@node 4.4BSD Scheduler
@appendix 4.4@acronym{BSD} Scheduler

@iftex
@macro tm{TEX}
@math{\TEX\}
@end macro
@macro nm{TXT}
@end macro
@macro am{TEX, TXT}
@math{\TEX\}
@end macro
@end iftex

@ifnottex
@macro tm{TEX}
@end macro
@macro nm{TXT}
@w{\TXT\}
@end macro
@macro am{TEX, TXT}
@w{\TXT\}
@end macro
@end ifnottex

@ifhtml
@macro math{TXT}
\TXT\
@end macro
@end ifhtml

@macro m{MATH}
@am{\MATH\, \MATH\}
@end macro

The goal of a general-purpose scheduler is to balance threads' different
scheduling needs.  Threads that perform a lot of I/O require a fast
response time to keep input and output devices busy, but need little CPU
time.  On the other hand, CPU-bound threads need to receive a lot of
CPU time to finish their work, but have no requirement for fast response
time.  Other threads lie somewhere in between, with periods of I/O
punctuated by periods of computation, and thus have requirements that
vary over time.  A well-designed scheduler can often accommodate threads
with all these requirements simultaneously.

For task 1, you must implement the scheduler described in this
appendix.  Our scheduler resembles the one described in @bibref{McKusick},
which is one example of a @dfn{multilevel feedback queue} scheduler.
This type of scheduler maintains several queues of ready-to-run threads,
where each queue holds threads with a different priority.  At any given
time, the scheduler chooses a thread from the highest-priority non-empty
queue.  If the highest-priority queue contains multiple threads, then
they run in ``round robin'' order.

Multiple facets of the scheduler require data to be updated after a
certain number of timer ticks.  In every case, these updates should
occur before any ordinary kernel thread has a chance to run, so that
there is no chance that a kernel thread could see a newly increased
@func{timer_ticks} value but old scheduler data values.

The 4.4@acronym{BSD} scheduler does not include priority donation.

@menu
* Thread Niceness::             
* Calculating Priority::        
* Calculating recent_cpu::      
* Calculating load_avg::        
* 4.4BSD Scheduler Summary::    
* Fixed-Point Real Arithmetic::  
@end menu

@node Thread Niceness
@section Niceness

Thread priority is dynamically determined by the scheduler using a
formula given below.  However, each thread also has an integer
@dfn{nice} value that determines how ``nice'' the thread should be to
other threads.  A @var{nice} of zero does not affect thread priority.  A
positive @var{nice}, to the maximum of 20, decreases the priority of a 
thread and causes it to give up some CPU time it would otherwise receive.
On the other hand, a negative @var{nice}, to the minimum of -20, tends
to take away CPU time from other threads.

The initial thread starts with a @var{nice} value of zero.  Other
threads start with a @var{nice} value inherited from their parent
thread.  You must implement the functions described below, which are for
use by test programs.  We have provided skeleton definitions for them in
@file{threads/thread.c}.

@deftypefun int thread_get_nice (void)
Returns the current thread's @var{nice} value.
@end deftypefun

@deftypefun void thread_set_nice (int @var{new_nice})
Sets the current thread's @var{nice} value to @var{new_nice} and
recalculates the thread's priority based on the new value
(@pxref{Calculating Priority}).  If the running thread no longer has the
highest priority, yields.
@end deftypefun

@node Calculating Priority
@section Calculating Priority

Our scheduler has 64 priorities and thus 64 ready queues, numbered 0
(@code{PRI_MIN}) through 63 (@code{PRI_MAX}).  Lower numbers correspond
to lower priorities, so that priority 0 is the lowest priority
and priority 63 is the highest.  Thread priority is calculated initially
at thread initialization.  It is also recalculated for each thread (if necessary) 
on every fourth clock tick.  
In either situation, it is determined by the formula:

@center @t{@var{priority} = @code{PRI_MAX} - (@var{recent_cpu} / 4) - (@var{nice} * 2)},

@noindent where @var{recent_cpu} is an estimate of the CPU time the
thread has used recently (see below) and @var{nice} is the thread's
@var{nice} value.  The result should be rounded down to the nearest
integer (truncated).
The coefficients @math{1/4} and 2 on @var{recent_cpu}
and @var{nice}, respectively, have been found to work well in practice
but lack deeper meaning.  The calculated @var{priority} is always
adjusted to lie in the valid range @code{PRI_MIN} to @code{PRI_MAX}.

This formula gives a thread that has received CPU time recently lower priority 
for being reassigned the CPU the next time the scheduler runs.  
This is key to preventing starvation: 
a thread that has not received any CPU time recently will have a
@var{recent_cpu} of 0, which barring a high @var{nice} value should
ensure that it receives CPU time soon.
This technique is sometimes referred to as "aging" in the literature.

@node Calculating recent_cpu
@section Calculating @var{recent_cpu}

We wish @var{recent_cpu} to measure how much CPU time each process has
received ``recently.'' Furthermore, as a refinement, more recent CPU
time should be weighted more heavily than less recent CPU time.  One
approach would use an array of @var{n} elements to
track the CPU time received in each of the last @var{n} seconds.
However, this approach requires O(@var{n}) space per thread and
O(@var{n}) time per calculation of a new weighted average.

