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Scheduling algorithms in optical packet switches with input wavelength conversion
Computer Communications 28 (2005) 1456–1467 www.elsevier.com/locate/comcom
Scheduling algorithms in optical packet switches with input wavelength conversion V. Eramo*, M. Listanti, A. Valletta INFOCOM Department, University of Roma ‘La Sapienza’, Via Eudossiana 18, 00184 Roma, Italy Received 12 March 2004; revised 28 January 2005; accepted 11 February 2005 Available online 19 March 2005
Abstract The objective of this study is to propose a new Optical Packet Switching architecture in which the wavelength converters, needed to solve output packet contentions, are shared per input line; according to this sharing strategy the packets arriving at a given input fiber shares a converter pool that can be accessed when wavelength conversions are required. In the paper, we propose analytical and simulation models able to evaluate the performances of the proposed architecture when control algorithms with different complexity, are adopted. Under a unicast traffic scenario, the obtained performances are compared to the ones of the architecture in which the wavelength converters are shared per output line. The carried out comparison shows that, with respect to the architecture with wavelength converters shared per output line, the proposed architecture allows for a 30% saving of wavelength converters when a simple control algorithm is adopted; the saving can reach 50% if an optimized control algorithm is used. In the paper, we also define a heuristic algorithm able to reach in low computation cost, performance near to the one of the optimum algorithm. q 2005 Elsevier B.V. All rights reserved. Keywords: Optical packet switching; Synchronous switching; Wavelength converter; Dimensioning; Share-per-input-link architecture; Share-per-output-link architecture
1. Introduction The transmission capacity of optical fibers has been increasing at a tremendous rate as a result of DWDM technology. Although terabit capacity IP routers are now starting to appear, the mismatch between the transmission capacity of DWDM and the switching capacity of electronic routers remains remarkable. Emerging all-optical switching technologies will enable packets to traverse nodes transparently without conversion and processing at each node. The limited optical processing power and the lack of optical buffering techniques are the main obstacles in the realization of all-optical networks in which both the payload and the header of the packets remain in the optical domain from the source node to the destination node. Although some primitive header processing forms begin to appear [1–4], the current approach to optical packet * Corresponding author. Tel.: C39 06 47852311; fax: C39 06 4744481. E-mail address: [email protected] (V. Eramo).
0140-3664/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2005.02.005
switching is to keep the payload in the optical domain and convert the header to the electronic domain for processing [5–8]. A critical issue in photonic packet switching is the contention resolution. Contention occurs when two or more packets contend for the same output port at the same time. In traditional electronic switches, packet contentions are handled through buffering; unfortunately, this is not feasible in the optical domain, since there is no optical equivalent of electronic random-access memories. Today, optical buffering can be only achieved through the use of fiber delay lines [9–15], however, the buffer size is severely limited, not only by signal quality concerns, but also by physical space limitations; as a matter of example, to delay a single packet for 5 ms requires over 1 km of fiber. In order to avoid optical buffering, Wavelength Conversion approach has been widely investigated to handle packet contentions in the optical domain [16–18]. The basic principle is very simple: packets contending for the same output are converted to different wavelengths by means of tunable optical wavelength converters (TOWC). So, the reference packet switch architecture foresees a TOWC for each output wavelength channel. As the number of TOWCs
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467
represents a critical cost element for a photonic packet switch, various Optical Packet Switching architectures, based on various forms of TOWCs sharing, have been proposed. In the Share-Per-Node (SPN) [19–22] architecture, all of the TOWCs are collected in a converter pool. This pool can be accessed by the packets requiring wavelength conversions. The SPN architecture allows for the greatest saving of TOWCs to be obtained, in some cases even near to 90% [19] with respect to an architecture using a TOWC for each output wavelength channel. The disadvantage of the SPN architecture is to increase the complexity of the switching matrix in terms of needed number of semiconductor optical amplifier (SOA), this increase can be in the order of 40% [19]. In the Share-Per-Output-Link (SPOL) [19] architecture, each output fiber is provided with a dedicated converter pool which can be accessed only by those packets directed to that particular output fiber. The SPOL architecture allows for a TOWC saving smaller than SPN, about 50% [13], but the complexity of its switching matrix is not increased with respect to the architecture having one TOWC for each output wavelength channel. In this paper, we investigate a new Optical Packet Switch architecture, referred to as Shared-Per-Input-Link (SPIL), where a dedicated wavelength converter pool is located for each input fiber. As it will be shown, SPIL architecture provides the same performance of SPOL architecture with a lower number of TOWCs. This is due to a more efficient sharing of TOWCs, in fact, packets directed to a given output fiber can use all of the TOWCs of the switch, sharing them with packets destined to other output fibers; on the contrary, in the SPOL architecture packets can use only the TOWCs placed on the output fiber which they are directed to. The performance of the SPIL architecture heavily depends on the adopted contention resolution control algorithm, deciding which packets have to be wavelength shifted. In this paper two objectives are pursued. The first aim is to compare the SPIL architecture performance with those offered by SPOL architecture when the same control algorithm, referred to as random algorithm, is adopted. The second item is to define an ‘optimum’ control algorithm so as to evaluate the best performance of the SPIL architecture is discussed. As far as the first item an analytical model able to evaluate the packet loss probability (PLP) in the SPIL architecture as a function of the needed number of TOWCs has been developed. This model provides an exact expression of PLP and enhances the model proposed in [23,24] where an upper bound of PLP is only evaluated. By using the models proposed in [20] for the SPOL architecture we will be able to compare the performance of the two architectures in terms of needed number of TOWCs when various traffic and system parameters are considered. As far as the second item, because the random algorithm is optimum for the SPOL architecture, but this is not true for
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the SPIL architecture, we will propose an integer linear programming (ILP) formulation of the packet scheduling problem allowing us to obtain, for a given number of TOWCs, the lowest value of PLP. This approach, though unfeasible in a practical context, provides a useful performance bound of the SPIL architecture, demonstrating the remarkable gain potentially obtainable with this kind of architecture. Finally, the best performance of the optimum algorithm, suggested to define a heuristic algorithm able to reach in low computation cost, performance near to the one of the optimum algorithm. The organisation of the paper is as follows. In Section 2, we describe the considered switching architectures; the definition of the control algorithms, under which the SPIL, SPOL architectures are compared, are discussed in Section 3; the analytical model, needed to evaluate the performance of the SPIL architecture when the random algorithm is adopted, is introduced in Section 4. Finally Section 5 illustrates the obtained results and, in particular, the number of TOWCs needed in the various proposed switching architectures, are evaluated and compared. The main conclusions and the further research topics are treated in Section 6.
2. Architectures of the WDM packet optical switches The two optical packet switches, that we want to analyse and compare, are illustrated in this section. They allow for a saving of TOWCs with respect to an architecture in which one TOWC is used for each output wavelength channel [19]. This is accomplished by sharing the TOWCs and allowing the access to them only to packets needing wavelength conversion. The proposed architectures adopt different sharing techniques, in particular in the SPOL and SPIL architectures the TOWCs are shared per output/input fiber, respectively. In Sections 2.1 and 2.2, the two architectures will be described. The architectures have N input and output fibers, each fiber supports a WDM signal with M wavelengths, so an input (or output) channel is characterized by the couple (i,lj) wherein i {iZ1,.,N} identifies the input/output fiber and lj {jZ1,.,M} identifies the wavelength. We assume that the operation mode of the architectures is synchronous, that is the arriving packets have a fixed size and their arrivals on each wavelength are synchronized on a time-slot basis [6,7] where a time-slot is the time needed to transmit a single packet. We also assume to have rl converters per input/output fiber in the SPOL/SPIL architectures. 2.1. SPOL architecture In the SPOL architecture, shown in Fig. 1, the TOWCs are shared per OL. There are rl TOWCs shared among the packets directed to a specific output fiber. The packets,
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Fig. 1. Single-Per-Output-Line (SPOL) architecture.
needing wavelength conversion are sent to the output branches where there are TOWCs. On the contrary, if a packet does not need a TOWC it is sent to an output branch in which there are not TOWCs. Note that the selection of an output branch of OL #i is accomplished by tuning on a SOA gate of the switching matrix S1,i. On each output branch one optical filter is used in order to select the wavelength on which the packet is arriving. Because the optical filtering is to be accomplished packet-by-packet and hence in a fast way, the optical filter is realised by means of one multiplexer, one demultiplexer and a bank of M SOAs, each one able to select a specific wavelength. For example, the circles in Fig. 1 denote the SOA gates to be turned on in order to wavelength shift a packet directed to OL #N and arriving at IL #1 on the wavelength lM.
can use all of the TOWCs of the switch, sharing them with packets destined to other output fibers; on the contrary, in the SPOL architecture packets can use only the TOWCs placed on the output fiber which they are directed to. The operation mode of the SPIL architecture is as follows. The arriving packets are first wavelength filtered, converted if it is needed and finally routed to the OL which they are directed to. The selection of OL #i is accomplished by turning on a SOA gate of the switching matrix S1,i. For example, the circles in Fig. 2 denote the SOA gates to be turned on in order to wavelength shift a packet directed to OL #1 and arriving at IL #N on the wavelength l1.
