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Matrix Analytic Methods - Some Real Life Applications

Matrix Analytic Methods - Some Real Life Applications. David Lucantoni DLT Consulting, L.L.C www.DLTconsulting.com. Objectives of this Talk. Review Matrix Analytic Methods pioneered by Marcel Neuts and others

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Matrix Analytic Methods - Some Real Life Applications

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  1. Matrix Analytic Methods -Some Real Life Applications David Lucantoni DLT Consulting, L.L.C www.DLTconsulting.com Queueing Theory and Network Applications, Korea University, 2011

  2. Objectives of this Talk • Review Matrix Analytic Methods pioneered by Marcel Neuts and others • Present real life examples where Matrix Analytic Methods have been used successfully in industry • Congestion due to Internet Dial Up • Application to Common Channel Signaling (SS7) • Modeling WiFi Hotspot Traffic • Multiplexing bursty traffic in an ATM or IP network • Call Acceptance for a VoIP System • Transient Results and applications Queueing Theory and Network Applications, Korea University, 2011

  3. Objectives of this Talk (Cont’d) • No new mathematical analyses • A detailed description of any of these models would take the whole talk • Will only give general description of model parameters • Detailed definitions and analyses are widely available in the literature • Will give some matrix results along with the corresponding scalar (exponential or Poisson) results Queueing Theory and Network Applications, Korea University, 2011

  4. Analysis of Matrix Analytic Models • Models are analyzed by purely probabilistic arguments (e.g., Markov Renewal Theory) • Explicit expressions lead to very stable numerical algorithms for computation of performance metrics • Matrix analytic methods avoid • the complex analysis used in previous approaches • the necessity of numerically computing the roots of an analytic function • Numerical instability when roots are close to each other or multiple Queueing Theory and Network Applications, Korea University, 2011

  5. Phase-Type (PH) Distributions • Will only consider continuous-time distributions here • Phase-type distributions contain many well-known and popular distributions • Exponential distribution, (m=1) • Erlang distribution • Hyper-exponential distribution Queueing Theory and Network Applications, Korea University, 2011

  6. Phase-Type Distributions (Cont’d) Defined as the time till absorption in an (m+1)-state Markov process with 1 absorbing state Let T be the m×m matrix of infinitesimal rates between transient states, α, the vector of probabilities of starting in a particular transient state and T0 be the rates into the absorbing state Let X be a phase-type random variable and F(x)=P(X≤x) Then Queueing Theory and Network Applications, Korea University, 2011

  7. Assuring Emergency Services Access (e.g., 911) in the presence of long holding times due to Internet dial-up calls • The Problem • No dial-tone condition - “Customers hear thin air.” • VERY SERIOUS – e.g., 90/day life threatening emergency calls . • Priorities non-implementable • Preliminary suspicion on maintenance activities. • Could it be congestion ? Red flag – internet usage ! • Model solved using a finite quasi-birth-and-death process (QBD) • [Ramaswami, et al., 2005] Queueing Theory and Network Applications, Korea University, 2011

  8. ADM Cable Telephony Proprietary Protocol GR-303 GR-303 Backbone Transport OC-48+ SS7 CLASS -5 SWITCH DWDM HDT DACS TDM/T-1 ADM CMTS IP DWDM Backbone Master Headend or Primary Hub (100K - 200K HHP) Each PSTN Network Interface Unit EMS Systems DHCP/DNS TFTP Servers DOCSIS Protocol Cable Modem Fiber Node  16 Per Secondary Hub Central Office Transport Zone 5 Zone 4 Zone 3 Zone 2 Zone 1 Queueing Theory and Network Applications, Korea University, 2011

  9. Effect of Long Holding Times (Cont’d) • Preliminary Analysis • problem in evening hours 7 PM – 11 PM • no discernible spatial pattern • Push-backs • Engineering followed “standard” procedures • Only 5-8% of calls are internet dial-ups • Field measurements “do not support” congestion hypothesis Queueing Theory and Network Applications, Korea University, 2011

  10. Phase Type Model(using the EM-algorithm) ; fit uses ~4M data points Queueing Theory and Network Applications, Korea University, 2011

  11. The Length Biasing Effect Remaining holding time distribution: [1-F(x)] / μ Queueing Theory and Network Applications, Korea University, 2011

  12. Empirical Validation Empirical data on residuals (~11 K observations) vs. Model Queueing Theory and Network Applications, Korea University, 2011

