Traffic modelling of asynchronous bufferless optical packet switched networks

Computer Communications 30 (2007) 1229–1243 www.elsevier.com/locate/comcom

Traffic modelling of asynchronous bufferless optical packet switched networks Harald Øverby

*

Department of Telematics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Received 16 August 2005; received in revised form 1 December 2006; accepted 7 December 2006 Available online 3 January 2007

Abstract Analytical models based on stochastic processes have been widely employed in order to assess the performance of optical packet switched (OPS) networks. A crucial issue with this approach is how to model packet arrivals to an output port in an optical packet switch, and how to evaluate the blocking probability. This article presents various Markovian arrival models for asynchronous bufferless OPS networks. In addition to the well-known Erlang and Engset arrival models, we present two novel Engset based arrival models, i.e. the Engset Asymmetric arrival model and the Engset Non-looping arrival model. We consider optical packet switches with and without wavelength conversion. Analytical expressions for the time-, call-, and traffic congestion are derived for each arrival model. A numerical evaluation shows that the choice of arrival model and performance metric significantly influences the observed blocking probabilities under certain parameter settings. The major aim with this article is to contribute to establish a theoretical framework for teletraffic analysis in asynchronous bufferless OPS networks.  2006 Elsevier B.V. All rights reserved. Keywords: Optical packet switching; Teletraffic modelling

1. Introduction Wavelength division multiplexing (WDM) has emerged as the most promising technology to increase available capacity in future optical core networks [1]. Today, WDM is utilized in a point-to-point architecture, where optical fibers are terminated by electronic switches [2]. In this architecture, referred to as first-generation optical networks, packets are switched in the electronic domain and thus undergo optical–electrical–optical (O/E/O) conversions at every switching point. At the same time we see an explosive growth of the data traffic in the core networks, which results from the increasing number of Internet users combined with the increased access network capacity [3]. In order to cost-efficiently support this capacity demand, today’s WDM point-to-point architecture is expected to

*

Tel.: +47 73 55 10 87; fax: +47 73 59 69 73. E-mail address: [email protected]

0140-3664/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2006.12.005

evolve into all-optical network architectures, such as wavelength routed optical networks (WRON) [4], optical burst switching (OBS) [5–7] and optical packet switching (OPS) [8–10]. These all-optical network architectures can provide core network transport services more cost efficiently and with higher data transparency compared to first-generation optical networks, since the O/E/O bottlenecks at the switching points are removed [1]. Among the all-optical network architectures considered in recent literature, OPS and OBS are viewed as the most promising, at least in the long run. This is because OPS/OBS benefit from statistical resource sharing and are thus better suited to provide transport services for a packet oriented Internet compared to WRON [1]. Analytical models based on stochastic processes are widely employed in order to assess the performance of OPS/OBS. The basis for these models has been developed by teletraffic scientists several decades ago [11], but are now being renewed and put in the context of OPS/OBS [12–14]. Two important issues related to these models are

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H. Øverby / Computer Communications 30 (2007) 1229–1243

how packet arrivals to an output port in an optical packet switch are modelled, and how to evaluate the blocking probability. In this article, we address these issues by presenting various Markovian arrival models for asynchronous OPS without buffering capabilities. In particular, we present two novel arrival models, i.e. the Engset Asymmetric arrival model and the Engset Non-looping arrival model. For each arrival model, we evaluate the blocking probability using the time-, call-, and traffic congestion [11]. The aim with this article is to contribute to establish a theoretical framework for teletraffic analysis in asynchronous bufferless OPS networks. This is crucial in order to better understand the underlying effects influencing the performance in OPS networks, as well as to develop efficient tools for assessing the performance of such networks. A similar study for slotted OPS has been performed in [13], where it was shown that the choice of arrival model impacts the observed blocking probability significantly under certain parameter settings. Unlike the work in [13], where the presented models are based on discrete-time Markov chains, the models presented in this article are based on continuous-time Markov chains, since the switches operate in asynchronous mode. The authors of [14] examined the error introduced by the Erlang and Engset traffic model in OBS, and it was shown that the Engset traffic model was a better approximation to a realistic scenario than the Erlang traffic model. Unlike the work in [14], we evaluate the Engset traffic model using the time-, call-, and traffic congestion. We also present the Engset Asymmetric arrival model, which, combined with the proposed general switch architecture, can be used to study the effects of non-uniform traffic. The rest of this article is organized as follows: Section 2 presents the general switch architecture. Section 3 presents arrival models for asynchronous OPS without wavelength conversion. Section 4 presents arrival models for asynchronous OPS with full-range output wavelength conversion. A numerical evaluation and comparison of the presented arrival models can be found in Section 5, followed by a conclusion in Section 6.

2. General switch architecture Consider a generic non-blocking asynchronous optical packet core switch with F input and output fibers, as depicted in Fig. 1. Each fiber provides N wavelengths by using WDM. Due to the asynchronous mode of operation, packets may arrive to the switch on the input wavelengths at any instant. When a packet arrives to the switch, the header is extracted and processed electronically by the control module, while the payload is delayed in the optical domain using input Fiber Delay Lines. Based on the destination information extracted from the header, the control module decides which output wavelength/fiber the arriving packet is switched to, and configures the switch fabric

control module

IF1

IW1

OW1

IWN

OWN

IW(a-1)N+1 IFa

IWaN

OF1

OW(c-1)N+1 OWcN

OFc

IW(F-1)N+1 OW(F-1)N+1 IFF

IWFN

OWFN

OFF

Fig. 1. An asynchronous optical packet core switch. Each input/output fiber (IF/OF) provides N input/output wavelengths (IW/OW), respectively.

accordingly. This is equal to the bufferless OPS switch architecture presented in [15]. A major concern in this type of switches is packet loss at the network layer [15]. Such packet loss occurs due to contention when a packet is destined to a wavelength/fiber that is busy, i.e. for switches without wavelength conversion, contention occurs when a packet is destined to a busy output wavelength, while in switches with full-range output wavelength conversion, contention occurs when a packet is destined to an output fiber where all wavelengths are busy. In order to resolve contentions, the wavelength domain, time domain and/or space domain may be utilized [15]. This article considers switches where only the wavelength domain is utilized, i.e. full-range wavelength converters are employed at each output wavelength (Section 4), but also switches where no contention resolution strategy is applied (Section 3). Denote input fiber a (1 6 a 6 F) and output fiber c (1 6 c 6 F) as IFa and OFc, respectively. Further, denote input wavelength b (1 6 b 6 N) on input fiber IFa as IW(a1+b) (1 6 (a  1)N + b 6 FN), and output wavelength d (1 6 d 6 N) on output fiber OFc as OW(c1)N+d (1 6 (c  1)N + d 6 FN). In order to clarify later discussions, let the fibers originate from unique physical locations. Also assume that two adjacent switches are connected by a fiber pair, where the traffic is transmitted in both directions. Packets transported on IW(a1)N+b and OW(c1)N+d are transmitted on the physical wavelengths hb and hd, respectively. This means that packets switched from input wavelength IW(a1)N+b to output wavelength OW(c1)N+d are transmitted on the same physical wavelength if b = d. On the other hand, if b „ d, packets switched from input wavelength IW(a1)N+b to output wavelength OW(c1)N+d must be converted from wavelength hb to wavelength hd. Further, assume the following: • The packet service time is exponential i.i.d. with mean service time equal to l1.

