bounds of the overflow priority classification for blocking...

Bounds of the Overflow PriorityClassification for Blocking Probability

Approximation in OBS NetworksShuo Li, Meiqian Wang, Eric W. M. Wong, Vyacheslav Abramov, and Moshe Zukerman

Abstract—It has been demonstrated that the overflowpriority classification approximation (OPCA) is an accu-ratemethod for blocking probability evaluation for variousnetworks and systems, including optical burst switchednetworks with deflection routing. OPCA is a hierarchicalalgorithm that requires fixed-point iterations in each layerof its hierarchy. This may imply a long running time. Weprove here that the OPCA iterations alternately produceupper and lower bounds that consistently become closerto each other as more fixed-point iterations in each layerare used, andwe demonstrate numerically that only a smallnumber of iterations per layer are required for the boundsto be sufficiently close to each other. This behavior is dem-onstrated for various system parameters including offeredload, number of channels per trunk, and maximum allow-able number of deflections.

Index Terms—Deflection routing; Loss networks; Lowerand upper bounds; Nonhierarchical networks; Opticalburst switching (OBS); Optical packet switching (OPS);Overflow priority classification approximation; Perfor-mance analysis.

I. INTRODUCTION

O ptical burst switching (OBS) [1–7] is an opticalnetworking technique where packets are aggregatedinto data bursts at the network edge and each burst istransmitted as one unit. It aims to achieve faster connec-tion time than optical circuit switching [8–11] and to avoidenergy consuming processing of individual packets and ex-cessive overhead due to guard-band provision betweenpackets, as in optical packet switching [6,12–14].

An important consideration in OBS networks is burstcontention that may lead to burst dumping and conse-quently loss of data [15–18]. Given that buffering datain the optical domain is difficult, especially for large bursts,one of the key contention resolution options is deflectionrouting [18]. As such, OBS deflection routing and itsperformance analysis has attracted significant attention

[19–32]. However, most of the existing performance studieswere either based on simulations or were limited to a singleOBS node. In [32], the blocking probability of an OBS net-work with deflection was evaluated using the Erlang fixed-point approximation (EFPA) [33]. Then, in [34], a recentlydeveloped overflow priority classification approximation(OPCA) [35] was used in combination with the EFPA toobtain a blocking probability approximation for this prob-lem that, as demonstrated there, is significantly moreaccurate than merely the approximation using the EFPA.Given the important role of the OPCA in accurately evalu-ating blocking probability of telecommunications networksand systems as demonstrated in [34] for OBS networks,and in [35] and [36] for other networks, the main focus ofthis paper is not on the accuracy and performance of theOPCA; instead, it provides new information about theproperties of the OPCA algorithm applied to bufferlessOBS networks based on just-enough-time (JET) signaling[37]. This new information has the potential for widerapplicability, as it can be used for further developmentof the OPCA in other applications.

A weakness of both the EFPA and the original OPCAapplied to OBS networks is that they require a fixed-pointsolution, which may require a large number of iterations.Because of the fixed-point iterations, analytical results forthe complexities of both the EFPA and OPCA are unattain-able. However, numerical studies presented in this paperindicate that the OPCA consumes less time than the EFPA,and that the advantage of the OPCA increases with thecapacity of the network (see Subsection V.B). For example,the EFPA requires 3006 s, compared to 397 s for the OPCA,to evaluate blocking probability for a National ScienceFoundation Network (NSFNET) topology with 10,000channels per trunk (such a number is not unreasonable[38–40], especially if subwavelength channels are consid-ered [41]). The advantage of the OPCA in running timeis probably due to the fact that the OPCA is based on a hier-archical structure with a finite number of layers, where ateach layer a separate set of fixed-point iterations are per-formed. Experience shows that this divide and rule ap-proach tends to reduce the total number of iterations.Although accuracy is not the main topic of this paper, wedo provide some new (to our knowledge) numerical resultsthat complement the running time comparison and demon-strate that for the cases of OBS/JET considered, within apractical traffic loading range, i.e., so that an acceptableDigitial Object Identifier 10.1364/JOCN.5.000378

Manuscript receivedMay 8, 2012; revised January 9, 2013; accepted Feb-ruary 14, 2013; published March 29, 2013 (Doc. ID 168224).

Shuo Li (e-mail: [email protected]), MeiqianWang, EricWong, andMoshe Zukerman are with the Department of Electronic Engineering, CityUniversity of Hong Kong, Hong Kong, China.

Vyacheslav Abramov is with the Center for Advanced Internet Architec-tures, Swinburne University of Technology, John Street, P.O. Box 218,Hawthorn 3122, Australia.

378 J. OPT. COMMUN. NETW./VOL. 5, NO. 4/APRIL 2013 Li et al.

1943-0620/13/040378-16$15.00/0 © 2013 Optical Society of America

grade of service is met, the OPCA is faster than and at leastas accurate as the EFPA.

We discover in this paper a new important property ofthe OPCA algorithm. In particular, we show that we canfind upper and lower bounds for the blocking probabilityevaluated by the OPCA such that they draw nearer to eachother with an increasing number of iterations. Specifically,in each iteration, the distance between them is never largerthan in the previous iteration. The effects of the designparameters (such as maximum allowable number of deflec-tions and number of channels per trunk) on the behavior ofthe OPCA blocking probability bounds are also discussed.It is important to clarify that the bounds discussed in thispaper are always the bounds of the OPCA result and not thebounds of the exact blocking probability result.

The remainder of the paper is structured as follows. Thedescription of the model is given in Section II. In Section IIIwe recall basics of the OPCA method, and in Section IV weprovide bounds of the OPCA results. To support the resultsof Section IV, in Section V numerical results for the OPCAbounds for the 13-node NSFNET are provided, as well asthe effects of the number of channels per trunk and themaximum allowable number of deflections. In Section VI,we conclude the paper.

II. MODEL

We consider an OBS network described by a graphG�N;E�, where N is a set of n nodes and E is the set ofe arcs. The nodes are designated 1; 2;…; n, each of whichis either an optical cross connect or an edge router. Thee arcs represent trunks, where trunk i ∈ E is composedof f i fibers, each of which supports wi wavelengths. Weassume full wavelength conversion in this paper, so accord-ingly, trunk i ∈ E carries Ci � f iwi unidirectional wave-length channels, which are called channels. Note thatour model is also applicable for networks where OBS usessubwavelength channels [41], in which case the term chan-nel represents a subwavelength channel. If all trunks havethe same number of channels, then Cj � C for all j. How-ever, we note that the results presented in this paper areequally applicable to networks with no wavelength conver-sion, which have f j, instead of f jwj, channels on each inter-mediate trunk (excluding the first trunk) in a route.

Each unique pair of origin and destination nodes forms adirectional source–destination (SD)pair,m. The set of all SDpairs in the network is denoted β � f1; 2;…; N�N − 1�g.Thus, m � fx; yg ∈ N represents traffic composed of burstssent from node x to node y. These bursts arrive at node xaccording to a Poisson process with parameter ρm. For trac-tability, the burst lengths are assumed to be exponentiallydistributed with a unit mean. The effect of this exponentialassumption has been numerically studied in [34], and it hasbeen demonstrated by comparison to scenarios involvingheavy tailed bursts that the performance results are onlyto a small extent sensitive to the shape of the burst lengthdistribution. It is very likely that for a directional SD pairm ∈ β, there is more than a single route between the sourceand the destination. We designate a route with the least

number of hops as the primary path of the SD pair.Then all the other routes are ranked alternative paths.

It is convenient to maintain the set

fUm�0�;Um;j1�1�;Um;j2 �1�;…;Um;jn�Tm�g

of the alternative routes for the directional SD pair m ∈ β.In this set, Um�0� denotes the primary path, and Um;j�d�denotes the alternative path with traffic deflected fromtrunk j, which including this deflection has already beendeflected d times. Tm is the maximum number of availablealternative paths for the directional SD pair m. Note thatTm is based on the network topology, which limits the num-ber of available alternative paths, for example, Tm � 0 inthe trivial example of a network of two nodes and twoopposite-directional trunks that connect the two nodes.

In our model, the ranking of alternative paths is basedon the number of hops, and in the case of equality in thenumber of hops, the rank is chosen randomly. However,in practice, various cost functions (e.g., geographic dis-tance) can also be used for ranking. If capacity is availableon all trunks of the primary path, then it will be used forthe transmission of a burst from the source node x to thedestination node y. However, if all the channels are occu-pied on at least one of the trunks of the primary route, thena burst will be deflected to the first trunk of the first alter-native path of that blocked trunk. If there is a free channelon this trunk, then the burst is transmitted on it; other-wise, the burst is deflected to the first trunk of the secondalternative path.

