1 / 58

Time-Dependent Infinite-Server General Queueing Networks

Time-Dependent Infinite-Server General Queueing Networks. Michael R. Taaffe University of Minnesota November 12, 2001 OR590 OPERATIONS RESEARCH COLLOQUIUM SERIES Penn State University. umn.edu/~taaffe. Time-Dependent Processes. Managing Time-Dependent Processes.

zeheb
Download Presentation

Time-Dependent Infinite-Server General Queueing Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Time-Dependent Infinite-Server General Queueing Networks Michael R. Taaffe University of Minnesota November 12, 2001 OR590 OPERATIONSRESEARCH COLLOQUIUM SERIES Penn State University umn.edu/~taaffe

  2. Time-Dependent Processes

  3. Managing Time-Dependent Processes

  4. Wireless Cellular-Phone Communications

  5. Iowa State University Wireless Network Access Points

  6. Time-dependent Ph-type queues • Time-dependent Pht arrival process • Time-dependent Pht service process • Time-dependent Markovian routing

  7. Time-dependent Ph-type queues - Today • Time-dependent Pht arrival process • Time-dependent Pht service process • Time-dependent Markovian routing

  8. Background… Mt / Mt / ∞

  9. Mt / Mt / ∞Moment Dif’l Equations (MDE’s)

  10. E[# busy servers] Mt / Mt / ∞Moment Dif’l Equations (MDE’s) System Departure rate System Arrival Rate Closed set of MDE’s!

  11. Background Cont’d… Mt / Mt / 1

  12. Mt / Mt / 1Moment Dif’l Equations (MDE’s)

  13. E[# busy servers] Mt / Mt / 1Moment Diffr’l Equations (MDE’s) System Departure rate System Arrival Rate Not Closed!

  14. Mt / Mt / 1Moment Diffr’l Equations (MDE’s) System Departure rate System Arrival Rate Pseudo-Closed!

  15. E[# busy servers] Mt / Mt / 1 / KMoment Dif’l Equations (MDE’s) System Arrival Rate System Departure rate Pseudo-Closed!

  16. E[# busy servers] Mt / Mt / C / CMoment Diffr’l Equations (MDE’s) System Departure rate System Arrival Rate Pseudo-Closed!

  17. Modeling Limitations with the Exponential

  18. Analysis Problems withNon-Exponential • Simple, first-order, linear differential equations have “no memory” (exponential) • “General” distributions have memory! • Not clear how to represent or interpret time-dependent “general” distributions

  19. Initial-state distribution: New initial-state distribution: state-occupation times … … … … … … jumps around guided by A until absorbed Ph-type process … Markov the lion … … … Absorbing state

  20. … … … … … … jumps around guided by A until absorbed Initial-state distribution: Ph-type process … Markov the lion … … … Absorbing state

  21. The (A,λ)Pht-type Process

  22. New initial-state distribution: Initial-state distribution: state-occupation times … … … … … … jumps around guided by A(t) until absorbed Pht-type process … Markov the lion … … … Absorbing state

  23. The (A(t),λ(t)) Pht-type Process

  24. The Pht / Pht / ∞ } -The Random Process

  25. Kolmogorov Forward Equations – Stationary Case

  26. Kolmogorov Forward Equations – Nonstationary case

  27. Moment Differential Equations (MDE’s) Not closed!

  28. Notation

  29. Partial-Moment Differential Equations (PMDE’s) Not closed!

  30. Arrival Differential Equations (ADE’s) Closed!

  31. {{First PMDE’s} U {ADE’s}} Closed!

  32. Matrix form

  33. Stationary System at Equilibrium

  34. Need higher-order moments; e.g., Variance… more notation

  35. Higher-Order PMDE’s Not closed!

  36. 2cdPMDE’s Not closed!

  37. Cross-Product PMDE’s (CPMDE’s)

  38. The Pht / Pht / ∞CPMDE’s Not closed!

  39. Pht / Pht / ∞ {ADE’s U 1st PMDE’s U CPMDE’s} is Closed! Quasi-Closure!

  40. Pht / Pht / ∞ MAPLE Code www.iems.nwu.edu/~nelsonb/PhPhInf

  41. Pht / Pht / ∞ Two-Node Network

  42. Pht Arrival Process

  43. Node-1 Service Process

  44. Node-2Pht Service Process

  45. 2-Node Service Network

  46. Single-Node Equivalent

  47. The [Pht / Pht / ∞]KNetworkMAPLE Code www.iems.nwu.edu/~nelsonb/PhPhInf

  48. The Expected Number at each Node and in the Network

  49. The Variance of the Number at Each Node and in the Network

  50. cdf of the time spent in the system for entities arriving at time t

More Related