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Wavelength decomposition approach for computing blocking probabilities in WDM optical networks without wavelength conversions
Computer Networks 49 (2005) 727–742 www.elsevier.com/locate/comnet
Wavelength decomposition approach for computing blocking probabilities in WDM optical networks without wavelength conversions Anwar Alyatama
*
Department of Computer Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 12 May 2004; received in revised form 20 December 2004; accepted 25 February 2005 Available online 23 March 2005 Responsible Editor: A. Gencata
Abstract We present an approximate analytical method to evaluate the blocking probabilities in Wavelength Division Multiplexing (WDM) networks without wavelength converters. Our approach assumes fixed routing with Random or FirstFit wavelength assignment. The new approach views the WDM network as a set of different layers (colors) in which, blocked traffic in one layer is overflowed to another layer. Analyzing blocking probabilities in each layer of the network is derived from an exact approach. A moment matching method is then used to characterize the overflow traffic from one layer to another. The results indicate that our approach is more accurate than previous works. 2005 Elsevier B.V. All rights reserved. Keywords: Blocking probability; Routing and wavelength assignment; WDM
1. Introduction Wavelength Division Multiplexing (WDM) optical network has emerged as the best solution to face the rapid explosion of bandwidth demand. An aggregated throughput in the order of Terabits
*
Tel.: +965 9865046; fax: +965 5341850. E-mail address: [email protected]
per second range can be achieved by supporting multiple simultaneous transmission, each utilizing a different lightpath. The lightpath is a connected sequence of dedicated wavelengths on each link between the source and the destination nodes. We assume that the network does not have conversion capabilities therefore, the same wavelength must be available on all links belonging to a predetermined path (static routing). Wavelengths are dynamically assigned to the users of the network
1389-1286/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.comnet.2005.02.003
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A. Alyatama / Computer Networks 49 (2005) 727–742
on the predetermined path. In this paper, we explore two wavelength assignment techniques: • Random wavelength assignment: the technique chooses a random wavelength from the set of all wavelengths that are available along the path. • First-Fit wavelength assignment: the wavelengths are searched in a fixed order. Some lightpath requests are denied (blocked) due to unavailable network resources. It is important to estimate the call blocking probabilities based on the expected traffic demands on the network. The problem of evaluating call-blocking probabilities under static (fixed or alternate) routing, has been analyzed in several studies. They differ in their underlying assumptions and have varying computational complexities and levels of accuracy [1–3]. Many researchers have developed analytical techniques to calculate the call blocking probabilities for random wavelength assignment [2–6]. The work in [2] assumes that link loads have Markovian spatial correlation, i.e., the wavelengths used on a link are assumed to depend only on the wavelengths used on the previous link of a route under consideration. In [3], the path decomposition technique is used, where longer paths are divided into small segments (such as two, three and four links). An approximate Markov process, which has a closed-form solution, is obtained to solve for the small segments in isolation. The individual solutions are appropriately combined to obtain a solution for the original path. However, blocking probabilities in WDM networks with First-Fit wavelength assignment WA are harder to analyze [3]. In First-Fit WA, wavelengths are searched in a fixed order and traffic that are blocked from using a wavelength are offered to the next wavelength in line [7]. Using the layered graph approach (wavelength decomposition) to calculate the blocking probability is the natural choice (Fig. 1). However, most studies use the link independence assumption and the mean value of the overflow traffic [8]. It has been shown by many, that the link independence assumption overestimates the call blocking probabilities, especially when the nodal degree of the
Fig. 1. The network is decomposed into different layers (wavelengths).
WDM network is small [3,9,10]. On the other hand, using only the mean value of the overflow traffic, underestimates the call blocking probabilities (see Section 2). The calculations may produce satisfactory results in certain situations, when the overestimating effect of the link independence assumption is canceled by the underestimating effect of using the mean value of the overflow traffic. In [9], the link independence assumption is replaced by the object independence assumption, where the objects are free links and paths. The analytical models in [10–13] use layered graph approach (wavelength decomposition). The study in [10] uses the link independence assumption with the Equivalent Random Theory ERT. By using two moments, the ERT method will fix the underestimating part. However, fixing the underestimating part of the problem may make the impact of the overestimated part even larger. [11] has extended the link independence assumption with ERT to calculate the blocking probabilities with fixed alternate routing. In [12], the overflow traffic is characterized as a Bernoulli– Poisson–Pascal BPP process and each link is considered as BPP/M/1/1. The study in [13], analyzes the optimal routing and wavelength problem by assigning a virtual path between two nodes. The purpose of this paper is to derive an iterative model to calculate the path blocking probabilities for fixed routing in WDM networks with arbitrary topologies and without conversions. Our approach uses the wavelength independence assumption. It is observed by many, that the wave-
A. Alyatama / Computer Networks 49 (2005) 727–742
length independence assumption holds naturally and the WDM network can be regarded as an aggregation of disjoint single wavelength sub-networks with a common physical topology [9]. We analyze a given wavelength-routing network by dividing it into layers (colors) as shown in Fig. 1. The analysis of each layer is derived from an exact approach. The overflow traffic from one layer to another is characterized by a moment matching method (Section 2). An equivalent path method is used to calculate the overflow moments. These moments are used to calculate the equivalent Poisson overflow load used in the calculation of the path blocking probabilities. The rest of the paper is organized as follows. the next section reviews basic concepts in the Teletraffic theory used to derive our method. Section 3 describes our proposed solution for First-Fit WA. Section 4 describes our proposed solution for Random WA. Section 5 introduces some numerical results. In Section 6, we present our conclusions.
2. Teletraffic theory For completeness, we review the basic Teletraffic concepts that will be used to derive our method. There has been a significant amount of work done in analyzing the overflow traffic in traditional circuit switching network. Let us consider a primary system with N channels (as shown in Fig. 2). The offered load is a Poissonian traffic with rate k. The holding time is assumed to be one unit. New calls will overflow to the secondary system from
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the primary system if all N channels in the primary system are occupied. Let Ns be the number of channels in the secondary system. The blocking probability of the primary system is given by the Erlang-B formula Er ðk; N Þ ¼
kN PNN ! ki i¼0 i!
.
ð1Þ
One of the nice properties of the Erlang-B formula is that it can be extended to non-integral N variable [14], as follows: kN ek kN ek Er ðk; N Þ ¼ R 1 N t ¼ ; t e dt CðN þ 1Þ½1 CðN þ 1; kÞ k ð2Þ where C(N + 1, k) is the incomplete Gamma function. Although the offered traffic to the primary system is assumed to be Poisson, the overflow traffic to the secondary system is bursty (non-Poisson). Therefore, the assumption that the overflow traffic offered to the secondary system is Poisson underestimates the blocking probability in the secondary system. The characteristics of the overflow traffic had been extensively studied [16]. However, the mean M, the variance V and the peakedness Z are of particular interest. The mean M is given by M ¼ k Er ðk; N Þ.
ð3Þ
The variance of the overflow traffic V is calculated using RiordanÕs formula [15,16] k V ¼M 1M þ . ð4Þ N þ1kþM The peakedness Z is defined as ratio between the variance and the mean value i.e., V ð5Þ Z¼ . M
Fig. 2. A primary system with N channels, offered a Poissonian traffic with rate k. The overflow traffic is offered to a secondary system with Ns channels.
