Performance optimization with propagation delay analysis in WDM networks

Available online at www.sciencedirect.com

Computer Communications 30 (2007) 3572–3585 www.elsevier.com/locate/comcom

Performance optimization with propagation delay analysis in WDM networks P.A. Baziana *, I.E. Pountourakis School of Electrical and Computer Engineering, Department of Communications Electronic & Information Engineering, National Technical University of Athens, 157 73 Zografou, Athens, Greece Available online 1 September 2007

Abstract In this study we attempt to analyze a synchronous transmission WDMA protocol for passive star topology based on the propagation delay latency effect. The introduction of propagation delay constitutes a serious parameter for WDM networks performance behavior and a realistic basis for the analysis. The proposed WDM network uses a Multi-channel Control Architecture (MCA) for two reasons: First to reduce the electronic processing bottleneck and second to optimize the performance measures by dynamically dividing the control channels into two groups according to the knowledge of the stations status (free or backlogged). In this way we develop a Markovian model for finite population with receiver collisions evaluation.  2007 Elsevier B.V. All rights reserved. Keywords: Wavelength Division Multiplexing (WDM); Multi-channel Control Architecture (MCA); Asymmetric access rights; Propagation delay latency; Receiver collisions

1. Introduction In WDMA protocols, a fundamental parameter that plays key role in the performance evaluation is the round trip propagation delay latency. In the literature, few studies consider its effect to the network efficiency. In [1], a passive star network with a centralized master/slave scheduler located at the hub is used and the effect of propagation delay is overcome by measuring the delays between the stations and the hub and taking that delay into account when scheduling transmissions. In case of WDMA protocols with no pre-transmission coordination, the impact of propagation delay latency is avoided prior to the transmission phase, as there is no control information exchange. In [2], two synchronous access protocols requiring no pre-transmission penalty for WDM passive star network architecture are presented and the performance measures are evaluated through an approximate analytical model. On the other hand, in case *

Corresponding author. Tel.: +30 210 7722145; fax: +30 210 7722534. E-mail address: [email protected] (P.A. Baziana).

0140-3664/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2007.08.032

of WDMA protocols with pre-transmission coordination schemes the presence of propagation delay is critical when scheduling transmissions at the control coordination phase. In [3], a collision-free pre-transmission coordination protocol is given to predict the transmission requests and to eliminate the scheduling delay time in WDM passive star architectures. Also, in [4] the effect of the propagation delay on the performance of two reservation based access protocols is given and the average packet delay is estimated in case of non zero propagation delay value and uniform traffic. A class of WDMA protocols based on pre-transmission coordination employs a single separate common shared channel to exchange control information and the remainder channels are used as data channels [5–8]. A fundamental problem in this protocol class is the ability of a station to receive and process all control packets that are transmitted over the single control channel. This causes the electronic processing bottleneck [9]. In this paper, the network configuration adopts the Multi-channel Control Architecture (MCA) [10] which provides significantly less processing overhead for control

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

information exchange and faces the electronic processing bottleneck problem, while it efficiently improves the network performance. The proposed access scheme belongs to the synchronous transmission protocols. We apply a similar to [11] ‘‘tell and wait’’ algorithm to access the control and data channels, given that round trip propagation delay is longer than data packet transmission time. Thus, the developed access algorithm avoids the data channel collisions improving even more the performance. On the contrary to the data channel collisions avoidance method, the system suffers from the control channel collisions that essentially deteriorate the effectiveness of the system. This fact has been proven in [11] where the stations have symmetric access rights to the MCA for control packet exchange. In the present study, we extend the analysis of [11] exploiting the benefit of the round trip propagation delay as acknowledgment time to reduce the channel collisions and to improve the network efficiency. Moreover, in order to improve even more the system performance and to manage optimum throughput, we introduce a dynamic division of the MCA into two groups of channels based on the knowledge about the stations status (free or backlogged). In the first group of control channels the free stations compete to gain access, while in the second group the backlogged stations compete. The dynamic procedure of control channels separation and the stations asymmetric access rights for (re)transmission over the two groups of control channels achieve maximum MCA utilization. This fact consists the novelty of our study in order to reduce the control channel collisions. Despite the performance improvement, these systems are affected by the receiver collisions phenomenon [12,13]. In this paper, we develop a rigorous queuing analysis without approximations based on Markovian models for a system with finite number of stations. In order to quantify the impact of receiver collisions in the proposed network architecture, we develop two Markovian models

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and we study two protocol cases: (a) the protocol case without receiver collisions, where the impact of receiver collisions is ignored. In this protocol case, packet loss may occur because of either the control channel collisions or the data channel collisions avoidance algorithm and (b) the protocol case with receiver collisions, where the effect of receiver collisions is taken into account. In this protocol case, packet loss may additionally occur because of the conflicts at destination [14–16]. The innovation of our study is the exploitation of the propagation delay latency and the introduction of the dynamic separation of the MCA into two groups of channels, in order to reduce the control channel collisions and to increase the MCA effectiveness achieving optimum performance, in conjunction to the proposed data channel collisions avoidance algorithm. Also, the Markovian performance measures analysis without simplifications taking into account the receiver collisions phenomenon is a rigorous way in which we face the evaluations giving advantages in our attempt. Our investigation is carried out as follows: The network model and the assumptions are described in Section 2. In Section 3, the protocol case without receiver collisions is extensively described and the performance measures are derived. In Section 4, the protocol case with receiver collisions is examined and the performance measures are given. In Section 5, the performance optimization is presented in both protocol cases. Comparative numerical results and comments are discussed in Section 6. Finally, the concluding remarks are outlined in Section 7. 2. Network model The considered system is a passive star network, as Fig. 1 shows. The system uses v + N wavelengths kc1, . . ., kcv, kd1, . . ., kdN to serve a finite number M of stations. The multi-channel system at wavelengths kc1, . . ., kcv forms the MCA and operates as the control multi-channel

Fig. 1. Passive star multi-wavelength architecture.