H. Øverby / Computer Communications 30 (2007) 1229–1243

• For the Engset Asymmetric arrival models (Sections 3.1 and 4.1), the Engset arrival models (Sections 3.2 and 4.2) and the Engset Non-looping arrival models (Sections 3.3 and 4.3), we assume that packets arrive on input wavelength IW(a1)N+b (1 6 (a  1)N + b 6 FN) according to Pure Chance Traffic type Two (PCT-II) [11]. Hence, input wavelength IW(a1)N+b changes between the states idle and busy with exponential holding time with intensities u(a1)N+b and l, as shown in Fig. 2. We assume that the intensity u(a1)N+b is independent of arrival intensities in other input wavelengths. Denote b(a1)N+b = u(a1)N+b/l (1 6 (a1)N + b 6 FN) as the offered traffic for input wavelength IW(a1)N+b in idle state. In the PCT-II model, the offered traffic for input wavelength IW(a1)N+b is given as [11] uða1ÞN þb bða1ÞN þb ¼ l þ uða1ÞN þb 1 þ bða1ÞN þb

1231

As will be shown later in this article, the general model allows for different normalized system loads on the various output wavelengths/fibers. • In the case of switches without wavelength conversion, let rij (1 6 i 6 FN, 1 6 j 6 FN, 0 6 rij 6 1) denote the routing probability from input wavelength IWi to output wavelength OWj, as illustrated in Fig. 3. Note that P FN j¼1 rij ¼ 1, since packets arriving on an input wavelength must be routed somewhere. In the case of switches with wavelength conversion, the routing probabilities will be slightly modified, as we will show in Section 4. • The Engset Asymmetric arrival model, the Engset arrival model and the Engset Non-looping arrival model are all based on the Engset Lost Calls Cleared Traffic model [11].

ð1Þ

In analytical modelling of OPS, the blocking probability may be expressed in the following ways [11]:

• For the Erlang arrival models (Sections 3.4 and 4.4), we assume that packets arrive on input wavelength IW(a1)N+b (1 6 (a  1)N + b 6 FN) according to a pure Poisson process with constant arrival intensity equal to t(a1)N+b. This is Pure Chance Traffic type One (PCTI) [11]. We further assume that the arrival intensity t(a1)N+b is independent of arrival intensities in other input wavelengths. In this case, the offered traffic for input wavelength IW(a1)N+b is t(a1)N+b/l. Note that t(a1)N+b 6 l since on average maximum 1 packet may be transmitted on a single input wavelength at the same time. • The effect of the switching time is ignored. This is shown to be a reasonable assumption if the switching time is less than 10% of the mean packet length [16]. • The normalized system load in the switch for the Engset Asymmetric arrival models, the Engset arrival models and the Engset Non-looping arrival models, is given as (note that i = (a  1)N + b in order to clarify notations):

• Time congestion (E): The relative share of the time the output wavelength/fiber is busy. This is the operator perceived QoS. • Call congestion (B): The relative share of arrivals that are generated when the output wavelength/fiber is busy. This is the user perceived QoS. • Traffic congestion (C): The relative share of offered traffic that is not carried by the output wavelength/fiber. This is the system perceived QoS.

aða1ÞN þb ¼

A¼

FN 1 X ai FN i¼1

ð2Þ

• The normalized system load in the switch for the Erlang arrival models is given as: A¼

FN 1 X ti FN l i¼1

ð3Þ

3. Arrival models for switches without wavelength conversion We consider a certain output wavelength OW(c1)N+d (1 6 (c  1)N + d 6 FN), where packets are transmitted on the physical wavelength hd. Since we assume no wavelength conversion, only input wavelengths IW(a1)N+d (1 6 a 6 F) may route packets to output wavelength OW(c1)N+d. For the Engset Asymmetric arrival model (NOWC-EAAM), the Engset arrival model (NOWCENAM) and the Engset Non-looping arrival model (NOWC-ENLAM), the traffic offered by input wavelength IW(a1)N+d to output wavelength OW(c1)N+d is j½ða1ÞN þd½ðc1ÞN þd ¼ aða1ÞN þd r½ða1ÞN þd½ðc1ÞN þd

ð4Þ

where a(a1)N+d is given in Eq. (1). Note that xij = x[i][j] in order to clarify notations. This means that the normalized system load on output wavelength OW(c1)N+d is

(a-1)N+b

Idle

Further on, in Sections 3 and 4, we present various arrival models for this switch architecture, where our aim is to provide explicit results for the time-, call-, and traffic congestion for each arrival model. The parameters used throughout this article are summarized in Table 1.

Busy

AOWðc1ÞNþd ¼

F X

j½ða1ÞN þd½ðc1ÞN þd ð0 6 AOWðc1ÞN þd 6 F Þ

a¼1

Fig. 2. Input wavelength IW(a1)N+b modelled as an on/off source.

ð5Þ

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H. Øverby / Computer Communications 30 (2007) 1229–1243 OW1 OF1 [(a-1)N+b][1]

IW(a-1)N+1

OWN [(a-1)N+b][N]

IFa

[(a-1)N+b][(c-1)N+1]

IW(a-1)N+b

OW(c-1)N+1

IWaN OFc

[(a-1)N+b][cN]

OWcN

Fig. 3. Routing probabilities in the considered optical packet switch. Note that r[i][j] = rij in order to clarify illustrations.

Table 1 The notations used in this article Parameter

Description

IFj OFj IWj OWj F N hj l1 uj bj aj tj rij ES BS CS A AOWj AOFj jij kij

Input fiber j Output fiber j Input wavelength j Output wavelength j Number of input/output fibers Number of wavelengths per fiber Physical wavelength j Mean packet service time Arrival intensity for packets arriving on IWj (Engset) Offered traffic for packets arriving on IWj in idle state (Engset) Offered traffic for packets arriving on IWj (Engset) Arrival intensity for packets arriving on IWj (Erlang) Routing probability from IWi to OWj (case without wavelength conversion), or from IWi to OFj (case with wavelength conversion) Time congestion for arrival model S Call congestion for arrival model S Traffic congestion for arrival model S Normalized system load Normalized system load on OWj Normalized system load on OFj Offered traffic imposed by IWi on OWj (case without wavelength conversion), or by IWi on OFj (case with wavelength conversion) (Engset) Arrival intensity for IWi that routes packets to OWj (case without wavelength conversion) or OFj (case with wavelength conversion) (Engset) Offered traffic for IWi in idle state that routes packets to OWj (case without wavelength conversion) or OFj (case with wavelength conversion) (Engset) Arrival intensity for packets routed from IWi to IWj (case without wavelength conversion) or from IWi to OFj (case with wavelength conversion) (Erlang) Carried traffic

gij xij Y

The arrival intensity for packets routed from input wavelength IW(a1)N+d to output wavelength OW(c1)N+d is k½ða1ÞN þd½ðc1ÞN þd

j½ða1ÞN þd½ðc1ÞN þd l ¼ 1  j½ða1ÞN þd½ðc1ÞN þd

g½ða1ÞNþd½ðc1ÞN þd ¼ ¼

ð6Þ

In particular, we obtain the offered traffic for input wavelength IW(a1)N+d in idle state that routes packets to output wavelength OW(c1)N+d as:

k½ða1ÞN þd½ðc1ÞNþd j½ða1ÞNþd½ðc1ÞN þd ¼ l 1  j½ða1ÞNþd½ðc1ÞN þd b½ða1ÞN þd r 1þb½ða1ÞNþd ½ða1ÞNþd½ðc1ÞN þd b

