Abstract

Service differentiation is a crucial issue in the next -generation Optical Packet Switched networks. In this paper we examine how bursty traffic influences the performance of a service differentiated Optical Packet Switched network. By using time -continuous Markov chains, we derive explicit results for the packet loss rates in the case of a bursty hyper-exponential arrival process. Results indicate that the performance is degraded as the burstiness of the arrival process increases.

©2004 Optical Society of America

1. Introduction

In order to increase available capacity in future core networks, Wavelength Division Multiplexing (WDM) has emerged as a viable technology [1]. Among the different WDM switching technologies proposed in recent literature, Optical Packet Switching (OPS) has emerged as the most promising, at least in the long run, due to the benefits from statistical multiplexing and adaptability to changes in the network infrastructure and traffic pattern [1].

Today’s Internet provides only the best-effort service, where each packet is getting the same treatment and there are no guarantees to the end-to-end delay or the packet loss rate (PLR) [2]. However, the increasing number of real-time applications and interactive Internet applications demand a stricter Quality of Service (QoS) than the current best-effort service can offer. Although the best-effort service is not suited to carry real-time or interactive applications, it is well suited for web browsing, e-mail and other relaxed services. Hence, in order to give different applications their needed level of QoS and to utilize network resources optimally, service differentiation should be present in future core networks [2].

Existing service differentiation schemes for traditional store-and-forward networks mandate the use of electronic buffers to isolate the different traffic classes, i.e. by the use of active queue management algorithms [3]. Here, all packet arrivals to a switch are stored in the buffer and managed according to an active queue management algorithm, which isolate the different service classes by marking, dropping or reordering packets in the queue [3]. However, as pointed out in [4], such schemes are not suitable for the WDM layer. First, electronic buffering necessitates the use of optical-electrical converters, which results inn a significant increase in the cost of the switch. Second, although optical buffering can be realized by utilizing Fibre Delay Lines (FDLs), they can only give limited buffering capabilities compared to electronic buffering, because data is delayed by traversing a fixed length optical fibre (i.e. the FDL is not a Random Access Memory). Hence, performing active queue management on FDL buffers is not an easy task, as complex management is required. Also, using FDLs makes the switches bulky and costly, especially when large amounts of data need to be buffered. Hence, as pointed out in [4], a viable service differentiation scheme for future optical core networks is to utilize bufferless methods to isolate the different service classes. One such bufferless method is the Wavelength Allocation algorithm (WA) [5,6].

According to [7], the Internet traffic shows a self-similar pattern, which means that the traffic is equally bursty when viewed over different time scales. Regarding arrivals to an optical switch, this results in periods with many arrivals and periods with little or no arrivals. The widely used Poisson process does not model such traffic very well, which means that other arrival processes should be considered, e.g. the hyper-exponential arrival process [8,9].

In earlier works we have presented an analytical model of the WA in the case of a non-bursty Poisson arrival process [5]. This paper presents an analytical model of the WA in the case of a bursty hyper-exponential arrival process and further considers the impacts of bursty traffic on the network performance. The rest of the paper is organized as follows. Section 2 presents the system model. Section 3 presents the WA, accompanied by analytical models. Section 4 presents simulation and analytical results, while Section 5 concludes the paper.

2. System model

We consider a network with two service classes, where service class 0 is given higher priority than service class 1 (the use of two service classes only in the core networks is argued by the authors of [10]). Since the switches in the network have no buffers for contention resolution, the aggregated delay will be very small compared to the delay from the access network, and can thus be ignored [10]. However, a major concern is such networks is packet losses, which is caused by external blocking (i.e. contentions at an output port). Hence, in this paper we focus on the PLR as the sole QoS parameter, which means that the different service classes will be isolated from each other based on different PLRs. When a packet arrives at the switch, the packet header is extracted and processed electronically by the control module. While the header is processed, the packet payload is buffered using FDL processing buffers (as seen in Fig. 1(a) [11]. Based on the destination information extracted from the packet header, the control module decides which output fibre the packet is switched to and configures the switch fabric accordingly. For the rest of the paper we focus on a single optical core switch and assume the following:

• The optical switch operates in asynchronous mode.

