Performance model of an optical switch using fiber ... - Semantic Scholar

Performance Model of an Optical Switch using Fiber Delay Lines for Resolving Contentions Ayman G. Fayoumi and Anura P. Jayasumana Department of Electrical and Computer Engineering Colorado State University Fort Collins, Colorado 80523 [email protected], [email protected]

Abstract The performance of an optical switch that handles contention resolution using a fiber delay line (FDL) is modeled and evaluated. We propose using a simple optical buffer that contains only a single FDL for contention resolution. The analytical model derived for the performance of an optical switch based on this buffer is verified using simulations. The analytical model can be utilized with both packet and burst switching schemes to characterize the performance of switches augmented with this FDL architecture.

1

Introduction

In all optical networks, when two or more packets contend for the same output wavelength simultaneously, one or more contending packets get dropped. Achieving an effective resolution mechanism for such contention is crucial for minimizing losses. In burst switching, for example, dropping a burst means dropping multiple packets or a large stream of data. In electronic packet switching environments, contention resolution is carried out using electronic store-and-forward buffers, implemented using random-access memory (RAM). However, an all-optical RAM is not available to utilize the same resolution approach in the optical domain. Additionally, implementing this type of memory would be very costly [8]. Therefore, other contention resolution schemes have been proposed that exploit wavelength dimension [4], space dimension [3], and time dimension [8]. In wavelength dimension based resolution schemes, a wavelength converter is utilized such that one of the contending packets is assigned the desired wavelength while the other gets converted to an unused wavelength. In space dimension based resolution schemes, either multiple output fibers or deflection routing schemes

can be utilized. The approach of utilizing multiple output fibers is similar to that in the wavelength dimension scheme; however, in this case, physical output fibers are utilized instead of multiple wavelengths. The assumption here is that multiple fibers connect the nodes instead of a single fiber. Deflection routing assigns the preferred output wavelength or port to one of the contending packets while deflecting the other to an unused wavelength or a non-preferable output port. The drawback of this approach is that the deflected packet may have to traverse a longer path to reach the destination, which can be of the order of kilometers in WAN networks, and hence a significant crosstalk may be encountered by the optical signals [6]. Additionally, the network becomes more congested as more deflections take place. A time-domain contention resolution scheme is implemented by utilizing a fixed-length Fiber Delay Line (FDL) on a first-in-first-out basis [11]. An optical packet propagates through an FDL for a fixed amount of time. This propagation delay time is usually chosen to be a multiple of the mean transmission time of the packet. Optical buffering schemes that use FDL for contention resolution can be divided into two categories: forward FDL buffering, and feedback FDL buffering. In forward FDL buffering, the packet gets delayed at the output port and leaves the node after the propagation delay of the FDL [2]. In feedback FDL buffering, the packet gets delayed and re-enters the node after the propagation delay of the FDL [5]. The feedback FDL approach is more costly than the forward FDL scheme since it contributes to higher complexity of input routing logic. In this paper, we study the performance of an optical switch that utilizes forward FDL buffering. An analytical model for a forward FDL buffering scheme is presented in [2]. However, it approximates the summation of non-exponentially distributed variables (FDL propagation delay and the packet transmission time) to be a Gammadistributed variable. The model proposed to analyze the FDL’s in [12] holds only for fixed packet size. In [7], a

Proceedings of the 28th Annual IEEE International Conference on Local Computer Networks (LCN’03) 0742-1303/03 $ 17.00 © 2003 IEEE