2.2. SPIL architecture
The SPN, SPOL and SPIL architectures will be firstly evaluated and compared under the hypothesis that the same control algorithm, proposed in [19] is used. The algorithm, referred to as random control algorithm, has low computation complexity because it can be accomplished in parallel on the various OLs of the switch. The operation mode of the algorithm is illustrated in Section 3.1. As the random algorithm is optimum for the SPOL architecture, but this is not true for SPIL architecture, we will propose in Section 3.2 an integer linear programming (ILP) formulation of
The new Optical Packet Switching architecture, investigated in this paper and referred to as SPIL, is shown in Fig. 2. In this architecture, a dedicated TOWCs pool is located for each IL. The SPIL architecture allows for a saving of number of TOWCs greater than the SPOL architecture, preserving the same level of complexity of the switching matrix. This is due to a more efficient sharing of TOWCs, in fact, packets directed to a given output fiber
3. Control algorithm for SPOL and SPIL architectures
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Fig. 2. Shared-Per-Input-Line (SPIL) architecture.
the optimum packet scheduling problem allowing the number of packets lost to be minimized once a particular pattern of arrivals is given. The optimum control algorithm is unfeasible in a practical context, for this reason in Section 3.3 we define a heuristic algorithm able to reach, in low computation cost, performance near to the one of the optimum control algorithm. 3.1. Random control algorithm The switch control unit adopts a simple and fair technique in assigning in each time-slot the TOWCs to the various arriving packets. Let Ii;lj be the set of the packets contending for the output wavelength channel (i,lj), that is the packets arriving at wavelength lj (jZ1,.,M) and directed to OL #i (iZ1,.,N). Further we denote as L and U the sets of the wavelengths and the OLs, respectively, that is Lh{l1,.,lM} and Uh{1,.N}. Further we denote as: (i) Li, the set of the free wavelengths in the course of the algorithm for the OL #i; (ii) G, the set of the OLs not considered yet in the course of the algorithm. In the following, we report the considered control algorithm referred to as random control algorithm:
*/Initialisation Phase-1*/ for (1%i%N) { Set LiZL; GZU for (1%j%M) Determine the set Ii,lj; } */Phase-2 of forwarding without wavelength conversion*/ for (1%i%N) for (1%j%M) if (Ii,lj is not empty) { Select randomly a packet b from Ii,lj; Remove the packet b from Ii,lj; Schedule the packet b for the output wavelength channel (i,lj); Remove lj from Li; } */Phase-3 of forwarding with wavelength conversion */ while (G is not empty) { Select randomly an OL #i; Remove i from G; Obtain IiZgj Ii,lj; while (Ii is not empty) {
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Select randomly a packet b2Ii; Remove b from Ii; If ((L i is not empty) and (TOWCs are available)) { Select randomly a wavelength lp from Li; Remove the wavelength lp from Li; Schedule the packet b for the output wavelength channel(i,lp); } else Discard the packet b; } } The algorithm is composed of three phases. In the phase1, the set Ii;lj (iZ1,.,N; jZ1,.,M) is evaluated, the set Li (iZ1,.N) and G are initialised. In the phase-2, the control unit, for each output wavelength channel (i,lj), determines a packet, if there is one, to be transmitted without wavelength conversion. If more than one packet is addressed to a specific output channel, the control unit randomly selects one of them; moreover for each OL #i, the set Li (iZ1,.,M) of free wavelengths is also updated. In the phase-3, the scheduling of the packets to be wavelength shifted is accomplished. The various OLs are randomly selected and, for each of them, the control unit tries, by means of wavelength conversions, to forward the packets belonging to the set Ii (iZ1,.,N), that is the packets addressed to the OL considered and which have not been selected in the phase-2 due to contentions. Obviously, these packets are forwarded only when both free wavelengths and TOWCs are available. The availability of the TOWCs refers to the availability of one TOWCs in the pool shared per output/input line in the SPOL/SPIL architectures, respectively. The random algorithm minimises the needed number of wavelengths conversions, since it is assured that one of the packets contending for each output wavelength channel is forwarded without wavelength conversion. However, different from the SPOL architecture case, in
the SPIL architecture, the random algorithm is not able to minimize the PLP; to achieve this result a suitable selection of the packets needing wavelength conversion should be accomplished. This is shown in Fig. 3, where we have reported an optical switch with 2 IL/OL, each one carrying two wavelengths l1, l2. The switch shares the TOWCs per IL and it is equipped with one TOWC per IL. We have denoted as ai,j (i,jZ1,2) the packet arriving at IL #i and on wavelength lj. If the packets are directed as shown in Fig. 3, the random control algorithm may choose a1,2 and a1,1 as packets to be wavelength shifted and a2,2 and a2,1 as packets to be forwarded without wavelength conversion. Now, note as this choice would need two TOWCs from IL#1 and, since only one TOWC is available for each IL, one packet should be lost. On the contrary, a packet scheduling without loss may be accomplished by wavelength shifting packets a1,1 and a2,2, so that one TOWC for both the ILs is needed and no packet loss occurs. 3.2. Optimum control algorithm for SPIL switches The optimum scheduling algorithm in SPIL architecture can be formulated as an integer linear programming (ILP) problem. The objective of the optimum scheduling problem is to minimize the number of packets lost once a particular pattern of arrivals is given. In the following, we illustrate the ILP formulation of the packet scheduling problem. Let fahi;j g be binary data variables that take the value 1 if a packet is arriving at input wavelength channel (i,lj) (iZ1,.,N; jZ1.,M) and it is directed to OL #h (hZ 1,.,N). Let fbki;j g be binary optimisation variables that take the value 1 if one packet arriving on input wavelength channel (i,lj) (iZ1,.,N; jZ1,.,M) is scheduled to be transferred on output wavelength lk (kZ1,.M). Obviously if ljZlk the packet is forwarded without wavelength conversion, otherwise the use of a TOWC is needed.
Fig. 3. Optimization of the used number of TOWC in a 2!2 SPIL switch with two wavelengths per IL/OL and one TOWC shared per IL. : denotes the empty set.
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With this definition, the ILP formulation of the problem can be expressed as follows: Subject to: M X
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Set LiZL; RiniZrl; GZU for (1%j%M) Determine the set Ii,lj; }
bki;j % 1
ði Z 1; .; N; j Z 1; .; MÞ
(3.1)
kZ1
XX i
(3.2)
bki;j % rl
(3.3)
j
XX j
bki;j ahi;j % 1 ðh Z 1; .; N; k Z 1; .; MÞ
maximise: XXXX i
ði Z 1; .; NÞ
ksj
j
k
bki;j ahi;j
(3.4)
h
The constraint (3.1) states that a packet, arriving at an input wavelength channel, can be forwarded on one output wavelength only; that follows from the unicast traffic assumption. The constraint (3.2) states that on each output wavelength channel (h,lk) cannot be forwarded more than one packet. The constraint (3.3) states that in any IL #i, the number of packets that can be wavelength shifted have to be fewer than the available number rl of TOWCs per IL. Finally the expression (3.4) is the forwarded number of packets and it is the term that we want to maximise. 3.3. Heuristic control algorithm for SPIL switches We propose for SPIL architecture a heuristic control algorithm able to obtain performance near to the one of the optimum algorithm. In order to define a heuristic algorithm we note from the example of Fig. 3 that the poor performance of the random algorithm is due to the not balanced use of the TOWCs located in the ILs. For this reason we based the heuristic algorithm on two rules aiming at balancing the use of the TOWCs. The first rule, applied in the phase of forwarding without wavelength conversion, consists in choosing the packet coming from the most congested IL when more packets contend for a same output wavelength channel. In this way the number of conversions needed on the IL with highest load is reduced. The second rule, applied in the phase offorwarding with wavelength conversion, consists in choosing, among the packets needing wavelength conversion, one coming from the less congested IL. In this way, if there is a loss due to the unavailability of wavelengths, the packets that are lost are the ones coming from the IL with heaviest traffic, thus relieving some more of the traffic burden from them. In the following we report the heuristic control algorithm: */Initialisation Phase-1*/ for (1%i%N){
*/Phase-2 of forwarding of non-contending packets */ for (1%i%N) for (1%j%M) if (Ii,lj contains one packet only) { Select the packet b in Ii,lj; Remove the packet b from Ii,lj; Schedule the packet b for the output wavelength channel (i,lj); Remove lj from Li; }
*/Phase-3 offorwarding without wavelength conversion*/ for (1%i%N) for (1%j%M) if (Ii,lj is not empty) { Obtain the set ji,lj4Ii,lj containing the packets of Ii,lj coming from the most congested IL Select randomly a packet b in ji,lj; Remove the packet b from Ii,lj; Schedule the packet b for the output wavelength channel (i,lj); Remove lj from Li; }
*/Phase-4 of forwarding with wavelength conversion */ while (G is not empty) { Select randomly an OL i; Remove i from G; Obtain IiZgj Ii,lj; while (Ii is not empty) { Obtain the set 2i 4Ii containing the packets of Ii coming from the less congested IL Select randomly a packet b22i; Remove b from Ii; Let k be the IL which the packet b arrives at If ((Li is not empty) and (Rinks0)) { Select randomly a wavelength lp from Li; Remove the wavelength lp from Li; Schedule the packet b for the output wavelength channel(i,lp); RinkZRinkK1; } else Discard the packet b; } }
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The algorithm is composed of four phases. In phase-1, it determines the sets Ii,lj and initialises each variable. With respect to the random algorithm the initialisation of the Rini (iZ1,.N) variables to the number rl of TOWCs available for each IL is added. The variables Rini denote the number of TOWCs not yet used for the OL #i in the course of the algorithm. In phase-2, the packets that does not collide are forwarded. They are treated as first because otherwise they could influence the next step of the algorithm aiming at balancing the use of TOWCs. In phase-3, the forwarding without conversion is accomplished. For each Output Wavelength Channel, the choice of the packet to be forwarded is carried out in the set of packets directed to that Output Wavelength Channel and coming from the more congested IL. In this way the load of the more congested IL is reduced. In this phase the set ji,lj is introduced containing the packets of the set Ii,lj, coming from the most congested IL. In phase-4, the forwarding with wavelength conversion is accomplished. An OL is selected and the choice of the packet to be forwarded is carried out in the set of packets directed to that OL and coming from the less congested IL. In phase-4, the set 2i is introduced containing the packets of the set Ii, coming from the less congested IL. Finally note as the introduced heuristic control algorithm allows the packets in the example of Fig. 3 to be scheduled in an optimum way. In fact for example whether packet a2,2 is scheduled to be directed without wavelength conversion in OL #1, according to the phase-3 of the control algorithm, packet a1,1 will be chosen to be forwarded without wavelength conversion in OL #2 because it is coming from the most congested IL. So one TOWC is needed for each IL and no packet loss occurs.
