Connection-dependent threshold model: a generalization of the Erlang multiple rate loss model

Performance Evaluation 48 (2002) 177–200

Connection-dependent threshold model: a generalization of the Erlang multiple rate loss model夽 Ioannis D. Moscholios, Michael D. Logothetis∗ , George K. Kokkinakis Wire Communications Laboratory, Department of Electrical and Computer Engineering, University of Patras, 265 00 Patras, Greece

Abstract In this paper first, we review two extensions of the Erlang multi-rate loss model (EMLM), whereby we can assess the call-level quality-of-service (QoS) of ATM networks. The call-level QoS assessment in ATM networks remains an open issue, due to the emerged elastic services. We consider the coexistence of ABR service with QoS guarantee services in a VP link and evaluate the call blocking probability (CBP), based on the EMLM extensions. In the first extension, the retry models, blocked calls can retry with reduced resource requirements and increased arbitrary mean residency requirements. In the second extension, the threshold models, for blocking avoidance, calls can attempt to connect with other than the initial resource and residency requirements which are state dependent. Secondly, we propose the connection-dependent threshold model (CDTM), which resembles the threshold models, but the state dependency is individualized among call-connections. The proposed CDTM not only generalizes the existing threshold models but also covers the EMLM and the retry models by selecting properly the threshold parameters. Thirdly, we provide formulas for CBP calculation that incorporate bandwidth/trunk reservation schemes, whereby we can balance the grade-of-service among the service-classes. Finally, we investigate the effectiveness of the models applicability on ABR service at call set-up. The retry models can hardly model the behavior of ABR service, while the threshold models perform better than the retry models. The CDTM performs much better than the threshold models; therefore we propose it for assessing the call-level performance of ABR service. We evaluate the above-mentioned models by comparing each other according to the resultant CBP in ATM networks. For the models validation, results obtained by the analytical models are compared with simulation results. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Quality-of-service; Call blocking probability; Recurrent formula; ATM networks; ABR service

1. Introduction ATM networks are designed to accommodate a variety of service-classes with different traffic description parameters and quality-of-service (QoS) requirements. We study the QoS assessment of call-level traffic in the multi-service environment of ATM networks. This assessment is very important for ATM networks because it is a basic requirement for several network/traffic controls like connection admission 夽

This work was supported by the Research Programme Caratheodory of the Research Committee, University of Patras, Greece. Corresponding author. Tel.: +30-61-996-166; fax: +30-61-991-855. E-mail address: [email protected] (M.D. Logothetis). ∗

0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 0 3 7 - 8

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control (CAC) [21], virtual path bandwidth (VPB) control [12,13] and network planning [14]. Nevertheless, the call-level QoS assessment still remains an open issue in ATM networks because of the accommodated elastic services [7]. In this study, we consider the coexistence of ABR service together with CBR service in an ATM network and in order to assess the call-level QoS of each service-class, we evaluate the call blocking probability (CBP), which is the call-level performance index. On the call-level, ATM networks that accommodate QoS guarantee services (CBR and VBR services) can be modeled as multi-rate circuit switched networks [20]. By adopting the notion of equivalent bandwidth [3,6], we can face CBR and VBR services in the same way and manage the call-level traffic by using the Erlang multi-rate loss model (EMLM) [9,22]. This model is attractive because the accuracy of CBP calculation is absolutely satisfactory, especially when the differences between the holding times of service-classes are small [15]. However, the notion of equivalent bandwidth is not directly applicable to ABR service. In the case of ABR service the notion of CBP needs to be reconsidered because no resource allocation is made prior to the information transfer phase [8]. The CBP strongly depends on the required holding (service) time of an ABR call-connection, while the holding time depends on the available bandwidth of the network and it is not known in advance. If we devote long service time in order to convey an ABR call-connection, eventually the required bandwidth will be found and no blocking will occur, while if we restrict the service time to a certain period, it is probable that no call completion will occur and the call will be blocked. Nevertheless, it is still possible to calculate CBP of ABR and QoS guarantee services in a common way, since there is an amount that is constant for both service categories; this is the “total” offered traffic-load [1]. The notion of time is inherently included in the traffic-load, as traffic-load is the product of arrival rate by the holding time. Total offered traffic-load is the product of traffic-load by the required bandwidth per call. Or by incorporating the arrival rate into the required bandwidth per call, we can assume the total offered traffic-load as the product of the holding time by the required bandwidth per call. Fig. 1 shows that an ABR call could be seen either as a concatenation of three CBR calls, ABR (a), or four CBR calls, ABR (b), or as a single CBR call in the case that it is serviced by the minimum bit rate it requires. In each case of Fig. 1, the holding time is worth noticing; ABR calls require the same holding time, while the CBR call requires more holding time than the double holding time of each ABR call. However, all calls have the same “total” offered traffic-load. The traditional models for calculation of CBP pay no attention to congestion/feedback control, which is inherently included in ABR service. Fortunately, in the literature we have found models, which comprise some kind of feedback control and therefore they can model the ABR service. These are the retry and

Fig. 1. ABR call (a) and (b) and a CBR call that all have the same total offered traffic-load.