A given burst is permitted to be deflected at most Dtimes. A burst is blocked, that is, dumped and cleared fromthe network, if it arrives at a given node where all outputtrunks are busy or if, while trying alternative trunks, theburst reaches the limit D of allowable number of deflec-tions. Setting the limit D implies that a burst in the direc-tional SD pair m can be deflected no more than

T�m� � minfTm;Dg

times.

This paper does not consider the use of trunk reservationas in [34]. Obtaining OPCA bounds for an OBS/JET net-work with trunk reservation is still an open problem.

III. OVERFLOW PRIORITY CLASSIFICATION APPROXIMATIONFOR OBS NETWORKS

A detailed description of the OPCA is given in [34] and[35]. To be self-contained, the paper repeats this definitionusing the earlier notation of [34].

For each SD pairm, let ρm be the offered traffic load. Theterm k-deflection burst is used to represent a burst that hasbeen deflected k times (k ∈ f1;…; T�m�g). Original burstsare the bursts that have not been deflected. In other words,they are 0-deflection bursts. Let akj �m� be the k-deflectionbursts’ offered traffic load of SD pairm on trunk j ∈ E, andlet bkj denote the probability that a k-deflection burst is

Li et al. VOL. 5, NO. 4/APRIL 2013/J. OPT. COMMUN. NETW. 379

blocked on trunk j. If the first trunk of the primary routebetween SD pair m is i1, then the offered load to this trunkis equated to the offered load of the SD pair, i.e.,a0i1�m� � ρm. By the carried load of the first trunk i1, wemean the proportion of the offered load to trunk i1 thatis not blocked in trunk i1. Then the offered load in the sec-ond trunk i2 is equated to the carried load of trunk i1, i.e.,

a0i2�m� � a0i1�m��1 − b0i1� � ρm�1 − b0i1�: (1)

When the network is congested and the trunk of a givenroute is fully occupied, then the bursts that initially triedto use the original trunk are deflected onto alternativetrunks and routes. Let trunk i be the busy trunk that trans-mits k-deflection bursts. Then there is a deflection on thepresent route being caused by this trunk i. The load offeredto the first trunk l1 of the first choice alternative route isrelated to the load offered to trunk i by

ak�1l1 �m� � aki �m�bki ; (2)

where k is the number of deflections prior to the latest de-flection. Similarly, due to the deflection from l1, the loadoffered to the first trunk l2 of the second choice alternativeroute is

ak�2l2 �m� � ak�1l1

�m�bk�1l1 � aki �m�bki bk�1l1

: (3)

Let akj be the total offered load of k-deflection bursts ontrunk j. The variables akj and a

kj �m� for k � 0; 1;…; D are

related by

akj �Xm∈β

akj �m�: (4)

Assume

I�i; j;Um;p�k��

8>><>>:

1; if i; j∈E and trunk i strictly precedes�not necessarily immediately� trunk jalong k deflection route Um;p�k�

0; otherwise:

Equation (4) can also be written as

akj �X

m∈β; j∈Um;p�k�ρkm;p

Yi∈E

�1 − I�i; j;Um;p�k��bki �; (5)

where ρkm;p is the offered load from trunk p to the kth de-flection route of SD pair m for k > 1, and

a0j �X

m∈β; j∈Um�0�ρm

Yi∈E

�1 − I�i; j;Um�0��b0i � (6)

for the primary layer.

In addition, let ~akj be the offered load of bursts that havebeen deflected up to k times, i.e.,

~akj �Xkh�0

ahj : (7)

The averaged blocking probability b̄kj on trunk j ∈ E forbursts with deflections up to k times is equal to

b̄kj � E� ~akj ; Cj�; (8)

where E�x; C� � �xC∕C!�∕�PCn�0 xn∕n!� is the Erlang-B for-mula with offered load x and the number of channels pertrunk is C. The blocking probability for k-deflection burst,k ∈ f0;…; Dg on trunk j is estimated by

bkj �8<:

b̄0j ; k � 0b̄kj ~a

kj −b̄

k−1j ~a

k−1j

akj1 ≤ k ≤ D

: (9)

Note that the blocking probability of undeflected bursts iscalculated by using the Erlang-B formula.

To obtain the OPCA blocking probability estimates, westart with the primary traffic, i.e., k � 0. Then, we solve thefixed-point equations described by Eqs. (6)–(9), with the aidof the successive substitution method in order to obtain thevalues a0j for j ∈ E and b̄

0j � b0j � E�a0j ; Cj�. These calcula-

tions for the primary traffic and blocking probabilities aredefined as layer 0 calculations.

Next, having completed the layer 0 calculations to obtainthe parameters related to the primary traffic (k � 0), weprogress to compute the parameters associated with thefirst deflection traffic (k � 1). Similarly, we solve the fixed-point equations, Eqs. (5) and (7)–(9) (existence of a solutionfollows from Brouwer’s theorem [42]; e.g., see [43], whereBrouwer’s theorem [42] was used to prove the existenceof a solution of the fixed-point equation used in that paper),to obtain the values a1j for j ∈ E, as well as b̄

1j and b

1j using

Eqs. (8) and (9), respectively, for every j ∈ E, where ~a1j isgiven by Eq. (7).

Then, having completed the layer 0 and layer 1 calcula-tions to obtain the parameters related to the primary andthe first deflection traffic (k � 0 and k � 1), we compute theparameters associated with the second deflection traffic(k � 2), which we call the layer 2 calculations.

The process of deriving the parameters for k > 1 repeatsitself until we have all the parameter values for all k ∈f1;…; Dg layers.

IV. BOUNDS OF TRUNK BLOCKING PROBABILITIES OF OPCA

In this section we derive the OPCA upper and lowerbounds and some properties of the bounds.

Let bksj denote the blocking probability obtained in thesth iteration for k-deflection bursts (kth layer) on trunkj ∈ E, and let aksj be the offered traffic load obtained inthe sth iteration for k-deflection bursts on trunk j ∈ E.Let fak�j ; bk

�j g be a set of pairs of the fixed-point solutions

of Eqs. (5–9). Let us denote by ak�minj and a

k�maxj the minimum

and maximum values, respectively, among the set of valuesfak�j g and denote by b

k�minj and b

k�maxj the minimum and maxi-

mum values, respectively, among the set of values fbk�j g.Denote the upper and lower bounds of the set fbk�j g bybkUj and b

kLj , respectively, and let the upper and lower

bounds of the set fak�j g be akUj and akLj , respectively.


Consider now the network model presented in Fig. 1(a).There are three SD pairs: A to D, D to H, and C to B, withtraffic in the network, where only primary paths areallowed. In Fig. 1(b), triangles are used to represent atrunk, and arrows from any trunk i to trunk j are usedin the cases where traffic that passes through trunk i fol-lows directly to trunk j. Trunks from 2 to 6 in that figureform a closed loop.

For trunks that are inside a closed loop, such as trunks2–6 in Fig. 1(b), and for trunks that receive any traffic thatpassed through the trunks that are inside a closed loop,such as trunk 7 in the same figure, their offered loadand trunk blocking probability are calculated on the basisof the successive substitution method, leading finally to thefixed-point solution. For any layer k, we will define a set oftrunks that include all such trunks.

Definition: A set of trunks Lk is a loop based trunk(LBT) set in layer k if the following hold:

1) Lk has at least one closed loop of trunks in layer k.2) If in layer k, trunk α receives traffic from any trunk in

Lk, then α ∈ Lk.

If a trunk belongs to an LBT set, the trunk is designatedas an LBT.

For simplicity, in the example presented in Fig. 1, weconsider only one trunk (trunk 7) that is not included inthe closed loop, but it is part of the loop tree. In general,we can have a large set of trunks (which may even includeother loops) that receive traffic from the LBT, and such aset of trunks is included in the LBT set.

All the trunks that are not LBTs in layer k are calleddirect trunks in layer k. Following are three examples (non-exhaustive) for the direct trunks in layer k:

1) A direct trunk where there is no traffic in a trunk inlayer k, so there is no offered load there. In this case,the trunk blocking probability in layer k is set to zero.

2) A set of trunks that form a tree structure (namely, traf-fic is directed toward the leaves of the trees) in layer k.In this case, all the bounds of the offered load and trunkblocking probabilities for all trunks can be calculatedone by one directly without iterations in that layer, fromtop to bottom of the tree.

3) A set of trunks that feed traffic to a closed loop in layerk—such trunks are not in the loop, and they do notreceive traffic from the loop. For instance, trunk 1 inFig. 1(b). In this case, the bounds of the offered loadand trunk blocking probability are calculated one byone directly without iterations as well. These trunkscan also form one or more tree structures.