• The switch utilizes full wavelength conversion.

• The switch has F input- and output fibres, as shown in Fig. 1(a).

• By utilizing WDM, each fibre provides N wavelengths (channels) to transport data, each wavelength with a specified capacity C bps.

• We assume a uniform traffic pattern, which means that we have an equal load on every output fibre and can restrict our study to consider the PLR on a single output fibre, marked red in Fig. 1(a).

• Packets arrive to the output fibre according to either a Poisson process (see Section 3.1) or a 2-stage hyper-exponential process. The hyper-exponential arrival process is depicted in Fig. 1(b) and will be discussed in Section 3.2.

• The relative share of class 0 traffic is S0, while the relative share of class 1 traffic is S1.

• The packet lengths for both class 0 and class 1 traffic are exponential independent identical distributed (i.i.d.) with mean service time µ-1.

• The system load is defined as A=l/(N·m), where l is the offered traffic intensity.

• The effects of the switching time are ignored.

 figure: Fig. 1.

Fig. 1. An optical packet switch (a) and the 2-stage hyper-exponential arrival process with priorities (b).

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3. The Wavelength Allocation algorithm (WA)

In order to isolate the two service classes, we utilize the WA. In the WA, a total of N available wavelengths at the output fibre are divided into two pools according to priority, i.e. a class 0 pool with W0 wavelengths and a class 1 pool with W1 wavelengths [6]. Incoming packets to the output fibre can access these wavelengths only if they have the necessary priority level. This means that class 1 traffic can access wavelengths only from the class 1 pool (total of n=W1 wavelengths), while class 0 traffic can access wavelengths from both pools (total N=W1+W0 wavelengths). Incoming packets which fails to acquire a wavelength is immediately dropped at the node. By adjusting the variable n, we achieve any desired level of the PLR for class 0 traffic.

3.1 Poisson arrival process

In the case of Poisson arrival process, the output fibre is modeled according to a modified M/M/N/N queuing model. Explicit results for the PLRs have been derived in earlier works (see [5]), and will not be detailed any further in this paper.

3.2 Two-stage hyper-exponential arrival process

We now turn our attention to a 2-stage hyper-exponential (H2) arrival process with parameters a, θ0 and θ1, as seen in Fig. 1(b) [8]. Due to the memory-less property of the exponential distribution, the H2 arrival process is not long range dependent, but still bursty. We see from Fig. 1(b) that there is a probability a that the intensity of the next inter-arrival time is θ0 (marked red) and a probability 1-a that the intensity of the next inter-arrival time is θ1 (marked dotted red). We denote these arrivals as type 0 and type 1 arrivals, respectively. Furthermore, regardless of the arrival type, we see that the probability for a class 0 (blue) and class 1 (green) arrival is S0 and S1, respectively. Hence, the intensity of a type k arrival with priority l(k,l=0,1) is φl,k. We also see that qi=j 0i,+j 1,i(i=0,1). The burstiness of this process can be characterized according to the squared coefficient of variation, given as [9,12]:

c2=(σm1)2=m2m121=ε1=2a+k2ak2(a+kak)21

Here, m1 is the i’th moment of the H2 process, s is the variance, e is Palms form factor [12] and k=q 0/q 1.

 figure: Fig. 2.

Fig. 2. Time continuous Markov chain of the WA with H2-arrival process.

