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PDQ is a library of functions that helps you to express and solve performance questions about computer systems using the abstraction of queues. The queueing paradigm is a natural choice because, whether big (a web site) or small (a laptop), all computer systems can be represented as a network or circuit of buffers and a buffer is a type of queue.

As a performance analyst, there are several things I really like about using PDQ in R; as opposed to the other programming languages: C, Perl, Python, etc. It enables you to:

- easily import (large) data with a variety formats
- perform sophisticated statistical analysis
- extract input parameters for a PDQ model
- construct and execute the PDQ model within R
- plot the PDQ output and compare it with the original data
- test your ideas in the R console and save the best into a script

In applying this approach, you could find yourself using a number of R library packages. To improve clarity in your modeling script, you might like to identify clearly which routines belong to PDQ; especially if you’re new to PDQ and not familiar with all the functions.

R syntax for naming function dependency is the same as Perl. The :: operator is used for explicitly exported names. It also avoids conflict between packages the export different functions with the same name. The ::: operator is used for access to functions that are not exported in the package namespace.

Let’s look at the above steps in the context of an example based on load testing data. A key point to observe here is how the performance data and the performance model play together to provide validation of the measurements.

#### Performance data

We begin by importing the load test data from measurements of an application intended for a three-tier architecture.

library(ineq)

library(pdq)

# Read in the performance measurements

gdat <- read.csv("/Users/njg/.../gcap.dat",header=TRUE)

Even though the *ineq* package is part of base R functionality, I’ve loaded it explicitly so as to name its functions explicitly. This will also provide a contrast with explicitly named functions from the PDQ package.

> gdat

Vusr Xgps Rms Uweb Uapp Udbm

1 1 24 26.0 0.21 0.08 0.04

2 2 48 26.0 0.41 0.13 0.05

3 4 85 29.3 0.74 0.20 0.05

4 7 100 44.7 0.95 0.23 0.05

5 10 115 66.0 0.96 0.22 0.06

6 20 112 140.0 0.97 0.22 0.06

The columns are respectively the client load, measured throughput, response time (in milliseconds), and system utilization on each of the three tiers.

#### Statistical analysis

We can now perform various kinds of statistical analysis on these data.

> summary(gdat)

Vusr Xgps Rms Uweb Uapp

Min. : 1.000 Min. : 24.00 Min. : 26.00 Min. :0.2100 Min. :0.0800

1st Qu.: 2.500 1st Qu.: 57.25 1st Qu.: 26.82 1st Qu.:0.4925 1st Qu.:0.1475

Median : 5.500 Median : 92.50 Median : 37.00 Median :0.8450 Median :0.2100

Mean : 7.333 Mean : 80.67 Mean : 55.33 Mean :0.7067 Mean :0.1800

3rd Qu.: 9.250 3rd Qu.:109.00 3rd Qu.: 60.67 3rd Qu.:0.9575 3rd Qu.:0.2200

Max. :20.000 Max. :115.00 Max. :140.00 Max. :0.9700 Max. :0.2300

Udbm

Min. :0.04000

1st Qu.:0.05000

Median :0.05000

Mean :0.05167

3rd Qu.:0.05750

Max. :0.06000

More significantly, we can use R statistical functions to derive appropriate parameters for a PDQ model.

# Apply Little's law to get mean service times + CoVs

Sweb <- mean(gdat$Uweb/gdat$Xgps)

Sapp <- mean(gdat$Uapp/gdat$Xgps)

Sdbm <- mean(gdat$Udbm/gdat$Xgps)

Csw <- ineq::var.coeff(gdat$Uweb/gdat$Xgps)

Csa <- ineq::var.coeff(gdat$Uapp/gdat$Xgps)

Csd <- ineq::var.coeff(gdat$Udbm/gdat$Xgps)

s1 <- sprintf("System: %6s %6s %6s\n", "Web","App","DBMS")

s2 <- sprintf("Mean S: %6.4f %6.4f %6.4f\n", Sweb, Sapp, Sdbm)

s3 <- sprintf("CoV S: %6.4f %6.4f %6.4f\n", Csw, Csa, Csd)

cat("\n",s1,s2,s3)

In particular, we calculate the average service times on each tier (second row) by applying Little’s law.

System: Web App DBMS

Mean S: 0.0088 0.0024 0.0008

CoV S: 0.0411 0.1989 0.5271

#### PDQ model

As shown in Figure 1, the service times for each of the three tiers in the load-test platform can be represented as queueing resources in PDQ.