4. Performance evaluation of the SPIL architecture In this section we present the models we have used to obtain the performances of the SPIL architecture with unicast traffic. We consider a traffic per input wavelength channel equal to p and we assume that the traffic is symmetric: that is a arriving packet has the same probability to be directed to any output fiber. In Sections 4.1 and 4.2, we illustrate the models allowing us to evaluate the performance of the SPIL architecture when the random and ‘optimised’ control algorithms are adopted, respectively. The performance of the SPIL architecture when the heuristic algorithm is adopted will be evaluated by means of simulations because we were not able to evaluate the performance of the switch when the packets are scheduled according to the heuristic control algorithm reported in Section 3.3.
the packet scheduling according to the random control algorithm mentioned in Section 3. In the analytical model we propose, the packet loss is attributed to two factors: one is the lack of wavelength channels on the output fibers, the other is the lack of TOWCs needed to wavelength shift the arriving packets. According to this remark the PLPSPIL of the SPIL architecture can be calculated as follows PLPSPIL Z
E½Nl E½Nwl C E½Ncl Z E½No E½No
Z PLPwl C aTOWC
(4.1)
PLPwl Z
E½Nwl E½No
(4.2)
aTOWC Z
E½Ncl E½No
(4.3)
wherein E[x] No Nl Nwl and Ncl
denoted the expected value of a random variable x is the number of packets offered to the switch is the number of packets lost denote the number of packets lost because of the wavelength and TOWCs lack, respectively.
Now the first loss term PLPwl in (4.1) due to wavelength lack is calculated. Keeping in mind the symmetric traffic assumption, we can remark that E[Nwl] and E[No] are equal to N times the average number of packet lost from and offered to any output fiber. Hence PLPwl can be expressed as follows: ! NM NM p i 1 X p NMKi wl PLP Z ði K MÞ 1K Mp iZMC1 N N i (4.4) In the expression (4.4), the term Mp is the average number of packets offered to any output fiber. On the contrary, the sum term is the average number of packets lost because of the wavelength lack and it is evaluated taking into account the fact that the loss occurs only when more than M packets are directed to the output fiber [20]. Due to the symmetric traffic assumption, the second loss term aTOWC in (4.1), due to the TOWCs lack, can be evaluated according to the following expression aTOWC Z
E½Ncli E½Noi
(4.5)
4.1. Performance evaluation in the case of random control algorithm
wherein
Our aim is to evaluate the PLP as a function of the used number of TOWCs when the control unit accomplishes
E½Noi Z Mp is the average number of packets offered to any IL
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467
E½Ncli
is the average number of packets lost because of TOWCs lack in any IL
The term E½Ncli in (4.5) is evaluated using the total probability law after defining the random variable C, representing the number of conversions required by the packets arriving at any IL. With this definition we obtain the following expression for E½Ncli : E½Ncli Z
M X
E½Ncli jC Z jProbfC Z jg
(4.6)
jZ0
The evaluation of E½Ncli jCZ j in (4.6) can be accomplished by taking into account that when the number j of conversions needed in any IL is smaller than the number rl of TOWCs available in any IL, no packets are lost because of TOWCs lack; otherwise a number of packets equal to jKrl is lost due to the unavailability of a sufficient number of TOWCs. Hence the term E½Ncli jCZ j can be expressed as: ( 0 if 0% j% rl i E½Ncl jC Z j Z (4.7) j K rl otherwise According to (4.7), expression (4.6) can be simply rewritten as: E½Ncli Z
M X
ðj K rl ÞProbfC Z jg
(4.8)
jZ0
Denoting b as the probability that a wavelength conversion is needed on an input wavelength channel of an IL, we can express the probabilities of the random variable C as follows: ! M j PrfC Z jg Z (4.9) b ð1 K bÞMKj j To evaluate b, we observe that according to the rules of the random control algorithm, a wavelength conversion occurs on an input wavelength channel when: (i) a packet arrives at that input wavelength channel; (ii) the arriving packet is not lost because of the lack of free wavelengths on the OL which the packet is directed to; (iii) the packet is not selected by the control unit to be forwarded to the OL without wavelength conversion. The last event S occurs with probability PrfSg Z
N K1 X kZ0
qðkÞ Z
k qðkÞ k C1
! N K 1 p k p NK1Kk 1K N N k
(4.10)
(4.11)
being q(k) the probability that k packets are directed to the same OL and arrive on the same wavelength of the considered packet.
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According to the earlier remarks and the expressions (4.10) and (4.11), we can express b as follows: b Z pð1 K PLPwl Þ !
NK1 X kZ0
k k C1
! N K 1 p k p NK1Kk 1K N N k
(4.12)
Finally the terms from (4.4) to (4.12), once inserted in the expression (4.1) allows us to evaluate the PLPSPIL as a function of the needed number of TOWCs and hence to perform the required dimensioning. 4.2. Performance evaluation in the case of ‘optimum’ control algorithm In this section the methodology is presented to evaluate the PLP when the ‘optimum’ control algorithm presented in Section 3.2 is used in each time-slot. We have, in each time-slot, as input data a particular pattern of arrivals and, by means of an ILP formulation of the scheduling problem reported in Section 3.2, the best way of wavelength shifting the arriving packets is determined so that the number of packets lost is minimised. The final results are obtained by simulation and they have a certain degree of uncertainty due to the pseudo-random procedure used to generate traffic in each time-slot. The solution of the ILP problem has been obtained using the ILOG CPLEX [25] software package; this tool allows us to obtain the optimal solution in each time-slot to be found and, after a suitable simulation time, we can estimate the PLP of the switch.
5. Numerical results This section reports results concerning the comparison among the various proposed switching architectures. In particular: (i) we evaluate the goodness of the analytical model, introduced in Section 4.1, by means of simulation results; (ii) the SPOL and SPIL architectures will be compared in terms of number of TOWCs needed in order to guarantee a specific value of PLP. Fig. 4 plots the analytical and simulation values of the PLP as a function of the total number rZNrl of TOWCs, when the random control algorithm is adopted. The curves are obtained for NZ16, MZ16, pZ0.2, 0.4, 0.6, and 0.8. As it can be seen our model gives results very close to the simulation ones. We have verified, by means of other results not shown in this paper, that this good agreement is confirmed for a large range of the values of the system parameter, so, results arising from the analytical model will be exclusively used in the following. The comparison among the SPOL and SPIL architectures in terms of number of TOWCs used is reported in
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1.00E+00
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
1.00E-01 1.00E-02 N=16 M=16
1.00E-03 1.00E-04 1.00E-05 1.00E-06
p=0.2 (sim.)
p=0.2 (anal.)
p=0.4 (sim.)
p=0.4 (anal.)
p=0.6 (sim.)
p=0.6 (anal.)
p=0.8 (sim.)
p=0.8 (anal.)
packet loss probability
packet loss probability
1.00E-02
SPOL (M=16) SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-04
1.00E-06
1.00E-08
N=16 p=0.4 1.00E-10
1.00E-07 1.00E-08
1.00E-12 0
32
64
96
128
160
192
224
256
0
48
96 144 192 240 288 336 384 432 480 528 576
number of TOWCs used (r)
number of TOWCs used (r) Fig. 4. Comparison between the analytical and simulation models of the SPIL architecture when the packets are scheduled according the random control algorithm. The traffic and system parameters are NZ16, MZ16 and pZ0.2, 0.4, 0.6, and 0.8.
Fig. 6. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.4, NZ16 and MZ16, 32, 48, and 64.
Figs. 5–8. In particular, we show the PLP as a function of r when NZ16, MZ16, 32, 48, and 64 and pZ0.2, 0.4, 0.6, and 0.8. The switching architectures are compared when the packets are scheduled according to the random control algorithm. The reported results are obtained by means of the analytical model proposed in Section 4.1 for the SPIL architecture, on the other hand the results of the SPOL architecture are obtained by means of the models proposed in [20–22].