simulation model is presented to study the performance of an optical switch augmented with a multiple FDL based buffer as that shown in Figure 1. The queuing model of the FDLs was assumed to be an M/M/k with balking in [10]. It considers an optical buffer that imitates the traditional electronic RAM as it assumes that a packet whose length exceeds the maximum FDL of the buffer is discarded. A FDL provides a packet with a certain propagation delay regardless of its transmission time. Therefore considering an optical buffer that functions as an electronic RAM is not accurate. In all of these studies, the optical buffer studied consists of B FDLs, each having a certain length. The assumption in these papers was that once the desired fiber delay line is free, it can be used regardless of the state of the other FDLs. However, one needs to consider the fact that contention may take place at the end of different FDLs between packets leaving these FDLs, when their wavelengths are the same. Therefore, the algorithms for forwarding the packets to the different FDLs of an optical buffer should be considered in order to capture all the capabilities of the switch. In this work, a scheme is presented by considering a variable-length packet size to reduce the Probability of Blocking (PB ) of the optical switch. A model is developed to predict the performance of an all-optical network where the packet size is exponentially distributed. Since the algorithm for forwarding the packets to the FDLs of the optical buffer has a great impact on the performance of the optical switch as mentioned, the analytical model presented here characterizes that algorithm as well. To buffer a single packet optically, for example to delay a packet of length 1 kb considering a 2.5 Gbps optical channel, an 80 meter long FDL is required. The situation would be worse when considering an optical buffer that consists of multiple levels of FDLs, each of which is a multiple of the packet size. More hardware complexity as well as the management and control overhead is required in this situation. Therefore, it is crucial to use as few FDLs as possible. In this work, a single FDL per output is considered to resolve the contention.

2

Optical switch with output buffers

0 µ sec (1−p) λ pλ

Routing and Control Logic

p λ (1−β) Bx

x

A

FDL D µ sec

OCX

Figure 1. Multiple fiber delay line buffer

OCX

.... B

The rest of the paper is organized as follows. Section 2 describes the output optical buffering mechanism and the related forwarding scheduler algorithm. In Section 3, the main analytical model of the optical buffer is presented. Results from a simulation model is used in Section 4 to evaluate the packet switch as well as to validate the analytical model. The conclusions are in Section 5.

OCX

0 1 2 3

Figure 2. Optical packet switch with an output FDL

Figure 2 shows the architecture of the optical switch augmented with output FDL. The enlarged segment in the figure represents the output buffer of one of the output links of the optical switch. Figure 3 shows a simplified version of this architecture with a 2x2 optical cross connects. The forwarding scheduler algorithm for forwarding an incoming packet to either the output port directly, i.e., to port A which has no FDL, or to port B, the FDL port, of the output buffer is shown in Figure 4. It guarantees that if the output port, port A, is busy, the incoming packet is buffered in the FDL until the port becomes available. An advantage of this algorithm is that it reduces the conflicts at the output link due to those packets that are coming from port B and those that are directly forwarded to port A.


State "0": FREE state

0 µ sec 1/2 λ

IN 1

λ

(1−p) λ

(1−p) λ p λ (1−β) pλ

Bx

x

A

State "1" BUSY state

pλ

OUT 1

FDL

1

λ 1/2

0

µ

µ

(a)

(b)

λ

D µ sec

1

0

1/2

0 µ sec

IN 2

1/2 λ

λ

(1−p) λ p λ (1−β) pλ

Bx

x

A

OUT 2

Figure 5. State diagram of (a) port A and (b) port B

FDL D µ sec

3.1 Figure 3. A logical representation depicting arrival rates from different inputs at output FDLs

If port A is FREE and no packet exists in the FDL forward the incoming packet to port A else if port B is FREE froward the incoming packet to port B else drop the packet

Figure 4. The forwarding scheduler algorithm

3

Analytical model

The model presented in this section characterizes an n × n optical switch. Each output link is augmented with a single optical buffer to resolve the contention that may take place at an output link. Due to the symmetry of the optical switch, the analytical model will consider only one output buffer and the rest of the links can be modeled similarly. A Poisson packet arrival to the switch is considered, and the packet destinations are assumed to be uniformly distributed among the output ports of the switch. The length of a packet is assumed to be exponentially distributed, with a mean transmission time of 1/µ. Let the rate at which data is injected into the switch from each input be λ packets per unit time. After multiplexing the data from all input links to all output links, each output buffer is assumed to receive a data rate of λ. The load is considered to be the ratio of the arrival rate to the service rate of the packets. Let p be the probability of forwarding an incoming data packet to port B. Therefore, as shown in Figure 3, data is forwarded at the rate of pλ to port B while data is directly forwarded to port A at the rate of (1 − p)λ. If a packet is forwarded to port A, it encounters zero delay whereas, if it is forwarded to port B, it encounters a delay of D, where D is the propagation delay of the FDL.

