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Multihour Design of Multi-Hop Virtual Path based Wide-Area ATM Networks (Technical Subject Category: Network Design and Routing)

Thomas Bauschert Lehrstuhl fur Kommunikationsnetze Technische Universitat Munchen 80290 Munchen, Germany Phone: +49 89 289-23500 Fax: +49 89 289-23523 E-mail: [email protected] Abstract The cost ecient design of ATM networks is considered to be a challenging problem due to the heterogenity of services, the statistical muliplexing gain resulting from the resource sharing of variable bitrate (VBR) connections and the possibility of resource separation through virtual paths (VPs). Most proposals in literature are well suited for small problems only. For that before developing a new algorithm we take a look at large scale design techniques known from telephone network dimensioning. It turns out that the well known Uni ed Algorithm (UA) of G. Ash [1] is a good basis for an ATM network design method. So we extend and improve the UA in order to cope with the following multihour multiservice ATM network design problem: Find the minimum cost VP network structure on top of a given physical network and the design hour individual optimal VC routing sequences according to end-to-end call blocking constraints. Beside transmission costs also virtual path/virtual channel switching costs are included in our problem formulation. Moreover a simpli ed procedure for single hour design is presented. The new design procedures have been applied successfully to several network examples including a German and an American wide-area ATM network carrying up to ve trac classes.

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1 Introduction For the single hour design of a VP network on top of a physical network a number of dierent methods have been proposed in the literature, see e.g. [4], [5], [6], [7], [9], [10], [11], [12], [13] and [14]. Most authors, except [6], [8], [9], [11], [12], [13] and [14] assume that for each end-to-end trac relation one or more separate VPs are established, and that the end-to-end trac streams are not allowed to mix. This VP operation mode is denoted Single-Hop-VP (SHVP) concept because every destination can be reached over one hop. In contrary the Multi-Hop-VP (MHVP) principle allows virtual channels (VCs) to be routed over multiple VPs. The pure SHVP concept seems to be advantageous because of its operational simplicity but suers from a very poor bandwidth utilization [15]. On the other hand, a MHVP solution with few VPs implies a larger amount of switched trac resulting in increased call processing eort. Hence a good MHVP network design algorithm has to consider the virtual path/virtual channel switching tradeo. This is included only in the methods proposed by Kim, [6], Cavallero, [14] and Aneroussis, [9]. Future broadband networks are supposed to show larger trac variations than todays single service telephone networks. In order to ensure some robustness the load variations should be considered in the network design. This can be done by using a multihour design procedure. For the multihour design of VP based ATM networks only the method of Medhi [4] is available so far. However this algorithm assumes SHVP operation and determines the optimal VP network structure and VP capacities for each design hour separately. In contrast to that we present a multihour design algorithm for MHVP networks with nonhierarchical VC routing. It is assumed that the VP network structure and VP capacities remain unchanged in each design hour but that the VC routing scheme is adapted to compensate the load variations. We formulate the VP network design problem as a cost minimization problem with end-to-end GoS constraints taking into account the virtual path/virtual channel switching tradeo. When developing a new design algorithm one usually takes a look on some well-proven classical single service design procedures. The multihour engineering of telephone networks with nonhierarchical routing has been adressed by Ash, [1] and Pioro, [3]. Pioros method is based on the solution of a mathematical program and therefore seems to be unsuited for larger networks. However the Uni ed Algorithm (UA) method of Ash was developed to cope with very large networks. Because of that we decided to take the UA as the basis for our multihour MHVP design procedure.

2 Modeling Assumptions At the network level we assume, that all nodes are of the same type having both switching and crossconnecting functionality. Concerning the VP level one VP per end-to-end node pair (demand pair) is

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allowed. The VPs may carry an arbitrary trac mix. Only point to point VC connections between demand pairs are supported. All connections are bidirectional with the same bandwidth requirements in both directions.

2.1 Virtual Channel Routing For design purposes the assumption of a simple alternate call routing scheme is sucient. This is due to the fact, that for design load and given end-to-end blocking requirements, few dierences in terms of network cost are observed between alternate routing and more sophisticated routing strategies. According to the MHVP principle the alternate paths may consist of multiple VPs. Moreover the set of alternate paths can be dierent for calls of each trac class and design hour.

