- Email: [email protected]
Analysis of optical packet loss rate under asymmetric traffic distribution in multi-fiber synchronous OPS switches
Optik 124 (2013) 769–772
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of optical packet loss rate under asymmetric traffic distribution in multi-fiber synchronous OPS switches Akbar Ghaffarpour Rahbar ∗ Computer Networks Research Lab, Sahand University of Technology, Sahand New Town, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 6 September 2011 Accepted 13 January 2012
Keywords: Multi-fiber OPS switch Loss rate analysis Asymmetric traffic distribution
a b s t r a c t In this article, optical packet loss rate (PLR) is analyzed for an all-optical buffer-less OPS (optical packet switched) switch located in a multi-fiber OPS network, where the multifiber OPS (MOPS) switch uses only shared-per-link wavelength converters for contention resolution purposes. The analysis is provided to compute PLR under asymmetric traffic. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Optical packet contention is a major problem in an OPS network, one promising switching technique with potential to utilize the huge bandwidth offered by all-optical networks. Researchers are interested in multi-fiber OPS (MOPS) network because this network architecture can provide lower optical packet contention compared to a single-fiber OPS network, e.g. [1,2]. Other benefits of MOPS network over single-fiber OPS network have been detailed in [1]. Different node architectures have been studied for MOPS switch in [8]. Synchronous OPS can also be a good candidate for future networks because a synchronous OPS encounters lower traffic loss compared to an asynchronous OPS network [3]. In synchronousOPS, the same-size user packets together with a header are placed inside fixed time-slots and sent to the OPS network. Different works have analytically computed optical packet loss rate (PLR) for synchronous MOPS networks. The analysis of PLR in [2] considers no wavelength conversion used inside a MOPS switch. The analysis of PLR in MOPS switch with shared-per-node wavelength conversion architecture has been performed in [4,5]. PLR analysis for a MOPS switch with shared-per-link wavelength conversion has been provided in [6].1 All the above analyses consider symmetric traffic distribution (i.e. equal traffic distribution from an input wavelength to all output links of the MOPS switch) because of the complexity of PLR analysis under asymmetric traffic
∗ Tel.: +98 412 3459367; fax: +98 412 3444322. E-mail addresses: akbar [email protected], [email protected] 1 Based on author’s investigations, two corrections should be applied to the symmetric analysis in paper [6]. The sigma ranges in Eq. (7) of [6] must be set to 1 to w-ny , instead of 0 to w. In addition, the condition y > 0 must be held in Eq. (3) of [6]. 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2012.01.011
distribution, where different output ports find different probabilities of having contentions, and therefore, different probabilities for resolving contentions by wavelength converters (WC) [5]. However, traffic distribution is not symmetric in practice in any type of switches including OPS switches, and therefore, the symmetric traffic assumption is not realistic. Since no work can be found in literature, the motivation of this paper is to provide loss rate analysis in a MOPS switch under asymmetric traffic distribution. The contribution of this paper is to provide an analysis to compute PLR in a buffer-less synchronous MOPS switch with shared-per-link wavelength conversion, where traffic distribution to output links of the MOPS switch is asymmetric. 2. Node model We consider an all-optical synchronous buffer-less MOPS switch with n × f input ports and n × f output ports (see Fig. 1), where the MOPS switch uses W wavelength-selective cross-connect nonblocking switches inside. Since each connection link has f fibers in each direction, an input link is connected to f input ports and an output link is connected to f output ports of the MOPS switch. Hence, there are n input links and n output links. There are W wavelength channels on each fiber. Each output link of the MOPS switch uses NWC shared-per-link wavelength converters among the f output ports of the output link. Multi-fiber architectures can provide a number of benefits. A multi-fiber WDM network architecture without using full wavelength converters inside optical switches is cheaper than a single-fiber architecture with using full wavelength converters [9]. In addition, devices such as dispersion compensators, used for a fiber with a higher number of wavelengths can be more expensive than the devices used for a fiber with a lower number of
770
A.G. Rahbar / Optik 124 (2013) 769–772
Fig. 2. Illustration of same-wavelength groups.
