Dimensioning ATM Networks with Uncertain Traffic Demands L.T.M. Berry +, R.J. Harris++ + RMIT University CATT Centre, Box 2476V, Melbourne, Australia
[email protected]
++ RMIT University CATT Centre, Box 2476V, Melbourne, Australia
[email protected]
Abstract – We consider a network planning method that takes into consideration the uncertainties associated with offered traffics when dimensioning ATM networks for prescribed grades of service. Assuming that each reference traffic stream offered between a pair of nodes in a network can only be specified within a lower and upper bound, together with a best guess value, we demonstrate a method for transforming these 3 values into a single traffic figure that takes uncertainty into consideration for a specified confidence level. We show that by using these new traffic values in the dimensioning process, specified call-based grades of service can be achieved.
given. For simplicity, we suppose that the distribution of the traffic value is a triangle distribution [6].
I. INTRODUCTION Telecommunications administrations have a long history of using formal measurement-based procedures for estimating and forecasting voice traffic demands, from which reference traffic values are derived. These representative offered traffic values (one for each origin to destination pair of exchanges) may be used to dimension voice networks for specified end-to-end grades of service [1, 2 ]. At the present time, there has been a limited opportunity to establish adequate databases of measured traffic values that would allow useful reference traffics to be deduced for networks that carry a heterogeneous mix of service types (such as ATM networks),. Indeed the concept of “reference traffic” for multi-rate networks still requires clear definition. We propose a method for dimensioning ATM networks when limited information is available on the traffic demands to be made on a network. Specifically, in order to demonstrate the method, we suppose that each origindestination pair of nodes (O-D pair) is offered a number of multi-slot streams. That is, a stream actualisation is defined by a call arrival rate, a call service rate, and an effective bandwidth (expressed in slots, units of which are, say, 0.064 Mbps). Only the ratio of the arrival and service rates is needed; this is the Erlang offered traffic, t. At the call level, it is assumed that call arrivals follow a Poisson process. It is supposed that each value t may be estimated within a lower and upper value L ≤ t ≤ U and that a “best guess” (most likely value) for t, within this range, Q, can be
If the slot size for calls from this stream is c, the offered “load” is the product tc. To use this mean value as a reference traffic value would be highly inaccurate if the difference between L and U were significant. Intuitively a value greater than m is needed and the “inflation” above m should depend upon the range L to U and the bias in the triangle distribution (i.e. the location of Q). In the next section, we demonstrate how this inflation is achieved in order to obtain a confidence bound on the uncertainty associated with drawing traffic values from this distribution Thus, the reference traffic to be used for each stream is no longer a single value defining a representative offered traffic load (such as that used in voice networks), but rather a 4-tuple (L, U, Q, c). The values L, U and Q, together with a confidence level are used to derive a
Height h =
L
Q
2 (U − L)
U
Figure 1: Triangle Distribution pdf The mean value m is readily obtained from
m=
L +U + Q 3
*
(1)
traffic value m for each traffic stream. These values, together with the slot sizes c are used in the dimensioning process (see section 2.2). It is supposed that a number of different multi-rate stream types are offered to each O-D pair in the network. A literature search revealed that few papers have been written dealing with capacity planning under uncertain traffic demand. The PhD thesis [6] focuses on the design of corporate networks under uncertainty (a formal classification of uncertainty is given) and provides a decision theoretic framework for static, incremental and evolutionary design. The volatility of demand for capacity is taken into consideration within a financial option pricing method in [7]. Uncertainty of demand is addressed in the context of virtual networks in [8] and a
framework for optical network planning under uncertainty is presented in [9]. Our approach may be seen as an extension of work summarised in [5] to take into account both multi-rate traffic streams and the possibility of allowing for alternative routing. II. UNCERTAINTY PLANNING Planning of a communication network typically involves using forecast traffic demands that are subject to uncertainty. This uncertainty arises from: • Errors and uncertainties in the measurement process when traffic conditions are being monitored in the network. • Errors and uncertainties associated with the traffic forecasting process. • Uncertainties concerning external factors influencing the demands being made upon the network such as technical and political factors. A. Incorporating Uncertainty into the Reference Traffic Values In [5] we considered the case of a single service network (the PSTN) and demonstrated that it