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Tracking in a Spaghetti Bowl: Monitoring Transactions Using Footprints Anima Anandkumar 1 , 2 Chatschik Bisdikian 2 Dakshi Agrawal 2 1 ECE Dept., Cornell University, Ithaca, NY 14853. 2 Networking Tech., IBM Watson Research, Hawthorne, NY 10532.

  1. Problem Formulation ? ? ? ? ? S 2 S 3 S 1 1 ? 2 S 0 S 4 3 ? � � � � Information in Footprint State & time stamp, optionally tokens (identifiers) Real-time match: all footprints may not yet have been generated Problem Statement Given footprints and transaction model, find maximum likelihood match between footprints and transactions Questions & Issues Complexity of maximum likelihood? Decentralized implementation? Probabilistic bounds on accuracy? Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 4 / 20

  2. Problem Formulation ? ? ? ? ? S 2 S 3 S 1 1 ? 2 S 0 S 4 3 ? � � � � Information in Footprint State & time stamp, optionally tokens (identifiers) Real-time match: all footprints may not yet have been generated Problem Statement Given footprints and transaction model, find maximum likelihood match between footprints and transactions Questions & Issues Complexity of maximum likelihood? Decentralized implementation? Probabilistic bounds on accuracy? Effect of transaction model ? Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 4 / 20

  3. Problem Formulation Transaction Model ? ? ? ? ? S 2 S 3 S 1 1 ? 2 S 0 S 4 3 ? � � � � Information in Footprint State & time stamp, optionally tokens (identifiers) Real-time match: all footprints may not yet have been generated Problem Statement Given footprints and transaction model, find maximum likelihood match between footprints and transactions Questions & Issues Complexity of maximum likelihood? Decentralized implementation? Probabilistic bounds on accuracy? Effect of transaction model ? Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 4 / 20

  4. Problem Formulation Transaction Model ? ? ? ? ? IID transitions of transactions S 2 S 3 S 1 1 ? 2 S 0 S 4 3 ? � � � � Information in Footprint State & time stamp, optionally tokens (identifiers) Real-time match: all footprints may not yet have been generated Problem Statement Given footprints and transaction model, find maximum likelihood match between footprints and transactions Questions & Issues Complexity of maximum likelihood? Decentralized implementation? Probabilistic bounds on accuracy? Effect of transaction model ? Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 4 / 20

  5. Problem Formulation Transaction Model ? ? ? ? ? IID transitions of transactions S 2 S 3 S 1 1 ? 2 S 0 Acyclic semi-Markov process S 4 3 ? � � � � Information in Footprint State & time stamp, optionally tokens (identifiers) Real-time match: all footprints may not yet have been generated Problem Statement Given footprints and transaction model, find maximum likelihood match between footprints and transactions Questions & Issues Complexity of maximum likelihood? Decentralized implementation? Probabilistic bounds on accuracy? Effect of transaction model ? Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 4 / 20

  6. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  7. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Maximum Likelihood for Two-state Systems S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  8. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Maximum Likelihood for Two-state Systems Transaction 1 Transaction 2 Y 0 W (1 , 1) . W (2 , 2) . S 0 S 1 W (2 , 1) . W (1 , 2) . Y 1 Footprint Node ccdf Node Reduction to bipartite minimum weight perfect matching Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  9. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Maximum Likelihood for Two-state Systems Transaction 1 Transaction 2 Y 0 W (1 , 1) . W (2 , 2) . S 0 S 1 W (2 , 1) . W (1 , 2) . Y 1 Footprint Node ccdf Node Reduction to bipartite minimum weight perfect matching Reduction to FIFO for a class of transition-time pdfs Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  10. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Maximum Likelihood for Two-state Systems Transaction 1 Transaction 2 Y 0 W (1 , 1) . W (2 , 2) . S 0 S 1 W (2 , 1) . W (1 , 2) . Y 1 Footprint Node ccdf Node Reduction to bipartite minimum weight perfect matching Reduction to FIFO for a class of transition-time pdfs Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  11. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Maximum Likelihood for Two-state Systems Transaction 1 Transaction 2 Y 0 W (1 , 1) . W (2 , 2) . S 0 S 1 W (2 , 1) . W (1 , 2) . Y 1 Footprint Node ccdf Node Reduction to bipartite minimum weight perfect matching Reduction to FIFO for a class of transition-time pdfs Worst Case Analysis of Maximum Likelihood Performance Uniform and exponential transition times are worst-case distributions Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  12. Summary of Results Problem Statement Maximum likelihood match between footprints & transactions Maximum Likelihood for Two-state Systems Transaction 1 Transaction 2 Y 0 W (1 , 1) . W (2 , 2) . S 0 S 1 W (2 , 1) . W (1 , 2) . Y 1 Footprint Node ccdf Node Reduction to bipartite minimum weight perfect matching Reduction to FIFO for a class of transition-time pdfs Worst Case Analysis of Maximum Likelihood Performance Uniform and exponential transition times are worst-case distributions Equal to reciprocal of number of unique perfect matchings Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 5 / 20