2.2 Trac Model At the call level we use the equivalent bandwidth connection acceptance algorithm of Tidblom and Lindberger [17] which is also proposed by RACE. The bandwidth requirement of each trac class s is characterized by two parameters: peak cell rate (PCR) and sustainable cell rate (SCR). From that an equivalent bandwidth rs is calculated which depends only on the VP capacity Np and not on the trac mix itself. This property allows [17] to be conveniently included into the optimization model. The trunk reservation policy of Roberts [18] is applied on each VP in order to achieve equal blocking for all services. This is because the ful llment of prede ned end-to-end blocking criteria sometimes could not be realized in a cost-eective manner if complete sharing is used at the VP level. Large dierences regarding the number of over ow paths for each service class would be the consequence. In addition the following common assumptions are made:

 the stream of calls oered to each demand pair is Poissonian with xed mean interarrival time. These streams are independent of each other.

 call holding times are independent and exponentially distributed randomvariables of equal mean.  blocked calls are cleared and do not return.  the reduced load approximation is applied [16] For optimization purposes over ow trac streams are treated as Poissonian too (one moment model). However in the VP dimensioning stage of our procedure the peakedness of such streams is included (two moment model) in order to enhance the accuracy.

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2.3 Multiservice Call Blocking Approximation For the optimization of ATM networks a fast calculation of the multiservice call blocking probabilities and their derivatives is essential. We modi ed the approximation of Labourdette [19] in order to get a continuous expression for the blocking probability and extended it to include the fair trunk reservation mechanism: 0 Vp 1 N ? max [ r ( N )] + p s s p M M pA Bp = E~ (Np ; Ap ) = E @ p ; (1) 

p

p

Here Mp , Vp denote mean and variance of the VP oered trac volume, p  is the bitrate of the trac in Labourdette's equivalent single server system and E () is the continuous extension of Erlang's loss function. Figure 9 shows the dependency of the fair VP blocking probability on the VP bandwidth calculated from (1) for xed oered multiservice trac. The trac is composed of ve service classes whose characteristics are shown in table 1. It can be seen that the the correspondence with the simulation results is very good.

3 Problem Formulation 3.1 Cost Model The total cost of a VP is assumed to consist of two components:

 transmission cost (hour-independent): C1p = CTp Np

 switching cost:

C2hp =

X s

(2)

COp ws ahps

(3)

where CTp means the cost per bandwidth unit of VP p and Np is its bandwidth. C2hp represents the switching cost caused by the trac on VP p in hour h. It is assumed that there exists a linear relationship between cost C2hp and weighted carried trac ws ahps . The factor COp may be expressed as follows:

COp = 21 (CSWin + CSWout ) +

X

nodes

i 2 VP p

CCCi

(4)

where CSWin=out represents the switching cost occuring at the start/end node of the VP and CCCi the cost for performing the crossconnecting function in node i. These costs could be real costs - e.g. switched trac is more costly in terms of node processing load than crossconnected trac.

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3.2 Mathematical Formulation Our problem can be formulated as follows: Given the structure of the physical network and the cost parameters nd:

 the node pairs that should be connected with VPs  the VP routes in the physical network  the bandwidth of each VP  the VC routing sequence for each demand pair, trac class and design hour so that:

W=

X p

[max (C + KC2hp )] = h 1p

X

X

p

s

[max (CTp Np + K h

is minimized subject to the following constraints:

COp ws ahps (Bh ))]

(5)

h (Bh )  GoS h 8 k; s; h EEBks ks

(6)

ah (Bh) Np ; 1p? B h p

(7)

Bph = E~

!

8 p; h

where factor K is used to control the impact of VP/VC switching costs on optimization output. h is the class s blocking Equation (6) represents the end-to-end blocking probability constraints: EEBks h its design value. The set of equations (7) probabilty of end-to-end node pair k in hour h and GoSks are called the Erlang xed-point equations. They describe the dependence of VP blockings and VP bandwidths given the VC routing plan. Note that the VP carried tracs in hour h depend only on the end-to-end oered loads and the VP blockings in the same hour via the VC routing plan.

4 Description of the Multi-Hour Optimization Procedure Because of its complexity it is impossible to solve the problem by searching the optimal VP routes, the VP bandwiths and the VC routing plan simultaneously. A natural solution is to decompose it into the three subproblems and solve one subproblem at a time while keeping the others constant. This process is iterated until convergence is reached. The idea to iterate between routing optimization and optimal trunk dimensioning is also used in the UA method [1]. In the following we describe the main modules of the multi-hour optimization procedure (see Figures 1, 1a and 1b).

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Generate Initial VP Routes For each end-to-end node pair exactly one VP is provided which carries only fresh oered trac of this node pair. First suciently high VP bandwidths have to be determined so that the GoS objective is met in all hours and for every service class:

n h

?

h = E~ Np ; Ah Np = max arg GoSks k s;h


io

with p = k

(8)

where Ahk;s denotes the oered trac on end-to-end node pair k. The period in which the VP bandwidth is the largest for VP p is denoted as its peak hour h^ p . Knowing the bandwidths Np and carried trac values ahps = Ahk;s (1 ? Bph ) the initial VP routes are determined so that for each VP the maximum cost P maxh (CTp Np + K s COp ws ahps ) is minimized.