Calculation of the bounds for the trunk blocking proba-bilities starts from layer 0. After s ≥ 2 iterations, for trunkj ∈ E, we obtain the upper bound b0Uj and the lower bound

b0Lj for the trunk blocking probability and the upper bound

a0Uj and the lower bound a0Lj for the trunk offered load. We

prove that b0Uj � b0Lj and a0Uj � a0Lj for the direct trunks.For the LBTs, when s increases, b0Uj and b

0Lj become closer

to each other and the set of fixed-point solutions for trunkblocking probability b0�j are always between b

0Uj and b

0Lj ;

a0Uj and a0Lj also become closer to each other, and the set

of fixed-point solutions for trunk offered load a0�j are always

between a0Uj and a0Lj . Then the upper and lower bounds for

the traffic load offered from SD pair m to trunk j is calcu-lated based on the bounds of the trunk blocking probabil-ities. After that, we calculate the upper bound a0Uq �m�b0Uqand the lower bound a0Lq �m�b0Lq of the overflowed trafficto layer 1 from SD pairm on the congested trunk q in layer0 for m ∈ β, q ∈ Um�0�.

After obtaining the bounds for the offered load and trunkblocking probabilities for all trunks, including directtrunks and LBTs in layer 0, we then perform the calcula-tions for layer 1. After s ≥ 2 iterations, for trunk j ∈ Ewe obtain the upper bound b1Uj and lower bound b

1Lj for

the trunk blocking probability and the upper bound a1Ujand the lower bound a1Lj for the trunk offered load. Fordirect trunks, we obtain either a1U1j � a1

�max

j � a1�minj � a1Lj

and b1Uj � b1� max

j � b1�minj � b1Lj or a1Uj > a1

�max

j ≥ a1�minj > a

1Lj

and b1Uj > b1�maxj ≥ b

1�minj > b

1Lj . For LBTs, the bounds

have the same properties as those of the LBTs in layer 1.Then we calculate the bounds of the overflowed traffic tolayer 2.

The procedure repeats itself until the bounds of theoffered load and blocking probabilities in each trunk forall layers are found. In each layer, we first calculate thebounds for the offered load and trunk blocking probabilitiesfor the direct trunks, and then we iteratively calculate thebounds for the LBTs. Then, based on the bounds of thetrunk offered load and blocking probabilities, we obtainthe bounds of the network blocking probabilities estimatedby the OPCA and prove that the fixed-point solutions arealways between these upper and lower bounds.

The following subsections provide the equations usedfor obtaining the bounds for the blocking probabilityand the offered load estimated by the OPCA in eachtrunk for all the layers and the OPCA network blockingprobability, as well as the proof of the behavior of thebounds.

Fig. 1. Model for LBT: (a) a networkmodel and (b) an example fordirect trunks and LBT in the network shown in (a).


A. Bounds of Trunk Blocking Probabilities forDirect Trunks

If the primary layer (layer 0) is a layer with directtrunks, then its upper and lower bounds for the offered loadand trunk blocking probabilities are calculated from top tobottom of the tree by the following equations:

a0Uj �X


Yi∈E

�1 − I�i; j;Um�0��b0Li �; (10)

a0Lj �X


Yi∈E

�1 − I�i; j;Um�0��b0Ui �; (11)

b0Lj � E�a0Lj ; Cj�; (12)

b0Uj � E�a0Uj ; Cj�: (13)

The calculations start from trunk j on the top of thetree. The offered load of this trunk does not depend onthose in other trunks, so the bounds of its offered loadare calculated directly using Eqs. (10) and (11). We obtaina0Uj � a0

�max

j � a0�minj � a0Lj . This indicates uniqueness of the

fixed-point solution for this trunk j. Then, substitutingthe bounds for the offered load into Eqs. (12) and (13),we obtain the bounds of the trunk blocking probability ofthis trunk, b0Uj � b

0�Uj � b

0�Lj � b0Lj . After that, we pass to

the next trunk, that below the top trunk in the tree. Thebounds of the offered load of this trunk depend only onthose of the top trunk, so substituting the bounds of thetrunk blocking probability of the top trunk into Eqs. (10)and (11), we obtain the bounds of the offered load for thissecond highest trunk. Then, substituting these boundsinto Eqs. (12) and (13), we obtain the bounds of the trunkblocking probability of this second highest trunk. Repeat-ing the steps from top to bottom in the tree, we obtainall the bounds of the offered load and trunk blockingprobability one by one. They are a0Uj � a0

�max

j � a0�minj � a0Lj

and b0Uj � b0�max

j � b0�minj � b0Lj .

For the same primary layer, the lower and upper boundsof the offered load for SD pair m are calculated by theformulas

a0Lj �m� � ρmYi∈E

�1 − I�i; j;Um�0��b0Ui �; (14)

a0Uj �m� � ρmYi∈E

�1 − I�i; j;Um�0��b0Li �: (15)

Note that the notation a0Lj �m� and a0Uj �m� given by Eqs. (14)and (15) is not the same as the previously defined notationa0Lj , and the value a

0Lj is the lower bound of the total offered

load to trunk j in layer 0, while a0Lj �m� is the lower boundof the offered load to trunk j by SD pair m. The differencebetween the notation a0Uj and a

0Uj �m� is explained similarly.

For layer k, we assume that the offered load of bursts de-flected less than k times is ~ak−1j . If the layer having direct

trunks is not the primary layer (that is, it is layer k > 0),then the overflowed traffic from SD pairm in the congestedtrunk q in layer k − 1 forms the traffic to the paths inlayer k. Then, the bounds of the offered load of the directtrunks in layer k are calculated by the equations

akUj �X

m∈β; q∈E;j∈Um;q�k�fak−1Uq �m�bk−1Uq

×Yi∈E

�1 − I�i; j;Um;q�k��bkLi �g; (16)

akLj �X

m∈β; q∈E;j∈Um;q�k�fak−1Lq �m�bk−1Lq

×Yi∈E

�1 − I�i; j;Um;q�k��bkUi �g: (17)

The equation for the trunk blocking probability is

bkj �E� ~ak−1j � akj ; Cj�� ~ak−1j � akj � − E� ~ak−1j ; Cj�� ~ak−1j �

akj: (18)

The left hand side of Eq. (18) increases when ~ak−1j or akj

increases (see Appendix A for the proof), and hence,

E� ~ak−1Lj � akLj ; Cj�� ~ak−1Lj � akLj � − E� ~ak−1Lj ; Cj�� ~ak−1Lj �akLj

≤E� ~ak−1Uj � akLj ; Cj�� ~ak−1Uj � akLj � − E� ~ak−1Uj ; Cj�� ~ak−1Uj �

akLj

≤E� ~ak−1Uj � akUj ; Cj�� ~ak−1Uj � akUj � − E� ~ak−1Uj ; Cj�� ~ak−1Uj �

akUj;

E� ~ak−1Lj � akLj ; Cj�� ~ak−1Lj � akLj � − E� ~ak−1Lj ; Cj�� ~ak−1Lj �akLj

≤E� ~ak−1Lj � akUj ; Cj�� ~ak−1Lj � akUj � − E� ~ak−1Lj ; Cj�� ~ak−1Lj �

akUj

≤E� ~ak−1Uj � akUj ; Cj�� ~ak−1Uj � akUj � − E� ~ak−1Uj ; Cj�� ~ak−1Uj �

akUj:

Thus, the upper and lower bounds of the trunk blockingprobabilities of direct trunks that are not in the primarylayer are calculated by the formulas

bkUj �E� ~ak−1Uj � akUj ; Cj�� ~ak−1Uj � akUj � − E� ~ak−1Uj ; Cj�� ~ak−1Uj �

akUj;

(19)

bkLj �E� ~ak−1Lj � akLj ; Cj�� ~ak−1Lj � akLj � − E� ~ak−1Lj ; Cj�� ~ak−1Lj �

akLj:

(20)

The upper and lower bounds of the offered load to eachdirect trunk in SD pair m in layer k are determined by theformulas


akLj �m� �X

q∈E; j∈Um;q�k�ak−1Lq �m�bk−1Lq

×Yi∈E

�1 − I�i; j;Um;q�k��bkUi �; (21)

akUj �m� �X

q∈E; j∈Um;q�k�ak−1Uq �m�bk−1Uq

×Yi∈E

�1 − I�i; j;Um;q�k��bkLi �: (22)