  13. Summary of Results (cont.) Maximum Likelihood for Multi-state Systems Series of bipartite matchings in high-level two-state systems Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 6 / 20

  14. Summary of Results (cont.) Maximum Likelihood for Multi-state Systems Series of bipartite matchings in high-level two-state systems Constructive proof of optimality: Decentralized rules Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 6 / 20

  15. Summary of Results (cont.) Maximum Likelihood for Multi-state Systems Series of bipartite matchings in high-level two-state systems Constructive proof of optimality: Decentralized rules Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 6 / 20

  16. Summary of Results (cont.) Maximum Likelihood for Multi-state Systems Series of bipartite matchings in high-level two-state systems Constructive proof of optimality: Decentralized rules 3 4 2 1 5 1 3 2 ({1,3},{2,4,5}) ({2},{3}) 4 2 5 3 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 6 / 20

  17. Summary of Results (cont.) Maximum Likelihood for Multi-state Systems Series of bipartite matchings in high-level two-state systems Constructive proof of optimality: Decentralized rules 3 4 2 1 5 1 3 2 ({1,3},{2,4,5}) ({2},{3}) 4 2 5 3 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 6 / 20

  18. Summary of Results (cont.) Maximum Likelihood for Multi-state Systems Series of bipartite matchings in high-level two-state systems Constructive proof of optimality: Decentralized rules 3 4 2 1 5 1 3 2 ({1,3},{2,4,5}) ({2},{3}) 4 2 5 3 Presence of Tokens For linear model, ML rule is still decentralized bipartite matching Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 6 / 20

  19. Outline Introduction 1 Two-state System 2 Multi-state System 3 Conclusion 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 7 / 20

  20. Two-state System S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  21. Two-state System S 0 S 1 1 2 Time S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  22. Two-state System S 0 S 1 1 2 Time S 0 ? S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  23. Two-state System S 0 S 1 1 2 Time S 0 ? S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  24. Two-state System S 0 S 1 1 2 3 Time S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  25. Two-state System S 0 S 1 1 2 3 Time S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  26. Two-state System S 0 S 1 1 2 3 Time S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  27. Two-state System S 0 S 1 1 2 3 Time S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  28. Two-state System S 0 S 1 1 2 3 Time S 0 S 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  29. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  30. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Y 0 Y 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  31. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 Y 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  32. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 Y 1 Footprint Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  33. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 Y 1 Footprint Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  34. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ f T ( Y 1 (1) − Y 0 (1))] . Y 1 Footprint Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  35. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ f T ( Y 1 (1) − Y 0 (2))] . − log[ f T ( Y 1 (1) − Y 0 (1))] . Y 1 Footprint Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  36. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ f T ( Y 1 (1) − Y 0 (2))] . − log[ f T ( Y 1 (1) − Y 0 (1))] . Y 1 Footprint Node ccdf Node Real Time Matching: All Footprints Not Yet Available Add virtual node: event that transaction is still resident at S 0 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  37. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ f T ( Y 1 (1) − Y 0 (2))] . − log[ f T ( Y 1 (1) − Y 0 (1))] . Y 1 Footprint Node ccdf Node Real Time Matching: All Footprints Not Yet Available Add virtual node: event that transaction is still resident at S 0 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  38. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (2))] . F T ( Y 1 (1) − Y 0 (2))] − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (1))] . F T ( Y 1 (1) − Y 0 (1))] Y 1 Footprint Node ccdf Node Real Time Matching: All Footprints Not Yet Available Add virtual node: event that transaction is still resident at S 0 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  39. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (2))] . F T ( Y 1 (1) − Y 0 (2))] − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (1))] . F T ( Y 1 (1) − Y 0 (1))] Y 1 Footprint Node ccdf Node Real Time Matching: All Footprints Not Yet Available Add virtual node: event that transaction is still resident at S 0 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  40. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (2))] . F T ( Y 1 (1) − Y 0 (2))] − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (1))] . F T ( Y 1 (1) − Y 0 (1))] Y 1 Footprint Node ccdf Node Real Time Matching: All Footprints Not Yet Available Add virtual node: event that transaction is still resident at S 0 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  41. Two-state System S 0 S 1 1 2 3 Time Count S 0 Time S 1 Batch Transaction 1 Transaction 2 Y 0 − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (2))] . F T ( Y 1 (1) − Y 0 (2))] − log[ ¯ − log[ f T ( Y 1 (1) − Y 0 (1))] . F T ( Y 1 (1) − Y 0 (1))] Y 1 Footprint Node ccdf Node Real Time Matching: All Footprints Not Yet Available Add virtual node: event that transaction is still resident at S 0 Maximum Likelihood ≡ Minimum Weight Perfect Match Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 8 / 20