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system, while the remaining N channels at wavelengths kd1, . . ., kdN constitute the data multi-channel system. The MCA is divided into two groups of channels, v = vf + vb. It means that vf control channels are used by free stations, while vb control channels are used by backlogged stations. The proposed MCA network model is described as [CC]vTT-[FR]v-[TR]. Thus, there are v control channels and each station has a tunable transmitter that can be tuned to a wavelength in the set kc1, . . ., kcv, kd1, . . ., kdN. The outcoming traffic of a station is connected to an input of the passive star coupler. Each station uses v fixed tuned receivers one for each control channel and one tunable receiver to any of data channels kd1, . . ., kdN. The incoming traffic to a station is splitted into v + 1 portions by an 1 · (v + 1) WDMA splitter, as Fig. 1 presents. The fixed size control packet transmission time is used as time unit (control slot) and the data packet transmission time normalized in time units is L (data slot). The control packet consists of the transmitter address, the receiver address and the data wavelength kk that belongs to the set of kd1, . . ., kdN and has been chosen for the data packet transmission. The normalized round trip propagation time between any station to the star coupler hub and to any other station is equal to R data slots (R · L time units) for all stations. Both control and data channels use the same time reference which we call cycle. We define as cycle the time interval that includes one time unit for control packets transmissions plus the normalized round trip propagation time R and the data packet transmission time L. Thus, the cycle time duration is C = 1 + (R + 1)L time units. We assume a common clock to all stations. Time axis is divided into contiguous cycles of equal length and stations are synchronized for transmission on the control and data channels during a cycle. At any point in time each station is able to transmit at a given wavelength kT and simultaneously receive at a wavelength kR. Finally, we assume negligible tuning times and very large tunable bandwidths. At the beginning of each cycle, all stations know the number of backlogged stations in the system. This knowledge defines the optimum division of the MCA into the groups of vf and vb control channels to obtain the optimum control channels utilization.

2.1. Access mode At the beginning of each cycle if a station has to send a data packet to another, it first chooses randomly a data wavelength on which the packet will be transmitted. Then, it informs the other stations by sending a control packet choosing randomly one of the vf or the vb control channels depending on its state (free or backlogged). The control packets compete according to the Slotted Aloha scheme to gain access. The station continuously monitors the MCA with its fixed tuned receivers. The outcome of its control packet will be known R · L time units later

(acknowledgment period of time) because of the broadcast nature of the control channels. After the end of this period, the station is aware about the data channel claims for transmission of all stations. Especially, if its control packet has been successfully transmitted over the MCA and the same data channel has been selected from some other stations for data transmission, a data channel collisions avoidance algorithm is applied (we can imagine several arbitration rules, as the age of the packet, priority etc.). In this case, only one among the competed the same data channel stations gains access and starts transmission immediately while the others are getting backlogged. Also, after the end of the same period the station knows the number of backlogged stations for the next cycle period. In this way, it knows the number of vf and vb control channels for the next cycle that provides the optimum throughput. 2.2. Reception mode After the data packet transmission, the destination waits R · L time units and then it adjusts its tunable receiver to the channel specified in the control packet for data packet reception. In the protocol case of receiver collisions consideration, if two or more packets from different data channels are addressed to the same destination station only one of them is correctly received according to specified arbitration rules for this case and the others are aborted. This phenomenon is called receiver collisions [12,13]. We assume that every station is equipped with a transmitter buffer with capacity of one data packet. If the buffer is empty the station is said to be free, otherwise it is backlogged. Packets are generated independently at each station following a geometric distribution, i.e. a packet is generated at each cycle with probability p. A backlogged station retransmits the unsuccessfully transmitted packet following a geometric distribution with probability r and defers the transmission by one cycle with probability (1  r). If a station is backlogged and generates a new packet, the packet is lost. Free stations that unsuccessfully transmit either on control or data channels during a cycle are getting backlogged on the next cycle. A backlogged station is getting free at the next cycle if it manages to retransmit without collision over a control channel and to gain access over a data channel. Additionally to the above requirements in the protocol case of receiver collisions consideration, a backlogged station becomes free if its data packet retransmission is not aborted due to receiver collisions. Also in this case, a free station becomes backlogged in case of rejection at destination due to receiver collisions. 3. Protocol case without receiver collisions In this section we develop the Markovian model of a system with finite population ignoring the impact of the receiver collisions phenomenon.

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

3.1. Model analysis The examined system performance can be described by a discrete time Markov chain. We denote the state of the system by Xt, t = 1, 2, . . . where Xt = 0, 1, . . ., M is the number of backlogged stations at the beginning of a cycle. Let: Ht = The number of new control packets arrivals at the first control slot of a cycle, t = 0, 1, 2, . . . At = The number of successfully (re)transmitted data packets over the N data channels during a cycle, t = 0, 1, 2, . . . SBk = The number of successfully retransmitted control packets over the vb control channels, given that k backlogged stations retransmit during a cycle, 0 6 SBk 6 min(vb, k). SFm = The number of successfully transmitted control packets over the vf control channels, given that m free stations attempt transmission during a cycle, 0 6 SFm 6 min(vf, m). An = The number of successful data packet (re)transmissions over the N data channels, conditional that n successful (re)transmissions occurred over the v control channels during a cycle, SBk + SFm = n for every SBk + SFm > 0. The probability Pr[SBk = n] of n successes from k retransmissions over the vb control channels during a cycle is given by [15,16]:

¼

qin ¼ binðn; i; rÞ

ð5Þ

Similar, the conditional probability Qin that i out of (M  n) free stations attempt to transmit with probability p during the cycle, is defined as: Qin ¼ binðM  n; i; pÞ   i j ij where binði; j; pÞ ¼ p ð1  pÞ ; j

ð6Þ iPj

minðv ;kÞ ð1Þn vb !k! Xb k

ðvb Þ n!

j¼n

ð1Þj ðvb  jÞðkjÞ ðj  nÞ!ðvb  jÞ!ðk  jÞ!

ð1Þ

P ij ¼ 0

ð8Þ

Case B: j = i  N (decrease of the number of backlogged stations equal to N), then: minði;v Xb Þ

minðv ;mÞ s j ðmjÞ ð1Þ vf !m! Xf ð1Þ ðvf  jÞ m ðvf Þ s! ðj  sÞ!ðvf  jÞ!ðm  jÞ! j¼s

ð2Þ

and 0 6 s 6 min(vf, m). The probability Pr[An = r] of r successful transmissions over the N data channels given that n successful (re)transmissions occurred over the v control channels during a cycle is given by [11]:  X n   r N ri i r Pr½An ¼ r ¼ ð1Þ ð3Þ N r i i¼0 and 1 6 r 6 min(N, n) for every n P 1. We define the function U(x, y, z, w, s) as the product of the probability of y successes from x retransmissions over the vb control channels, times the probability of w successes from z transmissions over the vf control channels, times the probability of s successfully transmitted data packets over the N data channels, during a cycle. It is: Uðx; y; z; w; sÞ ¼ Pr½SBx ¼ yPr½SFz ¼ wPr½Ayþw ¼ s

þ

i X n¼N þ2

Pr½SFm ¼ s

ð4Þ

ð7Þ

The Markov chain Xt, t = 1, 2, . . . is homogeneous, aperiodic and irreducible. The one step transition probabilities are given by: Pij = (Xt+1 = jjXt = i) where: Case A: j < i  N (decrease of the number of backlogged stations more than N), then:

qni Uðn; n; 0; 0; N Þ

n¼N

and 0 6 n 6 min(vb, k). The probability Pr[SFm = s] of s successes from m transmissions over the vf control channels during a cycle is given by [15,16]:

¼

Also, we define the conditional probability qin that i out of n backlogged stations attempt to retransmit with probability r during the cycle. qin is given by:

P ij ¼ Q0i

Pr½SBk ¼ n

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qni

minðn2;v X b 1Þ

! Uðn; y; 0; 0; N Þ

ð9Þ

y¼N

Case C: i  N < j < i (decrease of the number of backlogged stations less than N), then: 8 minði;v Þ minðMi;N iþj;v Þ f P Pb > > qni Qmi > > > n¼ij m¼0 > > > > > Uðn;n; m;m;m þ i  jÞ > > > > minði;v > P b Þ minðMi;N P iþjÞ > > > þ q Qmi > ni > > n¼ijþ2 m¼2 > > > > minðm2;v > P f 1Þ > > >  Uðn;n;m;x;m þ i  jÞ > > > x¼maxð0;mþijnÞ > > > < minðMi;N i P Piþj;vf Þ ð10Þ P ij ¼ þ qni Qmi > m¼0 n¼ijþ2 > > > > minðn2;v > P b 1Þ > > >  Uðn; y;m;m; m þ i  jÞ > > > y¼ij > > > > minðMi;N i > P P iþjÞ > > > qni Qmi þ > > > m¼2 n¼ijþ4 > > > > minðn2;v > P b 1Þ minðm2;v P f 1Þ > >  > > > y¼ij x¼maxð0;mþijyÞ > > : Uðn;y;m; x;m þ i  jÞ Case D: j = i (stable number of backlogged stations), then:

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P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

8 minði;v P Þ P b Þ minðMi;N > > > Q Uð0;0;0;0;0Þ þ q Qmi q > 0i 0i ni > > n¼2 m¼2 > > > > minðm2;v > P f 1Þ > >  Uðn;n;m;x;mÞ > > > x¼maxð0;mnÞ > > > > > minðMi;N ;vf Þ i i > P P P > > Qmi þQ0i qni Uðn;0;0;0;0Þ þ qni > > > m¼1 n¼2 n¼2 > > > < minðn2;v P b 1Þ P ij ¼  Uðn;y;m;m;mÞ > y¼0 > > > > > minði;v > P ;vf Þ P b Þ minðMi;N > > qni Qmi Uðn;n;m;m;mÞ > þ > > n¼0 m¼1 > > > > minðMi;N i > P P Þ > > Qmi > þ qni > > m¼2 n¼4 > > > > minðn2;v > P f 1Þ P b 1Þ minðm2;v > > >  Uðn;y;m;x;mÞ : y¼0

ð11Þ

x¼maxð0;myÞ

Case E: j > i (increase of the number of backlogged stations), then:

P ij ¼

8 minði;v i P bÞ P > > > q Uðn; 0; j  i; 0; 0Þ þ qni Q > ji;i ni > > n¼0 n¼0;n6¼1 > > > > minðMi;v > P fÞ > > >  Qmi Uðn; n; m; m; m þ i  jÞ > > > m¼jiþ1 > > > > minði;v > P b Þ minðMi;N P iþjÞ > > þ qni Qmi > > > n¼0 m¼jiþ1 > > > > minðm2;v > P f 1Þ > > > > <  x¼maxð0;mþijnÞ Uðn; n; m; x; m þ i  jÞ > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :

þ

i P

qni

minðMi;v P fÞ



yPkx

The steady state average throughput S is given by: ð12Þ

Qmi

m¼jiþ1

n¼2

minðn2;v P b 1Þ

þ

qni

n¼2



minðMi;N P iþjÞ m¼jiþ1

minðm2;v P f 1Þ

Qmi

minðn2;v P b 1Þ

M L L X E½SðiÞ ¼ SðiÞpi C C i¼0

B ¼ E½i ¼

y¼0

ð16Þ

M X

ð17Þ

ipi

i¼0

The conditional input rate Sin(i) is the expected number of arrivals during a cycle, given that the number of backlogged stations at the beginning of a cycle is i. It is:

Uðn; y; m; x; m þ i  jÞ

x¼maxð0;mþijyÞ

3.2. Performance measures

S in ðiÞ ¼ E½H t jX t ¼ i ¼ ðM  iÞp

Since the Markov chain Xt, t = 1, 2, . . . is ergodic, the steady state probabilities are given by solving the system of the following linear equations: p ¼ pP M X pi ¼ 1 and

S¼

The steady state average number B of backlogged stations is given by:

Uðn; y; m; m; m þ i  jÞ

y¼0 i P

The conditional throughput S(i) is the expected value of the output rate during a cycle conditional that the number of backlogged stations at the beginning of the cycle is i, i.e. S(i)=E[AtjXt = i] and is given by: 8 minði;v minðMi;v N > P P bÞ P fÞ > > > k qni Qmi : > > > m¼0 > k¼1 n¼maxðk;MiÞðMiÞ > mþnPk > > > > > Uðn; n; m; m; kÞ > > > > minði;v Mi > P bÞ P > > þ q Qmi > ni > > m¼0 > n¼maxðk;MiÞðMiÞ > > mþnPk > > > > minðm2;v f 1Þ > P > > > Uðn; n; m; x; kÞ  > > > > x ¼ 0 > > > > > xPkn < minðMi;v ð15Þ SðiÞ ¼ i P P fÞ > > þ qni Qmi > > > m¼0 n¼maxðk;MiÞðMiÞ > > mþnPk > > > > minðn2;v > P b 1Þ > > >  Uðn; y; m; m; kÞ > > > y¼0 > > yPkm > > > > i Mi > P P > > þ qni Qmi > > > m¼0 > n¼maxðk;MiÞðMiÞ > > mþnPk > > ! > > > minðm2;v 1Þ minðn2;v 1Þ f b P P > > >  Uðn; y; m; x; kÞ > > > x¼0 y¼0 :

ð13Þ ð14Þ

i¼0

where P is the transition matrix with elements the probabilities Pij and p is a row vector with elements the steady state probabilities pi.

