1. IEEE ICCA 2010 – Xiamen, June 11, 2010 On Closed Form Solutions for Equilibrium Probabilities in the Closed Lu-Kumar Network under Various Buffer Priority Policies Seunghwan Jung and James R. Morrison KAIST, Department of Industrial and Systems Engineering IEEE ICCA 2010 Xiamen, China June 11, 2010

2. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 2 Presentation Overview Introduction System Description Equilibrium Probabilities Under the LBFS Policy Equilibrium Probabilities Under the FBFS Policy Conclusion

3. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 3 Introduction Server 1Server 2 Customers arrive Customers exit Jackson network is one of the rare class of network that possess closed form equilibrium probability distributions.

4. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 4 Introduction Except for some classes of networks, few networks possess closed form equilibrium probability distributions. [1] James R. Morrison, “Implementation of a Fluctuation Smoothing Production Control Policy in IBM’s 200mm Wafer Fab”, European Control Conference, pp. 7732-7737, 2005.

5. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 5 Introduction Obtain closed form equilibrium probabilities. Allows complete characterization of the steady state behavior.

6. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 6 System Description: Network Model Two stations : σ 1 and σ 2 Buffers : b 1, b 2, b 3, b 4 Service time for a customer in buffer b i : exponential with rate μ i N trapped customers circulate within the network A closed reentrant queueing network

7. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 7 System Description: Last Buffer First Served Non-idling, preemptive Gives priority b 1 over b 4 and b 3 over b 2 A closed reentrant queueing network

8. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 8 System Description: First Buffer First Served Non-idling, preemptive Gives priority b 4 over b 1 and b 2 over b 3 A closed reentrant queueing network

9. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 9 Equilibrium Probabilities under LBFS System state at time t : S(t)={w(t),x(t),y(t),z(t)} w(t),x(t),y(t),z(t) : Number of customers in buffers b 1, b 2, b 3, b 4 at time t Uniformization : Get Discrete time Markov chain Steady state probability of state S : Π s A closed reentrant queueing network Transition diagram under LBFS 1 N-1 00

10. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 10 Equilibrium Probabilities under LBFS Transition diagram under LBFS To find equilibrium probability : Balance equations Π = Π P “Flow in” = “Flow out” So, assuming that we know, we can obtain. So we can express in terms of Recursively, we can express whole steady state probabilities in terms of initial condition.

11. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 11 Equilibrium Probabilities under LBFS To specify our main idea, we redefine the state as below :

12. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 12 Equilibrium Probabilities under LBFS Overall steps for obtaining closed form solutions Step 1: We make the equation involving only one type of signal by combining given equations Step 2: Taking z-transform and inverting it give a closed form solution for the signal Step 3: Plugging the closed form solution into the other balance equations gives closed form solutions for them

13. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 13 Equilibrium Probabilities under LBFS Overall steps for obtaining closed form solutions (continued) Step 4: Using the balance equations, all X k [n] are expressed in terms of X 0 [0] Step 5: Summing all probabilities and setting them equal to 1 to get X 0 [0]

14. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 14 Equilibrium Probabilities under FBFS A closed reentrant queueing network Transition diagram under FBFS System state at time t : S(t)={w(t),x(t),y(t),z(t)} w(t),x(t),y(t),z(t) : Number of customers in buffers b 1, b 2, b 3, b 4 at time t Uniformization : Get Discrete time Markov chain Steady state probability of state S : Π s

15. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 15 Equilibrium Probabilities under FBFS Transition diagram under FBFS Recursively, we can express whole steady state probabilities in terms of initial conditions. To find equilibrium probability : Balance equations Π = Π P “Flow in” = “Flow out” Initial conditions So, assuming that we know, we can obtain.

16. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 16 Equilibrium Probabilities under FBFS To specify our main idea, we redefine the state as below :

17. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 17 Equilibrium Probabilities under FBFS Overall steps for obtaining closed form solutions Step 1: Investigating X 0 [n], we obtain relationship below: Step 2: Using relationship between X k [m] and X k-1 [n], we obtain X 1 [n]. Step 3: Recursively, we can obtain

18. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 18 Equilibrium Probabilities under FBFS Step 4: By symmetry, we get the inverse transforms for the lower region Step 5: Using remaining balance equations, we express all X k [n] in terms of X 0 [0].(Toeplitz matrix structure)

19. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 19 Equilibrium Probabilities under FBFS Step 5: Summing all probabilities and setting them equal to 1 to get X 0 [0] Note: Not a complete closed form

20. IEEE ICCA 2010 – Xiamen, June 11, 2010 - 20 Concluding Remarks LBFS : Indeed obtained a closed form solution FBFS : Enough structure to reduce the computational complexity To obtain equilibrium probabilities by “Π = Π P ”, we have to inverse (N+1) 2 ╳ (N+1) 2 matrix. Future works Attempting to obtain a closed-form expression for the inverse of the Toeplitz matrix from the FBFS case. Extend the structure to more general cases.