Analytical Modeling of Parallel Systems

Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar

To accompany the text ``Introduction to Parallel Computing'',

Addison Wesley, 2003.
 Topic Overview
• Sources of Overhead in Parallel Programs

• Performance Metrics for Parallel Systems

• Effect of Granularity on Performance

• Scalability of Parallel Systems

• Minimum Execution Time and Minimum Cost-Optimal

Execution Time

• Asymptotic Analysis of Parallel Programs

• Other Scalability Metrics

 Analytical Modeling - Basics
• A sequential algorithm is evaluated by its runtime (in general,
asymptotic runtime as a function of input size).

• The asymptotic runtime of a sequential program is identical on any

serial platform.

• The parallel runtime of a program depends on the input size, the

number of processors, and the communication parameters of the
machine.
 Analytical Modeling - Basics
• An algorithm must therefore be analyzed in the context of the
underlying platform.

• A parallel system is a combination of a parallel algorithm and an

underlying platform.

• Wall clock time - the time from the start of the first processor to the
stopping time of the last processor in a parallel ensemble.
Sources of Overhead in Parallel Programs
• If I use two processors, shouldnt my program run twice as fast?

• No - a number of overheads, including wasted computation,

communication, idling, and contention cause degradation in
performance.

The execution profile of a hypothetical parallel program

executing on eight processing elements. Profile indicates times
spent performing computation (both essential and excess),
communication, and idling.
Sources of Overheads in Parallel Programs

• Interprocess interactions: Processors working on any non-

trivial parallel problem will need to talk to each other.

• Idling: Processes may idle because of load imbalance,

synchronization, or serial components.

• Excess Computation: This is computation not performed by

the serial version. This might be because the serial algorithm
is difficult to parallelize, or that some computations are
repeated across processors to minimize communication.
Performance Metrics for Parallel Systems:
Execution Time
• Serial runtime of a program is the time elapsed between the
beginning and the end of its execution on a sequential
computer.

• The parallel runtime is the time that elapses from the moment
the first processor starts to the moment the last processor
finishes execution.

• We denote the serial runtime by TS and the parallel runtime

by TP .
Performance Metrics for Parallel Systems:
Total Parallel Overhead
• Let Tall be the total time collectively spent by all the processing
elements.

• TS is the serial time.

• Observe that Tall - TS is then the total time spend by all processors
combined in non-useful work. This is called the total overhead.

• The total time collectively spent by all the processing elements

Tall = p TP (p is the number of processors).

• The overhead function (To) is therefore given by

To = p TP - TS
(1)
Performance Metrics for Parallel Systems:
Speedup
• What is the benefit from parallelism?

• Speedup (S) is the ratio of the time taken to solve a problem

on a single processor to the time required to solve the same
problem on a parallel computer with p identical processing
elements.

S = Ts / Tp
 Performance Metrics: Example

• Consider the problem of adding n numbers by using n

processing elements.

• If n is a power of two, we can perform this operation in log n

steps by propagating partial sums up a logical binary tree of
processors.
 Performance Metrics: Example

Computing the globalsum of 16 partial sums using 16 processing

elements . Σji denotes the sum of numbers with consecutive labels from i to j.
Performance Metrics: Example (continued)

• If an addition takes constant time, say, tc and communication

of a single word takes time ts + tw, we have the parallel time
TP = Θ (log n)

• We know that TS = Θ (n)

• Speedup S is given by S = Θ (n / log n)

 Performance Metrics: Speedup

• For a given problem, there might be many serial algorithms

available. These algorithms may have different asymptotic
runtimes and may be parallelizable to different degrees.

• For the purpose of computing speedup, we always consider

the best sequential program as the baseline.
Performance Metrics: Speedup Example
• Consider the problem of parallel bubble sort.

• The serial time for bubble sort is 150 seconds.

• The parallel time for odd-even sort (efficient parallelization of

bubble sort) is 40 seconds.