The overflow traffic has V > M (or Z > 1), which indicates that the overflow traffic is bursty (peaked). Hence, using only the mean M to characterize the overflow traffic will underestimate the blocking probability in the secondary system. Fredericks and HaywardÕs approximation is used to account for the burstiness (non-Poissonian) of overflow traffic [15,16]. It attempts to describe non-Poisson traffic Z 5 1 by an equivalent
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A. Alyatama / Computer Networks 49 (2005) 727–742
Poisson traffic Z = 1. Mainly, the blocking probability of the secondary system with Ns channels offered non-Poisson traffic with rate M and peakedness Z 5 1 has the same blocking probability for a system with Ns/Z channels, offered M/Z traffic, and peakedness Z = 1 (Poisson). Therefore, the blocking probability of the secondary system is given by M Ns ; Er . ð6Þ Z Z Since, the overflow traffic is bursty (Z > 1), then M Ns ; Er > Er ðM; N s Þ. Z Z On the right, only the mean value of the overflow traffic is being used which underestimates the blocking probability as mentioned before. Meanwhile, the mean m, the variance v and the peakedness z of the carried traffic in the primary system are given by [15,16] m ¼ kð1 Er ðk; N ÞÞ; v ¼ m kðN mÞEr ðk; N Þ. The carried traffic always has v < m (or z < 1), which indicates that the carried traffic is smoothed in contrast with the peaked overflow traffic. Therefore, if the carried traffic is offered to a second link and we only use the mean m to describe the second link offered load then, we will overestimate the second link blocking probability. This explains why the link independence assumption overestimates the path blocking probability, in which the offered load to a link is only characterized by the first moment (mean) of the carried traffic from the previous link(s). In general, the traffic is smoothed by the blocking on each link, and therefore experience less blocking on the last link of the path [15]. 3. Proposed solution for First-Fit WA In this section, we present the analytical tools that are used in our approach to analyze the blocking probabilities using wavelength decomposition. Let the order of wavelengths be numbered w = 1, 2, 3, . . . , W. Upon the arrival of a call to des-
tination d, the source s will offer the call to the first wavelength (layer) w = 1 on the predetermined fixed path from node s to node d. The call is accepted if the wavelength w is available on all links belonging to the predetermined fixed path. Otherwise, the call is offered to the next wavelength (w = 2). Thus, the traffic which cannot be carried by a wavelength w is offered to the next wavelength w + 1, and so on until the call is either accepted or blocked. The path blocking probability from source s to destination d for a given wavelength w is denoted by P ws;d . The following notations and assumptions will be used in this section and following sections: • A k 1 hop path r(1, k) shown in Fig. 3 consists of the set of k 1 links {(1, 2),(2, 3), . . . , (i, i + 1), . . . ,(j 1, j), . . . ,(k 1, k)}. In general, a path from node s to node d is denoted by r(s, d). • A path (or a segment) r(i, j) in Fig. 3 is a subset of path r(1, k) i.e., r(i, j) r(1, k). • The expression r(i, j) \ r(l, m) = ; denotes that there is no common subset between path r(i, j) and path r(l, m). • ks,d is the Poisson arrival rate for calls originated at source s to destination d. • l = 1 is the mean of the (exponentially distributed) call holding time. • Aws;d is the equivalent Poisson offered load to wavelength w and originated at source s to destination d, A1s;d ¼ ks;d . • awi;j is the total equivalent Poisson offered load to wavelength w from node i to node j. • nwi;j is the number of calls using the segment r(i, j), that are currently active in layer (wavelength) w. It is obvious that nwi;j 2 f0; 1g. • Cs,d is the path capacity from node s to node d. We now introduce the four steps used in our approach to calculate the blocking probabilities in WDM networks with fixed routing and First-Fit wavelength assignment. 3.1. The path blocking probability in a single layer Consider the k 1 hop route shown in Fig. 3, denoted as r(1, k). Let the state of a single wave-
A. Alyatama / Computer Networks 49 (2005) 727–742 w
a1w, j
a1w, 2 a1,3 1
aiw, k −1 aiw, j
a2w,3
2
hop 1
3
i
731
a wj ,k j
k
k-1
hop 2
hop k-1
Fig. 3. A k 1 hop path r(1, k).
length (layer) w at time t be described by the k(k 1)/2 dimensional process X wrð1;kÞ ðtÞ ¼ ðnw1;2 ðtÞ; nw1;3 ðtÞ; . . . ; nwk1;k ðtÞÞ.
a1w, 2
ð7Þ
The state of the k 1 hop path r(1, k) is thus denoted by the number of calls in progress for each segment r(i, j), 1 6 i < k, 1 < j 6 k, i < j, where nwi;j
þ
nwl;m
61
a2w,3
a1w,3
1
2
hop 1
8rði; jÞ \ rðl; mÞ 6¼ ; and
3
hop 2
Fig. 4. A two-hop path.
ð8Þ
1 6 l < k; 1 < m 6 k; l < m.
Process X wrð1;kÞ ðtÞ is time-reversible Markov process and the stationary vector p is given by [17] pðnw1;2 ; nw1;3 ; . . . ; nwk1;k Þ i 1 h w nw1;2 nw nw ða1;2 Þ ðaw1;3 Þ 1;3 ðawk1;k Þ k1;k ; ¼ w Grð1;kÞ
P wrð1;3Þ ¼ 1 P
h
1 w
w n nw þnw 61; ða1;2 Þ 1;2 1;2 1;3 w w n2;3 þn1;3 61
w
nw þnw 61 i;j l;m
rði;jÞ rð1;kÞ
8rði;jÞ\rðl;mÞ6¼; rði;jÞ rð1;kÞ;rðl;mÞ rð1;kÞ
The normalization constant Gwrð1;kÞ can be calculated recursively as Gwrð1;kÞ ¼ Gwrð1;k1Þ þ
k1 X
Gwrð1;iÞ awi;k ;
ð11Þ
i¼1
where Gwrð1;1Þ ¼ 1. Thus, path r(1, k) blocking probability P wrð1;kÞ (or P w1;k for short) in a single layer w 6 C1,k is calculated as P wrð1;kÞ ¼ 1 pð0; 0; . . . ; 0Þ ¼ 1
1 . Gwrð1;kÞ
ð12Þ
For example, P wrð1;3Þ for the two-hop path shown in Fig. 4 is
i.
ð13Þ
ð9Þ where Gwrð1;kÞ is the normalization constant for wavelength w on the path r(1, k) and is given by Y X w Gwrð1;kÞ ¼ ðawi;j Þni;j . ð10Þ
w
ðaw2;3 Þn2;3 ðaw1;3 Þn1;3
Or, simply P wrð1;3Þ ¼ 1
1þ
aw1;2
þ
aw2;3
1 . þ aw1;2 aw2;3 þ aw1;3
ð14Þ
Recall that, awi;j is the sum of all equivalent Poisson offered load from all source/destination pairs s, d on segment i, j at layer w, given that • segment i, j is a subset of r(1, k) path, i.e., r(i, j) r(1, k), • segment i, j is also a subset of s,d path, i.e., r(i, j) r(s, d), • traffic belonging to segment r(i, j) is unique, i.e., if rði; jÞ Aws;d then rðm; lÞ8Aws;d 8rðl; mÞ
rð1; kÞ, • the offered load to segment i, j is conditioned on the availability of other links belonging to the path s, d and not part of segment i, j. Hence, the reduced blocking path model is used [16].
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A. Alyatama / Computer Networks 49 (2005) 727–742
Thus, awi;j Õs for the path r(1, k) are calculated as X Aws;d ð1 P ws;d Þ awi;j ¼ . ð15Þ 1 P wi;j rði;jÞ rðs;dÞ then rðm;lÞ8Aw rði;jÞ Aw s;d s;d 8rðl;mÞ rð1;kÞ
3.2. Calculating the moments of the overflow traffic As mentioned earlier, the traffic which cannot be carried by a wavelength w is offered to the next wavelength w + 1. We need to calculate both the first and the second overflow traffic moments wþ1 and variance V wþ1 ) to the next layer (mean A w + 1. To accomplish this, we replace the path r(s, d) from source s to destination d at layer (wavelength) w by an equivalent single-link system with N ws;d 6 w such that, the blocking of the Poisson traffic in this system will approximate the blocking on the path r(s, d) (see Fig. 5). We know that the total offered load to the path is ks,d and the overflow mean, up to the current wavelength w is Aws;d P ws;d . Hence, ks;d Er ðks;d ; N ws;d Þ ¼ Aws;d P ws;d ;
ð16Þ
where, Er is the generalized (not integral) Erlang-B formula. The overflow mean to wavelength w + 1 is wþ1 ¼ ks;d Er ðks;d ; N w Þ. A s;d s;d
ð17Þ wþ1 V s;d
The variance of the overflow traffic calculated using RiordanÕs formula, Eq. (4) wþ1 V s;d
¼
wþ1 A s;d
1
wþ1 A s;d
þ
N ws;d
is
! ks;d . wþ1 ks;d þ1þA s;d
ð18Þ
The peakedness is calculated using Eq. (5), wþ1 V s;d wþ1 . Z s;d ¼ wþ1 As;d
3.3. Calculating the equivalent Poisson traffic The path blocking probability calculated in Eq. wþ1 ¼ V wþ1 . (12) assumes Poisson traffic with A s;d s;d However, the overflow traffic from Eqs. (17) and wþ1 (18) is in general non-Poisson Z s;d 6¼ 1. Again, we use an equivalent single-link system with wþ1 6 1 wavelengths to find an equivalent PoisN s;d son traffic with mean Awþ1 and Z wþ1 s;d s;d ¼ 1, that wþ1 and matches the overflow traffic with mean A s;d wþ1 variance V s;d . Fredericks and HaywardÕs approximation (Section 2) is used for this purpose where, it is proposed that a system with Non-Poisson traffic Z 5 1 has the same blocking probability as a system with N/Z channels, offered traffic A/Z and peakedness value = 1 (Poisson). Therefore, ! wþ1 N wþ1 A wþ1 s;d s;d wþ1 wþ1 wþ1 As;d Er As;d ; N s;d A ; wþ1 . s;d E r wþ1 Z Z s;d
ð19Þ wþ1 is calculated from The value of N s;d wþ1 wþ1 Er ðAwþ1 s;d ; N s;d Þ ¼ P s;d .
ð20Þ
Eqs. (12), (19) and (20), are solved iteratively for wþ1 which overeach layer. In general, Awþ1 s;d P As;d comes the underestimating effect of using one moment. Although, we used Fredericks and HaywardÕs approximation, other moment matching methods can be used to estimate the equivalent Poisson traffic. Such examples are SanderÕs method, the
λ s,d
λ s, d
w
w s
i
w
w j
A path up to wavelength
s;d
m
N sw,d d
w
Fig. 5. The equivalent path method.
s
d Equivalent Single-Link System
A. Alyatama / Computer Networks 49 (2005) 727–742
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Equivalent Random Traffic ERT method, BerkeleyÕs ERT approximation method, RappÕs ERT approximation method and the Bernoulli–Poisson–Pascal BPP method. Future research will explore these and other techniques.