From the figures we note that all of the curves have the same tend: they are a decreasing function of the number of LRWCs used up to a threshold value where the PLP saturates. The saturation value is the packet loss probability due to the limited number of output wavelengths. It represents the packet loss probability of an architecture using all of the wavelength converters. The threshold value of a curve represents the minimum number of LRWCs needed in order to obtain the same packet loss probability of
1.00E+00
1.00E+00
packet loss probability
1.00E-03 1.00E-06
SPOL (M=16)
SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-09 1.00E-12 1.00E-15 1.00E-18
N=16 p=0.2
1.00E-21
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
1.00E-01
packet loss probability
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
SPOL (M=16) SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-02
1.00E-03
1.00E-04
N=16 p=0.6 1.00E-05
1.00E-24 1.00E-27 0
48
96 144 192 240 288 336 384 432 480 528 576
number of TOWCs used (r) Fig. 5. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.2, NZ16 and MZ16, 32, 48, and 64.
1.00E-06
0
48
96 144 192 240 288 336 384 432 480 528 576
number of TOWCs used (r) Fig. 7. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.6, NZ16 and MZ16, 32, 48, and 64.
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467 1.00E+00
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
SPOL (M=16) SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-01
1.00E-02
N=16 p=0.8
M =3 (ILP) M =3 (Ran) M =3 (Heu) M =5 (ILP) M =5 (Ran) M =5 (Heu) M =7 (ILP) M =7 (Ran) M =7 (Heu)
1.00E-01
Packet Loss Probability
packet loss probability
1.00E+00
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1.00E-02
1.00E-03
1.00E-04
q=0.8 N=16 1.00E-05
1.00E-06 48 96 144 192 240 288 336 384 432 480 528 576 number of TOWCs used (r)
Fig. 8. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.8, NZ16 and MZ16, 32, 48, and 64.
an architecture using all of the wavelength converters. The results reported in the Figs. 5–8 show that the SPIL architecture allows for a greater saving of TOWCs with respect to the SPOL architecture even when the random algorithm is adopted. For example, for MZ64 the SPIL architecture needs a number of LRWCs equal to 384, 368, 384, 432 for pZ0.2, 0.4, 0.6, and 0.8, respectively. On the contrary, we can note that the SPOL architecture for the same traffic and switch parameters needs a greater number of LRWCS equal to 496, 464, 448, 480. Hence the SPIL architecture allows for a saving of TOWCs equal to 23, 21, 14, and 10% when MZ64 and pZ0.2, 0.4, 0.6, and 0.8, respectively. The better performance obtainable with the SPIL architecture are due to a more efficient sharing of TOWCs, in fact, packets directed to a given output fiber can use all of the TOWCs of the switch, sharing them with packets destined to other output fibers; on the contrary, in the SPOL architecture packets can use only the TOWCs placed on the output fiber which they are directed to. Now we provide for SPIL architecture a performance comparison when the packets are scheduled according to the random and optimum control algorithms, respectively. In order to obtain the PLP in the case of optimum control algorithm, we use the simulation/analytical hybrid model illustrated in Section 4.2. The obtained results are reported in Figs. 9 and 10 for MZ3, 5, and 7 and MZ2, 4, and 6, respectively. In both the figures, the switch dimension is NZ16 and a traffic qZM!pZ0.8 is offered to each fiber. For the SPIL architecture we report also the values of PLP in the case in which the packet scheduling is accomplished according to the heuristic control algorithm introduced in
0
16
32
48
64
80
96
112
number of converters (r) Fig. 9. Performance evaluation of the SPIL architecture when the packets are scheduled according to the random, optimum and heuristic control algorithms. The traffic and system parameters are qZ0.8, NZ16 and MZ3, 5, and 7.
Section 3.3. As it can be seen from Figs. 9 and 10 the adoption of the ‘optimum’ control algorithm allows dimensioning values of the TOWCs lower to be obtained and hence a TOWCs saving greater with respect the random control algorithm. For example, for MZ4–6 the percentage gain is in the order of 50%. Hence, with respect to the random control algorithm, an optimised scheduling of the packets allows for a further saving of TOWCs.
1.00E+00 M=2 (ILP) M=2 (Ran) M=2 (Heu) M=4 (ILP) M=4 (Ran) M=4 (Heu) M=6 (ILP) M=6 (Ran) M=6 (Heu)
1.00E-01
Packet Loss Probability
1.00E-03 0
1.00E-02
1.00E-03
q=0.8 N=16 1.00E-04
1.00E-05
0
16
32
48
64
80
96
number of converters (r )
Fig. 10. Performance evaluation of the SPIL architecture when the packets are scheduled according to the random, optimum and heuristic control algorithms. The traffic and system parameters are qZ0.8, NZ16 and MZ2, 4, and 6.
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Finally we show the goodness of the heuristic control algorithm proposed in Section 3.3, comparing its results with those obtained by adopting the optimum algorithm. From Figs. 9 and 10 we can note that the performance of the introduced heuristic control algorithm is near to the one of the optimum algorithm. In fact for M equal to or smaller than 6 the algorithms provides the same performance while for MZ7 the algorithms provide lightly different performance when the Optical Packet Switch is equipped with one TOWC per IL. Because of the good behaviour of the heuristic algorithm in future complexity issues of the random and heuristic algorithms will be carried out in order to understand the price to be paid for the better performance of the heuristic algorithm with respect to the random algorithm.
6. Conclusions In this paper, we have discussed a WDM Optical Packet Switching architecture denoted as SPIL and equipped with TOWCs, shared per IL. In order to compare the proposed architecture with other ones already proposed in literature, some analytical and simulation models have been developed that allowed us to evaluate the needed number of TOWCs as a function of the main system and traffic parameters (offered traffic, number of IL/OL, number of wavelength,.). The performance evaluation of the SPIL architecture has been carried out when two control algorithms are adopted in order to schedule the arriving packets; the former one, simple computationally but not able to reach the best dimensioning values of TOWCs, the latter one allowing to schedule the packets in an optimised way but needing a computation effort greater and according to a procedure still to be determined. With respect to an architecture sharing the TOWC per OL, the proposed models allowed us to evaluate for example for MZ16, 32, 48, and 64, a saving of TOWCs equal to 23, 21, 14, and 10% for the first above control algorithm, respectively. The second algorithm allows for a greater saving that for MZ7 reaches the 50%. The best performance involved when an optimised scheduling of the arriving packets is accomplished, suggested to define a heuristic control algorithm able to obtain, in low computation cost, performance near to the one of the optimum algorithm. The introduced algorithm was able to reach this aim and in particular the obtained results showed that the same performance can be reached for M equal to or smaller than 7. In future complexity issues of the random and heuristic algorithms will be carried out in order to understand the price to be paid for the better performance of the heuristic algorithm with respect to the random algorithm.
References [1] M. Murata, K. Kitayama, Ultrafast photonic label switch for asynchronous packets of varaiable length, IEEE Infocom, 2002 June 23–27, New York. [2] S. Sulur, T. Koonen, H. De Waardt, I.T. Monroy, IM/FSK format for payload/orthogonal labeling IP packets in IP over WDM networks supported by GMPLS-based LOBS Proceedings of ONDM 2003, The Seventh IFIP Working Conference on Optical Design and Modelling, February 3–5, Budapest, Hungary vol. II (2003) pp. 703–716. [3] K. Kitayama, M. Murata, Ultrafast phtonic label switch: architecture and its building blocks Proceedings of ONDM 2003, The Seventh IFIP Working Conference on Optical Design and Modelling, February 3–5, Budapest, Hungary vol. II (2003) pp. 717–736. [4] S. Aleksic, V. Krajinovic, Comparison of optical code correlators for all-optical MPLS networks, 28th European Conference on Optical Communication (ECOC ’02), September 8–12, Copenhagen Denmark (2002). [5] C. Qiao, M. Yoo, Optical burst switching (OBS)—a new paradigm for an optical internet 8 (1999) 69–84. [6] M. Renaud, C. Janz, P. Gambini, C. Guillemot, Transparent optical packet switching: The European ACTS KEOPS project approach, in Proc. LEOS ’99 vol. 2 (1999) pp. 401–402. [7] C. Guillemot, et al., Transparent optical packet switching: the European ACTS KEOPS project approach, IEEE Journal of Lightwave Technology 16 (12) (1998). [8] W. Vanderbauwhede, D.A. Harle, Novel design for an asynchronous optical packet switch Proceedings of ONDM 2003, The Seventh IFIP Working Conference on Optical Design and Modelling, February 3–5, Budapest, Hungary vol. II (2003) pp. 737–754. [9] M. Listanti, R. Sabella, V. Eramo, Architectural and technological issues for future optical internet networks, IEEE Communications MagazineSeptember2000; 82–92. [10] B. Li, Y. Qin, X. Cao, K. Sivalingam, Photonic packet switching: architectures and performance, Optical Network MagazineJanuary/February2001; 27–39. [11] L. Xu, H.G. Perros, G. Rouskas, Techniques for optical packet switching and optical burst switching, IEEE Communication Magazine January (2001). [12] S. Yao, B. Mukherjee, S. Dixit, Advances in photonic packet switching: an overview, IEEE Communication Magazine February (2000). [13] M. Mahony, D. Simeoniddu, D. Hunter, A. Tzanamaki, The application of optical packet switching in future communications networks, IEEE Communication Magazine March (2001). [14] C.S. Chang, D.S. Lee, C.K. Tu, Using switched delay for exact simulation of FIFO multiplexers with variable length bursts, INFOCOM 2003, 2003 San Francisco, March 30–April 3. [15] K. Laevens, H. Bruneel, Analysis of a single wavelength optical buffer, INFOCOM, 2003 San Francisco, March 30–April 3. [16] S. Yao, S.J.B. Yoo, B. Mukherjee, S. Dixit, All-optical packet switching for metropolitan area networks: opportunities and challenges, IEEE Communication Magazine March (2001). [17] S.L. Danielsen, C. Joergensen, B. Mikkelsen, K.E. Stubkyaer, Optical packet switched network layer without optical buffers, IEEE Photonic Technology Letters 10 (6) (1998). [18] S.L. Danielsen, C. Jorgensen, B. Mikkelsen, K.E. Stubbkyaer, Analysis of a WDM packet switch with improved performance under bursty traffic conditions due to tuneable wavelength converters, IEEE Journal of Lightwave Technology 16 (5) (1998). [19] V. Eramo, M. Listanti, Comparison of unicast/multicast optical packet switching architectures using wavelength conversion, Optical Network Magazine 3 (March–April) (2002) 63–75.