  13. Impact of Long Holding Times Insensitivity relates only to blocking over an infinite horizon. Heavy tails  PERSISTENCE of congestion. So, short term performance is quite different.“Standard” procedures do not work ! Bad periods compensated by long very good periods. Good situation for control !!! (See the paper and patents for possible controls.) Queueing Theory and Network Applications, Korea University, 2011

  14. Several Other Real Life Applications • Excessive Link Oscillations in the Common Channel Signaling (SS7) Network [Ramaswami and Wang, 1993] • Solved using Phase-type distributions • Modeling and characterization of large-scale Wi-Fi traffic in public hot-spots [Ghosh, et al., 2011] • Solved using an M/G/∞ where G is ln PH distribution Queueing Theory and Network Applications, Korea University, 2011

  15. The Batch Markovian Arrival Process (BMAP) • Natural generalization of the Poisson process • Contains many well-known arrival processes (in all cases, correlated batch sizes are easily handled) • Poisson process (m=1) • PH renewal process • Markov modulated Poisson process (MMPP) • Superposition of BMAPs is again a BMAP • Useful for modeling arrivals to an internal node in a network Queueing Theory and Network Applications, Korea University, 2011

  16. The Batch Markovian Arrival Process (Con’t) Consider an m-state Markov process and assume that at each transition of this process, there is a probability of a batch arrival that depends both on the state before and after the transition Let Dn, n≥1, denote the infinitesimal rate of a batch arrival of size n, keeping track of the underlying states before and after the transition The generating function D(z) is given by Queueing Theory and Network Applications, Korea University, 2011

  17. BMAP/G/1 Queue • The BMAP/G/1 queue is extremely useful for modeling the performance of broadband packet networks • Traffic is bursty • Service times are NOT phase-type • Asynchronous Transfer Mode (ATM) • Fixed length cells result in deterministic service times • IP networks • Finite packet sizes  finite mixture of deterministic service times • BMAP models very bursty traffic and is closed under superposition, i.e., modeling the output of nodes entering another node Queueing Theory and Network Applications, Korea University, 2011

  18. Busy Period of the BMAP/G/1 Queue Let h(s) be the Laplace-Stieltjes transform of the service time distribution Let G(z) be the probability generating function of the number of customers served during a busy period (keeping track of the arrival phase). Then we have Queueing Theory and Network Applications, Korea University, 2011

  19. Busy Period of the BMAP/G/1 Queue (Cont’d) Define G to be G(1) and g to be the stationary probability vector of G It is easy to show that g is also the stationary probability vector of the phase of the arrival process during idle periods For M/G/1, the probability that the system is empty is 1-ρ; for BMAP/G/1, the probability that the system is empty and that the arrival process is in phase j is the jth element of (1-ρ)g Queueing Theory and Network Applications, Korea University, 2011

  20. Virtual Waiting Time of the BMAP/G/1 Queue Let W(s) be the Laplace-Stieltjes transform of the virtual waiting time distribution (keeping track of the arrival phase) Let g be the stationary probability vector of the stochastic matrix G Then we have Queueing Theory and Network Applications, Korea University, 2011

  21. Stationary Queue Length at Departures Let Xn be the stationary probability that there are n customers in the queue at departures Let , then Queueing Theory and Network Applications, Korea University, 2011

  22. Computing these Distributions We compute these distributions by numerically inverting the transforms [Choudhury, Lucantoni and Whitt, 1994] These algorithms are for inverting multi-dimensional transforms which we use later for computing transient distributions Queueing Theory and Network Applications, Korea University, 2011

  23. Known Results for the BMAP/G/1 Queue • Stationary distributions [Lucantoni, 1993] • Queue length distribution at arrivals, departures, arbitrary time • Waiting time distributions • Transient distributions (given the appropriate initial conditions) [Lucantoni, Choudhury, and Whitt,1994], [Lucantoni, 1998] • P(system empty at time t) • Workload at time t • Queue length at time t • Delay of nth arrival • Queue length at nth departure • P(nth departure occurs less than or equal to time x) Queueing Theory and Network Applications, Korea University, 2011

  24. Modeling the Input to an Interior Network Node Consider the output from several nodes which all go to the same node Ideal candidate for the MAP/G/1 queue Queueing Theory and Network Applications, Korea University, 2011