Fig. 1. Node model for a MOPS switch.
wavelengths [1,9]. As another advantage, a multi-fiber architecture with a fewer number of wavelength channels per fiber has a smaller loss rate than a single-fiber architecture with a large number of wavelength channels per fiber [2], as it will be showed in this article. The MOPS switch first receives the header of optical packets at the same time in its input ports. After arriving the relevant header for an optical packet, the MOPS switch evaluates the potential contention, then resolves the contention by using NWC shared-per-link WCs, and finally makes its W switching structures ready to switch the incoming optical packets toward their desired output ports. Note that optical packet contention happens when more than f optical packets are attempting to reach to the same output link on the same wavelength and at the same time-slot. The MOPS switch needs to resolve the contention at any output link. Synchronization is an important aspect in a synchronous OPS network. The MOPS switch must receive all the optical packets coming from all its input links at the same time. However, the arrival of optical packets at the MOPS switch may be misaligned with each other due to the chromatic dispersion and temperature variation. To achieve proper synchronization, finely calibrated set of optical delay lines [7] must be used at all input ports of the MOPS switch. 3. Optical packet loss rate analysis In this section, optical packet loss rate is computed in a MOPS switch with NWC shared-per-link WCs. In the following computations, optical packet loss rate is obtained for a given output link within a time-slot in steady state. In general, asymmetric traffic transmission to all output links is assumed in the following computations. Let the set of traffic distribution probabilities to output links of the MOPS switch be P = {p1 , p2 , p3 ,. . ., pn }, where
n
pd = 1,
d=1
and pd denotes the probability of forwarding optical packets to output link d. Clearly, pd = 1/n (where d = 1, 2. . ., n) leads to a symmetric traffic distribution. Consider a given time-slot, where there are n × f time slots on wavelength w at the input ports of the MOPS switch. Define L to be the probability of the arrival of optical packets on each wavelength channel at the input ports of the MOPS switch. Let qd = L × pd denote the probability that an optical packet arriving at an input wavelength channel is destined to output link d (where d = 1, 2. . .,
n). In the following, PLR is first computed for a given output link d. Then, PLR is calculated for the MOPS switch. Recall that output link d has f output ports, where each output port has W wavelength channels 0 to W−1 . A set of f the same wavelength channels at output link d is referred to as the samewavelength group from now on. For example at f = 3 and W = 4, Fig. 2 shows the same-wavelength groups 2 and 4 . Therefore, there are W the same-wavelength groups at output link d. In the following analysis, W the same-wavelength groups can be divided into two categories as: 1) j the same-wavelength groups on which no optical packets have arrived on any of the wavelengths of each group; and 2) u = W − j the same-wavelength groups on which at least one optical packet has arrived on one wavelength channel of each group. For simplicity in our computations, the wavelength channels in the u groups are assigned alias names from 1 to u . Now define X as a random variable to be the number of the samewavelength groups on which no optical packets have arrived on any of the wavelengths of each group. Let q0,d be the probability of having no optical packets arriving on the wavelength channels of a given same-wavelength group at output link d. Considering the fact that there are n × f same wavelength channels at the input ports of the MOPS switch, we can compute q0,d = (1 − qd )n×f . Thus, the probability of having X = j on output link d is computed by:
Prob.{X = j}d =
W j
(q0,d )j (1 − q0,d )W −j ,
(1)
where j ∈ {0, 1, 2,. . .,W−1}. In addition, define Yi as a random variable to be the number of optical packets (at least one optical packet) arriving on the wavelength channels of the same-wavelength group i (i.e. the group of f wavelengths i ). The probability of having y > 0 optical packets on f wavelengths w on output link d is computed by:
Prob.{Yi = y}d =
n×f y
(qd )y (1 − qd )n×f −y 1 − q0,d
,
if y > 0.