was possible to treat uncertainty by simply inflating the normal design traffic (mean value) and then dimensioning the network using this inflated value and the normal design grade of service standard. We considered two different distributions to model the uncertainty associated with the offered traffic demand, viz: Normal (truncated) and a Beta distribution. The latter distribution was chosen for its simplicity and practicality. In particular, it is relatively straightforward to ask a planner (or review traffic data records) to nominate lower, upper and most likely traffic values for a given network or traffic stream. It may also be possible to use the results of a survey to ascertain these three numerical values. In the present study we have chosen to use the triangle distribution – it uses the same three traffic values. Once these values have been determined, the methodology is quite simple to apply. We determine the traffic Aconf that satisfies the requirement that P( A ≤ Aconf ) ≤ 0.95 , where we have selected a 95%
confidence interval to determine this traffic Aconf. For *
example, the inflated traffic ( m ) for a 95% confidence limit is obtained by solving the quadratic: (U − m* ) 2 = 0.05 (U − L)(U − M )
(2)
B. Dimensioning Method Outline The values m* and slot sizes c are transformed using the loss-equivalent single-slot transformation (see [4]) to obtain dimensioning parameters M & V for each stream. The multi-slot Poisson call arrival problem is converted into a single slot “rough traffic” arrival process. Each stream is defined by a mean M and variance V; this then permits the application of the chain flow dimensioning
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method [1, 2]. This transformation ensures that each stream is within a specified common GOS bound. The dimensioning method has been extensively investigated [10] and shown to give a good level of accuracy - in spite of simplifying assumptions. It also has an important advantage over alternative methods because it permits the modelling of overflow traffic in multi-rate networks. To isolate the factors related to the approach we take for dealing with uncertainty, minor changes to the link capacities have been made to obtain agreement for the dimensioned network with simulation. This is achieved by noting the largest slot size using a given link and ensuring that the allocated capacity is a multiple of this largest slot size. III. EXAMPLE NETWORK In order to demonstrate the above solution procedure and the applicability of our approach, we have used the network of Figure 1. Six Origin-Destination pairs (OD pairs) were selected for this network. Associated with each of these OD pairs are three traffic streams that are characterised by different slot sizes. A total of 18 traffic streams were modelled in this example network. 5
7
4 3 6
1
2
8
Figure 2: Sample 8 Node Network Used to Test the Modelling
Each of the three traffic streams for an OD pair used the same set of available paths (chains). The OD pairs, with their chains are summarised in the following table: O D
Chains st
1 Choice
nd
2
Choice
3rd Choice
1
2
{1,2}
{1,3,2}
{1,4,3,2}
5 6 7 7 8
3 3 3 5 2
{5,3} {6,3} {7,5,3} {7,5} {8,2}
{5,4,3} {6,5,3} {7,6,3} {7,6,5} {8,6,3,2}
-
Table 1 : OD Pair and Chain Data
Table 2 shows the traffic streams and associated traffic data. It shows, for each OD pair and associated traffic stream, the 4-tuple mentioned in previous sections. In the column headed “95% Conf” we have calculated the inflated traffic value associated with the 4-tuple. The final two columns list the loss-equivalent single slot traffic computed for the 95% confidence traffic value and associated slot sizes. This calculated traffic ensures exact
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O D TS # 1 2
5 3
6 3
7 3
7 5
8 2
Ref Traffic 4-tuple Slot
L
Q
95% Equiv Single Slot
U
Conf
Mean
Var
1
1
3
5
7
6.37
6.37
7.51
2
5
1
2
4
3.45
17.26
101.78
3
8
0.5
1
3
2.50
20.00
188.69
1
1
2
4
6
5.37
5.37
6.51
2
6
0.05 0.1 0.3
0.25
1.50
10.92
3
18
0.1 0.2 0.4
0.35
6.21
135.71
1
1
1
4
6
5.29
5.29
6.36
2
2
0.5
2
4
3.41
6.82
16.37
3
10
0.2
1
3
2.47
24.71
296.74
1
1
6
7
12 10.78
10.78
12.07
2
2
8
14
15 14.41
28.82
64.56
3
4
9
16
18 17.05
68.21
305.60
1
1
4
8
2
4
8
12
3
12
0.6
3
5
1
1
1
3
5
2
3
5
7
3
9
0.2
1
10
9.23
9.23
11.15
14 13.23
52.90
255.85
4.34
52.04
755.04
4.37
4.37
5.30
9
8.37
25.10
91.47
3
2.47
22.24
243.09
Table 2: Traffic Stream Slot Sizes and Equivalent Parameters
The network data was selected in a manner that tested different degrees of skewness for the triangle distribution and ensured that traffic with different and competing slot sizes was mixed together in the network. Using the data from Table 2 and the methodology outlined in Section 2.2, we have dimensioned the example network to achieve an average grade of service standard of 1.3% for the inflated traffic. This grade of service value is verified by simulation as described in Section 5. Connection Average 95% Confidence From To Capacity Slots Capacity Slots 1 2 1.6 25 2.56 40 1 3 1.216 19 2.048 32 1 4 0.704 11 1.024 16 3 2 3.392 53 4.608 72 4 3 1.408 22 2.304 36 5 3 4.864 76 6.912 108 5 4 1.216 19 2.304 36 6 3 5.376 84 7.04 110 6 5 5.312 83 6.912 108 7 5 7.36 115 9.216 144 7 6 7.232 113 9.216 144 8 2 1.856 29 2.88 45 8 6 2.304 36 3.456 54 Table 3: Summary of capacity requirements for average and 95% confidence limits