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Figure 2 shows a time-continuous Markov chain of the WA with H2-arrival process. The states Pi and Ri (0≤iN) denote the number of packets currently in transmission at the output fibre, e.g. in state Pi, i packets are currently in transmission. We define pi and ri as the state probabilities for being in state Pi and Ri, respectively. In the Pi states (not dotted), the next arrival is a type 0 arrival, while in the Ri states (dotted) the next arrival is a type 1 arrival. Furthermore, regardless of state, we see from Fig. 2 that the probability to end in state Pi (0≤iN) equals a, i.e. the probability that the next arrival is a type 0 arrival. On the opposite, the probability to end up in state Ri (0≤iN) is 1-a, which is in accordance with the model in Fig. 1(b). Furthermore, in the blue and black states, only class 0 traffic is accepted. Hence, in the case of a class 1 arrival, the number of packets currently in transmission does not change, although the arrival type of the next arrival may change. In the black states, both class 0 and class 1 traffic is discarded. From Fig. 2 we can easily derive the node equations as follows:

p0θ0=p1μ,r0θ1=r1μ
pi(iμ+θ0)=a(pi1θ0+ri1θi)+pi+1(i+1)μ1in1
pn(nμ+θ0)=a(pn1θ0+pnφ1,0+rn1θ1+rnφ1,1)+pn+1(n+1)μ
pi(iμ+θ0)=a(pi1φ0,0+piφ1,0+ri1φ0,1+riφ1,1)
+pi+1(i+1)μn+1iN1
PNθ0=a(pN1φ0,0+pNθ0+rN1φ0,1+rNθ1)
ri(iμ+θ1)=(1a)(ri1θ1+pi1θ0)+ri+1(i+1)μ1in1
rn(nμ+θ1)=(1a)(rn1θ1+rnφ1,1+pn1θ0+rnφ1,0)+rn+1(n+1)μ
ri(iμ+θ1)=(1a)(ri1φ0,1+riφ1,1+pi1φ0,0+piφ1,0)
+ri+1(i+1)μn+1iN1
rNθ1=(1a)(rN1φ0,1+rNθ1+PN1φ0,0+pNθ0)
i=0Npi+i=0Nri=1

The PLR for class 0 traffic equals the number of lost class 0 arrivals divided by the total number of class 0 arrivals. We know that class 0 arrivals are discarded only in states PN and RN (black states). In state PN the arrival intensity for class 0 traffic is φ 0,0, while in state RN the arrival intensity is φ 0,1. Class 1 traffic is lost in the states {Pn,…,PN} and {Rn,…,RN} (blue and black states). This leads us to the PLR for class PH20 and class PH21 traffic in Eq. (12). At last, the average PLR is given in Eq. (13).

PH20=pNφ0,0+rNφ0,1φ0,0i=0Npi+φ0,1i=0Nri,PH21=φ1,0i=nNpi+φ1,1i=nNriφ1,0i=0Npi+φ1,1i=0Nri
PH2av=pNθ0+rNθ1+φ1,0i=nN1pi+φ1,1i=nN1riθ0i=0Npi+θ1i=0Nri

4. Results

For the rest of the paper, unless stated differently, we have used N=4 wavelengths, C=2,5 Gbps, A=0,4, S0=0,2 and n=3. The mean packet length equals 472 bytes, which means that the mean duration of a packet is µ-1=1,5104 μs. Several simulation runs have been performed until the desired level of confidence has been achieved. First, simulations have been performed in order to verify the analytical model presented in Section 3.2. Regarding the arrival process, we have used a=0,2 and θ0=50θ1. The results can be seen in Table 1.

Tables Icon

Table 1. Simulation and analytical results.

From Table 1 we see that the simulation results matches the analytical results closely, which means that the proposed analytical model is very accurate. This is an improvement compared to the analytical models proposed in [9], which are very inaccurate, especially in the case of low system loads. Further in the paper, in order to determine the variables for the H2 arrival process, we have utilized the balanced mean fit distribution and the gamma fit distribution [9]. The parameters for the balanced mean fit distribution are seen in Eq. (14), and the parameters for the gamma fit distribution are seen in Eq. (15).

a=(1+(c21)(c2+1))2,θ0=2aλ,θ1=2λ(1a)
θ0=2λ(1+(c212)(c2+1)),θ1=4λθ0,a=(θ0(θ111))(θ1θ0)
 figure: Fig. 3.

Fig. 3. The PLR as a function of the system load (a) and the average PLR as a function of the coefficient of variation (b).

