Virtua trunk simuation Samui Aato * Laboratory of Teecommunications Technoogy Hesinki University of Technoogy Sivia Giordano Laboratoire de Reseaux de Communication Ecoe Poytechnique Federae de Lausanne 26 August 1997 Abstract One of the activities of the ACTS project EXPERT is to perform trias demonstrating the use of advanced Resource Management and Routing (RM&R) agorithms in ATM networks. The RM&R mode chosen for those trias is based on the Virtua Trunk (VT) concept. In ATM networks, a VT is a virtua path connection setup by the network for reducing connection awareness at the transit nodes. A virtua trunk is considered as a connection by the network supporting it (the VP network), and as a ogica trunk by the connections supported. In this context, VTs are considered to be VBR connections. In order to adapt the VT eve to the changes in traffic, we use dynamic virtua trunks. In this paper, we compare the VBR-over-VBR approach to a more traditiona VBR-over-CBR approach. The comparison is based on the simuations performed in the EXPERT project by WP3.2. In addition, a dynamic VP bandwidth aocation method is demonstrated by simuations. 1 Introduction One of the main difficuties in designing and operating connection oriented networks, such as ATM networks, is the compexity required for supporting connections [GGL95]. Every connection estabished through the network is associated at every node with abe swapping tabes, capacity reserved in the queues and connection contro bocks (or equivaent denominations) used by the signaing or contro protoco. The connection estabishment and maintenance require considerabe processing [GG93]. Moreover the management of each singe connection requires maintaining a arge amount of structures. The consequence is that the connection handing capabiities of the network nodes are imited. Such imits depend strongy on the node design and configuration (such as the amount of memory instaed, or the number of ports or port cards [GGL95]). Additiona factors, such as the ink capacity are responsibe of those imits [FOR95, ALL95]. In particuar the resource management in ATM networks starts to be difficut when ink resources are aocated to individua connections. In the iterature, severa approaches use the VP concept to bunde individua connections. The approach deveoped in [GLOR97] was chosen as a basis for the simuations performed in the EXPERT project by WP3.2. This approach is based on the Virtua Trunk (VT) concept. In ATM networks, a VT is a Virtua Path Connection (VPC) setup by the network for reducing the connection awareness in transit nodes. A VPC is a connection for the network supporting it (the VP network), and it is considered as a ogica trunk by the Virtua Channe Connections (VCC) supported. Here, for each VPC, we have a tunneing scenario where a number of VBR VCCs are mutipexed onto a VBR VPC at a node that acts as a genera shaper, and the VBR VPCs are mutipexed on the network. This is caed the VBR-over-VBR approach. In the more traditiona VBR-over-CBR approach, VBR VCCs are mutipexed onto a CBR VPC. In this case, it is not any more possibe to mutipex the VPCs on the network. * This work was competed whie the first author was at VTT Information Technoogy. 1
To better adapt to the changes in traffic, the resources shoud not be aocated to the VPs staticay but in a dynamic way. In the VBR-over-VBR approach, as specified in [GLOR97], the resources are aocated on-demand, reaizing the Compete Sharing (CS) poicy. Another possibiity is to reaocate the resources at reguar intervas. With a short time interva, these two methods are quite simiar but, with a ong interva, the atter method becomes rather static corresponding to the Compete Partitioning (CP) poicy. In this paper, we compare the nove VBR-over-VBR approach to the more traditiona VBR-over-CBR approach. The comparison is based on the simuations performed in the EXPERT project by WP3.2. In these simuations, the dynamicity in resource aocation is achieved by a periodic bandwidth aocation scheme originay deveoped in [BAMS94], [BMPS95], [MPS95] and [MPS96]. The simuation resuts presented here show that the VBR VPCs perform better than the CBR VPCs, even if the gain we obtained was not very high. We beieve that this is mainy due to the rather inefficient method used to mutipex the VPCs on the network. 