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MULTIUSER WIRELESS COMMUNICATION (EE381K) CLASS PROJECT, FALL 20021

Optical CDMA with Optical Orthogonal CodeSANGWOOK HAN

[email protected] of Electrical and Computer Engineering

The University of Texas at AustinAustin, TX 78712

Abstract - This report examines optical CDMA communi-cation techniques with optical orthogonal codes. Simulationsthat show the desired properties of theses codes and their usein optical CDMA are reported. Based on the simulations, weinvestigate the properties of optical CDMA. Probability of error is also evaluated.

I. INTRODUCTION

There have been many efforts to take the full advantageof fiber-optic signal processing techniques to establish anall optical CDMA communication systems since CDMAwas first applied to the optical domain in the mid-1980sby Prucnal, Salehi, and others [1-3]. Traditional fiberoptic communication systems use either TDMA orWDMA schemes to allocate bandwidth among multiple

asynchronously, without centralized control, and it doesnot suffer from packet collisions. As a result, opticalCDMA systems have lower latencies than TDMA orWDMA. Furthermore, since time and frequency (orwavelength) slots do not need to be allocated to eachindividual user, significant performance gains can beachieved through multiplexing. Also, TDMA and

WDMA systems are limited by hardware because of theslot allocation requirements. In contrast, CDMA systemsare only limited the tolerated bit error rate relationship tothe number of users, affording the designer a much moreflexible network design [4] .

To establish the optical CDMA, we have to overcomethe code orthogonality problem. Many researchers have

A property of MVG_OMALLOOR

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g y p y


the following two properties [5].1) The Auto-Correlation Property:

at

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for any x C and any integer t, 0

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can be discarded by choosing relevant threshold value.But when total number of users goes up, the cross-correlation due to interfering users adds up quickly toseverely degrade the system performance. For instance,when w is 4 like this case, accommodating 4 users make i tpossible to have 4 as the correlation value even if theintended transmitter s information bit is 0. To avoid this

phenomenon, both high w which can be considered as thesum of 1s in the sequence, and long n are required. If weincrease only w fixing n, cross-correlation value due tothe interference can be lowered. However, OOC has verysparse marks to keep the cross-correlation low, i.e. anumber of zeros is much higher than that of ones in thesequence. It means that cross-correlation increases byitself by increasing only w. Therefore, both w and n should be increased simultaneously. But this solution alsoreveals a drawback, long signal processing time due tolong n.

Finally the transmitted information is extracted bythresholding the correlator output in Fig. 4 (d) and (e). Inthis case, they are successfully recovered as intended.Here, 3 is chosen as the threshold values. Threshold issuewill be covered in detail in the section II.2.

scheme for synchronization. Therefore, now we can saythat optical CDMA does need no network synchronization.

II.2. K -User Synchronous Channel

In the previous section, we investigated a simple 2-userchannel to understand the optical CDMA. In this section,

we explore problems faced by increasing K (Fig. 6 a).Here we choose 7 as K , and C is (43, 3, 1, 1) having 7 set s,{{0, 1, 19}, {0, 2, 22}, {0, 3, 15}, {0, 4, 13}, {0, 5, 16},{0, 6, 14}, {0, 7, 17}}.

The first issue is a threshold value. As we saw in theprevious section, interfering signal can be effectivelydiscarded by setting a relevant threshold value. Thethreshold value can be chosen under the followingcondition.

wthreshold 0 (4)

Fig. 6 (d) through (g) show results from several thresholdvalues, 1, 2, 3, and 5. As the threshold value goes up, therecovered signal gets similar to the ideal signal in Fig. 6(c). However, the threshold value can not be over w. If theintended information bit is 1, the correlation value is w.Therefore, the threshold over w incorrectly converts it to


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IV. PROBABILITY OF E RROR

The probability of error per bit is defined as

PE = p ( LI th | b=0)? p (b=0)+ p ( LI < th | b=1) ? p(b=1) (6)

where PE , LI , th , and b are the probability of error, lightintensity, threshold, and intended binary information bit,respectively. Here, an interesting point is that p ( LI < t h |b=1) is always equal to zero because p ( LI < t h |b=1)= p (w+I < t h) = p (w+ I -th

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Interference Avoidance in Optical CDMA Networks