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467 [20] V. Eramo, M. Listanti, P. Pacifici, A comparison study on the wavelength converters number needed in synchronous and asynchronous all-optical switching architectures, IEEE Journal of Lightwave Technology February (2003). [21] V. Eramo, M. Listanti, P. Pacifici, Dimensioning of the wavelength converters in synchronous and asynchronous switching architecture, International Conference on Communications (ICC) 2003, May 11–15, Anchorage Alaska (2003).
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[22] V. Eramo, M. Listanti, Resources sharing in optical packet switching architectures, QoS in Multiservice Networks, February 24–26, Milan, 2003. [23] V. Eramo, M. Listanti, Input wavelength conversion in optical packet switches, IEEE Communication Letters 7 (6) (2003). [24] V. Eramo, M. Listanti, Advantages of input wavelength conversion in optical packet switches, Globecom 2003, December 1–5, San Francisco, 2003. [25] http://www.ilog.com/products/ampl.
Scheduling algorithms in optical packet switches with input wavelength conversion V. Eramo*, M. Listanti, A. Valletta INFOCOM Department, University of Roma ‘La Sapienza’, Via Eudossiana 18, 00184 Roma, Italy Received 12 March 2004; revised 28 January 2005; accepted 11 February 2005 Available online 19 March 2005
Abstract The objective of this study is to propose a new Optical Packet Switching architecture in which the wavelength converters, needed to solve output packet contentions, are shared per input line; according to this sharing strategy the packets arriving at a given input fiber shares a converter pool that can be accessed when wavelength conversions are required. In the paper, we propose analytical and simulation models able to evaluate the performances of the proposed architecture when control algorithms with different complexity, are adopted. Under a unicast traffic scenario, the obtained performances are compared to the ones of the architecture in which the wavelength converters are shared per output line. The carried out comparison shows that, with respect to the architecture with wavelength converters shared per output line, the proposed architecture allows for a 30% saving of wavelength converters when a simple control algorithm is adopted; the saving can reach 50% if an optimized control algorithm is used. In the paper, we also define a heuristic algorithm able to reach in low computation cost, performance near to the one of the optimum algorithm. q 2005 Elsevier B.V. All rights reserved. Keywords: Optical packet switching; Synchronous switching; Wavelength converter; Dimensioning; Share-per-input-link architecture; Share-per-output-link architecture
1. Introduction The transmission capacity of optical fibers has been increasing at a tremendous rate as a result of DWDM technology. Although terabit capacity IP routers are now starting to appear, the mismatch between the transmission capacity of DWDM and the switching capacity of electronic routers remains remarkable. Emerging all-optical switching technologies will enable packets to traverse nodes transparently without conversion and processing at each node. The limited optical processing power and the lack of optical buffering techniques are the main obstacles in the realization of all-optical networks in which both the payload and the header of the packets remain in the optical domain from the source node to the destination node. Although some primitive header processing forms begin to appear [1–4], the current approach to optical packet * Corresponding author. Tel.: C39 06 47852311; fax: C39 06 4744481. E-mail address: [email protected] (V. Eramo).
0140-3664/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2005.02.005
switching is to keep the payload in the optical domain and convert the header to the electronic domain for processing [5–8]. A critical issue in photonic packet switching is the contention resolution. Contention occurs when two or more packets contend for the same output port at the same time. In traditional electronic switches, packet contentions are handled through buffering; unfortunately, this is not feasible in the optical domain, since there is no optical equivalent of electronic random-access memories. Today, optical buffering can be only achieved through the use of fiber delay lines [9–15], however, the buffer size is severely limited, not only by signal quality concerns, but also by physical space limitations; as a matter of example, to delay a single packet for 5 ms requires over 1 km of fiber. In order to avoid optical buffering, Wavelength Conversion approach has been widely investigated to handle packet contentions in the optical domain [16–18]. The basic principle is very simple: packets contending for the same output are converted to different wavelengths by means of tunable optical wavelength converters (TOWC). So, the reference packet switch architecture foresees a TOWC for each output wavelength channel. As the number of TOWCs
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467
represents a critical cost element for a photonic packet switch, various Optical Packet Switching architectures, based on various forms of TOWCs sharing, have been proposed. In the Share-Per-Node (SPN) [19–22] architecture, all of the TOWCs are collected in a converter pool. This pool can be accessed by the packets requiring wavelength conversions. The SPN architecture allows for the greatest saving of TOWCs to be obtained, in some cases even near to 90% [19] with respect to an architecture using a TOWC for each output wavelength channel. The disadvantage of the SPN architecture is to increase the complexity of the switching matrix in terms of needed number of semiconductor optical amplifier (SOA), this increase can be in the order of 40% [19]. In the Share-Per-Output-Link (SPOL) [19] architecture, each output fiber is provided with a dedicated converter pool which can be accessed only by those packets directed to that particular output fiber. The SPOL architecture allows for a TOWC saving smaller than SPN, about 50% [13], but the complexity of its switching matrix is not increased with respect to the architecture having one TOWC for each output wavelength channel. In this paper, we investigate a new Optical Packet Switch architecture, referred to as Shared-Per-Input-Link (SPIL), where a dedicated wavelength converter pool is located for each input fiber. As it will be shown, SPIL architecture provides the same performance of SPOL architecture with a lower number of TOWCs. This is due to a more efficient sharing of TOWCs, in fact, packets directed to a given output fiber can use all of the TOWCs of the switch, sharing them with packets destined to other output fibers; on the contrary, in the SPOL architecture packets can use only the TOWCs placed on the output fiber which they are directed to. The performance of the SPIL architecture heavily depends on the adopted contention resolution control algorithm, deciding which packets have to be wavelength shifted. In this paper two objectives are pursued. The first aim is to compare the SPIL architecture performance with those offered by SPOL architecture when the same control algorithm, referred to as random algorithm, is adopted. The second item is to define an ‘optimum’ control algorithm so as to evaluate the best performance of the SPIL architecture is discussed. As far as the first item an analytical model able to evaluate the packet loss probability (PLP) in the SPIL architecture as a function of the needed number of TOWCs has been developed. This model provides an exact expression of PLP and enhances the model proposed in [23,24] where an upper bound of PLP is only evaluated. By using the models proposed in [20] for the SPOL architecture we will be able to compare the performance of the two architectures in terms of needed number of TOWCs when various traffic and system parameters are considered. As far as the second item, because the random algorithm is optimum for the SPOL architecture, but this is not true for
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the SPIL architecture, we will propose an integer linear programming (ILP) formulation of the packet scheduling problem allowing us to obtain, for a given number of TOWCs, the lowest value of PLP. This approach, though unfeasible in a practical context, provides a useful performance bound of the SPIL architecture, demonstrating the remarkable gain potentially obtainable with this kind of architecture. Finally, the best performance of the optimum algorithm, suggested to define a heuristic algorithm able to reach in low computation cost, performance near to the one of the optimum algorithm. The organisation of the paper is as follows. In Section 2, we describe the considered switching architectures; the definition of the control algorithms, under which the SPIL, SPOL architectures are compared, are discussed in Section 3; the analytical model, needed to evaluate the performance of the SPIL architecture when the random algorithm is adopted, is introduced in Section 4. Finally Section 5 illustrates the obtained results and, in particular, the number of TOWCs needed in the various proposed switching architectures, are evaluated and compared. The main conclusions and the further research topics are treated in Section 6.