2 Virtua Trunk mode The VT approach uses the concepts of arriva curve and extended service curve, defined in the Network Cacuus theory [LEB96]. An arriva curve is an upper bound to the traffic sent by an input stream. Thus, as defined in [CRU95], given a wide-sense increasing function α, we say that a fow with arriva function R has an arriva curve α, if and ony if, for a t and s t, Rt () Rs () α ( t s). An extended service curve is a ower bound to the service offered at a system S. Thus, as defined in [LEB96], the system S with output function R* offers an extended service curve σ to a fow with arriva function R, if and ony if, for a t there exists some t0 0, t0 t, such that R ( t) R( t0) σ ( t t0). Starting from these two definitions, the Network Cacuus theory provides powerfu toos to manage guaranteed services. B α σ Figure 1 VT Reference Mode: the input fows with aggregate arriva curve α are mutipexed onto a VT with arriva curve σ. The shaper node offers to the input fow service described by the service curve σ. The maximum backog is indicated by B. The VT is defined in terms of two sets of parameters, a connection descriptor and a trunk state [GLOR97]. The trunk state describes the characteristics of the traffic mutipexed onto the VT. It is given in terms of an aggregate arriva curve, α, of the fows incuded. The connection descriptor describes the characteristic of the VT. It is given in terms of a shaping curve, σ, of the shaper. Theorem 6 of [LEB96] says that a shaper node with shaping curve σ (that is the VT arriva curve) offers to the input fow a service curve equa to σ. Thus, by Theorem 1 of [netcac97], the backog R(t) - R*(t) has the foowing upper bound for any t: 2
(1) Rt () R*() t sup s 0{ α() s σ ()}. s The mutipexing node acts as a genera shaper with shaping curve σ, and generates a fow described by the output function R*, as in the basic mode depicted in Figure 1. With a buffer of size given by (1), there wi be no osses in the mutipexing node. 3 Simuation scenario The simuation scenario consists of two overaid networks, the physica network and the ogica (VP) network. The underying physica network is assumed to consist of nodes (ATM-switches), which are competey connected by identica physica inks. The physica inks are characterized by giving the capacity (bandwidth) of the ink and the size of the ink buffer. In the simpest case, there are three nodes connected together as a triange, see figure 2. Figure 2 Three nodes connected together as a triange. A the traffic is modeed to arrive from reguated VBR sources. Each connection is assumed to be symmetric (with identica traffic parameters for forward and backward streams) beonging to one of the traffic casses. Each traffic cass is characterized by its traffic descriptor (incuding PCR, SCR and BT) and the statistics characteristics (the mean hoding time, the arriva rate of connection requests). The atter are used for the generation of traffic. The traffic sources are reguated so that they conform to GCRA(1/PCR,0) and GCRA(1/SCR,BT). The VCC connection requests are assumed to arrive according to a stationary Poisson process. Thus, no transient effects due to variations in the traffic oad are taken into account. The hoding times are samped independenty from an exponentia distribution. In addition, the traffic pattern is thought to be even, i.e. the source and the destination of a VCC connection request are samped from a uniform distribution. Each traffic cass is assumed to be served by its own ogica network consisting of VPC inks. Thus, we have taken the traffic separation approach. Aso the ogica networks are modeed to be competey connected. So, each VPC traverses through exacty one physica ink, and each physica ink conveys as many VPCs as there are different traffic casses. In figure 2, there are two traffic casses, and, thus, two ogica triange networks. The structure of the ogica networks is assumed to be stabe. Thus, no new VPCs are estabished nor any of the existing VPCs are torn down during the simuation. 3