Purushotham KamathAdvisors: Joseph D. Touch and Joseph A. Bannister

USC/ISIMarina del Rey, CA 90292

{pkamath, touch, joseph }@isi.edu

Abstract Interference Avoidance is a media access controlmechanism that prevents throughput degradation in a broadcast

all optical Local Area Network (LAN) built on optical CodeDivision Multiple Access (CDMA) technology. Optical CDMA isa multiplexing technology that allows the utilization of the largetransmission capacity of an optical ber. However, the throughputof an optical CDMA broadcast LAN network degrades quicklydue to multiuser interference. Interference Avoidance controlsinterference and prevents throughput degradation due to interfer-ence at high loads. It is based on two mechanisms: state estimationand transmission scheduling . Algorithms for state estimation andtransmission scheduling are proposed and evaluated. Analysisand simulation show how they prevent degradation of throughputat high loads. A testbed implementation of state estimation andtransmission scheduling hardware is in progress.

I. INTRODUCTION

Optical CDMA is a code division multiplexing technologythat allows several transmitters to transmit simultaneously on

i l b N d i i d d f O

Fig. 1. Interference avoidance

If the receiver is currently receiving a packet, this kind of false


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III . S TATE ESTIMATION

Assume that several nodes on an optical CDMA LANare transmitting simultaneously on different codewords. Each

codeword is shifted by a different phase shift that depends onthe exact instant it was transmitted. The state of the line ata point on the line is the sum of the codewords taken over awindow of N chips. E.g. , in Figure 1(c), the state of the lineis [0120111].

Arrivals and departures of packets continuously changethis state of the line. The on-off keying of the codewordsalso changes the state of the line. The objective of stateestimation is to determine the true state , i.e. the sum of the

codewords being transmitted with appropriate phase shifts. Awindow based state estimation algorithm can give a reasonableestimate of the true state of the line. The node determines thestate by sampling the line. This is called the sensed state Thisis repeated for a window of time, say W bits. The estimatedstate is twice the average of the sensed states taken overover the window. State estimation can be either performedcontinuously or on-demand , i.e. when a packet arrives.

The performance of an on-demand window based state es-

timation algorithm has been measured through simulation andthe results of system performance are described in Section V.State estimation hardware needs to be implemented using

sampling over several bits because chips are arriving fasterthan the maximum electronic processing rate. The hardwarefor state estimation is currently being implemented.

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No transmission schedulingPseudo-cooperative

SelfishCooperative

Fig. 2. Throughput vs. offered load for different transmission schedulingalgorithms for simulation of a star network using a (10, 3, 2) codeset. Acontinuous window based state estimation algorithm was used with sensingwindow = 10 bits. Four transmission scheduling algorithms were evaluated:Sel sh, Pseudo-cooperative, Cooperative and None (Aloha-CDMA). A 1persistent defer algorithm was used with a retry limit of 10.

transmission scheduling hardware is currently being imple-mented.

V. S YSTEM PERFORMANCEThe performance of an Interference Avoidance system is

dependent on a large number of factors such as trafc patternsand network topology. An analysis based on Poisson arrivals,exponentially distributed packet lengths random codewordassignment, perfect state estimation may be found in [2].Fi 2 h h l f li i i l i h


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ISI-TR-2006-617 1

Algorithms for Transmission Scheduling

in Optical CDMA NetworksPurushotham Kamath, Joseph D. Touch, Joseph A. BannisterUSC/ISI

4676 Admiralty WayMarina del Rey, CA 90292

{pkamath, touch, joseph }@isi.edu22nd May 2006

Abstract Transmission scheduling is a media access controlmechanism that prevents degradation of throughput in opticalCDMA Local Area Networks (LANs) at high offered load. OpticalCDMA is a multiple access technique for broadcast opticalLocal Area Networks. The throughput of an optical CDMALAN at high offered load is limited by multi-user interference. Interference Avoidance , a distributed, contention based mediaaccess control mechanism, can prevent throughput degradation

at high loads. Interference avoidance consists of state estimationand transmission scheduling . This work analyzes algorithms fortransmission scheduling under perfect state estimation. The anal-ysis shows that transmission scheduling under specic conditionscan provide upto 30% network throughput at high offered load.This compares well to non scheduled systems which have closeto zero throughput under the same conditions. Simulations showthat the performance of transmission scheduling is independent of

The throughput of an optical CDMA LAN is limited bymulti-user interference. When several users transmit simul-taneously, their packets and hence their codewords overlap.When the optical pulses in the codeword overlap, their opticalpower is added. Optical pulses from one codeword can bedetected by receivers tuned to other codewords. As a resultreceivers may falsely detect their codewords resulting in packeterrors. These false positive errors increase with offered load,resulting in throughput collapse.