For the kth layer, we first determine the bounds of the of-fered load in trunk j1, which is on the top of the tree. Thebounds of the offered load depend only on the overflowedtraffic in the k − 1 layer. From Eqs. (16) and (17) we obtain

akUj1 �ak�maxj1

�ak�min

j1�akLj1 , provided that a

k−1Uq �m� � ak−1Lq �m�

is satisfied for all overflowed traffic into top trunk j; inthe opposite case, if ak−1Uq �m� � ak−1Lq �m� is not satisfiedat least for one instance of the available overflowed traffic,

then we obtain akUj1 > ak�maxj1

≥ ak�min

j1> akLj1 . [The equation

ak−1Uq �m� � ak−1Lq �m� is satisfied if for the upper layer weobtain the exact value of the offered load]. Substitutingthese bounds of the offered load into Eqs. (19) and (20),we obtain the bounds for the trunk blocking probabilityin the top trunk. If ak−1Uq �m� � ak−1Lq �m� for all overflowedtraffic into the top trunk, then bkUj1 � b

k�maxj1

� bk�min

j1� bkLj1 ;

otherwise, we obtain bkUj1 > bk�maxj1

≥ bk�min

j1> bkLj1 . In the next

step, we calculate the bounds for the offered load and block-ing probability for each of the trunks that are the secondfrom the top. The calculation for each one these trunks isidentical to the others and depends only on the trunk block-ing probability in the top trunk j1 and overflowed trafficfrom the k − 1 layer. Therefore, without loss of generality,let j2 denote any of these trunks. We obtain a

kUj2

� ak�maxj2 �ak�minj2

� akLj2 and bkUj2

� bk� maxj2 � bk�minj2

� bkLj2 provided thatak−1Uq �m� � ak−1Lq �m� is satisfied for all overflowed trafficinto trunk j1 and trunk j2. Otherwise, we have a

kUj2

> ak�max

j2≥

ak�minj2

> akLj2 and bkUj2

> bk�max

j2≥ bk

�min

j2> bkLj2 , respectively. Re-

peating these steps recurrently to the other direct trunks inthe kth layer from top to bottom of the tree, we obtainall bounds for the offered load and trunk blockingprobabilities.

B. Bounds of Trunk Blocking Probabilitiesfor LBTs

In Proposition 1 below and in all other statements fol-lowing, bk1j denotes the initial (setup) value of the blockingprobability for the kth layer on trunk j.

Proposition 1: Assume that bk1j � 0 (k � 0;1;…; D), j ∈ Ewith a nonzero offered load and trunk j is an LBT. Then,for any positive integer z, we have the following inequal-ities for the lower and upper bounds of ak�j and b

k�j :

ak2z−1j > ak2z�1j > a

k�maxj ;

ak2z−2j < ak2zj < a

k�minj ;

bk2z−2j > bk2zj > b

k�maxj ;

bk2z−1j < bk2z�1j < b

k�minj :

Proof: The computation starts from layer 0 of the pri-mary path bursts. First we calculate lower and upperbounds of the offered load and trunk blocking probabilitiesfor the direct trunks. Then, setting b01j � 0, we provide cal-culations for other trunks, where j belongs to the LBT setand 01 denotes the first iteration in the primary path.

For the offered load on trunk j, we have

a01j �X


Yi∈E

�1 − I�i; j;Um�0��1 −H�0; i��b01i �

× �1 − I�i; j;Um�0��H�0; i�b0Li �; (23)

where

I�i; j;Um�0��

8>><>>:

1; if i; j ∈ E and trunk i strictly precedes�not necessarily immediately� trunk jalong primary route of SD pair m

0; otherwise;

H�k; i� ��1; if trunk i is a LBT in layer k0; if trunk i is a direct trunk in layer k:

Hence, the blocking probability b02j is obtained by theErlang-B formula,

b02j � E�a01j ; Cj�: (24)

According to assumption, b01j � 0 and a01j > 0 are true forall trunks j belonging to the LBT set in layer 0. Hence, com-paring b01j with b

02j yields b

02j > b

01j for all j belonging to the

LBT set in layer 0.

By the successive substitution method, we are to replaceb01j in Eq. (23) by b

02j , and then the load is updated by the

formula

a02j �X


Yi∈E

�1 − I�i; j;Um�0��1 −H�0; i��b02i �

× �1 − I�i; j;Um�0��H�0; i�b0Ui �: (25)

Taking into account b02j > b01j for all j belonging to the LBT

set in layer 0 and b0Ui � b0Li whenH�0; i� � 1, by comparingEqs. (23) and (25), we arrive at the inequality a02j �m� <a01j �m� for all j belonging to the LBT set in layer 0.

Repeating the steps above, for the value b03j , we obtain

b03j � E�a02j ; Cj�: (26)

Since E�x; C� is an increasing function in x, we have0 � b01j < b03j < b02j .


In the third iteration, we have the formula

a03j �X


Yi∈E

�1 − I�i; j;Um�0��1 −H�0; i��b03i �

× �1 − I�i; j;Um�0��H�0; i�b0Li �; (27)

and then we arrive at

b04j � E�a03j ; Cj�: (28)

Since 0 � b01j < b03j < b02j , we have the property a02j < a03j <a01j and 0 � b01j < b03j < b04j < b02j for all j ∈ L.

Since a04j �P

m∈β;j∈Um�0�ρmQ

i∈E�1 − I�i; j;Um�0��b04i �, weobtain a02j < a

04j < a

03j < a

01j for all j belonging to the LBT

set in layer 0.

In a similar way, we define the recurrent relationsfor s > 1:

b0sj � E�a0s−1j ; Cj�; (29)

a0sj �X

m∈β;j∈Um�0�ρm

Yi∈E

�1 − I�i; j;Um�0��1 −H�0; i��b0si �

× �1 − I�i; j;Um�0��1 −H�0; i��b0Li : (30)

Notice that b0Li � b0Ui when H�0; i� � 0.Repeating the same procedure, we obtain all the neces-

sary relations for a0sj and b0sj for given integer values s.

We also have the inequalities

a01j > a03j > a

05j > � � � > limz→∞a

02z−1j � a0

�max

j ; (31)

a02j < a04j < a

06j < � � � < limz→∞a

02zj � a

0�minj : (32)

Let S be the number of iterations. We will write S � 2z inthe case of an even number of iterations and S � 2z� 1 inthe case when the number of iterations is odd.

In the case of S � 2z, the upper bound of the set of valuesfa0sj ; s � 1;2;…;2zg is a0Uj � a02z−1j and the lower bound isa0Lj � a02zj . In the case of S � 2z� 1, the upper bound ofthe set of these values is a0Uj � a02z�1j and the lower boundis a0Lj � a02zj .

For the set of the values fb0sj ; s � 1; 2;…; Sg, the boundsare defined similarly:

b02j > b04j > b

06j > � � � > limz→∞b

02zj � b0

�max

j ; (33)

b01j < b03j < b

05j < � � � < limz→∞b

02z−1j � b

0�minj : (34)

In the case of S � 2z, the upper and the lower bounds of theset of values fb0sj ; s � 1;2;…;2zg are b0Uj � b02zj andb0Lj � b02z−1j , respectively. In the case of S � 2z� 1, theupper and the lower bounds of the set of these valuesare b0Uj � b02zj and b0Lj � b02z�1j , respectively.

Equations (31)–(34) follow by induction.

Indeed, we earlier proved Eqs. (31)–(34) for z � 1. Hence,assuming that in the case z � i the inequalities

a02i−1j > a02i�1j > a

02i�2j > a

02ij ; (35)

are satisfied, we are to prove that the inequalities

a02i�1j > a02i�3j > a

02i�4j > a

02i�2j (36)

are true as well (case z � i� 1). At the next step, on thebasis of Eq. (35), we prove that

b02ij > b02i�2j > b

02i�3j > b

02i�1j : (37)

Indeed, since the function E�x;Cj� is increasing in x, thenEq. (37) follows from Eq. (35) by direct substitution of thevalues into Eq. (29).

Now on the basis of Eq. (37) we prove Eq. (36). Thefunction F�x� � Qi−1n�1�1 − xn� is a decreasing function invector x � �x1; x2;…; xn�. That is, for any two vectors x�1� ��x�1�1 ; x�1�2 ;…; x�1�n � and x�2� � �x�2�1 ; x�2�2 ;…; x�2�n � satisfying thecomponentwise inequalities x�1�i ≤ x

�2�i , i � 1;2;…; n, we

have F�x�1�� ≥ F�x�2��. The strong inequality F�x�1�� >F�x�2�� holds if, in addition, x�1�i < x�2�i is satisfied at leastfor one of indices i.

