1. CSE 807 Bounds on Performance1

2. 2 Significance of Bounds Provide valuable insight into the primary factors affecting the performance of computer system. Can be computed quickly and therefore serve as a first cut modeling technique. Several alternatives can treated together.

3. CSE 807 Bounds on Performance3 Model Parameters K, The number of service centers. D max, Max service demand at any server. D, Sum of the service demands at the centers. Type customer (batch, terminal, and transaction) Z, Average think time.

4. CSE 807 Bounds on Performance4 Asymptotic Bounds Requests may be served by one or more service centers Finite population model (Closed system)

5. CSE 807 Bounds on Performance5 Trans. Workloads Recall: U k = X k S k, and if we denote arrival rate as, then X k = V k => U k = D k, where D k = V k S k So, throughput bound is the smallest arrival rate sat at which any center saturates.

6. CSE 807 Bounds on Performance6 Trans. Workloads (cont’d) => U max ( ) = D max, < 1 => sat = 1/ D max Note: System is unstable if > sat For response time: D < R( )

7. CSE 807 Bounds on Performance7 Two Extreme Cases Best: No customer ever interferes with any other. So, System response time of each customer = D. Worst: n customers arrive together every n/ time units. Customers must Q and thus experience large response time. Note: For any pessimistic bound forecasted, it is possible to pick a batch size n sufficiently large that the bound is exceeded, regardless of how small the arrival rate is.

8. CSE 807 Bounds on Performance8 Batch and Terminal Workloads Consider the heavy load case: U k (N) = X (N) D k < 1 => X (N) < 1 / D max Now, consider the light load Case: At the Extreme, a single customer alone in system attains a throughput of 1/(D+Z) As more Customers added to the system, there are 2 boundaries situations:

9. CSE 807 Bounds on Performance9 Batch and Terminal Workloads (cont’d) Smallest possible throughput: For each customer is 1/(ND+Z) ; for N customers. We have N / (ND+Z) Largest possible throughput occurs when no time is spent queueing: For each customer is 1/(D+Z), and N customers We have N / (D+Z)

10. CSE 807 Bounds on Performance10 Batch and Terminal Workloads (cont’d) Note: Asymptotic Bounds on system throughput summarized: N * (population size) crossover Pt. If N < N *, Optimistic Bound applies. If N > N *, Pessimistic (Heavy Load) Bounds Applies

11. CSE 807 Bounds on Performance11 Batch and Terminal Workloads (cont’d) We can obtain bounds on response time R(N) by transforming our throughput bounds using Little’s law. We begin by rewriting the previous equation: Inverting each component to express the bounds on R(N) yields:

12. CSE 807 Bounds on Performance12 Workload Type bounds batch terminal transaction batch terminal transaction X R Summary of Asymptotic Bounds

13. CSE 807 Bounds on Performance13 Asymptotic Bounds on Performance ND X(N)X(N) N*N* N Batch throughput: 1

14. CSE 807 Bounds on Performance14 Asymptotic Bounds on Performance (cont’d) D ND R(N)R(N) N*N* N ND max Batch Response Time: 1

15. CSE 807 Bounds on Performance15 X(N)X(N) 1N*N* Terminal Throughput: N Asymptotic Bounds on Performance

16. CSE 807 Bounds on Performance16 D ND R(N)R(N) N*N* N ND max -Z Terminal Response Time: 1 Asymptotic Bounds on Performance

17. CSE 807 Bounds on Performance17 Example of a Modeling Study: IBM Equip. Through a combination of this information, “live” measurements of existing 3790 systems, and benchmark experiments on two of the systems (3790 and 8140), the following service demand were determined: Service demands, seconds SystemCPUdisk 3790 (observed) 8130 (estimated) 8140 (estimated) 4.6 5.1 3.1 4.0 1.9

18. CSE 807 Bounds on Performance18 Example of a Modeling Study: IBM Equip. (cont’d) Terminals CPUDisk Case Study Model

19. CSE 807 Bounds on Performance19 Example of a Modeling Study: IBM Equip. (cont’d) K, the number of service centers (2); D max, the largest service demand (4.6 seconds for the 3790, 5.1 for the 8130) and 3.1 for the 8140); D, the sum of the service demands (8.6, 7.0, and 5.0, respectively); the type of customer class (terminal); Z, the average think time (an estimate of 60 seconds was used).

20. CSE 807 Bounds on Performance20 N Throughput: 5 1015202530 0.10 0.20 0.30 X(N)X(N) 8130 3790 8140 Asymptotic Bounds in the Case Study

21. CSE 807 Bounds on Performance21 N Response Time: 5 1015202530 10 20 30 40 R(N)R(N) 8130 3790 8140 Asymptotic Bounds in the Case Study

22. CSE 807 Bounds on Performance22 48121620 N X(N)X(N) Throughput: 0.10 0.20 0.30 Secondary and Tertiary Asymptotic Bounds

23. CSE 807 Bounds on Performance23 48121620 N R(N)R(N) 10 20 30 40 Response Time: D Secondary and Tertiary Asymptotic Bounds

24. CSE 807 Bounds on Performance24 48121620 N X(N)X(N) Throughput: 0.10 0.20 0.30 Improving primary Original Improving secondary Relative Effects of Reducing Various Service Demands

25. CSE 807 Bounds on Performance25 Relative Effects of Reducing Various Service Demands 48121620 N R(N)R(N) 10 20 30 40 Response Time: Improving primary Original Improving secondary