Initialization An initial set of peak hour VP blocking probabilities B^p = Bphp is computed in the Initialize VP Blockings module using a modi ed version of Truitts ECCS method [20]. For each demand pair and trac class only the least-cost over ow path in the logical VP network is considered assuming a marginal occupancy (1:25 PCR)=Erl. With the new peak hour blockings new initial VP bandwidths Np , side hour blocking probabilities and initial VP cost metrics MCps are determined as follows: ^

h

i

?
Np = arg B^p = E~ Np ; Ahk ?

where:




with p = k

Bph = E~ Np ; Ahk 8 h 6= h^ p ; p = k MCps = CTp @Nh^pp + KCOp ws @aps @Np = ? ^p @ahps

@ E~ (Np ;Ahp^ p ) ^p @Ahps
1 @ E~ (Np ;Ahp^ p ) (1 ? B^p ) @Np

(9) (10) (11)

(12)

VC Routing Optimization The goal of the routing optimization is to generate the alternate routing plan for each demand pair, trac class and hour under the assumption that the peak hour VP blocking probabilities are kept xed. At ^ constant blockings, the relationship between capacity Np and carried trac ahpsp is approximately linear and the cost metrics MCps are constant. First in the Generate Initial Flowpaths Module a set of cheapest alternate paths ( owpaths) are searched for each demand pair, trac class and hour. The cost metric of a owpath is simply the sum of the cost metrics of its VPs. If possible, more owpaths than neccessary to satisfy the end-to-end GoS constraints should be included in the owpath set. Then a linear multicommodity ow model (LP) is

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hf ). Finally the routing plan used to determine the optimal amount of trac carried on each owpath (rks h is generated by sorting the owpaths according to owpath oered trac values Ahps = 1?apsBph . The linear optimization model is as follows: Find: X min Cp (13)

Subject to:

2 0 X 4X @ s

X

k

owpaths f (k;s;h)

p

X

owpaths f (k;s;h)

2 VP p

13

hf A5 MC  C rks ps p

?

8 p; h


8 k; s; h

hf = Gh = Ah 1 ? max(ERh (Bh ); GoS h ) rks ks ks k;s k;s

(14) (15)

hf  UPBDhf 8 k; s; h; f (k; s; h) 0  rks ks

(16)

Cp  0 8 p

(17)

hf and the values C which represent the maxiOptimization variables are the owpath carried tracs rks p mum cost of VP p over all hours due to inequality (14). Constraint (15) relates the s-type total carried load for demand pair k in hour h { Ghk;s (which can be divided among the owpaths) to the total oered h unless the smallest achievable route set blocking probability load Ahk;s . Ghk;s is determined by GoSks h (Bh ) is greater than the GoS objective. ERks hf have to be introduced (constraint (16)) to Upper bounds on the owpath carried tracs UPBDks force ow feasibility. This is because an unconstrained set of trac ows may be not realizable for any arrangement of alternate routing between the available owpaths. Initial upper bounds are choosen so as to constrain the LP as little as possible. As at this stage the optimal amount of ow on any owpath is unknown, the initial upper bounds are set according to the following formula: hf = Ah (1 ? Dh (Bh )) UPBDks k;s f

(18)

where Dfh(k;s;h) (Bh ) is the blocking of owpath f related to demand pair k, trac class s and hour h. Because the VP carried tracs ahps are modi ed as a result of the LP optimization also a change of the peak hour h^ p may occur. Under the assumption (see above) that the peak hour VP blocking value must not change, a new peak hour and new side hour VP blockings are determined as follows: Fictitious VP capacities for all hours are calculated based on the peak hour blocking probability B^p :

Nph = arg

(

a B^p = E~ Np ; 1 ?pB h p h

!)

(19)

and the period h for which the VP cost expression:

CTp Nph + K

X s

COp ws ahps

(20)

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takes its maximum value becomes the new peak hour h^ p . With Np = Nphp new side hour VP blockings are calculated: h ! a h ~ Bp = E Np ; 1 ?pB h 8 h 6= h^ p (21) ^

p

After the VP blockings have been updated the upper bounds also have to be recalculated based on the current LP ows in order to obtain a more feasible solution. For this like in the original UA the skip-one- owpath method is used. In the current implementation the LP loop is carried out only a few times (typically 5 iterations). By our experience the resulting VC routing plan will not change much if more iterations are performed. Moreover this measure helps to save computation time as the LP solution is the most time consuming part of the design algorithm.

Optimal VP Dimensioning Under the assumption that the VC routing plan will not be changed in the Optimal VP-Dimensioning module new VP bandwidths are determined so as to minimize the objective function (5) subject to the constraints (6) and (7). A straightforward solution of this nonlinear programming problem is impossible for networks of realistic size so that again heuristics have to be used. The key submodules of the Optimal VP Dimensioning block are explained below.

