Advanced Internet Technologies

Transcription

1 Advanced Internet Technologies Chapter 3 Performance Modeling Dr.-Ing. Falko Dressler Chair for Computer Networks & Internet Wilhelm-Schickard-Institute for Computer Science University of Tübingen dressler@informatik.uni-tuebingen.de Advanced Internet Technologies, SS

2 Chapter 3 Performance Modeling Queuing Analysis Introduction Queuing Models Self-Similar Traffic Advanced Internet Technologies, SS

3 Queuing Analysis Introduction Goals Prediction of network/system behavior Performance modeling and estimation Benefits Real-world tests are expensive and often infeasible Analytic models based on queuing theory often provide the needed answers Queue Server Advanced Internet Technologies, SS

4 How Queues Behave A Simple Example Consider a system with the following capabilities Capacity of input and output: 1000 packets/s Processing time for an average request: 1ms Unlimited queue size Non-uniform arrival rate Three experiments Arrival rate of 500 requests/s (50% server capacity) Arrival rate of 950 requests/s (95% server capacity) Arrival rate of 990 requests/s (99% server capacity) Advanced Internet Technologies, SS

5 Queue Behavior with Normalized Arrival Rate of 0.5 Results Avg buffer size: 43 Peak: >600 Advanced Internet Technologies, SS

6 Queue Behavior with Normalized Arrival Rate of 0.95 Results Avg buffer size: 1859 Peak: >4200 Advanced Internet Technologies, SS

7 Queue Behavior with Normalized Arrival Rate of 0.99 Results Avg buffer size: 2583 Peak: >5300 Advanced Internet Technologies, SS

8 Performance Evaluation 1. Do an after-the-fact analysis based on actual values 2. Make a simple projection by scaling up from existing experience to the expected future environment 3. Develop an analytic model based on queuing theory 4. Program and run a simulation model Advanced Internet Technologies, SS

9 Queuing Models Single-Server Queue Theoretical maximum input rate that can be handled by the system (at utilization ρ=1): λ max = 1 T s Advanced Internet Technologies, SS

10 Single-Server Queue Model Characteristics (Assumptions) Item population Items arrive from an infinite source population Thus, the arrival rate is not altered as items enter the system (If the population is finite, then the population available for arrival is reduced by the number of items currently in the system; this would typically reduce the arrival rate proportionally) Queue size The queue size is infinite Thus, the queue can grow without bound (With a finite queue, items can be lost from the system; that is, if the queue is full and additional items arrive, some items must be discarded) Dispatching discipline When the server becomes free, and if there is more than one item waiting, a decision must be made as to which item to dispatch next Examples: FIFO, FCFS, LIFO,... Advanced Internet Technologies, SS

11 Example of a Queuing Process Advanced Internet Technologies, SS

12 Notations Advanced Internet Technologies, SS

13 Queuing Models Multiserver Queue Theoretical maximum input rate that can be handled by the system (at utilization Nρ, which is also called the traffic intensity, u): λ max = N Ts Advanced Internet Technologies, SS

14 Queuing Models Multiple Single-Server Queues Advanced Internet Technologies, SS

15 Queuing Analysis - Basics Input information Output information Arrival rate: λ Items waiting: w Service time: T s Number of servers: N Waiting time: T w Items queued: r Residence time: T r Average values (w, T w, r, T r ) Standard deviation (σ w, σ Tw, σ r, σ Tr ) Advanced Internet Technologies, SS

16 Queuing Analysis Basics II Distribution of the interarrival times Typically, we assume that the interarrival time is exponential, which is equivalent to saying that the number of arrivals in a period t obeys the Poisson distribution, which is equivalent to saying that the arrivals occur randomly and independent of one another. Kendall s notation X/Y/N X refers to the distribution of the interarrival times Y refers to the distribution of service times N refers to the number of servers Distributions G... General distribution of interarrival times GI... General distribution of interarrival times, interarrival times are independed M... Poisson distribution D... Deterministic distribution Example M/M/1 refers to a single-server queuing model with Poisson arrivals Advanced Internet Technologies, SS