First layer (Wavelength) w=1
Assume initial values for the path blocking probabilities Pw
3.4. Calculating the overall path blocking probability The overall path blocking probability is calculated as C
P s;d
Calculate the Equivalent Poisson traffic to layer w (Eq. 19, 20) w = w+1
C
As;ds;d P s;ds;d ¼ . ks;d
ð21Þ
Calculate new values for the path blocking probabilities new_Pw (Eq. 12)
Fig. 6 shows the main steps used to calculate the blocking probabilities for WDM networks with First-Fit wavelength assignment algorithm. Pw=new_Pw
No
|new_Pw-Pw| < ε
4. Proposed solution for random WA
Yes Calculate the moments of the overflow traffic to layer w+1 (Eq. 16, 17, 18)
The solution presented in Section 3, can also be used to analyze blocking probabilities when Random Wavelength Assignment technique is chosen. In this case, all wavelengths will be offered the same traffic on average. The average offered traffic to a wavelength, is the sum of offered traffic randomly selected as a first choice, second choice, etc. Thus, the average offered load to a wavelength for each source/destination pair is AAvg s;d ¼
C s;d X
Ats;d ;
ð22Þ
t¼1
where Ats;d is the portion of the equivalent Poisson load being offered to a wavelength on the t attempt and A1s;d ¼ ks;d =C s;d . Consequently, the average offered load to each segment is calculated as X aAvg i;j ¼ rði;jÞ
rði;jÞ rðs;dÞ Avg Avg As;d then rðm;lÞ8As;d
Avg AAvg s;d ð1 P s;d Þ
1 P Avg i;j aAvg i;j Õs
.
ð23Þ
The new are used to calculate the average path blocking probabilities P Avg s;d as
Last layer? Yes
Calculate the overall path blocking probabilities (Eq. 21)
Fig. 6. A block diagram showing the main steps used to calculate the blocking probabilities for WDM networks with First-Fit wavelength assignment algorithm.
P Avg s;d ¼ 1
1
ð24Þ
GAvg rðs;dÞ
and the normalization constant is given by, GAvg rðs;dÞ ¼
8rðl;mÞ rð1;kÞ
No
X
Y
ni;j þnl;m 61 rði;jÞ rðs;dÞ 8rði;jÞ\rðl;mÞ6¼; rði;jÞ rðs;dÞ;rðl;mÞ rðs;dÞ
ðaAvg i;j Þ ni;j !
ni;j
.
ð25Þ The normalization constant lated recursively as in Eq. (11)
GAvg rðs;dÞ
can be calcu-
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A. Alyatama / Computer Networks 49 (2005) 727–742
Avg GAvg rðs;dÞ ¼ Grðs;d1Þ þ
d1 X
Avg GAvg rðs;iÞ ai;d ;
ð26Þ
i¼s
GAvg rðs;sÞ
¼ 1. The next attempt (level) mean where offered load is tþ1 ¼ At P Avg . A s;d s;d s;d
ð27Þ
The next attempt (level) variance for the offered tþ1 load V s;d and equivalent Poisson offered load Atþ1 s;d are calculated using Eqs. (18) and (19), respec-
Assume initial values for the average path blocking probabilities Pavg
t=1
Calculate the mean and the variance of the overflow traffic after t attempts (Eq. 17, 18, 27)
t = t+1
Calculate the Equivalent Poisson overflow traffic after t attempts (Eq. 19, 20)
No
t=C Yes
tively. Finally, the overall path blocking probability is calculated as Cs;d . ð28Þ P s;d ¼ P Avg s;d Fig. 7 shows the main steps used to calculate the blocking probabilities for WDM networks with Random wavelength assignment algorithm.
5. Numerical results First, we present the two-hop path shown in Fig. 4, as an illustrative example for First-Fit WA. Let k1,2 = k2,3 = k,13 = 0.5 calls/unit time. Link capacities are C1,2 = C2,3 = 4. The load will be offered to the first wavelength w = 1, thus A11;2 ¼ k1;2 ; A12;3 ¼ k2;3 ; A11;3 ¼ k1;3 . The path r(1,3) blocking probability P w1;3 is given in Eq. (14) where the traffic are calculated as aw1;2 ¼ Aw1;2 ; aw2;3 ¼ Aw2;3 ; aw1;3 ¼ Aw1;3 ; w Þ ¼ Pw ; Er ðAw1;3 ; N 1;3 1;3 ! w N w A w w 1;3 1;3 w w A1;3 Er A1;3 ; N ; 1;3 A1;3 E r w ; w Z 1;3 Z 1;3 wþ1 ¼ Aw P w ; A 1;3 1;3 1;3 Er ðk1;3 ; N w1;3 Þ ¼
wþ1 A 1;3 ; k1;3
wþ1 wþ1 1 A wþ1 þ V 1;3 ¼ A 1;3 1;3
Calculate the average offered load to a wavelength (Eq. 22)
Pavg = new_Pavg
Calculate new values for the average path blocking probabilities new_Pavg (Eq. 24)
No
| new_Pavg - Pavg | < ε
Yes Calculate the overall path blocking probabilities (Eq. 28)
Fig. 7. A block diagram showing the main steps used to calculate the blocking probabilities for WDM networks with Random wavelength assignment algorithm.
N w1;3
! k1;3 . wþ1 k1;3 þ1þA 1;3
The single-link path r(1, 2) blocking probability P w12 ¼ 1 1þa1 w , however the single-link path r(2, 3) 1;2
blocking probability P w2;3 ¼ 1 1þa1 w . The traffic 2;3 are calculated as 1 P w1;3 ; 1 P w1;2 1 P w1;3 ¼ Aw2;3 þ Aw1;3 . 1 P w2;3
aw12 ¼ Aw1;2 þ Aw1;3 aw2;3
The complete results for each wavelength are given in Tables 1 and 2. Now, we validate our analytical method by comparing it to simulation. Simulation results
A. Alyatama / Computer Networks 49 (2005) 727–742
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Table 1 Traffic and blocking probability calculation for path 1, 3 w
Aw1;3
aw1;2
aw2;3
aw1;3
P w1;3
w N 1;3
N w1;3
wþ1 A 1;3
wþ1 V 1;3
wþ1 Z 1;3
1 2 3 4
0.5 0.3254 0.1621 0.0490
0.5 0.2390 0.0888 0.0202
0.5 0.2390 0.0888 0.0202
0.5 0.3254 0.1621 0.0490
0.6364 0.4621 0.2568 0.0818
0.4520 0.5903 0.7220 0.8348
0.4520 1.0794 1.9518 3.2296
0.3182 0.1504 0.0416 0.0040
0.3422 0.1712 0.0482 0.0045
1.0755 1.1387 1.1589 1.1299
Table 2 Traffic and blocking probability calculation for path 1, 2 w
Aw1;2
aw1;2
P w1;2
w N 1;2
N w1;2
wþ1 A 1;2
wþ1 V 1;2
wþ1 Z 1;2
1 2 3 4
0.5 0.2383 0.0886 0.0204
0.8335 0.5003 0.2358 0.0675
0.4546 0.3335 0.1908 0.0632
0.7488 0.7010 0.6915 0.7273
0.7488 1.5359 2.4766 3.7748
0.2273 0.0795 0.0169 0.0013
0.2526 0.0919 0.0195 0.0014
1.1114 1.1569 1.1501 1.1156
are plotted along with 95% confidence intervals estimated by the method of replications. The number of replications is 30, with each simulation run having a holding time of 2000. Each run starts with a different random seed. In the case of First-Fit WA, we also compare our technique to the method in which the link independence assumption and the mean value of the overflow traffic are used. Whereas, in the case of random WA, we compare our technique to the correlation model of [2] and the path decomposition model of [3]. 5.1. Linear topology In this section, we consider a seven-hop linear path with uniform load and uniform link capacities. In Fig. 8, we plot the overall blocking probability against the link capacity (number of wavelengths w) for First-Fit WA. The first curve is obtained from using the link independence assumption with the mean value for the overflow traffic while the second curve is plotted from our new approach. The third curve represents simulation results with the confidence intervals. The results show that, our proposed technique is better than using the link independence assumption with the mean value for the overflow traffic. Fig. 9 has four curves for Random WA in the seven-hop path with uniform load and uniform
capacities. The first curve is plotted from our new approach, the second and the third curves represent numerical results from using the correlation model of [2] and the path decomposition model of [3], respectively. The fourth curve represents the simulation results with the confidence intervals. In this case, the results show that our approximation is more accurate than other techniques. 5.2. Ring topology We plot in Figs. 10 and 11 the overall blocking probability for a ring with 101 nodes using FirstFit WA, against different values for the link capacities. Fig. 10 presents the ring with uniform load and uniform link capacities whereas, Fig. 11 presents the ring with non-uniform load and non-uniform link capacities. Both figures show the results from simulation with confidence intervals, our technique and the link independence assumption with the mean value for the overflow traffic. We observed in both figures that, our analytical method is so close to simulation results while the link independence assumption with the mean value for the overflow traffic underestimates the blocking probabilities. For Random WA, Figs. 12 and 13 show the overall blocking probability for the 101-node ring with uniform load/link capacities and non-uniform load/link capacities, respectively. Both figures
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A. Alyatama / Computer Networks 49 (2005) 727–742 100
Blocking Probabilities
10-1
10-2
10-3
10-4
Link Independence Wavelength Decomposition Simulation 10
20
30
40
50
60
70
80
90
100
Link Capacity (wavelength) Fig. 8. Overall end-to-end blocking probability for First-Fit wavelength assignment in a seven-hop linear path with uniform load and uniform capacities.