 VP Dimensioning and Multihour Adjustment The rst step of the heuristic solution is to nd a consistent vector of VP bandwidths Np and VP blocking probabilities Bph so that the constraints (6) and (7) are satis ed and the peak hour blocking probabilities remain unchanged. This is done by an iteration scheme (see dashed box in Figure 1b): The VP network is loaded using the present estimates of the VP blocking probabilities Bph , the VC routing scheme and the external tracs Ahks . Then a new peak hour h^ p , new VP capacities and side hour VP blockings are calculated from (19), (20) and (21) like above (Update VP Blocking). Due to the new VP blockings the load proportions on the VPs may be dierent compared to the last iteration which in turn results in new VP blockings and so on. The iteration is continued until the sum of the relative blocking changes is less than a prescribed convergence threshold. Because of the VP blocking update procedure (or if not enough owpaths could be found in the Flowpath Generation Modules) it may happen that some EEB-constraints (6) are violated. This is veri ed in the Check EEB-constraints submodule. The demand pair, trac class and hour which suers the highest blocking violation is selected and the Correct VP Blockings module is activated. It decreases the blocking of the rst owpath (related to the selected demand pair, hour and trac class) until the desired GoS is reached. If the owpath consists of multiple VPs all VP blockings are reduced equally. Thus the additional trac which must be carried to reduce the blocking will be

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carried on the owpath with minimum incremental cost. Care has to be taken if the corrected VP blockings are not related to the peak hour of the respective VP. The subsequent execution of the Load VP network and Update VP Blockings cycle would reset the corrected VP blockings leading to an in nite loop. Therefore, in this case also the peak hour VP blockings have to be corrected based on the corrected side hour VP blocking probabilities (the peak hours itself do not change): For VP p let hp jcorr the hour for which the correction was performed and Bp jcorr the corrected blocking value. Calculate new VP bandwidths by:



n

Np = arg Bp jcorr = E~ Np ; Aphp jcorr

o

(22)

h

using the old estimates of VP oered tracs Apsp jcorr in hour hp jcorr . Then, with the new Np value, determine new VP blockings for each hour (including the peak hour):

?

Bph = E~ Np ; Ahp


8h

(23)

The Blocking Correction cycle is repeated until all GoS constraints are satis ed. As the determination of the VP dimensions Np is performed with a one-moment trac model so far, the resulting VP network may be underdimensioned. Therefore the VP bandwidths are recalculated by inversion of a Hayward-like extension of the modi ed Labourdette call blocking formula taking into account the mean peakedness Zp of the normalized VP oered trac Mpp . Here the following additional assumption is made: the call congestion of a VP is the same for all oered trac parcels and is set to the current VP blocking probability. It should be pointed out, that there are other methods [21], [22] to cope with multiservice peaked trac streams which may be more accurate. But in our opinion they are computationally too complex to be included in large scale dimensioning methods. If the resulting bandwidth of a VP is less than some minimum level, the VP is eliminated (i.e. its capacity is set to 0 and its blocking is set to 1). Therefore, the ful llment of the EEB constraints has to be checked again. In case of EEB violations the Correct VP Blockings module is reentered and the procedure continues.

 Cost Evaluation, Convergence Check and Update VP Cost Metrics After the VP Dimensioning and Multihour Adjustment Module has converged, the VP network cost W and the number of VPs (with bandwidth 6= 0) are evaluated. If the number of VPs and/or the network cost are still decreasing, the inner iteration is continued and the VP cost metrics are recalculated according to formula (11) in the Update VP Cost Metrics module.

 Update Optimal VP Blockings

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The next step of the Optimal VP Dimensioning module is to calculate new VP blocking probabilities and VP bandwidths respectively, so as to minimize the objective function (5) without considering the constraints (6), (7). This simpli cation is permissible because the constraints are enforced in the next inner iteration by the VP Dimensioning and Multihour Adjustment Module. In the original UA a modi cation of Truitts ECCS method is used to determine new optimal VP blockings: each link is considered in its peak hour assuming xed oered trac and an equivalent alternate routing cost is determined taking into account only the rst owpath. Our approach is somewhat dierent: First the derivatives of the objective function (5) referring to the peak hour VP blockings are determined for each VP j : ^p X @ahjs^ j X X d W = CT
1 @ahps + MC + K CO w j ~ ps ^ j s ^ ^ d B^j @ Bj @ Bj @ E(Nj ;Ahj j ) s pnj s @Nj

(24)

^p @ahps = 0 if h^ p 6= h^ j @ B^j

(25)

where:

All expressions in (24) can be obtained analytically which speeds up the calculation considerably. h^ p ps Moreover in (24) the VC routing plan of the whole network is included via the derivatives @a ^ @ Bj . The dimension of the gradient vector is independent of the number of trac classes (because of fair trunk reservation on the VPs) and load periods. After the objective gradient has been determined the peak hour VP blockings are modi ed dependent on its direction i.e.:

8 > < B^j ^
B j = > : ? B^j

if if

dW d B^j dW d B^j

0

(26)

It is convenient to set to a small value (e.g. 5 %) in order to limit possible violations of the constraints (6), (7). This also stabilizes the inner iteration loop. The value is adapted in subsequent inner iterations according to following rule: let constant if the network cost is decreasing, otherwise set = 2 . Finally new side hour VP blockings are generated based on new VP bandwidth values which are calculated by using the new peak hour blockings and the old estimates of VP oered trac: o  n (27) Np = arg B^p = E~ Np ; Ahp^ p

?

Bph = E~ Np ; Ahp


8 h 6= h^ p

(28)

 Update Flowpaths The Update Flowpaths Module plays an important role in the multihour design algorithm. New

owpaths for each demand pair, trac class and hour are searched taking into account spare VP capacities not used by the actual owpaths in side hours. Such capacities can then be utilized by

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shifts of ow from actual owpaths to low loaded spare owpaths during the LP ow optimization. In order to recognize hour-dependent spare capacities during the path search, modi ed VP cost h metrics MCps jmod (which are only valid in the Update Flowpaths Module) are de ned as follows:

" !# (B^p ? Bph ) h MCps jmod = MCps
max 0:01; 1 ? ^ Bp

(29)

using the fact that the amount of spare capacity in a side hour is proportional to (B^p ? Bph ). Note that it is convenient to supply more owpaths than neccessary to meet the GoS constraints so that the LP has some freedom in choosing the optimal trac ows.

Optimize VP Routes If the inner iteration has converged, the Update VP Routes module can be activated, which for each VP searches an improved least cost path through the physical network given the current VP dimension and trac ow. At this stage additional constraints on the VP network structure can be considered. If the VP network cost is still decreasing, the outer iteration is continued by calculating new VP cost metrics according to (11) in the Update VP Cost Metrics module and reentering the VC Routing Optimization module. Otherwise a stable VP network design is achieved and the design algorithm terminates with the determination of the ATM link bandwidths in the ATM Link Dimensioning Module. The link capacities are simply obtained through summing up the capacities of all VPs running along each link.

5 Single-Hour Optimization Procedure For single-hour network design the optimization procedure can be simpli ed considerably (see [23]). There is only one set of blocking variables Bp which coincides with the VP bandwidth vector. Therefore no multihour adjustment is required in all modules and in particular the dashed box within the VP Dimensioning and Multihour Adjustment module can be omitted. The most important modi cation applies to the VC routing optimization: just as many initial owpaths are searched as needed to best satisfy the GoS constraints. The optimal VC routing plan then can be generated without the need of solving an LP optimization problem simply by sorting the owpaths according to increasing metrics. In subsequent inner iterations (Update Flowpaths module) it is convenient to keep a certain number of

owpaths xed. This reduces the computation time and stabilizes the inner iteration loop. Because of the absence of the LP loop the single hour design algorithm is computationally much faster than the multihour procedure and requires much less storage memory.

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6 Numerical Studies and Discussion Multi-Hour Results The multihour optimization procedure has been applied to a 23 node network example which could form the ATM backbone network for the wide-area trac in Germany. Figure 2 shows the physical topology with 23 nodes and 41 ATM links. Assuming bidirectional VPs the maximum number of VPs in the network is equal to 253. The network oered trac is composed of ve trac classes - two variable bitrate (VBR) classes and three constant bitrate (CBR) classes. Their characteristics are shown in Table 1. Two design hours are regarded: in the second hour, one half of the demand pairs are assumed to generate up to 20% additional trac compared to the rst hour and the other demand pairs shall generate up to 20% less trac. The required end-to-end blocking probability is assumed to be 1% for all services, demand pairs and design hours (total fairness). The cost factors CTp are set to the number of physical links involved in VP p. Ten alternate paths per demand pair and service class are allowed at maximum. Figures 3 and 4 illustrate the improvement of total network cost and the decrease of the number of VPs as a function of the number of (inner) iterations of the multihour design algorithm. In this example the network is designed without regarding VP/VC switching costs (K = 0) so that the total network cost equals the transmission cost. The decrease of the number of VPs is due to the fact, that VPs which are recognized to be uneconomical by means of their cost metric, are assigned no trac ow in further iterations and will be eliminated. After the 20'th inner iteration almost no improvement in terms of network cost and VP number is possible. Compared to the initial value 3.5% cost savings have been obtained. In other network examples typical cost improvements in the range of 2% to 15% have been found. This con rms our opinion, that the extended ECCS method yields a very good starting point. The optimization procedure was enforced to carry out 30 (inner) iterations. It can be seen that the objective value slightly increases after the 25'th iteration. This is due to the fact that we have not to do with a rigorous optimization algorithm but rather a set of heuristics. In reality the procedure would recognize the increasing objective value, terminate the inner iteration loop and continue by joining the outer loop in order to nd better VP routes. In our case the execution of an additional outer iteration has shown to give no further cost improvement. Finally gures 5a and 5b show the end-to-end blocking (EEB) probability distribution of the designed network for both design hours obtained from call by call simulations. Here the EEB of a demand pair is regarded as the mean value of the individual EEBs suered from all trac classes. It can be seen that the quality of the design result is fairly good. The most time and memory consuming parts of the design algorithm are the execution of the large