  25. Multiplexing Bursty Traffic in an ATM Network • ATM requirement is that the probability of blocking must be less than 10-9 • Tails of distributions are the most important performance measure • Superposition of 64 on-off sources each modeled as a 2-state MMPP • Exact Analysis • Far from Poisson • Exponential tail does not become relevant until blocking is 10-40 Queueing Theory and Network Applications, Korea University, 2011

  26. Advantage of MAP/G/1 Approach over Matrix-Geometric Approach When service time has Many Phases We computed the tail of the waiting time distribution in an MMPP64/E1024/1 queue by an by a distribution that is close to an E1024 and matches the first three moments of the deterministic distribution The matrix-geometric approach would require using matrices of order 65,536 The MAP/G/1 approach looks at the process embedded at departures; since we don’t have to keep track of the phase of service, the matrices are only of order 64 Queueing Theory and Network Applications, Korea University, 2011

  27. Advantage of MAP/G/1 (Cont’d) G satisfies When H is E1024, the integral on the right hand side is computed by inverting a 64×64 matrix 10 times Much easier than computing the matrix exponential Queueing Theory and Network Applications, Korea University, 2011

  28. Packet Counts 5000 4500 4000 3500 3000 Packets 2500 2000 1500 1000 500 0 0 1 2 3 4 Hours IP Data Traffic is Bursty Queueing Theory and Network Applications, Korea University, 2011 Sample Measurements from a previous assignment Sampled every second Highly bursty Mean = 720 pkts/sec

  29. Packet Counts - Poisson 800 780 760 740 Packets 720 700 680 660 640 0 5 10 15 Minutes Simulated Poisson Traffic Queueing Theory and Network Applications, Korea University, 2011 Poisson traffic looks bursty However, note the scale of the y-axis

  30. Packet Counts - Poisson Packet Counts - Measurements 3500 3500 3000 3000 2500 2500 2000 2000 Packets Packets 1500 1500 1000 1000 500 500 0 0 0 5 10 15 0 5 10 15 Minutes Minutes Poisson Process Compared to Data Queueing Theory and Network Applications, Korea University, 2011

  31. Fit an D-MMPP to Data [Heyman, Lucantoni, 2003] Choose λ1 so that the peak is at the 95th percentile of a Poisson distribution, i.e., Then Queueing Theory and Network Applications, Korea University, 2011

  32. Fit an MMPP to Data (Cont’d) Lower bound of data covered by λ1 is Let this be the upper bound covered by λ2, i.e., Then Continue… Queueing Theory and Network Applications, Korea University, 2011

  33. Fit an MMPP to Data (Cont’d) Compute the transition probabilities by For performance computations convert to a continuous time MMPP (see paper) Queueing Theory and Network Applications, Korea University, 2011

  34. Fit an MMPP to Data Q-Q plot: plots quantiles Should be linear if both come from same distribution Very good fit Queueing Theory and Network Applications, Korea University, 2011

  35. Packet Counts - MMPP Packet Counts - Measurements 3500 3500 3000 3000 2500 2500 2000 2000 Packets Packets 1500 1500 1000 1000 500 500 0 0 0 5 10 15 0 5 10 15 Minutes Minutes Simulated MMPP Traffic Queueing Theory and Network Applications, Korea University, 2011

  36. Simulated MMPP Packet Counts Measurement Packet Counts 600 600 500 500 400 400 Packets Packets 300 300 200 200 100 100 0 0 0 5 10 15 0 5 10 15 Minutes Minutes Another Example Queueing Theory and Network Applications, Korea University, 2011

  37. Call Acceptance for a VoIP System A startup company was offering VoIP service to clients by buying a fixed bandwidth virtual circuit (“fat pipe”) through the Internet Wanted to know how big the fat pipe should be to handle a certain number of voice calls and still meet the Quality of Service (QoS) e.g., tail of the delay distribution Measurements were taken for IP voice calls for different codecs, e.g., G.723, G.729, etc., and different measurement intervals We developed a program where the user inputs the file containing the measurements, the measurement interval and the target delay Queueing Theory and Network Applications, Korea University, 2011

  38. Call Acceptance for a VoIP System (Cont’d) We fit an MMPP to the measurement data and by solving the delay using an MMPP/D/1 queue If the computed QoS is greater or less than the targeted QoS, we adjust the service rate and re-compute; continue adjusting the rate using a binary search until the computed QoS is within a given tolerance The program then outputs the size of the required fat pipe and the required buffer size Queueing Theory and Network Applications, Korea University, 2011