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Figure 3(a) shows the PLR as a function of the system load for different values of the squared coefficient of variation (c2). Here, we have utilized the balanced mean fit distribution. First, regardless of traffic -class, we see that the PLR increases when c2 increases. This is expected since an increase in c2 means that the traffic is increasingly bursty. Furthermore, we see that the difference in the PLR decreases as a function of an increasing system load. This is because at high loads, packet losses are dominated by congestion caused by high system load and not bursty traffic. Figure 3(b) shows the PLR as a function of the coefficient of variation for the balanced mean fit and the gamma fit distribution. First, we see that the balanced mean fit distribution show better performance compared to the gamma fit distribution. Second, we see that the increase in the PLR decreases when the coefficient of variation increases, which means that increasing the burstiness has greater impact on the PLR when the arrival process is initially less bursty.

5. Conclusion

This paper presented an analytical model of a service differentiated bufferless OPS network with a bursty hyper-exponential arrival process. Although not as general as the models proposed in [9], the proposed model is more accurate, especially at low system loads (i.e. A <0,8). Results obtained from the model indicate that the PLR increases as the burstiness of the arrival process increases, which is in accordance with results from previous research [4,9].

References and links

1. M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001). [CrossRef]  

2. X. Xiao and L. M. Ni, “Internet QoS: A Big Picture,” IEEE Network 13, 8–18 (1999). [CrossRef]  

3. B. Wydrowski and M. Zukerman, “QoS in Best -Effort Networks,” IEEE Communications Magazine 40, 44–49 (2002). [CrossRef]  

4. M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000). [CrossRef]  

5. H. Øverby and N. Stol, “A Teletraffic Model for Service Differentiation in OPS networks,” in Proceedings of IEEE 8th Optoelectronics and Communications Conference (Institute of Electrical and Electronic Engineers, 2003), pp. 677–678.

6. Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002). [CrossRef]  

7. W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994). [CrossRef]  

8. A.L. Schmid et al., “Impact of VC merging on buffer requirements in ATM networks,” Proceedings of Eighth International Conference on High Performance Networking, 203–217 (1998).

9. Deepankar Medhi, “Some Results for Renewal Arrival to a Communication Link,” Probability Models and Statistics: A J. Medhi Festschrift, 85–107 (1996).

10. S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).

11. E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

12. Leonard Kleinrock, Queueing Systems Volume I: Theory (John Wiley & Sons, 1975).

References

  • View by:

  1. M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
    [Crossref]
  2. X. Xiao and L. M. Ni, “Internet QoS: A Big Picture,” IEEE Network 13, 8–18 (1999).
    [Crossref]
  3. B. Wydrowski and M. Zukerman, “QoS in Best -Effort Networks,” IEEE Communications Magazine 40, 44–49 (2002).
    [Crossref]
  4. M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000).
    [Crossref]
  5. H. Øverby and N. Stol, “A Teletraffic Model for Service Differentiation in OPS networks,” in Proceedings of IEEE 8th Optoelectronics and Communications Conference (Institute of Electrical and Electronic Engineers, 2003), pp. 677–678.
  6. Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
    [Crossref]
  7. W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
    [Crossref]
  8. A.L. Schmid et al., “Impact of VC merging on buffer requirements in ATM networks,” Proceedings of Eighth International Conference on High Performance Networking, 203–217 (1998).
  9. Deepankar Medhi, “Some Results for Renewal Arrival to a Communication Link,” Probability Models and Statistics: A J. Medhi Festschrift, 85–107 (1996).
  10. S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).
  11. E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).
  12. Leonard Kleinrock, Queueing Systems Volume I: Theory (John Wiley & Sons, 1975).

2002 (3)

B. Wydrowski and M. Zukerman, “QoS in Best -Effort Networks,” IEEE Communications Magazine 40, 44–49 (2002).
[Crossref]

Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
[Crossref]

S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).

2001 (1)

M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
[Crossref]

2000 (1)

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000).
[Crossref]

1999 (1)

X. Xiao and L. M. Ni, “Internet QoS: A Big Picture,” IEEE Network 13, 8–18 (1999).
[Crossref]

1994 (1)

W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
[Crossref]

Ackaert, A.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Alnuweiri, Hussein

Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
[Crossref]

Bjørnstad, S.