Forwarding and port busy probabilities

We model the behavior of ports A and B with the state transition graphs shown in Figure 5. State “0” represents the FREE state, state “1” represents the BUSY state, and P [i] is the probability of being at state i. Let α be the probability that port A is busy and β be the probability that port B is busy. The only stream of packets that affects the state of port B is of rate pλ. By considering the fact that P [0]+P [1] = 1 and solving the two-state transition probabilities shown in Figure 5 (b), pλ (1) β= pλ + µ There are two packet streams that affect the state of port A. One of these streams has an arrival rate (1 − p)λ, which is the stream that is being directly forwarded to port A. This stream affects the state of port A according to the two-state transmission probabilities shown in Figure 5 (a). Let α be the probability that port A is busy due to this stream. Then, α =

(1 − p)λ (1 − p)λ + µ

(2)

The other stream that affects the state of port A corresponds to the data packets that is coming from port B. This stream is of rate (1 − β)pλ. Let α be the probability of port A being busy as a result of (1 − β)λ rate stream. α is considered to be the load of (1 − β)λ rate stream at port A, i.e., α

= =

(1 − β)pλ µ pλ pλ + µ

(3)

The events that “port A is being busy by (1 − p)λ rate stream” and “port A is being busy by (1−β)λ” are mutually exclusive, i.e., these events cannot happen simultaneously. Therefore, α

= α + α pλ (1 − p)λ + = (1 − p)λ + µ pλ + µ


(4)

Note that we model the two ports using two independent Markov chains. This is an approximation that allows us develop this model. The simulation results presented in Section 4 evaluate the accuracy of the model, and indicate this to be a reasonable approximation. Based on the algorithm shown in Figure 4, p defines the probability that either the FDL has at least one packet or port A is busy. Let Γ be the event that one or more packets are present in the FDL, and γ be its probability. Then, p=γ+α−η

(5)

be the rate of arrival of packets entering the FDL, Let λ therefore, Ψ is assumed to be exponentially distributed with rate λ, λψ − (8) fΨ (ψ) = λe where λ

fΘ (θ)

µe−µ(θ−D)

(6)

P [Ψ < Θ] =

D

D

1−

θ

0

Ψ

Figure 6. Timing diagram of packet arrivals at the FDL when the inter-arrival time is greater than D

fΘ (θ)fΨ (ψ)

µ+λ

P [Ψ > Θ] =

T

θ ∈ [D, ∞)

µ

and time

X

∞

.. . =

1111111 0000000 0000000 1111111 0000000 1111111

(9)

= fT (θ − D) =

Let ω be the probability that there is no packet in the FDL, then γ =1−ω (7)

11111 00000 00000 11111 00000 11111

pλ(1 − β) pλµ = pλ + µ

Let Θ = T + D, then

where η is the probability that port A is busy while there are one or more packets present in the FDL, i.e., η = P [port A is busy ∩ Γ)

=

dψ d θ

λD e−

µ µ+λ

λD e−

The distribution of X , the time during which the FDL is free of packets

Figure 6 shows the relationship between different parameters that constitute the inter-arrival time of the packet flow entering the FDL when the inter-arrival time, represented by the random variable Ψ, is greater than D. These parameters are 1. The random variable X, which represents the idle time between the packet arrivals. It corresponds to the time during which the FDL contains no packet. 2. The delay D, which represents the propagation delay of the FDL.