Instead, we use an @dfn{exponentially weighted moving average}, which
takes this general form:

@center @tm{x(0) = f(0),}@nm{x(0) = f(0),}
@center @tm{x(t) = ax(t-1) + (1-a)f(t),}@nm{x(t) = a*x(t-1) + (1-a)*f(t),}
@center @tm{a = k/(k+1),}@nm{a = k/(k+1),}

@noindent where @math{x(t)} is the moving average at integer time @am{t
\ge 0, t >= 0}, @math{f(t)} is the function being averaged, and @math{k
> 0} controls the rate of decay.  We can iterate the formula over a few
steps as follows:

@center @math{x(0) = f(0)},
@center @am{x(1) = af(0) + (1-a)f(1), x(1) = a*f(0) + (1-a)*f(1)},
@center @am{\vdots, ...}
@center @am{x(4) = a^4f(0) + a^3(1-a)f(1) + a^2(1-a)f(2) + a(1-a)f(3) + (1-a)f(4), x(4) = a**4*f(0) + a**3*(1-a)*f(1) + a**2*(1-a)*f(2) + a*(1-a)*f(3) + (1-a)*f(4)}.

@noindent The value of @math{f(t)} has a weight of @math{(1-a)} at time @math{t}, 
a weight of @math{a(1-a)} at time @math{t+1}, @am{a^2(1-a), a**2(1-a)} at time
@math{t+2}, and so on.  We can also relate @math{x(t)} to @math{k}:
@math{f(t)} has a weight of approximately @math{1/e} at time @math{t+k},
approximately @am{1/e^2, 1/e**2} at time @am{t+2k, t+2*k}, and so on.
From the opposite direction, @math{f(t)} decays to weight @math{w} at around
time @am{t + \log_aw, t + ln(w)/ln(a)}.

The initial value of @var{recent_cpu} is 0 in the first thread
created, or the parent's value in other new threads.  Each time a timer
interrupt occurs, @var{recent_cpu} is incremented by 1 for the running
thread only, unless the idle thread is running.  In addition, once per
second the value of @var{recent_cpu}
is recalculated for every thread (whether running, ready, or blocked),
using this formula:

@center @t{@var{recent_cpu} = (2*@var{load_avg})/(2*@var{load_avg} + 1) * @var{recent_cpu} + @var{nice}},

@noindent where @var{load_avg} is a moving average of the number of
threads ready to run (see below).  If @var{load_avg} is 1, indicating
that a single thread, on average, is competing for the CPU, then the
current value of @var{recent_cpu} decays to a weight of .1 in
@am{\log_{2/3}.1 \approx 6, ln(.1)/ln(2/3) = approx. 6} seconds; if
@var{load_avg} is 2, then decay to a weight of .1 takes @am{\log_{3/4}.1
\approx 8, ln(.1)/ln(3/4) = approx. 8} seconds.  The effect is that
@var{recent_cpu} estimates the amount of CPU time the thread has
received ``recently,'' with the rate of decay inversely proportional to
the number of threads competing for the CPU.

Assumptions made by some of the tests require that these recalculations of
@var{recent_cpu} be made exactly when the system tick counter reaches a
multiple of a second, that is, when @code{timer_ticks () % TIMER_FREQ ==
0}, and not at any other time.

The value of @var{recent_cpu} can be negative for a thread with a
negative @var{nice} value.  Do not clamp negative @var{recent_cpu} to 0.

You may need to think about the order of calculations in this formula.
We recommend computing the coefficient of @var{recent_cpu} first, then
multiplying.  In the past, some students have reported that multiplying
@var{load_avg} by @var{recent_cpu} directly can cause overflow.

You must implement @func{thread_get_recent_cpu}, for which there is a
skeleton in @file{threads/thread.c}.

@deftypefun int thread_get_recent_cpu (void)
Returns 100 times the current thread's @var{recent_cpu} value, rounded
to the nearest integer.
@end deftypefun

@node Calculating load_avg
@section Calculating @var{load_avg}

Finally, @var{load_avg}, often known as the system load average,
estimates the average number of threads ready to run over the past
minute.  Like @var{recent_cpu}, it is an exponentially weighted moving
average.  Unlike @var{priority} and @var{recent_cpu}, @var{load_avg} is
system-wide, not thread-specific.  At system boot, it is initialized to
0.  Once per second thereafter, it is updated according to the following
formula:

@center @t{@var{load_avg} = (59/60)*@var{load_avg} + (1/60)*@var{ready_threads}},

@noindent where @var{ready_threads} is the number of threads that are
either running or ready to run at time of update (not including the idle
thread).

Because of assumptions made by some of the tests, @var{load_avg} must be
updated exactly when the system tick counter reaches a multiple of a
second, that is, when @code{timer_ticks () % TIMER_FREQ == 0}, and not
at any other time.

You must implement @func{thread_get_load_avg}, for which there is a
skeleton in @file{threads/thread.c}.