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threshold models [10,11], which are the extensions of the EMLM. An extension of the retry models, the partially blocking loss model (PBLM), has already been proposed to model the ABR service at the call-level [2,4,5]. However, the PBLM refers to a simple case of ABR traffic that corresponds to the single retry model (see Section 2.1). Besides, the CBP formulas proposed in [2] for the PBLM are computationally inefficient because they are not recurrent and contain exponentials and factorials. In this paper, by exploiting the notion of the aforementioned total offered traffic-load, we propose the threshold models to assess the behavior of ABR service at call set-up and calculate CBP. In order for the threshold models to perform best, the CAC should adopt the same thresholds scheme as a CAC scheme. We investigate the effectiveness of the applicability of both the retry and threshold models on describing the call-level traffic of ABR service and show that the threshold models perform better than the retry models, which can hardly model the ABR traffic. Moreover, we propose a generalization of the threshold models, named connection-dependent threshold model (CDTM), whereby we can describe accurately the behavior of ABR service at call set-up. Therefore, the obtained CBP by the CDTM is more reliable. Furthermore, by selecting properly the threshold parameters of the CDTM, the EMLM and the retry models result. For all models, we provide formulas of CBP calculation that incorporate bandwidth/trunk reservation schemes, whereby we can balance the grade-of-service among the service-classes. A CAC provided with a proper bandwidth/trunk reservation scheme can considerably save network bandwidth [13,16]. With application examples, we reveal the superiority of the CDTM, in respect of CBP, over the other models. Certainly, for models validation, the results obtained by the analytical models are compared with simulation results. The organization of this paper is as follows: in Section 2, we review the two extensions of the EMLM; in Sections 2.1 and 2.2 we present the retry and threshold models, respectively, and give formulas for the final CBP which is of practical value. In Section 3, we introduce the proposed CDTM; in Section 3.1, we present the analytical model and in Section 3.2, the applicability of the model to the call-level traffic of ABR service is discussed. Section 4 shows the incorporation of bandwidth/trunk reservation into CBP formulas, for all models. In Section 5, based on case studies of ABR traffic, we present comparatively numerical results of the models along with simulation results, for evaluation. In a preliminary Section 5.1, we describe the considered service-classes accommodated in a VP link. In Section 5.2, we show the results (CBPs) when the EMLM is applied. The results of the multi-retry model (MRM) are shown in Section 5.3. Section 5.4 contains the results of the multiple threshold model (MTM) and CDTM. The models comparison is shown in Section 5.5. We conclude in Section 6. Finally, in appendix, sufficient notes/hints for software implementation of the CDTM are found.

2. Review of the extensions of the EMLM 2.1. The single and MRMs The EMLM copes with multi-dimensional traffic, i.e. calls with different characteristics sharing a resource. More precisely, considering K (k = 1, 2, . . . , K) independent service-classes of Poisson arriving calls, which offer traffic-load ak and request an integer valued bk bandwidth per call, and a transmission link (virtual path, VP) of bandwidth C (C servers), if at least bk of the C bandwidth units (b.u.) are available when a call of service-class k arrives in the link, bk b.u. are seized for an exponential holding time of mean µ−1 k to convey the call, and then are released simultaneously. When less than bk b.u. are available, the call is blocked and loses the links immediately.

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In the single retry model, a blocked call of service-class k can specify “retry parameters” (bk r , µ−1 kr ), −1 −1 where bkr < bk and µkr > µk , and immediately reattempts to be connected (requesting bk r b.u. per call with a mean holding time µ−1 kr ). According to [11], the following recurrent formulas are proposed for calculating CBP:  1 for j = 0,      K K 1 1 G(j ) = (1) ak bk G(j − bk ) + akr bkr γk (j )G(j − bkr ) for j = 1, . . . , C, j j k=1  k=1     0 otherwise, where γk (j ) = 1 when j > C − (bk − bkr ), otherwise γk (j ) = 0, akr = λk µ−1 (λk is the arrival rate of kr calls). By convention, if bkr = 0, it will be understood to mean that calls of service-class k do not retry. The probability of blocking Bk∗ of service-class k in a link (VP), for the first time, is defined as Bk∗ = Pr{j > C − bk }: Bk∗

=

C 

−1

G G(j ),

j =C−bk +1

C  where G = G(j ).

(2)

j =1

However, it is not the final CBP because calls reattempt to be connected. The blocking probability of retry calls of service-class k, Bk r , is worth mentioning. It is defined by the conditional probability Bk r , as Bkr = Pr{j > C − bkr |j > C − bk } (when bkr > 0). Bk r is given by C −1 j =C−bkr +1 G G(j ) Bkr = , (3) Bk∗ for all k such that bkr > 0, k = 1, 2, . . . , K. The final CBP, Bk , is defined as Bk = Pr{j > C − bkr } and is given by Bk = Bk∗ Bk r , i.e. Bk =

C 

G−1 G(j ).

(4)

j =C−bkr +1

In general, we may have multiple retrials (Fig. 2), i.e. a call of service-class k, which requires bk bandwidth and µ−1 k mean holding time and offers traffic-load ak , may retry s(k) times for connection, with “retry −1 1 parameters” (bkrl , µ−1 krl ) for l = 1, . . . , s(k), where bkrs(k) < . . . < bkrl < bk and µkrs(k) > . . . > µkrl > −1 µk . The following recurrent formula is found in [11] for calculating CBP:  1 for j = 0,      K K s(k) 1 1  G(j ) = (5) ak bk G(j − bk ) + akr bkr γk (j )G(j − bkrl ) for j = 1, . . . , C,  j k=1 j k=1 l=1 l l l      0 otherwise, where γkl (j ) = 1 when j > C − (bkrl−1 − bkrl ), otherwise γkl (j ) = 0 and akrl = λk µ−1 krl (λk is the arrival rate of calls).

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Fig. 2. Principles of the multiple retry model.

The probability of blocking Bkrl of service-class k in a link (VP), at lth retrial, given that the call has been blocked in the previous attempt of connection, is defined as Bkrl = Pr{j > C − bkrl |j > C − bkrl−1 }, for l = 1, . . . , s(k) and it is given by C j =C−bkrl +1 G(j ) Bkrl = C (6) for l = 1, . . . , s(k) and bkrl > 0, j =C−bkr −1 +1 G(j ) l

whereby convention bkr0 = bk . The final conditional CBP, Bkrs , that is the probability of call blocking at the last retrial s(k) of service-class k, given that the call initially had been blocked, is calculated by C s(k)  j =C−bkrs +1 G(j ) Bkrs = Bkrl = C , (7) j =C−bk +1 G(j ) l=1 where the denominator is GBk∗ (see Eq. (2)). The final unconditional CBP, Bk , is defined as Bk = Pr{j > C − bkrs } and is given by Bk = Bkrs Bk∗ , i.e. Bk =

C 

G−1 G(j ).