þd 1  1þb½ða1ÞN r½ða1ÞNþd½ðc1ÞN þd ½ða1ÞN þd r½ða1ÞN þd½ðc1ÞNþd ¼ 1 þ 1=b½ða1ÞNþd  r½ða1ÞN þd½ðc1ÞN þd

ð7Þ

For the Erlang arrival model (NOWC-ERAM), the arrival intensity for packets routed from input wavelength IW(a1)N+d to output wavelength OW(c1)N+d is

H. Øverby / Computer Communications 30 (2007) 1229–1243

x½ða1ÞN þd½ðc1ÞN þd ¼ t½ða1ÞN þd r½ða1ÞN þd½ðc1ÞN þd

ð8Þ

According to the splitting theorem, packets are offered to output wavelength OW(c1)N+d as a Poisson process where the arrival intensity equals the sum of the arrival intensities from input wavelengths IW(a1)N+d (1 6 a 6 F) [11]. The normalized system load on output wavelength OW(c1)N+d for the Erlang arrival model is: AOWðc1ÞN þd ¼

F 1X x½ða1ÞN þd½ðc1ÞN þd ð0 6 AOWðc1ÞN þd 6 F Þ l a¼1

ð9Þ 3.1. The Engset Asymmetric arrival model (NOWCEAAM) The Engset Asymmetric arrival model for switches without wavelength conversion (NOWC-EAAM) considers an output wavelength OW(c1)N+d, and takes into account the unique arrival intensities and routing probabilities for each input wavelength. In order to calculate the blocking probability on output wavelength OW(c1)N+d, we set up a F-dimensional Markov chain as shown in Fig. 4. The output wavelength OW(c1)N+d is idle in state S0. In state Sa (1 6 a 6 F), OW(c1)N + d is busy transmitting a packet routed from input wavelength IW(a1)N+d. The transition intensities from state S0 to state Sa are equal to the arrival intensities for packets arriving on input wavelength IW(a1)N+d and are routed to output wavelength OW(c1)N+d, as given in Eq. (6). Let Qi denote the probability of being in state Si (0 6 i 6 F). From the state diagram we set up the node equations:

Q0

F X

k½ði1ÞN þd½ðc1ÞN þd ¼ l

S1

F X

ð11Þ

Qi ¼ 1

ð12Þ

i¼0

The corresponding normalized system load on output wavelength OW(c1)N+d is given in Eq. (5). The state probabilities are found by solving the linear equation set with F + 1 unknown variables and F + 1 equations. We obtain the time congestion (ENOWC-EAAM) at OW(c1)N+d by summing over all the states where OW(c1)N+d is busy: ENOWC-EAAM ðc; dÞ ¼

F X

ð13Þ

Qi

i¼1

In order to calculate the call congestion (BNOWC-EAAM), we must consider packet arrivals when OW(c1)N+d is busy, i.e. in states Sa ("a P 1). Note that input wavelength IW(a1)N+d does not generate packets when OW(c1)N+d is in state Sa. F P

Qi

i¼1

F P

k½ðj1ÞN þd½ðc1ÞN þd

j¼1 j 6¼ i

BNOWC-EAAM ðc; dÞ ¼ Q0

F P

k½ðj1ÞNþd½ðc1ÞN þd þ

j¼1

F P i¼1

Qi

F P

k½ðj1ÞN þd½ðc1ÞNþd

j¼1 j 6¼ i

ð14Þ

We obtain the traffic congestion (CNOWC-EAAM) by considering the traffic not carried by OW(c1)N + d [11]: AOWðc1ÞNþd  Y AOWðc1ÞN þd F P

¼ a¼1

j½ða1ÞN þd½ðc1ÞN þd 

F P

Qi

i¼1 F P

ð15Þ

j½ða1ÞN þd½ðc1ÞN þd

a¼1 [(i-1)N+d][(c-1)N+d]

S0

ð10Þ

Qi

i¼1

Qi l ¼ Q0 k½ði1ÞN þd½ðc1ÞN þd ð1 6 i 6 F Þ

[N+d][(c-1)N+d]

[d][(c-1)N+d]

F X

i¼1

C NOWC-EAAM ðc; dÞ ¼

S2

1233

Here, Y is the carried traffic. Si

3.2. The Engset arrival model (NOWC-ENAM) [(F-1)N+d][(c-1)N+d]

SF

Fig. 4. Markov chain of the NOWC-EAAM. In state S0 the output wavelength OW(c1)N+d is idle. In states Sa (a P 1), OW(c1)N+d is transmitting a packet originating from input wavelength IW(a1)N+d.

In the NOWC-ENAM, we assume a uniform traffic pattern and equal arrival intensities in every input wavelength, i.e. ui = u (1 6 i 6 FN), bi = b = u/l (1 6 i 6 FN), and r[(a-1)N+d][(c-1)N+d] = r = 1/F (1 6 a, c 6 F) (1 6 d 6 N). We consider a tagged output wavelength, which may be any output wavelength. The traffic imposed per input wavelength on the tagged output wavelength is j ¼ ar ¼

a b ¼ F F ð1 þ bÞ

ð16Þ

1234

H. Øverby / Computer Communications 30 (2007) 1229–1243

Since F input wavelengths may route packets to the tagged output wavelength, we obtain the normalized system load on the tagged output wavelength as: A ¼ Fj ¼

b 1þb

ð17Þ

The offered traffic per idle input wavelength that routes packets to the tagged output wavelength is g¼

j 1 ¼ 1  j F þ F =b  1

ð18Þ

The arrival intensity per idle input wavelength that routes packets to the tagged output wavelength is k ¼ gl

ð19Þ

Also note that g j¼ 1þg

ð20Þ

Since there is no distinction between the input wavelengths, the resulting Markov chain has 2 states only, as seen in Fig. 5. This is the Engset traffic model with F sources and 1 server [11]. Let Q0 and Q1 denote the probability for being in state S0 and S1, respectively. The state probabilities are found using cut-operations: Q0 F k ¼ Q1 l

ð21Þ

Q0 þ Q1 ¼ 1

ð22Þ

which results in: Q0 ¼

l 1 ¼ l þ Fk 1 þ Fg

ð23Þ

Q1 ¼

Fk Fg ¼ l þ Fk 1 þ Fg

ð24Þ

The time congestion is ENOWC-ENAM ¼ Q1 ¼

Fg 1 ¼ 1 þ F g 1 þ F1g

ð25Þ

The call congestion is BNOWC-ENAM ¼ ¼

ðF  1ÞkQ1 1 ¼ FQ0 kðFQ0 þ ðF  1ÞQ1 Þ 1 þ ðF 1ÞQ 1

1

ð26Þ

1 1 þ ðF 1Þg

Fλ S0

S1

Fig. 5. Markov chain of the NOWC-ENAM. In state S0 the tagged output wavelength is idle, while in state S1 the tagged output wavelength is busy.

