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 Tandem fluid queues with On/Off source 11 22 nn 0 Exp(  ) X ~ G >  1 >  2 > … >  n 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 1.Non-iterative method for Laplace transform of fluid level at each queue 2.Closed-form exact expressions for moments of fluid level A’: # moments of X needed = ☺

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

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

6 Fluid level in a queue with On/Off source  =1 0 Exp(  ) X ~ G Fluid level Time L OFF = Fluid level | source OffL ON = Fluid level | source On L = L OFF 1 { source Off} + L ON 1 {source On}

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

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

9 Finally…  =1 0 Exp(  ) X ~ G = L OFF 1 {source Off} + L ON 1 {source On} = L OFF + ( -1)X e. 1 {source On} = M(  )/G(( -1)X)/1 workload + ( -1)X e.1 {source On} L d d First k moments of L completely determined by first (k+1) moments of X

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

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

12 Busy period analysis [Boxma,Dumas 98]  =1 0 Exp(  ) X ~ G  0 Exp(  ) B Fluid level Time U1U1 U2U2 UnUn V1V1 V2V2 VnVn = (U 1 +U 2 +…+U n )+(V 1 +V 2 +…+V n ) = [1/( -1)+1](V 1 +V 2 +…+V n ) = [1/( -1)+1] M(  )/G(( -1)X)/1 busy period B d First k moments of B completely determined by first k moments of X

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

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

15 Putting it together 11 22 nn 0 Exp(  ) X ~ G (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 X completely determine first k moments of fluid level at each queue

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

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

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

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

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