(2)
The probability of having y1 > 0 optical packets arriving on f wavelengths of the same-wavelength group 1, y2 > 0 optical packets on f wavelengths of the same-wavelength group 2, till yu > 0 optical packets on f wavelengths of the same-wavelength group u can be computed by: u
Prob.{Yk = yk }d .
k=1
Of yk optical packets arriving on f wavelengths of the samewavelength group k, min (f, yk ) of them can ube easily switched on f wavelengths k . In other words, of y optical packk=1 k
u
ets arriving at output link d, min(f, yk ) optical packets k=1 can be switched. In addition, some of the remaining optical
A.G. Rahbar / Optik 124 (2013) 769–772
packets can then be switched after using NWC shared-per-link WCs located at output link d. First, recall that there are f × j wavelength channels on which no optical packets have arrived at output d. In addition, among the u same-wavelength groups, there link u are max(0, f − yk ) wavelength channels on which no optik=1 cal packets have arrived. Therefore, there are ns = min(NWC , f × j + u max(0, f − yk )) optical packets in total that can be switched k=1 using NWC shared-per-link WCs. Thus, for a given state (j, y1 , y2 ,. . .,yu ), the number of optical packets that cannot be switched (i.e. lost) is computed by: ˛j, y1 , y2 , y3 ,...,yu = max
u
(yk − min(f, yk ))
− ns , 0
.
k=1
771
develop the simulation models in this section, and obtained 95% level of confidence interval, at worst within 1% of the mean values shown. Fig. 3 shows PLR for the MOPS switch under different scenarios with L ∈ {0.1,0.2,0.3,. . .,0.9}, NWC ∈ {0,2,4,6}, f ∈ {1,2,4}, and f × W = 8. According to Fig. 3, analysis results match well with simulation results. By increasing f or NWC or both, PLR decreases. However, there is no significant reduction on PLR at NWC = 6. At L = 0.9, the smallest PLR is 0.296 at f = 2 and NWC = 6. This high PLR is due to the asymmetric traffic distribution in which many packets are lost, especially at output links 3, 5, and 6, respectively, with traffic probability distributions of 0.29, 0.19, and 0.22. Fig. 4 shows PLR for the MOPS switch with symmetric traffic distribution at f = 2, n = 8, NWC ∈ {0,2,4}, where our analysis results
The average number of lost optical packets at output link d for a given j can then be computed by:
u
Zj,d =
Prob.{Yk = yk }d × ˛j,y1 ,y2 ,...,yu
1 ≤ y1 ≤ n × f ... 1 ≤ yu ≤ n × f
,
k=1
where the upper limit n × f for yk (k = 1, 2,. . ., u) denotes the total number of optical packets that could be destined to output link d on f wavelengths k in the worst case. Hence, at output link d, the average number of lost optical packets on all the same-wavelength groups can be computed by:
W −1
nloss (d) =
(Prob.{X = j}d × Zj,d ),
j=0
where Prob.{X = j}d is computed from Eq. (1), and Prob.{Yk = yk }d is given in Eq. (2). Of n × f × W × L optical packets arriving at n input links of the MOPS switch, on average ndlv (d) = n × f × W × qd optical packets are delivered to output link d. Hence, the average PLR at output link d can be given by: Loss(d, n, f, W, qd ) =
nloss (d) ndlv (d)
1 (Prob.{X = j}d × Zj,d ). n × f × W × qd W −1
=
j=0
Finally, average PLR in the MOPS switch is computed by total number of lost optical packets on all output links divided by total number of optical packets forwarded to all output links of the MOPS switch, i.e.
n
PLR(n, f, W ) =
d=1
n =
n × f × W × qd × Loss(d, n, f, W, qd )
q d=1 d
n
d=1
n × f × W × qd
× Loss(d, n, f, W, qd ) L
4. Performance analysis Based on the MOPS model described in Section 2, consider a MOPS switch with n = 8 connected to eight edge switches, where the asymmetric traffic distribution probabilities to output links of the MOPS switch is P = {0.10, 0.03, 0.29, 0.01, 0.19, 0.22, 0.11, 0.05}. User packets arrive at each edge switch according to a Poisson process and saved in electronic buffers, and then sent as optical packets at time-slot boundaries to the MOPS switch. We have used C to
Fig. 3. PLR under different MOPS switches with asymmetric traffic. (a) f = 1, (b) f = 2, (c) f = 4.
772
A.G. Rahbar / Optik 124 (2013) 769–772 1.0E+00
5. Conclusion
Packet Loss Rate (PLR)
1.0E-01
An analysis has been provided to compute optical packet loss rate in a multi-fiber OPS switch with shared-per-link wavelength conversion under asymmetric traffic distribution. As proved, the analysis can accurately obtain packet loss rate under both asymmetric and symmetric traffic distributions. The performance evaluation results show that the symmetric traffic can significantly reduce PLR compared to the asymmetric traffic case.