Interference Avoidance is a contention media access controlmechanism that prevents throughput collapse in optical LANsnetworks at high offered load. It consists of state estimationand transmission scheduling State estimation is a mechanism


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Receive fiberTransmit fiber

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M13101021 (Line)

0 1 0 0

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Fig. 1. Typical optical CDMA LAN topology. Nodes are connected bytransmit and receive bers to a passive optical coupler in a star topology.

I I . B ACKGROUND

This section provides background on optical CDMA LANarchitecture, codeset design and receiver design.

A. Optical CDMA LAN architecture

The optical CDMA network considered in this work is ashared medium, packet switched, multiple access LAN. Thephysical layer is optical CDMA that uses unipolar encodingand intensity modulation over a single wavelength.

allows the node to identify that a frame destined for it hasarrived and where the rst bit of the frame begins.

B. Optical CDMA codeset design

An Optical Orthogonal Codeset (OOC) is a set of (0,1)sequences of length N that satises correlation constraints [3].The term codeset is used to refer to the set of sequences, andthe term codeword is used for a member of the set. Each 0 or1 of a sequence is called a chip , and the codeword represents adata bit . For any two codewords in the codeset, the correlationconstraints are:

N 1

n =0 s ( i,n + ) s ( j,n )= w when i = j, = 0 otherwise

where s ( i,n ) is the n th chip of the i th codeword, additionis modulo N and 0 N 1. is called the Maximum Collision Parameter . The number w of 1 chipsof a codeword of the codeset is called its weight. A particularcodeset is specied by the parameters (N,w, ). The size Sof the codeset is the number of codewords in the codeset.Codesets with all codewords having the same weight are called

constant weight codesets. [3] and [4] describe several codesetconstruction methods. The codesets used in this work areconstant weight codesets generated by the greedy constructionmethod [3]. The rate at which individual chips are transmittedis called the chipping rate B . The rate at which the data bitsare transmitted is called the data rate . The chipping rate isN times the data rate The codewords are pseudo orthogonal


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Control

TDL

TDL

TDLSplitter Coupler

Thresholddetector

Photo-detector

Hard-limiter

Receivefiber

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11

111

323

1 11101111 1

Fig. 2. Optical CDMA receiver: The gure shows a hard-limiting correlationdetector that consists of a hard-limiter, decoder, photo-detector and a thresholddetector. The receiver is tuned to the codeword 1110000. The power in the1st , 2nd and 3rd chip positions is summed by the decoder. The photo-detectorconverts the signal to an electrical signal and the threshold detector detects a1 bit.

III . I NTERFERENCE AVOIDANCE

This section denes the transmission scheduling problemand discusses algorithms for transmission scheduling. Firstit discusses the the problem of interference, the need for

(a)

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00110001 C0 (1 bit)

00100110 C3

Fig. 3. A bit error. In (a) C0 is OFF. Codewords C1 and C2 have chip overlapswith the 1 chips of C0 . A false positive error will occur at the receiver. In(b) C0 is ON, so no false positive error will occur.

from an (8, 3, 3) codeset 2 . The gure is a snapshot of data

bits on an optical ber sent by four nodes. Their combinedsignal on the line is indicated below the codewords. C0 ithe codeword being received. C1 and C2 have 1 chips thatoverlap with C0 s 1 chips. Figure 3(a) shows the case when a0 data bit is transmitted by the node sending C0 . Figure 3(b)shows the case when a 1 data bit is transmitted. In (a)the receiver will erroneously detect a codeword ( C0 ) because


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ISI-TR-2006-617 4

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Fig. 4. An interference error is caused in C0 in (a). No error is caused whenthe same codewords are sent with a different set of chip offsets. (b) showscodeword C0 delayed by 1 chip time

B. Interference Avoidance media access control

Interference Avoidance is a contention media access con-trol (MAC) protocol. Each node on the network contendsfor access to the medium using the Interference Avoidanceprotocol. Figure 5 shows a block digram of an InterferenceAvoidance Network Interface Card. It consists of an optical

Bus(from node

processor)

Optical CDMAReceiver

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fiber(to coupler)

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processor)

Fig. 5. Block diagram of an Interference Avoidance Network Interface Card.

