Ó 1998 Menascé & Almeida. All Rights Reserved.1 Part VI System-level Performance Models for the Web (Book, Chapter 8)

Ó 1998 Menascé & Almeida. All Rights Reserved.2 Configuration Plan Investment Plan Personnel Plan Understanding the Environment Workload Characterization Workload Model Validation and Calibration Workload Forecasting Performance Prediction Cost Prediction Valid Model Cost Model Developing a Cost Model Performance Model Cost/Performance Analysis

Ó 1998 Menascé & Almeida. All Rights Reserved.3 Part VI: Learning Objectives Characterize system-level models Present State Transition Diagram (STD) technique Show solution for simple models Show general solution to STDs Show how to obtain performance metrics from the solution of STDs

Ó 1998 Menascé & Almeida. All Rights Reserved.4 System-level Models System is seen as a black box. Only its input-output characteristics are considered. Inputs: arrivals of requests Output: throughput. Throughput, X 0 (k) (requests/sec) requests in the system, k

Ó 1998 Menascé & Almeida. All Rights Reserved.5 System-level Example A Web server receives 10 requests/sec. The maximum number of requests in the server is 3. Requests that arrive and find three requests being processed are rejected.

Ó 1998 Menascé & Almeida. All Rights Reserved.6 System-level Example The measured throughput as a function of the number of requests is:

Ó 1998 Menascé & Almeida. All Rights Reserved.7 System-level Example: a few questions Q1: What is the probability that an incoming request is rejected? Q2: What is the average number of requests in execution? Q3: What is the average throughput of the Web server? Q4: What is the average time spent by an HTTP request in the Web server?

Ó 1998 Menascé & Almeida. All Rights Reserved.8 System-level Example Characterize the Web server by its state, i.e., the number k of requests in the Web server. Assumptions made: –homogeneous workload (or single class): all requests are statistically equivalent, only the number of req. counts –memoryless (markovian): how the system arrived at state k does not matter. –operational equilibrium: no. requests at beginning of interval = no. request at the end.

Ó 1998 Menascé & Almeida. All Rights Reserved.9 System-level Example req/sec 12 req/sec 15 req/sec 16 req/sec state arrivals completions of requests State Transition Diagram (STD)

Ó 1998 Menascé & Almeida. All Rights Reserved.10 System-level Example Assume we are able to find the values of: –p k = probability that (or fraction of time where) there are k requests in the Web server. Question: can we answer all the questions posed before as a function of the p k ’s?

Ó 1998 Menascé & Almeida. All Rights Reserved.11 System-level Example: a few questions Q1: What is the probability that an incoming request is rejected? A: It is the probability that an arriving HTTP request finds 3 requests already being processed. The answer is then p 3.

Ó 1998 Menascé & Almeida. All Rights Reserved.12 System-level Example: a few questions Q2: What is the average number of requests in execution? A: using the definition of average: N req = 0 x p x p x p x p 3

Ó 1998 Menascé & Almeida. All Rights Reserved.13 System-level Example: a few questions Q3: What is the average throughput of the Web server? A: again, using the definition of average: X = 0 x p x p x p x p 3 throughput value at each state

Ó 1998 Menascé & Almeida. All Rights Reserved.14 System-level Example: a few questions Q4: What is the average time spent by an HTTP request in the Web server? A: It will be a function of the average number of requests, N req, and the average throughput X. Can be computed by applying Little’s law.

Ó 1998 Menascé & Almeida. All Rights Reserved.15 System-level Example: computing the p k ’s req/sec 12 req/sec 15 req/sec 16 req/sec use the flow in = flow out principle: the flow into a set of states is equal to the flow out of this set of states in equilibrium.