(8)

j =C−bkrs +1

In this paper, we are concerned with the final unconditional CBP. It is worth noticing in Eqs. (1) and (5) that if all the retry parameters are zero, the retry models are transformed to the EMLM. 2.2. The single and multi-threshold models In the so-called single threshold model, instead of waiting for calls to be blocked and then retry, it may be preferable, bandwidth and service time requests to be dependent on the total number of occupied b.u., denoted by j. In other words, a call of service-class k has a resource requirement (bk , µ−1 k ) that will be −1 satisfied if j ≤ J0 , and a contingency resource requirement (bk c , µkc ) that will be used if the occupied

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bandwidth is high enough, i.e. when j > J0 , where J0 is an arbitrary, unique bandwidth threshold. The following recurrent formula for G(j)/G (probability of j out of C trunks to be seized) is proposed in [10]:  1 for j = 0,      K K 1 1 G(j ) = (9) ak bk δk (j )G(j − bk ) + akc bkc δkc (j )G(j − bkc ) for j = 1, . . . , C,  j k=1 j k=1      0 otherwise, where δk (j ) = 1 when 1 ≤ j ≤ C and bkc = 0 or, when j ≤ J0 + bk and bkc > 0, otherwise δk (j ) = 0; δkc (j ) = 1 when j > J0 + bkc , otherwise δkc (j ) = 0; akc = λk µ−1 kc . Based on Eq. (9) and following the analysis of [10], we propose the following approximate formula for calculating the conditional CBP, Bk c , of service-class k, given that the number of seized trunks j in the link of capacity C exceeds the threshold J0 , (j > J0 ), and therefore the call uses the alternative resource requirement (Bkc = Pr{j > C − bkc |j > J0 }): C j =C−b +1 G(j ) Bkc = C kc . (10) j =J0 +1 G(j ) The final unconditional CBP, Bk , is approximated as Bk = Pr{j > C − bkc } = Pr{j > C − bkc |j > J0 } ∗ Pr{j > J0 } and is given by Bk =

C 

G−1 G(j ).

(11)

j =C−bkc +1

An extension of this model exists with more than one contingency resource requirements and thresholds; this is the multiple threshold (state dependent) model (MTM). A call of service-class k, which requires bk bandwidth and µ−1 k mean holding time and offers traffic-load ak , may attempt n + 1 times to be connected; one initial attempt with parameters (bk c , µ−1 kc ) and n more attempts, with parameters (bkcl , −1 −1 −1 −1 µkcl ), l = 1, . . . , n, where bkcn < . . . < bkcl < bk and µ−1 kcn > . . . > µkcl > µk ; the pair (bkcl , µkcl ) is used when Jl < j ≤ Jl+1 , Jn+1 = C, and the highest possible threshold Jn = C − bkcn (the minimum value obtained among all service-classes). Fig. 3 portrays the MTM. −1 By convention, bk = bkc0 and µ−1 k = µkc0 . The following recurrent formula for G(j) is found in [10]:  1 for j = 0,      K K n 1 1  G(j ) = ak bk δk (j )G(j − bk ) + akc bkc δkc (j )G(j − bkcl ) for j = 1, . . . , C,  j k=1 j k=1 l=1 l l l      0 otherwise, (12) where δk (j ) = 1 when 1 ≤ j ≤ C and bkc = 0, or, when j ≤ J1 + bk and bkc > 0, otherwise δk (j ) = 0; δkcl (j ) = 1 when Jl+1 + bkcl ≥ j > J1 + bkcl , otherwise δkcl (j ) = 0; akcl = λk µ−1 kcl (λk is the arrival rate of calls).

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Fig. 3. Principles of the MTMs.

Based on Eq. (12), we propose the following formula for calculating the conditional CBP, Bk c , of service-class k, given that the number of seized trunks j in the link of capacity C exceeds the last threshold Jn , (j > Jn ), and therefore the call uses the last contingency resource requirement, bkcn (Bkc = Pr{j > C − bkc |j > Jn }). C j =C−b +1 G(j ) . (13) Bkc = C kcn j =Jn +1 G(j ) The final unconditional CBP, Bk , is defined as Bk = Pr{j > C − bkc } = Pr{j > C − bkc |j > Jn } ∗ Pr{j > Jn } and is given by Bk =

C 

G−1 G(j ).

(14)

j =C−bkcn +1

3. The CDTM 3.1. The analytical model As in the case of the threshold models, we consider calls of different service-classes that have contingency resource requirements, which depend on the so called thresholds. However, these thresholds are not common for all service-classes but, as we propose, they are individualized among the call-connections (Fig. 4). Therefore, we call the proposed model, CTDM. So, a call of service-class k, which requires bk bandwidth and µ−1 k mean holding time and offers traffic-load ak , may attempt n + 1 times to be connected; one initial attempt with parameters (bk c , µ−1 kc ) −1 and n more attempts, with parameters (bkcl , µkcl ), l = 1, . . . , n, where bkcn < · · · < bkcl < bk and −1 −1 −1 µ−1 kcn > . . . > µkcl > µk . The pair (bkcl , µkcl ) is used for service-class k when Jkl < j ≤ Jkl+1 , Jkn+1 = C and the highest possible (“real”) threshold Jkn = C − bkcn . Following the analysis of the existing threshold models, in the proposed CDTM, we conclude that we can use Eq. (12), by introducing the necessary modifications on the threshold parameters, as the recurrent formula for calculating G(j)/G, the probability j out of C trunks to be seized:

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Fig. 4. Principles of the CTDM.

• δk (j ) = 1 when 1 ≤ j ≤ C and bkc = 0, or, when j ≤ Jkl + bk and bkc > 0, otherwise δk (j ) = 0; • δkcl (j ) = 1 when Jkl+1 + bkcl ≥ j > Jkl + bkcl , otherwise δkcl (j ) = 0; • akcl = λk µ−1 kcl (λk is the arrival rate of calls). The conditional CBP, Bk c , given that the number of seized trunks j in the link of capacity C exceeds the last threshold Jkn , (j > Jkn ), for service-class k, and therefore the call uses the last contingency resource requirement, bkcn , is calculated by the following formula. It results by modifying the limit of the summation in the denominator of Eq. (13): C j =C−b +1 G(j ) Bkc = C kcn . (15) j =Jkn +1 G(j ) The final unconditional CBP, Bk , is calculated by Eq. (14). It is not necessary to distinguish the single CDTM from the multiple one because the single model is a direct simplification of the multiple model. In the single CDTM, the number of thresholds is limited to one per each service-class, whereas if all service-classes have the same single threshold, the existing single threshold model results (Section 2.2). Besides, the existing MTM is a special case of the CDTM, where the thresholds of different service-classes coincide. In other words, the proposed CDTM generalizes the existing threshold models. In Appendix A, sufficient hints are found for the computer implementation of the CDTM. It is worth mentioning that the final (unconditional) CBP obtained by the CDTM coincide with that of the retry models, if we choose the thresholds Jkn , for each service-class k, so that Jkn = C − bkn−1 . By the previously existing threshold models, due to the necessity that all service-classes meet the same threshold(s), it is not always possible to produce the same results with the retry models. Furthermore, if we set all the thresholds parameters of the CDTM to zero, the EMLM results. 3.2. Applicability of the CDTM to the call-level traffic of ABR service The “standard” behavior of ABR calls at the call set-up phase, where call blocking (not cell loss) occurs, is as follows. An ABR call declares to the CAC, two bandwidth parameters: the minimum and the maximum required bandwidth. When the network (CAC) cannot provide the minimum required bandwidth to the ABR call, call blocking occurs and the ABR call leaves immediately from the network. This is the case where the minimum required bandwidth is not zero. If the minimum required bandwidth