C. Optical CDMA State and State Estimation

The signal at any time at any point on the receive ber of an optical CDMA LAN is a multilevel signal due to the sumof the codewords. The state of the line is a vector of lengthN equal to the sum of the codewords at output of the couplerassuming that all nodes are transmitting 1 bits.

S (t) = [ s0 s1s2 ....s N 1] =M

i =0

rot (C i , i )


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ISI-TR-2006-617 5

ctx Codeword to be transmittedstate State estimatehstate hardlimit (state )t d 0for offset = 0 to N

if (hstate & ctx = ctx ) thenmark offset as a feasible offset

rotate ctx to the right by one chipt d any feasible offset

TABLE I

T HE P URE SELFISH TRANSMISSION SCHEDULING ALGORITHM

transmission can be scheduled, then the scheduling algorithmreturns an offset k such that 0 k < N . The offset isthe number of chips that the packet transmission should bedelayed. The offset is measured with respect to the estimatedstate of the line. If transmission is not possible, then the packettransmission is deferred by returning it to a higher layer fora retransmission attempt. Other defer mechanisms such as1, non and p-persistent sensing [7] may also be used. Thetransmitting node does not have a receiver to detect errors in

its transmitted packet during transmission. Therefore packetswhich experience interference errors during transmission aretransmitted until completion. Transmission scheduling is doneon a per packet basis.

This work assumes perfect state estimation by all nodes onthe network. In perfect state estimation, every node knows thestate of the line. All nodes see the same state at the same time.


if (hstate & ctx = ctx ) thennewstate = state + ctxnumoverlaps = overlaps (newstate )if (numoverlaps < threshold )

mark offset as a feasible offsetrotate ctx to the right by one chip

t d any feasible offset

TABLE II

T HE T HRESHOLD TRANSMISSIONSCHEDULING ALGORITHM

erative strategy if the node schedules its transmission only if it Preserves self/Preserves others or Destroys self/Preservesothers . Simple implementations of cooperative strategy areeither not feasible for all codesets or result in low throughput.A pseudo-cooperative strategy attempts to reduce the proba-bility of destroying other packets on the line. It is a best effortstrategy which increases the probability of the events Preserves

self/Preserves others or Destroys self/Preserves others .The following sections discuss three transmission schedul-ing algorithms: Pure selsh, Threshold and Overlap sectionscheduling . The transmission scheduling algorithms imple-ment either selsh or pseudo-cooperative strategies or both.The section compares their performance to Aloha-CDMA i.eoptical CDMA without any media access control


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if (hstate & ctx = ctx ) thennewstate = state + ctxnumoverlaps = overlaps (newstate )numones = ones (newstate )if (numoverlaps < numones )

mark offset as a feasible offsetrotate ctx to the right by one chip

t d any feasible offset

TABLE III

T HE OVERLAP SECTION TRANSMISSION SCHEDULING ALGORITHM

selsh algorithm. A simulation study is used to validate themathematical analysis.

The metric used to evaluate performance is the normalizednetwork throughput at different values of the normalizedoffered load. The normalized offered load is the arrival rate(in packets/s) expressed as a fraction of the maximum possiblearrival rate (in packets/s) of the network when it is used asa single channel network 3 . The arrival rate is dened as theaggregate rate at which packets arrive to all the nodes fortransmission on the network. The normalized network throughput is the ratio of the number of packets that are transmittedover the network without error to the total number of packetsoffered for transmission multiplied by the normalized offered

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Threshold schedulingOverlap section scheduling

Pure selfish schedulingAloha-CDMA

Fig. 6. Comparison of the performance of the transmission schedulingalgorithms based on analysis. The trafc model is Poisson arrivals with expo-nentially distributed packet lengths. The codeset is (10 , 3, 3) and codewordsare chosen uniform randomly. For the threshold scheduling algorithm, thethreshold parameter was set to 0.5

of Poisson arrivals and exponentially distributed packetsizes, the state transition diagram can be viewed as a

Markov chain. The Markov chain is solved for equi-librium state probabilities at a particular offered load(Appendix III).