S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).

Breusegem, E. V.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Cheyns, J.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Demeester, P.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Dixit, S.

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000).
[Crossref]

Hjelme, D. R.

S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).

Hunter, D.

M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
[Crossref]

Kaheel, Ayman

Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
[Crossref]

Khattab, Tamer

Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
[Crossref]

Kleinrock, Leonard

Leonard Kleinrock, Queueing Systems Volume I: Theory (John Wiley & Sons, 1975).

Lannoo, B.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Leland, W.

W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
[Crossref]

Medhi, Deepankar

Deepankar Medhi, “Some Results for Renewal Arrival to a Communication Link,” Probability Models and Statistics: A J. Medhi Festschrift, 85–107 (1996).

Mohamed, Amr

Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
[Crossref]

Ni, L. M.

X. Xiao and L. M. Ni, “Internet QoS: A Big Picture,” IEEE Network 13, 8–18 (1999).
[Crossref]

O’Mahony, M.

M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
[Crossref]

Øverby, H.

H. Øverby and N. Stol, “A Teletraffic Model for Service Differentiation in OPS networks,” in Proceedings of IEEE 8th Optoelectronics and Communications Conference (Institute of Electrical and Electronic Engineers, 2003), pp. 677–678.

Pickavet, M.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Qiao, C.

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000).
[Crossref]

Schmid, A.L.

A.L. Schmid et al., “Impact of VC merging on buffer requirements in ATM networks,” Proceedings of Eighth International Conference on High Performance Networking, 203–217 (1998).

Simeonidou, D.

M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
[Crossref]

Stol, N.

S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).

H. Øverby and N. Stol, “A Teletraffic Model for Service Differentiation in OPS networks,” in Proceedings of IEEE 8th Optoelectronics and Communications Conference (Institute of Electrical and Electronic Engineers, 2003), pp. 677–678.

Taqqu, M.

W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
[Crossref]

Tzanakaki, A.

M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
[Crossref]

Willinger, W.

W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
[Crossref]

Wilson, D.

W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
[Crossref]

Wydrowski, B.

B. Wydrowski and M. Zukerman, “QoS in Best -Effort Networks,” IEEE Communications Magazine 40, 44–49 (2002).
[Crossref]

Xiao, X.

X. Xiao and L. M. Ni, “Internet QoS: A Big Picture,” IEEE Network 13, 8–18 (1999).
[Crossref]

Yoo, M.

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000).
[Crossref]

Zukerman, M.

B. Wydrowski and M. Zukerman, “QoS in Best -Effort Networks,” IEEE Communications Magazine 40, 44–49 (2002).
[Crossref]

IEEE Communications Magazine (3)

M. O’Mahony, D. Simeonidou, D. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39, 128–135 (2001).
[Crossref]

B. Wydrowski and M. Zukerman, “QoS in Best -Effort Networks,” IEEE Communications Magazine 40, 44–49 (2002).
[Crossref]

Ayman Kaheel, Tamer Khattab, Amr Mohamed, and Hussein Alnuweiri, “Quality-of-Service Mechanisms in IP-over WDM Networks,” IEEE Communications Magazine 40, 38–43 (2002).
[Crossref]

IEEE Journal on Selected Areas in Communications (1)

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP -Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18, 2062–2071 (2000).
[Crossref]

IEEE Network (1)

X. Xiao and L. M. Ni, “Internet QoS: A Big Picture,” IEEE Network 13, 8–18 (1999).
[Crossref]

IEEE/ACM Transactions on Networking (1)

W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking 2, 1–15 (1994).
[Crossref]

in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE (1)

S. Bjørnstad, N. Stol, and D. R. Hjelme, “Quality of service in optical packet switched DWDM transport networks,” in Asia -Pacific Optical and Wireless Communications Conference, Proc. SPIE 4910, 63–74 (2002).

Other (5)

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2002).

Leonard Kleinrock, Queueing Systems Volume I: Theory (John Wiley & Sons, 1975).

