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Multireceiver architecture analysis of a synchronous WDMA protocol
Journal of the Franklin Institute 339 (2002) 145–159
Multireceiver architecture analysis of a synchronous WDMA protocol Ioannis E. Pountourakis* Department of Electrical and Computer Engineering, National Technical University of Athens, Zographou, 157 73 Athens, Greece
Abstract In this paper, we propose the design and analysis of a reservation-based protocol for synchronous WDM multi-channel optical networks. The network architecture is based on a passive star topology and a new architecture for the network interface per station. The main objective of the scheduling algorithm and network interface is to maximize the performance measures by studying the problem of receiver collision phenomena at destination that multichannel nature of WDM networks introduces. We develop an analytical model based on a finite number of tunable receivers and a finite number of stations, following the ‘‘tell and go’’ policy for the access to communication system. Numerical results are showing the performance behavior for various number of channels, stations, and tunable receivers. Also, simulation results are presented for comparison with the results obtained by the performance analysis. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Wavelength division multiple access WDMA; Multichannel; Receiver collision; Multireceiver architecture; Average rejection probability
1. Introduction The wavelength division multiplexing (WDM) deployment is being driven by the increasing demands on communication bandwidth. WDM is an effective technique for utilizing the large bandwidth ðB50 THz) of an optical fiber. WDM divides the available bandwidth of the optical fiber into several channels with transmission rates (1–10 Gb=s) which are then shared amongst the various nodes of the network. In addition, WDM networks offer potential advantages, including compatibility with *Tel.: +30-1-772-2426; fax: +30-1-772-2534. E-mail address: [email protected] (I.E. Pountourakis). 0016-0032/02/$ 22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 8 - 2
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current electronic processing speeds of end stations, noise immunity, better network configurability and survivability. WDM based networks are possible in two architectures: (a) single-hop (direct communication between nodes and (b) multihop (communication through intermediate nodes). Many MAC protocols for single-hop WDM networks have been proposed and investigated in the resent past. These protocols were mainly used on passive star topology due to its relatively low losses and its inherent broadcast capability. A passive star topology also simplifies the addition of new stations, if compared with other highly interconnected topologies. In the single-hop configuration, the communications between each pair of stations are all-optical without passing through intermediate stations, thus the packet transmission between transmitter and receiver must occur at the same wavelength. The performance features of protocols proposed for a WDM network architecture depend on a number of key factors. Among them, the most important are tuning times, tuning range, processing requirements, propagation delay with respect to the packet transmission time, waiting time before packet transmission, whether or not a protocol requires network synchronization, channel collision, and receiver collision. These parameters set an upper bound on the maximum utilization of such protocols. A comprehensive survey of the above parameters for various protocols developed for the star-coupler networks are presented in [1]. A class of WDMA protocols based on pretransmission coordination employ commonly shared channel(s) to exchange control information while the remainder of the channels are used as data channels [2–15]. Thus, for each packet on a data channel, a control packet is transmitted on the shared control channel(s), and each station is required to process all the control packets on the control channel(s). In this protocol category, the phenomenon of receiver collision occurs when a collision-free packet transmission cannot be received by the intended destination since the destination’s receiver has been tuned to some other channel for receiving a data packet from some other source. Analytic approaches taking into account the effect of receiver collision in the performance evaluation has been done in [6–16]. In this paper, which is an extension of the study in [14], we propose a new network architecture where the network interface unit (NIU) of each station has more than one tunable receivers. The proposed protocol belongs to the WDMA synchronous transmission protocols in which data channels are slotted and stations are obligated to transmit at the beginning of each time interval denoted as data slot. We follow the ‘‘tell and go’’ procedure for the access to the medium in which a data packet is send irrespective of whether the corresponding control packet is successful or not. We examine the effect of receiver collision in performance measures using a passive star topology and a finite number of stations. For the proposed network architecture and protocol, we expect more efficient use of data channel bandwidth, thus improving the throughput of the system and providing lower packet delays due to the multiplicity of tunable receivers of the NIU. We also evaluate the rejection probability at a destination as a function of the number of tunable receivers, which substantiates the throughput loss in a quantitative fashion. Critical components for the implementation of the proposed protocol are the tunable receivers. Currently, tuning times for receivers are still relatively long compared to
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the packet transmission times and their tuning range is small. Under the assumption that these technological limitations will be solved in the near future, it is worth to explore the performance and complexity involved in designing such networks. For the analysis, we assume fast-tunable transmitters and fast-tunable receivers, with tuning times considerably shorter than the packet transmission times. The paper is organized as follows: Section 2 describes the network architecture. The basic assumptions about the proposed protocol are introduced here. In Section 3, the analysis of the throughput evaluation considering the receiver collision phenomenon is presented. In Section 4, the delay analysis is presented. In Section 5, numerical results and comments on the numerical results are given and an explanation of the behavior is discussed. Finally, some concluding remarks are made.
2. Network model and assumptions The system under consideration is a passive star network shown in Fig. 1. The system uses ðv þ 1Þ wavelengths, fl0 ; l1 ; y; lv g; to serve a finite number MðM > vÞ of stations. The channel at wavelength l0 operates as control channel while the remaining v channels at wavelengths fl1 ; y; lv g constitute the data multichannel system. The NIU can be described as a CC-TT-FR-TRF structure. It has a common control channel CC: TT means that each station has a tunable transmitter to be tuned to any of the channels fl0 ; l1 ; y; lv g: The outgoing traffic from each station is connected to one input of the passive star coupler. Additionally, every station uses one fixed tuned receiver FR for the control channel and F tunable receivers to any of the data channel wavelengths fl1 ; y; lv g indicated by TRF : The incoming traffic to a user station is splitted into v þ 1 wavelengths by a WDMA splitter from which F of the v wavelengths can be selected by the tunable receivers, as Fig. 1 indicates.
Fig. 1. Passive star multiwavelength architecture.