  42. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  43. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  44. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 Y 1 Footprint Node ccdf Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  45. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 Y 1 Footprint Node ccdf Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  46. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 Y 1 Footprint Node ccdf Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  47. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 No. of unique matches =2 Prob. of correct ML match = 1 2 Y 1 Footprint Node ccdf Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  48. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 No. of unique matches =2 Prob. of correct ML match = 1 2 Y 1 Footprint Node ccdf Node Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  49. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 No. of unique matches =2 Prob. of correct ML match = 1 2 Y 1 Footprint Node ccdf Node Partial Batch Exponential = Worst Case Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  50. Worst Case Maximum Likelihood Performance Definition Minimum probability of correct match over transition time pdfs 1 Worst Case ML Performance= No. of Unique Matches Transaction 1 Transaction 2 Y 0 No. of unique matches =2 Prob. of correct ML match = 1 2 Y 1 Footprint Node ccdf Node Partial Batch Complete Batch Exponential = Worst Case Uniform, Exponential = Worst Case Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 9 / 20

  51. Comparison of Maximum Likelihood with FIFO Time S 0 S 1 FIFO = Match footprint with earliest unmatched transaction Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 10 / 20

  52. Comparison of Maximum Likelihood with FIFO Time S 0 S 1 FIFO = Match footprint with earliest unmatched transaction Always valid, Distribution free, Simpler rule than maximum likelihood Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 10 / 20

  53. Comparison of Maximum Likelihood with FIFO Time S 0 S 1 FIFO = Match footprint with earliest unmatched transaction Always valid, Distribution free, Simpler rule than maximum likelihood Can FIFO and maximum likelihood coincide? Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 10 / 20

  54. Comparison of Maximum Likelihood with FIFO Time S 0 S 1 FIFO = Match footprint with earliest unmatched transaction Always valid, Distribution free, Simpler rule than maximum likelihood Can FIFO and maximum likelihood coincide? Yes: Quadrangle Inequality 0.6 0.4 f T ( t 1 ) f T ( t 2 ) ≥ f T ( t 1 − τ ) f T ( t 2 − τ ) 0.2 0 0 1 2 3 4 5 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 10 / 20

  55. Outline Introduction 1 Two-state System 2 Multi-state System 3 Conclusion 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 11 / 20

