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).

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

 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

). 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

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

, λ

,.... λ

}. The channel λ

is called control channel and it transmits the control

packets, while the remaining channels {λ

, λ

,.... λ

}

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

, λ

,....

}, 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

that belongs to the set of {λ

, λ

,.... λ

} 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

is:

=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

and simultaneously receive

from a set of wavelengths {λ

R1,

R2,…

}. 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

FiberLans

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

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

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:

suc

GeP

−

(2)

Let S

be a random variable representing the

number of successful data packet transmissions over

the N data channels during a frame, 0≤S

≤N.

The probability

]sSPr[

= 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):

suc

sucN

)P1(P

]sSPr[

−













(3)

Also, let A

(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,

S)s(A1 ≤≤ for

> .

The probability

]r)s(APr[

= 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

)!z(!kM

!s!M

]r)s(APr[

(4)

where: the sets of integers

{}

M210

k,...,k,k,k

{}

M,...,2,1,0ki,

∈ are defined in the Appendix.

Thus, the probability S

(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



===

)N,Mmin(

NNrc

]r)s(APr[]sSPr[)r(S

(5)

It is obvious that:

)Mx,smin(r)x,smin( ≤≤

(6)

We define the throughput S

as the average

number of correctly received data packets at

destination during a frame in steady state. Thus:



)N,Mmin(

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:

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):

max

(10)

Finally, we define the normalized system

throughput during a frame S

nor

as:

nor

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

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

FiberLans

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

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

FiberLans

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Baziana P.A. and Pountourakis I.E. (2007) ‘Performance

Optimization with Propagation Delay Analysis in

WDM Networks’, Computer Communications, 30(18),

pp. 3572–3585.

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Metropolitan WDM Ring Architecture for a Slotted

Transmission Technique’, IEEE Journal of Lightwave

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Baziana P.A. and Pountourakis I.E. (2010) ‘An Input

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Transmission Technique for Collisions-Free WDM

Ring MANs’, Optical Fiber Technology, 16(5), pp.

279-291.

Baziana P.A. and Pountourakis I.E. (2012) ‘A

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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[

= 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

M possible arrangements of the

s successfully transmitted data packets to the M

destination stations, each with equal and constant

probability:

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

of M destination stations,

{}

M,...,2,1,0k

∈ : for each of them there is

no successfully transmitted data packet

destined to it,

 there are k

of M destination stations,

{}

s,...,2,1,0k

∈ : for each of them there is 1

successfully transmitted data packet destined

to it, and so on. In general, there are k

of M

OPTICS2015-InternationalConferenceonOpticalCommunicationSystems

destination stations,

{}

s,...,2,1ki,

∈ : for each

of them there are i successfully transmitted

data packets destined to it.

It is obvious that:



(12)

and:

sik



(13)

Since each destination station is capable of

receiving up to x data packets per frame, it is:

rxiik

0k,xi



≠=

−

(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

that:

no data packet is destined to k

stations, one data

packet is destined to k

stations, and so on; and

generally, i data packets are destined to k

stations is

given by Wentzel and Ovcharov (1986):

∏∏

)!z(!kM

!s!M

(15)

Thus, the probability

]r)s(APr[

= is defined as

the sum of the probabilities P

, 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

)!z(!kM

!s!M

]r)s(APr[

(16)

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