Hence, substituting the values of Eq. (37) into Eq. (30),we arrive at the inequality

a02i�1j > a02i�3j > a

02i�2j > a

02ij : (38)

From Eq. (38), it is easy to obtain the desired inequality[Eq. (36)]. To this end, we first substitute Eq. (38) intoEq. (29). This yields

b02i�2j > b02i�4j > b

02i�3j > b

02i�1j : (39)

Finally, substituting Eq. (39) into Eq. (30) once again, wearrive at Eq. (36).

For layer 0, the lower and upper bounds of the offeredload to each trunk of SD pair m are also calculated byEqs. (14) and (15).

For layer 1, the bounds of overflowed traffic from SD pairm caused by the congestion in trunk q in layer 0 is calcu-lated by the equation

a1j �X

m∈β;q∈E; j∈Um;q�1�a0q�m�b0q

Yi∈E

�1 − I�i; j;Um;q�1��

× �1 −H�1; i��b1i ��1 − I�i; j;Um;q�1��H�1; i�b1i �: (40)

On the basis of the lower and upper values for each a0i �m�and bi for direct trunks i, we have the following trans-formed formulas of Eq. (40):


a1Uj �X

m∈β;q∈E;j∈Um;q�1�a0Uq �m�b0Uq

Yi∈E

�1 − I�i; j;Um;q�1��

× �1 −H�1; i��b1i ��1 − I�i; j;Um;q�1��H�1; i�bLi �; (41)

a1Lj �X

m∈β;q∈E;j∈Um;q�1�a0Lq �m�b0Lq

Yi∈E

�1 − I�i; j;Um;q�1��

× �1 −H�1; i��b1i ��1 − I�i; j;Um;q�1��H�1; i�bUi �: (42)

Since a1j increases when the term a0q�m�b0q increases, then

substituting the same value of b1j into Eqs. (41) and (42), weobtain the inequality a1Uj > a

1Lj .

Setting b11j � 0, for upper and lower bounds of a1j and b1j ,we have the following relations:

a12z−1j �X

m∈β;q∈E;j∈Um;q�1�a0Uq �m�b0Uq

Yi∈E

f1 − I�i; j;Um;q�1��

× �1 −H�1; i��b12z−1i gf1 − I�i; j;Um;q�1��H�1; i�b1Li g;(43)

a12zj �X

m∈β;q∈E;j∈Um;q�1�a0Lq �m�b0Lq

Yi∈E

f1 − I�i; j;Um;q�1��

× �1 −H�1; i��b12zi gf1 − I�i; j;Um;q�1��H�1; i�b1Ui g;

b12zj �E�a0Uj � a12z−1j ; Cj��a0Uj � a12z−1j � − E�a0Uj ; Cj��a0Uj �

a12z−1j;

b12z�1j �E�a0Lj � a12zj ; Cj��a0Lj � a12zj � − E�a0Lj ; Cj��a0Lj �

a12zj: (44)

Then we have the inequalities

a11j > a13j > a

15j > � � � > limz→∞a

12z−1j ≥ a

1�maxj ; (45)

a12j < a14j < a

16j < � � � < limz→∞a

12zj ≤ a

1�minj ; (46)

b12j > b14j > b

16j > � � � > limz→∞b

12zj ≥ b

1�maxj ; (47)

b11j < b13j < b

15j < � � � < limz→∞b

12z−1j ≤ b

1�minj : (48)

In the case S � 2z, the upper and lower bounds of a1j area1Uj � a12z−1j and a1Lj � a12zj , respectively, and the upperand lower bounds of a1j are a

1Uj � a12z−1j and a1Lj � a12zj ,

respectively. In the case S � 2z� 1, the upper bound ofa1j is changed to a

1Uj � a12z�1j , and the lower bound of b1j

is changed to b1Lj � b12z�1j .The proof of Eqs. (45)–(48) is similar to that of

Eqs. (31)–(34).

The lower and upper bounds of the offered load to eachtrunk of SD pair m in layer 1 are determined by theequations

a1Lj �m� �X

q∈E;j∈Um;q�1�a0Lq �m�b0Lq

Yi∈E

�1 − I�i; j;Um;q�1��b1Ui �;

(49)

a1Uj �m� �X

q∈E;j∈Um;q�1�a0Uq �m�b0Uq

Yi∈E

�1 − I�i; j;Um;q�1��b1Li �:

(50)The upper and lower bounds of ~ak−1j are ~a

�k−1�Uj �

Pk−1i�0 a

iUj

and ~a�k−1�Lj �Pk−1

i�0 aiLj .

The values of aksj and bksj are calculated by the formulas

ak2z−1j �X

m∈β;q∈E;j∈Um;q�k�a�k−1�Uq �m�b�k−1�Uq

×Yi∈E

f1 − I�i; j;Um;q�k��1 −H�k; i��bk2z−1i g

× f1 − I�i; j;Um;q�k��H�k; i�bkLi g; (51)

ak2zj �X

m∈β;q∈E;j∈Um;q�k�a�k−1�Lq �m�b�k−1�Lq

×Yi∈E

f1 − I�i; j;Um;q�k��1 −H�k; i��bk2zi g

× f1 − I�i; j;Um;q�k��H�k; i�bkUi g; (52)

bk2zj �E� ~a�k−1�Uj � ak2z−1j ; Cj�� ~a�k−1�Uj � ak2z−1j �

ak2z−1j

−

E� ~a�k−1�Uj ; Cj�� ~a�k−1�Uj �ak2z−1j

; (53)

bk2z�1j �E� ~a�k−1�Lj � ak2zj ; Cj�� ~a�k−1�Lj � ak2zj �

ak2zj

−

E� ~a�k−1�Lj ; Cj�� ~a�k−1�Lj �ak2zj

: (54)

Then we obtain the inequalities

ak1j > ak3j > a

k5j > � � � > limz→∞a

k2z−1j ≥ a

k�maxj ; (55)

ak2j < ak4j < a

k6j < � � � < limz→∞a

k2zj ≤ a

k�minj ; (56)

bk2j > bk4j > b

k6j > � � � > limz→∞b

k2zj ≥ b

k�maxj ; (57)

bk1j < bk3j < b

k5j < � � � < limz→∞b

k2z−1j ≤ b

k�minj : (58)

In the case S � 2z, the upper and lower bounds of akj areakUj � ak2z−1j and akLj � ak2zj , respectively, and the upperand lower bounds of bkj are b

kUj � bk2zj and bkLj � bk2z−1j , re-

spectively. In the case S � 2z� 1, the upper bound of akjis changed to akUj � ak2z�1j , and the lower bound of bkj ischanged to bkLj � bk2z�1j .

The upper and lower bounds of the offered load toeach trunk of SD pair m in layer k are determined bythe formulas


akLj �m� �X

q∈E;j∈Um;q�k�fak−1Lq �m�bk−1Lq

Yi∈E

�1

− I�i; j;Um;q�k��bkUi �g; (59)

akUj �m� �X

q∈E;j∈Um;q�k�fak−1Uq �m�bk−1Uq

Yi∈E

�1 − I�i; j;Um;q�k��bkLi �g: (60)

Corollary 1: Consider two OPCA runs. Let the number ofiterations in layer i in the first run be Ii, and let the numberof iterations in layer i in the second run be I0i. Let S

ij and S

0ij

be the distance between the bounds for the blocking prob-ability for trunk j in layer i in the first and second OPCAruns, respectively. If I0i > Ii for one layer and the numbers ofiterations in other layers are the same, and there are LBTsin the layer, then Skj > S

0kj for the trunks with the following

situations:

1) if trunk j is a loop trunk, then for all k ≥ i, Skj > S0kj ;

2) if in a layer k0 > i, a trunk j that receives traffic thatpasses through or overflows from a trunk j1 in theupper layer satisfies the inequality Sk0−1j1 > S

0k0−1j1

, thenfor all k0 ≤ k ≤ D, Skj > S

0kj .

Proof: According to the construction, for layers k < i, thebounds of the trunk blocking probability for j ∈ E are thesame in two OPCA runs.

Since there are LBTs, then for layer i byEqs. (57) and (58),for LBTs we obtain inequalities Sizj < S

iz−1j < � � � < Si1j , in

which Sizj � jbiz�1j − bizj j. Thus, if I0i > Ii, then Sij > S0ij .Also, if I0i > Ii, then by Eqs. (21), (22), and (55)–(58), we

obtain a0iUj �m� < aiUj �m�, a0iLj �m� > aiLj �m�, b0iUj < biUj , andb0iLj > b

iLj for j ∈ E and m ∈ β.

