1. Asaf Cohen (joint work with Rami Atar) Department of Mathematics University of Michigan Financial Mathematics Seminar University of Michigan March 11, 2015 1

2. Contents The Differential Game The Model Varadhan’s Lemma The Approximation 2

3. Contents The Differential Game The Model Varadhan’s Lemma The Approximation 2

4. The Model Basic definitions 3 size of the i-th buffer arrival process finite 2 nd moment controlled service process finite 2 nd moment controlled rejection process buffers

5. The Model Basic definitions 3 Moderate-deviation rate parameters Arrival rates Departure rates The system is critically loaded size of the i-th buffer arrival process finite 2 nd moment controlled service process finite 2 nd moment controlled rejection process buffers Proportion of time that server i is busy

6. The Model Scaling 4

7. (1) and (2) and 5 The sequence satisfies the moderate-deviation principle with rate parameters and rate function. That is, for every and every closed and open sets one has The Model Sufficient condition:

8. The Model Cost and value function 6 Alternative costs: Scaling implies Technical difficulties Non-stationary solutions Value: Cost: Leads to a stationary robust control discount factor

9. The Model Why Moderate-Deviation? 7 While there are many diffusion-scaling papers, there are very few papers with the large/moderate-deviations scaling. In diffusion scaling the probability of overflow is of order 1, while in the large/moderate deviation the order is very low. In the large/moderate deviation we get a robust control. Unlike the large-deviation, which suffers from high complexity. Under the moderate-deviation, we solve the problem completely. TO ADD COLORS LATER ON AND TO SIMPLIFY THE POLICY LIKE IN THE 20MIN

10. The Model State space collapse 8 where is called the workload. Cost

11. The Model Varadhan’s Lemma (intuition for the differential game) 9 Suppose that (with support ) satisfies the large deviation principle with the good rate function and let be a continuous function. Assume further (to simplify the proof) that is compact. Then, Intuition:

12. The Model State space collapse 10 where Cost Deterministic differential game Cost: Value: Anticipating Elliot-Kalton is related to

13. 11 The Model Assume that and that Assumption 1 holds. Then, where To his end, we analyze the differential game… (We will refer only to the one dimensional case here and work with. )

14. Contents The Differential Game The Model Varadhan’s Lemma The Approximation 12

15. The Differential Game Intuition 1313 Cost: Value: Anticipating Elliot-Kalton

16. Properties of the game 14 Explicit expression for the value function. Optimal strategy for the minimizer The Differential Game

17. Properties of the game 15 Simple controls for the maximizer that achieves : - Stopping immediately as the dynamics hits zero,. - until the first time the dynamics drop below and then decreases until the dynamics hit zero (solves PDE, independent of the minimizer choice). Also, the terminal time is smaller than. Under this control, if then the minimizer would prefer to reject immediately! The Differential Game

18. Contents The Differential Game The Model Varadhan’s Lemma The Approximation 16

19. Varadhan’s Lemma 17 Suppose that (with support ) satisfies the Large deviation principle with the good rate function and let be a continuous function. Assume further (to simplify the proof) that is compact. Then, Intuition:

20. Varadhan’s Lemma (proof) 18 Lower bound: Fix. There is s.t. So, and Since is arbitrary Varadhan’s Lemma

21. Varadhan’s Lemma (proof) 19 Upper bound: For every there is s.t. Since is compact there is a finite cover ( independent of ) Varadhan’s Lemma

22. Contents The Differential Game The Model Varadhan’s Lemma The Approximation 20

23. The Approximation Deterministic differential game 21 Cost: Value: anticipating

24. 22 We prove it by showing that for every sequence one has and that, there is a sequence of policies for which The Approximation We will refer only to the one dimensional case here, and work with … Assume that and that Assumption 1 holds. Then, where

25. Lower bound 2323 Notice that depends on the control. Now we consider a sequence of controls. So, we consider. But how? We would like to center around. The Approximation less than

26. Lower bound 24 Take (3) Fix. There is an interval s.t. (1) s.t. stopping afterwards is too expensive. (2) Divide into small intervals: (4) Divide, the interval into small time intervals. Recall that termination time of the game is smaller than. (5) Now, continue centering around the second part of. There is an interval s.t. The Approximation

27. Lower bound 25 We focus on ‘s for which for a specific. Recall that there are only finitely many such. The Approximation is linear

28. Lower bound 26 So, The Approximation

29. Service (from workload to buffers): W.o.l.g. assume that Asymptotically optimal policy 27 Rejections: If overflow occurs (happens with low prob.) Otherwise, reject from buffer. Low priority: let the cheapest buffer, which is not almost full (up to ) to be filled. Serve the others with rate higher than The Approximation

30. Asymptotically optimal policy 28 Assume and the buffers have the same sizes The Approximation cheapest buffer

31. 29

32. The Model State space collapse 3 where Cost Deterministic differential game Cost: Value: anticipating is related to