1. Methods of Stochastic Performance Analysis
Perry Lea
Hewlett Packard - Core Technology Laboratory

2. Why Am I Talking About This
Currently I’m an ES working on future architectures and use these techniques with frequency to aid my design decisions
Used these techniques to discover good designs
Used these techniques to stop bad designs

3. Who Uses This Stuff System Architects / Engineers / Scientists
Transportation systems
Industrial and production analysis
Behavorialists
Communication systems
and of course the general engineering community.

4. Typical Problems Addressed
How fast will it print or copy?
How do we distribute various tasks to various processors?
What is the bus contention?
Will we meet the real-time requirements?

5. What Is Performance Analysis (Jain Book Definition)
Specifying performance requirements
Comparing 2 or more systems
Determining optimal/achievable parameters
Finding bottlenecks
Capacity planning
Workload planning
This talk is about:
Stochastic Analysis
Operational Analysis
Discrete Event Simulation

6. The Stochastic Process
Anything that is observed as a random process in time is a stochastic process.
A stochastic process can be discrete (good for CS folks) or continuous (good for signals folks)
If a future state of your system is not dependent on a previous state, the process can be called a Markov process. This is good because Markov processes are easy to model.
Memoryless property
If the arrival times of “data” to the different states are independent of each other – AND – the arrival times have the same distribution (hopefully exponentially distributed), then you can describe the process as a Poisson process.
IID: Independent and identically distributed

7. Queue Rules
 the interarrival rate, time between 2 successive arrivals (Random Variable)
 the arrival rate
s the service time
 the mean service rate per server, or 1/E[s]
n the number of jobs in the system
r the response time (includes time waiting in queue and actual cpu time)
w the waiting time

8. Some Math l = arrival rate (going in) m = service rate (going out)
r = l / m is the traffic intensity or density
The probability that there are n tasks waiting in the queue is:

9. Birth Death Process The Birth – Death Process
Each state represents the queue depth 1 2
j-1 j
j+1

10. Queues Kendall Notation A/B/C/D/E Types of processes
A = arrival process
B = service process
C = number of servers
D = maximum number of jobs for which the queue has room.
E = maximum number of jobs being simulated
Types of processes
M = (Memoryless) Poisson arrival process or exponentially distributed service time
D = deterministic arrival process (each job has the same service time) Aha! Here’s the deterministic part of all this.
G = general distribution arrival process
E = Erlang distribution
Hyperexponential

11. Typical Queues M/M/1 M/G/1 Others
Jobs arrive according to a Poisson process
Service time of each job is exponentially distributed.
There is one server
There is infinite waiting room and no limit to the number of jobs
M/G/1
Service time is generally distributed (no restriction of the form the distribution may take)
Others
M/D/1 (D denotes each jobs take the same amount of time to be service)
M/G/ (Infinite amount of servers available)
G/M/1
G/G/m

12. M/M/1 Model Analysis Traffic Intensity (Utilization): r = l/m
Condition of Stability: r < 1
Probability queue empty: p0 = 1 – r
Probability queue = n: pn = (1-r)rn *
Mean queue depth: E[n]=r/(1 - r)
Mean waiting time: E[w]=r ((1/m) / (1-r))
Mean response time (E[n] = /E[r]) *
E[r] = (1/m) / (1-r)
Other models have similar equations (See Jain Book)

13. M/M/1 Example
A printer processes 125 packets per second and takes 2 ms to process them. When will the printer overflow with 13 buffers? How many buffers are needed to keep packet loss below one in a million?
Arrival rate l = 125 pps
Service rate m = 1/.002 = 500 pps
Utilization r = 0.25 or 25%
Probability of n packets in gateway = (1-r)rn = 0.75(0.25)n
Mean number of packets in gateway = r/1-r = 0.33
Probability of buffer overflow -> P(more than 13 packets)
13 = = 15 packets in 1 billion
To limit the probability loss to 10-6
n <= 10-6
n > log(10-6) / log(0.25) = 9.96 buffers

14. Operational Laws (Denning and Buzen) (Simple formulas for finite observable time)
If you observe a system for a finite amount of time, you can skip the distributions of service and interarrival times.
You can use a set of simple (directly measurable) equations to predict performance
Arrival Rate : # arrivals / time : Ai/T
Throughput : # completions / time : Ci/T
Busy Time : Bi
Mean Service Time : total time served / # served : Bi / Ci
Utilization Law : U = Bi/T (T is the observed time)
Forced Flow Law : system throughput : X = C0/T
Little’s Law : mean # in device = arrival rate * mean time
Qi=i*Ri
Bottleneck Identification : The device with the largest Di

15. Queuing Networks Open Network Closed Network
Jackson product-form solution
Closed Network
Buzen convolution algorithms

16. Large Queuing Networks
Large systems can be modeled as a collection of queues.
Bounding algorithms
Find upper and lower bounds only.
Mean value analysis (MVA)
Iterative method to find response times for each device in the system. This will sum to the system response time and overall queue lengths.
Convolution algorithms
Array convolutions
Can answer more detailed questions than MVA, i.e. are 2 IO devices busy at the same time?
For large networks, we use software simulators

17. Discrete Event Simulations
Why use a simulator?
Hard to build analytic models for:
Virtual memory
Resource sharing (CPU and IO)
Mutual exclusion to resources
Bulk packet arrivals (self similarity)
Non-exponential service times
Can run many variants and perturbations of a system
Can build stochastic and deterministic models and intermix them.
To automate the mathematical process
Allows you to study the problem from many angles with ease compared to analytical models

18. Modeling Tips Understand the architecture in detail first
Decide on how detailed your model should be
The more detailed the more time, and possibly the more bugs you will introduce
you don’t want to build the final product.
Do you need to model the whole system, or just a part?
Get the abstraction defined before you start!!
Chose the right tool and language
Verify and test the models
State what the goal of the model is, know when to stop
Don’t let mysterious results pass – understand them.
Review the model design – like any other piece of code.

19. Good Simulation Tools http://www.idsia.ch/~andrea/simtools.html C++SIM
Good site with links to many modeling tools
C++SIM
C Libraries
Do it yourself standalone, no GUI, good for academics
Opnet Modeler
MIL3, mainly for network and data flow
CSIM
Lockheed Martin – good for computer and electronic systems
Discrete and continuous models
NT and Unix flavors
SES Workbench
A generic modeling tool
GUI, object orientated, NT and Unix

20. Recommended Reading
“The Art of Computer Systems Performance Analysis”, Raj Jain, Wiley & Sons 1991
“Performance Modeling for Computer Architects”, C. Krishna, IEEE Computer Society Press 1996
“Telecommunication Networks”, Mischa Shwartz, Addison Wesley 1987