1.0E-02 Ana1(Nwc=0) 1.0E-03
Ana1(Nwc=2) Ana1(Nwc=4)
1.0E-04
Ana2(Nwc=0) Ana2(Nwc=2) Ana2(Nwc=4)
1.0E-05
Sim(Nwc=0) Sim(Nwc=2)
1.0E-06
Sim(Nwc=4)
References
1.0E-07 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Load(L)
Fig. 4. PLR under symmetric traffic distribution.
(showed with Ana1) well match with the analysis results of [6] (showed with Ana2). The analysis results also follow well the simulation results. As expected, we have the lowest PLR under symmetric traffic distribution when pk = 1/8 for k ∈ {1,2,. . .n} compared to the asymmetric traffic distribution. For example, according to our simulation and analysis results at L = 0.9, the smallest PLR of 0.092 is obtained under symmetric traffic when f = 2 and NWC = 6 (comparable to PLR of 0.296 under asymmetric traffic). The worst case analysis (i.e. the whole traffic is destined only to one output link of the MOPS switch) shows a PLR of 0.821 at f = 2 and NWC = 6.
[1] O. Gerstel, H. Raza, Merits of low-density WDM line systems for long-haul networks, IEEE J. Lightw. Technol. 21 (November (11)) (2003) 2470–2475. [2] Y. Li, G. Xiao, H. Ghafouri-Shiraz, On the benefits of multifiber optical packet switch, Micrw. Opt. Technol. Lett. 43 (December (5)) (2004) 376–378. [3] C. Papazoglou, G. Papadimitriou, A. Pomportsis, Design alternatives for opticalpacket-interconnection network architectures, J. Opt. Netw. 3 (November (11)) (2004) 810–825. [4] V. Eramo, An analytical model for TOWC dimensioning in a multifiber opticalpacket switch, IEEE J. Lightw. Technol. 24 (12) (2006) 4799–4810. [5] L. Yi, X. Gaoxi, H. Ghafouri-Shiraz, Performance evaluation of multi-fiber optical packet switches, Comput. Netw. 51 (March (4)) (2007) 995–1012. [6] A.G. Fayoumi, F.A. Al-Zahrani, A.A. Habiballa, A.P. Jayasumana, Performance analysis of multi-fiber synchronous photonic share-per-link packet switches, in: Proceedings of IEEE LCN 2005, November, Sydney, 2005. [7] B. Bostica, M. Burzio, P. Gambini, L. Zucchelli, Synchronization issues in optical packet switched networks, in: G. Prati (Ed.), Photonic Networks, Springer Verlag, London, UK, 1997, pp. 362–376. [8] L. Yi, X. Gaoxi, H. Ghafouri-Shiraz, On the performance of different node configurations in multi-fiber optical packet-switched networks, Photon. Netw. Commun. 14 (August (1)) (2007). [9] A.K. Somani, M. Mina, L. Li, On trading wavelengths with fibers: a cost performance based study, IEEE/ACM Trans. Network 12 (5) (2004) 944–951.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of optical packet loss rate under asymmetric traffic distribution in multi-fiber synchronous OPS switches Akbar Ghaffarpour Rahbar ∗ Computer Networks Research Lab, Sahand University of Technology, Sahand New Town, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 6 September 2011 Accepted 13 January 2012
Keywords: Multi-fiber OPS switch Loss rate analysis Asymmetric traffic distribution
a b s t r a c t In this article, optical packet loss rate (PLR) is analyzed for an all-optical buffer-less OPS (optical packet switched) switch located in a multi-fiber OPS network, where the multifiber OPS (MOPS) switch uses only shared-per-link wavelength converters for contention resolution purposes. The analysis is provided to compute PLR under asymmetric traffic. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Optical packet contention is a major problem in an OPS network, one promising switching technique with potential to utilize the huge bandwidth offered by all-optical networks. Researchers are interested in multi-fiber OPS (MOPS) network because this network architecture can provide lower optical packet contention compared to a single-fiber OPS network, e.g. [1,2]. Other benefits of MOPS network over single-fiber OPS network have been detailed in [1]. Different node architectures have been studied for MOPS switch in [8]. Synchronous OPS can also be a good candidate for future networks because a synchronous OPS encounters lower traffic loss compared to an asynchronous OPS network [3]. In synchronousOPS, the same-size user packets together with a header are placed inside fixed time-slots and sent to the OPS network. Different works have analytically computed optical packet loss rate (PLR) for synchronous MOPS networks. The analysis of PLR in [2] considers no wavelength conversion used inside a MOPS switch. The analysis of PLR in MOPS switch with shared-per-node wavelength conversion architecture has been performed in [4,5]. PLR analysis for a MOPS switch with shared-per-link wavelength conversion has been provided in [6].1 All the above analyses consider symmetric traffic distribution (i.e. equal traffic distribution from an input wavelength to all output links of the MOPS switch) because of the complexity of PLR analysis under asymmetric traffic