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linear program in the VC Routing Optimization module and the gradient computation in the Update Optimal VP blockings module. In our implementation the MINOS package [24] is used to solve the LP. With that networks up to 40 nodes can be designed for several trac classes and design hours on a standard workstation requiring some hours computation time. The use of an interior-point algorithm may further speed up the LP solution. For real large scale applications (> 100 nodes) the use of a LP solution heuristic (cf. [1]) at the cost of optimality is essential. Regarding the optimal VP blocking update, the question arises, what in uence it has on optimization performance and what happens if it is omitted i.e if the ECCS blocking values generated by the initialization module are used throughout the optimization. By our experience the design output is more sensitive regarding the blocking update in case of smaller networks and low oered trac. The reason is, that in such situations the variation of one VP blocking value may have consequences on larger portions of trac streams (and network elements) and that for low oered trac ecient trac concentration is important in order to obtain a cost-economical design result. But in the case of larger networks with suciently large oered trac in general good suboptimal results can be achieved despite of omitting the recalculation of the VP blockings. Obviously this was also observed by other authors [5], [2] which proposed very simple algorithms based on only one LP execution for problems of similar nature. In our example a total transmission capacity value of 702.1 Gbit/s compared to 687.5 Gbit/s in the case with activated Update Optimal VP Blocking module has been found.

Single-Hour Results Finally, to complete the discussion, some additional results for single hour network design are presented. Now an wide-area network example with 84 nodes and 115 ATM links which was taken from [2] is considered (see Figure 8). There can be 3486 VPs at maximum. The trac classes and the GoS objective remain unchanged compared to above. In order to evaluate the impact of service integration and VP/VC switching cost parameters on the design output we studied several cases (Table 2). In the following we shortly describe the main results. First it can be observed, that the consideration of VP/VC switching costs (Examples 2, 4 and 6) only slightly aects the transmission cost but leads to signi cant dierences regarding the the number of VPs. Obviously many economical VP network structures exist at almost the same transmission cost. With decreasing CCC=CSW ratio (i.e. crossconnecting becomes cheaper) more VPs are generated and the portion of switched trac decreases. In the limit CCC=CSW ! 0 the design result would be an optimal Single-Hop-VP (SHVP) network. With VBR trac and statistical multiplexing (Examples 3 to 6) the optimal number of VPs turns out to be smaller compared to the CBR examples. Also the transmission cost is more sensitive to the CCC=CSW ratio. Compared with optimal SHVP networks the highest transmission cost savings are

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found in the pure VBR case. It may surprise that the integration of CBR and VBR tracs rather than to carry them over separately optimized VP networks yields no much dierences in terms of network cost (Examples 5 and 6). This is because the total oered trac for the US network is so high that bundling gain savings from integration are hardly possible. Finally Figures 6 and 7 show optimization results for Example 6. It can be seen that the behaviour of the optimization algorithm is similar to the multihour case. The optimal SHVP transmission cost would be 29.698 Tbit/s for Example 6. It should be noted that the above mentioned tendency to increase the objective value during the course of the inner iteration loop was not observed in the single hour case.

7 Conclusion In this contribution the single and multihour design of MHVP based ATM networks is considered. The presented design methods were developed based on the well known Uni ed Algorithm. Key features of the algorithm are the decomposition in smaller subproblems which are solved repeatedly, the integration of the VP routing optimization and the consideration of statistically multiplexed multiservice trac streams. The enhanced problems of modular design and of nding optimal VP routes separately for every design hour (in addition to the optimal adaption of the VC routing scheme) will be tackled in near future.