For layer i� 1, let us first consider a trunk j for whichSij > S

0ij in layer i is satisfied. Then we have a

0iUj < a

iUj

and a0iLj > aiLj . Substituting these inequalities into

Eqs. (19)–(22), we obtain the inequalities b0i�1Uj < bi�1Uj ,

b0i�1Lj > bi�1Lj , a

0i�1Uj �m� < ai�1Uj �m�, and a0i�1Lj �m� >

ai�1Lj �m�. Thus, if trunk j is a loop trunk in layer i, thenwe have Si�1j > S

0i�1j .

Consider now a direct trunk j receiving traffic that over-flowed from trunks j1, for which Sij1 > S

0ij1

in layer i, orpassed through a trunk j2 in layer i� 1, for whichSi�1j2 > S

0i�1j2

. Substituting the inequalities a0iUj1 �m� <aiUj1 �m� and a

0iLj1�m� > aiLj1 �m�, b

0iUj1

< biUj1 and b0iLj1

> biLj1 , or

b0i�1Uj2 < bi�1Uj2

and b0i�1Lj2 > bi�1Lj2

into Eqs. (17) and (16),

we obtain a0i�1Uj < ai�1Uj and a

0i�1Lj > a

i�1Lj . Then, substitut-

ing the inequalities into Eqs. (19)–(22), we obtain theinequalities b0i�1Uj < b

i�1Uj , b

0i�1Lj > b

i�1Lj , a

0i�1Uj �m� <

ai�1Uj �m�, and a0i�1Lj �m� > ai�1Lj �m�. Thus, the inequalitySi�1j > S

0i�1j is satisfied if trunk j is a direct trunk receiving

traffic that overflowed from LBTs in layer i.

Let us consider now LBTs receiving traffic that over-flowed from the LBTs in layer i or passing through a direct

trunk for which the inequality Si�1j > S0i�1j is satisfied in

layer i� 1. We start from the setup value bi�11j � 0. Accord-ing to Eq. (51) for layer i� 1, we obtain the inequalityai�11j > a

0i�11j . Substituting this inequality into Eq. (53),

we in turn obtain bi�12j > b0i�12j . Repeating the procedure,

we again substitute the inequality obtained for Eq. (52)and now obtain ai�11j > a

0i�11j > a

0i�12j > a

i�12j . Further sub-

stitution of the inequality obtained for Eq. (54) yieldsbi�12j > b

0i�12j > b

0i�11j > b

i�11j . After a number of repetitions

of these steps for any integer z, we finally arrive at the fol-lowing inequalities:

ai�12z−1j > a0i�12z−1j > a

0i�12zj > a

i�12zj ;

bi�12zj > b0i�12zj > b

0i�12z−1j > b

i�12z−1j :

By the same method in layer i, we obtain Si�1j > S0i�1j . In

addition, we obtain ai�1Uj �m� > a0i�1Uj �m� > a0i�1Lj �m� >ai�1Lj �m� and bi�1Uj > b0i�1Uj > b0i�1Lj > bi�1Lj . Repeating thesame steps as in layer i� 1, we obtain the same solutionfor layer k > i� 1.

C. Bounds for Network Blocking Probability of theOPCA

Proposition 2: Let BU�m� and BL�m� denote the upperand the lower bounds, respectively, of the network blockingprobability for SD pair m. We have

BU�m� � 1 − ρmQ

i∈Um�0��1 − b0Ui �

ρm

−

Pq∈E

PTk�1�m�ρLUm;q�k�

Qp∈Um;q�k��1 − b

kUp �

ρm; (61)

where

ρLUm;q�k� ��a�k−1�Lq �m�b�k−1�Lq ; if path Um;q�h� exists;0; otherwise;

and we have

BL�m� � 1 − ρmQ

i∈Um�0��1 − b0Li �

ρm

−

Pq∈E

PTh�1�m�ρUUm;q�k�

Qp∈Um;q�k��1 − b

kLp �

ρm; (62)

where

ρUUm;q�k� ��a�k−1�Uq �m�b�k−1�Uq ; if path Um;q�k� exists;0; otherwise:

Proof: In order to calculate the blocking probability B�m�for SD pairm, we are required first to calculate the receivedload from every path by the destination node. Then, we cal-culate the probability that a message will be served and theblocking probability


B�m� � 1 − ρmQ

i∈Um�0��1 − b0i �ρm

−

Pq∈E

PTk�1�m�ρUm;q�h�

Qp∈Um;q�k��1 − bkp�

ρm; (63)

where ρUm;q�k� is the offered load to the pathUm;q�k�, and it iscalculated by the formula

ρUm;q�k� ��ak−1q �m�bk−1q ; if path Um;q�k� exists0; otherwise:

After calculation of the lower and upper bounds for akj andbkj (k � 0;1;…; T�m�), we derive Eqs. (61)–(63). For moreiterations, the sequence BU�m� is not increasing. If thereis at least one layer with LBTs, then for more iterationsthe sequence bhUi is decreasing and ρ

LUm;q�k� is a decreasing

sequence in m ∈ β and k � 0; 1;…; T�m� as well. Hence,BU�m� is a decreasing sequence. Unlike BU�m�, BL�m� isnot an increasing sequence. When there is at least onelayer with LBTs, the sequence BL�m� is decreasing formore iterations of the OPCA algorithm.

Substituting akUj �m� ≥ ak�max

j �m� ≥ ak�minj �m� ≥ akLj �m� and

bkUj ≥ bk�maxj ≥ b

k�minj ≥ b

kLj into Eqs. (61)–(63), we arrive at

the inequality BUm ≥ B�max�m� ≥ B�min�m� ≥ BLm, whereB�max�m� and B�min�m� denote the maximum and minimumvalue in the set fB��m�g of the fixed-point solutionsfor B�m�.

Proposition 3: Let fB�g be the set of the network blockingprobabilities obtained by the fixed-point solutions for trunkblocking probabilities in trunks j ∈ E. Let B�min and B

�max

denote the minimum and maximum values among theset of values fB�g. Let BU and BL denote the upper andlower bounds of the network blocking probabilities. Wehave

BU ≥ B�max ≥ B�min ≥ BL:

BU and BL are calculated by the equations

BU �P

m∈βBU�m�

#SD; (64)

BL �P

m∈βBL�m�

#SD; (65)

where #SD is the number of SD pairs in the network.

Proof: Substituting BUm ≥ B�max�m� ≥ B�min�m� ≥ BLm intoEquations (64) and (65), we obtain the bounds for thenetwork blocking probability

BU ≥ B�max ≥ B�min ≥ BL:

Corollary 2: Consider an OPCA run. Let M be the dis-tance between the bounds of the network blocking proba-bility. Increasing the number of iterations in one or morelayers, among which at least one of those layers has LBTs,and obtaining the new distance between the bounds of thenetwork blocking probability M0, then M0 < M.

Proof: Corollary 1 shows that when the number of iter-ations increases in one layer having trunk loops, then thedistance between the upper and lower bounds of the trunkblocking probability for some trunks decreases. If there isat least one trunk j, in which the distance between theupper and lower bounds of the trunk blocking probabilitydecreases in layer i, then for SD pair m having trafficpassing through trunk j in layer i we have the inequalitiesBU�m� > B0U�m� and BL�m� < B0L�m�. Then, we arrive atBU > B0U and BL < B

0L. Hence, for M � BU − BL and

M0 � B0U − B0L, we arrive at M0 < M.If the trunks in all layers are direct trunks, then the sol-

ution obtained by the OPCA is not a fixed-point solution,and it is obtained in only a finite number of steps. Thisnumber of steps is bounded by J × �D� 1�, where J isthe total number of trunks and D is the maximum numberof allowable deflections in the network. If there are LBTs atleast in one layer, then the bounds of the OPCA fixed-pointsolutions always become closer when the numbers of iter-ations in those layers are increased.

Thus, either the OPCA finds its solution in a finitenumber of steps, or the bounds of its fixed-point solu-tions always become closer with increasing number ofiterations.

V. NUMERICAL RESULTS

In this section we provide numerical results for OBS/JET without trunk reservation over a 13-node NSFNETin order to illustrate the behavior of the bounds of theOPCA algorithm. In particular, we focus on illustratinghow the bounds become closer to each other with an in-crease of the number of iterations. The numerical resultsin this section demonstrate that the bounds become closerto each other even after a small number of iterations (perlayer). We will illustrate here the behavior of the OPCAblocking probability bounds for the aforementioned net-work considering a wide range of parameters and designfactors, such as the number of channels per trunk andthe maximum allowable number of deflections. We will alsocompare the running time and accuracy of the EFPA andOPCA algorithms.

In all the scenarios considered, the arrival process ofcalls for each directional SD pair follows a Poisson process.The shortest path is set to be the primary route for each SDpair, and the alternative routes are preassigned, ordered bytheir length. For those routes with the same lengths, theorder is chosen randomly and remains unchanged after-ward. All the results in this section are obtained usingMATLAB software executed on a laptop with an Intel Corei7-3520M CPU at 2.96 GHz with 8 GHz RAM and a 64 bitoperating system.