10
Blocking Probabilities
10
10
10
10
0
-1
-2
-3
-4
Wavelength Decomposition Correlation Model Path Decomposition Si mu lation 10
20
30
40
50
60
70
80
90
100
Link Capacity (wavelength)
Fig. 9. Overall end-to-end blocking probability for Random wavelength assignment in a seven-hop linear path with uniform load and uniform capacities.
A. Alyatama / Computer Networks 49 (2005) 727–742
737
Blocking Probabilities
100
10-1
10-2
Link Independence Wavelength Decomposition Simulation 5
10
15
20
25
30
Link Capacity (wavelength) Fig. 10. Overall end-to-end blocking probability for First-Fit wavelength assignment in a 101 nodes ring with uniform load and uniform capacities.
Blocking Probabilities
100
10-1
10-2
Link Independence Wavelength Decomposition Simulation
5
10
15
20
25
30
Link Capacity (wavelength) Fig. 11. Overall end-to-end blocking probability for First-Fit wavelength assignment in a 101 nodes ring with non-uniform load and non-uniform capacities.
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Blocking Probabilities
10
10
10
0
-1
-2
Wavelength Decomposition Simulation Correlation Model Path Decomposition 5
10
15
20
25
30
Link Capacity (wavelength)
Fig. 12. Overall end-to-end blocking probability for Random wavelength assignment in a 101 nodes ring with uniform load and uniform capacities.
Blocking Probabilities
10
10
10
0
-1
-2
Wavelength Decomposition Simulation Correlation Model Path Decomposition 5
10
15
20
25
30
Average Link Capacity (wavelength)
Fig. 13. Overall end-to-end blocking probability for Random wavelength assignment in a 101 nodes ring with non-uniform load and non-uniform capacities.
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show the results from simulation with confidence intervals, our technique, the correlation model of [2] and the path decomposition model of [3]. The simulation results show that, the performance of the proposed technique (wavelength decomposition) is comparable to that of the correlation model of [2] and the path decomposition model of [3] for a ring with Random WA.
739 7
14 10 8 13
1
11
3
2
5.3. Mesh topology
5
9
6
12 4
We used the NSFNET network topology (Fig. 14), which has 14 nodes and 20 links as a network model. Two traffic conditions will be studied, the realistic and the uniform case. The traffic, in the first case, is given in [18] and it presents some realistic observations. Moreover, the network link capacities in this case, are an outcome of the shortest path dimensioning based on the given static traffic matrix [18]. In the second case, the link capacities are uniform and the end-to-end traffic is also uniform. Different load factors are considered where, they range from 1 (light traffic) to 12 (heavy traffic).
Fig. 14. The 14-nodes NSFNET network topology.
We show first, the networkÕs overall blocking probability under static routing and First-Fit WA. In Fig. 15, we plot the overall blocking probability in the network against the load factor for the realistic case. The uniform case is shown is Fig. 16. In both figures, the first curve is obtained from using the link independence assumption with the mean value for the overflow traffic while the second curve is plotted from our new
0
10
Link Independence Wavelength Decomposition Simulation
-1
Blocking Probabilities
10
-2
10
-3
10
-4
10
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 15. Overall end-to-end blocking probability for static routing and First-Fit wavelength assignment in the 14-nodes NSFNET network topology with realistic load and capacities.
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0
Link Independence Wavelength Decomposition Simulation
Blocking Probabilities
10
10
10
10
-1
-2
-3
-4
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 16. Overall end-to-end blocking probability for static routing and First-Fit wavelength assignment in the 14-nodes NSFNET network topology with uniform load and uniform capacities.
0
10
Wavelength Decomposition Simulation Correlation Model Path Decomposition -1
Blocking Probabilities
10
-2
10
-3
10
-4
10
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 17. Overall end-to-end blocking probability for static routing and Random wavelength assignment in the 14-nodes NSFNET network topology with realistic load and capacities.
A. Alyatama / Computer Networks 49 (2005) 727–742 10
741
0
Wavelength Decomposition Simulation Correlation Model Path Decomposition
Blocking Probabilities
10
10
10
10
-1
-2
-3
-4
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 18. Overall end-to-end blocking probability for static routing and Random wavelength assignment in the 14-nodes NSFNET network topology with uniform load and uniform capacities.
approach. The third curve represents simulation results with the confidence intervals. The results show that, our proposed technique performs significantly better than the link independence assumption with the mean value for the overflow traffic. Next, we show the network performance under static routing with Random wavelength assignment. Fig. 17 shows the overall blocking probability for the mesh network with the realistic load and the dimensioned link capacities. Whereas, Fig. 18 shows the overall blocking probability for the mesh network with the uniform load and uniform link capacities. In both figures, the first curve is plotted from our new approach, the second and the third curves represent numerical results from using the correlation model of [2] and the path decomposition model of [3], respectively. The fourth curve represents the simulation results with the confidence intervals. Again, our approximation preforms better than other techniques for the mesh topology with Random wavelength assignment.
6. Conclusion We have presented a new analytical approach to evaluate more accurately the call blocking probabilities of wavelength-routing networks with an arbitrary topology. Our method is based on fixed routing with First-Fit/Random wavelength assignment. Our iterative algorithm analyzes the network by splitting it into layers (colors) and uses a moment matching method to calculate an equivalent Poisson overflow traffic to each layer. The analysis of blocking probabilities in each layer is derived from an exact approach. Our numerical results include linear, ring and mesh topologies with uniform and non-uniform traffic. For First-Fit wavelength assignment, the results show that our approach performs better than the link independence assumption with the mean value for the overflow traffic. Furthermore, the results show the accuracy of our approach for Random wavelength assignment. Exploring other techniques to calculate the equivalent Poisson traffic and applying our new method to sparse conversion, multiple
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classes and multicast networks will be considered in future research. References [1] A. Kamal, A. Alyatama, Blocking probabilities in circuitswitched wavelength division multiplexing networks under multicast service, Performance Evaluation 47 (1) (2002) 43–71. [2] S. Subramaniam, M. Azizoglu, A. Somani, All-optical networks with sparse wavelength conversion, IEEE/ACM Transactions on Networking 4 (4) (1996). [3] S. Ramesh, G. Rouskas, H. Perros, Computing blocking probabilities in multiclass wavelength routing network, IEEE Journal on Selected Areas in Communications 20 (January) (2002). [4] A. Birman, Computing approximate blocking probabilities for a class of all optical networks, IEEE Journal on Selected Areas in Communications 13 (June) (1996) 852–857. [5] R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Prospective, second ed., Morgan Kaufmann, 2002. [6] Y. Zhu, G. Rouskas, H. Peros, A path decomposition approach for computing blocking probabilities in wavelength-routing networks, IEEE/ACM Transactions on Networking 8 (6) (2000). [7] H. Zang, J. Jue, B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM network, Optical Networks Magazine (August, 1999). [8] C. Xin, C. Qiao, S. Dixit, Analysis of single-hop traffic grooming in mesh WDM optical networks, Optics Communications (2003) 91–101. [9] H. Waldman, D. Campelo, R. Almeida, A new analytical approach for the estimation of blocking probabilities in wavelength routing networks, Optics Communications (2003) 324–335. [10] E. Karasan, E. Ayanoglu, Effects of wavelength routing and selection algorithms on wavelength conversion gain in WDM optical networks, IEEE/ACM Transactions on Networking 6 (2) (1998).
[11] J. Yates, M. Rumsewicz, J. Lacey, D. Everitt, Modelling blocking probabilities in WDM network with fixed alternate routing, in: International Conference on Telecommunications, April 1997. [12] H. Harai, M. Murata, H. Miyahara, Performance analysis of wavelength assignment policies in all-optical networks with limited-range wavelength conversion, IEEE Journal on Selected Areas in Communications 16 (September) (1998) 1051–1060. [13] C. Chen, S. Banerjee, A new model for optimal routing and wavelength assignment in wavelength division multiplexed optical networks, INFOCOM (1996). [14] D. Jagerman, Some properties of the Erlang loss function, Bell System Technical Journal 53 (1974) 525–551. [15] ITU-D Study Group 2, Teletraffic Engineering Handbook, Geneva, December 2002. [16] A. Girard, Routing and Dimensioning in Circuit-switched Networks, Addison-Wesley, 1990. [17] C. Lea, A. Alyatama, Bandwidth quantization and states reduction in the broadband ISDN, IEEE/ACM Transactions on Networking 3 (3) (1995). [18] R. Hulsermann, M. Jager, S. Krumke, D. Poensgen, J. Ramboui, A. Turchscherer, Dynamic routing algorithms in transparent optical networks, in: Proceedings of ONDM, 2003, pp. 293–312.
Anwar Alyatama received his Ph.D. from Georgia Institute of Technology. He is now an Assistant Professor in the Computer Engineering Department at Kuwait University. He received the Teaching Excellence Award, College of Engineering, Kuwait University in July 1997. He also received the Kuwait Electronic Excellence Award, at the second E-Government Conference, April 2002. He also selected as the Top Ten Chief Information Officers (CIO) within the Middle East for the year 2002, Arabian Computer News (ACN). His research interests include wireless and broadband networks, queuing Analysis and Simulation. He is a member of IEEE.