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A station will hear the result of a transmission of its control and data packets by listening to the star coupler multichannel system since it operates as a broadcast medium. We assume that the total offered traffic from newly generated packets and retransmitted packets obeys to Poisson statistics according to Bertsekas’s assumptions [17]. That is, if a newly generated packet is waiting for transmission or colliding with another packet, then the new arrivals at the station are lost and never return. A station involved in an unsuccessful transmission is called backlogged. The packet of a backlogged station must be retransmitted in some later time slot according to a retransmission policy, with possibly further such retransmissions until successful reception. In the proposed protocol, the control channel and data channels are slotted. The control channel is slotted with the fixed size of the control packet that is called minislot. Slots on data channels fit to the fixed size of data packets. The transmission time of a control packet is used as time unit. Thus, the data packet transmission time normalized to control minislot time units is L ðL > 1Þ: In our analysis, the access methods to the control channel and data multichannel system are based on the slotted Aloha protocol. Both control and data channels use the same time reference and are organized in cycles. We define the time interval that includes w time units for control packet transmissions followed by a data packet transmission period as cycle. Thus, the cycle time duration is C ¼ w þ L time units as Fig. 2 shows. The time axis is divided into contiguous cycles of equal length. The stations are synchronized for the transmission of the control and data packet during a cycle. Furthermore, propagation delays and tuning times are neglected and all stations have an equal distance from the star coupler. A station generating a data packet waits for the beginning of the next cycle, selects randomly one of the w contiguous minislots and sends a control packet on the control channel to compete according to the slotted Aloha protocol to gain access. The control packet, as Fig. 2 displays, consists of the transmitter address, the
Fig. 2. Cycle and structure of the control packet.
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receiver address, and the wavelength lk of the data channel. The lk is selected randomly among the v data channels. After the end of the first w time units of the cycle according to ‘‘tell and go’’ policy, the station transmits the data packet over the lk data channel. In high speed networks, this transmission policy is preferable when the round trip delays are more much longer than reservation part of the cycle. Thus, there are two causes of unsuccessful transmissions over the multichannel system. (a) If two or more stations choose the same minislot to transmit their control packets and they therefore overlap in time, a control channel collision occurs. (b) When more than one data packets select the same data channel for transmission, a collision will occur which is called a multichannel collision. In the receiving part, a station’s fixed tuned receiver monitors the control channel examining the control packets to listen for its address. If a station identifies its address as being announced in a control packet, it immediately adjusts a receiver to the wavelength channel specified in the control packet for data packet reception. We say that a receiver is active if it is tuned for receiving a data packet from a data channel li ði ¼ 1; y; vÞ: Data channels are slotted so that data packets are synchronized for transmission at the beginning of data slots. It is possible that more than F data packets on different data channels are transmitted successfully to the same destination station during a given cycle. In this case, all the F tunable receivers of the station are tuned to F data channels with incoming successful transmissions and rejects the others. This phenomenon is called receiver collision [10,11,16], see Fig. 3. There are different policies to select which packets are to be received correctly at the destination. As an example, we may decide that data packets that are transmitted on the lower data channels numbers win the competition. If a newly generated and successfully transmitted data packet is involved in a receiver collision, the packet is destroyed and the transmitting station becomes
Fig. 3. Receiver collision in synchronous transmission protocols.
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backlogged. A backlogged station waits some random time before retransmission after a collision, depending upon the particular backoff strategy being used. We assume that the random time-delay introduced between two consecutive retransmissions is uniformly distributed from 1 to K time units. We consider that at any point in time, each station is capable of transmitting at a particular wavelength and simultaneously receiving at F different wavelengths that belong to the set l1 ; y; lv :
3. Throughput analysis We use the following steady state notations:
G ¼ average number of transmitted control packets per minislot on the control channel GT ¼ average number of transmitted control packets during a cycle GT ¼ wG
ð1Þ
ScT ¼ Ps ¼
average number of successfully transmitted control packets during a cycle the probability of successful transmission of a control packet during a cycle Psuc ¼ the probability of success of a station on both the control channel and a data channel SdT ¼ the average number of successful transmitted data packets during a cycle S¼ the throughput of the system, defined as the average number of successful transmitted data packets normalized to the cycle duration time Srej ¼ average rate of destination conflicts of data packets during a cycle Src ¼ the average rate of the correctly received packets at the destination during a cycle, conditioning on the receiver collision effect Src;nor ¼ the throughput per data channel of the correctly received packets at the destination D¼ interval between the generation time of a data packet and the time of successful reception at the destination Dr ¼ delay from the transmission of a packet until the beginning of successful reception at the destination Dw ¼ waiting delay from the generation of a packet or from the time instant that a backlogged station decides to retransmit until the beginning of the next cycle. Now we analyze the throughput performance of the proposed protocol and evaluate the conditions for maximum throughput per data channel. The probability that a given station k transmits during a cycle in a minislot j is Gk and obeys to the binomial probability law.
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The probability that none of the other M 1 stations transmit in the same j minislot is given by M Y
ð1 Gl Þ:
ð2Þ
l¼1; lak
The probability that station k transmits successfully in minislot j can then be evaluated as Sck ðjÞ ¼ Gk
M Y
ð1 Gl Þ:
ð3Þ
l¼1; lak
Next, we suppose that stations operate independently and statistically identical. So in steady state Gl ¼ G=M 8lAf1; y; Mg: Thus, substituting in the above equation, we get the mean probability of a control packet being transmitted by a station in minislot j G G M1 k 1 Sc ðjÞ ¼ E½Sc ðjÞ ¼ : ð4Þ M M We define ScT ðj; MÞ; as the mean rate of successful transmissions of control packets in steady state during a given jth minislot. For large number of stations, the control channel throughput can be approximated by a Poisson distribution. According to the approximation ð1 xÞy Eexy for small x; we get, G M1 ScT ðj; MÞ ¼ G 1 EGeGðM1Þ=M EGeG : ð5Þ M Also ScT ¼
w X
ScT ðj; MÞ ¼ GT eG ;
ð6Þ
j¼1
Ps ¼
ScT ¼ eG : GT
ð7Þ