ð18Þ

The steady state average input rate Sin is given by: S in ¼

M X

pðM  iÞpi

ð19Þ

i¼0

The delay D is defined as the average number of time units that a data packet has to wait until its successful transmission. Delay is calculated by means of the Little’s formula, as follows: D ¼ f1 þ ðR þ 1ÞLg þ f1 þ ðR þ 1ÞLg

B S in

ð20Þ

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

We define the throughput per data channel Sd in steady state as the number of the successfully transmitted data packets per data channel during a cycle. It is given by: Sd ¼

S N

P ij ¼ Q0i 

minðn2;v P b 1Þ

!

qni

n¼N þ2

ð25Þ

Urc ðn; y; 0; 0; N ; N Þ

y¼N

Case C: i  N < j < i (decrease of the number of backlogged stations less than N), then:

In this section we develop the Markovian model of a system with finite population considering the effect of the receiver collisions phenomenon. 4.1. Model analysis As in Section 3, the examined system performance can be described by a discrete time Markov chain. We denote the state of the system by Xt, t = 1, 2, . . . where Xt = 0, 1, . . ., M is the number of backlogged stations at the beginning of a cycle. We consider the random variables: Ht, At, SBk, SFm and An as in Section 3. Also, let: Ct = The number of correctly received packets at destination during a cycle, t = 0,1,2. . . Cr = The number of correctly received packets at destination given that r successful (re)transmissions occurred over the N data channels during a cycle, 16Cr6An for every An > 0. The probability Pr[Cr = u] of u correctly received data packets at destination given that r successful (re)transmissions occurred over the N data channels during a cycle is given by [12]:  X r   u M ui i u Pr½C r ¼ u ¼ ð1Þ ð22Þ M u i i¼0 and 1 6 u 6 min(r, M) for every r P 1. We define the function Urc(x, y, z, w, s, r) as the product of the function U(x, y, z, w, s) times the probability of r correct data packet receptions at destination, during a cycle. Considering (4), it is: ð23Þ

where: x, y, z, w, s are defined in (4) and r is the number of correctly received data packets at destination, during a cycle. The Markov chain Xt, t = 1, 2,. . . is homogeneous, aperiodic and irreducible. The one step transition probabilities are given by: Pij = (Xt+1 = jjXt = i) where: Case A: j < i  N (decrease of the number of backlogged stations more than N), then: P ij ¼ 0

i P

qni Urc ðn; n; 0; 0; N ; N Þ þ

n¼N

ð21Þ

4. Protocol case with receiver collisions

Urc ðx; y; z; w; s; rÞ ¼ Pr½SBx ¼ yPr½SFz ¼ w  Pr½Ayþw ¼ sPr½C s ¼ r

minði;v P bÞ

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P ij ¼

8 minði;v P b Þ minðMi;N Piþj;vf Þ > > > q Qmi > ni > > n¼ij m¼0 > > > > minðmþn;NÞ > P > >  Urc ðn;n;m;m;l;m þ i  jÞ > > > l¼mþij > > > > > minði;v > P iþjÞ P b Þ minðMi;N > > þ qni Qmi > > > n¼ijþ2 m¼2 > > > > minðm2;v > P f 1Þ minðxþn;N P Þ > > >  > < x¼maxð0;mþijnÞ l¼mþij Urc ðn;n;m;x;l;m þ i  jÞ > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :

þ

i P

qni

minðMi;N Piþj;vf Þ

n¼ijþ2



Qmi

ð26Þ

y¼ij

m¼0

minðyþm;N P Þ

minðn2;v P b 1Þ

Urc ðn;y;m;m;l;m þ i  jÞ

l¼mþij

þ

i P

qni

minðMi;N P iþjÞ

n¼ijþ4



minðm2;v P f 1Þ

Qmi

m¼2 minðxþy;N P Þ

minðn2;v P b 1Þ y¼ij

Urc ðn;y;m;x;l;m þ i  jÞ

x¼maxð0;mþijyÞ l¼mþij

Case D: j = i (stable number of backlogged stations), then: 8 q0i Q0i Urc ð0;0;0; 0;0;0Þ > > > > minði;v > P b Þ minðMi;N P ;vf Þ > > qni Qmi > þ > > n¼0 m¼1 > > > > minðnþm;N > P Þ > >  Urc ðn; n;m; m; l;mÞ > > > l¼m > > > > minði;v i > P P bÞ > > > þQ0i qni Urc ðn;0;0; 0;0;0Þ þ qni > > > n¼2 n¼2 > < minðMi;N minðm2;v P Þ P f 1Þ minðxþn;N P Þ P ij ¼  Q Urc ðn; n;m;x;l;mÞ > mi > > l¼m m¼2 x¼maxð0;mnÞ > > > > minðMi;N minðn2;v i > P P ;vf Þ P b 1Þ minðmþy;N P Þ > > > þ qni Qmi > > > l¼m m¼1 y¼0 n¼2 > > > > minðMi;N minðn2;v i > P Þ P b 1Þ > > Urc ðn;y;m; m;l; mÞ þ P q Qmi > ni > > m¼2 y¼0 n¼4 > > > > > minðm2;v Þ f 1Þ minðxþy;N > P P > >  Urc ðn;y;m; x;l;mÞ : x¼maxð0;myÞ

l¼m

ð27Þ

ð24Þ

Case B: j = i  N (decrease of the number of backlogged stations equal to N), then:

Case E: j > i (increase of the number of backlogged stations), then:

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P ij ¼

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

8 i P > > > qni Urc ðn; 0;j  i; 0;0; 0Þ > Qji;i > > n¼0;n6¼1 > > > > > minði;v > P b Þ minðMi;v P fÞ > > > þ qni Qmi > > > n¼0 m¼jiþ1 > > > > > minðnþm;NÞ > P > > >  Urc ðn; n; m; m;l; m þ i  jÞ > > > l¼mþij > > > > > minði;v > P b Þ minðMi;N P iþjÞ > > > þ q Qmi > ni > > n¼0 m¼jiþ1 > > > > > minðm2;v > P Þ P f 1Þ minðnþx;N > > <  > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :

The number Brc of backlogged stations, the conditional input rate Sin,rc(i) and the steady state input rate Sin,rc of the protocol case with receiver collisions are calculated by (17)–(19), considering the steady state probabilities pi of the receiver collisions Markovian model. The delay Drc is defined as the average number of time units that a packet has to wait until its successful transmission and is calculated through Little’s formula: Drc ¼ f1 þ ðR þ 1ÞLg þ f1 þ ðR þ 1ÞLg ð28Þ

x¼maxð0;mþijnÞ l¼mþij

Urc ðn;n; m; x; l;m þ i  jÞ þ

i P

qni

n¼2



minðMi;v P fÞ

Qmi

minðn2;v P b 1Þ

m¼jiþ1

minðmþy;N P Þ

Urc ðn;y;m;m; l;m þ i  jÞ

i P n¼2



qni

minðMi;NiþjÞ P

Qmi

minðn2;v P b 1Þ

m¼jiþ1

minðm2;v P f 1Þ

ð32Þ

We define the rejection probability Prej of a data packet at destination as the ratio of the expected number of packet rejections due to receiver collisions, to the expected number of successfully transmitted packets over the N data channels per cycle, in steady state. It is:

y¼0

minðxþy;NÞ P

S rc N

4.3. Rejection probability Prej

l¼mþij

þ

ð31Þ

Finally, we define the throughput per data channel Sd,rc in steady state as the number of the correctly received data packets at destination per data channel during a cycle. It is: S d;rc ¼

y¼0

Brc S in;rc

Urc ðn; y; m; x; l; m þ i  jÞ

x¼maxð0;mþijyÞ l¼mþij

P rej ¼ 4.2. Performance measures

S  S rc S

ð33Þ

where: S and Src are given by (16), (30), respectively. The steady state probabilities can be estimated by solving the system of the linear equations (13), (14) of the Markovian model with the receiver collisions consideration. The conditional throughput Src(i) is the expected value of the output rate during a cycle given that the number of backlogged stations at the beginning of the cycle is i, i.e. Src(i) = E[CtjXt = i] and is given by:

