Seunghwan Jung and James R. Morrison KAIST , Department of Industrial and Systems Engineering

1. 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. Presentation Overview • Introduction • System Description • Equilibrium Probabilities Under the LBFS Policy • Equilibrium Probabilities Under the FBFS Policy • Conclusion

3. Introduction • Jackson network is one of the rare class of network that possess closed • form equilibrium probability distributions. Server 1 Server 2 Customers arrive Customers arrive Customers exit Customers exit < Jackson network >

4. Introduction • Except for some classes of networks, few networks possess closed form • equilibrium probability distributions. < General reentrant network [1] > • [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. Introduction • Obtain closed form equilibrium probabilities. • Allows complete characterization of the steady state behavior. • < Closed Lu-Kumar network >

6. System Description: Network Model • Two stations : σ1 and σ2 • Buffers : b1, b2 , b3 , b4 • Service time for a customer in buffer bi : exponential with rate μi • N trapped customers circulate within the network • A closed reentrant queueing network

7. System Description: Last Buffer First Served • Non-idling , preemptive • Gives priority b1 over b4 and b3 over b2 • A closed reentrant queueing network

8. System Description: First Buffer First Served • Non-idling , preemptive • Gives priority b4 over b1 and b2over b3 • A closed reentrant queueing network

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 b1, b2, b3, b4 at time t • Uniformization : Get Discrete time Markov chain • Steady state probability of state S : Πs 1 N-1 0 0 • A closed reentrant queueing network Transition diagram under LBFS

10. Equilibrium Probabilities under LBFS • To find equilibrium probability : Balance equations Π=ΠP • “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 . Transition diagram under LBFS

11. Equilibrium Probabilities under LBFS To specify our main idea, we redefine the state as below :

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. Equilibrium Probabilities under LBFS • Overall steps for obtaining closed form solutions (continued) Step 4: Using the balance equations, all Xk[n] are expressed in terms of X0[0] Step 5: Summing all probabilities and setting them equal to 1 to get X0[0]

14. Equilibrium Probabilities 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 b1, b2, b3, b4 at time t • Uniformization : GetDiscrete time Markov chain • Steady state probability of state S : Πs Transition diagram under FBFS • A closed reentrant queueing network

15. Equilibrium Probabilities under FBFS • To find equilibrium probability : Balance equations Π=ΠP • “Flow in” = “Flow out” Initial conditions So, assuming that we know , we can obtain . Recursively, we can express whole steady state probabilities in terms of initial conditions. Transition diagram under FBFS

16. Equilibrium Probabilities under FBFS To specify our main idea, we redefine the state as below :

17. Equilibrium Probabilities under FBFS • Overall steps for obtaining closed form solutions Step 1: Investigating X0[n], we obtain relationship below: Step 2: Using relationship between Xk[m] and Xk-1[n], we obtain X1[n]. Step 3: Recursively, we can obtain

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 Xk[n] in terms of X0[0].(Toeplitz matrix structure)

19. Equilibrium Probabilities under FBFS Step 5: Summing all probabilities and setting them equal to 1 to get X0[0] • Note: Not a complete closed form

20. Concluding Remarks • LBFS : Indeed obtained a closed form solution • FBFS : Enough structure to reduce the computational complexity •To obtain equilibrium probabilities by “Π=ΠP”, 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.