• The speedup would appear to be 150/40 = 3.75.

• But is this really a fair assessment of the system?

• What if serial quick sort only took 30 seconds? In this case,

the speedup is 30/40 = 0.75. This is a more realistic
assessment of the system.
Performance Metrics: Speedup Bounds
• Speedup can be as low as 0 (the parallel program never
terminates).

• Speedup, in theory, should be upper bounded by p - after all,

we can only expect a p-fold speedup if we use times as many
resources.

• A speedup greater than p is possible only if each processing

element spends less than time TS / p solving the problem.

• In this case, a single processor could be time slided to

achieve a faster serial program, which contradicts our
assumption of fastest serial program as basis for speedup.
 Performance Metrics:
Superlinear Speedups
One reason for superlinearity is that the parallel version
does less work than corresponding serial algorithm.

Searching an unstructured tree for a node with a given

label, `S', on two processing elements using depth-first traversal.
The two-processor version with processor 0 searching the left
subtree and processor 1 searching the right subtree expands only
the shaded nodes before the solution is found. The corresponding
serial formulation expands the entire tree. It is clear that the serial
algorithm does more work than the parallel algorithm.
 Performance Metrics:
Superlinear Speedups
Resource-based superlinearity: The higher aggregate
cache/memory bandwidth can result in better cache-hit ratio, and
therefore superlinearity.
 Performance Metrics: Efficiency
• Efficiency is a measure of the fraction of time for which a
processing element is usefully employed

• Mathematically, it is given by

=
(2)

• Following the bounds on speedup, efficiency can be as low as 0

and as high as 1.
Performance Metrics: Efficiency Example
• The speedup of adding numbers on processors is given by

• Efficiency is given by

=
 Cost of a Parallel System
• Cost is the product of parallel runtime and the number of
processing elements used (p x TP ).

• Cost reflects the sum of the time that each processing

element spends solving the problem.

• A parallel system is said to be cost-optimal if the cost of

solving a problem on a parallel computer is asymptotically
identical to serial cost.

• Since E = TS / p TP, for cost optimal systems, E = O(1).

• Cost is sometimes referred to as work or processor-time

product.
 Cost of a Parallel System: Example
Consider the problem of adding numbers on processors.

• We have, TP = log n (for p = n).

• The cost of this system is given by p TP = n log n.

• Since the serial runtime of this operation is Θ(n), the algorithm

is not cost optimal.
 Impact of Non-Cost Optimality
Consider a sorting algorithm that uses n processing elements to sort
the list in time (log n)2.

• Since the serial runtime of a (comparison-based) sort is n log n, the

speedup and efficiency of this algorithm are given by n / log n and 1
/ log n, respectively.

• The p TP product of this algorithm is n (log n)2.

• This algorithm is not cost optimal but only by a factor of log n.

• If p < n, assigning n tasks to p processors gives TP = n (log n)2 / p .

• The corresponding speedup of this formulation is p / log n.

• This speedup goes down as the problem size n is increased for a

given p !
 Effect of Granularity on Performance
• Often, using fewer processors improves performance of parallel
systems.

• Using fewer than the maximum possible number of processing

elements to execute a parallel algorithm is called scaling down a
parallel system.

• A naive way of scaling down is to think of each processor in the

original case as a virtual processor and to assign virtual processors
equally to scaled down processors.

• Since the number of processing elements decreases by a factor of

n / p, the computation at each processing element increases by a
factor of n / p.

• The communication cost should not increase by this factor since

some of the virtual processors assigned to a physical processors
might talk to each other. This is the basic reason for the
improvement from building granularity.
 Building Granularity: Example
• Consider the problem of adding n numbers on p processing
elements such that p < n and both n and p are powers of 2.

• Use the parallel algorithm for n processors, except, in this

case, we think of them as virtual processors.

• Each of the p processors is now assigned n / p virtual

processors.