(11)

(12)

The values associated with the random variable X satisfies, 0 if Ψ ≤ Θ X= (13) > 0 if Ψ > Θ The pdf for X can be written as fX (x) = gX (x) + hX (x)

3.2

(10)

(14)

where gX (x) and hX (x) are contributions corresponding to the cases Ψ < Θ and Ψ > Θ respectively. Consider the case when Ψ < Θ. In this case, X = 0 as, from Figure 6, there will be one or more packets existing in the FDL. Therefore, gX (x) = δ(x) · P [Ψ < Θ] µ − e λD ) = δ(x)(1 − µ+λ

(15)

where δ(x) is the delta function. Consider the other case when Ψ > Θ. Here, X = Ψ − Θ. The conditional CDF of X given that Ψ > Θ is given by HX (x |Ψ > Θ ) = P [X ≤ x | Ψ > Θ]

3. The random variable T , which represents the transmission time of a packet. This random variable is assumed to be exponentially distributed with a rate of µ.


=

P [Ψ − Θ ≤ x | Ψ > Θ] P [Ψ ≤ x + Θ, Ψ > Θ] = P [Ψ > Θ P [Θ ≤ Ψ ≤ x + Θ] (16) = P [Ψ > Θ]

3.3

Now, P [Θ ≤ Ψ ≤ x + Θ]

=

∞

D

.. .

x+θ

θ

λ ) (1 − e−x

=

fΘ (θ)fΨ (ψ)dψ dθ

µ µ+λ

λD e− (17)

Therefore, λx HX (x |Ψ > Θ ) = 1 − e−

(18)

Therewhich implies that, when Ψ > Θ, X ∼ Exp(λ). fore, λx − hX (x) = λe · P [Ψ > Θ] µ − − λ x e λD = λe µ+λ

= gX (x) + hX (x) µ − e λD ) + = δ(x)(1 − µ+λ µ − λx − e λD λe µ+λ

λD e− λ µ+λ

µ

Since the system is assumed to be stable [1], then, ω

E(X) E(Ψ) µ − e λD µ+λ

=

(19) and γ =1−

µ µ+λ

(23)

λD e−

(24)

Using the value of λ, γ =1−

µ e−pλ(1−β)D µ + pλ(1 − β)

(25)

By definition, (20)

From fX (x), the expected value of X is found to be E(X) =

From Figure 6, the probability that no packet exists in the FDL, ω, can be approximated by n xi ω = limn→∞ ni=1 ψ i i=1 n ( i=1 xi )/n = limn→∞ n (22) ( i=1 ψi )/n

=

Using equation (14), the distribution of X in the whole domain; i.e., when Ψ < Θ and Ψ > Θ, is given by, fX (x)

The forwarding probabilities to the ports of output buffer

(21)

η = P [Γ|A is busy] × α

(26)

The probability P [Γ|A is busy] defines the probability of Γ given that port A is busy. When port A is known to be busy, all the data traffic will certainly be forwarded to port B, i.e., p = 1. The derivation of γ will be the same as before except that p = 1. Therefore, P [Γ|A is busy] = 1 −

f (ψ) Ψ

µ µ + λ(1 − β)

where β =

e−λ(1−β)D

λ λ+µ

(27)

(28)

Therefore, η

= (1 −

µ e−λ(1−β)D ) ) µ+λ(1−β

pλ ( pλ+µ + fX(x) = δ(x)

f

X

^x −λ (x) = ^ λ e

K

Figure 7. The relation between the density functions of X and Ψ

A

×

(1−p)λ (1−p)λ+µ )

(29)

Now, from Equations 1, 4, 25, and 27, the equation p = γ + α × (1 − P [Γ|port A is busy])

(30)

can be solved by means of iteration for different values of λ, µ, and D. Now to predict the Probability of Blocking (PB ), consider the optical switch that is shown in Figure 3. When port


B is kept busy by a packet that has arrived from the input link IN 1, the only packets that will be dropped, i.e. contribute to PB , are those coming from the other input link, link IN 2. Since the packets that are being injected from the input link IN 1 form a flow of packets in a sequence, no packet of this flow will cause a drop to a subsequent packet. Similarly, when port B is kept busy by a packet that came from input link IN 2, the only packets that will cause the PB are those that are coming from the input link IN 1. Therefore, in general, PB will be due to packets that are coming from all input links except the link from which the packet causing port B to be busy has come from. On the other hand, port B can be busy as a result of packets that are coming from all input links. Thus, PB is calculated as n−1 ×β PB = n

(31)

where n is the number of input links to the switch. The factor n−1 n represents the fraction of input links that causes the PB .