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is zero, the call waits for any available bandwidth. (We do not refer to the zero-bandwidth, non-blocking, case.) Otherwise, the ABR call receives from the network any available bandwidth between the declared values, and the ABR connection is established. This very simple CAC scheme fulfils the assumptions of the proposed CDTM, as far as the CBP is concerned. That is, the minimum and the maximum bandwidth requirements are taken into account by the CDTM in the calculation of CBP no matter what the thresholds are. The blocking condition results from the minimum bandwidth (resource) requirements of the ABR call. If CAC ignores the threshold scheme of the ABR call, the (available) bandwidth that gives to it does not necessarily coincide with a value of its resource requirements. This means that the CBP calculation through the CDTM is an approximate one, because either the holding time determined through the total offered traffic-load is incorrect, or the thresholds assumption of the CDTM is violated by the CAC. As an example, suppose that the following thresholds scheme is applied to an ABR call for a transmission link with bandwidth capacity of 19.2 Mbps: maximum resource requirement of 1.536 Mbps when the available link bandwidth is 6.4 Mbps at least (first threshold at 12.8 Mbps), resource requirement of 768 kbps when the available link bandwidth is 3.2 Mbps at least (second threshold at 16.0 Mbps), and minimum resource requirement of 384 kbps when the available link bandwidth is less than 3.2 Mbps (last threshold at 19.2 Mbps or the real last threshold at 19.2 − 0.384 = 18.816 Mbps). Assume that the CAC knows only the minimum and maximum resource requirements of the ABR call and offers to it (a) 700 kbps when the available bandwidth is 4.0 Mbps and (b) 1.536 Mbps when the available bandwidth is 4.0 Mbps. In the first case, the holding time will be determined incorrectly (by taking into account 768 kbps instead of 700 kbps), while in the second case, although the holding time will be determined correctly, the thresholds assumption has been violated. However, when we propose the threshold models and especially the CDTM, for ABR traffic, we imply that the CAC adopts the thresholds scheme of the model as a CAC scheme. This means that the holding time of an ABR connection will be determined correctly and the thresholds assumption will not be violated; therefore, the CBP calculation will be accurate through the CDTM. The logic behind the thresholds scheme is that even if the available bandwidth of a link is large enough, a CAC does not always waste it on one call-connection only, but saves a part of it for sharing it with next calls. Conclusively, the assumptions of the proposed CDTM correspond completely to the behavior of ABR calls at call set-up. In practice, between the minimum and the maximum bandwidth requirements, several (and not infinite) bandwidth values exist, since bandwidth is quantized and provided as a group of trunks. Thus, in a realistic network environment the number of thresholds is manageable [17]. This situation can be accurately described (by using the thresholds of the CDTM) and taken into account by the CAC; thus, the CBP obtained by the CDTM is reliable, given that we have validated the model through simulation. On the contrary, the MTM has the strict condition that the service-classes must have common thresholds, while the MRM assumes call blocking before bandwidth reduction occurs, which is unrealistic for ABR at the phase of call set-up.

4. Incorporating bandwidth reservation into the models In ATM networks cells of different service-classes, which have different bandwidth requirements per call are integrated and commonly share a VP. Therefore, the CBP of service-classes with higher bandwidth requirements (high-speed calls) becomes worse than that of service-classes requiring lower bandwidth (low-speed calls). To decrease this imbalance of the CBP, we have to reserve some fraction of the shared

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VP bandwidth to benefit the high-speed calls. More precisely, calls of the service-class k are rejected (by CAC), when less than t(k) b.u. remain available in the VP. The reserved bandwidth t(k) can be defined so that the sum of the required bandwidth per call of the service-class k and the t(k) to be the same for all service-classes. In this way, all service-classes meet the same grade-of-service. To incorporate bandwidth reservation (BR) in the calculation of CBP for EMLM, a good approach is found in [18,19]. This approach can readily be applied to the above models. To incorporate BR into the retry models, we introduce the following modifications to the first and the second sum of expression (1) or (5):  1 for j = 0,      K   1   ak Dk (j − bk )G(j − bk )    j k=1 G(j ) = (16) s(k)  K    1    + akr Dkr (j − bkrl )γkl (j )G(j − bkrl ) for j = 1, . . . , C,   j k=1 l=1 l      0 otherwise, where



Dk (j − bk ) =

bk

for j ≤ C − t (k),

0

for j > C − t (k),

 Dkr (j − bkrl ) =

bkrl

for j ≤ C − t (k),

0

for j > C − t (k).

(17)

To calculate CBP taking into consideration BR, we have to modify properly the limits of the summations in the corresponding expressions. For example, Eq. (2) should become Bk∗ =

bk +t (k)−1 

G−1 G(C − j ).

(18)

j =0

It is worth noticing that the first sum of the expression (16), together with (17) and (18) compose the EMLM with BR. As with the retry models, we can incorporate BR into the threshold/CDTM models. To achieve it, we have to substitute in the expression (12), bk with Dk (j − bk ) and bkcl with Dkcl (j − bkcl ), which is defined similarly to Dk r in the expression (17):  1 for j = 0,      K   1   ak Dk (j − bk )δk (j )G(j − bk )   j k=1 (19) G(j ) = n K     1   + akc Dkcl (j − bkcl )δkcl (j )G(j − bkcl ) for j = 1, . . . , C,    j k=1 l=1 l      0 otherwise,

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where Dk (j − bk ) =



bk

for j ≤ C − t (k),

0

for j > C − t (k),

 Dkcl (j − bkcl ) =

bkcl

for j ≤ C − t (k),

0

for j > C − t (k).