Wavelength decomposition approach for computing blocking probabilities in WDM optical networks without wavelength conversions Anwar Alyatama
*
Department of Computer Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 12 May 2004; received in revised form 20 December 2004; accepted 25 February 2005 Available online 23 March 2005 Responsible Editor: A. Gencata
Abstract We present an approximate analytical method to evaluate the blocking probabilities in Wavelength Division Multiplexing (WDM) networks without wavelength converters. Our approach assumes fixed routing with Random or FirstFit wavelength assignment. The new approach views the WDM network as a set of different layers (colors) in which, blocked traffic in one layer is overflowed to another layer. Analyzing blocking probabilities in each layer of the network is derived from an exact approach. A moment matching method is then used to characterize the overflow traffic from one layer to another. The results indicate that our approach is more accurate than previous works. 2005 Elsevier B.V. All rights reserved. Keywords: Blocking probability; Routing and wavelength assignment; WDM
1. Introduction Wavelength Division Multiplexing (WDM) optical network has emerged as the best solution to face the rapid explosion of bandwidth demand. An aggregated throughput in the order of Terabits
*
Tel.: +965 9865046; fax: +965 5341850. E-mail address: [email protected]
per second range can be achieved by supporting multiple simultaneous transmission, each utilizing a different lightpath. The lightpath is a connected sequence of dedicated wavelengths on each link between the source and the destination nodes. We assume that the network does not have conversion capabilities therefore, the same wavelength must be available on all links belonging to a predetermined path (static routing). Wavelengths are dynamically assigned to the users of the network
1389-1286/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.comnet.2005.02.003
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on the predetermined path. In this paper, we explore two wavelength assignment techniques: • Random wavelength assignment: the technique chooses a random wavelength from the set of all wavelengths that are available along the path. • First-Fit wavelength assignment: the wavelengths are searched in a fixed order. Some lightpath requests are denied (blocked) due to unavailable network resources. It is important to estimate the call blocking probabilities based on the expected traffic demands on the network. The problem of evaluating call-blocking probabilities under static (fixed or alternate) routing, has been analyzed in several studies. They differ in their underlying assumptions and have varying computational complexities and levels of accuracy [1–3]. Many researchers have developed analytical techniques to calculate the call blocking probabilities for random wavelength assignment [2–6]. The work in [2] assumes that link loads have Markovian spatial correlation, i.e., the wavelengths used on a link are assumed to depend only on the wavelengths used on the previous link of a route under consideration. In [3], the path decomposition technique is used, where longer paths are divided into small segments (such as two, three and four links). An approximate Markov process, which has a closed-form solution, is obtained to solve for the small segments in isolation. The individual solutions are appropriately combined to obtain a solution for the original path. However, blocking probabilities in WDM networks with First-Fit wavelength assignment WA are harder to analyze [3]. In First-Fit WA, wavelengths are searched in a fixed order and traffic that are blocked from using a wavelength are offered to the next wavelength in line [7]. Using the layered graph approach (wavelength decomposition) to calculate the blocking probability is the natural choice (Fig. 1). However, most studies use the link independence assumption and the mean value of the overflow traffic [8]. It has been shown by many, that the link independence assumption overestimates the call blocking probabilities, especially when the nodal degree of the
Fig. 1. The network is decomposed into different layers (wavelengths).
WDM network is small [3,9,10]. On the other hand, using only the mean value of the overflow traffic, underestimates the call blocking probabilities (see Section 2). The calculations may produce satisfactory results in certain situations, when the overestimating effect of the link independence assumption is canceled by the underestimating effect of using the mean value of the overflow traffic. In [9], the link independence assumption is replaced by the object independence assumption, where the objects are free links and paths. The analytical models in [10–13] use layered graph approach (wavelength decomposition). The study in [10] uses the link independence assumption with the Equivalent Random Theory ERT. By using two moments, the ERT method will fix the underestimating part. However, fixing the underestimating part of the problem may make the impact of the overestimated part even larger. [11] has extended the link independence assumption with ERT to calculate the blocking probabilities with fixed alternate routing. In [12], the overflow traffic is characterized as a Bernoulli– Poisson–Pascal BPP process and each link is considered as BPP/M/1/1. The study in [13], analyzes the optimal routing and wavelength problem by assigning a virtual path between two nodes. The purpose of this paper is to derive an iterative model to calculate the path blocking probabilities for fixed routing in WDM networks with arbitrary topologies and without conversions. Our approach uses the wavelength independence assumption. It is observed by many, that the wave-
A. Alyatama / Computer Networks 49 (2005) 727–742
length independence assumption holds naturally and the WDM network can be regarded as an aggregation of disjoint single wavelength sub-networks with a common physical topology [9]. We analyze a given wavelength-routing network by dividing it into layers (colors) as shown in Fig. 1. The analysis of each layer is derived from an exact approach. The overflow traffic from one layer to another is characterized by a moment matching method (Section 2). An equivalent path method is used to calculate the overflow moments. These moments are used to calculate the equivalent Poisson overflow load used in the calculation of the path blocking probabilities. The rest of the paper is organized as follows. the next section reviews basic concepts in the Teletraffic theory used to derive our method. Section 3 describes our proposed solution for First-Fit WA. Section 4 describes our proposed solution for Random WA. Section 5 introduces some numerical results. In Section 6, we present our conclusions.
2. Teletraffic theory For completeness, we review the basic Teletraffic concepts that will be used to derive our method. There has been a significant amount of work done in analyzing the overflow traffic in traditional circuit switching network. Let us consider a primary system with N channels (as shown in Fig. 2). The offered load is a Poissonian traffic with rate k. The holding time is assumed to be one unit. New calls will overflow to the secondary system from
729
the primary system if all N channels in the primary system are occupied. Let Ns be the number of channels in the secondary system. The blocking probability of the primary system is given by the Erlang-B formula Er ðk; N Þ ¼
kN PNN ! ki i¼0 i!
.
ð1Þ
One of the nice properties of the Erlang-B formula is that it can be extended to non-integral N variable [14], as follows: kN ek kN ek Er ðk; N Þ ¼ R 1 N t ¼ ; t e dt CðN þ 1Þ½1 CðN þ 1; kÞ k ð2Þ where C(N + 1, k) is the incomplete Gamma function. Although the offered traffic to the primary system is assumed to be Poisson, the overflow traffic to the secondary system is bursty (non-Poisson). Therefore, the assumption that the overflow traffic offered to the secondary system is Poisson underestimates the blocking probability in the secondary system. The characteristics of the overflow traffic had been extensively studied [16]. However, the mean M, the variance V and the peakedness Z are of particular interest. The mean M is given by M ¼ k Er ðk; N Þ.
ð3Þ
The variance of the overflow traffic V is calculated using RiordanÕs formula [15,16] k V ¼M 1M þ . ð4Þ N þ1kþM The peakedness Z is defined as ratio between the variance and the mean value i.e., V ð5Þ Z¼ . M
Fig. 2. A primary system with N channels, offered a Poissonian traffic with rate k. The overflow traffic is offered to a secondary system with Ns channels.
The overflow traffic has V > M (or Z > 1), which indicates that the overflow traffic is bursty (peaked). Hence, using only the mean M to characterize the overflow traffic will underestimate the blocking probability in the secondary system. Fredericks and HaywardÕs approximation is used to account for the burstiness (non-Poissonian) of overflow traffic [15,16]. It attempts to describe non-Poisson traffic Z 5 1 by an equivalent
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Poisson traffic Z = 1. Mainly, the blocking probability of the secondary system with Ns channels offered non-Poisson traffic with rate M and peakedness Z 5 1 has the same blocking probability for a system with Ns/Z channels, offered M/Z traffic, and peakedness Z = 1 (Poisson). Therefore, the blocking probability of the secondary system is given by M Ns ; Er . ð6Þ Z Z Since, the overflow traffic is bursty (Z > 1), then M Ns ; Er > Er ðM; N s Þ. Z Z On the right, only the mean value of the overflow traffic is being used which underestimates the blocking probability as mentioned before. Meanwhile, the mean m, the variance v and the peakedness z of the carried traffic in the primary system are given by [15,16] m ¼ kð1 Er ðk; N ÞÞ; v ¼ m kðN mÞEr ðk; N Þ. The carried traffic always has v < m (or z < 1), which indicates that the carried traffic is smoothed in contrast with the peaked overflow traffic. Therefore, if the carried traffic is offered to a second link and we only use the mean m to describe the second link offered load then, we will overestimate the second link blocking probability. This explains why the link independence assumption overestimates the path blocking probability, in which the offered load to a link is only characterized by the first moment (mean) of the carried traffic from the previous link(s). In general, the traffic is smoothed by the blocking on each link, and therefore experience less blocking on the last link of the path [15]. 3. Proposed solution for First-Fit WA In this section, we present the analytical tools that are used in our approach to analyze the blocking probabilities using wavelength decomposition. Let the order of wavelengths be numbered w = 1, 2, 3, . . . , W. Upon the arrival of a call to des-
tination d, the source s will offer the call to the first wavelength (layer) w = 1 on the predetermined fixed path from node s to node d. The call is accepted if the wavelength w is available on all links belonging to the predetermined fixed path. Otherwise, the call is offered to the next wavelength (w = 2). Thus, the traffic which cannot be carried by a wavelength w is offered to the next wavelength w + 1, and so on until the call is either accepted or blocked. The path blocking probability from source s to destination d for a given wavelength w is denoted by P ws;d . The following notations and assumptions will be used in this section and following sections: • A k 1 hop path r(1, k) shown in Fig. 3 consists of the set of k 1 links {(1, 2),(2, 3), . . . , (i, i + 1), . . . ,(j 1, j), . . . ,(k 1, k)}. In general, a path from node s to node d is denoted by r(s, d). • A path (or a segment) r(i, j) in Fig. 3 is a subset of path r(1, k) i.e., r(i, j) r(1, k). • The expression r(i, j) \ r(l, m) = ; denotes that there is no common subset between path r(i, j) and path r(l, m). • ks,d is the Poisson arrival rate for calls originated at source s to destination d. • l = 1 is the mean of the (exponentially distributed) call holding time. • Aws;d is the equivalent Poisson offered load to wavelength w and originated at source s to destination d, A1s;d ¼ ks;d . • awi;j is the total equivalent Poisson offered load to wavelength w from node i to node j. • nwi;j is the number of calls using the segment r(i, j), that are currently active in layer (wavelength) w. It is obvious that nwi;j 2 f0; 1g. • Cs,d is the path capacity from node s to node d. We now introduce the four steps used in our approach to calculate the blocking probabilities in WDM networks with fixed routing and First-Fit wavelength assignment. 3.1. The path blocking probability in a single layer Consider the k 1 hop route shown in Fig. 3, denoted as r(1, k). Let the state of a single wave-
A. Alyatama / Computer Networks 49 (2005) 727–742 w
a1w, j
a1w, 2 a1,3 1
aiw, k −1 aiw, j
a2w,3
2
hop 1
3
i
731
a wj ,k j
k
k-1
hop 2
hop k-1
Fig. 3. A k 1 hop path r(1, k).