3.4

µ

P [T > D] =

(32)

where is very small value. Therefore, D=

ln( ) µ

(33)

For example, if is chosen to be 0.05, then D ≈ µ3 . This relation can be used to determine the approximate length of the FDL.

Results

A discrete-event simulator was developed using C code to evaluate the performance, in terms of the blocking probability at each of the output buffers, of the n × n optical switch, similar to the switch that is shown in Figure 2.

OUT 1

IN 1

2x2 ROBS λ

µ

OUT 2

IN 2

Figure 8. Source model for optical switch (i+3) arrival

(i+2) arrival 1/λ

(i+1) arrival 1/λ

ith arrival 1/λ

(i+2) burst 111111 000000 000000 111111 (i+1) burst 000000 111111 111111 000000 00000000000000 11111111111111 00000000000000 11111111111111 11111111111111 00000000000000

(i+3) burst

11111111 00000000 00000000 11111111 11111111 00000000

ith busrt

111111 000000 000000 111111 111111 000000

time

(a)

(i+3) arrival

(i+2) arrival 1/λ

(i+3) burst

11111111 00000000 00000000 11111111 00000000 11111111

(i+1) arrival 1/λ

(i+2) burst

(i+1) burst

111111 000000 00000000000000 11111111111111 000000 111111 00000000000000 11111111111111 000000 111111 00000000000000 11111111111111

Determining the propagation delay of FDL

Next, we characterize the propagation delay of the FDL, D. If two packets arrive simultaneously and both are intended for the same output link, then one of them will be forwarded to port A, while the other packet will be forwarded to port B. Losses at port A will take place if the packet that is coming from port B arrives at port A while the packet that has been forwarded to port A is still leaving port A. To avoid this situation, the propagation delay of the FDL, D, should be large enough so that the packet that has been forwarded to port A leaves this port before the packet that is coming from port B arrives at port A. This means that, with high probability, the transmission time of a packet should be lower than D. In other words, given that the length of a packet, T , is exponentially distributed,

4

λ

ith arrival 1/λ ith busrt

111111 000000 000000 111111 000000 111111

time

(b)

Figure 9. (a) Poisson arrival process and (b) Arrival process to the switch

Since, at each of the input links, the inter-arrival times and the transmission times of the packet generator are assumed to be independent, each of the input ports to the optical switch was modeled as an M/M/1/∞ queue to emulate a source of packets as shown in Figure 8. By this approach, the generated packets were re-arranged from a conflicting state, as in Figure 9 (a), to a non-conflicting state, as in Figure 9 (b). Now, we present analytical and simulation results for an output link of an n × n optical switch for different values of n. Unless otherwise mentioned, the mean transmission time of a packet is considered to be 100 × 103 bit times units, where a bittime is assumed to be the transmission time of a single bit. In addition, a minimum value of the FDL propagation delay, as explained in Section 3.4, is also considered in this section. The results shown in Figure 10 correspond to the simulation and the analytical blocking probabilities for a 2x2 optical switch. There is a close match between the result from the analytical model and the simulation model. The difference shown between two results is partly due to the approximation used in developing the model and to the fact that, in the simulation model, the injection rate into the switch is the departure rate from the M/M/1/∞ queues, as explained above. The accuracy achieved is similar to those of models evaluating other architectures based on delay lines [9, 10].


0

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Load

Figure 10. Blocking probability for a 2x2 optical switch

0.7

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Analytical result at load = 30% Simulation result at load = 30% Analytical result at load = 50% Simulation result at load = 50% Analytical result at load = 80% Simulation result at load = 80%



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Packet Mean Transmission Time

Load

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Figure 15. The blocking probability for different values of µ at a 30%, 50%, and 80% load