  56. Recap of Transaction Model Transaction Model ? ? ? ? ? S 2 S 3 Acyclic semi-Markov process S 1 1 ? 2 S 0 � � S 4 ? 3 � � � � Definition of Semi-Markov Process Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  57. Recap of Transaction Model Transaction Model ? ? ? ? ? S 2 S 3 Acyclic semi-Markov process S 1 1 ? 2 IID transitions of transactions S 0 � � S 4 ? 3 � � � � Definition of Semi-Markov Process Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  58. Recap of Transaction Model Transaction Model ? ? ? ? ? S 2 S 3 Acyclic semi-Markov process S 1 1 ? 2 IID transitions of transactions S 0 � � S 4 ? 3 � � � � Definition of Semi-Markov Process Sequence of states visited is a Markov chain 1 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  59. Recap of Transaction Model Transaction Model ? ? ? ? ? S 2 S 3 Acyclic semi-Markov process S 1 1 ? 2 IID transitions of transactions S 0 � � S 4 ? 3 � � � � Definition of Semi-Markov Process Sequence of states visited is a Markov chain 1 Transition time depends only on states involved in transition 2 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  60. Recap of Transaction Model Transaction Model ? ? ? ? ? S 2 S 3 Acyclic semi-Markov process S 1 1 ? 2 IID transitions of transactions S 0 � � S 4 ? 3 � � � � Definition of Semi-Markov Process Sequence of states visited is a Markov chain 1 Transition time depends only on states involved in transition 2 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  61. Recap of Transaction Model Transaction Model ? ? ? ? ? S 2 S 3 Acyclic semi-Markov process S 1 1 ? 2 IID transitions of transactions S 0 � � S 4 ? 3 � � � � Definition of Semi-Markov Process Sequence of states visited is a Markov chain 1 Transition time depends only on states involved in transition 2 Maximum Likelihood Rule Maximize probability that each footprint is matched correctly to the unique transaction that generated it 1 , . . . , Y π Ns π ML π ML P ( Y π 1 [ˆ 1 , . . . , ˆ N s ] := arg max | Y 0 ) . N s π 1 ,..., π Ns Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  62. Recap of Transaction Model Transaction Model π 2 π 3 π 1 S 2 S 3 Acyclic semi-Markov process S 1 1 2 IID transitions of transactions S 0 S 4 π 4 3 � � � � Definition of Semi-Markov Process Sequence of states visited is a Markov chain 1 Transition time depends only on states involved in transition 2 Maximum Likelihood Rule Maximize probability that each footprint is matched correctly to the unique transaction that generated it 1 , . . . , Y π Ns π ML π ML P ( Y π 1 [ˆ 1 , . . . , ˆ N s ] := arg max | Y 0 ) . N s π 1 ,..., π Ns Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 12 / 20

  63. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  64. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  65. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model 1 2 3 4 S 0 S 1 S 2 S 3 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  66. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model 1 2 3 4 S 0 S 1 S 2 CCDF Node S 3 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  67. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model 1 2 3 4 S 0 S 1 S 2 S 3 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  68. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model 1 2 3 4 S 0 S 1 S 2 S 3 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  69. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Tree Model 1 2 3 4 S 0 S 1 S 2 S 3 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  70. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Tree Model 1 2 3 4 S 0 4 3 S 1 5 1 S 2 6 S 3 2 7 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  71. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Tree Model 1 2 3 4 S 0 4 3 S 1 5 1 S 2 6 S 3 2 7 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  72. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Tree Model 1 2 3 4 S 0 4 3 S 1 5 1 S 2 6 S 3 2 7 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  73. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Tree Model 1 2 3 4 S 0 4 3 S 1 5 1 S 2 6 S 3 2 7 S 4 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  74. Special Cases: Linear & Tree Models Linear Model With Snapshot of Footprints 4 3 2 1 ? ? ? ? ... S 0 S 1 S 3 S 4 Maximum Likelihood in Linear Model Tree Model 1 2 3 4 S 0 4 3 S 1 5 1 S 2 6 S 3 2 7 S 4 ML ≡ Series of Bipartite Matchings Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 13 / 20

  75. Acyclic Semi-Markov Process Acyclic Semi Markov Process: Two-Stage Systems No common imm. predecessor: States in end stage of different systems Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 14 / 20

  76. Acyclic Semi-Markov Process Acyclic Semi Markov Process: Two-Stage Systems No common imm. predecessor: States in end stage of different systems 3 4 2 1 5 ({2},{3}) 4 2 5 3 Anandkumar, Bisdikian, Agrawal Monitoring Transactions Using Footprints ACM SIGMETRICS ‘08 14 / 20

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