A.L. Schmid et al., “Impact of VC merging on buffer requirements in ATM networks,” Proceedings of Eighth International Conference on High Performance Networking, 203–217 (1998).

Deepankar Medhi, “Some Results for Renewal Arrival to a Communication Link,” Probability Models and Statistics: A J. Medhi Festschrift, 85–107 (1996).

H. Øverby and N. Stol, “A Teletraffic Model for Service Differentiation in OPS networks,” in Proceedings of IEEE 8th Optoelectronics and Communications Conference (Institute of Electrical and Electronic Engineers, 2003), pp. 677–678.

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Figures (3)

Fig. 1.
Fig. 1. An optical packet switch (a) and the 2-stage hyper-exponential arrival process with priorities (b).
Fig. 2.
Fig. 2. Time continuous Markov chain of the WA with H2-arrival process.
Fig. 3.
Fig. 3. The PLR as a function of the system load (a) and the average PLR as a function of the coefficient of variation (b).

Tables (1)

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Table 1. Simulation and analytical results.

Equations (17)

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c 2 = ( σ m 1 ) 2 = m 2 m 1 2 1 = ε 1 = 2 a + k 2 a k 2 ( a + k a k ) 2 1
p 0 θ 0 = p 1 μ , r 0 θ 1 = r 1 μ
p i ( i μ + θ 0 ) = a ( p i 1 θ 0 + r i 1 θ i ) + p i + 1 ( i + 1 ) μ 1 i n 1
p n ( n μ + θ 0 ) = a ( p n 1 θ 0 + p n φ 1 , 0 + r n 1 θ 1 + r n φ 1 , 1 ) + p n + 1 ( n + 1 ) μ
p i ( i μ + θ 0 ) = a ( p i 1 φ 0 , 0 + p i φ 1 , 0 + r i 1 φ 0 , 1 + r i φ 1 , 1 )
+ p i + 1 ( i + 1 ) μ n + 1 i N 1
P N θ 0 = a ( p N 1 φ 0 , 0 + p N θ 0 + r N 1 φ 0 , 1 + r N θ 1 )
r i ( i μ + θ 1 ) = ( 1 a ) ( r i 1 θ 1 + p i 1 θ 0 ) + r i + 1 ( i + 1 ) μ 1 i n 1
r n ( n μ + θ 1 ) = ( 1 a ) ( r n 1 θ 1 + r n φ 1 , 1 + p n 1 θ 0 + r n φ 1 , 0 ) + r n + 1 ( n + 1 ) μ
r i ( i μ + θ 1 ) = ( 1 a ) ( r i 1 φ 0 , 1 + r i φ 1 , 1 + p i 1 φ 0 , 0 + p i φ 1 , 0 )
+ r i + 1 ( i + 1 ) μ n + 1 i N 1
r N θ 1 = ( 1 a ) ( r N 1 φ 0 , 1 + r N θ 1 + P N 1 φ 0 , 0 + p N θ 0 )
i = 0 N p i + i = 0 N r i = 1
P H 2 0 = p N φ 0 , 0 + r N φ 0 , 1 φ 0 , 0 i = 0 N p i + φ 0 , 1 i = 0 N r i , P H 2 1 = φ 1 , 0 i = n N p i + φ 1 , 1 i = n N r i φ 1 , 0 i = 0 N p i + φ 1 , 1 i = 0 N r i
P H 2 a v = p N θ 0 + r N θ 1 + φ 1 , 0 i = n N 1 p i + φ 1 , 1 i = n N 1 r i θ 0 i = 0 N p i + θ 1 i = 0 N r i
a = ( 1 + ( c 2 1 ) ( c 2 + 1 ) ) 2 , θ 0 = 2 a λ , θ 1 = 2 λ ( 1 a )
θ 0 = 2 λ ( 1 + ( c 2 1 2 ) ( c 2 + 1 ) ) , θ 1 = 4 λ θ 0 , a = ( θ 0 ( θ 1 1 1 ) ) ( θ 1 θ 0 )

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