187

(20)

5. Numerical examples—evaluation 5.1. Preliminaries We consider a VP link in ATM network that accommodates four service-classes with bandwidth capacity 300 b.u. The first two service-classes, s1 and s2 are CBR services, which require 1 and 6 b.u. per call, respectively. For example, assuming that 1 b.u. = 64 Kbps, s1 could correspond to the telephony, and s2 to the videophony (of 384 Kbps). The other two service-classes, s3 and s4 , are ABR services. s3 requires 6 b.u. per call (as s2 ), but since it is an ABR service, it can reduce it to 2 b.u., unit by unit, according to the amount of bandwidth that is available in the VP link (Fig. 5). We choose s3 from a set of seven ABR services, which all have the same behavior, but they are distinguished from the first point (threshold) of

Fig. 5. ABR services for service-class s3 and s4 .

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available bandwidth at which they change the required bandwidth per call (see Fig. 5, horizontal axis). Two out of the seven ABR services, the ABR 11 and ABR 17 , are shown in Fig. 5, for s3 . s4 requires 24 b.u. (1.536 Mbps) per call and it can reduce it to 8 b.u. per call, in steps of 4 b.u. Alternatively, s4 can reduce the required bandwidth per call to 11 b.u., unit by unit, according to the available VP bandwidth. So, we choose s4 from two sets of seven ABR services per set, which are distinguished by a) the contingency minimum required bandwidth per call, b) the step whereby the bandwidth per call is reduced, c) the first point (threshold) of available bandwidth at which they change the bandwidth per call. Fig. 5, for s4 , portrays two ABR services from each set: ABR 21 and 27 (first points for resize = 24 and 84 b.u., respectively, minimum bandwidth per call = 8 b.u., step = 4 b.u.) from the set ABR 2 = {ABR 21 , . . . , 27 }, ABR 31 and 37 (first points for resize = 24 and 84 b.u., respectively, minimum bandwidth per call = 11 b.u., step = 1 b.u.) from the set ABR 3 = {ABR 31 , . . . , 37 }. We evaluate the call-level QoS of each service-class accommodated in the VP, i.e. we calculate the CBP by applying the formulas of Sections 2–4 and compare their results. To this end, we consider the following traffic-loads offered to the VP: 100 erl for s1 , 12 erl for s2 and s3 , and 1 erl for s4 . When calls of s3 and s4 reduce the initially required bandwidth, they prolong their holding times, so that the total offered traffic-load is kept constant. For example, the offered traffic-load of s3 changes to 14.4, 18.0, 24.0 and 36.0 erl, when the required bandwidth per call reduces to 5, 4, 3 and 2 b.u., respectively. 5.2. Application of the EMLM First, we apply the EMLM formula. Of course, since we cannot apply it for ABR services, we consider s3 and s4 as CBR services, i.e. we assume that they have the traffic characteristics described above, but not the ability to change the initially required bandwidth per call. The resultant CBP, without BR and with BR is shown in Table 1. When we apply BR, we use the BR scheme that equalizes the CBP of the service-classes (see column 2 of Table 1). Although these results do not refer to ABR services, they are useful in our evaluation because we get lower bounds of CBP for s1 and s2 , and upper bounds of CBP for s3 and s4 . It can be understood intuitively, since ABR services, s3 and s4 , expect to meet a better QoS than CBR services, because of their ability of reducing bandwidth requirements in case of lack of bandwidth; this in turn will deteriorate the QoS for s1 , s2 . 5.3. Application of the MRM Secondly, we consider s3 and s4 as any combination of ABR services shown in Table 2, and we apply the MRM. The retry models cannot distinguish ABR services that belong to the same set; i.e. they meet Table 1 Results of the EMLM, with or without BR Service-classes

BR per service-class (b.u.)

Without BR CBP (%)

With BR CBP (%)

s1 : telephony (CBR) s2 : videophony (CBR) s3 : CBR s4 : CBR

1/23 6/18 6/18 24/0

0.825 5.219 5.219 24.477

8.194 8.194 8.194 8.194

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Table 2 ABR services for service-classes s3 and s4

1 2 ... 7

(s3 :ABR 1, s4 :ABR 2)

(s3 :ABR 1, s4 :ABR 3)

(s3 :ABR 1, s4 :ABR 27 )

(ABR 11 , ABR 21 ) (ABR 12 , ABR 22 ) (. . . , . . . ) (ABR 17 , ABR 27 )

(ABR 11 , ABR 31 ) (ABR 12 , ABR 32 ) (. . . , . . . ) (ABR 17 , ABR 37 )

(ABR 11 , ABR 27 ) (ABR 12 , ABR 27 ) (. . . , . . . ) (ABR 17 , ABR 27 )

no differences, e.g. between ABR 11 and ABR 12 or ABR 17 ; therefore, we apply the MRM only once for each combination of ABR service-classes, in order to get approximated CBP. This approximation is much better than the previous one, achieved by the EMLM, since it takes into account that calls decrease their bandwidth, while at the same time, they increase their holding time. Nevertheless, this mechanism is not incorporated into the retry models with accuracy. The retry models assume that retry parameters change in a fixed sequence that is not always true for ABR services, despite the fact that ABR services do not experience actually blocking in order to modify their bandwidth. Therefore, the retry models hardly model the call-level behavior of ABR service. According to Fig. 5, since the points (24, 20, 16, 12) of the available VP bandwidth at which both ABR 11 and 21 change bandwidth requirements are closer to the points (6, 5, 4, 3) and (24, 20, 16, 12) used by the MRM for the set ABR 1 and 2, respectively, than to the points (84, 80, 76, 72) at which both ABR 17 and 27 change bandwidth requirements, the results of the MRM will approximate better ABR 11 and 21 than ABR 17 and 27 . The CBP obtained by the MRM are presented in Tables 3 and 4. We have also obtained results through simulation. These are in an acceptable range of around 2% from the values presented in Tables 3 and 4. Simulation results of the MRM are not presented, since the validation of MRM is already known [10]. Table 3 Results of the MRM for s3 :ABR 1 and s4 :ABR 2 or ABR 27 Service-classes