4 Resource Management agorithms used in simuations In the setting presented in section 3, the purpose of the resource management is to aocate bandwidth to the VPCs. The periodic aocation scheme used in the simuations is described beow in section 4.1. Once aocated an amount of bandwidth, it is possibe to cacuate how many VCC connections (beonging to the cass considered) the VPC can support. This is the task of the CAC function. As mentioned in the introduction, two different approaches to the connection admission contro have been impemented: VBR-over-CBR and VBR-over-VBR. These are described more detaied in section 4.2. We note that these approaches assume that at each period the resources (shaping buffer, maximum burst toerance) are competey avaiabe, ike at the initia time. This is, of course, not true in a rea scenario, but we beieve that, in the scenario under consideration, we can make this assumption. In fact we introduce severa factors of overestimation: the input fows are described by means of their arriva curves (VBR traffic descriptors), which are upper bounds to the generated input traffic; the VT is described by means of its service curve (a CBR or a VBR traffic descriptor), which is a ower bound to the service offered; and, finay, we use a worst case approach to optimize the VT parameters. A these steps add some approximation to agorithms used in simuations. 4.1 Dynamic, periodic bandwidth aocation scheme In this centraized RM method, VPC bandwidths are reaocated at periodic intervas. In the seque, the updating interva is denoted by t u. The objective is to aocate for each VPC just as much bandwidth as needed to satisfy the stationary bocking probabiity target, which may be cass-specific. Thus, it may happen that not a the capacity of the physica inks is aocated to VPCs. The origina references are [BDMS94], [BMPS95], [MPS95] and [MPS96]. The method is based on the knowedge of the number of active connections per each VPC. In addition, the average ca arriva intensities and the mean hoding times for a traffic streams are needed. The method incudes the foowing three phases. 1st phase Aocation is first made for each VPC separatey. Consider a VPC. Let n denote the number of active VCC connections conveyed by the VPC. Denote by λ and T the arriva rate and the mean hoding time of such VCC connection requests, respectivey. In the simuations, the two statistica parameters λ and T are assumed to be known. Let ρ = λt. What is first cacuated is the maximum number, N, of connections the VPC shoud support during the next updating interva in order that the bocking probabiity during the interva be ess than ε/2, 1 where ε is the target bocking probabiity for the cass. For this we need the foowing function: N = transienterangrequirement( n, ρ, ε / 2, t / T). In principe, there are precise numerica methods to impement this function [VA97]. However, these methods are far too time-consuming for our purposes. Thus a simpe approximation is needed. In the simuations, the foowing (rather crude) approximation is used: 2 where pt () = exp( t) and N= npt ( / T) + N ( 1 pt ( / T)), u N = stationaryerangrequirement( ρ, ε / 2 ). u u 1 The (vague) heuristic behind this is as foows: the upper imit N for N is chosen so that the proportion of time when there are N active connections is (approximatey) ε/2. In this state, a the incoming connection requests are rejected. On the other hand, when there are ess than N active connections, the proportion of time for which is 1-ε/2, the dimensioning is made so that the proportion of rejected connection requests woud be ε/2. Thus, the overa bocking probabiity becomes (approximatey) ε / 21 + ( 1 ε / 2) ε / 2 ε. 2 According to the simuations made, this dimensioning formua seems to function when t u /T is great enough, say 1. However, with smaer vaues, the aocations seem to be too sma. 4