There is a finite number of requests allowed in the system corresponding to the load clients or virtual users that range between N = 1 and N = 20 Vusers, represented by the octagonal box in Figure 1. Using the diagram, we set up the following PDQ model. Note the use of explicitly named functions from the PDQ library

# Plotting variables

xc <- 0 # Vuser loads

yc <- 0 # PDQ throughputs

rc <- 0 # PDQ response times

# Define and solve the PDQ model

for(n in 1:max(gdat$Vusr)) {

pdq::Init("Three-Tier Model")

pdq::CreateClosed("httpGETs", TERM, as.numeric(n), 0.028)

pdq::CreateNode("WebServer", CEN, FCFS)

pdq::CreateNode("AppServer", CEN, FCFS)

pdq::CreateNode("DBMServer", CEN, FCFS)

pdq::SetDemand("WebServer", "httpGETs", Sweb)

pdq::SetDemand("AppServer", "httpGETs", Sapp)

pdq::SetDemand("DBMServer", "httpGETs", Sdbm)

pdq::Solve(EXACT)

xc[n] <- n

yc[n] <- pdq::GetThruput(TERM, "httpGETs")

rc[n] <- pdq::GetResponse(TERM, "httpGETs") * 10^3

}

In the above PDQ model, we’ve selected the predicted throughput and the predicted response times to compare with the original load-test data.

#### Plot PDQ results

# Plot throughput and response time models

par(mfrow=c(2,1))

plot(xc, yc, type="l", lwd=1, col="blue", ylim=c(0,120), main="PDQ Throughput Model", xlab="Vusers (N)", ylab="Gets/s X(N)")

points(gdat$Vusr, gdat$Xgps)

plot(xc, rc, type="l", lwd=1, col="blue", ylim=c(0,220), main="PDQ Response Time Model", xlab="Vusers (N)", ylab="ms R(N)")

points(gdat$Vusr, gdat$Rms)

The above R code produces the following plot array:

We see that the data and PDQ model are in good agreement with the throughput saturating above N = 5 vusers with the corresponding response time rising up the proverbial “hockey stick” handle.

#### PDQ report

Optionally, we can produce a formal PDQ report to examine the performance of each of the three tiers, even if we don’t have any corresponding performance measurements from the load-test platform. This is one way by which bottlenecks can be predicted and checked before deploying into production.

> pdq::Report()

***************************************

****** Pretty Damn Quick REPORT *******

***************************************

*** of : Sun May 15 18:26:21 2011 ***

*** for: Three-Tier Model ***

*** Ver: PDQ Analyzer v5.0 030211 ***

***************************************

***************************************

=======================================

****** PDQ Model INPUTS *******

=======================================

Node Sched Resource Workload Class Demand

---- ----- -------- -------- ----- ------

CEN FCFS WebServer httpGETs TERML 0.0088

CEN FCFS AppServer httpGETs TERML 0.0024

CEN FCFS DBMServer httpGETs TERML 0.0008

Queueing Circuit Totals:

Streams: 1

Nodes: 3

WORKLOAD Parameters:

httpGETs 20.00 0.0120 0.03

=======================================

****** PDQ Model OUTPUTS *******

=======================================

Solution Method: EXACT

****** SYSTEM Performance *******

Metric Value Unit

------ ----- ----

Workload: "httpGETs"

Mean concurrency 16.8004 Users

Mean throughput 114.2725 Users/Sec

Response time 0.1470 Sec

Round trip time 0.1750 Sec

Stretch factor 12.2633

Bounds Analysis:

Max throughput 114.2725 Users/Sec

Min response 0.0120 Sec

Max Demand 0.0088 Sec

Tot demand 0.0120 Sec

Think time 0.0280 Sec

Optimal clients 4.5696 Clients

****** RESOURCE Performance *******

Metric Resource Work Value Unit

------ -------- ---- ----- ----

Throughput WebServer httpGETs 114.2725 Users/Sec

Utilization WebServer httpGETs 100.0000 Percent

Queue length WebServer httpGETs 16.3144 Users

Waiting line WebServer httpGETs 15.3144 Users

Waiting time WebServer httpGETs 0.1340 Sec

Residence time WebServer httpGETs 0.1428 Sec

Throughput AppServer httpGETs 114.2725 Users/Sec

Utilization AppServer httpGETs 27.7529 Percent

Queue length AppServer httpGETs 0.3841 Users

Waiting line AppServer httpGETs 0.1066 Users

Waiting time AppServer httpGETs 0.0009 Sec

Residence time AppServer httpGETs 0.0034 Sec

Throughput DBMServer httpGETs 114.2725 Users/Sec

Utilization DBMServer httpGETs 9.2447 Percent

Queue length DBMServer httpGETs 0.1019 Users

Waiting line DBMServer httpGETs 0.0094 Users

Waiting time DBMServer httpGETs 0.0001 Sec

Residence time DBMServer httpGETs 0.0009 Sec

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