Without BR CBP (%)

With BR1 CBP (%)

With BR2 CBP (%)

s1 : telephony s2 : videophony s3 : ABR 1

2.318 7.886 First: 7.886 Final: 3.627 First: 26.602 Final: 9.716

8.724 8.724 First: 8.724 Final: 4.588 First: 8.724 Final: 1.544

4.635 4.635 First: 8.713 Final: 4.635 First: 22.472 Final: 4.635

s4 : ABR 2, or ABR 27

Table 4 Results of the MRM for s3 :ABR 1 and s4 :ABR 3 Service-classes

Without BR CBP (%)

With BR1 CBP (%)

With BR2 CBP (%)

s1 : telephony s2 : videophony s3 : ABR 1

2.775 7.083 First: 7.083 Final: 3.601 First: 25.962 Final: 11.825

8.693 8.693 First: 8.693 Final: 4.556 First: 8.693 Final: 1.890

4.989 4.989 First: 9.396 Final: 4.989 First: 20.222 Final: 4.989

s4 : ABR 2, or ABR 27

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We consider the existence or not of BR schemes. We consider two BR schemes: (1) The first BR scheme, BR1, which is that of Fig. 1, equalizes the first CBP (before the retrials) and further benefits s3 and s4 . (2) The second BR scheme, BR2, equalizes the final CBP taking into account the bandwidth per call that eventually (after the retrials) the service-classes require. For example, when eventually s4 and s3 require 8 and 2 b.u. per call, respectively, the BR units of s1 , s2 , s3 and s4 are 7, 2, 6 and 0, respectively. 5.4. Application of the MTM and CDTM Then, we calculate CBP by applying the MTM and CDTM. The MTM is not possible to be applied to the combinations of service-classes shown on Table 2, but ABR 1 and 2. This is because not all service-classes have the same thresholds (e.g. ABR 11 and ABR 21 have common thresholds at 276, 280, 284 and 288 b.u., while ABR 31 has a different set of thresholds: 276, 277, 278, . . . , 289 b.u.). For the rest of the combinations, we have to apply the CDTM. The resultant CBPs, versus group-of-thresholds (GoTs), are presented in Figs. 6–8. Figs. 6b and 7b present simulation results of the MTM and the CDTM with BR1, respectively, while the numerical results are presented in Figs. 6a and 7a. The error bars of Figs. 6b and 7b represent reliability ranges of 95%. The analysis of the simulation results is based on the batch mean method [1]. In the horizontal axis of all figures, the seven GoT correspond to the seven rows of Table 2. For example, the first GoT of Fig. 6 refers to ABR 11 and 21 and has the values (276, 280, 284, 288 b.u.) for both ABR services, the second GoT refers to ABR 12 and 22 and has the values (266, 270, 274, 278 b.u.) for both ABR services, and so on. The first threshold, J1 , in the ith group is defined by J1 = C−(point i) where (point i), i = 1, 2, . . . , 7, appears in horizontal axis of Fig. 5. Similarly, the GoT of Fig. 7 refers to ABR 11 and 31 , and has the values (276, 280, 284, 288 b.u.) for ABR 11 , and (276, 277, 278, . . . , 287, 288 b.u.) for ABR 31 . We do not mention the last “threshold” which is always 300 b.u. The CBPs shown in Figs. 6a, 7a and 8 by bars, present final, unconditional CBPs, without BR. When BR is used, we show (lines) three CBPs only, because telephony and videophony meet the same QoS, which is shown in the figure by “CBR/res”. As far as the ABR services are concerned, since some CBPs with BR are even greater than CBPs without BR, we have to make it clear, that we apply the first BR scheme, BR1, which equalizes only the probabilities of call resizing for the first time, with the CBP of the CBR services. As an example of which probabilities appear in Figs. 6a, 7a and 8, we consider the highest CBP of Fig. 6a, which is 10.298% (corresponds to GoT 3, CBR/res), and we present in Table 5 Table 5 Example of what probabilities appear in Fig. 6a Service-classes

Without BR

With BR1

Shown in figures

s1 : telephony s2 : videophone s3 : ABR 13

CBP = 0.985% CBP = 6.450% Pr{J > J11 } = 6.450% CBP = 2.003% Pr{J > J21 } = 33.655% CBP = 8.907%

10.298% 10.298% Pr{J > J11 } = 10.298% CBP = 3.133% Pr{J > J21 } = 10.298% CBP = 0.064%




s4 : ABR 23





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Fig. 6. (a) CBP obtained by MTM for s3 :ABR 1 and s4 :ABR 2; (b) CBP obtained by simulating the MTM for s3 :ABR 1 and s4 :ABR 2.

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Fig. 7. (a) CBP obtained by the CDTM for s3 :ABR 1, s4 :ABR 3; (b) CBP obtained by simulating the CDTM with BR1, for s3 :ABR 1, s4 :ABR 3.

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Fig. 8. CBP obtained by the CDTM for s3 :ABR 1, s4 :ABR 27 .

the probabilities that appear in Fig. 6a for GoT 3. The equalized CBPs of all ABR services (s3 , s4 ) when we apply the BR2 scheme, are presented in Fig. 9. 5.5. Comparison The comparison of Figs. 6a and 7a with Figs. 6b and 7b, respectively, proves the validity of the MTM and the CDTM. Since we assume that the thresholds scheme is taken into account by the CAC, the

Fig. 9. Equalized CBP, obtained by the CDTM, for s3 and s4 .

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Fig. 10. Models comparison based on Table 3 and Fig. 6a (without BR).

simulation of the CDTM coincides with the actual simulation of ABR traffic at call set-up; therefore, the obtained CBP from the CDTM is more reliable. Finally, in order to facilitate the evaluation of all models, we present in the same figure comparative results of the analytical models. Figs. 10–12 show the results of the EMLM, MRM and CDTM (or MTM) when the GoT is the first or the seventh. The first GoT is chosen because the resultant CBP of the MTM/CDTM is close to MRM. The seventh GoT is chosen not only because bigger differences between CBPs result but also in order to show that as the set of thresholds are away from the VP capacity C, the CBP converge to values very close to the right-hand-side CBP of Figs. 6–8. Figs. 10–12 clearly show that the results of EMLM can be seen as lower bounds for s1 , s2 and as upper bounds for s3 , s4 , i.e. exactly as it was anticipated. Also, a “clear” difference of about 2% in CBP between the MRM and the CDTM exists for s3 , s4 . This difference increases, as the difference of the required bandwidth per call

Fig. 11. Models comparison based on Table 4 and Fig. 7a (without BR).