The function stationaryerangrequirement utiizes the ordinary Erang bocking formua, stationaryerangrequirement( ρ, ε) = min{ N Erang( N, ρ) ε}. 2nd phase In the second phase aocation is made for each physica ink separatey. Consider a physica ink. Let C denote its capacity (bandwidth), and denote by K the number of VPCs conveyed. The VPCs are indexed by k. As the resut of the first phase we have the maximum number N k of connections to be supported by each individua VPC k. This is converted into a bandwidth requirement C k. For this we need the CAC function caed requiredbandwidth (see section 4.2), C = requiredbandwidth ( N ). k k k After the bandwidth requirements are cacuated we have to check whether the ink capacity is sufficient, i.e. Ck C. k If this is true, we can step into the fina phase. Otherwise the aocations must be adjusted not to exceed the capacity avaiabe. In the atter case, we first cacuate the bandwidth requirements c k of the existing connections. For this we need the number n k of active connections and (again) the CAC function requiredbandwidth (see section 4.2), The remaining capacity is denoted by R, c = requiredbandwidth ( n ). k k k R C ck k =. It is shared as fairy as possibe, the fair share for VPC k defined by Thus, we have the foowing adjusted capacities: Note that Ck ck C c i i ~ C c R C k c k k = k +. Ci ci i k ~ C k i. = C. By using the (inverse) CAC function caed aowednrcas (see section 4.2), we may cacuate the adjusted maximum number ~ N k of connections to be supported by VPC k, ~ N = aowednrcas ( C ~ ). k k k After this, the (rea) capacity aocations are as foows: ~ C = requiredbandwidth ( N ~ ), k k k which is ess than or equa to ~ C k. Thus, there may sti remain some capacity eft over, namey 5
~ ~ R C = C k. This remaining capacity may sti be utiized by traffic casses with ower bandwidth demands. 3rd phase Finay, VPC bandwidths are adjusted at the network eve. However, since in our setting each VPC traverses exacty one physica ink, no more adjustments are needed. k 4.2 CAC functions The purpose of the CAC functions is to cacuate the required bandwidth given the number of homogeneous connections and their cass, or to cacuate the aowed number of homogeneous connections given the bandwidth avaiabe and the traffic cass. The former function is caed requiredbandwidth and the atter one aowednrcas. The mode is defined in section 2. Beow we describe the two approaches to the connection admission contro used in the simuations, first the traditiona VBR-over-CBR approach and then the nove VBR-over-VBR approach. VBR-over-CBR approach This is a modified peak rate aocation method that takes into account the shaping buffers of VPCs when avaiabe. The node reference configuration used in the simuation is shown in figure 3. A mutipexer, fed with a number of input connections of VBR type, mutipexes them into one CBR virtua trunk, using a shaping buffer of size B. The shaper guarantees that the buffer output conforms to GCRA(1/R 0,0). Figure 3 CBR Node Reference Configuration. Denote by N the number of homogeneous VBR VCC connections (with peak rate R, sustainabe ce rate m and maximum burst ength t) sharing the VPC. The connection between the maximum burst ength t and the burst toerance τ is as foows (see [GLOR97]): t = τ m /( R m). Denote further by C the bandwidth avaiabe for the VPC, and by B the size of the shaping buffer connected to the VPC. The VT attributes are defined by: Trunk state z = ( N, R, m, τ ). Connection descriptor y = R 0. To get the aggregate arriva curve α corresponding to the trunk state z, we assume (as in [GLOR97]) that a the VBR sources are of the deterministic on-off type with active and ide periods of ength t and τ, respectivey. The worst case is that the active periods of the sources start at the same time. As a consequence, we have Note that α s { NRs Nm τ + s } [ ) [ ) ( τ), ( + τ), ( + τ) +, ( + ) ( + τ) + ( + )( + τ) NR s k s k t k t t α() s =. NR k 1 t, s k t t, k 1 t. () min, ( ). 6