5

x 10

Figures 11, 12, 13, and 14 show the blocking probabilities of simulation and analytical models for 4x4, 8x8, 16x16, and 32x32 optical switches respectively. As explained in Section 3.4, the FDL propagation delay has to be selected relative to the mean transmission time of packets. As a result, despite the mean transmission time of the packet, the simulation and analytical models show that, for a given load, optical switch performs similarly in terms of the blocking probability as long as the length of the FDL satisfies Equation 33. For example, Figure 15 shows a constant blocking probability of a 4x4 optical switch at 80% load for different values of µ. We conducted further 1

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Analytical result at load = 30% Simulation result at load = 30% Analytical result at load = 50% Simulation result at load = 50% Analytical result at load = 80% Simulationl result at load = 80%

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[1] D. Bertsekas, R. Gallager, Data Networks, Prentice Hall, Second Edition, New Jersey, 1992 [2] F. Callegati, “Optical buffers for variable length packets ,” IEEE Communication Letters, Vol. 4, No. 9, September 2000, pp. 292-294 [3] G. Castanon, “Design-dimensioning model for transparent WDM packet-Switched irregular networks,” Journal of Lightwave Technology, Vol. 20, No. 1, January 2002, pp. 1-9 [4] S. Danielsen, B. Mikkelsen, C. Joergensen, T. Durhuus, K. Stubkjaer, “WDM packet switch architectures and analysis of the influence of tunable wavelength converters on the performance,” Journal of Lightwave Technology, Vol. 15, No. 2, February 1997, pp. 219-27

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References

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FDL Propagation Delay in bit time

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8

5

x 10

Figure 16. The blocking probability for different values of D at a 30%, 50%, and 80% load analysis and simulation of a 4x4 optical switch for different values of FDL propagation delays considering a mean packet transmission time of 100 × 103 bit times. The minimum length of FDL was three times this packet mean transmission time in order avoid contention at the buffer output ports as explained in Section 3.4. As shown in Figure 16, the results confirm a close match between the results from analytical model and those from the simulation model.

5 Conclusion In this paper, we derive an analytical model to evaluate the performance of optical switches augmented with output fiber delay lines. The model has been verified using simulations. We present results for different switching nodal degrees, different FDL propagation delays, and different mean packet transmission times. The models can be utilized to evaluate the effects of the different parameters on the performance of the optical switch. The future work include the derivation of models based on different output forwarding algorithms in order to achieve an optimal routing algorithm for optical switching.

[5] A. Fayoumi, A. Jayasumana, “Effect of optical buffering on the performance of Manhattan-Street networks,” Special issue on IP over WDM and Optical Packet Switching, Photonic Network Communications, Vol. 3, No. 1, January 2001, pp. 161-171 [6] A. Fayoumi, A. Jayasumana, J. Sauer, “Performance of Multi-hop Networks using Optical Buffering and Deflection Routing,” Proceedings of the 25th Annual IEEE Conference on Local Computer Networks (LCN), Tampa, FL, November 2000, pp. 548-555 [7] H. Harai, N. Wada, F. Kubota, W. Chujo, “Contention resolution using multi-stage fiber delay line buffer in a photonic packet switch,” Proceedings of the IEEE International Conference on Communications, ICC 2002, Vol. 5 , 2002, pp. 2843-2847 [8] D. Hunter, M. Chia, I. Andonovic, “Buffering in optical packet switches,” Journal of Lightwave Technology, Vol. 16, No. 12, December 1998, pp. 2081 -2094 [9] X. Lu, B. Mark, “A new performance model of optical burst switching with fiber delay lines,” Proceedings of the IEEE International Conference on Communications, Vol. 2, May 2003, 1365-1369 [10] X. Lu, B. Mark, “Analytical modeling of optical burst switching with fiber delay lines,” Proceedings of the 10th IEEE International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunications Systems, MASCOTS 2002, , 2002, pp. 501-506


[11] F. Masatti, P. Morin, D. Chiaroni, G. Loura, “Fiber delay lines optical buffer for ATM photonic switching applications,” Proceedings of the IEEE Infocom, Vol. 3, 1993, pp. 935-942 [12] X. Zhu, J. Kahn, “Queueing models of optical delay lines in synchronous and asynchronous optical packetswitching networks,” Optical Engineering, Vol. 42, No. 6, June 2003, pp. 1741-1748