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Fig. 12. Models comparison, based on Table 3 and Fig. 8 (without BR).

Fig. 13. Influence of the bandwidth-per-call difference on CBP difference between CDTM(1) and MRM.

of the service-classes increases. Although it can be understood intuitively, Fig. 13 shows an example on how this difference increases, according to the difference of the required bandwidth per call of the service-classes.

6. Conclusion Thanks to the notion of total offered traffic-load, we can evaluate the call-level QoS of both CBR and ABR services in a common way. To this end, we review the retry and threshold models and investigate

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their applicability on ABR services. We show that the retry models can hardly model the ABR traffic at call set-up, while the threshold models perform better. In order for the threshold models to be applicable to a wide range of ABR traffic, we extend them to the CTDM, which eventually includes not only the threshold models, but also the retry models and the initial EMLM. We reveal the superiority of the proposed CDTM against the other models, in respect of its applicability and the resultant CBP reliability, by applying all models to realistic service-classes accommodated in a VP link and comparing the results. The MRM can provide approximate CBP only (because hardly models the ABR traffic at call set-up), the MTM may not be applicable at all (when not all service-classes have common thresholds), while when it is applicable it provides better results than the MRM. The CDTM is always applicable and provides reliable results. The CBP obtained from the threshold models is accurate when the thresholds scheme is considered to be the CAC scheme. Besides, we show that it is possible to incorporate bandwidth/trunk reservation schemes in all models.

Appendix A. Software implementation of the CDTM This appendix contains guidelines for the software implementation of the CDTM. The following pseudo-code implements the CDTM in an easy to understand programming pseudo-language. A.1. Declarations Variable Matrix Matrix

K a[k,l] a[k,0] b[k,l]

Matrix Array Variable

b[k,0] C t[k] n[k] thr[l,k] thr[0,k] cbp[k,l] G[j] GG

Parameter Parameter

G0 G1

Variable Array Array Matrix

Number of service-classes Offered traffic-load of service-class k for the lth threshold Initial offered traffic-load of service-class k Bandwidth per call requirement of service-class k for the lth threshold Initial bandwidth per call requirement of service-class k Bandwidth capacity of transmission link (VP) Bandwidth/trunk reservation number of service-class k Number of thresholds of service-class k The lth threshold of service-class k The first threshold of service-class k (note: thr(n(k), k) = C) Matrix to store the CBP of service-class k for the lth threshold Array of size C to store the values of the function G(j) Variable to store the sum of all G(j). Since this sum is a tremendous high number, we need to reduce it and because of this we introduce the following two parameters A very large number (computer/machine dependent, e.g. G0 = 1E + 30) A very small positive number (e.g. G1 = 1E − 20)

A.2. Instructions Comment: the values of function G(j) are determined.

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GG = 1 For j = 1 to C If (GG > G0) then GG = 1 For i = 1 to j − 1 G(i) = G(i) ∗G1 GG = GG + G(i) Next i End if For k = 1 to K If (j > C − it(k)) go to 1 For l = 1 to n(k) delta = 0 If (j = (thr(l, k) + b(k, l)) and j > (thr(l − 1, k) + b(k, l))) delta = 1. If (j − b(k, 0) < 0) then If (j − b(k, l) < 0) then Go to 1 Else if (j − b(k, l) = 0) then G(j ) = G(j ) + (1/j )∗a(k,l)∗b(k,l)∗delta Else G(j ) = G(j ) + (1/j )∗a(k,l)∗b(k,l)∗delta∗ G(j − b(k, l)) End if Else if (j − b(k, 0) = 0) then If (j − b(k, l) > 0)G(j ) = G(j ) + (1/j )∗a(k,l)∗ delta∗G(j − b(k, l)) If (j − b(k, l) = 0)G(j ) = G(j ) + (1/j )∗a(k,l)∗delta Else G(j ) = G(j ) + (1/j )∗a(k,l)∗b(k,l)∗delta∗ G(j − b(k, l)) End if Next l If (n(k) = 0) then delta = 1 Else if (j ≤ thr(0, k) + b(k, 0)) then delta = 1 Else delta = 0 End if If (j − b(k, 0) = 0)G(j ) = G(j ) + (1/j )∗a(k,0)∗b(k,0)∗delta If (j − b(k, 0) > 0)G(j ) = G(j ) + (1/j )∗a(k,0)∗b(k,0)∗delta∗G(j − b(k, 0)) 1 continue Next k GG = GG + G(j ) Next j

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Comment: the “initial” CBP is determined for every service-class (it is stored in cbp[k,1]). For k = 1 to K

cbp(k, 1) = 0 For i = 1 to (b(k, 0) + t (k)) cbp(k, 1) = cbp(k, 1) + G(C − i + 1) Next i cbp(k, 1) = cbp(k, 1)/GG

Next k Comment: the “rest” CBPs are determined for service-classes who resize (the final CBP is stored in cbp[k, n(k) + 1]). For k = 1 to K

2 Next k

For l = 1 to n(k) cbp(k, l + 1) = 0 If (b(k, l) = 0) go to 2 For i = 1 to (b(k, l) + t (k)) cbp(k, l + 1) = cbp(k, l + 1) + G(C − i + 1) Next i cbp(k, l + 1) = cbp(k, l + 1)/G Next l continue