Note that BNOWC-ENAM equals ENOWC-ENAM with one less input wavelength. This is known as the arrival theorem [11]. The traffic congestion is: A  Y F j  Q1 ðF  1Þg ¼ ¼ A 1 þ Fg Fj F 1 ¼ ENOWC-ENAM F

C NOWC-ENAM ¼

ð27Þ

Note that Eq. (27) is in accordance with the relation between ENOWC-ENAM and CNOWC-ENAM derived in [11]. Also, note that: oENOWC-ENAM > 0 ðF P 1Þ ðg > 0Þ oF oBNOWC-ENAM > 0 ðF P 2Þ ðg > 0Þ oF oC NOWC-ENAM > 0 ðF P 2Þ ðg > 0Þ oF

ð28Þ ð29Þ ð30Þ

3.3. The Engset Non-looping arrival model (NOWCENLAM) We define looping as when packets are routed to the same physical location as they originated from, that is, packets arriving on input wavelength IWg (1 6 g 6 FN) and leaving on output wavelength OWh (1 6 h 6 FN) are denoted as looping packets if g 2 [(i  1)N+1, iN] and h 2 [(i  1)N+1, iN] ("i) (note that we assumed a unique location for each fiber pair). In some networks looping is defined as an error-state because looped packets have consumed link resources without getting any closer to their destination [17]. Looping may be caused by errors in the routing tables or other logical errors in the switch [17]. Note that looping is an error state in some networks only, e.g. optical networks employing deflection routing allows looping. In this case, contended packets may be routed back to where they came from in order to find an alternative route to their destination [18]. Further on we focus on networks where looping should not occur, which means that {rij = 0ji 2 [(j  1)N+1, jN]} ("i"j) and hence {kij = 0ji 2 [(j-1)N+1, jN]} ("i"j). We still assume equal arrival intensities on each input wavelength and a uniform traffic pattern as in the NOWC-ENAM. This means that the maximum number of simultaneous arrivals to a tagged output wavelength has been reduced from F to F  1 compared to the NOWC-EAAM and NOWC-ENAM. Hence, we obtain the following expressions for the routing probability (Eq. (31)), the traffic imposed per input wavelength on the tagged output wavelength (Eq. (32)), the normalized system load on the tagged output wavelength (Eq. (33)), the offered traffic per idle input wavelength that routes packets to the tagged output wavelength (Eq. (34)), and the corresponding arrival intensity per idle input wavelength (Eq. (35)):

H. Øverby / Computer Communications 30 (2007) 1229–1243

r¼

1 F 1

ω

ð31Þ

a b j ¼ ar ¼ ¼ F  1 ðF  1Þð1 þ bÞ b A ¼ ðF  1Þj ¼ 1þb j 1 ¼ g¼ 1  j F þ ðF  1Þ=b  2

Q0 ðF  1Þk ¼ Q1 l Q0 þ Q1 ¼ 1

Fig. 7. Markov chain of the NOWC-ERAM. In state S0 output wavelength OW(c1)N+d is idle, while in state S1 output wavelength OW(c1)N+d is busy.

ð34Þ

In particular, note the following properties:

ð35Þ

oENOWC-ENLAM > 0 ðF P 2Þ ðg > 0Þ oF oBNOWC-ENLAM > 0 ðF P 3Þ ðg > 0Þ oF oC NOWC-ENLAM > 0 ðF P 3Þ ðg > 0Þ oF lim BNOWC-ENLAM ¼ 0

ð36Þ ð37Þ

which results in: l 1 ¼ l þ ðF  1Þk 1 þ ðF  1Þg ðF  1Þk ðF  1Þg Q1 ¼ ¼ l þ ðF  1Þk 1 þ ðF  1Þg

ð38Þ

Q0 ¼

ð39Þ

F !2

lim C NOWC-ENLAM ¼ 0

F !2

ð43Þ ð44Þ ð45Þ ð46Þ ð47Þ

3.4. The Erlang arrival model (NOWC-ERAM) In the NOWC-ERAM, we assume an infinite number of input wavelengths, and that packets arrive to output wavelength OW(c1)N+d according to a pure Poisson process with intensity: x¼

F X

x½ða1ÞN þd½ðc1ÞN þd

ð48Þ

a¼1

The time congestion is ðF  1Þg 1 ¼ 1 1 þ ðF  1Þg 1 þ ðF 1Þg

ð40Þ

The call congestion is: ðF  2ÞkQ1 kðQ0 ðF  1Þ þ Q1 ðF  2ÞÞ 1 1 ¼ ¼ 1 Q0 ðF 1Þ 1 þ Q ðF 2Þ 1 þ ðF 2Þg

BNOWC-ENLAM ¼

ð41Þ

1

Note that the BNOWC-ENLAM equals ENOWC-ENLAM with one less input wavelength. The traffic congestion is:

Here, x[(a1)N+d][(c1)N+d] is given in Eq. (8). The normalized system load on output wavelength OW(c1)N+d is given in Eq. (9). This is the Erlang traffic model with 1 server, as seen in Fig. 7 [11]. Let Q0 and Q1 denote the state probabilities for S0 and S1, respectively. The state probabilities are obtained from cut-operations: l ð49Þ Q0 ¼ lþx x ð50Þ Q1 ¼ lþx The time-, call-, and traffic congestion at output wavelength OW(c1)N+d are all equal according to the Poisson Arrivals See Time Averages (PASTA) property [11]:

A  Y ðF  1Þj  Q1 ¼ A ðF  1Þj

¼

ðF 1Þg ðF 1Þg  1þðF gþ1 1Þg ðF 1Þg gþ1

¼

ðF  2Þg F 2 ¼ ENOWC-ENLAM 1 þ ðF  1Þg F  1

ð42Þ

(F-1)λ S0

S1

ð33Þ

Note that the normalized system load on a tagged output wavelength is equal for the NOWC-ENAM and NOWCENLAM. The resulting Markov chain for the NOWC-ENLAM is depicted in Fig. 6. This is the Engset traffic model with F  1 sources and 1 server [11]. Let Q0 and Q1 denote the probability for being in state S0 and S1, respectively. The balance equations are obtained from cut-operations:

C NOWC-ENLAM ¼

S0

ð32Þ

k ¼ gl

ENOWC-ENLAM ¼ Q1 ¼

1235

S1

Fig. 6. Markov chain of the NOWC-ENLAM. In state S0 the tagged output wavelength is idle, while in state S1 the tagged output wavelength is busy.