length (layer) w at time t be described by the k(k 1)/2 dimensional process X wrð1;kÞ ðtÞ ¼ ðnw1;2 ðtÞ; nw1;3 ðtÞ; . . . ; nwk1;k ðtÞÞ.
a1w, 2
ð7Þ
The state of the k 1 hop path r(1, k) is thus denoted by the number of calls in progress for each segment r(i, j), 1 6 i < k, 1 < j 6 k, i < j, where nwi;j
þ
nwl;m
61
a2w,3
a1w,3
1
2
hop 1
8rði; jÞ \ rðl; mÞ 6¼ ; and
3
hop 2
Fig. 4. A two-hop path.
ð8Þ
1 6 l < k; 1 < m 6 k; l < m.
Process X wrð1;kÞ ðtÞ is time-reversible Markov process and the stationary vector p is given by [17] pðnw1;2 ; nw1;3 ; . . . ; nwk1;k Þ i 1 h w nw1;2 nw nw ða1;2 Þ ðaw1;3 Þ 1;3 ðawk1;k Þ k1;k ; ¼ w Grð1;kÞ
P wrð1;3Þ ¼ 1 P
h
1 w
w n nw þnw 61; ða1;2 Þ 1;2 1;2 1;3 w w n2;3 þn1;3 61
w
nw þnw 61 i;j l;m
rði;jÞ rð1;kÞ
8rði;jÞ\rðl;mÞ6¼; rði;jÞ rð1;kÞ;rðl;mÞ rð1;kÞ
The normalization constant Gwrð1;kÞ can be calculated recursively as Gwrð1;kÞ ¼ Gwrð1;k1Þ þ
k1 X
Gwrð1;iÞ awi;k ;
ð11Þ
i¼1
where Gwrð1;1Þ ¼ 1. Thus, path r(1, k) blocking probability P wrð1;kÞ (or P w1;k for short) in a single layer w 6 C1,k is calculated as P wrð1;kÞ ¼ 1 pð0; 0; . . . ; 0Þ ¼ 1
1 . Gwrð1;kÞ
ð12Þ
For example, P wrð1;3Þ for the two-hop path shown in Fig. 4 is
i.
ð13Þ
ð9Þ where Gwrð1;kÞ is the normalization constant for wavelength w on the path r(1, k) and is given by Y X w Gwrð1;kÞ ¼ ðawi;j Þni;j . ð10Þ
w
ðaw2;3 Þn2;3 ðaw1;3 Þn1;3
Or, simply P wrð1;3Þ ¼ 1
1þ
aw1;2
þ
aw2;3
1 . þ aw1;2 aw2;3 þ aw1;3
ð14Þ
Recall that, awi;j is the sum of all equivalent Poisson offered load from all source/destination pairs s, d on segment i, j at layer w, given that • segment i, j is a subset of r(1, k) path, i.e., r(i, j) r(1, k), • segment i, j is also a subset of s,d path, i.e., r(i, j) r(s, d), • traffic belonging to segment r(i, j) is unique, i.e., if rði; jÞ Aws;d then rðm; lÞ8Aws;d 8rðl; mÞ
rð1; kÞ, • the offered load to segment i, j is conditioned on the availability of other links belonging to the path s, d and not part of segment i, j. Hence, the reduced blocking path model is used [16].
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Thus, awi;j Õs for the path r(1, k) are calculated as X Aws;d ð1 P ws;d Þ awi;j ¼ . ð15Þ 1 P wi;j rði;jÞ rðs;dÞ then rðm;lÞ8Aw rði;jÞ Aw s;d s;d 8rðl;mÞ rð1;kÞ
3.2. Calculating the moments of the overflow traffic As mentioned earlier, the traffic which cannot be carried by a wavelength w is offered to the next wavelength w + 1. We need to calculate both the first and the second overflow traffic moments wþ1 and variance V wþ1 ) to the next layer (mean A w + 1. To accomplish this, we replace the path r(s, d) from source s to destination d at layer (wavelength) w by an equivalent single-link system with N ws;d 6 w such that, the blocking of the Poisson traffic in this system will approximate the blocking on the path r(s, d) (see Fig. 5). We know that the total offered load to the path is ks,d and the overflow mean, up to the current wavelength w is Aws;d P ws;d . Hence, ks;d Er ðks;d ; N ws;d Þ ¼ Aws;d P ws;d ;
ð16Þ
where, Er is the generalized (not integral) Erlang-B formula. The overflow mean to wavelength w + 1 is wþ1 ¼ ks;d Er ðks;d ; N w Þ. A s;d s;d
ð17Þ wþ1 V s;d
The variance of the overflow traffic calculated using RiordanÕs formula, Eq. (4) wþ1 V s;d
¼
wþ1 A s;d
1
wþ1 A s;d
þ
N ws;d
is
! ks;d . wþ1 ks;d þ1þA s;d
ð18Þ
The peakedness is calculated using Eq. (5), wþ1 V s;d wþ1 . Z s;d ¼ wþ1 As;d
3.3. Calculating the equivalent Poisson traffic The path blocking probability calculated in Eq. wþ1 ¼ V wþ1 . (12) assumes Poisson traffic with A s;d s;d However, the overflow traffic from Eqs. (17) and wþ1 (18) is in general non-Poisson Z s;d 6¼ 1. Again, we use an equivalent single-link system with wþ1 6 1 wavelengths to find an equivalent PoisN s;d son traffic with mean Awþ1 and Z wþ1 s;d s;d ¼ 1, that wþ1 and matches the overflow traffic with mean A s;d wþ1 variance V s;d . Fredericks and HaywardÕs approximation (Section 2) is used for this purpose where, it is proposed that a system with Non-Poisson traffic Z 5 1 has the same blocking probability as a system with N/Z channels, offered traffic A/Z and peakedness value = 1 (Poisson). Therefore, ! wþ1 N wþ1 A wþ1 s;d s;d wþ1 wþ1 wþ1 As;d Er As;d ; N s;d A ; wþ1 . s;d E r wþ1 Z Z s;d
ð19Þ wþ1 is calculated from The value of N s;d wþ1 wþ1 Er ðAwþ1 s;d ; N s;d Þ ¼ P s;d .
ð20Þ
Eqs. (12), (19) and (20), are solved iteratively for wþ1 which overeach layer. In general, Awþ1 s;d P As;d comes the underestimating effect of using one moment. Although, we used Fredericks and HaywardÕs approximation, other moment matching methods can be used to estimate the equivalent Poisson traffic. Such examples are SanderÕs method, the
λ s,d
λ s, d
w
w s
i
w
w j
A path up to wavelength
s;d
m
N sw,d d
w
Fig. 5. The equivalent path method.
s
d Equivalent Single-Link System
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733
Equivalent Random Traffic ERT method, BerkeleyÕs ERT approximation method, RappÕs ERT approximation method and the Bernoulli–Poisson–Pascal BPP method. Future research will explore these and other techniques.