The analysis can be used to determine the normalizedthroughput at any normalized offered load. A graph of the nor-malized throughput vs. normalized offered load for different

h d li l ith i h i Fi 6 Th t f d l


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Fig. 7. Comparison of the performance of the transmission schedulingalgorithms based on simulation. The trafc model is Poisson arrivals withexponentially distributed packet lengths. The codeset is (10 , 3, 3) and code-words are chosen uniform randomly. For the threshold scheduling algorithm,the threshold parameter was set to 0.5

Figure 7 shows the results of simulation for the samecodeset as described in the analytical results. The results

are quite similar to the analytical results. All transmissionscheduling algorithms prevent throughput degradation. Alsothe overlap section and threshold scheduling show marginallyhigher throughput than pure selsh scheduling. The analyticalmodel over predicts the throughput for Aloha-CDMA. This isbecause the analysis is based on a nite state model. A nitestate model is suitable for transmission scheduling algorithms

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Fig. 8. Comparison of the codeword multiplexing of the different transmis-sion scheduling algorithms based on simulation. The trafc model is Poissonarrivals with exponentially distributed packet lengths. The codeset is (10 , 3, 3and codewords are chosen uniform randomly. For the threshold schedulingalgorithm, the threshold parameter was set to 0.5

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Fig. 11. Comparison of the performance of the transmission schedulingalgorithms as the codeset weight is varied (based on simulation). The trafcmodel is Poisson arrivals with exponentially distributed packet lengths. Thecodeset length is 100 and = 3 . Codewords are chosen uniform randomly forthe codeset. For the threshold scheduling algorithm, the threshold parameterwas set to 0.5

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Fig. 13. Comparison of the performance of the transmission schedulingalgorithms for a trimodal packet size distribution (based on simulation). Thetrafc model is Poisson arrivals with packet size distribution consisting of 70% 40 byte packets, 20% 1500 byte packets and 10% 500 byte packets. Thecodeset is (100 , 3, 3) and codewords are chosen uniform randomly. For thethreshold scheduling algorithm, the threshold parameter was set to 0.5

where short packets tend to experience lower error rates thanlong packets [11].

The squeeze through effect can be demonstrated analyticallyand through simulation for a network with a bimodal distri-bution of packet sizes which uses a pure selsh transmissionscheduling algorithm and a codeset with = w. Consider anetwork where the trafc has two packet types of sizes land l2 where l1 < l 2 . Let the fraction of packets of size


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SimulationAnalysis

Fig. 14. The squeeze through effect. The throughput is maximized whenthe fraction of short packets is 0.9. The graph shows both analytical andsimulation results. The transmission scheduling algorithm is pure selsh. Thepacket sizes are 50 bytes and 1000 bytes. The codeset is (100,3,3).

by packets that arrive during the packets transmission. Thetransmission scheduling algorithm allows only a fraction of thearriving packets (called colliding packets) to be transmitted.

The probability of packet error in a codeword on the line isthe probability that at least one of its colliding packets causesan interference error. If the number of colliding packets is n c ,then n c 1 chips are added to the state selshly (align with 0chips) and n c (w 1) 1 chips are added to the state in randompositions. These random positions are chosen from N 1possible choices (1 chip is chosen selshly) The probability

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Fig. 15. Comparison of the performance of the transmission schedulingalgorithms for a realistic trafc model (based on simulation). The trafcmodel was based on real network trafc traces (see description). The codesetis (100 , 3, 3) and codewords are allocated to addresses. For the thresholdscheduling algorithm, the threshold parameter was set to 0.5

low error rates. The throughput attains a maximum when thefraction of short packets reaches a particular value (around

0.9).The higher throughput of shorter packets may not be adesirable characteristic, because it is unfair to longer packets.Future work will address the issue of providing a uniformdropping probability to all trafc. Possible alternatives includeusing constant packet sizes or varying the codeset length.


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VI. R ELATED WORK

Work related to Interference Avoidance can be divided intofour areas: Bit error rate analysis of optical CDMA networks,

optical CDMA codeset design, FEC for optical CDMA andmedia access control in optical/wireless networks.Salehi [5], [8] analyzed an optical CDMA based network

and developed expressions for the bit error rate of a network that uses codesets with = 1 . The analysis also determinedthe bit error rate for codesets with different lengths andweights and with hard-limiting at low loads using Aloha-CDMA. This work examines these results in the context of transmission scheduling at high offered loads.