• The first log p of the log n steps of the original algorithm are
simulated in (n / p) log p steps on p processing elements.

• Subsequent log n - log p steps do not require any

communication.
Building Granularity: Example (continued)
• The overall parallel execution time of this parallel system is
Θ ( (n / p) log p).

• The cost is Θ (n log p), which is asymptotically higher than

the Θ (n) cost of adding n numbers sequentially. Therefore,
the parallel system is not cost-optimal.
Building Granularity: Example (continued)
Can we build granularity in the example in a cost-optimal
fashion?
• Each processing element locally adds its n / p numbers in time
Θ (n / p).
• The p partial sums on p processing elements can be added in time
Θ(n /p).

A cost-optimal way of computing the sum of 16 numbers using four

processing elements.
Building Granularity: Example (continued)
• The parallel runtime of this algorithm is

(3)

• The cost is

• This is cost-optimal, so long as !

 Scalability of Parallel Systems
How do we extrapolate performance from small problems and small
systems to larger problems on larger configurations?

Consider three parallel algorithms for computing an n-point Fast Fourier

Transform (FFT) on 64 processing elements.

A comparison of the speedups obtained by the binary-exchange, 2-D

transpose and 3-D transpose algorithms on 64 processing elements with tc
= 2, tw = 4, ts = 25, and th = 2.

Clearly, it is difficult to infer scaling characteristics from observations on

small datasets on small machines.
Scaling Characteristics of Parallel Programs
• The efficiency of a parallel program can be written as:

or (4)

• The total overhead function To is an increasing function of p .

Scaling Characteristics of Parallel Programs
• For a given problem size (i.e., the value of TS remains constant), as
we increase the number of processing elements, To increases.

• The overall efficiency of the parallel program goes down. This is the
case for all parallel programs.
 Scaling Characteristics of Parallel
Programs: Example
• Consider the problem of adding numbers on processing
elements.

• We have seen that:

=
(5)

=
(6)

=
(7)
 Scaling Characteristics of Parallel
Programs: Example (continued)
Plotting the speedup for various input sizes gives us:

Speedup versus the number of processing elements for adding a

list of numbers.

Speedup tends to saturate and efficiency drops.

Scaling Characteristics of Parallel Programs
• Total overhead function To is a function of both problem size Ts and
the number of processing elements p.

• In many cases, To grows sub-linearly with respect to Ts.

• In such cases, the efficiency increases if the problem size is

increased keeping the number of processing elements constant.

• For such systems, we can simultaneously increase the problem size

and number of processors to keep efficiency constant.

• We call such systems scalable parallel systems.

Scaling Characteristics of Parallel Programs
• Recall that cost-optimal parallel systems have an efficiency of Θ(1).

• Scalability and cost-optimality are therefore related.

• A scalable parallel system can always be made cost-optimal if the

number of processing elements and the size of the computation are
chosen appropriately.
 Isoefficiency Metric of Scalability
• For a given problem size, as we increase the number of processing
elements, the overall efficiency of the parallel system goes down for
all systems.

• For some systems, the efficiency of a parallel system increases if

the problem size is increased while keeping the number of
processing elements constant.
 Isoefficiency Metric of Scalability

Variation of efficiency: (a) as the number of processing elements

is increased for a given problem size; and (b) as the problem size
is increased for a given number of processing elements. The
phenomenon illustrated in graph (b) is not common to all parallel
systems.
 Isoefficiency Metric of Scalability
• What is the rate at which the problem size must increase with
respect to the number of processing elements to keep the efficiency
fixed?

• This rate determines the scalability of the system. The slower this
rate, the better.