First layer (Wavelength) w=1
Assume initial values for the path blocking probabilities Pw
3.4. Calculating the overall path blocking probability The overall path blocking probability is calculated as C
P s;d
Calculate the Equivalent Poisson traffic to layer w (Eq. 19, 20) w = w+1
C
As;ds;d P s;ds;d ¼ . ks;d
ð21Þ
Calculate new values for the path blocking probabilities new_Pw (Eq. 12)
Fig. 6 shows the main steps used to calculate the blocking probabilities for WDM networks with First-Fit wavelength assignment algorithm. Pw=new_Pw
No
|new_Pw-Pw| < ε
4. Proposed solution for random WA
Yes Calculate the moments of the overflow traffic to layer w+1 (Eq. 16, 17, 18)
The solution presented in Section 3, can also be used to analyze blocking probabilities when Random Wavelength Assignment technique is chosen. In this case, all wavelengths will be offered the same traffic on average. The average offered traffic to a wavelength, is the sum of offered traffic randomly selected as a first choice, second choice, etc. Thus, the average offered load to a wavelength for each source/destination pair is AAvg s;d ¼
C s;d X
Ats;d ;
ð22Þ
t¼1
where Ats;d is the portion of the equivalent Poisson load being offered to a wavelength on the t attempt and A1s;d ¼ ks;d =C s;d . Consequently, the average offered load to each segment is calculated as X aAvg i;j ¼ rði;jÞ
rði;jÞ rðs;dÞ Avg Avg As;d then rðm;lÞ8As;d
Avg AAvg s;d ð1 P s;d Þ
1 P Avg i;j aAvg i;j Õs
.
ð23Þ
The new are used to calculate the average path blocking probabilities P Avg s;d as
Last layer? Yes
Calculate the overall path blocking probabilities (Eq. 21)
Fig. 6. A block diagram showing the main steps used to calculate the blocking probabilities for WDM networks with First-Fit wavelength assignment algorithm.
P Avg s;d ¼ 1
1
ð24Þ
GAvg rðs;dÞ
and the normalization constant is given by, GAvg rðs;dÞ ¼
8rðl;mÞ rð1;kÞ
No
X
Y
ni;j þnl;m 61 rði;jÞ rðs;dÞ 8rði;jÞ\rðl;mÞ6¼; rði;jÞ rðs;dÞ;rðl;mÞ rðs;dÞ
ðaAvg i;j Þ ni;j !
ni;j
.
ð25Þ The normalization constant lated recursively as in Eq. (11)
GAvg rðs;dÞ
can be calcu-
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A. Alyatama / Computer Networks 49 (2005) 727–742
Avg GAvg rðs;dÞ ¼ Grðs;d1Þ þ
d1 X
Avg GAvg rðs;iÞ ai;d ;
ð26Þ
i¼s
GAvg rðs;sÞ
¼ 1. The next attempt (level) mean where offered load is tþ1 ¼ At P Avg . A s;d s;d s;d
ð27Þ
The next attempt (level) variance for the offered tþ1 load V s;d and equivalent Poisson offered load Atþ1 s;d are calculated using Eqs. (18) and (19), respec-
Assume initial values for the average path blocking probabilities Pavg
t=1
Calculate the mean and the variance of the overflow traffic after t attempts (Eq. 17, 18, 27)
t = t+1
Calculate the Equivalent Poisson overflow traffic after t attempts (Eq. 19, 20)
No
t=C Yes
tively. Finally, the overall path blocking probability is calculated as Cs;d . ð28Þ P s;d ¼ P Avg s;d Fig. 7 shows the main steps used to calculate the blocking probabilities for WDM networks with Random wavelength assignment algorithm.
5. Numerical results First, we present the two-hop path shown in Fig. 4, as an illustrative example for First-Fit WA. Let k1,2 = k2,3 = k,13 = 0.5 calls/unit time. Link capacities are C1,2 = C2,3 = 4. The load will be offered to the first wavelength w = 1, thus A11;2 ¼ k1;2 ; A12;3 ¼ k2;3 ; A11;3 ¼ k1;3 . The path r(1,3) blocking probability P w1;3 is given in Eq. (14) where the traffic are calculated as aw1;2 ¼ Aw1;2 ; aw2;3 ¼ Aw2;3 ; aw1;3 ¼ Aw1;3 ; w Þ ¼ Pw ; Er ðAw1;3 ; N 1;3 1;3 ! w N w A w w 1;3 1;3 w w A1;3 Er A1;3 ; N ; 1;3 A1;3 E r w ; w Z 1;3 Z 1;3 wþ1 ¼ Aw P w ; A 1;3 1;3 1;3 Er ðk1;3 ; N w1;3 Þ ¼
wþ1 A 1;3 ; k1;3
wþ1 wþ1 1 A wþ1 þ V 1;3 ¼ A 1;3 1;3
Calculate the average offered load to a wavelength (Eq. 22)
Pavg = new_Pavg
Calculate new values for the average path blocking probabilities new_Pavg (Eq. 24)
No
| new_Pavg - Pavg | < ε
Yes Calculate the overall path blocking probabilities (Eq. 28)
Fig. 7. A block diagram showing the main steps used to calculate the blocking probabilities for WDM networks with Random wavelength assignment algorithm.
N w1;3
! k1;3 . wþ1 k1;3 þ1þA 1;3
The single-link path r(1, 2) blocking probability P w12 ¼ 1 1þa1 w , however the single-link path r(2, 3) 1;2
blocking probability P w2;3 ¼ 1 1þa1 w . The traffic 2;3 are calculated as 1 P w1;3 ; 1 P w1;2 1 P w1;3 ¼ Aw2;3 þ Aw1;3 . 1 P w2;3
aw12 ¼ Aw1;2 þ Aw1;3 aw2;3
The complete results for each wavelength are given in Tables 1 and 2. Now, we validate our analytical method by comparing it to simulation. Simulation results
A. Alyatama / Computer Networks 49 (2005) 727–742
735
Table 1 Traffic and blocking probability calculation for path 1, 3 w
Aw1;3
aw1;2
aw2;3
aw1;3
P w1;3
w N 1;3
N w1;3
wþ1 A 1;3
wþ1 V 1;3
wþ1 Z 1;3
1 2 3 4
0.5 0.3254 0.1621 0.0490
0.5 0.2390 0.0888 0.0202
0.5 0.2390 0.0888 0.0202
0.5 0.3254 0.1621 0.0490
0.6364 0.4621 0.2568 0.0818
0.4520 0.5903 0.7220 0.8348
0.4520 1.0794 1.9518 3.2296
0.3182 0.1504 0.0416 0.0040
0.3422 0.1712 0.0482 0.0045
1.0755 1.1387 1.1589 1.1299
Table 2 Traffic and blocking probability calculation for path 1, 2 w
Aw1;2
aw1;2
P w1;2
w N 1;2
N w1;2
wþ1 A 1;2
wþ1 V 1;2
wþ1 Z 1;2
1 2 3 4
0.5 0.2383 0.0886 0.0204
0.8335 0.5003 0.2358 0.0675
0.4546 0.3335 0.1908 0.0632
0.7488 0.7010 0.6915 0.7273
0.7488 1.5359 2.4766 3.7748
0.2273 0.0795 0.0169 0.0013
0.2526 0.0919 0.0195 0.0014
1.1114 1.1569 1.1501 1.1156
are plotted along with 95% confidence intervals estimated by the method of replications. The number of replications is 30, with each simulation run having a holding time of 2000. Each run starts with a different random seed. In the case of First-Fit WA, we also compare our technique to the method in which the link independence assumption and the mean value of the overflow traffic are used. Whereas, in the case of random WA, we compare our technique to the correlation model of [2] and the path decomposition model of [3]. 5.1. Linear topology In this section, we consider a seven-hop linear path with uniform load and uniform link capacities. In Fig. 8, we plot the overall blocking probability against the link capacity (number of wavelengths w) for First-Fit WA. The first curve is obtained from using the link independence assumption with the mean value for the overflow traffic while the second curve is plotted from our new approach. The third curve represents simulation results with the confidence intervals. The results show that, our proposed technique is better than using the link independence assumption with the mean value for the overflow traffic. Fig. 9 has four curves for Random WA in the seven-hop path with uniform load and uniform
capacities. The first curve is plotted from our new approach, the second and the third curves represent numerical results from using the correlation model of [2] and the path decomposition model of [3], respectively. The fourth curve represents the simulation results with the confidence intervals. In this case, the results show that our approximation is more accurate than other techniques. 5.2. Ring topology We plot in Figs. 10 and 11 the overall blocking probability for a ring with 101 nodes using FirstFit WA, against different values for the link capacities. Fig. 10 presents the ring with uniform load and uniform link capacities whereas, Fig. 11 presents the ring with non-uniform load and non-uniform link capacities. Both figures show the results from simulation with confidence intervals, our technique and the link independence assumption with the mean value for the overflow traffic. We observed in both figures that, our analytical method is so close to simulation results while the link independence assumption with the mean value for the overflow traffic underestimates the blocking probabilities. For Random WA, Figs. 12 and 13 show the overall blocking probability for the 101-node ring with uniform load/link capacities and non-uniform load/link capacities, respectively. Both figures
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A. Alyatama / Computer Networks 49 (2005) 727–742 100
Blocking Probabilities
10-1
10-2
10-3
10-4
Link Independence Wavelength Decomposition Simulation 10
20
30
40
50
60
70
80
90
100
Link Capacity (wavelength) Fig. 8. Overall end-to-end blocking probability for First-Fit wavelength assignment in a seven-hop linear path with uniform load and uniform capacities.