Define as Aw ; a random variable representing the number of transmitted data packets, during the second part of the ith cycle. We assume that the examined problem corresponds to the uniform model of distribution of indistinguishable packets to data channels supposing that arrangements should have equal probabilities. Let Aw ¼ k packets be uniformly distributed among v data channels. The random distribution in v data channels gives vk arrangements each with probability vk : Pd ðkÞ is the conditional probability that only one of k packets is associated with a given data channel n; nAf1; y; vg in the data slot time of the ith cycle. The remaining k 1 packets are associated with the remaining ðv 1Þ data channels in ðv 1Þk1 different ways. Then, k 1 k1 1 Pd ðkÞ ¼ : ð8Þ v v
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The approximation for large numbers of v gives k Pd ðkÞE eðk1Þ=v : v
ð9Þ
In steady state, E½Aw ¼ k ¼ GT :
ð10Þ
Let Pd be the probability of a data packet transmission in a data slot time of a cycle in steady state, regardless of successful transmission of the corresponding control packet. Then we get, Pd ¼ E½Pd ðkÞ E
GT ðGT 1Þ=v GT GT =v e E e ; v v
Psuc ¼ Ps Pd ¼ eG
GT GT =v e : v
ð11Þ
ð12Þ
Let Av be a random variable representing the number of successfully transmitted data packets during the ith cycle. The probability of finding Av ¼ m successfully transmitted data packets obeys the binomial probability law. ! v vm Pm : ð13Þ Pr½Av ¼ m ¼ suc ð1 Psuc Þ m Substituting Eq. (12) into Eq. (13) we get (see Appendix A), SdT ¼ E½Av ¼
v X
mPr½Av ¼ m ¼ wGeGð1þw=vÞ :
ð14Þ
m¼1
Then, S¼
L SdT : C
ð15Þ
From the above equation and for a given v; w and L; we can find the optimum value of traffic G that maximize the throughput of the system without the effect of receiver collision. Gopt ¼
1 ; 1 þ w=v
ð16Þ
Smax ¼
L w 1 : C 1 þ ðw=vÞ e
ð17Þ
The corresponding Snor;max is Snor;max ¼
L w 1 Cwþve
ð18Þ
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for large values of w we get L 1 Smax E v ; C e Snor;max E
L1 : Ce
ð19Þ
3.1. Receiver collision Taking into account the receiver collision phenomenon we define G ¼ fAvg: no: of collisions at the multichannel systemg þ fAvg: no: of correctly received packets at the destinationg þ fAvg: no: of abortions at the destinationg: The finite number of tunable receivers causes the possibility of a packet to be rejected at the destination. Consider that SdT ðGÞ ¼ r packets are successfully transmitted from the data multichannel system during a given cycle conditioning on the G: Let UZ be an indicator function denoting whether a station Z ðZ ¼ 1; 2; y; MÞ is selected as destination of a packet, i.e. ( þ1 if station Z is selected as destination in the ith cycle; UZ ¼ 0 else: Let Uv ðrÞ be the number of transmissions with destination station Z ðUZ ¼ 1Þ conditional that r packets have been transmitted successfully, 1pUv ðrÞpv: The probability of finding k packets with UZ ¼ 1; is given by Pr½Uv ðrÞ ¼ k=SdT ðGÞ ¼ r :
ð20Þ
The examined problem corresponds to the occupancy problem of distributing indistinguishable balls to cells supposing that arrangements should have equal probabilities. We assume r indistinguishable packets transmitted to M indistinguishable stations using Maxwell–Boltzman statistics [18]. We assume that these packets have been uniformly distributed among M stations. For the sake of simplicity of the analysis, we assume that a station may send packets to itself. The random distribution of r packets in M stations gives M r arrangements each with probability ð1=MÞr : The probability that k packets are destined to the Z station can be found as follows: The k packets can be chosen in r!=ðr kÞ!k! ways and the r k packets are destined to the remaining M 1 stations in ðM 1Þrk different ways. It follows that ! r ð1=MÞk ð1 1=MÞrk : ð21Þ Pr½Uv ðrÞ ¼ k ¼ k Suppose that Z station is the source of one of the r successfully transmitted packets. We assume that Z station does not transmit to itself. The conditional Uv ðrÞ ¼ k; without the simplification of Eq. (21) is evaluated in Appendix B.
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Given the number of tunable receivers of a station is F ; we examine the rejection probability of a successful transmitted data packet destined to station Z in steady state. The probability Pcol ðmÞ that m packets destined to station Z are aborted due to a finite number of tunable receivers is given by Pcol ðmÞ ¼
minðM;vF X Þ
Pr½SdT ðGÞ ¼ F þ i Pr½Uv ðF þ iÞ ¼ F þ m :
ð22Þ
i¼m
The mean probability that a successfully transmitted data packet is to be aborted by station Z in steady state, given that station’s finite number of tunable receivers is F ; is defined as Pcol ¼
minðM;vF X Þ
mPcol ðmÞ:
ð23Þ
m¼1
The probability Wv ¼ j of finding j different stations as destination of successfully transmitted packets that are aborted due to receiver collision at the end of a cycle obeys to the binomial probability law given by ! M PrðWv ¼ jÞ ¼ Pjcol ð1 Pcol ÞMj : ð24Þ j With that (see Appendix A), M X Srej ¼ E½PrðWv Þ ¼ jPrðWv ¼ jÞ ¼ MPcol ;
ð25Þ
j¼1
Src ¼ S Srej ; Src;nor ¼
Src : v
ð26Þ ð27Þ
3.2. Average rejection probability The average rejection probability at the destination of a packet is evaluated as the ratio of the average number of packet rejections at the destination in steady state due to the finite number of tunable receivers, and the average number of successfully transmitted packets per cycle Prej ¼ Srej =S:
ð28Þ
4. Delay analysis Stations participating in unsuccessful transmissions, defer their retransmissions for a random time until retransmission. We assume an uniform distribution of retransmissions. The random time delay introduced between two consecutive
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retransmissions is uniformly distributed from 1 to K time units, with average value given by Db ¼ ðK þ 1Þ=2: The overall delay consists of the following terms: D ¼ Dw þ Dr þ C:
ð29Þ
The average rate of correctly received data packets per station per cycle in steady state is wG Gð1þw=vÞ Srd ¼ e ð1 Pcol Þ: ð30Þ M From Eq. (30), we get for the probability of successful reception at the destination of a data packet in steady state Fr ¼ eGð1þw=vÞ ð1 Pcol Þ:
ð31Þ
Let Qr ðnÞ be the probability of successful retransmission of a data packet after n trials. Assuming that the probability of success is the same on any try, n has a geometric distribution, i.e. Qr ðnÞ ¼ Fr ð1 Fr Þn1 :
ð32Þ
Then, the average number of trials for a successful transmission of a data packet is p X eGð1þw=vÞ R ¼ E½n ¼ nQr ðnÞ ¼ 1=Fr ¼ : ð33Þ 1 Pcol n¼1 The mean value of Dw is C=2 and the average retransmission delay per packet is given by, ðR 1Þ½Dw þ C þ Db : So the average packet delay D is E½D ¼ E½Dw þ E½Dr þ C ¼ C=2 þ ðR 1ÞE½Dw þ C þ Db þ C ¼ feGð1þw=vÞ =ð1 Pcol Þ 1gf3C=2 þ ðK þ 1Þ=2Þg þ 3C=2:
ð34Þ
5. Numerical results We apply the above analytical results to study the effect of the finite number of tunable receivers on the performance measures. To verify the analytical results of the proposed finite number of tunable receivers, we have used a simulation model of the proposed system. Fig. 4, illustrates the throughput Src versus the offered traffic G for v ¼ 30 data channels, w ¼ 30 minislots, M ¼ 50 stations with F ¼ 1; 2; 3 tunable receivers, and data packet transmission time L ¼ 100 time units. Fig. 5 depicts the corresponding average rejection probabilities Prej versus the offered traffic G: It is obvious from the figures that for low values of traffic G; the Src and Prej increase almost linearly with G (low values of throughput) for all values of F ¼ 1–3. As G increases approaching Gopt and throughput approaches Smax ; then Src and Prej begin to saturate increasing
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Fig. 4. The throughput of the correctly received packets Src versus the offered traffic G(packets/minislot) for v ¼ 30 channels, w ¼ 30 minislots and M ¼ 50 stations and F ¼ 1; 2; 3 tunable receivers. Analytical and simulation results.
