1 / 43

Exact solutions for first-passage and related problems in certain classes of queueing system

Exact solutions for first-passage and related problems in certain classes of queueing system. Michael J Kearney School of Electronics and Physical Sciences University of Surrey June 29 th 2006. Presentation outline. Introduction to the Geo/Geo/1 queue Some physical examples

varick
Download Presentation

Exact solutions for first-passage and related problems in certain classes of queueing system

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. Exact solutions for first-passage and related problems in certain classes of queueing system Michael J Kearney School of Electronics and Physical Sciences University of Surrey June 29th 2006

  2. Presentation outline • Introduction to the Geo/Geo/1 queue • Some physical examples • Mathematical analysis • Link to the Brownian motion problem • Further problems

  3. Queueing schematic Customers in Customers out Buffer Server Service protocol - First come, first served

  4. A discrete-time queueing systemGeo/Geo/1

  5. Small scale queue dynamics

  6. Large scale queue dynamics

  7. Brownian motion with drift

  8. Some questions of interest • Time until the queue is next empty • Busy period (first passage time) statistics • Probability that the busy period is infinite • Maximum queue length during a busy period • Extreme value statistics (correlated variables) • Cumulative waiting time during a busy period • Area under the curve

  9. Areas of application • Abelian sandpile model • Compact directed percolation • Lattice polygons • Cellular automaton road traffic model

  10. Cellular automaton model Queueing representation Nagel and Paczuski (1995) The link to road traffic

  11. The critical scalings

  12. The busy period (first passage time)

  13. Moments and ‘defectiveness’

  14. The probability distribution

  15. Maximum length L Lifetime T The maximum (extreme) length

  16. Two important consequences

  17. Mapping onto staircase polygons – the area problem

  18. Arrivals Departures

  19. A functional equation

  20. Three-fold strategy • A scaling approach based on the dominant balance method, following Richard (2002) • Consider the singularity structure of the generating function G(1,y) as y tends to unity, following Prellberg (1995) • Consider the equivalent problem for Brownian motion, following Kearney and Majumdar (2005)

  21. The scaling approach

  22. The q-series approach

  23. The Brownian motion approach

  24. The area distribution

  25. Guillemin and Pinchon (1998) The M/M/1 queue Taking the continuous time limit (but discrete customers)

  26. Rules Compact directed percolation time

  27. Critical condition Making the connection …

  28. Summary of key CDP results • Probability that the avalanches are infinite • critical condition • Distribution of avalanches by duration (perimeter) • Distribution of avalanches by size (area) Dhar and Ramaswamy (1989) Rajesh and Dhar (2005)

  29. Brownian motion

  30. Conclusions • New results for discrete and continuous-time queues, and possibly deeper results • Large area scaling behaviour for CDP determined exactly at all points in the phase diagram • Exact solution for the v = 1 cellular automaton traffic model of Nagel and Paczuski • A solvable model of extreme statistics for strongly correlated variables

  31. Departures Time Partition polygon queues Queue length N = 5 T = 7 Time

  32. State dependent queues (balking)

  33. Some references • On a random area variable arising in discrete-time queues and compact directed percolation • M J Kearney 2004 J.Phys. A: Math. Gen., 37 8421 • On the area under a continuous time Brownian motion • M J Kearney and S N Majumdar 2005 J.Phys. A: Math. Gen., 38 4097 • A probabilistic growth model for partition polygons and related structures • M J Kearney 2004 J.Phys. A: Math. Gen., 37 3749

More Related