10
Blocking Probabilities
10
10
10
10
0
-1
-2
-3
-4
Wavelength Decomposition Correlation Model Path Decomposition Si mu lation 10
20
30
40
50
60
70
80
90
100
Link Capacity (wavelength)
Fig. 9. Overall end-to-end blocking probability for Random wavelength assignment in a seven-hop linear path with uniform load and uniform capacities.
A. Alyatama / Computer Networks 49 (2005) 727–742
737
Blocking Probabilities
100
10-1
10-2
Link Independence Wavelength Decomposition Simulation 5
10
15
20
25
30
Link Capacity (wavelength) Fig. 10. Overall end-to-end blocking probability for First-Fit wavelength assignment in a 101 nodes ring with uniform load and uniform capacities.
Blocking Probabilities
100
10-1
10-2
Link Independence Wavelength Decomposition Simulation
5
10
15
20
25
30
Link Capacity (wavelength) Fig. 11. Overall end-to-end blocking probability for First-Fit wavelength assignment in a 101 nodes ring with non-uniform load and non-uniform capacities.
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Blocking Probabilities
10
10
10
0
-1
-2
Wavelength Decomposition Simulation Correlation Model Path Decomposition 5
10
15
20
25
30
Link Capacity (wavelength)
Fig. 12. Overall end-to-end blocking probability for Random wavelength assignment in a 101 nodes ring with uniform load and uniform capacities.
Blocking Probabilities
10
10
10
0
-1
-2
Wavelength Decomposition Simulation Correlation Model Path Decomposition 5
10
15
20
25
30
Average Link Capacity (wavelength)
Fig. 13. Overall end-to-end blocking probability for Random wavelength assignment in a 101 nodes ring with non-uniform load and non-uniform capacities.
A. Alyatama / Computer Networks 49 (2005) 727–742
show the results from simulation with confidence intervals, our technique, the correlation model of [2] and the path decomposition model of [3]. The simulation results show that, the performance of the proposed technique (wavelength decomposition) is comparable to that of the correlation model of [2] and the path decomposition model of [3] for a ring with Random WA.
739 7
14 10 8 13
1
11
3
2
5.3. Mesh topology
5
9
6
12 4
We used the NSFNET network topology (Fig. 14), which has 14 nodes and 20 links as a network model. Two traffic conditions will be studied, the realistic and the uniform case. The traffic, in the first case, is given in [18] and it presents some realistic observations. Moreover, the network link capacities in this case, are an outcome of the shortest path dimensioning based on the given static traffic matrix [18]. In the second case, the link capacities are uniform and the end-to-end traffic is also uniform. Different load factors are considered where, they range from 1 (light traffic) to 12 (heavy traffic).
Fig. 14. The 14-nodes NSFNET network topology.
We show first, the networkÕs overall blocking probability under static routing and First-Fit WA. In Fig. 15, we plot the overall blocking probability in the network against the load factor for the realistic case. The uniform case is shown is Fig. 16. In both figures, the first curve is obtained from using the link independence assumption with the mean value for the overflow traffic while the second curve is plotted from our new
0
10
Link Independence Wavelength Decomposition Simulation
-1
Blocking Probabilities
10
-2
10
-3
10
-4
10
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 15. Overall end-to-end blocking probability for static routing and First-Fit wavelength assignment in the 14-nodes NSFNET network topology with realistic load and capacities.
740
A. Alyatama / Computer Networks 49 (2005) 727–742 10
0
Link Independence Wavelength Decomposition Simulation
Blocking Probabilities
10
10
10
10
-1
-2
-3
-4
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 16. Overall end-to-end blocking probability for static routing and First-Fit wavelength assignment in the 14-nodes NSFNET network topology with uniform load and uniform capacities.
0
10
Wavelength Decomposition Simulation Correlation Model Path Decomposition -1
Blocking Probabilities
10
-2
10
-3
10
-4
10
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 17. Overall end-to-end blocking probability for static routing and Random wavelength assignment in the 14-nodes NSFNET network topology with realistic load and capacities.
A. Alyatama / Computer Networks 49 (2005) 727–742 10
741
0
Wavelength Decomposition Simulation Correlation Model Path Decomposition
Blocking Probabilities
10
10
10
10
-1
-2
-3
-4
1
2
3
4
5
6
7
8
9
10
11
12
Load Factor
Fig. 18. Overall end-to-end blocking probability for static routing and Random wavelength assignment in the 14-nodes NSFNET network topology with uniform load and uniform capacities.
approach. The third curve represents simulation results with the confidence intervals. The results show that, our proposed technique performs significantly better than the link independence assumption with the mean value for the overflow traffic. Next, we show the network performance under static routing with Random wavelength assignment. Fig. 17 shows the overall blocking probability for the mesh network with the realistic load and the dimensioned link capacities. Whereas, Fig. 18 shows the overall blocking probability for the mesh network with the uniform load and uniform link capacities. In both figures, the first curve is plotted from our new approach, the second and the third curves represent numerical results from using the correlation model of [2] and the path decomposition model of [3], respectively. The fourth curve represents the simulation results with the confidence intervals. Again, our approximation preforms better than other techniques for the mesh topology with Random wavelength assignment.
6. Conclusion We have presented a new analytical approach to evaluate more accurately the call blocking probabilities of wavelength-routing networks with an arbitrary topology. Our method is based on fixed routing with First-Fit/Random wavelength assignment. Our iterative algorithm analyzes the network by splitting it into layers (colors) and uses a moment matching method to calculate an equivalent Poisson overflow traffic to each layer. The analysis of blocking probabilities in each layer is derived from an exact approach. Our numerical results include linear, ring and mesh topologies with uniform and non-uniform traffic. For First-Fit wavelength assignment, the results show that our approach performs better than the link independence assumption with the mean value for the overflow traffic. Furthermore, the results show the accuracy of our approach for Random wavelength assignment. Exploring other techniques to calculate the equivalent Poisson traffic and applying our new method to sparse conversion, multiple
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classes and multicast networks will be considered in future research. References [1] A. Kamal, A. Alyatama, Blocking probabilities in circuitswitched wavelength division multiplexing networks under multicast service, Performance Evaluation 47 (1) (2002) 43–71. [2] S. Subramaniam, M. Azizoglu, A. Somani, All-optical networks with sparse wavelength conversion, IEEE/ACM Transactions on Networking 4 (4) (1996). [3] S. Ramesh, G. Rouskas, H. Perros, Computing blocking probabilities in multiclass wavelength routing network, IEEE Journal on Selected Areas in Communications 20 (January) (2002). [4] A. Birman, Computing approximate blocking probabilities for a class of all optical networks, IEEE Journal on Selected Areas in Communications 13 (June) (1996) 852–857. [5] R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Prospective, second ed., Morgan Kaufmann, 2002. [6] Y. Zhu, G. Rouskas, H. Peros, A path decomposition approach for computing blocking probabilities in wavelength-routing networks, IEEE/ACM Transactions on Networking 8 (6) (2000). [7] H. Zang, J. Jue, B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM network, Optical Networks Magazine (August, 1999). [8] C. Xin, C. Qiao, S. Dixit, Analysis of single-hop traffic grooming in mesh WDM optical networks, Optics Communications (2003) 91–101. [9] H. Waldman, D. Campelo, R. Almeida, A new analytical approach for the estimation of blocking probabilities in wavelength routing networks, Optics Communications (2003) 324–335. [10] E. Karasan, E. Ayanoglu, Effects of wavelength routing and selection algorithms on wavelength conversion gain in WDM optical networks, IEEE/ACM Transactions on Networking 6 (2) (1998).
[11] J. Yates, M. Rumsewicz, J. Lacey, D. Everitt, Modelling blocking probabilities in WDM network with fixed alternate routing, in: International Conference on Telecommunications, April 1997. [12] H. Harai, M. Murata, H. Miyahara, Performance analysis of wavelength assignment policies in all-optical networks with limited-range wavelength conversion, IEEE Journal on Selected Areas in Communications 16 (September) (1998) 1051–1060. [13] C. Chen, S. Banerjee, A new model for optimal routing and wavelength assignment in wavelength division multiplexed optical networks, INFOCOM (1996). [14] D. Jagerman, Some properties of the Erlang loss function, Bell System Technical Journal 53 (1974) 525–551. [15] ITU-D Study Group 2, Teletraffic Engineering Handbook, Geneva, December 2002. [16] A. Girard, Routing and Dimensioning in Circuit-switched Networks, Addison-Wesley, 1990. [17] C. Lea, A. Alyatama, Bandwidth quantization and states reduction in the broadband ISDN, IEEE/ACM Transactions on Networking 3 (3) (1995). [18] R. Hulsermann, M. Jager, S. Krumke, D. Poensgen, J. Ramboui, A. Turchscherer, Dynamic routing algorithms in transparent optical networks, in: Proceedings of ONDM, 2003, pp. 293–312.
Anwar Alyatama received his Ph.D. from Georgia Institute of Technology. He is now an Assistant Professor in the Computer Engineering Department at Kuwait University. He received the Teaching Excellence Award, College of Engineering, Kuwait University in July 1997. He also received the Kuwait Electronic Excellence Award, at the second E-Government Conference, April 2002. He also selected as the Top Ten Chief Information Officers (CIO) within the Middle East for the year 2002, Arabian Computer News (ACN). His research interests include wireless and broadband networks, queuing Analysis and Simulation. He is a member of IEEE.