1. Constructions of Optical Priority Queues Jay Cheng Department of Electrical Engineering & Institute of Communications Engineering National Tsing Hua University Hsinchu, Taiwan 30013, R.O.C. E-mail: jcheng@ee.nthu.edu.tw The slides are made by using MS PowerPoint 2007

2. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

3. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

4. Motivation As photons cannot be easily stopped, stored, and forwarded, the only known way to store optical packets without converting them into other media is to direct optical packets through optical Switches and fiber Delay Lines (SDL) so that packets appear at the right place and at the right time Recent advances in SDL constructions have shown that there exist systematic methods for constructing various types of optical queues, including buffered switches, FIFO multiplexers, FIFO queues, LIFO queues, priority queues, linear compressors, non-overtaking delay lines, and flexible delay lines The purpose of the slides is to help the audience understand our SDL constructions of priority queues (for simplicity, only the single-input single-output case is presented here). One of the well-recognized key problems for all-optical packet-switched networks is the lack of optical buffer for contention resolution among packets competing for the same resources The key idea in the SDL constructions is to design different types of optical buffers for different types of network elements that do not need buffers with random-access capability

5. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

6. Basic Assumptions Time in all optical links is slotted and synchronized so that a packet can be transmitted within a time slot. Packets are of the same size. A fiber delay line with delay d is an optical link that requires d time slots for a packet to traverse through. An M x M crossbar switch is a network element with M input links and M output links that realizes all the M! permutations between its inputs and outputs.

7. Control input link Arrival link Buffer size Departure link Loss link Definition of Priority Queues Priority queue with buffer B B a(t) c(t) d(t) (t) (P1) Flow conservation (P2) Non-idling (P3) Priority departure (P4) Maximum buffer usage (P5) Priority loss Arriving packets from the arrival link are either stored in the buffer or transmitted through the departure link or the loss link. If c(t)=1, then there is always a departing packet if there are packets in the buffer or there is an arriving packet. If there is a departing packet at time t, then the departing packet is the one with the highest priority among all the packets in the queue at time t. If c(t)=0, then there is a lost packet only when buffer is full and there is an arriving packet. If there is a lost packet at time t, then the lost packet is the one with the lowest priority among all the packets in the queue at time t.

8. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

9. The Operations of a Priority Queue (B=9) c(t) RAM

10. The Operations of a Priority Queue (B=9) t = 1 c(t)c(t)=0

11. The Operations of a Priority Queue (B=9) (P1) Flow conservation: Arriving packets from the arrival link are either stored in the buffer or transmitted through the departure link or the loss link c(t)=0 1 t = 1

12. The Operations of a Priority Queue (B=9) t = 1t = 2 1 2 (P2) Non-idling: If c(t)=1, then there is always a departing packet if there are packets in the buffer or there is an arriving packet (P3) Priority departure: If there is a departing packet at time t, then the departing packet is the one with the highest priority among all of the packets in the queue at time t c(t)=0c(t)=1

13. The Operations of a Priority Queue (B=9) t = 2t = 3 1 1 2 c(t)=1 c(t)=0

14. The Operations of a Priority Queue (B=9) t = 3t = 4 1 2 3 c(t)=0

15. The Operations of a Priority Queue (B=9) t = 4t = 5 1 2 3 4 c(t)=0

16. The Operations of a Priority Queue (B=9) t = 5t = 6 1 2 3 4 445 c(t)=0

17. The Operations of a Priority Queue (B=9) t = 6t = 7 1 2 3 1 5 4 2 3 4 6 5 c(t)=0

18. The Operations of a Priority Queue (B=9) t = 7t = 8 2 3 4 5 6 5 1 6 7 c(t)=0

19. The Operations of a Priority Queue (B=9) t = 8t = 9 2 3 4 8 7 6 1 5 c(t)=0

20. The Operations of a Priority Queue (B=9) t = 9t = 10 2 3 4 8 7 6 1 5 8 9 c(t)=0

21. The Operations of a Priority Queue (B=9) t = 10t = 11 2 3 4 5 7 6 1 5 9 8 6 10 9 (P4) Maximum buffer usage: If c(t)=0, then there is a lost packet only when buffer is full and there is an arriving packet (P5) Priority loss: If there is a lost packet at time t, then the lost packet is the one with the lowest priority among all of the packets in the queue at time t 7 8 c(t)=0

22. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

23. Our Previous SDL Construction Theorem If for all, then the following construction can be operated as a priority queue with buffer

24. An Example of Our Previous SDL Construction (B=9) 1 23 654 87 9 2 34 76 5 98 10 1 t = 0 t = 1 (P1) Flow conservation is satisfied c(t) = 0 The key is to view empty time slots as fictitious packets that have priorities lower than those of real packets. Moreover, the priorities among the fictitious packets are decreasing in the order of their arrival times. Note that both the packet with the highest priority and the packet with the lowest priority appear at the inputs of the sorter at the beginning of each time slot.