On the other hand, the service curve σ offered by the simpe shaper to the input fow is as foows: σ () s = R 0 s. Now it foows from equation (1) that R0 = max { NR B / t, Nm}. This is the minimum vaue of R 0 that guarantees no osses with a shaping buffer of size B. Finay we concude that the two CAC functions needed in the bandwidth aocation are as foows: { NR B/ t, Nm}, max{ nr B/ t, nm} requiredbandwidth( N, R, m, t, B) = max aowednrcas( C, R, m, t, B) = max{ n C}. Note that by omitting the shaping buffer (B = 0) we resut in the ordinary peak rate aocation method. VBR-over-VBR approach This is an advanced aocation method that takes into account both the shaping buffers of VPCs and the ink buffers of physica inks when avaiabe. In addition, the target ce oss probabiity is needed. The node reference configuration used in the simuation is shown in figure 4. A mutipexer, fed with a number of input connections of the VBR type, mutipexes them into one VBR connection (the VBR trunk), using a buffer of size B. The shaper used in the simuations is not a buffered eaky bucket reguator but just a simpe shaper guaranteeing that the buffer output conforms to GCRA(1/R 0, 0). However, due to the reguated nature of the input fow, it is possibe to find parameters m 0 and τ 0 such that the output conforms aso to GCRA(1/m 0,τ 0 ). Figure 4 VBR Node Reference Configuration. Denote by δ the target ce oss probabiity. In addition, et B denote the size of the ink buffer. In this case the VT attributes are defined by: Trunk state z = ( N, R, m, τ ). Connection descriptor y = ( R, m, τ ). 0 0 0 As above, to get the aggregate arriva curve α corresponding to the trunk state z, we assume that the VCC connections are of the same deterministic on-off type. Thus, [ ) [ ) ( τ), ( + τ), ( + τ) +, ( + ) ( + τ) + ( + )( + τ) NR s k s k t k t t α() s =. NR k 1 t, s k t t, k 1 t. Since we used a simpe shaper pus a buffered eaky bucket reguator in the simuations, the service curve σ is as foows: { τ } σ() s = min R0s, m0( 0 + s). So it foows (again) from equation (1) that R0 = max { NR B / t, Nm}. 7
Since we assumed that the service rate of the shaping buffer is R 0 and a the bursts of the underying VCC connections start at the same asting the maximum time t, the output from the shaper ooks ike another deterministic on-off source with sustainabe rate m 0 and burst ength t 0. The tripe (R 0,m 0,t 0 ) is further mapped to an equivaent capacity needed for the bandwidth aocation by using the function equivaentcapacity originay defined in [GAN91] as foows: ( ) / equivaentcapacity( R, m, t, X, ) R Y X + Y X + 4 XYm 0 R 0 0 0 0 δ = 0, 2Y where Y = n( δ ) t 0 ( R 0 m 0 ). This is the rate necessary for achieving a desired buffer overfow probabiity δ on a given physica ink, given a physica ink buffer of size X and the traffic descriptor (R 0,m 0,t 0 ). From that we derive given by: m0 = Nm, t0 = NRt / R0. Finay we concude that τ 0 = t 0 ( R 0 m 0 ) / m 0. Thus, the two CAC functions are in this case as foows: requiredbandwidth( NRmtBB,,,,,, δ) = equivaentcapacity( R0( N), m0( N), t0( N), B, δ), aowednrcas( CRmtBB,,,,,, δ) = max{ n equivaentcapacity( R( n), m( n), t( n), B, δ) C}. 2 0 0 0 Here R 0 (N) and R 0 (n) correspond to peak rates cacuated from the previous formua by assuming that the number of active connections is N and n, respectivey. The same is true aso for the functions m 0 and t 0. Note that by omitting the ink buffer (B = 0) we obtain the same CAC functions as in the CBR-over- VBR approach described above. 5 Simuation trias The foowing two simuation trias were performed: Tria 1: the nove VBR-over-VBR approach was compared to the traditiona VBR-over-CBR approach. Tria 2: the ength of the updating interva was varied. Two different (abeit rather artificia) traffic casses were considered, one with a high bandwidth demand and the other with a ow bandwidth demand. The constant parameters of the two casses are given in tabe 1. In particuar, we see from the definitions beow that the maximum burst ength (with fu ce rate) is 20 ce eve time units 3 for both casses. During a burst of a connection beonging to cass 1, ces may arrive at maximum rate 10 ces per ce eve time unit, impying that the maximum burst size is 200 ces. For cass 2 the corresponding vaues are 1 ce per ce eve time unit and 20 ces. Note further that the mean hoding time, which is the average ength of a connection, is chosen to be 1 ca eve time unit 4 for both casses. The network considered consists of three nodes connected together with identica physica inks as a triange. In fact, the network configuration is as aready presented in figure 2. The capacity (bandwidth) of physica inks is assumed to be 100 ces per ce eve time unit in every case. 3 The ce eve time unit can be chosen freey, e.g. a miisecond. 4 Aso the ca eve time unit can be chosen freey, e.g. a minute. In particuar, it does not need to be the same as the ce eve time unit. 8