ENOWC-ERAM ðc; dÞ ¼ BNOWC-ERAM ðc; dÞ ¼ C NOWC-ERAM ðc; dÞ x ð51Þ ¼ Q1 ¼ lþx Note that the blocking probability is independent of the parameter F. 4. Arrival models for switches with full-range output wavelength conversion This section presents arrival models for asynchronous bufferless OPS employing full-range output wavelength conversion. Packets arriving on a certain input wavelength

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H. Øverby / Computer Communications 30 (2007) 1229–1243

can now be converted to any output wavelength. This means that packets are indifferent on what wavelength they are transported on, and only the output fiber matters when it comes to routing decisions. Hence, the routing probabilities are changed from rij (1 6 i 6 FN) (1 6 j 6 FN) to rij (1 6 i 6 FN) (1 6 j 6 F). For the Engset Asymmetric arrival model (WC-EAAM), the Engset arrival model (WC-ENAM) and the Engset Non-looping arrival model (WC-ENLAM), the traffic offered by input wavelength IW(a-1)N+b to output fiber OFc is j½ða1ÞN þb½c ¼ aða1ÞN þb r½ða1ÞN þb½c

ð52Þ

where a(a-1)N+b is given in Eq. (1). This means that the normalized system load on output fiber OFc is AOFc ¼

FN 1 X j½i½c N i¼1

ð0 6 AOFc 6 F Þ

ð53Þ

The arrival intensity for input wavelength IW(a-1)N+b in idle state that routes packets to output fiber OFc is j½ða1ÞN þb½c l ð54Þ k½ða1ÞN þb½c ¼ 1  j½ða1ÞN þb½c In particular, we obtain the offered traffic for input wavelength IW(a-1)N+b in idle state that routes packets to output fiber OFc as: g½ða1ÞN þb½c ¼ ¼

probability for state Sj, and let kj = k[j][c] (1 6 j 6 FN) in order to simplify notations. Q0

X

b½ða1ÞN þb 1þb½ða1ÞNþb

r½ða1ÞN þb½c

b

ð58Þ

Qj

j¼1...FN

0

1

B C X B C Qj Bl þ kk C ¼ Q0 kj þ l @ A

0

ð55Þ

X

Qk1 k 2

k 1 ;k 2 ¼1...FN k 1 _k 2 ¼j k 1 6¼k 2

ð59Þ

1

B C X B C Qj1 j2 B2l þ kk C @ A k¼1...FN k6¼j1 ;j2

ð56Þ

Hence, the normalized system load on output fiber OFc for WC-ERAM is FN 1 X ¼ xic ð0 6 AOFc 6 F Þ N l i¼1

X

k¼1...FN k6¼j

For the Erlang arrival model (WC-ERAM), the arrival intensity for packets routed from input wavelength IW(a1)N+b to output fiber OFc is x½ða1ÞN þb½c ¼ tða1ÞN þb r½ða1ÞN þb½c

kj ¼ l

j¼1...FN

k½ða1ÞN þb½c j½ða1ÞN þb½c ¼ l 1  j½ða1ÞN þb½c

þb 1  1þb½ða1ÞN r½ða1ÞN þb½c ½ða1ÞNþb r½ða1ÞN þb½c ¼ 1 þ 1=b½ða1ÞN þb  r½ða1ÞN þb½c

AOFc

packet arrival from input wavelength i brings the output fiber OFc to state Si (1 6 i 6 FN). In state Si, the packet currently transmitting may finish or there may be a new packet arrival from another input wavelength IWj (j „ i). In the former case, the output fiber OFc returns to state S0. In the latter case, when there is a packet arrival on input wavelength IWj (j „ i), the output fiber changes from state Si to state Sij. Note that Sij = Sji ("i"j) due to the memory-less property of the exponential distribution. Also note that state Sii ("i) does not exist because there cannot be two simultaneous packet arrivals from the same input wavelength. From state Sij, a packet arrival on input wavelength IWk (k „ i, j) brings the output fiber OFc from state Sij to state Sijk. This may continue until all wavelengths at OFc are busy, i.e. in state  Sk1k2     kN PN FN (k1 „ k2 „ . . . „ kN). There are a total of i¼0 states i in the considered Markov chain. Let Qj denote the state

ð57Þ

X

¼ Qj1 kj2 þ Qj2 kj1 þ l

ð60Þ

Qk1 k2 k3

k 1 ;k 2 ;k 3 ¼1...FN k 1 _k 2 _k 3 ¼j1 ^j2 k 1 6¼k 2 6¼k 3

0

1

B B Qj1 j2 ...jm Bml þ @

X k¼1...FN k6¼j1 ;j2 ;...;jm

C C kk C A

¼ Qj2 j3 jm kj1 X þ Qj1 j3 jm kj2 þ    þ Qj1 j2 ...jm1 kjm þl Qk1 k2 ...kmþ1

ð61Þ

k 1 ;k 2 ;...;k mþ1 ¼1...FN k 1 _k 2 __k mþ1 ¼j1 ^j2 ^^jm k 1 6¼k 2 6¼6¼k mþ1

4.1. The Engset Asymmetric arrival model (WC-EAAM) The Engset Asymmetric arrival model takes into account that there is a finite number of input wavelengths with distinct arrival intensities and routing probabilities. In order to derive the blocking probability at output fiber OFc (0 6 c 6 F), we set up a FN-dimensional Markov chain, described by Eqs. ()(58)–(63). In state S0, the output fiber OFc is idle, i.e. none of the N wavelengths at OFc are busy. In this state packets may arrive from any input wavelength (FN in total) with dedicated arrival intensity ki for input wavelength IWi (1 6 i 6 FN). A

Qj1 j2 ...jN N l ¼ Qj2 j3 ...jN kj1 þ Qj1 j3 ...jN kj2 þ   þ Qj1 j2 ...jN 1 kjN ð62Þ FN FN X FN FN X FN FN X X X X Q0 þ Qk þ Qk 1 k2 þ    þ . .. Qk 1 k2 ...kN ¼ 1 k¼1

k 1 ¼1 k 2 ¼1

k 1 ¼1 k 2 ¼1

k N ¼1

ð63Þ The time congestion is the sum of all states where OFc is busy transmitting N packets, i.e.: X EWC-EAAM ðcÞ ¼ Qk1 k2 ...kN ð64Þ k 1 ;k 2 ;...;k N ¼1...FN k 1 6¼k 2 6¼6¼k N

The call congestion is obtained by considering the number of arrivals when output fiber OFc is busy:

H. Øverby / Computer Communications 30 (2007) 1229–1243

P k 1 ; k 2 ; . . . ; k N ¼ 1 . . . FN k 1 6¼ k 2 6¼    6¼ k N

BWC-EAAM ðcÞ ¼

Qk1 k2 ...kN

P

1237

kj

j ¼ 1 . . . FN j 6¼ k 1 ; k 2 ;    ; k N 2

3

7 6 7 6 P 7 6 kj þ    þ Q0 kj þ Qi kj 7 6Qk1 k2 ...kN 7 6 j¼1 i¼1 k 1 ; k 2 ; . . . ; k N ¼ 1 . . . FN 4 j ¼ 1 . . . FN j¼1 5 j 6¼ k 1 ; k 2 ; . . . ; k N j 6¼ i k 1 6¼ k 2 6¼    6¼ k N FN P

FN P

ð65Þ

P

FN P

The traffic congestion is given as: 3

2

7 6 7 6P P P 7 6 FN 2Qij þ    þ j½i½c  6 Qi þ NQk1 k2 ...kN 7 7 6 i¼1 i; j ¼ 1 . . . FN k 1 ; k 2 ; . . . ; k N ¼ 1 . . . FN 5 4i¼1 i 6¼ j k1 ¼ 6 k 2 6¼    6¼ k N C WC-EAAM ðcÞ ¼ FN P j½i½c FN P

ð66Þ

i¼1

4.2. The Engset arrival model (WC-ENAM)

g¼

In the WC-ENAM, we assume a uniform traffic pattern, as in the NOWC-ENAM, i.e. ui = u (1 6 i 6 FN), bi = b = u/l (1 6 i 6 FN), and rij = r = 1/F (1 6 i 6 FN) (1 6 j 6 F). We consider a tagged output fiber, which may be any output fiber. The traffic offered per input wavelength to the tagged output fiber is j ¼ ar ¼

b F ð1 þ bÞ

k ¼ gl

ð67Þ

b 1þb

ð70Þ

We model the tagged output fiber as a one-dimensional Markov chain as seen in Fig. 8. The states denote the number of wavelengths busy, e.g. in state S3 three wavelengths are busy. Let Qi denote the probability of being in state Si. This is the Engset traffic model with FN sources and N servers [11]. The state probabilities are given as:

ð68Þ

The offered traffic per idle input wavelength that routes packets to the tagged output fiber is

FN

(FN-i)

(FN-i+1)

S0

((FN-N+1)

Si i

ð69Þ

The arrival intensity per idle input wavelength that routes packets to the tagged output fiber is

The normalized system load on the tagged output fiber is A ¼ Fj ¼

j 1 ¼ 1  j F þ F =b  1

SN (i+1)

N

Fig. 8. Markov chain of the WC-ENAM. In state S0 the tagged output fiber is idle, while in states Si (1 6 i 6 N) i wavelengths at the tagged output fiber are busy.

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H. Øverby / Computer Communications 30 (2007) 1229–1243



FN



gi i  Qi ¼ N  P FN gj j j¼0

ð0 6 i 6 N Þ

This is the Engset traffic model with (F  1)N sources and N servers, and from [11] we have that the state probabilities are given as:   ðF  1ÞN gi i  ð0 6 i 6 N Þ Qi ¼ N  ð80Þ P ðF  1ÞN gj j j¼0

ð71Þ

The time congestion is the probability of finding all wavelengths at the tagged output fiber busy, i.e.:   FN gN N  EWC-ENAM ¼ QN ¼ N  ð72Þ P FN j g j j¼0

The time congestion is given as:   ðF  1ÞN EWC-ENLAM

The call congestion is [11] 

BWC-ENAM

 gN QN ðFN  N Þk N  ¼ N ¼ N  P P FN  1 j Qj ðFN  jÞk g j j¼0 j¼0 FN  1

ð73Þ

C WC-ENAM ¼

N P



 ðF  1ÞN  1 N g QN ððF  2ÞN Þk N  BWC-ENLAM ¼ N ¼ N  P P ðF  1ÞN  1 j Qj ððF  1ÞN  jÞk g j j¼0 j¼0 ð82Þ

iQi

i¼1

FN j

¼

F 1 EWCENAM F

The traffic congestion is given as:

ð74Þ

ðF  1ÞN j  4.3. The Engset non-looping arrival model (WC-ENLAM)

1 F 1

b ðF  1Þð1 þ bÞ b A ¼ ðF  1Þj ¼ 1þb j 1 ¼ g¼ 1  j F þ ðF  1Þ=b  2 j ¼ ar ¼

iQi

ðF  1ÞN j F 2 EWC-ENLAM ¼ F 1

ð83Þ

4.4. The Erlang arrival model (WC-ERAM)

ð75Þ

In the WC-ERAM, we assume an infinite number of input wavelengths, and that packets arrive to output fibre OFc according to a pure Poisson process with intensity:

ð76Þ

x¼

FN X

ð84Þ

x½i½c

i¼1

ð77Þ

where x[i][c] is given in Eq. (56). The normalized system load on output fibre OFc is given in Eq. (57). The output fibre OFc is modelled as a one-dimensional Markov chain as seen in Fig. 10. This is the Erlang traffic model with N servers. The state probabilities are given as [11]:

ð78Þ

k ¼ gl

ð79Þ

The resulting Markov chain is depicted in Fig. 9.

(F-1)N

N P i¼1

C WC-ENLAM ¼

The WC-ENLAM assumes a finite number of input wavelengths, equal arrival intensities and uniform traffic pattern, but unlike the WC-ENAM we now assume that looping is not allowed (see Section 3.3 for an explanation of looping). Now, only (F  1)N input wavelengths may route packets to the tagged output fiber. Hence, we have that: r¼

ð81Þ

The call congestion is given as:

As in Section 3.2, note that BWC-ENAM equals EWC-ENAM with one less input wavelength. The traffic congestion is given as [11]: FN j 

gN N  ¼ QN ¼ N  P ðF  1ÞN gj j j¼0

((F-1)N-i)

((F-1)N-i+1)

S0

(((F-2)N+1)

Si i

SN (i+1)

N

Fig. 9. Markov chain of the WC-ENLAM. In state S0 the tagged output fiber is idle, while in states Si (1 6 i 6 N) i wavelengths at the tagged output fiber are busy.

H. Øverby / Computer Communications 30 (2007) 1229–1243

ω

ω

ω

S0

1239

ω

Si

SN

i

(i+1)

N

Fig. 10. Markov chain of the WC-ERAM. In state S0 output fiber OFc is idle, while in states Si (1 6 i 6 N) i wavelengths at output fiber OFc are busy.

N P j¼0

0,5

ð0 6 i 6 N Þ

ð85Þ

ðx=lÞj j!

0,4

According to the PASTA property [11], the time-, call-, and traffic congestion are all given as: EWC-ERAM ðcÞ ¼ BWC-ERAM ðcÞ ¼ C WC-ERAM ðcÞ ¼ QN ¼

ðx=lÞN N! N P ðx=lÞj j! j¼0

ð86Þ

Call Congestion

Qi ¼

ðx=lÞi i!