• Before we formalize this rate, we define the problem size W as the

asymptotic number of operations associated with the best serial
algorithm to solve the problem.
 Isoefficiency Metric of Scalability
• We can write parallel runtime as:

(8)

• The resulting expression for speedup is

(9)

• Finally, we write the expression for efficiency as

 Isoefficiency Metric of Scalability
• For scalable parallel systems, efficiency can be maintained at a
fixed value (between 0 and 1) if the ratio To / W is maintained at a
constant value.
• For a desired value E of efficiency,

(11)

• If K = E / (1 – E) is a constant depending on the efficiency to be

maintained, since To is a function of W and p, we have

(12)
 Isoefficiency Metric of Scalability
• The problem size W can usually be obtained as a function of p by
algebraic manipulations to keep efficiency constant.

• This function is called the isoefficiency function.

• This function determines the ease with which a parallel system can
maintain a constant efficiency and hence achieve speedups
increasing in proportion to the number of processing elements.
 Isoefficiency Metric: Example
• The overhead function for the problem of adding n numbers on p
processing elements is approximately 2p log p .

• Substituting To by 2p log p , we get

=
(13)

Thus, the asymptotic isoefficiency function for this parallel system is

.

• If the number of processing elements is increased from p to p’, the

problem size (in this case, n ) must be increased by a factor of
(p’ log p’) / (p log p) to get the same efficiency as on p processing
elements.
 Isoefficiency Metric: Example
Consider a more complex example where
• Using only the first term of To in Equation 12, we get

=
(14)

• Using only the second term, Equation 12 yields the following

relation between W and p:

(15)

• The larger of these two asymptotic rates determines the

isoefficiency. This is given by Θ(p3)
Cost-Optimality and the Isoefficiency Function
• A parallel system is cost-optimal if and only if

(16)

• From this, we have:

(17)

(18)

• If we have an isoefficiency function f(p), then it follows that the

relation W = Ω(f(p)) must be satisfied to ensure the cost-optimality of
a parallel system as it is scaled up.
Lower Bound on the Isoefficiency Function
• For a problem consisting of W units of work, no more than W
processing elements can be used cost-optimally.

• The problem size must increase at least as fast as Θ(p) to maintain

fixed efficiency; hence, Ω(p) is the asymptotic lower bound on the
isoefficiency function.
 Degree of Concurrency and the
Isoefficiency Function
• The maximum number of tasks that can be executed simultaneously
at any time in a parallel algorithm is called its degree of
concurrency.

• If C(W) is the degree of concurrency of a parallel algorithm, then for

a problem of size W, no more than C(W) processing elements can
be employed effectively.
 Degree of Concurrency and the
Isoefficiency Function: Example
Consider solving a system of equations in variables by using
Gaussian elimination (W = Θ(n3))

• The n variables must be eliminated one after the other, and

eliminating each variable requires Θ(n2) computations.

• At most Θ(n2) processing elements can be kept busy at any time.

• Since W = Θ(n3) for this problem, the degree of concurrency C(W) is

Θ(W2/3) .

• Given p processing elements, the problem size should be at least

Ω(p3/2) to use them all.
Minimum Execution Time and Minimum
Cost-Optimal Execution Time
Often, we are interested in the minimum time to solution.

• We can determine the minimum parallel runtime TPmin for a given W

by differentiating the expression for TP w.r.t. p and equating it to
zero.

= 0
(19)

• If p0 is the value of p as determined by this equation, TP(p0) is the

minimum parallel time.
Minimum Execution Time: Example
Consider the minimum execution time for adding n numbers.

= (20)

Setting the derivative w.r.t. p to zero, we have p = n/ 2 . The

corresponding runtime is

=
(21)

(One may verify that this is indeed a min by verifying that the
second derivative is positive).

Note that at this point, the formulation is not cost-optimal.

 Minimum Cost-Optimal Parallel Time
• Let TPcost_opt be the minimum cost-optimal parallel time.

• If the isoefficiency function of a parallel system is Θ(f(p)), then a

problem of size W can be solved cost-optimally if and only if
W= Ω(f(p)) .

• In other words, for cost optimality, p = O(f--1(W)) .