2. Architectures of the WDM packet optical switches The two optical packet switches, that we want to analyse and compare, are illustrated in this section. They allow for a saving of TOWCs with respect to an architecture in which one TOWC is used for each output wavelength channel [19]. This is accomplished by sharing the TOWCs and allowing the access to them only to packets needing wavelength conversion. The proposed architectures adopt different sharing techniques, in particular in the SPOL and SPIL architectures the TOWCs are shared per output/input fiber, respectively. In Sections 2.1 and 2.2, the two architectures will be described. The architectures have N input and output fibers, each fiber supports a WDM signal with M wavelengths, so an input (or output) channel is characterized by the couple (i,lj) wherein i {iZ1,.,N} identifies the input/output fiber and lj {jZ1,.,M} identifies the wavelength. We assume that the operation mode of the architectures is synchronous, that is the arriving packets have a fixed size and their arrivals on each wavelength are synchronized on a time-slot basis [6,7] where a time-slot is the time needed to transmit a single packet. We also assume to have rl converters per input/output fiber in the SPOL/SPIL architectures. 2.1. SPOL architecture In the SPOL architecture, shown in Fig. 1, the TOWCs are shared per OL. There are rl TOWCs shared among the packets directed to a specific output fiber. The packets,
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Fig. 1. Single-Per-Output-Line (SPOL) architecture.
needing wavelength conversion are sent to the output branches where there are TOWCs. On the contrary, if a packet does not need a TOWC it is sent to an output branch in which there are not TOWCs. Note that the selection of an output branch of OL #i is accomplished by tuning on a SOA gate of the switching matrix S1,i. On each output branch one optical filter is used in order to select the wavelength on which the packet is arriving. Because the optical filtering is to be accomplished packet-by-packet and hence in a fast way, the optical filter is realised by means of one multiplexer, one demultiplexer and a bank of M SOAs, each one able to select a specific wavelength. For example, the circles in Fig. 1 denote the SOA gates to be turned on in order to wavelength shift a packet directed to OL #N and arriving at IL #1 on the wavelength lM.
can use all of the TOWCs of the switch, sharing them with packets destined to other output fibers; on the contrary, in the SPOL architecture packets can use only the TOWCs placed on the output fiber which they are directed to. The operation mode of the SPIL architecture is as follows. The arriving packets are first wavelength filtered, converted if it is needed and finally routed to the OL which they are directed to. The selection of OL #i is accomplished by turning on a SOA gate of the switching matrix S1,i. For example, the circles in Fig. 2 denote the SOA gates to be turned on in order to wavelength shift a packet directed to OL #1 and arriving at IL #N on the wavelength l1.
2.2. SPIL architecture
The SPN, SPOL and SPIL architectures will be firstly evaluated and compared under the hypothesis that the same control algorithm, proposed in [19] is used. The algorithm, referred to as random control algorithm, has low computation complexity because it can be accomplished in parallel on the various OLs of the switch. The operation mode of the algorithm is illustrated in Section 3.1. As the random algorithm is optimum for the SPOL architecture, but this is not true for SPIL architecture, we will propose in Section 3.2 an integer linear programming (ILP) formulation of
The new Optical Packet Switching architecture, investigated in this paper and referred to as SPIL, is shown in Fig. 2. In this architecture, a dedicated TOWCs pool is located for each IL. The SPIL architecture allows for a saving of number of TOWCs greater than the SPOL architecture, preserving the same level of complexity of the switching matrix. This is due to a more efficient sharing of TOWCs, in fact, packets directed to a given output fiber
3. Control algorithm for SPOL and SPIL architectures
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Fig. 2. Shared-Per-Input-Line (SPIL) architecture.
the optimum packet scheduling problem allowing the number of packets lost to be minimized once a particular pattern of arrivals is given. The optimum control algorithm is unfeasible in a practical context, for this reason in Section 3.3 we define a heuristic algorithm able to reach, in low computation cost, performance near to the one of the optimum control algorithm. 3.1. Random control algorithm The switch control unit adopts a simple and fair technique in assigning in each time-slot the TOWCs to the various arriving packets. Let Ii;lj be the set of the packets contending for the output wavelength channel (i,lj), that is the packets arriving at wavelength lj (jZ1,.,M) and directed to OL #i (iZ1,.,N). Further we denote as L and U the sets of the wavelengths and the OLs, respectively, that is Lh{l1,.,lM} and Uh{1,.N}. Further we denote as: (i) Li, the set of the free wavelengths in the course of the algorithm for the OL #i; (ii) G, the set of the OLs not considered yet in the course of the algorithm. In the following, we report the considered control algorithm referred to as random control algorithm:
*/Initialisation Phase-1*/ for (1%i%N) { Set LiZL; GZU for (1%j%M) Determine the set Ii,lj; } */Phase-2 of forwarding without wavelength conversion*/ for (1%i%N) for (1%j%M) if (Ii,lj is not empty) { Select randomly a packet b from Ii,lj; Remove the packet b from Ii,lj; Schedule the packet b for the output wavelength channel (i,lj); Remove lj from Li; } */Phase-3 of forwarding with wavelength conversion */ while (G is not empty) { Select randomly an OL #i; Remove i from G; Obtain IiZgj Ii,lj; while (Ii is not empty) {
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Select randomly a packet b2Ii; Remove b from Ii; If ((L i is not empty) and (TOWCs are available)) { Select randomly a wavelength lp from Li; Remove the wavelength lp from Li; Schedule the packet b for the output wavelength channel(i,lp); } else Discard the packet b; } } The algorithm is composed of three phases. In the phase1, the set Ii;lj (iZ1,.,N; jZ1,.,M) is evaluated, the set Li (iZ1,.N) and G are initialised. In the phase-2, the control unit, for each output wavelength channel (i,lj), determines a packet, if there is one, to be transmitted without wavelength conversion. If more than one packet is addressed to a specific output channel, the control unit randomly selects one of them; moreover for each OL #i, the set Li (iZ1,.,M) of free wavelengths is also updated. In the phase-3, the scheduling of the packets to be wavelength shifted is accomplished. The various OLs are randomly selected and, for each of them, the control unit tries, by means of wavelength conversions, to forward the packets belonging to the set Ii (iZ1,.,N), that is the packets addressed to the OL considered and which have not been selected in the phase-2 due to contentions. Obviously, these packets are forwarded only when both free wavelengths and TOWCs are available. The availability of the TOWCs refers to the availability of one TOWCs in the pool shared per output/input line in the SPOL/SPIL architectures, respectively. The random algorithm minimises the needed number of wavelengths conversions, since it is assured that one of the packets contending for each output wavelength channel is forwarded without wavelength conversion. However, different from the SPOL architecture case, in
the SPIL architecture, the random algorithm is not able to minimize the PLP; to achieve this result a suitable selection of the packets needing wavelength conversion should be accomplished. This is shown in Fig. 3, where we have reported an optical switch with 2 IL/OL, each one carrying two wavelengths l1, l2. The switch shares the TOWCs per IL and it is equipped with one TOWC per IL. We have denoted as ai,j (i,jZ1,2) the packet arriving at IL #i and on wavelength lj. If the packets are directed as shown in Fig. 3, the random control algorithm may choose a1,2 and a1,1 as packets to be wavelength shifted and a2,2 and a2,1 as packets to be forwarded without wavelength conversion. Now, note as this choice would need two TOWCs from IL#1 and, since only one TOWC is available for each IL, one packet should be lost. On the contrary, a packet scheduling without loss may be accomplished by wavelength shifting packets a1,1 and a2,2, so that one TOWC for both the ILs is needed and no packet loss occurs. 3.2. Optimum control algorithm for SPIL switches The optimum scheduling algorithm in SPIL architecture can be formulated as an integer linear programming (ILP) problem. The objective of the optimum scheduling problem is to minimize the number of packets lost once a particular pattern of arrivals is given. In the following, we illustrate the ILP formulation of the packet scheduling problem. Let fahi;j g be binary data variables that take the value 1 if a packet is arriving at input wavelength channel (i,lj) (iZ1,.,N; jZ1.,M) and it is directed to OL #h (hZ 1,.,N). Let fbki;j g be binary optimisation variables that take the value 1 if one packet arriving on input wavelength channel (i,lj) (iZ1,.,N; jZ1,.,M) is scheduled to be transferred on output wavelength lk (kZ1,.M). Obviously if ljZlk the packet is forwarded without wavelength conversion, otherwise the use of a TOWC is needed.
Fig. 3. Optimization of the used number of TOWC in a 2!2 SPIL switch with two wavelengths per IL/OL and one TOWC shared per IL. : denotes the empty set.