S rc ðiÞ ¼

5. Performance optimization In order to achieve the optimum system performance, the following considerations are made: The conditional throughput S vb ðiÞ from the vb control channels is given by [17]:

0 8 > minði;v minðMi;v minðnþm;N N > P bÞ P fÞ P Þ P > > k@ qni Qmi Urc ðn; n; m; m; l; kÞ þ > > > l¼k m¼0 n¼maxðk;MiÞðMiÞ > k¼1 > mþnPk > > > > minði;v minðm2;v Mi bÞ > P P P f 1Þ minðnþx;N P Þ > > qni Qmi Urc ðn; n; m; x; l; kÞ þ > > > l¼k x¼0 m¼0 > < n¼maxðk;MiÞðMiÞ mþnPk xPkn minðMi;v minðn2;v i P P fÞ P b lÞ minðmþy;N P Þ > > q Q Urc ðn; y; m; m; l; kÞ þ > ni mi > > l¼k m¼0 y¼0 n¼maxðk;MiÞðMiÞ > > mþnPk yPkm > > > 1 > > > > minðm2;v i Mi > P P f 1Þ minðn2;v P b 1Þ minðxþy;N P Þ P > C > > qni Qmi Urc ðn; y; m; x; l; kÞA > > : n¼maxðk;MiÞðMiÞ mþnPk l¼k m¼0 x¼0 y¼0

ð29Þ

yPkx



The steady state average throughput Src is given by: S rc ¼

L L E½S rc ðiÞ ¼ C C

M X i¼0

S rc ðiÞpi

r S vb ðiÞ ¼ ir 1  vb ð30Þ

i1 ð34Þ

Similar, we define the conditional throughput S vf ðiÞ from the vf control channels as:

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

 Mi1 p S vb ðiÞ ¼ ðM  iÞp 1  vf

ð35Þ

Finally, we define the conditional throughput Sv(i) from the v control packets as the sum of S vb ðiÞ and S vf ðiÞ, that is:  i1  Mi1 r p þ ðM  iÞp 1  ð36Þ S v ðiÞ ¼ ir 1  vb vf

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Finally, we define the conditional throughput SN(i) from the N data channels as: S N ðiÞ ¼ E½Pr½H N fS v ðiÞg ¼ x ¼

N X

xPr½H N fS v ðiÞg ¼ x

x¼1

¼ NP N ðiÞ Substituting (40)–(42), we get:  S ðiÞ ! 1 v S N ðiÞ ¼ N 1  1  N

ð42Þ

ð43Þ

5.1. An approximate analysis 5.1.1. Protocol case without receiver collisions We consider that SBk(i) + SFm(i) = n control packets are successfully transmitted over the v control channels during a cycle, given that the state of the system is i. We assume that the transmitted data packets are uniformly distributed among the N data channels. Thus, the random distribution in N data channels gives Nn arrangements, each with probability Nn. Let PN0(n, i) be the conditional probability that no one from the n data packets has selected the data channel Z for the transmission. Thus, the n data packets are transmitted over the remaining (N  1) data channels in (N  1)n different ways. Then, PN0(n, i) can be written as:  n 1 1 n P N 0 ðn; iÞ ¼ n ðN  1Þ ¼ 1  ð37Þ N N In steady state it is: EbSBk ðiÞ þ SFm ðiÞ ¼ nc ¼ S v ðiÞ Consequently, in steady state (37) is written as:  S ðiÞ 1 v P N 0 ðiÞ ¼ 1  N

ð38Þ

5.1.2. Protocol case with receiver collisions Additionally to the considerations of the previous Section 5.1.1, we assume that An = s data packets are successfully transmitted over the N data channels during a cycle, given that the state of the system is i. We assume that the data packets are uniformly distributed among the M stations (for sake of simplicity of the analysis, we consider that a station may send packets to itself). Thus, the random distribution in M stations gives Ms arrangements each with probability Ms. Let PM0(s, i) be the conditional probability that no one from the s successfully transmitted data packets has as destination the station X. Thus, the s data packets should be destined to the remaining (M  1) stations in (M  1)s different ways. Then, PM0(s, i) can be written as [12]:  s 1 1 s P M0 ðs; iÞ ¼ n ðM  1Þ ¼ 1  ð44Þ M M In steady state it is: EbAn ¼ sc ¼ S N ðiÞ

ð39Þ

We define the conditional probability PN(i) that one data packet is transmitted over the data channel Z during a cycle in steady state. In other words, PN(i) implies that at least one data packet has selected data channel Z and has won the data channel collisions avoidance competition. So, we get:  S ðiÞ 1 v P N ðiÞ ¼ 1  P N 0 ðiÞ ¼ 1  1  ð40Þ N Let HN{Sv(i)} be the random variable representing the number of different data channels selected for transmissions, given that Sv(i) is the conditional output rate of successful (re)transmitted control packets over the v control channels, during a cycle in steady state. We define the conditional probability Pr[HN{Sv(i)} = x] that x different data channels have been selected for transmissions during a cycle in steady state. It is:   N x N x Pr½H N fS v ðiÞg ¼ x ¼ ðP N ðiÞÞ ð1  P N ðiÞÞ ð41Þ x

Consequently, in steady state (45) is written as:  S ðiÞ 1 N P M0 ðiÞ ¼ 1  M

ð45Þ

ð46Þ

We define the conditional probability PM(i) that one data packet with destination X is received correctly without collisions during a cycle in steady state. It is:  S ðiÞ 1 N P M ðiÞ ¼ 1  P M0 ðiÞ ¼ 1  1  ð47Þ M Let HM{SN(i)} be the random variable representing the number of different stations selected as destination, given that SN(i) is the conditional output rate of successful (re)transmitted data packets over the N data channels, during a cycle in steady state. Also, we define the PrºHM{SN(i)} = yß as the conditional probability that y different stations have been selected as destination during a cycle in steady state. It is:   M y My Pr½H M fS N ðiÞg ¼ y ¼ ðP M ðiÞÞ ð1  P M ðiÞÞ ð48Þ y Finally, we define the conditional throughput SRC(i) at the destinations as:

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S RC ðiÞ ¼ EbPrbH M fS N ðiÞg ¼ ycc ¼ MP M ðiÞ Substituting (47)–(49), we get:  S ðiÞ ! 1 N S RC ðiÞ ¼ M 1  1  M

ð49Þ vf

Also, for the excluded values we define that the number vf_opt of the control channels is given by:  v; i¼0 vf opt ¼ ð58Þ v  1; i¼1

In this part we explore the best capabilities of the proposed protocol cases. For each cycle, the system state is denoted by i. 5.2.1. Optimum retransmission probability ropt The optimum retransmission probability ropt is obtained by setting (a) for the protocol case without receiver collisions: the first derivative of (43) with respect to r equal to zero and (b) for the protocol case with receiver collisions: the first derivative of (50) with respect to r equal to zero. It is: ð51Þ

Since 0 6 ropt 6 1, for both protocol cases we get: 8v b > i > vb > < i ; ð52Þ ropt ¼ 1; 0 < i 6 vb > > : 0; i¼0 5.2.2. Optimum rate vf/vb The optimum rate vvbf is obtained by setting (a) for the protocol case without receiver collisions: the first derivative of (43) with respect to vb equal to zero, and (b) for the protocol case with receiver collisions: the first derivative of (50) with respect to vb equal to zero. It is: oS RC ðiÞ oS N ðiÞ oS v ðiÞ ¼0) ¼0) ¼0 ovb ovb ovb Finally, for both protocol cases we get: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vf p ðM  iÞðM  i  1Þ ; i 6¼ 0; 1 ¼ iði  1Þ vb r

i 6¼ 0; 1 ð57Þ

ð50Þ

5.2. Optimum performance parameters

oS RC ðiÞ oS N ðiÞ oS v ðiÞ ¼0) ¼0) ¼0 or or or

opt

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ðM  iÞðM  i  1Þ ¼ IntegerPart p ; iði  1Þ

ð53Þ

In all cases, it is obvious that: vb

opt

¼ v  vf

opt

ð59Þ

6. Numerical results The numerical solution of the proposed Markovian queueing model is presented in this section. In order to verify the accuracy of the theoretical analysis, we developed a specific network simulator based on the C programming to simulate the proposed system performance. The developed simulator implements an extensive discrete-event simulation model and uses as confidence level (1  a) the value 99% that gives almost one to one correspondence to the actual system, as Figs. 4 and 8 illustrate. The numerical results have been evaluated for optimum protocol parameters (ropt, vf_opt, vb_opt). Especially for each cycle calculations, the optimum value for vf_opt, vb_opt has been determined by dynamically splitting the total number v of control channels into two groups according (57)–(59). In the presented numerical evaluations we assume that the value of the data packet length is: L = 10 time units. In the following figures we study the performance optimization achieved by the MCA division into the groups of vf_opt, vb_opt control channels. Thus, for the protocol case with receiver collisions we compare the performance of the proposed protocol with this of the system of [11] on

ð54Þ

For the excluded values, we define that the number vf and vb of the control channels are: vb ¼ 1 vb ¼ 0

and and

vf ¼ v  1; if i ¼ 1 vf ¼ v; if i ¼ 0

ð55Þ ð56Þ

5.2.3. Optimum MCA division into vf_opt, vb_opt control channels In order to obtain the optimum division of the MCA into the groups of vf_opt, vb_opt(vf_opt 6 v, vb_opt 6 v) control channels, we assume that the backlogged stations retransmit with the optimum retransmission probability ropt given by (52). In this case, the vf_opt is given by substituting (52) to (54)–(56), that is:

Fig. 2. Throughput per data channel Sd,rc versus birth probability p, for M = 50, N = 10, R = 5, ropt and v = 20, 25, 30.

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which all stations have symmetric access rights to the MCA. In this way, we representatively illustrate the significant performance improvement achieved by the asymmetric access rights to the MCA provided by the proposed protocol. Also, we study the effect of the receiver collisions phenomenon on the proposed architecture by comparing the performance of the protocol cases presented in Sections 3 and 4. Fig. 2 shows the throughput per data channel Sd,rc curves versus the birth probability p for M = 50 stations, R = 5 data slots, N = 10 data channels and ropt, for v = 20, 25, 30 control channels. It is shown that the proposed architecture with asymmetric access rights on the MCA essentially increases the Sd,rc as compared to the symmetric access rights protocol of [11] for all values of v. This is due to the fact that considering optimum protocol parameters ropt, vf_opt, vb_opt, the probability of a control channel collision decreases which causes significant increase of Sd,rc. Also, it is illustrated that the Sd,rc improvement due to the proposed asymmetric access rights protocol is an increasing function of v. The explanation comes from the fact that as v increases the probability of a control channel collision decreases which reduces the number of backlogged stations in the system. In this case, the value of retransmission probability ropt is higher according to (52), and the consequent value assignment of vf_opt, vb_opt for the control channels according to (57)–(59) provides higher values of Sd,rc improvement. This behavior is observed for example, for p = 0.18 where Sd,rc improves for v = 30 at 17%, for v = 25 at 13% and for v = 20 at 8%. Also, the significant performance improvement achieved by the proposed asymmetric access rights protocol is depicted in Fig. 3. This figure presents the delay Drc curves versus the throughput per data channel Sd,rc, for M = 50, R = 5, N = 10 and ropt, for v = 20, 25, 30 control channels. In fact, it is observed that the proposed protocol provides

much higher values of throughput per data channel Sd,rc while it keeps almost equal values of system delay Drc, for all number of control channels v. This is due to the fact that the optimum division of the MCA into the groups of the vf_opt, vb_opt channels aims at the throughput optimization, as (51), (53) show. Following the previous remarks, as the number of v increases the control channel collisions decreases, which consequently increases the number of free stations. In this case, the performance optimization is achieved by determining the value of the vf_opt that allows the successful control packets transmission by free stations, which significantly improves the Sd,rc values and does not affect to the values of Drc. The effect of the receiver collisions phenomenon on the system performance is illustrated in Fig. 4. This Figure presents the delay Dd, Dd,rc curves versus the throughput per data channel Sd, Sd,rc, for M = 50, N = 10, R = 5, ropt and v = 20, 25, 30 control channels for the proposed asymmetric access rights protocol for both the protocol cases with and without receiver collisions consideration, according to the analysis of Sections 3 and 4. It can be observed that for a given value of R and for fixed values of M, N and Delay, the difference between the Sd, Sd,rc is getting wider as v increases. The explanation comes from the fact that as v increases the probability of a successful data packet transmission increases too, giving rise to the receiver collisions phenomenon. The numerical results provided by the simulation are also given having almost one to one correspondence with the analysis results. The same behavior is noticed in Fig. 5 which presents the average rejection probability Prej curves versus the birth probability p for M = 50, N = 10, R = 5, ropt and v = 20, 25, 30 control channels. In fact, as v increases the throughput per data channel rises causing simultaneous increase of Prej. Similar, the variation of the number of data channel N causes the same behavior. In other words, the proposed

Fig. 3. Delay Drc versus throughput per data channel Sd,rc, for M = 50, N = 10, R = 5, ropt and v = 20, 25, 30.