25. An Example of Our Previous SDL Construction (B=9) 1 42 376 59 8 3 48 6 t = 1t = 2 (P2) Non-idling and (P3) Priority departure are satisfied c(t) = 0c(t) = 1 2 5 7 10 9 1

26. An Example of Our Previous SDL Construction (B=9) 1 24 637 85 9 5 74 9 t = 2t = 3 c(t) =1 c(t) = 0 2 3 8 6 10 1

27. An Example of Our Previous SDL Construction (B=9) 1 52 374 69 8 1 62 485 710 9 3 t = 3 t = 4 c(t) = 0

28. An Example of Our Previous SDL Construction (B=9) 1 23 548 67 9 1 23 659 78 10 4 t = 4t = 5 c(t) = 0

29. An Example of Our Previous SDL Construction (B=9) 1 32 465 87 9 1 32 576 98 10 4 t = 5t = 6 c(t) = 0

30. An Example of Our Previous SDL Construction (B=9) 1 23 457 69 8 2 34 568 710 9 1 t = 6t = 7 c(t) = 0

31. An Example of Our Previous SDL Construction (B=9) 1 42 356 87 9 1 42 367 98 10 5 t = 7t = 8 c(t) = 0

32. An Example of Our Previous SDL Construction (B=9) 1 24 536 79 8 1 24 536 710 9 8 t = 8t = 9 c(t) = 0

33. An Example of Our Previous SDL Construction (B=9) 1 42 653 87 9 1 42 653 98 10 7 t = 9t = 10 c(t) = 0

34. An Example of Our Previous SDL Construction (B=9) 1 23 465 79 8 1 23 476 810 9 5 t = 10t = 11 (P4) Maximum buffer usage and (P5) Priority loss are satisfied c(t) = 0

35. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

36. (A2) When a packet enters the i th delay line, there are at least d i -1 packets with priorities higher than its priority, and there are at least d i -1 packets with priorities lower than its priority An Improved SDL Construction The key to the success of our previous SDL construction is we make sure that the following condition is satisfied: (A1) Both the highest priority packet and the lowest priority packet appear at the inputs of the sorter at the beginning of each time slot Our improvement is made possible by the observation that the following condition (A2) implies that (A1) holds: In other words, all the packets stored in the fiber delay lines cannot be either the highest priority packet or the lowest priority packet until they appear at the inputs of the sorter

37. An Improved SDL Construction

39. The Concept of the Cells of the Fiber Delay Lines m m M-2m d 1 =1 d 2 =2 d m =m d M+1-m =m d M =1 d M-1 =2

40. The (i,j) th cell, i=1,2,…M, j=1,2,…,d i, is the j th cell of the i th fiber delay line. Example (M=7, m=3) (4,4) (1,1) (2,2)(2,1) (3,3)(3,2)(3,1) (4,3)(4,2)(4,1) (7,1) (6,2)(6,1) (5,3)(5,2)(5,1) i=1 i=2 i=3 i=4 i=5 i=6 i=7 m=3 M-2m=1 j=1 j=3 j=4 j=2 We view a fiber delay line with delay d as a sequential buffer that consists of d cells with each cell capable of holding a packet. A packet entering the i th delay line at time t will be stored in the (i,j) th cell at time t+j-1.

41. 0 1 11 111 011 0 00 000 1 2 4 8 13 14 16 3 6 7 12 5 10 11 9 15 Tagged packet We consider a packet that enters the i th delay line at some time t 0 and call it the tagged packet 0 Let p i,j (t)= 1, if the priority of the packet in cell (i,j) at time t is higher than or equal to the priority of the tagged packet 0, otherwise. Properties of p i,j (t)

42. At time t-1 Properties of p i,j (t) 0 1 11 111 011 0 00 000 1 2 4 8 13 14 16 3 6 7 12 5 10 11 9 15 0 1 11 111 011 0 00 000 1 2 4 8 13 14 16 3 6 7 12 5 10 11 9 15 1 3 5 8 10 12 16 At time t-1 At time t i=4 1 1 1 1 0 0 0 15 13 9 14 11 Monotonically decreasing

43. The total number of packets in the cells of the j th column that have properties higher than or equal to that of the tagged packet at time t

44. 0 1 11 111 111 0 00 000 1 3 5 8 10 12 16 2 4 9 14 6 7 13 11 15 0 1 11 111 011 0 00 000 1 2 4 8 13 14 16 3 6 7 12 5 10 11 9 15 At time t-1 At time t

45. The total number of packets in the cells of the j th column that have properties higher than or equal to that of the tagged packet at time t

46. 0 1 11 111 111 0 00 000 1 3 5 8 10 12 16 2 4 9 14 6 7 13 11 15 0 1 11 111 011 0 00 000 1 2 4 8 13 14 16 3 6 7 12 5 10 11 9 15 At time t-1 At time t

47. 0 1 11 111 111 0 00 000 1 3 5 8 10 12 16 2 4 9 14 6 7 13 11 15 0 1 11 111 011 0 00 000 1 2 4 8 13 14 16 3 6 7 12 5 10 11 9 15 At time t-1 At time t p 1 (t) is the total number of packets in the first column at time t that have priorities higher than or equal to that of the tagged packet These M packets can only come from the arriving packet at time t and those packets stored at time t-1 in the cells (i,i) and (M+1-i,i) for i=1,2,…,m, and (i,d i ) for i=m+1,m+2,…,M-m Arriving packet M-2m

50. Outline Motivation Definition of Priority Queues The Operations of a Priority Queue (Electronic Random Access Memory (RAM) is used to Buffer Packets) Our Previous SDL Construction An Improved SDL Construction Conclusion and Future Works

51. We have considered the construction of an optical priority queue with a single (M+1) × (M+1) switch and M fiber delay lines, and have shown that such a construction can be operated as a priority queue with O(M 3 ) buffer size Further studies are needed to go beyond the O(M 3 ) buffer size and march toward the exponential upper bound O(2 M ) for a construction of an optical priority queue with a single (M+1) × (M+1) switch and M fiber delay lines

52. Thank you!