∗ Tel.: +98 412 3459367; fax: +98 412 3444322. E-mail addresses: akbar [email protected], [email protected] 1 Based on author’s investigations, two corrections should be applied to the symmetric analysis in paper [6]. The sigma ranges in Eq. (7) of [6] must be set to 1 to w-ny , instead of 0 to w. In addition, the condition y > 0 must be held in Eq. (3) of [6]. 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2012.01.011
distribution, where different output ports find different probabilities of having contentions, and therefore, different probabilities for resolving contentions by wavelength converters (WC) [5]. However, traffic distribution is not symmetric in practice in any type of switches including OPS switches, and therefore, the symmetric traffic assumption is not realistic. Since no work can be found in literature, the motivation of this paper is to provide loss rate analysis in a MOPS switch under asymmetric traffic distribution. The contribution of this paper is to provide an analysis to compute PLR in a buffer-less synchronous MOPS switch with shared-per-link wavelength conversion, where traffic distribution to output links of the MOPS switch is asymmetric. 2. Node model We consider an all-optical synchronous buffer-less MOPS switch with n × f input ports and n × f output ports (see Fig. 1), where the MOPS switch uses W wavelength-selective cross-connect nonblocking switches inside. Since each connection link has f fibers in each direction, an input link is connected to f input ports and an output link is connected to f output ports of the MOPS switch. Hence, there are n input links and n output links. There are W wavelength channels on each fiber. Each output link of the MOPS switch uses NWC shared-per-link wavelength converters among the f output ports of the output link. Multi-fiber architectures can provide a number of benefits. A multi-fiber WDM network architecture without using full wavelength converters inside optical switches is cheaper than a single-fiber architecture with using full wavelength converters [9]. In addition, devices such as dispersion compensators, used for a fiber with a higher number of wavelengths can be more expensive than the devices used for a fiber with a lower number of
770
A.G. Rahbar / Optik 124 (2013) 769–772
Fig. 2. Illustration of same-wavelength groups.
Fig. 1. Node model for a MOPS switch.
wavelengths [1,9]. As another advantage, a multi-fiber architecture with a fewer number of wavelength channels per fiber has a smaller loss rate than a single-fiber architecture with a large number of wavelength channels per fiber [2], as it will be showed in this article. The MOPS switch first receives the header of optical packets at the same time in its input ports. After arriving the relevant header for an optical packet, the MOPS switch evaluates the potential contention, then resolves the contention by using NWC shared-per-link WCs, and finally makes its W switching structures ready to switch the incoming optical packets toward their desired output ports. Note that optical packet contention happens when more than f optical packets are attempting to reach to the same output link on the same wavelength and at the same time-slot. The MOPS switch needs to resolve the contention at any output link. Synchronization is an important aspect in a synchronous OPS network. The MOPS switch must receive all the optical packets coming from all its input links at the same time. However, the arrival of optical packets at the MOPS switch may be misaligned with each other due to the chromatic dispersion and temperature variation. To achieve proper synchronization, finely calibrated set of optical delay lines [7] must be used at all input ports of the MOPS switch. 3. Optical packet loss rate analysis In this section, optical packet loss rate is computed in a MOPS switch with NWC shared-per-link WCs. In the following computations, optical packet loss rate is obtained for a given output link within a time-slot in steady state. In general, asymmetric traffic transmission to all output links is assumed in the following computations. Let the set of traffic distribution probabilities to output links of the MOPS switch be P = {p1 , p2 , p3 ,. . ., pn }, where
