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 29 th 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 BufferServer Service protocol - First come, first served Customers inCustomers out

4. A discrete-time queueing system Geo/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

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

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

12. The critical scalings

13. The busy period (first passage time)

15. Moments and ‘defectiveness’

16. The probability distribution

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

21. Two important consequences

22. Mapping onto staircase polygons – the area problem

23. Arrivals Departures

24. A functional equation

27. 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)

28. The scaling approach

29. The q-series approach

30. The Brownian motion approach

32. The area distribution

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

36. time Rules Compact directed percolation

37. Critical condition Making the connection …

38. 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)

39. Brownian motion

40. 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

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

42. State dependent queues (balking)

43. 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