Parameter cass 1 cass 2 peakcerate 10.0 1.0 sustainabecerate 2.0 0.2 bursttoerance 80 80 maxburstlength 20 20 bockingthreshod 0.01 0.01 celossthreshod 0.00001 0.00001 meanhodingtime 1 1 Tabe 1 The two traffic casses used in the simuations. In the simuations we used the dynamic, periodic bandwidth aocation scheme by Mocci et. a. described earier in section 4.1. In addition, a connection requests accepted were routed aong the direct paths, which impies that, in fact, the resuts of the simuations are independent of the size of the network. In each tria, mutipe simuation runs were performed with varying offered traffic oad. The traffic oad of each cass was taken to be equa. The foowing parameters were considered as a resut of each simuation run: the percentage of average free capacity (i.e. the part of the capacity of physica inks not aocated to VPCs) in the physica network, the percentage of rejected cas (from a cas offered) for each traffic cass, In the appendices, where the resuts of the simuation runs are given, these parameters are presented as a function of the offered traffic oad. By the traffic oad we mean the ratio of the traffic offered (from a casses together) to a physica ink and the capacity of a physica ink (expressed in percents). Thus, if the offered traffic oad is said to be 50, it means that, on the average, the traffic offered requires haf of the capacity in each physica ink. In these figures, the percentage of average free capacity is potted in a norma inear scae, whereas the percentage of rejected cas is presented in a og-inear scae. 5.1 Tria 1: VBR-over-VBR vs. VBR-over-CBR In this tria the nove VBR-over-VBR approach was compared to the traditiona VBR-over-CBR approach. In both approaches we further studied the effect of a shaping buffer. Thus we had four aternatives to be compared. The parameters of these aternatives are given in tabe 2. Parameter VBR-over-CBR no shaping shaping VBR-over-VBR no shaping shaping shapingbuffer 0 200 0 200 inkbuffer 0 0 1000 1000 Tabe 2 Parameters for the four aternatives in tria 1. Shaping buffers are assumed to be identica for a VPCs. Correspondingy, ink buffers are assumed to be identica for a physica inks. The buffer sizes are given in number of ces. Note that a shaping buffer of 200 ces can incude 1 burst of cass 1 or 10 bursts of cass 2. Correspondingy, a ink buffer of 1000 ces can incude 5 bursts of cass 1 or 50 bursts of cass 2. In the simuations we used the 9
dynamic, periodic bandwidth aocation scheme by Mocci et. a. described earier in section 4.1 with updating interva 1 ca eve time unit. The resuts of the simuations are presented in Appendix A. As expected, the VBR-over-VBR approach resuts in a better performance. However, the difference between the two approaches does not seem to be very arge. This is party due to the rather inefficient method for cacuating the effective bandwidth. By introducing more advanced methods, better resuts may be achieved by the VBR-over-VBR approach. On the other hand, by introducing shaping buffers it is possibe to increase remarkaby the performance of both approaches. However, this requires that the traffic shaped is not critica for deays. In addition, the simuations show that the dynamic bandwidth aocation method functions as expected. With ight or medium traffic oad, the bocking probabiity is in the target area varying from 0.5 % to 2 %. The deviation from the exact target of 1 % is party due to random variations, which coud be diminshed by having onger simuation runs. Note that the stabiity in the bocking probabiity is achieved by an increasing use of network resources: the percentage of the average free capacity fas from 100 % down to 0 % when the traffic oad is increased. With heavy traffic oad, the bocking probabiity naturay grows because of the ack of network resources. 