Fig. 5. Average rejection probabilities Prej (Log. Scale) versus G(packets/minislot) for v ¼ 30 channels, w ¼ 30 minislots and M ¼ 50 stations with F ¼ 1; 2 tunable receivers. Analytical and simulation results.
slowly towards the maximum Src;max and Prej;max : The explanation is that as G grows, the throughput S increases so that the probability of receiver collision is large. For higher values of GðG > Gopt Þ; S is reduced due to channel(s) collision. So Prej decreases because the probability of collision at the destination is lower. From Table 1 it can be observed that increasing the number of tunable receivers by one unit reduces Prej;max or Prej by many orders of magnitude. Also, for a large number of tunable receivers F ; Prej;max or Prej decreases and allows to approach Src to the maximum throughput S: In addition, the values given in Table 1 denotes that as M increases, Prej;max decreases.
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Table 1 Prej;max ; in a WDM network as a function of v ¼ 20; w ¼ 30; L ¼ 100; F ¼ 1–3 tunable receivers and M ¼ 50; 100; 200 stations F 1 2 3
M
50 (%)
100 (%)
200 (%)
4.085 0.107 0.002
2.07 0.0272 0.00025
1.041 0.0068 0.000032
Fig. 6. The average delay E½D versus the throughput per data channel Src;nor for v ¼ 30 channels, M ¼ 100 stations and L ¼ 100; K ¼ 50; with F ¼ 2 tunable receivers and w ¼ 20; 30; 40 minislots.
Fig. 6 shows the average delay E½D versus the throughput per data channel Src;nor for v ¼ 30 data channels, w ¼ 20; 30; 40 minislots, M ¼ 100 stations K ¼ 50; with F ¼ 2 tunable receivers, and data packet transmission time L ¼ 100 time units. It can be observed that for fixed value of K; and in the lower part of the curves, the delay increases very slowly with the throughput showing the high throughput and low delay desirable regions. As traffic increases, approaching its optimum value Gopt which corresponds to Smax and the average number of retransmissions R becomes significant, the average delay Db ¼ ðK þ 1Þ=2 between successive retransmissions begins to make noticeable difference in the average delay E½D : Also for fixed value of v; as w increases, the performance behavior is getting better. Also Eq. (17) denotes that for fixed value of v as w increases, the numerator increases faster than denominator so Smax improves.
6. Conclusions The important problem of receiver collision is analyzed for the well known passive star WDM topology with pretransmisson coordination of a reservation based-
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protocol using the ‘‘tell and go’’ (re)transmission policy. We propose at first an approximate and then a rigourous (Appendix B) method for the analytic evaluation of the performance measures of the proposed algorithm based upon Poisson assumptions for the total offered traffic in which each station is equipped with a finite number 1pF pv of tunable receivers. The numerical results show that for fixed M and v and increasing number of tunable receivers, the probability of packet rejection Prej decreases. Also for fixed v and L and increasing M; Prej decreases, too. Numerical results prove that four parameters characterize the performance behavior of the WDM networks, fv; M; F ; Prej;max g: It was shown that for large population systems or for a number of tunable receivers F X3; the effect of receiver collision can be ignored with only a small loss of accuracy. In the opposite case, in smaller systems the influence of receiver collision is significant for the performance behavior and cannot be neglected. Finally, simulation results show excellent agreement with the theoretically calculated values.
Appendix A Let F ðx; yÞ ¼
v X k¼1
! v xk yvk ¼ ðx þ yÞv : k
ðA:1Þ
If we get the first derivative of the above equation with respect to x; and multiply the two parts by x; we obtain v X k¼1
! v k xk yvk ¼ vxðx þ yÞv1 : k
ðA:2Þ
Substituting x by P; and y by ð1 PÞ; we obtain v X k¼1
! v k Pk ð1 PÞvk ¼ vP: k
ðA:3Þ
Appendix B Let suppose station Z does not transmit to itself. The probability that station Z is also the source of one of the r transmitted packets is r=M: In this case we assume that r 1 packets have been uniformly distributed among M 1 stations. The random distribution of r 1 packets in M 1 stations gives ðM 1Þr1 arrangements each with probability ð1=ðM 1ÞÞr1 : The probability that k packets are destined to Z
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station can be found as follows: ! k rk1 r r1 1 1 Pr½Uv ðrÞ ¼ k ¼ 1 M M1 M 1 k ! rk k r r 1 1 þ 1 1 : M M1 M 1 k
159
ðB:1Þ
References [1] P.A. Humblet, R. Ramaswami, K.N. Sivarajan, An efficient communication protocol for high-speed packet switched multichannel networks, IEEE J. Select. Areas Commun. 11 (4) (1993) 568–578. [2] M.I. Habbab, M. Kavehrad, C.E.W. Sundberg, Protocols for very high-speed optical fiber local area networks using a passive star topology, J. Lightwave Technol. LT-5 (12) (1987) 1782–1794. [3] N. Mehravari, Performance and protocol improvements for very high-speed optical fiber local area networks using a passive star topology, J. Lightwave Technol. 8 (4) (1990) 520–530. [4] G.N.M. Sudhakar, N.D. Georganas, M. Kavehrad, Slotted aloha and reservation aloha protocols for very high-speed optical fiber local area networks using passive star topology, J. Lightwave Technol. 9 (10) (1991) 1411–1422. [5] J. Lu, L. Kleinrock, A wavelength division multiple access protocol for high-speed local area networks with a passive star topology, Performance Eval. 16 (1992) 223–239. [6] I.E. Pountourakis, Receiver collision in WDM networks with finite number of tunable receivers, Proceedings of European Conference on Networks and Optical Communications NOC’99, Vol. II, Delft, June 1999, pp. 178–181. [7] I.E. Pountourakis, Performance of multichannel control protocols for WDMA LAN’s with a passive star topology, Proceedings of the Fourth IEEE Mediterranean Symposium on New Directions in Control and Automation, Krete, Greece, June 1996, pp. 546–551. [8] I.E. Pountourakis, ALOHA-type multiwavelength control protocols for very high speed optical fiber networks, Proceedings of the International Workshop on Networking and Information Services in the CEE Countries, Blagoevgrad, Bulgaria, July 1996, pp. 198–206. [9] I.E. Pountourakis, WDMA-local lightwave networks with an efficient control channels architecture for electronic processing bottleneck reduction, Proceedings of the Second IEEE Symposium on Computers and Communications ISCC’97, Alexandria, Egypt, July 1997, pp. 574–578. [10] I.E. Pountourakis, Receiver collisions in very high-speed optical fiber local area networks using passive star topology, Proceedings of European Conference on Networks and Optical Communications NOC’96, Vol. ii, Heidelberg, Germany, June 1996, pp. 45–52. [11] I.E. Pountourakis, Performance evaluation with receiver collisions analysis in very high-speed optical fiber local area networks using passive star topology, J. Lightwave Technol. 16 (12) (1998) 2303–2310. [12] I.E. Pountourakis, A multiwavelength control architecture for electronic processing bottleneck reduction in WDMA lightwave networks, Comput. Commun. 12 (15–16) (1999) 1468–1480. [13] I.E. Pountourakis, Wavelength control protocols for WDM LAN’s with restricted data channel collisions, Int. J. Commun. Syst. 13 (4) (2000) 335–352. [14] I.E. Pountourakis, Analysis of synchronous tell and go WDMA protocol, Seventh Annual International Conference on Advances in Communication and Control Proceedings COMCON 71999, Athens, Greece, June 28–July 2, 1999, pp. 543–554. [15] I.E. Pountourakis, Design and protocol analysis for passive star topology of a WDM network, Microwave Opt. Technol. Lett. 25 (3) (2000) 216–220. [16] I.E. Pountourakis, Throughput evaluation of multichannel slotted ALOHA-type protocols with receiver collisions, Telecommun. Syst. J. 5 (4) (1996) 413–419. [17] D. Bertsekas, R. Gallager, Data Networks. Prentice-Hall, India, 1989, pp. 209–213. [18] William Feller, An Introduction to Probability Theory and its Applications, Vol. 1, Wiley, New York, 1957,1968.