A. Network Topology and SD Pairs

We now consider the NSF network with 13 nodes and 30directional trunks. The topology of the NSF network is


shown in Fig. 2. We randomly select a set of 12 SD pairsshown in Table I.

B. Comparison of the Computational Time of theEFPA and OPCA

Table II provides the running times used to calculate theblocking probability in NSFNET for different C values bythe EFPA and OPCA. The offered load to each SD pair is0.5C. In each iteration, we only consider four significantdigits of the fixed-point solutions for the trunks with block-ing probabilities larger than 10−50, whereas trunk blockingprobabilities lower than 10−50 are set equal to 0. FromTable II, we observe that the EFPA consumes much moretime than the OPCA for all three different C values, andwhen the C value increases, the computational time ofthe EFPA grows faster than that of the OPCA. To gainsome insight into the reason why the EFPA consumesmuchmore time than the OPCA, we count the total numberof iterations required by the EFPA, and by each layer of theOPCA, with C = 10,000. The results are shown in Table III.We observe that the EFPA requires 78 iterations to con-verge, but the first layer of the OPCA only requires 6 iter-ations, and the other layers require even fewer iterations.Layer 3 for the OPCA algorithm consumes only 0.0024 s,because there is no overflowed traffic from layer 2; there-

fore, the offered load to each trunk in layer 3 is 0, in whichcase all the layer 3 trunk blocking probabilities equal to 0without the need to run the Erlang-B formula.

C. Accuracy of OPCA and EFPA for the NSFNET

Having demonstrated that the OPCA converges fasterthan the EFPA, it is important to evaluate the accuracyof the two algorithms to see whether the longer runningtime enables the EFPA to provide more accurate resultsthan the OPCA. Note also that the results presented hereare for an OBS network without trunk reservation, so theseresults also complement those in [34], which apply to net-works with trunk reservation.

Results for the comparison of the accuracy of the OPCAand EFPA for the case C � 50 are presented in Fig. 3. Theresults are limited to the case C � 50, because simulationsfor large C values are computationally prohibitive. The re-sults are based on a comparison of the two approximationswith simulation results for the case of theNSFNETexampleof Fig. 2, with the 12 SD pairs shown in Table I setting themaximum allowable number of deflections to 3. Error barsfor 95% confidence intervals based on Student’st-distribution are provided for all the simulation results,although in many cases the intervals are too small tobe clearly visible. We observe that the OPCA generallyslightlyoverestimatestheblockingprobabilityforthisexam-ple,whiletheEFPAunderestimatesitwhenthetraffic is low;however, the OPCA turns to underestimate the blockingprobability when the traffic is high. Notice that when theoffered load is within 40–50, the results of the EFPA aremissing. This is because we cannot achieve a convergencein cases where the offered load to each SD pair draws nearthe number of channels per trunk, which is equal to 50. Itis known that evaluating blocking probability by the EFPAfor OBS may fail to converge in certain instances for unpro-tected deflection routing, as shown in [32].

It is observed from Fig. 3 that in the practical loadingrange, the EFPA does not perform better than the OPCA.The EFPA is only more accurate than the OPCA when theoffered load is within 35–40, where the blocking probabilityis above 3 × 10−1, which is way above what is considered anacceptable grade of service. Notice that these results areconsistent with the results in [34] (for cases with trunk res-ervations), where we observed that the EFPA is more ac-curate than the OPCA for high loads. Thus we observedin the present example that in the practical traffic loadingrange, normally used for dimensioning purposes, thelonger running time does not enable the EFPA to providemore accurate results than OPCA.

Fig. 2. (Color online) NSF network topology in which each solidline represents two unidirectional trunks in opposing directions.

TABLE IINGRESS AND EGRESS SD PAIRS

Ingress WA CA1 CA1 CA2 TX GAEgress MD IL MA MA CD MAIngress MD IL MA MA CD MAEgress WA CA1 CA1 CA2 TX GA

TABLE IICOMPARISON OF THE TIMES USED BY THE EFPA AND OPCA TO CALCULATE THE BLOCKING PROBABILITIES IN NSFNET

Calculation Task Running Time of EFPA in Seconds Running Time of OPCA in Seconds

Blocking probability of the whole network and C = 50 0.271 0.197Blocking probability of the whole network and C = 2000 64.45 12.91Blocking probability of the whole network and C = 10,000 3006 397Blocking probability of the whole network and C = 20,000 13,665 1232


D. OPCA Bounds Behavior

In Fig. 4 we present results for the bounds of the OPCAresult iterations as a function of the number of iterations ineach layer for different offered loads to each directional SDpair in the NSF network. The maximum number of deflec-tions is set to 3, and each trunk has 50 channels. Weobserve in Fig. 4(a) that the lower and upper bounds ofthe OPCA results for overall network blocking probabilitybecome closer to each other, and when six iterations aremade in each layer, the distance between the lower andupper bounds is less than 10−5% of the lower bound value.Similar results are observed in all four cases of differentoffered load alternatives presented in Fig. 4. Notice, how-ever, that as the offered load increases, the rate at whichthe bounds become closer to each other is somewhat re-duced. Still, in the cases presented in Figs. 4(a) and 4(b),the distance between the bounds is less than 10−5% ofthe lower bound value when six iterations per layer havebeen completed. However, when the offered load is 30 foreach SD pair, which is the case presented in Fig. 4(c), seveniterations per layer are required to achieve a distance be-tween the bounds of 10−5% of the lower bound value, andwhen the offered load increases to 55 [presented inFig. 4(d)], eight iterations per layer are required to achievethe same accuracy. This is due to the fact that in the NSFnetwork, there are LBTs in each layer. For the LBTs, whenthe offered load increases in all layers, the first upperbound of trunk blocking probability obtained by the firstiteration also increases. This fact can be observed byEqs. (24) and (53), which show that the trunk blockingprobability increases in offered load. Since the first lower

TABLE IIICOMPARISON OF THE TIMES USED BY THE EFPA AND OPCA INEACH LAYER TO CALCULATE THE BLOCKING PROBABILITIES IN

NSFNET WITH 10,000 CHANNELS PER TRUNK

AlgorithmLayerNumber

Number ofIterations

Total RunningTime in Seconds

EFPA only 1 layer 78 3006OPCA layer 0 6 177.9

layer 1 5 119.7layer 2 4 99.7layer 3 1 0.0024

15 20 25 30 35 40 45 5010

−5

10−4

10−3

10−2

10−1

100

offered load to each SD pair

aver

age

bloc

king

pro

babi

litie

s

simulationEFPAOPCA

Fig. 3. Blocking probabilities in the NSFNET.

2 4 6 8

10−3.886

10−3.885

(a) offered load = 15

Number of iterations (each layer)

Net

wor

k bl

ocki

ng p

roba

bilit

y

2 4 6 8

10−1.84

10−1.82

10−1.8

10−1.78

(b) offered load = 20


Net

wor

k bl

ocki

ng p

roba

bilit

y

2 4 6 810

−0.68

10−0.64

10−0.6

10−0.56

(c) offered load = 30


Net

wor

k bl

ocki

ng p

roba

bilit

y

2 4 6 8

10−0.28

10−0.25

10−0.22

10−0.19

(d) offered load = 55


Net

wor

k bl

ocki

ng p

roba

bilit

y

upper bound

lower bound

upper bound

lower bound

upper bound

lower bound

upper bound

lower bound

Fig. 4. Bounds of OPCA blocking probabilities in the NSFNET with a different offered load to each directional SD pair.


2 3 4 5 6 710

−1.55

10−1.54

10−1.53


Net

wor

k bl

ocki

ng p

roba

bilit

y

(a) D = 0

2 3 4 5 6 7

10−1.82

10−1.8

10−1.78

10−1.76


Net

wor

k bl

ocki

ng p

roba

bilit

y

(a) D = 1

2 3 4 5 6 7

10−1.84

10−1.82

10−1.8

10−1.78


Net

wor

k bl

ocki

ng p

roba

bilit

y

(a) D = 2

2 3 4 5 6 7

10−1.84

10−1.82

10−1.8

10−1.78


Net

wor

k bl

ocki

ng p

roba

bilit

y

(b) D = 3

upper boundupper bound

upper boundupper bound

lower bound lower bound

lower bound lower bound

Fig. 6. Bounds of OPCA blocking probabilities in the NSFNET with a different maximum allowable number of deflections (D), in whichthe offered load to each directional SD pair is 20 erlangs.