n
pd = 1,
d=1
and pd denotes the probability of forwarding optical packets to output link d. Clearly, pd = 1/n (where d = 1, 2. . ., n) leads to a symmetric traffic distribution. Consider a given time-slot, where there are n × f time slots on wavelength w at the input ports of the MOPS switch. Define L to be the probability of the arrival of optical packets on each wavelength channel at the input ports of the MOPS switch. Let qd = L × pd denote the probability that an optical packet arriving at an input wavelength channel is destined to output link d (where d = 1, 2. . .,
n). In the following, PLR is first computed for a given output link d. Then, PLR is calculated for the MOPS switch. Recall that output link d has f output ports, where each output port has W wavelength channels 0 to W−1 . A set of f the same wavelength channels at output link d is referred to as the samewavelength group from now on. For example at f = 3 and W = 4, Fig. 2 shows the same-wavelength groups 2 and 4 . Therefore, there are W the same-wavelength groups at output link d. In the following analysis, W the same-wavelength groups can be divided into two categories as: 1) j the same-wavelength groups on which no optical packets have arrived on any of the wavelengths of each group; and 2) u = W − j the same-wavelength groups on which at least one optical packet has arrived on one wavelength channel of each group. For simplicity in our computations, the wavelength channels in the u groups are assigned alias names from 1 to u . Now define X as a random variable to be the number of the samewavelength groups on which no optical packets have arrived on any of the wavelengths of each group. Let q0,d be the probability of having no optical packets arriving on the wavelength channels of a given same-wavelength group at output link d. Considering the fact that there are n × f same wavelength channels at the input ports of the MOPS switch, we can compute q0,d = (1 − qd )n×f . Thus, the probability of having X = j on output link d is computed by:
Prob.{X = j}d =
W j
(q0,d )j (1 − q0,d )W −j ,
(1)
where j ∈ {0, 1, 2,. . .,W−1}. In addition, define Yi as a random variable to be the number of optical packets (at least one optical packet) arriving on the wavelength channels of the same-wavelength group i (i.e. the group of f wavelengths i ). The probability of having y > 0 optical packets on f wavelengths w on output link d is computed by:
Prob.{Yi = y}d =
n×f y
(qd )y (1 − qd )n×f −y 1 − q0,d
,
if y > 0.
(2)
The probability of having y1 > 0 optical packets arriving on f wavelengths of the same-wavelength group 1, y2 > 0 optical packets on f wavelengths of the same-wavelength group 2, till yu > 0 optical packets on f wavelengths of the same-wavelength group u can be computed by: u
Prob.{Yk = yk }d .
k=1
Of yk optical packets arriving on f wavelengths of the samewavelength group k, min (f, yk ) of them can ube easily switched on f wavelengths k . In other words, of y optical packk=1 k
u
ets arriving at output link d, min(f, yk ) optical packets k=1 can be switched. In addition, some of the remaining optical
A.G. Rahbar / Optik 124 (2013) 769–772
packets can then be switched after using NWC shared-per-link WCs located at output link d. First, recall that there are f × j wavelength channels on which no optical packets have arrived at output d. In addition, among the u same-wavelength groups, there link u are max(0, f − yk ) wavelength channels on which no optik=1 cal packets have arrived. Therefore, there are ns = min(NWC , f × j + u max(0, f − yk )) optical packets in total that can be switched k=1 using NWC shared-per-link WCs. Thus, for a given state (j, y1 , y2 ,. . .,yu ), the number of optical packets that cannot be switched (i.e. lost) is computed by: ˛j, y1 , y2 , y3 ,...,yu = max
u
(yk − min(f, yk ))
− ns , 0
.