0,3

0,2

ERAM 0,1

ENAM

ENLAM

5. Numerical evaluation

0 2

0,5

3

4

5

6

7

8

10

Fig. 12. The Call congestion (B) for the NOWC-ERAM, NOWC-ENAM, and NOWC-ENLAM on a tagged output wavelength as a function of the number of input/output fibers (F). The normalized system load is A = 0.8.

1,0E-02

1,0E-03 ERAM

0,4

Call Congestion

9

F

Call Congestion

In this section, we present a numerical evaluation of the NOWC-ENAM, NOWC-ENLAM, NOWC-ERAM, WCENAM, WC-ENLAM, and WC-ERAM. The results are presented in Figs. 11–22. In order to calculate the blocking probabilities, the equations presented in Sections 2–5 are used, with the parameter settings indicated in the corresponding figure text. Furthermore, regarding the NOWC-ENAM, NOWC-ENLAM, WC-ENAM, and WC-ENLAM, we observe that the blocking formulas are given as a function of the parameter g. However, it is more convenient to express the blocking probability as a

ENAM ENLAM

0,3

1,0E-04 2

3

4

5

6

7

8

9

10

F 0,2

Fig. 13. The Call congestion (B) for the WC-ERAM, WC-ENAM and WC-ENLAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.5, and the number of wavelengths per fiber is N = 16.

ERAM ENAM

0,1

ENLAM

function of the normalized system load (A). In order to achieve this, we begin with

0 2

3

4

5

6

7

8

9

10

F Fig. 11. The Call congestion (B) for the NOWC-ERAM, NOWC-ENAM, and NOWC-ENLAM on a tagged output wavelength as a function of the number of input/output fibers (F). The normalized system load is A = 0.5.

Sar ¼ NA

ð87Þ

Here, S is the number of input wavelengths that may route packets to the tagged output wavelength/fiber, a is the offered traffic per input wavelength, r is probability for

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H. Øverby / Computer Communications 30 (2007) 1229–1243 1,0E -03

Call Congestion

1,0E+00

Call Congestion

1,0E -04

1,0E -05

1,0E-01

ERAM

1,0E-02

ERAM 1,0E -06

ENAM

ENAM

ENLAM

ENL AM

1,0E-03 2

1,0E -07 2

3

4

5

6

7

8

9

F Fig. 14. The Call congestion (B) for the WC-ERAM, WC-ENAM and WC-ENLAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.5, and the number of wavelengths per fiber is N = 32.

4

5

6

7

8

9

10

F Fig. 16. The Call congestion (B) for the WC-ERAM, WC-ENAM and WC-ENLAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.8, and the number of wavelengths per fiber is N = 32.

1,0E+00

0,5

0,4

Blocking probability

Call Congestion

3

10

1,0E-01

ERAM

1,0E-02

ENAM

0,3

0,2

ENLAM

Time Congestion (E) 0,1

Call Congestion (B)

1,0E-03 2

3

4

5

6

7

8

9

10

Traffic Congestion (C)

F

0

Fig. 15. The Call congestion (B) for the WC-ERAM, WC-ENAM and WC-ENLAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.8, and the number of wavelengths per fiber is N = 16.

routing a packet from an input wavelength to the tagged output wavelength/fiber, N is the number of wavelengths at the tagged output wavelength/fiber, and A is the desired normalized system load. The right side of Eq. (87) denotes how much traffic that should be offered to the tagged output wavelength/fiber when the normalized system load is A (0 6 A 6 1), while the left side of Eq. (87) denotes the offered traffic for all input wavelengths that may route packets to the tagged output wavelength/fiber. By inserting Eq. (16) into Eq. (18), we obtain: j ar ¼ 1  j 1  ar g a¼ r þ gr

g¼

ð88Þ

2

3

4

5

6

7

8

9

10

F Fig. 17. The time-, call-, and traffic congestion for the NOWC-ENAM on a tagged output wavelength as a function of the number of input/output fibers (F). The normalized system load is A = 0.5.

We substitute Eq. (88) into Eq. (87) and obtain: g r ¼ NA r þ gr Sg ¼ NA þ NAg NA g¼ S  NA S

ð89Þ

The values for S and N are given in Table 2 for the NOWCENAM, NOWC-ENLAM, WC-ENAM, and WC-ENLAM. Hence, we obtain the following expressions for g for the NOWC-ENAM and the WC-ENAM in Eq. (90), and for the NOWC-ENLAM and the WC-ENLAM in Eq. (91).

H. Øverby / Computer Communications 30 (2007) 1229–1243 0,6

1241

1,0E-04

Blocking probability

Blocking probability

0,5

0,4

0,3

0,2

1,0E-05

Time Congestion (E)

Time Congestion (E)

Call Congestion (B)

Call Congestion (B)

0,1

Traffic Congestion (C)

Traffic Congestion (C)

0

1,0E-06 2

3

4

5

6

7

8

9

10

2

3

4

5

F

6

7

8

9

10

F

Fig. 18. The time-, call-, and traffic congestion for the NOWC-ENAM on a tagged output wavelength as a function of the number of input/output fibers (F). The normalized system load is A = 0.8.

Fig. 20. The time-, call-, and traffic congestion for the WC-ENAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.5, and the number of wavelengths per fiber is N = 32.

1,0E-02

Blocking probability

Blocking probability

1,0E-01

1,0E-03

Time Congestion (E) Call Congestion (B)

Time Congestion (E)

Traffic Congestion (C)

Call Congestion (B)

1,0E-04 2

3

4

5

6

7

8

9

10

F

Traffic Congestion (C) 1,0E-02 2

Fig. 19. The time-, call-, and traffic congestion for the WC-ENAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.5, and the number of wavelengths per fiber is N = 16.

A F A A g¼ F A1 g¼

3

4

5

6

7

8

9

10

F Fig. 21. The time-, call-, and traffic congestion for the WC-ENAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.8, and the number of wavelengths per fiber is N = 16.

ð90Þ ð91Þ

Eqs. (90) and (91) should be substituted in the blocking formulas in order to evaluate the blocking probability as a function of the parameter A. Figs. 11–16 show the call congestion (B) as a function of the parameter F for the NOWC-ENAM, NOWCENLAM, NOWC-ERAM, WC-ENAM, WC-ENLAM, and WC-ERAM. For the NOWC-ERAM and WCERAM, we assume a uniform traffic pattern and equal

arrival intensities on each input wavelength, i.e. mi = m and ri = r = 1/F. Figs. 17, 18 and 19–22 show the blocking probability evaluated using the time-, call-, and traffic congestion for the NOWC-ENAM and WC-ENAM, respectively. The major findings include: • For any value of the parameters F and A, the NOWCERAM shows a higher blocking probability (i.e. call congestion) than the NOWC-ENAM, which in turn shows a higher blocking probability than the NOWC-ENLAM. This can be seen in Figs. 11 and 12.