Ó 1998 Menascé & Almeida. All Rights Reserved.16 System-level Example: computing the p k ’s req/sec 12 req/sec 15 req/sec 16 req/sec flow in = flow out 12 x p 1 = 10 x p 0

Ó 1998 Menascé & Almeida. All Rights Reserved.17 System-level Example: computing the p k ’s req/sec 12 req/sec 15 req/sec 16 req/sec flow in = flow out 15 x p 2 = 10 x p 1

Ó 1998 Menascé & Almeida. All Rights Reserved.18 System-level Example: computing the p k ’s req/sec 12 req/sec 15 req/sec 16 req/sec flow in = flow out 16 x p 3 = 10 x p 2

Ó 1998 Menascé & Almeida. All Rights Reserved.19 System-level Example: computing the p k ’s Putting it all together: 12 x p 1 = 10 x p 0 p 1 = 10/12 p 0 15 x p 2 = 10 x p 1 p 2 = 10/15 p 1 = 10x10 p 0 15x12 16 x p 3 = 10 x p 2 p 3 = 10/16 p 2 = 10x10x10 p 0 16x15x12

Ó 1998 Menascé & Almeida. All Rights Reserved.20 System-level Example: computing the p k ’s Putting it all together: p 1 = 10/12 p 0 ; p 2 = 10x10 p 0 ; and 15x12 p 3 = 10x10x10 p 0 16x15x12 But, the Web server has to be in one of the four states at any time. So, p 0 + p 1 + p 2 + p 3 = 1.

Ó 1998 Menascé & Almeida. All Rights Reserved.21 System-level Example: computing the p k ’s Solving for p 0 and then for the other p k ’s we get:

Ó 1998 Menascé & Almeida. All Rights Reserved.22 System-level Example: answering the questions Q1: What is the probability that an incoming request is rejected? A: It is the probability that an arriving HTTP request finds 3 requests already being processed. The answer is then p 3 = = 12.7%.

Ó 1998 Menascé & Almeida. All Rights Reserved.23 System-level Example: answering the questions Q2: What is the average number of requests in execution? A: using the definition of average: N req = 0 x x x x = requests

Ó 1998 Menascé & Almeida. All Rights Reserved.24 System-level Example: answering the questions Q3: What is the average throughput of the Web server? A: again, using the definition of average: X = 0 x x x x = requests/sec.

Ó 1998 Menascé & Almeida. All Rights Reserved.25 System-level Example: answering the questions Q4: What is the average time spent by an HTTP request in the Web server? A: It is a function of the average number of requests, N req, and the average throughput X. We need Little’s Law to answer this question.

Ó 1998 Menascé & Almeida. All Rights Reserved.26 Little’s Law PUB avg. number people in the pub = avg. departure rate from the pub X avg. time spent at the pub

Ó 1998 Menascé & Almeida. All Rights Reserved.27 Little’s Law Web server avg. number requests in the server = avg. departure rate from the server X avg. time spent at the server req/sec req?

Ó 1998 Menascé & Almeida. All Rights Reserved.28 System-level Example: answering the questions Q4: What is the average time spent by an HTTP request in the Web server? A: From Little’s Law, R = N req / X = / = sec.

Ó 1998 Menascé & Almeida. All Rights Reserved.29 Practice Drill Practice Drill Using Models for Decision Making What happens if the maximum number of allowed TCP connections changes from 3 to 10? What if the load on the server doubles? What is the impact of a threefold increase in the server’s capacity?

Ó 1998 Menascé & Almeida. All Rights Reserved.30 Types of System-level Models Population Size: –infinite –finite Service Rate: –fixed –variable Maximum Queue Size: –unlimited –limited

Ó 1998 Menascé & Almeida. All Rights Reserved.31 Types of System-level Models (population size) Infinite Population: the number of clients is very large. The rate at which requests arrive to the system does not depend on the number of requests being processed in the system. –e.g., requests arriving from the Internet to a public Web server.

Ó 1998 Menascé & Almeida. All Rights Reserved Types of System-level Models (infinite population) SERVER completed requests X 1 2 M infinite population arrival rate (requests/sec) internal requests N req Open Model

Ó 1998 Menascé & Almeida. All Rights Reserved.33 Your Web server is used for e-commerce. The company will announce a new product in the Fall and expects the number of current sales to increase by 50%. Will the server handle the load? –What is the new response time? –What is the server utilization? Practice Drill Practice Drill Using Models for Decision Making

Ó 1998 Menascé & Almeida. All Rights Reserved.34 What is the meaning of “ Will the server handle the load?” What is the new response time? What is the server utilization? Practice Drill Practice Drill Using Models for Decision Making

Ó 1998 Menascé & Almeida. All Rights Reserved.35 Types of System-level Models (population size) Finite Population: the number of clients M is limited. The rate at which requests arrive to the system depends on how many have already arrived. –e.g., requests arriving to an intranet Web server from a known number of clients within the organization (Intranet).

