Numerical on general pipelines, computation of reservation table, state diagram, simple cycle, greedy cycle, minimum, average latency

1. Numerical on General
Pipelines
Dr. Prasenjit Dey

2. Dynamic Pipeline
Paths = s1s2s3s2s3s1s3s1
Reservation table for the function X
1 2 3 4 5 6 7 8
S1 X X x
S2 X x
S3 X x X

3. Latency
Latencies causing collision = {(6-1),(8-1),(8-6),(4-2),(5-3),(7-5),(7-3)}
Forbidden latency = {5, 7, 2, 4} = {2,4,5,7}
Maximum forbidden latency = max(Forbidden latency) = 7
Permissible latency = latency sequence = {1,3,6,8+}
1 2 3 4 5 6 7 8
S1 X1 X1 x1
S2 X1 x1
S3 X1 X1 X1

4. Collision free scheduling
 To avoid collisions, all tasks should be scheduled properly
 The objective is to achieve shortest average latency
between initiations without causing collisions
 Steps to achieve shortest average latency are
 Computation of collision vector
 Formation of State diagram
 Computation of greedy cycles
 Computation of Minimum Average Latency (MAL)

5. Collision Vectors
 A vector of m-bits, where m is Max(forbidden latency)
 Length of collision vector = maximum forbidden latency = 7
 The bit position (Bi)= 1 if latency i causes collision
 The bit position (Bi)= 0 if latency i do not cause collision
 Collision vector is = (cm,cm-1, …, c2, c1) = 1011010
 This is the Initial Collision Vector (ICV)
1 2 3 4 5 6 7 8
S1 X1 X1 x1
S2 X1 x1
S3 X1 X1 X1

6. State Diagrams
 A state diagram is formed from the collision vector
 It shows are permissible state transitions among successive
initiations
 Right shift ICV, then OR it with ICV
1011010
1011011 1111111
8+
3 6 8+ 1* 8+
3* 6
1011010 >>1
0101101
0101101
OR
1011010
1111111

7. Greedy Cycles
 Simple cycles: a latency cycle in which each state appears only
once
 Greedy cycles: Is a simple cycle whose edges are all made with
minimum latencies from their respective starting states
 Their average latencies must be lower than those of other simple
cycles
 Simple cycles were (1,8),(3,6),(6,8)(8)(3),(6)
 Greedy cycles = (3),(1,8)
 Average Latency(3)=3
 Average Latency(1,8)=4.5
 MAL (Minimum Average Latency)= 3

8. Schedule Optimization
 As greedy cycle not sufficient for optimality of MAL, lower
bound on MAL is required
 Optimize the reservation table to obtain the lower bound

9. Thank you