5.2 Tria 2: Varying updating interva of VPC capacities In this tria the ength of the updating interva of VPC capacities, which reates to the dynamic, periodic bandwidth aocation scheme by Mocci et. a., was varied. The comparison was made between three different vaues of the ength parameter (updatinginterva): 1.0, 0.5 and 0.1 ca eve time units. A connection requests accepted were routed aong the direct paths. In the simuations we used the VBRover-CBR approach as regards the CAC functions. The resuts of the simuations are presented in Appendix B. The resuts show ceary that the approximative method for the bandwidth aocation used in the simuations functions ony if the updating interva is great enough (1 ca eve time unit or greater). With smaer vaues, the aocations are too sma. 6 Summary and discussion We presented two simuation trias of the RM&R agorithms impemented in the EXPERT project. In the first one, VBR-over-VBR vs. VBR-over-CBR, we found that the nove approach using VBR VTs performs better than the traditiona one using CBR VTs. The gain in our exampe is not very high, perhaps because of the rather inefficient method (Equivaent Capacity) that used for cacuating the effective bandwidth. By introducing more advanced methods, better resuts may be achieved by the VBR-over-VBR approach. The second tria was performed to evauate the accuracy of the approximative bandwidth aocation function presented in section 4.1. The resuts showed that the approximative method functions ony if the updating interva is great enough (typicay of the size of one average hoding time). Thus, further deveopment is needed. 7 Acknowedgements This work has been carried out as part of the ATCS project EXPERT, in which VTT Information Technoogy participated as subcontractor of NOKIA Corporation, and LRC-EPFL participated as subcontractor of ASCOM Technoogy. References [ALL95] A. Aes, Interworking with ATM, InterOp proceedings, 1995. 10
[BAMS94] C. Bruni, P. D Andrea, U. Mocci and C. Scogio, Optima capacity management of virtua paths in ATM networks, IEEE Gobecom 94, San Francisco, November 1994, vo. 1, pp. 207-211. [BMPS95] C. Bruni, U. Mocci, P. Pannunzi and C. Scogio, Efficient capacity assignment for ATM virtua paths, RACE Workshop, ATM hot topics on Traffic and Performance: from RACE to ACTS, Mian, June 1995, paper 11. [CRU95] D. Cruz, Quaity of service guarantees in virtua circuit switched networks, IEEE JSAC, pp. 1048-1056 August 1995. [FOR95] R. Forberg, Where is the atency in ATM?, Data Communications Magazine, 1995. [GAN91] R. Guerin, H. Ahmadi and M. Naghshineh, Equivaent capacity and its appication to bandwidth aocation in high-speed networks, IEEE JSAC 9, 7, 968-981, 1991. [GG93] L Gun and R. Guerin, Bandwidth management and congestion contro framework in broadband networks architecture, Computer Network and ISDN Systems, vo. 26 (1), 1993. [GGL95] E. Gauthier, S. Giordano, J.-Y. Le Boudec, Reduce Connection Awareness, High-Speed Networking for Mutimedia Appications, W. Effesberg, O. Spanio, A. Danthine, D. Ferrari (eds.), 1995. [GLOR97] S. Giordano, J.-Y. Le Boudec, P. Oechsin and S. Robert, VBR over VBR: the homogeneous, oss-free case, INFOCOM 97. [LEB96] J.-Y. Le Boudec, Network Cacus made easy, EPFL TR96-218. [MPS95] U. Mocci, P. Perfetti and C. Scogio, VP capacity management in ATM networks for short and ong term traffic variations, COST242 TD(95)59, September 1995. [MPS96] U. Mocci, P. Pannunzi and C. Scogio, Adaptive capacity management of Virtua Path networks, IEEE Gobecom 96, London, November 1996. [VA97] J. Virtamo and S. Aato, Bocking probabiities in a transient system, COST257 TD(97)14, January 1997. 11
Appendix A Resuts of tria 1: VBR-over-VBR vs. VBR-over-CBR Percentage of average free capacity vs. traffic oad (both casses) 100 80 60 40 20 0 20 40 60 80 100 120 100. 50. Percentage of rejected cas vs. traffic oad (cass 1) 10. 5 1 0.5 0.1 20 40 60 80 100 120 100. 50. Percentage of rejected cas vs. traffic oad (cass 2) 10. 5 1 0.5 0.1 20 40 60 80 100 120 12
Appendix B Resuts of tria 2: Varying updating interva Percentage of average free capacity vs. traffic oad (a casses) 100 80 60 40 20 0 20 40 60 80 100 120 100. 50. Percentage of rejected cas vs. traffic oad (cass 1) 10. 5 1 0.5 0.1 20 40 60 80 100 120 100. 50. Percentage of rejected cas vs. traffic oad (cass 2) 10. 5 1 0.5 0.1 20 40 60 80 100 120 13