Multireceiver architecture analysis of a synchronous WDMA protocol Ioannis E. Pountourakis* Department of Electrical and Computer Engineering, National Technical University of Athens, Zographou, 157 73 Athens, Greece
Abstract In this paper, we propose the design and analysis of a reservation-based protocol for synchronous WDM multi-channel optical networks. The network architecture is based on a passive star topology and a new architecture for the network interface per station. The main objective of the scheduling algorithm and network interface is to maximize the performance measures by studying the problem of receiver collision phenomena at destination that multichannel nature of WDM networks introduces. We develop an analytical model based on a finite number of tunable receivers and a finite number of stations, following the ‘‘tell and go’’ policy for the access to communication system. Numerical results are showing the performance behavior for various number of channels, stations, and tunable receivers. Also, simulation results are presented for comparison with the results obtained by the performance analysis. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Wavelength division multiple access WDMA; Multichannel; Receiver collision; Multireceiver architecture; Average rejection probability
1. Introduction The wavelength division multiplexing (WDM) deployment is being driven by the increasing demands on communication bandwidth. WDM is an effective technique for utilizing the large bandwidth ðB50 THz) of an optical fiber. WDM divides the available bandwidth of the optical fiber into several channels with transmission rates (1–10 Gb=s) which are then shared amongst the various nodes of the network. In addition, WDM networks offer potential advantages, including compatibility with *Tel.: +30-1-772-2426; fax: +30-1-772-2534. E-mail address: [email protected] (I.E. Pountourakis). 0016-0032/02/$ 22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 8 - 2
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current electronic processing speeds of end stations, noise immunity, better network configurability and survivability. WDM based networks are possible in two architectures: (a) single-hop (direct communication between nodes and (b) multihop (communication through intermediate nodes). Many MAC protocols for single-hop WDM networks have been proposed and investigated in the resent past. These protocols were mainly used on passive star topology due to its relatively low losses and its inherent broadcast capability. A passive star topology also simplifies the addition of new stations, if compared with other highly interconnected topologies. In the single-hop configuration, the communications between each pair of stations are all-optical without passing through intermediate stations, thus the packet transmission between transmitter and receiver must occur at the same wavelength. The performance features of protocols proposed for a WDM network architecture depend on a number of key factors. Among them, the most important are tuning times, tuning range, processing requirements, propagation delay with respect to the packet transmission time, waiting time before packet transmission, whether or not a protocol requires network synchronization, channel collision, and receiver collision. These parameters set an upper bound on the maximum utilization of such protocols. A comprehensive survey of the above parameters for various protocols developed for the star-coupler networks are presented in [1]. A class of WDMA protocols based on pretransmission coordination employ commonly shared channel(s) to exchange control information while the remainder of the channels are used as data channels [2–15]. Thus, for each packet on a data channel, a control packet is transmitted on the shared control channel(s), and each station is required to process all the control packets on the control channel(s). In this protocol category, the phenomenon of receiver collision occurs when a collision-free packet transmission cannot be received by the intended destination since the destination’s receiver has been tuned to some other channel for receiving a data packet from some other source. Analytic approaches taking into account the effect of receiver collision in the performance evaluation has been done in [6–16]. In this paper, which is an extension of the study in [14], we propose a new network architecture where the network interface unit (NIU) of each station has more than one tunable receivers. The proposed protocol belongs to the WDMA synchronous transmission protocols in which data channels are slotted and stations are obligated to transmit at the beginning of each time interval denoted as data slot. We follow the ‘‘tell and go’’ procedure for the access to the medium in which a data packet is send irrespective of whether the corresponding control packet is successful or not. We examine the effect of receiver collision in performance measures using a passive star topology and a finite number of stations. For the proposed network architecture and protocol, we expect more efficient use of data channel bandwidth, thus improving the throughput of the system and providing lower packet delays due to the multiplicity of tunable receivers of the NIU. We also evaluate the rejection probability at a destination as a function of the number of tunable receivers, which substantiates the throughput loss in a quantitative fashion. Critical components for the implementation of the proposed protocol are the tunable receivers. Currently, tuning times for receivers are still relatively long compared to
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the packet transmission times and their tuning range is small. Under the assumption that these technological limitations will be solved in the near future, it is worth to explore the performance and complexity involved in designing such networks. For the analysis, we assume fast-tunable transmitters and fast-tunable receivers, with tuning times considerably shorter than the packet transmission times. The paper is organized as follows: Section 2 describes the network architecture. The basic assumptions about the proposed protocol are introduced here. In Section 3, the analysis of the throughput evaluation considering the receiver collision phenomenon is presented. In Section 4, the delay analysis is presented. In Section 5, numerical results and comments on the numerical results are given and an explanation of the behavior is discussed. Finally, some concluding remarks are made.