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With this definition, the ILP formulation of the problem can be expressed as follows: Subject to: M X
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Set LiZL; RiniZrl; GZU for (1%j%M) Determine the set Ii,lj; }
bki;j % 1
ði Z 1; .; N; j Z 1; .; MÞ
(3.1)
kZ1
XX i
(3.2)
bki;j % rl
(3.3)
j
XX j
bki;j ahi;j % 1 ðh Z 1; .; N; k Z 1; .; MÞ
maximise: XXXX i
ði Z 1; .; NÞ
ksj
j
k
bki;j ahi;j
(3.4)
h
The constraint (3.1) states that a packet, arriving at an input wavelength channel, can be forwarded on one output wavelength only; that follows from the unicast traffic assumption. The constraint (3.2) states that on each output wavelength channel (h,lk) cannot be forwarded more than one packet. The constraint (3.3) states that in any IL #i, the number of packets that can be wavelength shifted have to be fewer than the available number rl of TOWCs per IL. Finally the expression (3.4) is the forwarded number of packets and it is the term that we want to maximise. 3.3. Heuristic control algorithm for SPIL switches We propose for SPIL architecture a heuristic control algorithm able to obtain performance near to the one of the optimum algorithm. In order to define a heuristic algorithm we note from the example of Fig. 3 that the poor performance of the random algorithm is due to the not balanced use of the TOWCs located in the ILs. For this reason we based the heuristic algorithm on two rules aiming at balancing the use of the TOWCs. The first rule, applied in the phase of forwarding without wavelength conversion, consists in choosing the packet coming from the most congested IL when more packets contend for a same output wavelength channel. In this way the number of conversions needed on the IL with highest load is reduced. The second rule, applied in the phase offorwarding with wavelength conversion, consists in choosing, among the packets needing wavelength conversion, one coming from the less congested IL. In this way, if there is a loss due to the unavailability of wavelengths, the packets that are lost are the ones coming from the IL with heaviest traffic, thus relieving some more of the traffic burden from them. In the following we report the heuristic control algorithm: */Initialisation Phase-1*/ for (1%i%N){
*/Phase-2 of forwarding of non-contending packets */ for (1%i%N) for (1%j%M) if (Ii,lj contains one packet only) { Select the packet b in Ii,lj; Remove the packet b from Ii,lj; Schedule the packet b for the output wavelength channel (i,lj); Remove lj from Li; }
*/Phase-3 offorwarding without wavelength conversion*/ for (1%i%N) for (1%j%M) if (Ii,lj is not empty) { Obtain the set ji,lj4Ii,lj containing the packets of Ii,lj coming from the most congested IL Select randomly a packet b in ji,lj; Remove the packet b from Ii,lj; Schedule the packet b for the output wavelength channel (i,lj); Remove lj from Li; }
*/Phase-4 of forwarding with wavelength conversion */ while (G is not empty) { Select randomly an OL i; Remove i from G; Obtain IiZgj Ii,lj; while (Ii is not empty) { Obtain the set 2i 4Ii containing the packets of Ii coming from the less congested IL Select randomly a packet b22i; Remove b from Ii; Let k be the IL which the packet b arrives at If ((Li is not empty) and (Rinks0)) { Select randomly a wavelength lp from Li; Remove the wavelength lp from Li; Schedule the packet b for the output wavelength channel(i,lp); RinkZRinkK1; } else Discard the packet b; } }
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The algorithm is composed of four phases. In phase-1, it determines the sets Ii,lj and initialises each variable. With respect to the random algorithm the initialisation of the Rini (iZ1,.N) variables to the number rl of TOWCs available for each IL is added. The variables Rini denote the number of TOWCs not yet used for the OL #i in the course of the algorithm. In phase-2, the packets that does not collide are forwarded. They are treated as first because otherwise they could influence the next step of the algorithm aiming at balancing the use of TOWCs. In phase-3, the forwarding without conversion is accomplished. For each Output Wavelength Channel, the choice of the packet to be forwarded is carried out in the set of packets directed to that Output Wavelength Channel and coming from the more congested IL. In this way the load of the more congested IL is reduced. In this phase the set ji,lj is introduced containing the packets of the set Ii,lj, coming from the most congested IL. In phase-4, the forwarding with wavelength conversion is accomplished. An OL is selected and the choice of the packet to be forwarded is carried out in the set of packets directed to that OL and coming from the less congested IL. In phase-4, the set 2i is introduced containing the packets of the set Ii, coming from the less congested IL. Finally note as the introduced heuristic control algorithm allows the packets in the example of Fig. 3 to be scheduled in an optimum way. In fact for example whether packet a2,2 is scheduled to be directed without wavelength conversion in OL #1, according to the phase-3 of the control algorithm, packet a1,1 will be chosen to be forwarded without wavelength conversion in OL #2 because it is coming from the most congested IL. So one TOWC is needed for each IL and no packet loss occurs.
4. Performance evaluation of the SPIL architecture In this section we present the models we have used to obtain the performances of the SPIL architecture with unicast traffic. We consider a traffic per input wavelength channel equal to p and we assume that the traffic is symmetric: that is a arriving packet has the same probability to be directed to any output fiber. In Sections 4.1 and 4.2, we illustrate the models allowing us to evaluate the performance of the SPIL architecture when the random and ‘optimised’ control algorithms are adopted, respectively. The performance of the SPIL architecture when the heuristic algorithm is adopted will be evaluated by means of simulations because we were not able to evaluate the performance of the switch when the packets are scheduled according to the heuristic control algorithm reported in Section 3.3.
the packet scheduling according to the random control algorithm mentioned in Section 3. In the analytical model we propose, the packet loss is attributed to two factors: one is the lack of wavelength channels on the output fibers, the other is the lack of TOWCs needed to wavelength shift the arriving packets. According to this remark the PLPSPIL of the SPIL architecture can be calculated as follows PLPSPIL Z
E½Nl E½Nwl C E½Ncl Z E½No E½No
Z PLPwl C aTOWC
(4.1)
PLPwl Z
E½Nwl E½No
(4.2)
aTOWC Z
E½Ncl E½No
(4.3)
wherein E[x] No Nl Nwl and Ncl
denoted the expected value of a random variable x is the number of packets offered to the switch is the number of packets lost denote the number of packets lost because of the wavelength and TOWCs lack, respectively.
Now the first loss term PLPwl in (4.1) due to wavelength lack is calculated. Keeping in mind the symmetric traffic assumption, we can remark that E[Nwl] and E[No] are equal to N times the average number of packet lost from and offered to any output fiber. Hence PLPwl can be expressed as follows: ! NM NM p i 1 X p NMKi wl PLP Z ði K MÞ 1K Mp iZMC1 N N i (4.4) In the expression (4.4), the term Mp is the average number of packets offered to any output fiber. On the contrary, the sum term is the average number of packets lost because of the wavelength lack and it is evaluated taking into account the fact that the loss occurs only when more than M packets are directed to the output fiber [20]. Due to the symmetric traffic assumption, the second loss term aTOWC in (4.1), due to the TOWCs lack, can be evaluated according to the following expression aTOWC Z
E½Ncli E½Noi
(4.5)
4.1. Performance evaluation in the case of random control algorithm
wherein
Our aim is to evaluate the PLP as a function of the used number of TOWCs when the control unit accomplishes
E½Noi Z Mp is the average number of packets offered to any IL
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467
E½Ncli
is the average number of packets lost because of TOWCs lack in any IL
The term E½Ncli in (4.5) is evaluated using the total probability law after defining the random variable C, representing the number of conversions required by the packets arriving at any IL. With this definition we obtain the following expression for E½Ncli : E½Ncli Z
M X
E½Ncli jC Z jProbfC Z jg
(4.6)
jZ0
The evaluation of E½Ncli jCZ j in (4.6) can be accomplished by taking into account that when the number j of conversions needed in any IL is smaller than the number rl of TOWCs available in any IL, no packets are lost because of TOWCs lack; otherwise a number of packets equal to jKrl is lost due to the unavailability of a sufficient number of TOWCs. Hence the term E½Ncli jCZ j can be expressed as: ( 0 if 0% j% rl i E½Ncl jC Z j Z (4.7) j K rl otherwise According to (4.7), expression (4.6) can be simply rewritten as: E½Ncli Z
M X
ðj K rl ÞProbfC Z jg
(4.8)
jZ0
Denoting b as the probability that a wavelength conversion is needed on an input wavelength channel of an IL, we can express the probabilities of the random variable C as follows: ! M j PrfC Z jg Z (4.9) b ð1 K bÞMKj j To evaluate b, we observe that according to the rules of the random control algorithm, a wavelength conversion occurs on an input wavelength channel when: (i) a packet arrives at that input wavelength channel; (ii) the arriving packet is not lost because of the lack of free wavelengths on the OL which the packet is directed to; (iii) the packet is not selected by the control unit to be forwarded to the OL without wavelength conversion. The last event S occurs with probability PrfSg Z
N K1 X kZ0
qðkÞ Z
k qðkÞ k C1
! N K 1 p k p NK1Kk 1K N N k
(4.10)
(4.11)
being q(k) the probability that k packets are directed to the same OL and arrive on the same wavelength of the considered packet.
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According to the earlier remarks and the expressions (4.10) and (4.11), we can express b as follows: b Z pð1 K PLPwl Þ !
NK1 X kZ0
k k C1
! N K 1 p k p NK1Kk 1K N N k
(4.12)
Finally the terms from (4.4) to (4.12), once inserted in the expression (4.1) allows us to evaluate the PLPSPIL as a function of the needed number of TOWCs and hence to perform the required dimensioning. 4.2. Performance evaluation in the case of ‘optimum’ control algorithm In this section the methodology is presented to evaluate the PLP when the ‘optimum’ control algorithm presented in Section 3.2 is used in each time-slot. We have, in each time-slot, as input data a particular pattern of arrivals and, by means of an ILP formulation of the scheduling problem reported in Section 3.2, the best way of wavelength shifting the arriving packets is determined so that the number of packets lost is minimised. The final results are obtained by simulation and they have a certain degree of uncertainty due to the pseudo-random procedure used to generate traffic in each time-slot. The solution of the ILP problem has been obtained using the ILOG CPLEX [25] software package; this tool allows us to obtain the optimal solution in each time-slot to be found and, after a suitable simulation time, we can estimate the PLP of the switch.
