Estimation of the Probability of Congestion using Monte Carlo method in OPS Networks

1. ISCC 2005 Estimation of the Probability of Congestion using Monte Carlo method in OPS Networks Anna Urra, Jose L Marzo, Mateu Sbert, Eusebi Calle Institute of Informatics and Applications (IIiA) eusebi@eia.udg.es

2. Universitat de Girona Contents Introduction Traffic and Congestion Control Calculation of the Bandwidth requirements Performance Evaluation Conclusions

3. 1. Introduction 1.1 Introduction Contention may arise when two or more packets demand the utilization of the same resources, for instance, when two or more packets have to be forwarded to the same output link at the same time. In traditional electronic packet switches, packet queues are implemented using buffers. In optical networks Fiber Delay Lines (FDLs) are used to implement optical packet queues. FDLs are long fiber lines used to delay the optical signal for a particular period of time in order to simulate a buffer.

4. 1. Introduction 1.2 Introduction WR presents low bandwidth utilization because it does not use statistical sharing of resources. On the other hand, OPS and OBS networks where packets streams can be multiplexed statistically, making a more efficient use of capacity.. • Optical switching methods: • Wavelength Routing (WR) • Optical Packet Switching (OPS) • Optical Burst Switching (OBS). This work focuses on OPS networks where statistical multiplexing for bandwidth sharing is used. Packets are aggregated at the edge of the network reducing the processing overhead, and then are routed over a bufferless core network (OPS). Major admission control and resource management methods have been addressed oriented to bursty traffic. However, our proposal focuses on heterogeneous traffic.

5. 2. Traffic and Congestion Control 2.1 Connection Admission Control Avoiding congestion -> Traffic control methods: Reactive control: methods regulate the traffic flow at the access points based on current traffic levels within the network. Preventive control: methods provide a fair allocation of bandwidth by requiring, at times of high network load, that each connection's traffic flow remains within specified bounds appropriate for the supported service. Due to real time constraints preventive control is more suitable than reactive control in high-speed networks. Connection Admission Control (CAC) is one of the traffic and congestion preventive control function used to maintain the QoS requested. CAC is the procedure responsible for determining whether a connection request is accepted or rejected.

6. 2. Traffic and Congestion Control 2.2 Stadistical Multiplexing Gain • Evaluation of the bandwidth demands: • Linear CAC • Fluid Flow Approximation • Heuristics methods. • Stationary approximations • In this case the effect of statistical multiplexing is the dominant factor. It considers that packets are lost when the instantaneous rate is greater than the bandwidth provided by the link. Stationary approaches are Binomial, Gaussian and Convolution. • In small buffer networks, Convolution Approach (CA) is the most accurate method used in CAC. But it has a considerable computational cost and a high number of accumulated calculations. Nevertheless, in critical near-congestion situations, the convolution is the only algorithm that gives enough accuracy.

7. 3. Calculation of the Bandwidth requirements 3.1 Model Formalization Types of sources : t Each source s emits in ms states Each state (i) has an associated rate r and probability p. For each source s there are ns connections. Link has a maximum capacity avalaible Cmax. Congestion is produced when the addition of connection rates is grater than Cmax

8. 3. Calculation of the Bandwidth requirements 3.2 Convolution Approach (CA) Probability density funcion for the offered system load, expressed as the probability that all traffic sources together are emitting at a giving rate. Y is the bandwidth requirement of the already established connections X is the bandwidth requirement of a new connection request b denotes the instantaneous required bandwidth. Convolution Approach 2n combinations 3 sources S1=0 S2=0 S3=0 ( C = 0 Prob=(0,9)3 ) S1=0 S2=0 S3=2 ( C = 2 Prob=(0.9)2(0.1)) S1=0 S2=2 S3=0 (C = 2 Prob=(0.9)2(0.1)) … S1=2 S2=2 S3=3 ( C = 6 Prob=(0.1)3) Source: Rate 2 Prob (0.1) Rate 0 Prob (0.9) Vp=0.2 Cmax = 5

9. 3. Calculation of the Bandwidth requirements 3.2 Convolution Approach (CA) Drawbacks: The size of the storage required 2n presents a dimensioning problem. This requirement increases with the number of connections and possible source states. In addition, the probability is the result of a large number of previous calculations.