References [1] H. Akimaru, K. Kawashima, Teletraffic—Theory and Applications, Springer, Berlin, 1993. [2] S. Blaabjerg, G. Fodor, A generalization of the multirate circuit switched loss model to model ABR services in ATM networks, in: Proceedings of the IEEE International Conference on Communications Systems, ICCS’96, Singapore, November 1996, pp. 487–491. [3] G. De Veciana, G. Kesidis, J. Walrand, Resource management in wide-area ATM networks using effective bandwidths, IEEE J. Select. Areas Commun. 13 (6) (1995) 1081–1089. [4] G. Fodor, S. Blaabjerg, A.T. Andersen, A partially blocking-queuing system with CBR/VBR and ABR/UBR arrival streams, in: Proceedings of the IFIP WG 7.3 Fifth International Conference on Telecommunications Systems Modelling and Performance Analysis, Nashville, TN, March 1997, pp. 411–424. [5] G. Fodor, A. Racz, S. Blaabjerg, Simulative analysis of routing and link allocation strategies in ATM networks supporting ABR services, IEICE Trans. Commun. (Special Issue on ATM Traffic Control and Performance Evaluation) E81-B (5) (1998) 985–995. [6] R. Guerin, H. Ahmadi, M. Naghshineh, Equivalent capacity and its application to bandwidth allocation in high-speed networks, IEEE J. Select. Areas Commun. 9 (7) (1991) 968–981. [7] Y.T. Hou, L. Tassiulas, H.J. Chao, Overview of implementing ATM-based enterprise local area network for desktop multimedia computing, IEEE Commun. Mag. 34 (4) (1996) 70–78. [8] R. Jain, S. Kalyanaraman, S. Fahmy, R. Goyal, Source behavior for ATM ABR traffic management: an explanation, IEEE Commun. Mag. 34 (11) (1996) 50–57. [9] J.S. Kaufman, Blocking in a shared resource environment, IEEE Trans. Commun. COM-29 (10) (1981) 1474–1481. [10] J.S. Kaufman, Blocking with retrials in a completely shared resource environment, Perform. Eval. 15 (1992) 99–113.

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[11] J.S. Kaufman, Blocking in a completely shared resource environment with state dependent resource and residency requirements, in: Proceedings of the IEEE Conference, INFOCOM’92, 1992, pp. 2224–2232. [12] M. Logothetis, S. Shioda, G. Kokkinakis, Optimal virtual path bandwidth management assuring network reliability, in: Proceedings of the Conference, ICC’93, Vol. 1, Geneva, May 1993, pp. 30–36. [13] M. Logothetis, S. Shioda, Medium-term centralized virtual path bandwidth control based on traffic measurements, IEEE Trans. Commun. 43 (10) (1995) 2630–2640. [14] M. Logothetis, G. Kokkinakis, Network planning based on virtual bath bandwidth management, Int. J. Commun. Syst. 8 (1995) 143–153. [15] M. Logothetis, S. Shioda, Centralized virtual path bandwidth allocation scheme for ATM networks, IEICE Trans. Commun. E75-B (10) (1992) 1071–1080. [16] M. Logothetis, Optimal resource management in ATM networks, in: D. Kouvatsos (Ed.), Tutorial Paper in Performance Evaluation and Application of ATM Networks, Kluwer Academic Publishers, New York, 2000, pp. 201–226. [17] I. Moscholios, M. Logothetis, Call-level QoS assessment in ATM networks supporting elastic traffic, in: Proceedings of the Conference, ICC 2001, Helsinki, June 2001. [18] J.W. Roberts, Teletraffic models for the telecom 1 integrated services network, in: Proceedings of the Conference, ITC-10, Montreal, 1982. [19] J.W. Roberts, A service system with heterogeneous user requirements, in: G. Pujolle (Ed.), Performance of Data Communications systems and their Applications, North Holland, Amsterdam, 1981, pp. 423–431. [20] K.W. Ross, Multiservice Loss Models for Broadband Telecommunication Networks, Springer, Berlin, 1995. [21] H. Saito, K. Shiomoto, Dynamic call admission control in ATM networks, IEEE J. Select. Areas Commun. 9 (7) (1991) 982–988. [22] M. Schwartz, B. Kraimeche, An analytic control model for an integrated node, in: Proceedings of the IEEE Conference, INFOCOM’83, 1983, pp. 540–545.

Ioannis D. Moscholios was born in Athens, Greece, in 1976. He received a Dipl.-Eng. degree in Electrical and Computer Engineering from the University of Patras, Patras, Greece, in 1999, and an M.Sc. degree in Spacecraft Technology and Satellite Communications from the University College London (UCL), London, UK, in 2000. Currently, he is a Ph.D. student at the Wire Communications Laboratory, Department of Electrical and Computer Engineering, University of Patras, Greece. Since he was an undergraduate student, he was involved in national research projects. His main research interest is on teletraffic engineering. He is a member of the Technical Chamber of Greece (TEE).

Michael D. Logothetis was born in Stenies, Andros, Greece, in 1959. He received his Dipl.-Eng. degree and Ph.D. in Electrical Engineering, both from the University of Patras, Patras, Greece, in 1981 and 1990, respectively. From 1982 to 1990, he was a Teaching and Research Assistant at the Laboratory of Wire Communications, University of Patras, and participated in many national and EC research programs, dealing with telecommunication networks, as well as with natural language processing. From 1991 to 1992 he was Research Associate in NTT’s Telecommunication Networks Laboratories (Tokyo). Afterwards, he was a Lecturer in the Department of Electrical and Computer Engineering of the University of Patras, and since 1996 he is an Assistant Professor in the same department. His research interests include traffic control, network management, simulation and performance optimization of telecommunications networks. He is a member of the IEEE (Commun. Society—CNOM), IEICE and the Technical Chamber of Greece (TEE).

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George K. Kokkinakis was born in Chios, Greece, in 1937. He received a Diploma in Electrical Engineering (Dipl.-Ing.) in 1961, the doctor’s degree in Engineering (Dr.-Ing.) in 1966, and the Diploma in Engineering Economics (Dipl. Wirt.-Ing.) in 1967, all from the Technical University of Munich, Germany. Since 1969 he has been with the Department of Electrical Engineering at the University of Patras, where he organized and he has been directing the Wire Communications Laboratory (WCL). His current activity in research and development coincides with the activity of WCL, and includes the design and optimization of telecommunication networks, and the analysis, synthesis, recognition and linguistic processing of the Greek language. He has published several books and over 120 technical papers, articles and reports on telecommunications, electrotechnology and speech technology. He has organized the International Conference EUROSPEECH’97. Prof. Kokkinakis is a Senior Member of IEEE and a Member of the Technical Chamber of Greece (TEE), the VDE (Verein Deutscher Elektrotechniker), the ESCA (European Speech Communication Association), the EURASIP (European Association for Signal Processing), the SEFI (Societe Europeene pour la Formation des Ingenieurs), the LSA (Linguistic Society of America) and the EEEE (Greek Operations Research Society).