5. Numerical results This section reports results concerning the comparison among the various proposed switching architectures. In particular: (i) we evaluate the goodness of the analytical model, introduced in Section 4.1, by means of simulation results; (ii) the SPOL and SPIL architectures will be compared in terms of number of TOWCs needed in order to guarantee a specific value of PLP. Fig. 4 plots the analytical and simulation values of the PLP as a function of the total number rZNrl of TOWCs, when the random control algorithm is adopted. The curves are obtained for NZ16, MZ16, pZ0.2, 0.4, 0.6, and 0.8. As it can be seen our model gives results very close to the simulation ones. We have verified, by means of other results not shown in this paper, that this good agreement is confirmed for a large range of the values of the system parameter, so, results arising from the analytical model will be exclusively used in the following. The comparison among the SPOL and SPIL architectures in terms of number of TOWCs used is reported in
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1.00E+00
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
1.00E-01 1.00E-02 N=16 M=16
1.00E-03 1.00E-04 1.00E-05 1.00E-06
p=0.2 (sim.)
p=0.2 (anal.)
p=0.4 (sim.)
p=0.4 (anal.)
p=0.6 (sim.)
p=0.6 (anal.)
p=0.8 (sim.)
p=0.8 (anal.)
packet loss probability
packet loss probability
1.00E-02
SPOL (M=16) SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-04
1.00E-06
1.00E-08
N=16 p=0.4 1.00E-10
1.00E-07 1.00E-08
1.00E-12 0
32
64
96
128
160
192
224
256
0
48
96 144 192 240 288 336 384 432 480 528 576
number of TOWCs used (r)
number of TOWCs used (r) Fig. 4. Comparison between the analytical and simulation models of the SPIL architecture when the packets are scheduled according the random control algorithm. The traffic and system parameters are NZ16, MZ16 and pZ0.2, 0.4, 0.6, and 0.8.
Fig. 6. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.4, NZ16 and MZ16, 32, 48, and 64.
Figs. 5–8. In particular, we show the PLP as a function of r when NZ16, MZ16, 32, 48, and 64 and pZ0.2, 0.4, 0.6, and 0.8. The switching architectures are compared when the packets are scheduled according to the random control algorithm. The reported results are obtained by means of the analytical model proposed in Section 4.1 for the SPIL architecture, on the other hand the results of the SPOL architecture are obtained by means of the models proposed in [20–22].
From the figures we note that all of the curves have the same tend: they are a decreasing function of the number of LRWCs used up to a threshold value where the PLP saturates. The saturation value is the packet loss probability due to the limited number of output wavelengths. It represents the packet loss probability of an architecture using all of the wavelength converters. The threshold value of a curve represents the minimum number of LRWCs needed in order to obtain the same packet loss probability of
1.00E+00
1.00E+00
packet loss probability
1.00E-03 1.00E-06
SPOL (M=16)
SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-09 1.00E-12 1.00E-15 1.00E-18
N=16 p=0.2
1.00E-21
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
1.00E-01
packet loss probability
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
SPOL (M=16) SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-02
1.00E-03
1.00E-04
N=16 p=0.6 1.00E-05
1.00E-24 1.00E-27 0
48
96 144 192 240 288 336 384 432 480 528 576
number of TOWCs used (r) Fig. 5. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.2, NZ16 and MZ16, 32, 48, and 64.
1.00E-06
0
48
96 144 192 240 288 336 384 432 480 528 576
number of TOWCs used (r) Fig. 7. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.6, NZ16 and MZ16, 32, 48, and 64.
V. Eramo et al. / Computer Communications 28 (2005) 1456–1467 1.00E+00
SPIL (M=16) SPIL (M=32) SPIL (M=48) SPIL (M=64)
SPOL (M=16) SPOL (M=32) SPOL (M=48) SPOL (M=64)
1.00E-01
1.00E-02
N=16 p=0.8
M =3 (ILP) M =3 (Ran) M =3 (Heu) M =5 (ILP) M =5 (Ran) M =5 (Heu) M =7 (ILP) M =7 (Ran) M =7 (Heu)
1.00E-01
Packet Loss Probability
packet loss probability
1.00E+00
1465
1.00E-02
1.00E-03
1.00E-04
q=0.8 N=16 1.00E-05
1.00E-06 48 96 144 192 240 288 336 384 432 480 528 576 number of TOWCs used (r)
Fig. 8. Comparison among the SPIL and SPOL architectures in terms of the total number of TOWCs used when the packets are scheduled according to the random control algorithm. The traffic and system parameters are pZ0.8, NZ16 and MZ16, 32, 48, and 64.
an architecture using all of the wavelength converters. The results reported in the Figs. 5–8 show that the SPIL architecture allows for a greater saving of TOWCs with respect to the SPOL architecture even when the random algorithm is adopted. For example, for MZ64 the SPIL architecture needs a number of LRWCs equal to 384, 368, 384, 432 for pZ0.2, 0.4, 0.6, and 0.8, respectively. On the contrary, we can note that the SPOL architecture for the same traffic and switch parameters needs a greater number of LRWCS equal to 496, 464, 448, 480. Hence the SPIL architecture allows for a saving of TOWCs equal to 23, 21, 14, and 10% when MZ64 and pZ0.2, 0.4, 0.6, and 0.8, respectively. The better performance obtainable with the SPIL architecture are due to a more efficient sharing of TOWCs, in fact, packets directed to a given output fiber can use all of the TOWCs of the switch, sharing them with packets destined to other output fibers; on the contrary, in the SPOL architecture packets can use only the TOWCs placed on the output fiber which they are directed to. Now we provide for SPIL architecture a performance comparison when the packets are scheduled according to the random and optimum control algorithms, respectively. In order to obtain the PLP in the case of optimum control algorithm, we use the simulation/analytical hybrid model illustrated in Section 4.2. The obtained results are reported in Figs. 9 and 10 for MZ3, 5, and 7 and MZ2, 4, and 6, respectively. In both the figures, the switch dimension is NZ16 and a traffic qZM!pZ0.8 is offered to each fiber. For the SPIL architecture we report also the values of PLP in the case in which the packet scheduling is accomplished according to the heuristic control algorithm introduced in
0
16
32
48
64
80
96
112
number of converters (r) Fig. 9. Performance evaluation of the SPIL architecture when the packets are scheduled according to the random, optimum and heuristic control algorithms. The traffic and system parameters are qZ0.8, NZ16 and MZ3, 5, and 7.
Section 3.3. As it can be seen from Figs. 9 and 10 the adoption of the ‘optimum’ control algorithm allows dimensioning values of the TOWCs lower to be obtained and hence a TOWCs saving greater with respect the random control algorithm. For example, for MZ4–6 the percentage gain is in the order of 50%. Hence, with respect to the random control algorithm, an optimised scheduling of the packets allows for a further saving of TOWCs.
1.00E+00 M=2 (ILP) M=2 (Ran) M=2 (Heu) M=4 (ILP) M=4 (Ran) M=4 (Heu) M=6 (ILP) M=6 (Ran) M=6 (Heu)
1.00E-01
Packet Loss Probability
1.00E-03 0
1.00E-02
1.00E-03
q=0.8 N=16 1.00E-04
1.00E-05
0
16
32
48
64
80
96
number of converters (r )
Fig. 10. Performance evaluation of the SPIL architecture when the packets are scheduled according to the random, optimum and heuristic control algorithms. The traffic and system parameters are qZ0.8, NZ16 and MZ2, 4, and 6.
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Finally we show the goodness of the heuristic control algorithm proposed in Section 3.3, comparing its results with those obtained by adopting the optimum algorithm. From Figs. 9 and 10 we can note that the performance of the introduced heuristic control algorithm is near to the one of the optimum algorithm. In fact for M equal to or smaller than 6 the algorithms provides the same performance while for MZ7 the algorithms provide lightly different performance when the Optical Packet Switch is equipped with one TOWC per IL. Because of the good behaviour of the heuristic algorithm in future complexity issues of the random and heuristic algorithms will be carried out in order to understand the price to be paid for the better performance of the heuristic algorithm with respect to the random algorithm.
6. Conclusions In this paper, we have discussed a WDM Optical Packet Switching architecture denoted as SPIL and equipped with TOWCs, shared per IL. In order to compare the proposed architecture with other ones already proposed in literature, some analytical and simulation models have been developed that allowed us to evaluate the needed number of TOWCs as a function of the main system and traffic parameters (offered traffic, number of IL/OL, number of wavelength,.). The performance evaluation of the SPIL architecture has been carried out when two control algorithms are adopted in order to schedule the arriving packets; the former one, simple computationally but not able to reach the best dimensioning values of TOWCs, the latter one allowing to schedule the packets in an optimised way but needing a computation effort greater and according to a procedure still to be determined. With respect to an architecture sharing the TOWC per OL, the proposed models allowed us to evaluate for example for MZ16, 32, 48, and 64, a saving of TOWCs equal to 23, 21, 14, and 10% for the first above control algorithm, respectively. The second algorithm allows for a greater saving that for MZ7 reaches the 50%. The best performance involved when an optimised scheduling of the arriving packets is accomplished, suggested to define a heuristic control algorithm able to obtain, in low computation cost, performance near to the one of the optimum algorithm. The introduced algorithm was able to reach this aim and in particular the obtained results showed that the same performance can be reached for M equal to or smaller than 7. In future complexity issues of the random and heuristic algorithms will be carried out in order to understand the price to be paid for the better performance of the heuristic algorithm with respect to the random algorithm.
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