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H. Øverby / Computer Communications 30 (2007) 1229–1243

and approaches the variance of NOWC-ERAM. Although not shown explicitly in the figures, the following property holds regarding the call congestion:

1,0E-01

lim BNOWC-ENAM ¼ lim BNOWC-ENLAM ¼ BNOWC-ERAM

Blocking probability

F !1

F !1

ð92Þ

Time Congestion (E) Call Congestion (B)

Traffic Congestion (C) 1,0E-02 2

3

4

5

6

7

8

9

10

F

Fig. 22. The time-, call-, and traffic congestion for the WC-ENAM on a tagged output fiber as a function of the number of input/output fibers (F). The normalized system load is A = 0.8, and the number of wavelengths per fiber is N = 32.

Table 2 The values of S and N for the various arrival models Arrival model

S

N

NOWC-ENAM NOWC-ENLAM WC-ENAM WC-ENLAM

F F1 FN (F  1)N

1 1 N N

• For any value of the parameter F, N and A, the WCERAM shows a higher blocking probability (i.e. call congestion) than the WC-ENAM, which in turn shows a higher blocking probability than the WC-ENLAM. This is because in the WC-ERAM, packets arrive to a tagged output fiber from 1 input wavelengths, while in the WC-ENAM and NOWC-ENLAM, packets arrive to a tagged output fiber from FN and (F  1)N input wavelengths, respectively. This is seen in Figs. 13–16. Hence, the WC-ERAM has a higher variance regarding the number of arrivals to a tagged output fiber, followed by the WC-ENAM and the WC-ENLAM. • The blocking probability for the WC-ENAM and WCENLAM increases as the parameter F increases. • The difference in the observed call congestion between WC-ENAM, WC-ENLAM, and WC-ERAM decreases as the parameter F increases. This is because as the parameter F increases, the variance in the WC-ENAM and WC-ENLAM increases and approaches the variance for the WC-ERAM. Although not shown explicitly in the figures, the following property holds regarding the call congestion: lim BWC-ENAM ¼ lim BWC-ENLAM ¼ BWC-ERAM

F !1

The reason for this is because in the NOWC-ERAM, packets arrive to a tagged output wavelength from 1 input wavelengths, while in the NOWC-ENAM and NOWC-ENLAM, packets arrive to a tagged output wavelength from F and F  1 input wavelengths, respectively. Hence, the NOWC-ERAM has a higher variance regarding the number of arrivals to a tagged output wavelength, followed by the NOWC-ENAM and the NOWC-ENLAM. • The blocking probability for the NOWC-ENAM and NOWC-ENLAM increases as the parameter F increases. This is seen in Figs. 11 and 12, and is in accordance with the results from Eqs. (29) and (44). This is expected, since an increase in the parameter F increases the variance, which leads to an increased blocking probability. However, observe that the blocking probability for the NOWC-ERAM is independent of the parameter F, which means that the NOWC-ERAM does not capture the effect of increased blocking probability due to an increase in the parameter F. • The difference in the observed call congestion between the NOWC-ENAM, NOWC-ENLAM, and NOWCERAM decreases as the parameter F increases. This is because as the parameter F increases, the variance in the NOWC-ENAM and NOWC-ENLAM increases

F !1

ð93Þ

• The observed difference in the call congestion between WC-ERAM, WC-ENAM and WC-ENLAM increases as the system load (A) decreases and as the number of wavelengths per fiber (N) increases. • For the NOWC-ENLAM and the WC-ENLAM we see that the blocking probability (i.e. the call or traffic congestion) is 0 when F = 2. This is because in networks where looping is not allowed, contentions cannot occur when F = 2. This can be seen from Eqs. (41), (42), (82) and (83). However, note that this effect is not captured by the time congestion in Eqs. (40) and (81), i.e. ENOWC-ENLAM > 0 and EWC-ENLAM > 0 when F = 2. • For any value of the parameter F, N and A, we see that the time congestion is higher than the call congestion, which in turn is higher than the traffic congestion, i.e. ES P BS P CS ("S). This is seen in Figs. 17–22, and, as stated in [11], is a general characteristic of the Engset traffic model. • In Figs. 17, 18, we see that the call-, and traffic congestion increases as the parameter F increases, while the time congestion decreases as the parameter F increases. The latter issue is not in accordance with our expectations, and indicates that the time congestion is not a proper performance metric for asynchronous bufferless OPS without wavelength conversion.

H. Øverby / Computer Communications 30 (2007) 1229–1243

• For all values of the parameter N and A, the difference between the time-, call-, and traffic congestion decreases as the parameter F increases. This is seen in Figs. 17–22. Ultimately, we have the following relationships: lim ENOWCENAM ¼ lim BNOWCENAM ¼ lim C NOWCENAM ¼ ENOWCERAM

ð94Þ

limF !1 EWCENAM ¼ limF !1 BWCENAM ¼ limF !1 C WCENAM ¼ EWCERAM

ð95Þ

F !1

F !1

F !1

6. Conclusion In this article, we have presented various arrival models for asynchronous bufferless OPS. For each arrival model, we have derived the time-, call-, and traffic congestion. Our major findings can be summarized as: • Among the considered arrival models, we should expect the EAAM, ENAM, and ENLAM to be most accurate, i.e. closest to a realistic scenario, since they take into account the limited number of input wavelengths in the optical packet switch. Whether the EAAM, ENAM or the ENLAM is the most accurate depends on the uniformity of the traffic pattern and whether looping is allowed or not. However, it should be noted that the ENAM and ENLAM is both simplifications of the EAAM. • A numerical evaluation of the analytical models shows that there is a significant deviation in the blocking probability between the presented arrival models depending on the chosen performance metric, system load, number of input/output fibers and the number of wavelengths per fiber. These results should be taken into account when choosing an appropriate arrival model and performance metric for analytical modelling of asynchronous OPS. • Using the time congestion for evaluating the blocking probability has several drawbacks. First, the time congestion does not capture the increased variance due to an increasing number of input wavelengths in OPS without wavelength conversion. Second, the time congestion does not capture that the blocking probability in the NOWC-ENLAM and WC-ENLAM should be zero when F = 2. Future works should consider how an output port may be modelled using the Engset overflow traffic model, which is assumed to be more accurate, but also more complex. The models presented in this article may be extended to OPS with buffering capabilities by following the methodology in [5]. However, this is left for future research. References [1] M.J. O’Mahony, D. Simeonidou, D.K. Hunter, A. Tzanakaki, The application of optical packet switching in future communication networks, IEEE Communications Magazine 39 (3) (2001) 128–135.

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Harald Øverby received his M.Sc. degree in Telematics in 2002 and his Ph.D. in Information and Communication Technology in 2005, both from the Norwegian University of Science and Technology (NTNU). His main research interests include traffic analysis, survivability and Quality of Service issues in optical networks. Harald has published over 40 papers in international conferences and journals, and is currently employed as a postdoctoral researcher at the Department of Telematics at NTNU.