  39. Notes on Transient Results • Generalized every result derived by Takács for the transient M/G/1 queue [Takács, 1962] • All results are direct matrix analogues of the scalar results • Using only probabilistic arguments, resulted in expressions that can be computed numerically without computing the roots of an analytic equation Queueing Theory and Network Applications, Korea University, 2011

  40. Sample Transient Result (emptiness function) Consider a MAP/G/1 queue (similar results hold for the BMAP/G/1 queue) Let Px0(t) be an m×m matrix where the (i,j) entry is the probability that the system is empty at time t with the arrival process is in phase j given the at time 0 the work in the system is x and the arrival process is in phase i Let px0(s) be the Laplace-Stieltjes transform of Px0(t) Queueing Theory and Network Applications, Korea University, 2011

  41. Emptiness Function (Cont’d) • Then Queueing Theory and Network Applications, Korea University, 2011

  42. Transient Performance Measures • We compute the virtual waiting time (work in system) at time t for a given queue length at time 0 by numerically inverting the 2-dimensional transform, W(s1,s2) Queueing Theory and Network Applications, Korea University, 2011

  43. Sample Transient Results -Virtual Delay Distribution • Superposition of four identical two-state MMPP’s • Service times are distributed as E16 • Utilization = 0.7 • queue length = 0 at time 0 • queue length = 32 at time 0 Queueing Theory and Network Applications, Korea University, 2011

  44. Sample Transient Results -Unstable System Utilization (ρ) = 2.0 Queue length = 0 at time 0 Queueing Theory and Network Applications, Korea University, 2011

  45. Potential Use of Transient Results • Call acceptance algorithms • Current call acceptance algorithms are based on stationary probabilities • Given the number of calls in the system, the time to reach a stationary distribution can be MUCH longer than the inter-arrival and inter-departure times • Call acceptance tables could be computed a-priori based on decision time intervals, the current state of the system, the average and peak bandwidth and burstiness parameter of the arriving call Queueing Theory and Network Applications, Korea University, 2011

  46. References • V. Ramaswami, D. Poole, S. Ahn, S. Byers, and A. Kaplan, "Assuring emergency services access: .providing dial tone in the presence of long holding time internet dial-up calls," Interfaces, vol. 35, pp. 411-22, 2005 • V. Ramaswami and J. L. Wang, "Analysis of the Link Error Monitoring Protocols in the Common Channel Signaling Network," IEEE/ACM Transactions on Networking, vol. 1, pp. 34-47, 1993 • Ghosh, R. Jana, V. Ramaswami, J. Rowland, and N. K. Shankaranarayanan, "Modeling and characterization of large-scale Wi-Fi traffic in public hot-spots," in IEEE INFOCOM, Shanghai, China, 2011 • D. M. Lucantoni, "The BMAP/G/1 queue: A tutorial," in Models and techniques for Performance Evaluation of Computer and Communications Systems, L. D. a. R. Nelson, Ed., ed: Springer Verlag, 1993, pp. 330-58 • D. M. Lucantoni, G. L. Choudhury, and W. Whitt, "The transient BMAP/G/1 queue," Stoch. Mod., vol. 10, pp. 145-82, 1994 • D. M. Lucantoni, "Further transient analysis of the BMAP/G/1 queue," Stoch. Mod., vol. 14, pp. 461-78, 1998 Queueing Theory and Network Applications, Korea University, 2011

  47. References (Cont’d) • Heymen and Lucantoni, “Modeling Multiple IP Traffic Streams with Rate Limits,” IEEE/ACM Trans. On Networking, 11, No. 6, 2003 • Choudhury, Lucantoni and Whitt, “Multidimensional transform inversion with applications to the transient M/G/1 queue," Ann. Appl. Prob ., 4, No. 3, 719-740, 1994 • L. Takács, Introduction to the Theory of Queues. New York: Oxford University Press, 1962 Queueing Theory and Network Applications, Korea University, 2011

  48. Where to Find Additional References • Some Lucantoni papers can be downloaded from www.DLTconsulting.com • Follow the links to my Publications page (or just Google my name) • More references are listed in those papers Queueing Theory and Network Applications, Korea University, 2011

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