The area of optical CDMA code design has focused on con-struction of codesets with large size. Chung et al. [3] describedseveral algorithms to construct OOCs. These constructions arefor codes with maximum crosscorrelation parameter = 1 .Chung and Kumar [13] described a method for constructionof codes with = 2 . Several construction methods for OOCsare described in [4] and [14] among others.

Efforts at reducing packet errors in optical CDMA havemostly focused on using error correcting codes on top of

optical CDMA. Hsu et al. [15] analyzed the performance of slotted and unslotted optical CDMA packet networks. Theydeveloped expressions for the throughput of the network andshowed performance can be improved using Forward ErrorCorrection (FEC) codes and hard limiters. Muckenheim etal. [16] studied the effect of bit error probability on the packeterror probability and suggested the use of block codes to

systems state is a scalar variable and media access controlis through admission control i.e. the number of simultaneoususers is controlled.

VII. C ONCLUSIONS AND FUTURE WORK

This work has presented an analysis of transmissionscheduling algorithms for optical CDMA media access control.The analysis quantied the difference between throughput of systems with and without transmission scheduling and showedthat transmission scheduling achieved 30% throughput whilenon scheduled systems had close to zero throughput. Simula-tions showed that the throughput of transmission scheduling isindependent of codeset length. It also showed that an increasein weight can lead to a degradation in the performance of thesealgorithms, although the degradation is not as bad as systemswithout transmission scheduling. Simulations also showed thattransmission scheduling prevents degradation when used witha realistic trafc model based on trafc obtained from a realnetwork.

Limitations of this work include the fact that it assumes per-fect state estimation and neglects errors due to synchronizationand receiver contention. Future work will explore the impactof realistic state estimation.

Work in progress includes a testbed implementation of thetransmission scheduling hardware. The testbed demonstratesa simplied form of threshold transmission scheduling bytransmitting bits such that the number of chip overlaps is


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[11] B. S. Bakshi, P. Krishna, N. H. Vaidya, and D. K. Pradhan, Improvingperformance of TCP over wireless networks, in International Confer-ence on Distributed Computing Systems (IDCS) , May 1997, pp. 365373.

[12] NLANR, Trafc traces from an OC48 link, www.nlanr.net , March

2005.[13] H. Chung and P. V. Kumar, Optical orthogonal codes - new boundsand an optimal construction, IEEE Transactions on Information Theory ,vol. 36, no. 4, pp. 866873, July 1990.

[14] R. Fuji-Hara and Y. Miao, Optical orthogonal codes: Their bounds andnew optimal constructions, IEEE Transactions on Information Theory ,vol. 46, no. 7, pp. 23962406, Nov. 2000.

[15] C. S. Hsu and V. O. K. Li, Performance analysis of slotted ber-optic code division multiple access (CDMA) packet networks, IEEE Transactions on Communications , vol. 45, no. 7, pp. 819 828, July1997.

[16] J. Muckenheim, K. Iversen, and D. Hampicke, Construction of high-

efcient optical CDMA computer networks: Statistical design, in IEEE International Conference on Communications , vol. 3, June 1998, pp.12891293.

[17] C. Chae, E. Wong, and R. Tuckker, Ethernet over passive opticalnetwork based on optical CSMA/CD media access technique, in In-ternational Symposium on Services and Local access , April 2002.

[18] Cable Television Labs Inc., Data Over Cable Service Interface Spec-ications (DOCSIS) 2.0, Radio Frequency Interface Specication, no.SP-RFIv2.0, April 2004.

[19] A. H. Abdelmonem and T. N. Saadawi, Performance analysis of spreadspectrum packet radio network with channel load sensing, IEEE Journalon Special Areas in Communications , vol. 7, no. 1, pp. 161166, Jan.

1989.[20] G. Judge and F. Takawira, Spread Spectrum CDMA Packet Radio MACProtocol using Channel Overload Detection and Blocking, Wireless Networks , vol. 6, pp. 467479, December 2000.

[21] K. Toshimitsu, T. Yamazoto, M. Katayama, and A. Ogawa, A novelspread slotted Aloha system with channel load sensing protocol, IEEE Journal on Special Areas in Communications , vol. 12, no. 4, pp. 665672, May 1994.

