================ Start Lecture #18 ================
4.4.8: The Working Set Page Replacement Problem (Peter Denning)
The working set policy (Peter Denning)
The goal is to specify which pages a given process needs to have
memory resident in order for the process to run without too many
But this is impossible since it requires predicting the future.
So we make the assumption that the immediate future is well
approximated by the immediate past.
We measure time in units of memory references, so t=1045 means the
time when the 1045th memory reference is issued.
In fact we measure time separately for each process, so t=1045
really means the time when this process made its 1045th memory
W(t,&omega) is the set of pages referenced (by the given process) from
time t-ω to time t.
That is, W(t,ω) is the set pages referenced during
the window of size ω ending at time t.
That is, W(t,ω) is the set of pages referenced by the last
ω memory references ending at reference t.
W(t,ω) is called the working set at time t
(with window ω).
w(t,ω) is the size of the set W(t,ω), i.e. is the
number of distinct pages referenced in the window.
The idea of the working set policy is to ensure that each process
keeps its working set in memory.
Allocate w(t,ω) frames to each process.
This number differs for each process and changes with time.
On a fault, one evicts a page not in the working set. But it is
not easy to find such a page quickly.
Indeed determining W(t,ω) precisely is quite time consuming
and difficult. It is never done in real systems.
If a process is suspended, it is often swapped out; the working
set then can be used to say which pages should be brought back
when the process is resumed.
The definition of Working Set is local to a process. That is, each
process has a working set; there is no system wide working set other
than the union of all the working sets of each process.
However, the working set of a single process has effects on the
demand paging behavior and victim selection of other processes.
If a process's working set is growing in size, i.e. w(t,ω) is
increasing as t increases, then we need to obtain new frames from
other processes. A process with a working set decreasing in size is a
source of free frames. We will see below that this is an interesting
local and global replacement policies.
Interesting questions concerning the working set include:
What value should be used for ω?
Experiments have been done and ω is surprisingly robust (i.e.,
for a given system, a fixed value works reasonably for a wide variety
of job mixes).
How should we calculate W(t,ω)?
Hard so do exactly so ...
... Various approximations to the working set, have been devised.
We will study two: using virtual time instead of memory references
(immediately below) and Page Fault Frequency (section 4.6).
In 4.4.9 we will see the popular WSClock algorithm that includes an
approximation of the working set as well as several other ideas.
Using virtual time
Approximate the working set by those pages referenced during the
last m milliseconds rather than the last ω memory references.
Note that the time is measured only while this process is running,
i.e., we are using virtual time.
Clear the reference bit every m milliseconds and set it on every
To choose a victim, we need to find a page with the R bit
Essentially we have reduced the working set policy to NRU.
4.4.9: The WSClock Page Replacement Algorithm
This treatment is based on one by Prof. Ernie Davis.
Tannenbaum suggests that the WSClock Page
Replacement Algorithm is a natural outgrowth of the idea of a working set.
However, reality is less clear cut.
WSClock is actually embodies several ideas,
one of which is connected to the idea of a working set.
As the name suggests another of the ideas is the clock implementation
of 2nd chance.
The actual implemented algorithm is somewhat complicated and not a
clean elegant concept.
It is important because
It works well and is in common use.
The embodied ideas are themselves interesting.
Inelegant amalgamations of ideas are more commonly used in real
systems than clean, elegant, one-idea algorithms.
Since the algorithm is complicated we present it in stages.
As stated above this is an important algorithm since it works well and
is used in practice. However, I certainly do not assume you remember
all the details.
We start by associating a node with every page loaded in memory
(i.e., with every frame given to this process).
In the node are stored R and M bits that we assume are set by the
(Of course we don't design the hardware so really the R and M bits
are set in a hardware defined table and the nodes reference the
entries in that table.)
Every k clock ticks the R bit is reset.
So far this looks like NRU.
To ease the explanation we will assume k=1, i.e., actions
are done each clock tick.
We now introduce an LRU aspect (with the virtual time
approximation described above for working set): At each clock
tick we examine all the nodes for the running process and store
the current virtual time in all nodes for which R is 1.
Thus, the time field is an approximation to the time of the
most recent reference, accurate to the clock period. Note that
this is done every clock tick (really every k ticks) and
not every memory reference. That is why it is feasible.
If we chose as victim the page with the smallest time field, we
would be implementing a virtual time approximation to LRU.
But in fact we do more.
We now introduce some working set aspects into the algorithm by
first defining a time constant τ (analogous to ω in the
working set algorithm) and consider all pages older than τ
(i.e., their stored time is smaller than the current time minus
τ) as candidate victims.
The idea is that these pages are not in the working set.
The OS designer needs to tune τ just as one would need to
tune ω and, like ω, τ is quite robust (the same
value works well for a variety of job mixes).
The advantage of introducing τ is that a victim search can
stop as soon as a page older than τ is found.
If no pages have a reference time older than Tau, then the page
with the earliest time is the victim.
Next we introduce the other aspect of NRU, preferring clean to
We search until we find a clean page older than τ, if
there is one; if not, we use a dirty page older than τ.
Now we introduce an optimization similar to prefetching (i.e.,
speculatively fetching some data before it is known to be needed).
Specifically, when we encounter a dirty page older than τ
(while looking for a clean old page), we write the dirty page back
to disk (and clear the M bit, which Tanenbaum forgot to mention)
without evicting the page, on the
presumption that, since the page is not in (our approximation to)
the working set, this I/O will be needed eventually.
The down side is that the page could become dirty again, rendering
our speculative I/O redundant.
Suppose we've decided to write out old dirty pages
D1 through Dd and to replace old clean page
C with new page N.
We must block the current process P until N is completely read
in, but P can run while D1 through Dd are
being written. Hence we would desire the I/O read to be done
before the writes, but we shall see later, when we study I/O, that
there are other considerations for choosing the order to perform
Similarly, suppose we can not find an old clean page and have
decided to replace old dirty page D0 with new page N,
and have detected additional old dirty pages D1 through
Dd (recall that we were searching for an old clean
page). Then P must block until D0 has been written
and N has been read, but can run while D1 through
Dd are being written.
We throttle the previous optimization to prevent overloading the
Specifically we set a limit on the number of dirty pages the
previous optimization can request be written.
Finally, as in the clock algorithm, we keep the data structure
(nodes associated with pages) organized as a circular list with a
single pointer (the hand of the clock).
Hence we start each victim search where the previous one left
4.4.10: Summary of Page Replacement Algorithms
|Random||Poor, used for comparison
|Optimal||Unimplementable, used for comparison
|FIFO||Not good ignores frequency of use
|Second Chance||Improvement over FIFO
|Clock||Better implementation of Second Chance
|LRU||Great but impractical
|NFU||Crude LRU approximation
|Aging||Better LRU approximation
|Working Set||Good, but expensive
|WSClock||Good approximation to working set
4.5: Modeling Paging Algorithms
4.5.1: Belady's anomaly
Consider a system that has no pages loaded and that uses the FIFO
Consider the following “reference string” (sequences of
0 1 2 3 0 1 4 0 1 2 3 4
If we have 3 frames this generates 9 page faults (do it).
If we have 4 frames this generates 10 page faults (do it).
Theory has been developed and certain PRA (so called “stack
algorithms”) cannot suffer this anomaly for any reference string.
FIFO is clearly not a stack algorithm. LRU is. Tannenbaum has a few
details, but we are skipping it.
Repeat the above calculations for LRU.