2. Network model and assumptions The system under consideration is a passive star network shown in Fig. 1. The system uses ðv þ 1Þ wavelengths, fl0 ; l1 ; y; lv g; to serve a finite number MðM > vÞ of stations. The channel at wavelength l0 operates as control channel while the remaining v channels at wavelengths fl1 ; y; lv g constitute the data multichannel system. The NIU can be described as a CC-TT-FR-TRF structure. It has a common control channel CC: TT means that each station has a tunable transmitter to be tuned to any of the channels fl0 ; l1 ; y; lv g: The outgoing traffic from each station is connected to one input of the passive star coupler. Additionally, every station uses one fixed tuned receiver FR for the control channel and F tunable receivers to any of the data channel wavelengths fl1 ; y; lv g indicated by TRF : The incoming traffic to a user station is splitted into v þ 1 wavelengths by a WDMA splitter from which F of the v wavelengths can be selected by the tunable receivers, as Fig. 1 indicates.
Fig. 1. Passive star multiwavelength architecture.
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A station will hear the result of a transmission of its control and data packets by listening to the star coupler multichannel system since it operates as a broadcast medium. We assume that the total offered traffic from newly generated packets and retransmitted packets obeys to Poisson statistics according to Bertsekas’s assumptions [17]. That is, if a newly generated packet is waiting for transmission or colliding with another packet, then the new arrivals at the station are lost and never return. A station involved in an unsuccessful transmission is called backlogged. The packet of a backlogged station must be retransmitted in some later time slot according to a retransmission policy, with possibly further such retransmissions until successful reception. In the proposed protocol, the control channel and data channels are slotted. The control channel is slotted with the fixed size of the control packet that is called minislot. Slots on data channels fit to the fixed size of data packets. The transmission time of a control packet is used as time unit. Thus, the data packet transmission time normalized to control minislot time units is L ðL > 1Þ: In our analysis, the access methods to the control channel and data multichannel system are based on the slotted Aloha protocol. Both control and data channels use the same time reference and are organized in cycles. We define the time interval that includes w time units for control packet transmissions followed by a data packet transmission period as cycle. Thus, the cycle time duration is C ¼ w þ L time units as Fig. 2 shows. The time axis is divided into contiguous cycles of equal length. The stations are synchronized for the transmission of the control and data packet during a cycle. Furthermore, propagation delays and tuning times are neglected and all stations have an equal distance from the star coupler. A station generating a data packet waits for the beginning of the next cycle, selects randomly one of the w contiguous minislots and sends a control packet on the control channel to compete according to the slotted Aloha protocol to gain access. The control packet, as Fig. 2 displays, consists of the transmitter address, the
Fig. 2. Cycle and structure of the control packet.
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receiver address, and the wavelength lk of the data channel. The lk is selected randomly among the v data channels. After the end of the first w time units of the cycle according to ‘‘tell and go’’ policy, the station transmits the data packet over the lk data channel. In high speed networks, this transmission policy is preferable when the round trip delays are more much longer than reservation part of the cycle. Thus, there are two causes of unsuccessful transmissions over the multichannel system. (a) If two or more stations choose the same minislot to transmit their control packets and they therefore overlap in time, a control channel collision occurs. (b) When more than one data packets select the same data channel for transmission, a collision will occur which is called a multichannel collision. In the receiving part, a station’s fixed tuned receiver monitors the control channel examining the control packets to listen for its address. If a station identifies its address as being announced in a control packet, it immediately adjusts a receiver to the wavelength channel specified in the control packet for data packet reception. We say that a receiver is active if it is tuned for receiving a data packet from a data channel li ði ¼ 1; y; vÞ: Data channels are slotted so that data packets are synchronized for transmission at the beginning of data slots. It is possible that more than F data packets on different data channels are transmitted successfully to the same destination station during a given cycle. In this case, all the F tunable receivers of the station are tuned to F data channels with incoming successful transmissions and rejects the others. This phenomenon is called receiver collision [10,11,16], see Fig. 3. There are different policies to select which packets are to be received correctly at the destination. As an example, we may decide that data packets that are transmitted on the lower data channels numbers win the competition. If a newly generated and successfully transmitted data packet is involved in a receiver collision, the packet is destroyed and the transmitting station becomes
Fig. 3. Receiver collision in synchronous transmission protocols.
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backlogged. A backlogged station waits some random time before retransmission after a collision, depending upon the particular backoff strategy being used. We assume that the random time-delay introduced between two consecutive retransmissions is uniformly distributed from 1 to K time units. We consider that at any point in time, each station is capable of transmitting at a particular wavelength and simultaneously receiving at F different wavelengths that belong to the set l1 ; y; lv :
3. Throughput analysis We use the following steady state notations:
G ¼ average number of transmitted control packets per minislot on the control channel GT ¼ average number of transmitted control packets during a cycle GT ¼ wG
ð1Þ
ScT ¼ Ps ¼
average number of successfully transmitted control packets during a cycle the probability of successful transmission of a control packet during a cycle Psuc ¼ the probability of success of a station on both the control channel and a data channel SdT ¼ the average number of successful transmitted data packets during a cycle S¼ the throughput of the system, defined as the average number of successful transmitted data packets normalized to the cycle duration time Srej ¼ average rate of destination conflicts of data packets during a cycle Src ¼ the average rate of the correctly received packets at the destination during a cycle, conditioning on the receiver collision effect Src;nor ¼ the throughput per data channel of the correctly received packets at the destination D¼ interval between the generation time of a data packet and the time of successful reception at the destination Dr ¼ delay from the transmission of a packet until the beginning of successful reception at the destination Dw ¼ waiting delay from the generation of a packet or from the time instant that a backlogged station decides to retransmit until the beginning of the next cycle. Now we analyze the throughput performance of the proposed protocol and evaluate the conditions for maximum throughput per data channel. The probability that a given station k transmits during a cycle in a minislot j is Gk and obeys to the binomial probability law.
