CSE 807 Bounds on Performance 1

CSE 807 Bounds on Performance 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.

CSE 807 Bounds on Performance 3

Model Parameters

• K, The number of service centers.

• Dmax, 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.

CSE 807 Bounds on Performance 4

Asymptotic Bounds

• Requests may be served by one or more service centers

• Finite population model (Closed system)

CSE 807 Bounds on Performance 5

Trans. Workloads

• Recall:

Uk= XkSk, and if we denote arrival rate as , then

Xk = Vk

=> Uk= Dk, where Dk= VkSk

So, throughput bound is the smallest arrival rate sat at which any center saturates.

CSE 807 Bounds on Performance 6

Trans. Workloads (cont’d)

=> Umax() = Dmax, < 1

=> sat = 1/ Dmax

Note: System is unstable if > sat

For response time: D < R()

CSE 807 Bounds on Performance 7

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.

CSE 807 Bounds on Performance 8

Batch and Terminal Workloads

Consider the heavy load case:Uk (N) = X (N) Dk < 1=> X (N) < 1 / Dmax

Now, consider the light loadCase:

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:

CSE 807 Bounds on Performance 9

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 customersWe have N / (D+Z)

CSE 807 Bounds on Performance 10

Batch and Terminal Workloads (cont’d)

Note: Asymptotic Bounds on system throughput summarized:

),min()( )(1

)( max ZDN

DZNDN NX

N* (population size) crossover Pt.

max

*

D

ZDN

If N < N*, Optimistic Bound applies.If N > N*, Pessimistic (Heavy Load) Bounds Applies

CSE 807 Bounds on Performance 11

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:

),min(max

1)( ZD

NDZNR

NZND

N

Inverting each component to express the bounds on R(N) yields:

NDNRZNDDor

D NZND

NZNR

NZD

)(),max(:

),max(

max

)(max

CSE 807 Bounds on Performance 12

),min()(max

11DD

ND NX

),min()(max

1DZD

NZND

N NX

WorkloadType

bounds

max

1)( DX

NDNRNDD )(),max( max

NDNRZNDD )(),max( max

)(RD

batch

terminal

transaction

batch

terminal

transaction

X

R

Summary of Asymptotic Bounds

CSE 807 Bounds on Performance 13

Asymptotic Bounds on Performance

ND

X(N)

N*

N

D1

max

1D

Batch throughput:

1

CSE 807 Bounds on Performance 14

Asymptotic Bounds on Performance (cont’d)

D

ND

R(N)

N*

N

NDmax

Batch Response Time:

1

CSE 807 Bounds on Performance 15

X(N)

D1

max

1D

1 N*

ZNDN

ZDNTerminal Throughput:

N

Asymptotic Bounds on Performance

CSE 807 Bounds on Performance 16

D

ND

R(N)

N*

N

NDmax-Z

Terminal Response Time:

1

Asymptotic Bounds on Performance

CSE 807 Bounds on Performance 17

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

System CPU disk

3790 (observed)8130 (estimated)8140 (estimated)

4.65.13.1

4.01.91.9

CSE 807 Bounds on Performance 18

Example of a Modeling Study:IBM Equip. (cont’d)

Terminals

CPU Disk

Case Study Model

CSE 807 Bounds on Performance 19

Example of a Modeling Study:IBM Equip. (cont’d)

•K, the number of service centers (2);•Dmax , 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).

CSE 807 Bounds on Performance 20

N

Throughput:

5 10 15 20 25 30

0.10

0.20

0.30

X(N)

8130

3790

8140

Asymptotic Bounds in the Case Study

CSE 807 Bounds on Performance 21

N

Response Time:

5 10 15 20 25 30

10

20

30

40

R(N)

8130

3790

8140

Asymptotic Bounds in the Case Study

CSE 807 Bounds on Performance 22

4 8 12 16 20

N

X(N)

Throughput:

0.10

0.20

0.30

1

1

D

2

1

D

3

1

D

ZD

N

Secondary and Tertiary Asymptotic Bounds

CSE 807 Bounds on Performance 23

4 8 12 16 20

N

R(N)

10

20

30

40

Response Time:

D

ZND 1 ZND 2

ZND 3

Secondary and Tertiary Asymptotic Bounds

CSE 807 Bounds on Performance 24

4 8 12 16 20

N

X(N)

Throughput:

0.10

0.20

0.30 Improving primary

Original

Improving secondary

Relative Effects of Reducing Various Service Demands

CSE 807 Bounds on Performance 25

Relative Effects of Reducing Various Service Demands

4 8 12 16 20

N

R(N)

10

20

30

40

Response Time:

Improving primary

Original

Improving secondary