station a network interface that consists of a number
of tunable receivers. The utilization of more than
one tunable receivers per station improves the
network performance, since it provides essential
rejection probability reduction at destination.
Also in this study, we provide an analytical
framework for the performance measures evaluation,
based on Poisson statistics. Thus, we derive
analytical formulas for the estimation of both the
system throughput and the rejection probability,
considering a finite number of tunable receivers per
station. The proposed protocol is general and
expands previous studies that consider a single
tunable receiver per station. Numerical results for
various numbers of stations, WDM data channels,
and tunable receivers per station depict that the
increase of the number of tunable receivers about
one significantly improves the total system
performance and reduces almost 100% the
probability of conflicts at destination. This result
offers additional insights in WDM high-speed
LANs.
PerformanceAnalysisofaWDMAProtocolwithaMultipleTunableReceiversNodeArchitectureforHigh-speedOptical
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REFERENCES
Baziana P.A. (2014) ‘An Approximate Protocol Analysis
with Performance Optimization for WDM Networks’,
Optical Fiber Technology, 20(4), pp. 414-421.
Baziana P.A. and Pountourakis I.E. (2007) ‘Performance
Optimization with Propagation Delay Analysis in
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APPENDIX
We assume the model that consists of N data
channels and M stations. We aim to analytically
describe the distribution of the successfully
transmitted data packets over the N data channels to
the M stations. This model corresponds to the
occupancy problem of the distribution of
indistinguishable balls (data packets) to cells
(destination stations), supposing that the
arrangements should have equal probabilities. We
consider indistinguishable packets transmitted to
indistinguishable destination stations using
Maxwell-Boltzman statistics (Feller, 1968).
We are interested in the probability
]r)s(APr[
N
= of r correctly received data packets at
destination when s data packets have been
successfully transmitted over the N-channel system,
during a time frame, 1
≤s .
Let’s suppose that each station may transmit to
any of the M stations (for the sake of generality we
suppose that a station may send to and receive from
itself). According to the Maxwell-Boltzman
statistics, there are
s
M possible arrangements of the
s successfully transmitted data packets to the M
destination stations, each with equal and constant
probability:
s
M/1 .
We consider that the distribution of s data
packets to M stations provides the following result
by the end of a frame:
there are k
0
of M destination stations,
{}
M,...,2,1,0k
0
: for each of them there is
no successfully transmitted data packet
destined to it,
there are k
1
of M destination stations,
{}
s,...,2,1,0k
1
: for each of them there is 1
successfully transmitted data packet destined
to it, and so on. In general, there are k
i
of M
OPTICS2015-InternationalConferenceonOpticalCommunicationSystems
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destination stations,
{}
s,...,2,1ki,
i
: for each
of them there are i successfully transmitted
data packets destined to it.
It is obvious that:
Mk
s
0i
i
=
=
(12)
and:
sik
s
0i
i
=
=
(13)
Since each destination station is capable of
receiving up to x data packets per frame, it is:
rxiik
s
0k,xi
1x
0i
i
i
=+
=
=
(14)
For each set of integers
{}
M210
k,...,k,k,k that
satisfy (12), (13), and (14) and it is
{}
M,...,2,1,0k,...k,k
M10
, the probability P
ki
that:
no data packet is destined to k
0
stations, one data
packet is destined to k
1
stations, and so on; and
generally, i data packets are destined to k
i
stations is
given by Wentzel and Ovcharov (1986):
==
=
s
1z
k
s
0i
i
s
ki
z
)!z(!kM
!s!M
P
(15)
Thus, the probability
]r)s(APr[
N
= is defined as
the sum of the probabilities P
ki
, for all possible sets
of integers
{}
M210
k,...,k,k,k that satisfy (12), (13),
and (14) and it is
{}
M,...,2,1,0k,...k,k
M10
, i.e.:
==
==
sets all
s
1z
k
s
0i
i
s
N
z
)!z(!kM
!s!M
]r)s(APr[
(16)
PerformanceAnalysisofaWDMAProtocolwithaMultipleTunableReceiversNodeArchitectureforHigh-speedOptical
FiberLans
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