[22] S A R dd d L T g E l iti g d t li d h l t t i f

For any other transmission scheduling algorithm, the sameaverage number of packets are offered for transmission. How-ever the transmission scheduling algorithm does not allow allof these packets to be transmitted. Let the average number

of packets on the line at any point of a receive ber beN online , of which a fraction P e are lost due to error. Theratio of the average number of error free packets transmittedto the average number of packets offered for transmission isN online (1 P e )/ (/ )

Therefore,

T hnorm = ( N online (1 P e )/ (/ ))= ( N online (1 P e )/ (/ ))( /N )

= N online (1

P e )/N

A PPENDIX IIN ORMALIZED NETWORK THROUGHPUT

This appendix derives an expression for the normalizednetwork throughput of an Interference Avoidance based opticalCDMA LAN. First, a concise representation of line state whichallows easy mathematical manipulation is dened. Using thisstate representation, expressions are derived for the number of codewords on the line and the probability of error when thesystem is in any state.

A. State representation

The state of the line can be represented by a pair ( n 0 , n1


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N, 0

N-w, w

N-w-1, w-1

N-w-2,w-2

N-w-1, w

N-w-1, w-1

Fig. 16. State transition diagram

Then,c0 = from n 0 ton 0c1 = tonoverlap from noverlapcoverlap = w (c0 + c1)

A valid transition is dened as a transition where the startstate and the destination state are valid, reachable states and

c0 0c1 0coverlap 0

An admissible transition is dened as a valid transitionwhich is permitted by the transmission scheduling algorithm.A same state transition is dened as a transition from a stateto itself A state transition diagram can be drawn based on

assumed to be ON i.e. transmitting 1 bits although that maynot necessarily be true. Therefore the calculated packet errorrate is the worst case packet error rate.

C. Number of codewords multiplexed at a point on the line

Consider a graph where the nodes of the graph are the statesand each state transition due to an arrival forms a directed edge(neglect same state transitions). For any selsh transmissionscheduling algorithm, this graph is a directed acyclic graph.In the graph, each edge represents the arrival of exactly onepacket. Therefore the number of codewords on the line for

a state ( n0 , n1 ) depends on the number of edges from theinitial state ( N, 0) to (n0 , n1 ). Both the shortest and the longestpath may be calculated in polynomial time. The length of theshortest and longest path from the initial state to that state arelower and upper bounds on the number of codewords on theline when the line is in that state 5 .

Therefore, for valid states,

N online (n0 , n 1) ShortestPath ((N, 0), (n 0 , n 1))

N online (n 0 , n 1) LongestPath ((N, 0), (n 0 , n 1))

For invalid states,

N online (n0 , n 1) = 0


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A. Assumptions

The analysis assumes perfect state estimation. The onlyreason for a state transition is an arrival or a departure of apacket. Packet arrivals are assumed to be Poisson arrivals andpacket lengths are exponentially distributed. The distributionof the destinations codeword is uniform over the codeset.Under this assumption, the probability of transitioning to aparticular state on an arrival is dependent only on the currentstate and not on the path taken to reach that state. Theprobability of departure to a state is assumed to be proportionalto the rate of arrival from that state. Then the next stateis dependent only on the current state and not on the pathtaken to reach that state. Under these circumstances, the statetransition diagram for arrivals and departures is a Markovchain. Equilibrium probabilities may be found by solving thebalance equations for the system.

B. Admissible transmissions

This subsection describes how to identify admissible statetransitions given a codeset and a transmission schedulingalgorithm. 6

1) Aloha-CDMA: For Aloha-CDMA, all transitions areadmissible.

2) Pure selsh scheduling: A transmission is admissible if,

c0 1

3) Threshold scheduling: A transmission is admissible if

c0 1

E. Balance equations

When the system is in equilibrium, the ow into any statemust equal the ow out of the state. Therefore for a valid,reachable state ( s0 , s1 ),

( + N online (s0 , s1))P state (s0 , s1) =N n 0 =0

N n 1 =0 P state (n 0 , n 1)P arr (n0 , n 1 , s0 , s1)+

N n 0 =0

N n 1 =0 P state (n 0 , n 1)N online (n0 , n 1)P dep (n 0 , n 1 ,

Also,N

n 0 =0

N

n 1 =0P state (n 0 , n 1) = 1;

These equations can be solved for the equilibrium stateprobabilities P state (s0 , s1).


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