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The probability that none of the other M 1 stations transmit in the same j minislot is given by M Y
ð1 Gl Þ:
ð2Þ
l¼1; lak
The probability that station k transmits successfully in minislot j can then be evaluated as Sck ðjÞ ¼ Gk
M Y
ð1 Gl Þ:
ð3Þ
l¼1; lak
Next, we suppose that stations operate independently and statistically identical. So in steady state Gl ¼ G=M 8lAf1; y; Mg: Thus, substituting in the above equation, we get the mean probability of a control packet being transmitted by a station in minislot j G G M1 k 1 Sc ðjÞ ¼ E½Sc ðjÞ ¼ : ð4Þ M M We define ScT ðj; MÞ; as the mean rate of successful transmissions of control packets in steady state during a given jth minislot. For large number of stations, the control channel throughput can be approximated by a Poisson distribution. According to the approximation ð1 xÞy Eexy for small x; we get, G M1 ScT ðj; MÞ ¼ G 1 EGeGðM1Þ=M EGeG : ð5Þ M Also ScT ¼
w X
ScT ðj; MÞ ¼ GT eG ;
ð6Þ
j¼1
Ps ¼
ScT ¼ eG : GT
ð7Þ
Define as Aw ; a random variable representing the number of transmitted data packets, during the second part of the ith cycle. We assume that the examined problem corresponds to the uniform model of distribution of indistinguishable packets to data channels supposing that arrangements should have equal probabilities. Let Aw ¼ k packets be uniformly distributed among v data channels. The random distribution in v data channels gives vk arrangements each with probability vk : Pd ðkÞ is the conditional probability that only one of k packets is associated with a given data channel n; nAf1; y; vg in the data slot time of the ith cycle. The remaining k 1 packets are associated with the remaining ðv 1Þ data channels in ðv 1Þk1 different ways. Then, k 1 k1 1 Pd ðkÞ ¼ : ð8Þ v v
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The approximation for large numbers of v gives k Pd ðkÞE eðk1Þ=v : v
ð9Þ
In steady state, E½Aw ¼ k ¼ GT :
ð10Þ
Let Pd be the probability of a data packet transmission in a data slot time of a cycle in steady state, regardless of successful transmission of the corresponding control packet. Then we get, Pd ¼ E½Pd ðkÞ E
GT ðGT 1Þ=v GT GT =v e E e ; v v
Psuc ¼ Ps Pd ¼ eG
GT GT =v e : v
ð11Þ
ð12Þ
Let Av be a random variable representing the number of successfully transmitted data packets during the ith cycle. The probability of finding Av ¼ m successfully transmitted data packets obeys the binomial probability law. ! v vm Pm : ð13Þ Pr½Av ¼ m ¼ suc ð1 Psuc Þ m Substituting Eq. (12) into Eq. (13) we get (see Appendix A), SdT ¼ E½Av ¼
v X
mPr½Av ¼ m ¼ wGeGð1þw=vÞ :
ð14Þ
m¼1
Then, S¼
L SdT : C
ð15Þ
From the above equation and for a given v; w and L; we can find the optimum value of traffic G that maximize the throughput of the system without the effect of receiver collision. Gopt ¼
1 ; 1 þ w=v
ð16Þ
Smax ¼
L w 1 : C 1 þ ðw=vÞ e
ð17Þ
The corresponding Snor;max is Snor;max ¼
L w 1 Cwþve
ð18Þ
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for large values of w we get L 1 Smax E v ; C e Snor;max E
L1 : Ce
ð19Þ
3.1. Receiver collision Taking into account the receiver collision phenomenon we define G ¼ fAvg: no: of collisions at the multichannel systemg þ fAvg: no: of correctly received packets at the destinationg þ fAvg: no: of abortions at the destinationg: The finite number of tunable receivers causes the possibility of a packet to be rejected at the destination. Consider that SdT ðGÞ ¼ r packets are successfully transmitted from the data multichannel system during a given cycle conditioning on the G: Let UZ be an indicator function denoting whether a station Z ðZ ¼ 1; 2; y; MÞ is selected as destination of a packet, i.e. ( þ1 if station Z is selected as destination in the ith cycle; UZ ¼ 0 else: Let Uv ðrÞ be the number of transmissions with destination station Z ðUZ ¼ 1Þ conditional that r packets have been transmitted successfully, 1pUv ðrÞpv: The probability of finding k packets with UZ ¼ 1; is given by Pr½Uv ðrÞ ¼ k=SdT ðGÞ ¼ r :
ð20Þ
The examined problem corresponds to the occupancy problem of distributing indistinguishable balls to cells supposing that arrangements should have equal probabilities. We assume r indistinguishable packets transmitted to M indistinguishable stations using Maxwell–Boltzman statistics [18]. We assume that these packets have been uniformly distributed among M stations. For the sake of simplicity of the analysis, we assume that a station may send packets to itself. The random distribution of r packets in M stations gives M r arrangements each with probability ð1=MÞr : The probability that k packets are destined to the Z station can be found as follows: The k packets can be chosen in r!=ðr kÞ!k! ways and the r k packets are destined to the remaining M 1 stations in ðM 1Þrk different ways. It follows that ! r ð1=MÞk ð1 1=MÞrk : ð21Þ Pr½Uv ðrÞ ¼ k ¼ k Suppose that Z station is the source of one of the r successfully transmitted packets. We assume that Z station does not transmit to itself. The conditional Uv ðrÞ ¼ k; without the simplification of Eq. (21) is evaluated in Appendix B.