17 Single-Server Queues Advanced Internet Technologies, SS

18 Single-Server Queues II Mean number of items in system for single-server queue Advanced Internet Technologies, SS

19 Single-Server Queues III Mean residence time for single-server queue Advanced Internet Technologies, SS

20 Single-Server Queues IV σ Ts /T s is also known as the coefficient of variation and gives a normalized measure of variability The meaning of σ Ts /T s Zero: constant service time, e.g. if all transmitted packets are of the same length Ratio less then 1: ratio better than the exponential case, the M/M/1 model would give answers on the safe side Ratio close to 1: common occurrence, corresponds to exponential service time Ratio greater than 1: the M/G/1 model is required Advanced Internet Technologies, SS

21 Multiserver Queues (M/M/N) Poisson ratio function Erlang C function (probability that all servers are busy) C( N, ρ) 1 K( N, ρ) = 1 ρk( N, ρ) Equation for single-server systems C(1, ρ ) = ρ Advanced Internet Technologies, SS

22 An Example Single-Server Model Engineers use PCs plus a single graphic workstation. On a typical 8- hour day, 10 engineers will use the workstation and spend an average of 30 minutes at a session. Average time an engineer spends waiting for the workstation T w = ρts = 1 ρ 50 minutes Average rate of engineers λ = 10 = 8* engineers/minute Average number of engineers waiting w = λtw = engineers Advanced Internet Technologies, SS

23 An Example Single-Server Model II Calculating percentiles mr y ln(1 ) ( y) = ln( ρ) 90th percentile waiting time mtw Tw ( 90) = *ln(10ρ) = minutes ρ Results Engineers wait almost an hour to use the workstation In 10% of the cases they wait longer than 2 hours Advanced Internet Technologies, SS

24 An Example Multiserver Model Probability that both servers are busy C(2, ρ) = C(2,0.3125) = Average time an engineer spends waiting for a workstation T w = CTs N(1 ρ) = minutes 90th percentile waiting time m Tw Ts ( 90) = *ln(10c) = 2(1 ρ) 8.67 minutes Average number of engineers waiting w = λtw = 0.07 engineers Advanced Internet Technologies, SS

25 An Example Single-Server vs. Multiserver Model Advanced Internet Technologies, SS

26 Network of Queues In a distributed environment, isolated queues are unfortunately not the only problem presented to the analyst. Often, the problem to be analyzed consists of several interconnected queues. Partitioning and merging of traffic (nodes 1 and 5) Queues in tandem, or series (nodes 3 and 4) Advanced Internet Technologies, SS

27 Network of Queues Partitioning and Merging Partitioning If the incoming distribution is Poisson, then the two departing traffic flows also have Poisson distributions, with mean rates Pλ and (1-P)λ. Merging If two Poisson streams with mean rates λ 1 and λ 2 are merged, the resulting stream is Poisson with a mean rate of λ 1 + λ 2. Advanced Internet Technologies, SS

28 Network of Queues Queues in Tandem Assume that the input to the first queue is Poisson. Then, if the service time of each queue is exponential and the queues are of infinite capacity, the output of each queue is a Poisson stream statistically identical to the input. Thus, the queues are independent and may be analyzed one at a time. Therefore, the mean total delay for the tandem system is equal to the sum of the mean delay at each stage. Advanced Internet Technologies, SS

29 Jackson s Theorem Assumptions The queuing network consists of m nodes, each of which provides an independent exponential service. Items arriving from outside the system to any one of the nodes arrive with a Poisson rate. Once served at a node, an item goes (immediately) to one of the other nodes with a fixed probability, or out of the system. Jackson s Theorem Jackson s theorem states that in such a network of queues, each node is an independent queuing system, with Poisson input determined by the principles of partitioning, merging, and tandem queuing. Thus, each node may be analyzed separately from the others using the M/M/1 or M/M/N model, and the results may be combined by ordinary statistical methods. Advanced Internet Technologies, SS

30 Self-Similar Traffic Queuing analysis depends on the Poisson nature of the data traffic It has shown that for some environments the traffic pattern is selfsimilar rather than Poisson Self-similarity is a concept related to two others Fractals Chaos theory Outline Phenomenon of self-similarity Performance implications Storage Model with Self-Similar Advanced Internet Technologies, SS

31 Self-Similarity Statement by Manfred-Schroeder: The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale or size, is an attribute of many laws in nature and innumerable phenomena in the world around us. Self-similarity is, in fact, one of the decisive symmetries that shape our universe and our effort to comprehend it. Advanced Internet Technologies, SS