Fig. 4. Asymmetric Access Rights Protocol: Delay D, Drc versus throughput per data channel Sd, Sd,rc, for M = 50, N = 10, R = 5, ropt and v = 20, 25, 30.

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Fig. 5. Asymmetric Access Rights Protocol: Average rejection probability Prej versus birth probability p, for M = 50, N = 10, R = 5, ropt and v = 20, 25, 30.

protocol with asymmetric access rights on the MCA essentially increases the Sd,rc as compared to the symmetric access rights protocol of [11] for all values of N. This is shown in Fig. 6 that presents the throughput per data channel Sd,rc curves versus the birth probability p for M = 50, v = 30, R = 5, ropt and N = 10, 15, 20 data channels. The explanation comes from the fact that the optimum parameters ropt, vf_opt, vb_ opt consideration decreases the probability of a control channel collision that consequently increases the probability of a data packet successful transmission over the data multi-channel system. This fact essentially increases the Sd,rc as compared to the symmetric access rights protocol of [11]. This behavior is observed for example, for p = 0.21 where Sd improves for N = 10 at 17%, for N = 15 at 18% and for N = 20 at 18%. Also, the performance optimization achieved by the proposed asymmetric access rights protocol is depicted in

Fig. 6. Throughput per data channel Sd,rc versus birth probability p, for M = 50, v = 30, R = 5, ropt and N = 10, 15, 20.

Fig. 7 which illustrates the delay Drc curves versus the throughput per data channel Sd,rc, for M = 50, R = 5, v = 30 and ropt, for N = 10, 15, 20 data channels. In fact, it is observed that the proposed protocol provides lower values of delay Drc while it reaches much higher values of throughput per data channel Sd,rc, for all number of data channels N. The explanation comes from the optimum division of the number v of control channels into the groups of the vf_opt, vb_opt channels. In this case, the probability of a control channel collision decreases which simultaneously causes essential decrease of the number of backlogged stations at this stage of transmission. This fact provides lower values of delay Drc while it guarantees the Sd,rc increase, as it is previously remarked. It is obvious that as N increases the probability of a data packet successful transmission over the data multi-channel system increases too. For this reason, the probability of a data packet rejection at destination due to the receiver collisions increases. Also, as N increases the number of backlogged stations Brc increases too, due to the rise of receiver collisions. This fact causes the significant increase of the delay Drc as N increases. This behavior can be noticed in Fig. 8 that depicts the delay D, Drc curves versus the throughput per data channel Sd, Sd,rc, for M = 50, v = 30, R = 5 and ropt, for N = 10, 15, 20 data channels for the proposed protocol for both the cases with and without receiver collisions consideration. Also, the numerical results are verified by those of the simulation. The above mentioned remarks are verified by the Fig. 9 that depicts the average rejection probability Prej curves versus the birth probability p for M = 50, v = 30, R = 5 and ropt, for N = 10, 15, 20 data channels. As it is presented, as N increases the throughput per data channel reduces causing simultaneous increase of Prej. On the other hand, the throughput optimization due to the proposed asymmetric access rights protocol is less noticeable for large population systems. This fact can be

Fig. 7. Delay Drc versus Throughput per data channel Sd,rc, for M = 50, v = 30, R = 5, ropt and N = 10, 15, 20.

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585

Fig. 8. Asymmetric Access Rights Protocol: Delay D, Drc versus throughput per data channel Sd, Sd,rc, for M = 50, v = 30, R = 5, ropt and N = 10, 15, 20.

Fig. 9. Asymmetric Access Rights Protocol: Average rejection probability Prej versus birth probability p, for M = 50, v = 30, R = 5, ropt and N = 10, 15, 20.

noticed in Fig. 10 that presents the throughput per data channel Sd,rc curves versus the birth probability p, for N = 10, v = 30, R = 5, ropt and M = 50, 100, 150 stations. Indeed, as M grows for a given value of R and for fixed v, N and p, the load increases and consequently the number of backlogged stations increases too. In this case, the protocol optimization reaches an upper limit while the optimum parameters ropt, vf_opt, vb_ opt influence less the system efficiency. This behavior is observed for example, for p = 0.1 where Sd,rc improves for M = 50 at 8%, for M = 100 at 7% and for M = 150 at 2%. Additionally, large population systems suffer less from receiver collisions. This fact is depicted in Fig. 11 that presents the average rejection probability Prej curves versus the

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Fig. 10. Throughput per data channel Sd,rc versus birth probability p, for N = 10, v = 30, R = 5, ropt and M = 50, 100, 150.

Fig. 11. Asymmetric Access Rights Protocol: Average rejection probability Prej versus birth probability p, for N = 10, v = 30, R = 5, ropt and M = 50, 100, 150.

birth probability p, for N = 10, v = 30, R = 5, ropt and M = 50, 100, 150 stations. Thus, as M grows the load rises which causes increase of the probability of control channel collisions. This fact results to the reduction of the throughput per data channel and consequent reduction of the rate of the receiver collisions. It is evident that the system performance optimization is essentially determined by the relation among v, N and M, for fixed values of R. The impact of the value variation of the propagation delay R on the performance optimization is representatively depicted in Fig. 12. This Figure presents the throughput per data channel Sd,rc curves versus the birth probability p, for M = 50, v = 30, ropt, N = 10 for R = 0, 5, 10 data slots. As it can be observed, as R increases the difference between the two curves is getting

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Fig. 12. Throughput per data channel Sd,rc versus birth probability p, for M = 50, N = 10, v = 30, ropt and R = 0, 5, 10.