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Table 3 summarises the capacities (Mbps) and slot sizes that are required to meet the grade of service standards if the average traffic was used for dimensioning or if the inflated 95% traffic was used (final two columns). For dimensioning purposes, the traffic was allocated to the chains in the proportion 60:39 for streams with 2 chains and 60:30:9 for streams with 3 chains. It should also be noted that the use of the 95% confidence traffic results in a capacity increase of the order of 60% for the link. It should be noted that for this example, the use of the average traffic is equivalent to a confidence level that is less than 40%. IV. SIMULATION STUDY In order to validate the approach outlined in earlier sections, an extensive set of simulations was performed using the sample 8 node network depicted in Figure 2. Approximately 2000 blocks of simulations were performed using a multi-rate simulation program contained in the APTNet™ planning tool. For each “block” we performed 8 separate simulation runs that generated 50,000 “calls” to the network across the 18 traffic streams. Traffic values selected for these 18 traffic streams were generated as random values from the triangle distribution. Figure 3 below, shows the method for generation of random values from the triangle distribution using the cumulative distribution function (cdf) and an associated inverse transform method. Cumulative Distribution Function for Triangle Probability 1.0000
0.9000
0.8000 Cumulative Probability Distribution
agreement with the well-known Kaufman and Roberts’s model for a single link.
0.7000
0.6000
0.5000
Area
Mode of the distribution 0.4000
0.3000
0.2000
0.1000
0.0000 9.00
9.50
10.00
10.50
11.00
11.50
12.00
x-value
Figure 3: Generation of Random Values from the Triangle Distribution
The simulator was capable of generating approximately 2.5 million multi-rate calls per minute and the overall study generated about 800 million calls on 1.2GHz Pentium computers. V. RESULTS After dimensioning the network to meet the required grade of service (1.3%) for the inflated traffic, a simulation was performed to validate this dimensioning.
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Ave O D Slots Traf 1 2 1 4.977 5 2.341 8 1.498 5 3 1 4.003 6 0.149 18 0.235 6 3 1 3.669 2 2.171 10 1.393 7 3 1 8.311 2 12.293 4 14.320 7 5 1 7.357 4 11.373 12 2.864 8 2 1 2.982 3 6.996 9 1.408
Max Gos 0.000 0.014 0.034 0.001 0.007 0.027 0.001 0.003 0.030 0.002 0.007 0.020 0.001 0.008 0.045 0.001 0.008 0.055
Ave Gos 0.0000 0.0009 0.0021 0.0001 0.0005 0.0053 0.0000 0.0001 0.0018 0.0001 0.0005 0.0015 0.0001 0.0006 0.0036 0.0001 0.0006 0.0048
Target % conf
99.25%
97.60%
99.58%
99.95%
98.05%
96.75%
Table 4: Summary of Results for 2000 Simulation Blocks
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Whilst multi-rate trunk reservation schemes may be suitable for such a task, they were not considered in this study. Finally, a brief study was performed to determine the penalty associated with dimensioning the network using the average traffic values instead of the inflated ones. A simulation involving 500 blocks was performed using the same approach as described above and it was found that the network grade of service satisfied the design grade of service on only 2.6% of occasions for this set of traffics. This illustrates the consequences of not allowing for uncertainty in the network planning process! Total Slots Required
1000 900 800 700
Slots
This simulation, using approximately 1 million calls demonstrated that the network achieved a traffic weighted grade of service equal to 1.3% and the 95% confidence limits for this grade of service were: (0.0126, 0.0134). When the above validation procedure was completed, the 2000 blocks of simulations, described in Section 4, were performed. The results of these simulations are summarised in Table 4 below. The column headed “Ave Traf” represents the average traffic generated for the traffic stream in the simulation. These figures can be compared with the expected average traffic computed using equation (1) and the values given in Table 2. It will be seen that the results match very closely to the expected traffic values. The column headed “Target %Conf” gives the percentage of times that the traffic stream congestion exceeded the design grade of service (1.3%) over the 2000 blocks from an OD pair viewpoint. It is interpreted as the percentage of occasions when the traffic stream remains under the target design grade of service. It can be seen that the approach of inflating the traffics has certainly ensured that the network is robust with respect to the uncertainty in the offered traffic values. However, it is well-known that multi-rate traffic streams experience different blocking probabilities and this can be seen by looking at the columns labelled “Ave Gos” and “Max Gos”. They show that the average grade of service for large slot size traffic is much higher than for the small slot size traffic streams. It should be noted that there is no mechanism deployed in the simulator (or in current practice) that provides equalisation of performance for the different streams.
