1. A gentle introduction to fluid and diffusion limits for queues Presented by: Varun Gupta April 12, 2006

2. Example : Tandem Queues Interarrival times at queue A are i.i.d. random variables Interarrival times at queue C are no more independent – they are ‘weakly’ dependent Very difficult to analyze queues with correlated input/service processes.

3. Example : Non-stationary Queues The arrival process fluctuates over a day – high load during day, low load during night Difficult to analyze queues with 2 environment states Numerical methods exist if the arrival process has certain Markovian properties Exact solutions for more complex environment processes are intractable

4. Example : Non-stationary Queues Observation: The soujorn times in environment states are much larger than the service and interarrival times Question: What is the limiting queue length distribution as the mean environment state sojourn times become infinity?

5. A model for non-stationary queues Define E, a reference environment process, as a random process taking values in {1,2,..,m}. E n are a family of slowly-changing environment process defined by time scaling E as N n  {N n (t): t  0} is the queue length process obtained by letting the system evolve as a GI/GI/1 queue with mean arrival rate 1/ i and mean service rate 1/  i when E n is in state i.

6. Fluid Approximation

8. The Functional SLLN X i : i.i.d. random variables with mean m, finite variance  2 Let Question: How does the plot of first n partial sums behave as n increases?

9. n=10

10. n=100

11. n=10000

12. The Functional SLLN Define the continuous parameter stochastic process Functional SLLN Note that while SLLN says that at each t, Y n (t) converges to mt, FSLLN says that entire sample paths of the sequences of stochastic processes Y n converge to the non-random process mt.

13. Fluid limit for the non-stationary queue Theorem : If then, where Y is the stochastic fluid process with environment process E, deterministic flow rate r i = i -  i in state i and initial content Y(0)=y.

14. Example: MMPP/M/1 queue Take the reference environment process, E, to be the following 2-state continuous time Markov chain In state H the queue behaves like an M/M/1 with service rate  and arrival rate H ( H >  ) In state L the queue behaves like an M/M/1 with service rate  and arrival rate L ( L <  ) Fluid limit:

15. n=10

16. n=100

17. n=1000

18. Problems with fluid limits

20. Functional Central Limit theorem Define the ‘centered’ partial sums of X i as Central Limit Theorem Define the continuous time process Question: How does Z n (t) behave as n increases?

21. n=100

22. n=1000

23. n=10000

24. n=1,000,000

25. Functional Central Limit theorem FCLT (Donsker’s Theorem) where B(t) is the standard Brownian motion (with drift coefficient 0 and diffusion coefficient 1) Brownian motion with drift coefficient  and diffusion coefficient  2 is a real valued stochastic process with stationary and independent increments having continuous sample paths where

26. Functional Central Limit theorem While CLT says that for any t, FCLT also shows that Z n (t) converges to an (a.s.) continuous stochastic process with independent increments. Note that just as CLT is a refinement of the SLLN, the FCLT is a refinement of the FSLLN and hence is more accurate.

27. Diffusion limit for the non-stationary queue Theorem: If then where Z is a zero-drift Brownian motion with diffusion coefficient  2 z depending on the limiting fluid process, Y, and environment process, E, as follows –If Y(t)=0, then  2 z = 0 –If Y(t)>0 and E(t)=I, then  2 z = i 3   i 2 +  i 3  Si 2

28. Diffusion limit for the non-stationary queue Proof: Lemma: Let X i be a sequence of positive random variables. Define Let denote the counting process with X i as the interarrival times. Then,

29. Diffusion limit for the non-stationary queue Proof contd. Using the lemma on last slide, the counting process of arrivals, V A n (t) in environment i converges to Similarly, the counting process for service completions converges to Taking the difference of the above Brownian motions gives the diffusion limit.

30. Some implications of fluid and diffusion limits 1.The fluid limits only depend on the means of the service and arrival processes. Therefore, the variability of the environment process affects the queues more than the variability of the arrival and service processes within each environment state. 2.The limiting distribution does not depends on moments higher than the second moments of arrival and service processes. 3.The fluid and diffusion limits still hold when the arrival and service processes are not i.i.d but weakly dependent. This is a consequence of the fact that FSLLN and FCLT hold under much weaker conditions.

31. Conclusions Fluid and Diffusion limits are powerful tools that produce asymptotically exact distributions by appropriately scaling time and/or space for otherwise intractable problems by stripping away unnecessary details of the statistical processes involved. Engineering Applications –Buffer Provisioning for Network Switches and Routers –Scheduling Service for Multiple Sources

32. Thank you! (Please register for Random Distance Run today! Early registration still on)