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Given the number of tunable receivers of a station is F ; we examine the rejection probability of a successful transmitted data packet destined to station Z in steady state. The probability Pcol ðmÞ that m packets destined to station Z are aborted due to a finite number of tunable receivers is given by Pcol ðmÞ ¼
minðM;vF X Þ
Pr½SdT ðGÞ ¼ F þ i Pr½Uv ðF þ iÞ ¼ F þ m :
ð22Þ
i¼m
The mean probability that a successfully transmitted data packet is to be aborted by station Z in steady state, given that station’s finite number of tunable receivers is F ; is defined as Pcol ¼
minðM;vF X Þ
mPcol ðmÞ:
ð23Þ
m¼1
The probability Wv ¼ j of finding j different stations as destination of successfully transmitted packets that are aborted due to receiver collision at the end of a cycle obeys to the binomial probability law given by ! M PrðWv ¼ jÞ ¼ Pjcol ð1 Pcol ÞMj : ð24Þ j With that (see Appendix A), M X Srej ¼ E½PrðWv Þ ¼ jPrðWv ¼ jÞ ¼ MPcol ;
ð25Þ
j¼1
Src ¼ S Srej ; Src;nor ¼
Src : v
ð26Þ ð27Þ
3.2. Average rejection probability The average rejection probability at the destination of a packet is evaluated as the ratio of the average number of packet rejections at the destination in steady state due to the finite number of tunable receivers, and the average number of successfully transmitted packets per cycle Prej ¼ Srej =S:
ð28Þ
4. Delay analysis Stations participating in unsuccessful transmissions, defer their retransmissions for a random time until retransmission. We assume an uniform distribution of retransmissions. The random time delay introduced between two consecutive
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retransmissions is uniformly distributed from 1 to K time units, with average value given by Db ¼ ðK þ 1Þ=2: The overall delay consists of the following terms: D ¼ Dw þ Dr þ C:
ð29Þ
The average rate of correctly received data packets per station per cycle in steady state is wG Gð1þw=vÞ Srd ¼ e ð1 Pcol Þ: ð30Þ M From Eq. (30), we get for the probability of successful reception at the destination of a data packet in steady state Fr ¼ eGð1þw=vÞ ð1 Pcol Þ:
ð31Þ
Let Qr ðnÞ be the probability of successful retransmission of a data packet after n trials. Assuming that the probability of success is the same on any try, n has a geometric distribution, i.e. Qr ðnÞ ¼ Fr ð1 Fr Þn1 :
ð32Þ
Then, the average number of trials for a successful transmission of a data packet is p X eGð1þw=vÞ R ¼ E½n ¼ nQr ðnÞ ¼ 1=Fr ¼ : ð33Þ 1 Pcol n¼1 The mean value of Dw is C=2 and the average retransmission delay per packet is given by, ðR 1Þ½Dw þ C þ Db : So the average packet delay D is E½D ¼ E½Dw þ E½Dr þ C ¼ C=2 þ ðR 1ÞE½Dw þ C þ Db þ C ¼ feGð1þw=vÞ =ð1 Pcol Þ 1gf3C=2 þ ðK þ 1Þ=2Þg þ 3C=2:
ð34Þ
5. Numerical results We apply the above analytical results to study the effect of the finite number of tunable receivers on the performance measures. To verify the analytical results of the proposed finite number of tunable receivers, we have used a simulation model of the proposed system. Fig. 4, illustrates the throughput Src versus the offered traffic G for v ¼ 30 data channels, w ¼ 30 minislots, M ¼ 50 stations with F ¼ 1; 2; 3 tunable receivers, and data packet transmission time L ¼ 100 time units. Fig. 5 depicts the corresponding average rejection probabilities Prej versus the offered traffic G: It is obvious from the figures that for low values of traffic G; the Src and Prej increase almost linearly with G (low values of throughput) for all values of F ¼ 1–3. As G increases approaching Gopt and throughput approaches Smax ; then Src and Prej begin to saturate increasing
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Fig. 4. The throughput of the correctly received packets Src versus the offered traffic G(packets/minislot) for v ¼ 30 channels, w ¼ 30 minislots and M ¼ 50 stations and F ¼ 1; 2; 3 tunable receivers. Analytical and simulation results.
Fig. 5. Average rejection probabilities Prej (Log. Scale) versus G(packets/minislot) for v ¼ 30 channels, w ¼ 30 minislots and M ¼ 50 stations with F ¼ 1; 2 tunable receivers. Analytical and simulation results.
slowly towards the maximum Src;max and Prej;max : The explanation is that as G grows, the throughput S increases so that the probability of receiver collision is large. For higher values of GðG > Gopt Þ; S is reduced due to channel(s) collision. So Prej decreases because the probability of collision at the destination is lower. From Table 1 it can be observed that increasing the number of tunable receivers by one unit reduces Prej;max or Prej by many orders of magnitude. Also, for a large number of tunable receivers F ; Prej;max or Prej decreases and allows to approach Src to the maximum throughput S: In addition, the values given in Table 1 denotes that as M increases, Prej;max decreases.
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Table 1 Prej;max ; in a WDM network as a function of v ¼ 20; w ¼ 30; L ¼ 100; F ¼ 1–3 tunable receivers and M ¼ 50; 100; 200 stations F 1 2 3
M
50 (%)
100 (%)
200 (%)
4.085 0.107 0.002
2.07 0.0272 0.00025
1.041 0.0068 0.000032
Fig. 6. The average delay E½D versus the throughput per data channel Src;nor for v ¼ 30 channels, M ¼ 100 stations and L ¼ 100; K ¼ 50; with F ¼ 2 tunable receivers and w ¼ 20; 30; 40 minislots.
Fig. 6 shows the average delay E½D versus the throughput per data channel Src;nor for v ¼ 30 data channels, w ¼ 20; 30; 40 minislots, M ¼ 100 stations K ¼ 50; with F ¼ 2 tunable receivers, and data packet transmission time L ¼ 100 time units. It can be observed that for fixed value of K; and in the lower part of the curves, the delay increases very slowly with the throughput showing the high throughput and low delay desirable regions. As traffic increases, approaching its optimum value Gopt which corresponds to Smax and the average number of retransmissions R becomes significant, the average delay Db ¼ ðK þ 1Þ=2 between successive retransmissions begins to make noticeable difference in the average delay E½D : Also for fixed value of v; as w increases, the performance behavior is getting better. Also Eq. (17) denotes that for fixed value of v as w increases, the numerator increases faster than denominator so Smax improves.
6. Conclusions The important problem of receiver collision is analyzed for the well known passive star WDM topology with pretransmisson coordination of a reservation based-
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protocol using the ‘‘tell and go’’ (re)transmission policy. We propose at first an approximate and then a rigourous (Appendix B) method for the analytic evaluation of the performance measures of the proposed algorithm based upon Poisson assumptions for the total offered traffic in which each station is equipped with a finite number 1pF pv of tunable receivers. The numerical results show that for fixed M and v and increasing number of tunable receivers, the probability of packet rejection Prej decreases. Also for fixed v and L and increasing M; Prej decreases, too. Numerical results prove that four parameters characterize the performance behavior of the WDM networks, fv; M; F ; Prej;max g: It was shown that for large population systems or for a number of tunable receivers F X3; the effect of receiver collision can be ignored with only a small loss of accuracy. In the opposite case, in smaller systems the influence of receiver collision is significant for the performance behavior and cannot be neglected. Finally, simulation results show excellent agreement with the theoretically calculated values.
Appendix A Let F ðx; yÞ ¼
v X k¼1
! v xk yvk ¼ ðx þ yÞv : k
ðA:1Þ
If we get the first derivative of the above equation with respect to x; and multiply the two parts by x; we obtain v X k¼1
! v k xk yvk ¼ vxðx þ yÞv1 : k
ðA:2Þ
Substituting x by P; and y by ð1 PÞ; we obtain v X k¼1
! v k Pk ð1 PÞvk ¼ vP: k
ðA:3Þ
Appendix B Let suppose station Z does not transmit to itself. The probability that station Z is also the source of one of the r transmitted packets is r=M: In this case we assume that r 1 packets have been uniformly distributed among M 1 stations. The random distribution of r 1 packets in M 1 stations gives ðM 1Þr1 arrangements each with probability ð1=ðM 1ÞÞr1 : The probability that k packets are destined to Z
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station can be found as follows: ! k rk1 r r1 1 1 Pr½Uv ðrÞ ¼ k ¼ 1 M M1 M 1 k ! rk k r r 1 1 þ 1 1 : M M1 M 1 k
159
ðB:1Þ
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