Ó 1998 Menascé & Almeida. All Rights Reserved.36 Types of System-level Models (finite population) SERVER completed requests X 1 2 M finite population M think time Z internal requests N req arrival rate (requests/sec) Closed Model

Ó 1998 Menascé & Almeida. All Rights Reserved.37 Company X’s intranet collects sales data from 100 stores and processes reorder data on hundreds of products electronically. The company will merge with company Y that has 150 stores. Does the existing Web site have enough capacity to handle the merged company’s demand? Practice Drill Practice Drill Using Models for Decision Making

Ó 1998 Menascé & Almeida. All Rights Reserved.38 Types of System-level Models (service rate) Fixed Service Rate: the throughput does not vary with the number of requests being processed. Throughput (requests/sec) requests in the system

Ó 1998 Menascé & Almeida. All Rights Reserved.39 Types of System-level Models (service rate) Variable Service Rate: the throughput depends on the number of requests being processed. Throughput (requests/sec) requests in the system

Ó 1998 Menascé & Almeida. All Rights Reserved.40 Types of System-level Models (maximum queue size) Unlimited Queue Size: all arriving requests are queued for service. No requests are rejected! Limited Queue Size: requests that find more than W requests waiting for service are rejected. n=W? Yes No

Ó 1998 Menascé & Almeida. All Rights Reserved.41 Types of System-level Models

Ó 1998 Menascé & Almeida. All Rights Reserved.42 System-level models: Methodology Determine proper representation for the state of system being modeled (numb. of requests in system, k) Determine set of feasible states Determine possible transitions between states, considering possible events (arrival, completion of request)

Ó 1998 Menascé & Almeida. All Rights Reserved.43 System-level models: Methodology (2) For each possible state transition, determine transition rate, looking at event that caused the transition Use flow in = flow out principle to write down equations relating values of p k ; remember that sum of all p k s is one.

Ó 1998 Menascé & Almeida. All Rights Reserved.44 System-level models: Methodology (3) Solve for p k s and use them to compute performance metrics: –utilization U –average throughput X –average number of requests N req –average response time R –fraction of requests that are lost because of queue overflow P loss

Ó 1998 Menascé & Almeida. All Rights Reserved.45 Generalized System-level Models      ..   Generalized System-level Models can be solved using the flow in = flow out principle

Ó 1998 Menascé & Almeida. All Rights Reserved.46 Generalized System-level Models

Ó 1998 Menascé & Almeida. All Rights Reserved.47 System-level Models Example A Web server receives 30 requests/sec. Its throughput function is given below. The server queue is limited to five requests. What is the server utilization, avg. throughput, avg. no. requests, avg. response time, and fraction of lost requests?

Ó 1998 Menascé & Almeida. All Rights Reserved.48 System-level Models Example (cont’d) Using the Generalized System-level model equations we get that From Little’s Law, avg. response time = avg. no requests/ avg. throughput = 1.85 / 28.4 = sec.

Ó 1998 Menascé & Almeida. All Rights Reserved.49 System-level Models Example (cont’d)

Ó 1998 Menascé & Almeida. All Rights Reserved.50 System-level Models Example (cont’d)

Ó 1998 Menascé & Almeida. All Rights Reserved.51 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.52 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.53 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.54 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.55 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.56 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.57 Practical computation tools

Ó 1998 Menascé & Almeida. All Rights Reserved.58 Part VI: Summary System-level models view a server as a black box. Only its arrival process and throughput functions are relevant. State Transition Diagrams (STDs) can be used to find the probability that k requests are in the server. Use the flow in = flow out principle. Little’s Law can be used to compute the response time from the average number of requests and from the throughput.