10. 3. Calculation of the Bandwidth requirements 3.3 Enhanced Convolution Approach (ECA) Based on Multinomial Distribution Function (MDF). Multinomial (Enhanced Convoltuion Approach) N+1 combinations C = 0, 1 combination -> 1 (0.9)3 C = 2, 3 combination -> 3 (0.9)2(0.1) C = 4, 3 combination -> 3 (0.9)(0.1)2 C = 6, 1 combination -> 1 (0.1)3 Cmax • PC is the Probability of Congestion • Y is the Bandwidth requirement of the already established connections • Xj is the bandwidth requirement of a new j-type connection • b is the instantaneous rate considered • Epsilon is the maximum accepted Probability of Congestion due to excess bandwidth required set according to the QoS requirements • Gamma is the load factor that one has to pay regard to in order to limit congestion arising from contention resolution on the packet level.

11. 3. Calculation of the Bandwidth requirements 3.3 Enhanced Convolution Approach (ECA) In CA case a), all the probabiliy density function is computed. On the other hand, in ECA case b) only the probability density function with final rate higher than Cmax is computed reducing the computational cost.

12. 3. Calculation of the Bandwidth requirements 3.4 Monte Carlo Methods Main Idea: Evaluate a circle area (painted on the floor) throwing little stones randomly. Approximated Area = Stones in the circle / Total stones. This involves an error (depending on the number of throwed stones).

13. 3. Calculation of the Bandwidth requirements 3.4 Monte Carlo Methods Uniform Monte Carlo Convolution Approach (UMCA) Generate one random number for connection. This number match on a probability range of transmitting to an specifyc rate. Adding all these values (rates) we obtain the first iteration (first stone). Estimated value of PC = Number of iterations (>Cmax) / Total number of iterations. Drawback : Large number of iterations. Importance Sampling Convolution Approach (ISCA) Idea: Reduce the number of iterations mofifying the rate probabilities. Assigning higher probabilities to high rates (causing more samples with >Cmax) (more stones in the cicle).

14. 4. Performance Evaluation 4.1 Model Adopted 5 types of sources (emission rates and probabilities). 4 scenarios varying proportionally the number of connections. (*) ICSA assigns a higher probability to the highest rates of each source. For this set of experiments we assign a prob. Of 0,5 to the highest rate a (1-05/ms-1) to the rest.

15. 4. Performance Evaluation 4.3.1 Computational Cost For 500000 iterations in different scenarios (varying the numer of connections). UMCA and ISCA sharply increase as the number of connections increase. On the contrary ECA dramatically increase. Computational cost (sec) varying Cmax. For 100000 iterations. in scenario D (high number of connections). ECA dramatically decrease. On the contrary, UMCS and ISCA present a similar behaviour.

16. 4. Performance Evaluation 4.3.2 Absolute Error of the Probability of Congestion (PC) Difference between the estimated value (ISCA and UMCA) and the exact value (ECA) in scenario A varying the number of iterations. ISCA performs better than UMCA. The absolute error odf UMCA, as expected, is higher than ISCA. In addiction, the large is the number of iterations, the less is the absolute error. 4.3.3 Relative Efficiency ISCA is around 50% more efficient than UMCA considering both the absolute error (expresed in terms of variance) and the computational cost (expressed in terms of time).

17. 5. Conclusions New mechanisms to evaluate the Probability of Congestion have been presented in a OPS network with heterogeneous traffic. The tradeoff between computational cost and accuracy of the estimation has been analyzed. The presented methods overcome previous proposals (CA and ECA) in terms of computational cost. The proposed mechanisms (UMCA and ISCA) compute an estimation of the Probability of Congestion. Simulation results show an improvement in performance of our mechanisms in comparison to Enhanced Convolution Approach of 3 orders of magnitude,at the penalty of introducing a small stochastic error in the probability estimation. This error is lower in ISCA case though it has a higher computational cost than UMCA. ISCA, as expected, is more efficient than UMCA in terms of accuracy and computational cost (50\%).

18. Thank you ! ISCC 2005 Estimation of the Probability of Congestion using Monte Carlo method in OPS Networks Anna Urra, Jose L Marzo, Mateu Sbert, Eusebi Calle Institute of Informatics and Applications (IIiA)