600
Slots
500 400 300 200 100 0 40%
50%
70%
80%
95%
Confidence Level
Figure 4: Total Network Capacity Required for Different Confidence Levels
Figure 4 illustrates the impact on the additional capacity required in the network as a result of selecting different confidence levels. As previously mentioned, when the network was dimensioned using traffic averages, the grade of service requirements were satisfied only 2.6% of the time. To achieve the improvement to a 95% guarantee, the total network capacity must be increased by a further 38%. For different networks, the percentage increase in network capacity will be a function of the degree of uncertainty in the traffic forecasts. Overall, it is clear that method outlined above can be used to dimension networks to meet uncertainty in offered traffic demands. It is clear that arbitrarily increasing the capacities on links can achieve greater robustness with regard to this uncertainty, but the methodology that has been proposed can be used to quantify the extent of this increase. VI. CONCLUSIONS In this paper, we have proposed a model which takes into consideration uncertainty in traffic demand. We have shown that the approach of inflating the traffic based on a confidence level from a triangle distribution is simple to implement and that the network designed in this way would ensure that the average traffic congestion experienced is under the target design grade of service in better than 95% of cases. The methodology is also applicable for other distributions describing the uncertainty. Further research is required to more accurately dimension to such a level of confidence. It
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should be noted that insisting upon confidence levels of 95% carries a penalty of additional capacity to be installed. Conversely, the use of the average traffic to dimension the network carries the obvious penalty that revenue might be lost due to under-provision of capacity. VII. REFERENCES 1.
Berry L.T.M., An application of mathematical programming to alternate routing. A.T.R., 4 (2):20-27, 1970.
2.
Berry L.T.M. and Harris R.J., Modular design of a large metropolitan telephone network: A case study. 11th I.T.C., Kyoto, 1985.
3.
Berry L.T.M. and Harris R.J. Optimal design of international communications networks which exploit time zone differences and uncertainties in traffic loads. Networks’96, Proceedings of 7th Int. Network Planning Symposium, Sydney, Nov. 1996 Session 5.
4.
Berry L.T.M, Harris R.J., Puah L.K., Methods for trunk dimensioning in a multi-service network. Globecom’98, Sydney Nov. 1998, Session S52.5. 5. Harris, R.J. and Berry, L.T.M., Network Dimensioning Under Uncertainty, In 5th Australian Teletraffic Research Seminar, Melbourne, 1990. 6. Fischer V. G. Evolutionary Design of Corporate networks under Uncertainty. Dr. rer. nat. Dissertation, Technical University of Munich 2000. K.R., 7. d’Halluin Y.,Forsyth P.A., and Vetzal Managing Capacity for Telecommunications Networks Under Uncertainty, IEEE /ACM Trans Networking, Aug 2002, Vol. 10, No.4 pp579 – 588. 8. Lisser A., Ouorou A., Vial J.-Ph. and Gond J., Capacity Planning under Uncertain Demand in Telecommunications Networks, Research Funded by CNET-France Telecom, 1999. 9. Geary N., Antonopoulos E., O’Reilly J.J., Mitchell J.E., A Framework for Optical Network Planning under Traffic Uncertainty, in Proc. 3rd International Workshop on the Design of Reliable Communication Networks (DRCN) October 2001, Budapest Hungary. 10. Puah L.K., Capacity Dimensioning Methods for Multi-service Networks, PhD thesis, RMIT, 1999. VIII. ACKNOWLEDGEMENT Some of the computations for this paper were made using APTNet™ and we wish to acknowledge our appreciation to Telekom Malaysia for approval to use this planning tool package for this research.
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