References [1] G. R. Ash, R. H. Cardwell, R. P. Murray, "Design and Optimization of Networks With Dynamic Routing," Bell System Technical Journal, vol. 60, no. 8, pp. 1787-1820, October 1981. [2] G. R. Ash, F. Chang, J-F. Labourdette, "Analysis and Design of Fully Shared Networks," Proc. of the ITC-14, 1994. [3] M. Pioro, B. Wallstrom, "Multihour Optimization of Non-Hierarchical Circuit Switched Communication Networks with Sequential Routing," Proc. of the ITC-11, 1985. [4] D. Medhi, "Multi-Hour, "Multi-Trac Class Network Design for Virtual Path-Based Dynamically Recon gurable Wide-Area ATM Networks," IEEE/ACM Transactions on Networking, vol. 3, no. 6, pp. 809-818, December 1995. [5] D. Medhi, S. Guptan, "Network Dimensioning and Performance of Multiservice, Multi-Rate Loss Networks with Dynamic Routing," Technical Report, University of Missouri-Kansas City, December 1995. [6] S-B. Kim, "An Optimal Establishment of Virtual Path Connections for ATM Networks," Proc. of the IEEE Infocom'95, pp. 72-244, 1995. [7] B. Ryu, H. Ohsaki, M. Murata, H. Miyahara, "Design Method for Virtual Path Based ATM Networks with Multiple Trac Classes," Proc. of the ICCC'95, pp. 336-341, 1995. [8] D. Mitra, J. A. Morrison, K. G. Ramakrishnan, "ATM Network Design and Optimization: A Multirate Loss Network Framework," Proc. of the IEEE Infocom'96, pp. 994-1003, 1996.

Bauschert: Multihour Design of Multi-Hop Virtual Path based Wide-Area ATM Networks

15

[9] N. G. Aneroussis, A. A. Lazar, "Virtual Path Control for ATM Networks with Call Level Quality of Service Guarantees," Proc. of the IEEE Infocom'96, pp. 312-319, 1996. [10] J. Chlamtac, A. Farago, T. Zhang, "How to Establish and Utilize Virtual Paths in ATM Networks," Proc. of the ICC'93, pp. 1368-1372, May 1993. [11] J. Chlamtac, A. Farago, T. Zhang, "Optimizing the System of Virtual Paths," IEEE/ACM Transactions on Networking, vol. 2, no. 6, pp. 581-587, December 1994. [12] A. Farago, S. Blaabjerg, L. Ast, G. Gordos, T. Henk, "A New Degree of Freedom in ATM Network Dimensioning: Optimizing the Logical Con guration," IEEE Journal on Selected Areas in Communications, vol. 13, no. 7, pp. 1199-1205, September 1995. [13] K-T. Cheng, F. Y-S. Lin, "On the Joint Virtual Path Assignment and Virtual Circuit Routing Problem in ATM Networks," Proc. of the IEEE Globecom'94, pp. 777-782, December 1994. [14] E. Cavallero, U. Mocci, C. Scoglio, A. Tonietti, "Optimization of Virtual Path/Virtual Circuit Management in ATM Networks," Proc. of Networks'92, pp. 153-158, May 1992. [15] Z. Dziong, K-Q. Liao, L. Mason, N. Tetreault, "Bandwidth Management in ATM Networks," Proc. of the ITC-13, North-Holland, pp. 821-827, 1991. [16] A. Girard, Routing and Dimensioning in Circuit-Switched Networks, Addison-Wesley Publishing Company, Reading, MA, 1990. [17] K. Lindberger, "Dimensioning and Design Methods for Integrated ATM Networks," Proc. of the ITC-14, North-Holland, pp. 897-906, 1994. [18] J. W. Roberts, "Teletrac Models for the Telecom 1 Integrated Services Network," Proc. of the ITC-10, Montreal, Canada, 1983. [19] J.-F. P. Labourdette, G. W. Hart, "Blocking Probabilities in Multitrac Loss Systems: Intensitivity, Asymptotic Behavior, and Approximations," IEEE Transactions on Communications, vol. 40, no. 8, pp. 1355-1366, August 1992. [20] C. J. Truitt, "Trac Engineering Techniques for Determining Trunk Requirements in Alternate Routed Networks," Bell System Technical Journal, vol. 33, no. 2, pp. 277-302, March 1954. [21] F. Machihara, "An Extended Equivalent Random Method For Engineering Networks with Heterogeneous Inputs," Proc. of OPERATIONAL RESEARCH'87, North-Holland, pp. 759-771, 1988. [22] T. Oda, Y. Watanabe, "A Method For Analyzing Circuit-Switched Networks With Multiple Bit Rate Classes," Proc. of the ITC-12, North-Holland, pp. 850-856, 1988. [23] T. Bauschert, "Design of VP-based ATM Networks with Nonhierarchical Routing," Proc. of the 7'th International Network Planning Symposium, NETWORKS'96, 1996. [24] B.A. Murtagh, M.A. Saunders, MINOS 5.4 User's Guide, Technical Report SOL 83-20R, Stanford University, March 1993.