2 3 4 5 6 7

10−1.26

10−1.23

10−1.2

10−1.17


Net

wor

k bl

ocki

ng p

roba

bilit

y

(a) number of channels per trunk = 20

2 3 4 5 6 7

10−1.84

10−1.82

10−1.8

10−1.78


Net

wor

k bl

ocki

ng p

roba

bilit

y

(b) number of channels per trunk = 50

2 3 4 5 6 7

10−2.57

10−2.56

10−2.55


Net

wor

k bl

ocki

ng p

roba

bilit

y

(c) number of channels per trunk = 100

2 3 4 5 6 7

10−3.785

10−3.783

10−3.781


Net

wor

k bl

ocki

ng p

roba

bilit

y

(d) number of channels per trunk = 200

lower bound

lower bound

upper bound

upper bound

lower bound

upper bound

lower bound

upper bound

Fig. 5. Bounds of OPCA blocking probabilities in the NSFNET with a different number of channels per trunk (C), in which the offeredload to each directional SD pair is 0.4C.


bound is 0 for all LBTs, and the first upper bound increasesas the traffic increases, the distance between the firstupper and lower bounds of trunk blocking probability ob-tained by the first iteration is larger when the offered loadis larger. These general assertions are consistent with ob-servations in [34]. We also observe in Fig. 4 that as the of-fered load to the network increases, which implies moreprimary and deflected bursts in the network, it apparentlymakes it somewhat more difficult for the bounds to becomecloser to each other.

1) Effect of the Number of Channels per Trunk: Figure 5shows the bounds of the OPCA blocking probability resultsfor the NSFNET considering four scenarios where in eachscenario there are different channels per trunk. In all thescenarios, the offered load to each directional SD pair is0.4C, where C is the number of channels per trunk andthe maximum number of deflections is set to 3. We observefrom the figure that when the number of channels pertrunk increases, the lower and upper bounds become closerto each other faster. In particular, when there are 20 chan-nels per trunk, six iterations per layer are required toachieve a distance between the lower and upper boundsof around 10−5% of the lower bound value, but when thereare 100 channels per trunk, in five iterations per layer weachieve a much lower distance between the bounds,namely, approximately 10−8% of the lower bound value.The results are related to the fact that with a larger num-ber of channels per trunk, the variance of the number ofbusy channels is lower, which implies less deflected burstsin the networks, as we have already observed. This makesit easier for the bounds to become closer to each other.

2) Effect of the Maximum Allowable Number of Deflec-tions: Figure 6 shows the bounds of the OPCA blockingprobability results for the NSFNET considering four sce-narios where in each scenario we set a different valuefor the maximum allowable number of deflections (D). Inall scenarios, the offered load to each directional SD pairis 20 erlangs, and each trunk has 50 channels. We observefrom the figure that whenD increases, the lower and upperbounds become closer to each other slightly more slowly,since the overflowed traffic increases with an increasing D.However, this effect does not seem to be significant whenD ≥ 2, because the traffic in layers k (for k ≥ 2) is very smalland its effect on the end-to-end blocking probability is neg-ligible. In particular, when D � 0 and six iterations perlayer are made, the distance between the lower and upperbounds is around 10−8% of the lower bound value, but whenD � 2 and D � 3, in six iterations per layer the distance

between the lower and upper bounds is only around 10−6%of the lower bound value.

VI. CONCLUSIONS

In this paper, we have presented the bounds of the OPCfor blocking probability approximation in OBS networkswith deflection routing. The bounds obtained by the itera-tions of the OPCA algorithm consistently become closerto each other, and after a small number of steps yield asatisfactory blocking probability approximation. We haveobserved that the speed of the bounds moving closerdecreases when the proportion of the overflowed traffic in-creases, due to the growth of the offered load or the maxi-mum allowable number of deflections, as well as thereduction of the number of channels per trunk. We havealso demonstrated in the case of NSFNETwith 50 channelsper trunk that the OPCA is faster and at least as accurateas the EFPA.

APPENDIX A

Let F�x� � xPk�x�, where Pk�x� � �xk∕k!�∕�Pk

i�0 xi∕i!� for

positive integer k and x ≥ 0. The meaning of Pk�x� is theloss probability in the M∕G∕k∕k queueing system with theoffered traffic load x. Pk�x� is an increasing function in x.

The challenge is to prove that

Q�x; a� � F�x� a� − F�x�a

(A1)

is an increasing function in a.

We can write Eq. (A1) as

Q�x; a� � F�x� a� − F�x�a

� 1a

Zx�a

xF0�y�dy: (A2)

The function F�x� � xPk�x� is a convex function (for directproof, see, e.g., [44,45]). Using the first mean value theoremfor integration, we have

Q�x; a� � F0�x� θa�;

where 0 < θ < 1.

Using the fact that F�x� is a convex function, we have thefollowing properties:

1) Q�x; a� increases when x increases.2) Let a1 < a2. Then, because F0�x� is an increasing

function, we have the inequality a1θ1 ≤ a2θ2, where θ1and θ2 are the constants belonging to the interval(0,1), for which Q�x; a1� � F0�x� θ1a1� and Q�x; a2� �F0�x� θ2a2� (as shown in Fig. 7).

ACKNOWLEDGMENTS

The work on this paper was done when V. Abramov waswith the Department of Electronic Engineering, City Uni-versity of Hong Kong.

Fig. 7. Positions of Q�x; a1� and Q�x; a2�.


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Shuo Li received a B.Sc. degree in elec-tronic and communication engineering fromCity University of Hong Kong, Hong Kong,in 2009. She is currently working towarda Ph.D. degree in the Department of Elec-tronic Engineering at City University ofHong Kong. Her research interests are per-formance evaluation of telecommunicationnetworks.

Meiqian Wang received her bachelor’s de-gree and master’s degree in the Departmentof Electronic Engineering at Harbin Insti-tute of Technology in 2006 and 2009, respec-tively. She is now working toward a Ph.D.degree at City University of Hong Kong.Her research interest lies in performanceevaluation in circuit switching and burstswitching networks.

Eric W. M. Wong (S’87–M’90–SM’00)received B.Sc. and M. Phil. degrees in elec-tronic engineering from the Chinese Univer-sity of Hong Kong, Hong Kong, in 1988 and1990, respectively, and a Ph.D. degree inelectrical and computer engineering fromthe University of Massachusetts, Amherst,in 1994. In 1994, he joined the City Univer-sity of Hong Kong, where he is now an Asso-ciate Professor with the Department ofElectronic Engineering. His research inter-

ests include the analysis and design of telecommunications net-works, optical switching, and video-on-demand systems. Hismost notable research work involved the first accurate and work-able model for state-dependent dynamic routing based on theErlang fixed-point approximation, presented at INFOCOM ’90.Since 1991, the model has been used by AT&T to design and di-mension its telephone network that uses real-time networkrouting.

Vyacheslav Abramov graduated fromTajik State University (former USSR) in1977. During 1977–1992 he worked at theResearch Institute of Economics under theState Planning Committee of Tajikistan.In 1992 he repatriated to Israel, where heworked in different institutions and soft-ware companies. He received a Ph.D. degreein mathematics from Tel Aviv Universityin 2005. During 2005–2011 he workedat the School of Mathematical Sciences in

Monash University (Australia). During 2011–2012 he worked atthe Department of Electronic Engineering in City University ofHong Kong as a Senior Research Fellow. Since August 2012 hehas been working at the Center of Advanced Internet Architec-tures (CAIA) in Swinburne University of Technology. He is the au-thor of two books and about 40 research papers in the areas ofapplied probability, statistics, queueing theory, and financial engi-neering.

Moshe Zukerman (M’87–SM’91–F’07) re-ceived a B.Sc. degree in industrial engineer-ing and management and a M.Sc. degree inoperations research from the Technion—Israel Institute of Technology, Haifa, Israel,and a Ph.D. degree in engineering from theUniversity of California, Los Angeles, in1985. He was an Independent Consultantwith the IRI Corporation and a PostdoctoralFellow with the University of California, LosAngeles, in 1985–1986. In 1986–1997, he

was with the Telstra Research Laboratories (TRL), first as a Re-search Engineer and, in 1988–1997, as a Project Leader. He alsotaught and supervised graduate students at Monash University in1990–2001. During 1997–2008, he was with the University of Mel-bourne, Victoria, Australia. In 2008 he joined City University ofHong Kong as a Chair Professor of Information Engineeringand a group leader. He has served on various editorial boards suchas Computer Networks, IEEE Communications Magazine, IEEEJournal of Selected Areas in Communications, IEEE Transactionson Networking, and the International Journal of CommunicationSystems.


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