wider. The reason is that the cycle duration C is an increasing function of R and Sd,rc is an inverse proportional function of C as (30), (32) denote. In other words, the time percent of the cycle duration that each data channel is occupied with a successful data packet transmission decreases. This fact causes that Sd,rc is getting lower and the system efficiency depends less on the optimum parameters ropt, vf_opt, vb_opt. 7. Conclusions The objective of this paper is to present a decentralized synchronous transmission WDMA protocol for passive star network architecture that exploits the MCA and introduces the MCA optimum division into two groups of channels providing significant throughput improvement. The proposed access scheme takes advantage of the round trip propagation delay latency as acknowledgment time to adopt a data channel collision avoidance algorithm and to increase the network efficiency, as in [11]. The innovation of the presented study is the provided asymmetric access rights to the MCA among the stations, as a result of the optimum MCA division, that ameliorate even more the system performance. In our study, we develop a Markovian model for finite population taking into account the receiver collisions. Our rigorous analysis provides the performance measures evaluation counting in the propagation delay latency effect. Moreover, we investigate through exhaustive statistical analysis the mathematical relations among the optimum performance parameters (optimum MCA division, optimum retransmission probability) in order to achieve optimum performance measures efficiency. This fact constitutes the main advantage of our work. The numerical results evaluation proves that the system performance depends on the following parameters: the

total number of control channels in the MCA and the optimum MCA division, the number of data channels, the system population, the average rejection probability Prej and the propagation delay latency value. Thus, is it shown that the effect of receiver collisions is more noticeable in large population systems. Also, it is illustrated that the behavior of the system deteriorates dramatically as the ratio of the round trip propagation delay to data packet transmission time (normalized propagation delay) increases. Finally, as the figures show and the numerical results denote the new asymmetric access rights algorithm achieves significant throughput improvement as compared with the protocol of case [11], which in many cases reaches values approaching to 18%. Our rigorous and real analysis gives a frame for the future researchers of the performance limitations for this protocol category. This paper provides a strong motive to exploit the powerful influence of both the propagation delay latency and the receiver collision phenomenon for the performance evaluation in conjunction with the optimization of the performance parameters values. The results of this paper could be a helpful tool for the analytical performance optimization study for many classes of multi-channel networks, like the wireless networks. References [1] E. Modiano, R. Barry, A medium access control protocol for WDMbased LAN’s and access networks using a master/slave scheduler, IEEE J. Lightwave Technol. 18 (2000) 461–468. [2] A. Ganz, Z. Koren, WDM passive star – protocols and performance analysis, in: Proceedings of IEEE INFOCOM, 1991, pp. 991–1000. [3] P.G. Sarigiannidis, G.I. Papadimitriou, A.S. Pomportsis, CS-POSA: a high performance scheduling algorithm for WDM star networks, Photonic Netw. Commun. 11 (2006) 211–227. [4] R. Chipalkatti, Z. Zhang, A.S. Acampora, Protocols for optical starcoupler network using WDM: performance and complexity study, IEEE J. Selected Areas Commun. 11 (1993) 579–589. [5] M.I. Habbab, M. Kavehrad, C.E.W. Sundberg, Protocols for very high-speed optical fiber local area networks using a passive star topology, IEEE J. Lightwave Technol. LT-5 (1987) 1782–1794. [6] N. Mehravari, Performance and protocol improvements for very high speed optical fiber local area networks using a passive star topology, IEEE J. Lightwave Technol. 8 (1990) 520–530. [7] 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, IEEE J. Lightwave Technol. 9 (10) (1991) 1411–1422. [8] J. Lu, L. Kleinrock, Wavelength division multiple access protocol for high-speed local area networks with a passive star topology, Perform. Eval. 16 (1992) 223–239. [9] P.A. Humblet, R. Ramaswami, K.N. Sivarajan, An efficient communication protocol for high-speed packet switched multichannel networks, IEEE J. Selected Areas Commun. 11 (1993) 568–578. [10] I.E. Pountourakis, A multiwavelength control architecture for electronic processing bottleneck reduction in WDMA lightwave networks, Comput. Commun. 22 (1999) 1468–1480. [11] I.E. Pountourakis, P.A. Baziana, A collision-free with propagation latency WDMA protocol analysis, Opt. Fiber Technol. 13 (2007) 160–169. [12] I.E. Pountourakis, Performance evaluation with receiver collisions analysis in very high-speed optical fiber local area networks using passive star topology, IEEE J. Lightwave Technol. 16 (1998) 2303– 2310.

P.A. Baziana, I.E. Pountourakis / Computer Communications 30 (2007) 3572–3585 [13] I.E. Pountourakis, A. Stavdas, A WDM control architecture and destination conflicts analysis, Int. J. Commun. Syst. 15 (2002) 161–174. [14] P.A. Baziana, I.E. Pountourakis, Multichannel MAC protocol: performance mathematical analysis and comparison, in: Proceedings of 12th International Conference on Software, Telecommunications & Computer Networks (SoftCom 2004), Dubrovnik-Split-Venice, 2004, pp. 146–150. [15] I.E. Pountourakis, P.A. Baziana, Markovian receiver collision analysis of high-speed multi-channel networks, Math. Methods Appl. Sci. J. 29 (2006) 575–593. [16] I.E. Pountourakis, P.A. Baziana, Multi-channel multi-access protocols with receiver collision Markovian analysis, WSEAS Trans. Commun. 4 (2005) 564–569. [17] I.E. Pountourakis, E.D. Sykas, Analysis, stability and optimization of Aloha-type protocols for multi-channel networks, Comput. Commun. 15 (1992) 619–629.

Peristera A. Baziana received the Diploma degree in Electrical and Computer Engineering from the University of Patras, Patras, Greece in 1998. From 1999 to 2002, she has participated in research programs of the Greek PTT Organization (O.T.E.) as a researcher of University of Patras. Since 2002 she has been working towards the Ph.D. degree at the Electronic and Information Engineering Division, School of Electrical and Computer Engineering of National Technical University of Athens, Athens, Greece at the field of architectures and protocols for optical networks. Her current research interests include optical communications, OBS networks, MAC protocols and queuing analysis. She is a member of the Technical Chamber of Greece.

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Ioannis E. Pountourakis is a Professor at the Communications, Electronic and Information Engineering Division, School of Electrical and Computer Engineering of National Technical University of Athens (N.T.U.A). His research interests include Optical Communication Networks, Network Architecture and Protocols, Performance Evaluation and Stability. He has taught several undergraduate and graduate courses at NTUA, supervised many doctoral students working in the areas of queuing analysis of contention resolution mechanisms in local area networks and optical networks, WDM network design, optical networks architectures, analysis of data link layer protocols, evaluation of performance of computer systems, etc and has been reviewer in International Journals, Conferences, and research project proposals. He has participated and organized many International Conferences. He has also participated in several RACE projects and in several national research programs dealing with communication networks. He is member of IEEE and the Greek society of Computer Science.