Generate Initial VPĆRoutes

VCĆRoutingOptimization

Initialize VPĆ Blockings with mod. ECCS

Initialize VPĆCostMetrics

Generate Initial Flowpaths

Update VPć CostćMetrics

Initialization ATMĆLinkDimensioning

Update Flowpaths

Conv. ?

VPĆDimenć sioning & Multihour Adjustment

Update Optimal VPćBlockings

CostEvaluation Optimize VP-Routes

Generate RoutingĆ Plan

LP

Update VPć CostćMetrics

Update VPĆRoutes

Conv. ?

CostEvaluation

Optimal VPDimensioning

Figure 1: Multihour Optimization Procedure

Initialize Upper Bounds

Linear Program

Conv. ? Load VPNetwork

Update Upper Bounds

LP

Update VPĆ Blockings

Update VPĆ Blockings

Correct VPĆ Blockings

Conv. ?

Figure 1a: LP Loop Detail Check EEBĆ Constraints

Blocking Probability

1

Conv. ?

VPĆDimensioning (2-Moment-Model)

0.1

VPĆElimination Check EEBĆ Constraints

0.01

Conv. ?

0.001 0

50

100

150

200

250

300

VP Capacity

Figure 9: Calculated and Simulated Blocking Probability (fair trunk reservation)

Figure 1b: VP Dimensioning and Multihour Adjustment Module

710000

242 240

Transmission Cost (Mbit/s)

Number of VPs

238 236 234 232 230 228

705000

700000

695000

690000

226 224

685000 0

5

10

15

20

25

0

30

5

10

15

20

Figure 3: Number of VPs vs. Iteration Number

30

Figure 4: Transmission Cost vs Iteration Number

70

70

60

60

mean EEB = 3.44 %

Number of Demand Pairs

mean EEB = 3.45 % Number of Demand Pairs

25

Iteration Number

Iteration Number

50

Design Hour 1 40 30 20 10

50

Design Hour 2

40 30 20 10

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

End-to-End Blocking Probability (EEB) (%)

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

End-to-End Blocking Probability (EEB) (%)

Figure 5a: EEB–Distribution (Hour 1)

Figure 5b: EEB–Distribution (Hour 2)

KIE ROS

HBG BRE

HAN BIE

ESS

BLN MAG

DOR

LPZ ERF

DUS

DRE

KLN

FFM

WBG

MAN

Traffic Class PCR (∼ peak bitrate) SCR (∼ mean bitrate) ȍ offered Traffic (GER) ȍ offered Traffic (USA) 1. (CBR)

1 Mbit/s

1 Mbit/s

42750 Erl

573330 Erl

2. (CBR)

3 Mbit/s

3 Mbit/s

19950 Erl

268162 Erl

3. (CBR)

10 Mbit/s

10 Mbit/s

11400 Erl

153724 Erl

4. (VBR)

5 Mbit/s

1 Mbit/s

19950 Erl

268162 Erl

5. (VBR)

10 Mbit/s

1.5 Mbit/s

11400 Erl

153724 Erl

Table 1: Traffic Classes

NBG

KLR STG AUG MCH

Figure 2: Physical Network (Germany)

1.4

2.86e+07

3200

2.84e+07

Transmission Cost (Mbit/s)

3400

Number of VPs

3000 2800 2600 2400 2200 2000

2.82e+07 2.8e+07 2.78e+07 2.76e+07 2.74e+07 2.72e+07

1800

2.7e+07 0

10

20

30

40

50

60

0

10

Iteration Number

20

30

40

50

Iteration Number

Figure 6: Number of VPs vs. Iteration Number

Figure 7: Transmission Cost vs Iteration Number

Figure 8: Physical Network (USA)

Ex.

Characteristics

Transm. Cost ∼ Tbit/s

Cost Savings Number No. of Altern. Paths acc. to SHVP of VPs min, max, mean

1.

only CBR Traffic 22.310 K=0

11.12 %

1067

1, 10, 4.32

2.

only CBR Traffic 22.757 K=1, CSW=1.0 CCC/CSW=0.25

9.34 %

3482

2, 10, 4.81

3.

only VBR Traffic 3.960 K=0

50.42 %

197

1, 3, 2.18

4.

only VBR Traffic 4.305 K=1, CSW=1.0 CCC/CSW=0.25

46.11 %

529

2, 4, 2.41

5.

CBR & VBR Tr. K=0

26.365

11.22 %

547

1, 5, 2.73 (CBR) 1, 4, 2.70 (VBR)

6.

CBR & VBR Tr. K=1, CSW=1.0 CCC/CSW=0.25

27.069

8.85 %

1917

2, 4, 2.42 (CBR) 2, 4, 2.46 (VBR)

Table 2: Results of Case Study

60