Fluid level in tandem queues with an On/Off source

1. Fluid level in tandem queues with an On/Off source VARUN GUPTA Carnegie Mellon University Joint work with PETER HARRISON Imperial College

2. Why fluid queues? • A simple model for shared resources with high arrival/service rates – e.g. telecommunication networks • Markov modulated fluid queues can describe correlated input processes – e.g. self-similar traffic • Often, the only tractable approximation

3. 1 2 n Tandem fluid queues with On/Off source X ~ G  0  > 1 > 2 > … > n Exp() • Q: First k moments of fluid level at queue n? • PRIOR WORK: Mostly numerical and iterative • Markov modulated queues – Martingale methods [Kella Whitt 92], Sample path SDEs [Brocket Gong Guo 99] • General On periods – approximate X by PH distribution and solve for moments of fluid level iteratively [Field Harrison 07] • Q’:How many moments of X do you need? (All? kn? k+n?) • OUR CONTRIBUTIONS • Non-iterative method for Laplace transform of fluid level at each queue • Closed-form exact expressions for moments of fluid level • A’: # moments of X needed = ☺

4. 1 2 n 1 1 Analysis Roadmap X ~ G  0 Exp() STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() STEP 3: Making the method non-iterative

5. 1 2 n 1 1 Analysis Roadmap X ~ G  0 Exp() STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() STEP 3: Making the method non-iterative

6. =1 Fluid level in a queue with On/Off source X ~ G  0 Exp() Fluid level Time LOFF = Fluid level | source Off LON = Fluid level | source On L = LOFF1{source Off} + LON1{source On}

7. =1 stationary M/G/1 workload with arrival rate  and job sizes ~ (-1)X d LOFF = X ~ G  STEP 1a: Analysis of LOFF 0 Exp() ~ (-1)X i.i.d.  = 1 Contract ON periods Contracted time Time Exp() Exp()

8. =1 X ~ G  STEP 1b: Analysis of LON 0 Exp() LON (By PASTA) LOFF Fluid level Time T stationary excess/age in a renewal process with i.i.d. renewals according to X LON = LOFF + (-1)T = LOFF + (-1)Xe d

9. =1 Finally… X ~ G  0 Exp() L = LOFF1{source Off} + LON1{source On} = LOFF + (-1)Xe . 1{source On} = M()/G((-1)X)/1 workload + (-1)Xe .1{source On} d d First k moments of Lcompletely determined by first (k+1) moments of X

10. 1 2 n 1 1 Analysis Roadmap X ~ G  0 Exp() STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level  STEP 3: Making the method non-iterative

11. 1 2 n 1 1 Analysis Roadmap X ~ G  0 Exp() STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level  STEP 3: Making the method non-iterative

12. =1 Busy period analysis [Boxma,Dumas 98] B X ~ G   0 0 Exp() Exp() Fluid level Time U1 V1 U2 V2 Un Vn B = (U1+U2+…+Un)+(V1+V2+…+Vn) = [1/(-1)+1](V1+V2+…+Vn) = [1/(-1)+1] M()/G((-1)X)/1 busy period d First k moments of Bcompletely determined by first k moments of X

13. 1 1 2 n 1 Analysis Roadmap X ~ G  0 Exp() STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level (k+1) moments of X (k+1) moments of busy period   STEP 3: Making the method non-iterative

14. 1 1 2 n 1 Analysis Roadmap X ~ G  0 Exp() STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level (k+1) moments of X (k+1) moments of busy period   STEP 3: Making the method non-iterative

15. 1 2 n Putting it together X ~ G  0 Exp() (k+1) moments of X give.. .. k moments of fluid level and (k+1) of busy period .. k moments of fluid level and (k+1) of busy period .. k moments of fluid level and (k+1) of busy Period. First (k+1) moments of Xcompletely determine first k moments of fluid level at each queue

16. 1 1 2 n 1 Analysis Roadmap X ~ G  0 Exp() (k+1) moments of X  k moments of fluid level at all queues STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level (k+1) moments of X (k+1) moments of busy period   STEP 3: Making the method non-iterative

17. 1 1 2 n 1 Analysis Roadmap X ~ G  0 Exp() (k+1) moments of X  k moments of fluid level at all queues STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level (k+1) moments of X (k+1) moments of busy period   STEP 3: Making the method non-iterative

18. n 1 n n-1 n-1 Bn-1 n-1 X ~ G Getting rid of busy period iteration 0  Exp() 0 Exp() Bn-1 = function of Bn-2 = … 1 > 2 > … > n-1Bn-1 is identical to: Bn-1 n-1 X ~ G 0  Exp() 0 Exp() Can obtain Bn-1 in one step (no need to iterate)

19. 1 1 2 n 1 Analysis Roadmap X ~ G  0 Exp() (k+1) moments of X  k moments of fluid level at all queues STEP 1: Fluid level at queue 1 STEP 2: Busy period analysis X ~ G X ~ G B 1   0 0 0 Exp() Exp() Exp() (k+1) moments of X k moments of fluid level (k+1) moments of X (k+1) moments of busy period   STEP 3: Making the method non-iterative

20. Contributions/Conclusion • Non-iterative method to obtain fluid level transform and moments in a tandem network • Show that first (k+1) moments of On period determine first k moments of fluid level at each queue • Method generalizes to a wider class of input processes