32 Self-Similarity An Example Network monitoring, analysis of the interarrival time of single frames Minimum transmission time for one frame: 4ms Recorded arrivals (ms): Clustering all samples with gaps smaller than 20ms: Clustering all samples with gaps smaller than 40ms: Advanced Internet Technologies, SS

33 Self-Similarity An Example II Repeating patterns: arrival, short gap, arrival, long gap, arrival, short gap, arrival) Gaps between single frames / clusters [ms] Reihe1 Reihe2 Reihe Advanced Internet Technologies, SS

34 Self-Similarity An Example III Repeating patterns: arrival, short gap, arrival, long gap, arrival, short gap, arrival) Advanced Internet Technologies, SS

35 Cantor Set Famous construct appearing in virtually every book on chaos, fractals, and nonlinear dynamics Construction rules: Begin with the closed interval [0,1], represented by a line segment Remove the open middle third of a line For each succeeding step, remove the middle third of the lines left by the preceding step Cantor set: S 0 = [0, 1] S 1 = [0, 1/3] U [2/3, 1] S 3 = [0, 1/9] U [2/9, 1/3] U [2/3, 7/9] U [8/9, 1] Advanced Internet Technologies, SS

36 Cantor Set II Properties of Cantor sets seen in all self-similar phenomena It has a structure at arbitrarily small scales. If we magnify part of the set repeatedly, we continue to see a complex pattern of points separated by gaps of various sizes. The process seems unending. In contrast, when we look at a smooth, continuous curve under repeated magnification, it becomes more and more featureless. The structure repeat. A self-similar structure contains smaller replicas of itself at all scales. For example, at every step, the left (and right) portion of the Cantor set is an exact replica of the full set in the preceding step. These properties do not hold indefinitely for real phenomena. At some point under magnification, the structure and the self-similarity break down. But over a large range of scales, many phenomena exhibit selfsimilarity. Advanced Internet Technologies, SS

37 Stochastical Self-Similarity So far, we examined exact self-similarity: A pattern is reproduced exactly at different scales Data traffic is a stochastic process, therefore we talk about statistical self-similarity. For a stochastic process, we say that the statistics of the process do not change with the change in the time scale. The average behavior of the process in the short-term is the same as it is in the long term. Examples Data traffic Earthquakes Ocean waves Fluctuations in the stock market Advanced Internet Technologies, SS

38 Self-Similar Stochastic Process Advanced Internet Technologies, SS

39 Examples of Self-Similar Data Traffic Ethernet Traffic W. Leland, M. Taqqu, W. Willinger: On the Self-Similar Nature of Ethernet Traffic. Proceedings of the SIGCOMM 93, September World-Wide Web Traffic M. Crovella, A. Bestavros: Self-Similarity in World-Wide Web Traffic: Evidence and Possible Causes. Proceedings of the ACL Sigmetrics Conference on Measurement and Modeling of Computer Systems, May Signaling System Number 7 (SS7) Traffic D. Duffy, A. McIntosh, M. Rosenstein, W. Willinger: Statistical Analysis of CCSN/SS7 Traffic from Working CCS Subnetworks. IEEE Journal on Selected Areas in Communication, April TCP, FTP, and TELNET Traffic V. Paxson, S. Floyd: Wide Area Traffic: The Failure of Poisson Modeling. IEEE/ACM Transactions on Networking, June Variable-Bit-Rate (VBR) Video M. Garnett, W. Willinger: Analysis, Modeling, and Generation of Self-Similar VBR Video Traffic. Proceedings of the SIGCOMM 94, August Advanced Internet Technologies, SS

40 Performance Implications Ethernet Traffic Advanced Internet Technologies, SS

41 Storage Model with Self-Similar Input Buffer requirement q = ρ (1 ρ) 1/ 2(1 H ) H /(1 H ) For H = 0.5, this relationship simplifies to q = ρ/(1- ρ), which is the classic queuing result of a system with exponential interarrival times and exponential service times (M/M/1). Hurst parameter H defines the degree of self-similarity, i.e. the higher the parameter H, the higher the self-similarity. Advanced Internet Technologies, SS

42 Storage Model with Self-Similar Input II Advanced Internet Technologies, SS