k=1
771
develop the simulation models in this section, and obtained 95% level of confidence interval, at worst within 1% of the mean values shown. Fig. 3 shows PLR for the MOPS switch under different scenarios with L ∈ {0.1,0.2,0.3,. . .,0.9}, NWC ∈ {0,2,4,6}, f ∈ {1,2,4}, and f × W = 8. According to Fig. 3, analysis results match well with simulation results. By increasing f or NWC or both, PLR decreases. However, there is no significant reduction on PLR at NWC = 6. At L = 0.9, the smallest PLR is 0.296 at f = 2 and NWC = 6. This high PLR is due to the asymmetric traffic distribution in which many packets are lost, especially at output links 3, 5, and 6, respectively, with traffic probability distributions of 0.29, 0.19, and 0.22. Fig. 4 shows PLR for the MOPS switch with symmetric traffic distribution at f = 2, n = 8, NWC ∈ {0,2,4}, where our analysis results
The average number of lost optical packets at output link d for a given j can then be computed by:
u
Zj,d =
Prob.{Yk = yk }d × ˛j,y1 ,y2 ,...,yu
1 ≤ y1 ≤ n × f ... 1 ≤ yu ≤ n × f
,
k=1
where the upper limit n × f for yk (k = 1, 2,. . ., u) denotes the total number of optical packets that could be destined to output link d on f wavelengths k in the worst case. Hence, at output link d, the average number of lost optical packets on all the same-wavelength groups can be computed by:
W −1
nloss (d) =
(Prob.{X = j}d × Zj,d ),
j=0
where Prob.{X = j}d is computed from Eq. (1), and Prob.{Yk = yk }d is given in Eq. (2). Of n × f × W × L optical packets arriving at n input links of the MOPS switch, on average ndlv (d) = n × f × W × qd optical packets are delivered to output link d. Hence, the average PLR at output link d can be given by: Loss(d, n, f, W, qd ) =
nloss (d) ndlv (d)
1 (Prob.{X = j}d × Zj,d ). n × f × W × qd W −1
=
j=0
Finally, average PLR in the MOPS switch is computed by total number of lost optical packets on all output links divided by total number of optical packets forwarded to all output links of the MOPS switch, i.e.
n
PLR(n, f, W ) =
d=1
n =
n × f × W × qd × Loss(d, n, f, W, qd )
q d=1 d
n
d=1
n × f × W × qd
× Loss(d, n, f, W, qd ) L
4. Performance analysis Based on the MOPS model described in Section 2, consider a MOPS switch with n = 8 connected to eight edge switches, where the asymmetric traffic distribution probabilities to output links of the MOPS switch is P = {0.10, 0.03, 0.29, 0.01, 0.19, 0.22, 0.11, 0.05}. User packets arrive at each edge switch according to a Poisson process and saved in electronic buffers, and then sent as optical packets at time-slot boundaries to the MOPS switch. We have used C to
Fig. 3. PLR under different MOPS switches with asymmetric traffic. (a) f = 1, (b) f = 2, (c) f = 4.
772
A.G. Rahbar / Optik 124 (2013) 769–772 1.0E+00
5. Conclusion
Packet Loss Rate (PLR)
1.0E-01
An analysis has been provided to compute optical packet loss rate in a multi-fiber OPS switch with shared-per-link wavelength conversion under asymmetric traffic distribution. As proved, the analysis can accurately obtain packet loss rate under both asymmetric and symmetric traffic distributions. The performance evaluation results show that the symmetric traffic can significantly reduce PLR compared to the asymmetric traffic case.
1.0E-02 Ana1(Nwc=0) 1.0E-03
Ana1(Nwc=2) Ana1(Nwc=4)
1.0E-04
Ana2(Nwc=0) Ana2(Nwc=2) Ana2(Nwc=4)
1.0E-05
Sim(Nwc=0) Sim(Nwc=2)
1.0E-06
Sim(Nwc=4)
References
1.0E-07 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Load(L)
Fig. 4. PLR under symmetric traffic distribution.
(showed with Ana1) well match with the analysis results of [6] (showed with Ana2). The analysis results also follow well the simulation results. As expected, we have the lowest PLR under symmetric traffic distribution when pk = 1/8 for k ∈ {1,2,. . .n} compared to the asymmetric traffic distribution. For example, according to our simulation and analysis results at L = 0.9, the smallest PLR of 0.092 is obtained under symmetric traffic when f = 2 and NWC = 6 (comparable to PLR of 0.296 under asymmetric traffic). The worst case analysis (i.e. the whole traffic is destined only to one output link of the MOPS switch) shows a PLR of 0.821 at f = 2 and NWC = 6.
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