• For cost-optimal systems, TP = Θ(W/p) , therefore,

=
(22)
 Minimum Cost-Optimal Parallel Time:
Example
Consider the problem of adding n numbers.

• The isoefficiency function f(p) of this parallel system is Θ(p log p).

• From this, we have p ≈ n /log n .

• At this processor count, the parallel runtime is:

= (23)

• Note that both TPmin and TPcost_opt for adding n numbers are
Θ(log n). This may not always be the case.
Asymptotic Analysis of Parallel Programs
Consider the problem of sorting a list of n numbers. The fastest
serial programs for this problem run in time Θ(n log n). Consider
four parallel algorithms, A1, A2, A3, and A4 as follows:

Comparison of four different algorithms for sorting a given list of

numbers. The table shows number of processing elements, parallel
runtime, speedup, efficiency and the pTP product.
Asymptotic Analysis of Parallel Programs
• If the metric is speed, algorithm A1 is the best, followed by A3, A4,
and A2 (in order of increasing TP).

• In terms of efficiency, A2 and A4 are the best, followed by A3 and

A1.

• In terms of cost, algorithms A2 and A4 are cost optimal, A1 and A3

are not.

• It is important to identify the objectives of analysis and to use

appropriate metrics!
 Other Scalability Metrics
• A number of other metrics have been proposed, dictated by specific
needs of applications.

• For real-time applications, the objective is to scale up a system to

accomplish a task in a specified time bound.

• In memory constrained environments, metrics operate at the limit of

memory and estimate performance under this problem growth rate.
Other Scalability Metrics: Scaled Speedup
• Speedup obtained when the problem size is increased linearly with
the number of processing elements.

• If scaled speedup is close to linear, the system is considered

scalable.

• If the isoefficiency is near linear, scaled speedup curve is close to

linear as well.

• If the aggregate memory grows linearly in p, scaled speedup

increases problem size to fill memory.

• Alternately, the size of the problem is increased subject to an upper-

bound on parallel execution time.
 Scaled Speedup: Example
• The serial runtime of multiplying a matrix of dimension n x n with a
vector is tcn2 .

• For a given parallel algorithm,

(24)

• Total memory requirement of this algorithm is Θ(n2) .

Scaled Speedup: Example (continued)
Consider the case of memory-constrained scaling.

• We have m= Θ(n2) = Θ(p).

• Memory constrained scaled speedup is given by

or

• This is not a particularly scalable system

 Scaled Speedup: Example (continued)

Consider the case of time-constrained scaling.

• We have TP = O(n2) .

• Since this is constrained to be constant, n2= O(p) .

• Note that in this case, time-constrained speedup is identical to

memory constrained speedup.
 Scaled Speedup: Example
• The serial runtime of multiplying two matrices of dimension n x n is
t cn3.

• The parallel runtime of a given algorithm is:

• The speedup S is given by:

(25)
Scaled Speedup: Example (continued)
Consider memory-constrained scaled speedup.
• We have memory complexity m= Θ(n2) = Θ(p), or n2=c x p .

• At this growth rate, scaled speedup S’ is given by:

• Note that this is scalable.

Scaled Speedup: Example (continued)
Consider time-constrained scaled speedup.

• We have TP = O(1) = O(n3 / p) , or n3=c x p .

• Time-constrained speedup S’’ is given by:

• Memory constrained scaling yields better performance.

 Serial Fraction f
• If the serial runtime of a computation can be divided into a totally
parallel and a totally serial component, we have:

• From this, we have,

(26)
 Serial Fraction f
• The serial fraction f of a parallel program is defined as:

• Therefore, we have:
 Serial Fraction
• Since S = W / TP , we have

• From this, we have:

(27)

• If f increases with the number of processors, this is an indicator of

rising overhead, and thus an indicator of poor scalability.
 Serial Fraction: Example
Consider the problem of extimating the serial component of the
matrix-vector product.

We have:

(28)

or

Here, the denominator is the serial runtime and the numerator is the
overhead.