@deftypefun int thread_get_load_avg (void)
Returns 100 times the current system load average, rounded to the
nearest integer.
@end deftypefun

@node 4.4BSD Scheduler Summary
@section Summary

The following formulas summarize the calculations required to implement the
scheduler.  They are not a complete description of the scheduler's requirements.

Every thread has a @var{nice} value between -20 and 20 directly under
its control.  Each thread also has a priority, between 0
(@code{PRI_MIN}) through 63 (@code{PRI_MAX}), which is recalculated (as necessary)
using the following formula:

@center @t{@var{priority} = @code{PRI_MAX} - (@var{recent_cpu} / 4) - (@var{nice} * 2)}.

@var{recent_cpu} measures the amount of CPU time a thread has received
``recently.''  On each timer tick, the running thread's @var{recent_cpu}
is incremented by 1.  Once per second, every thread's @var{recent_cpu}
is updated this way:

@center @t{@var{recent_cpu} = (2*@var{load_avg})/(2*@var{load_avg} + 1) * @var{recent_cpu} + @var{nice}}.

@var{load_avg} estimates the average number of threads ready to run over
the past minute.  It is initialized to 0 at boot and recalculated once
per second as follows:

@center @t{@var{load_avg} = (59/60)*@var{load_avg} + (1/60)*@var{ready_threads}}.

@noindent where @var{ready_threads} is the number of threads that are
either running or ready to run at time of update (not including the idle
thread).

Note that it is important that each of these calculations is based on up-to-date data values.
That is, the calculation of each thread's @var{priority} should be based on the most recent @var{recent_cpu} value
and, similarly, the calculation of @var{recent_cpu} should itself be based on the most recent @var{load_avg} value.
You should take these dependencies into account when implementing these calculations.

You should also think about the efficiency of your calculations.
The more time your scheduler spends working on these calculations, 
the less time your actual processes will have to run.
It is important, therefore, to only perform calculations when absolutely necessary. 

@node Fixed-Point Real Arithmetic
@section Fixed-Point Real Arithmetic

In the formulas above, @var{priority}, @var{nice}, and
@var{ready_threads} are integers, but @var{recent_cpu} and @var{load_avg}
are real numbers.  Unfortunately, Pintos does not support floating-point
arithmetic in the kernel, because it would
complicate and slow the kernel.  Real kernels often have the same
limitation, for the same reason.  This means that calculations on real
quantities must be simulated using integers.  This is not
difficult, but many students do not know how to do it.  This
section explains the basics.

The fundamental idea is to treat the rightmost bits of an integer as
representing a fraction.  For example, we can designate the lowest 14
bits of a signed 32-bit integer as fractional bits, so that an integer
@m{x} represents the real number
@iftex
@m{x/2^{14}}.
@end iftex
@ifnottex
@m{x/(2**14)}, where ** represents exponentiation.
@end ifnottex
This is called a 17.14 fixed-point number representation, because there
are 17 bits before the decimal point, 14 bits after it, and one sign
bit.@footnote{Because we are working in binary, the ``decimal'' point
might more correctly be called the ``binary'' point, but the meaning
should be clear.} A number in 17.14 format represents, at maximum, a
value of @am{(2^{31} - 1) / 2^{14} \approx, (2**31 - 1)/(2**14) =
approx.} 131,071.999.

Suppose that we are using a @m{p.q} fixed-point format, and let @am{f =
2^q, f = 2**q}.  By the definition above, we can convert an integer or
real number into @m{p.q} format by multiplying with @m{f}.  For example,
in 17.14 format the fraction 59/60 used in the calculation of
@var{load_avg}, above, is @am{(59/60)2^{14}, 59/60*(2**14)} = 16,110.
To convert a fixed-point value back to an
integer, divide by @m{f}.  (The normal @samp{/} operator in C rounds
toward zero, that is, it rounds positive numbers down and negative
numbers up.  To round to nearest, add @m{f / 2} to a positive number, or
subtract it from a negative number, before dividing.)

Many operations on fixed-point numbers are straightforward.  Let
@code{x} and @code{y} be fixed-point numbers, and let @code{n} be an
integer.  Then the sum of @code{x} and @code{y} is @code{x + y} and
their difference is @code{x - y}.  The sum of @code{x} and @code{n} is
@code{x + n * f}; difference, @code{x - n * f}; product, @code{x * n};
quotient, @code{x / n}.

Multiplying two fixed-point values has two complications.  First, the
decimal point of the result is @m{q} bits too far to the left.  Consider
that @am{(59/60)(59/60), (59/60)*(59/60)} should be slightly less than
1, but @tm{16,110\times 16,110}@nm{16,110*16,110} = 259,532,100 is much
greater than @am{2^{14},2**14} = 16,384.  Shifting @m{q} bits right, we
get @tm{259,532,100/2^{14}}@nm{259,532,100/(2**14)} = 15,840, or about 0.97,
the correct answer.  Second, the multiplication can overflow even though
the answer is representable.  For example, 64 in 17.14 format is
@am{64 \times 2^{14}, 64*(2**14)} = 1,048,576 and its square @am{64^2,
64**2} = 4,096 is well within the 17.14 range, but @tm{1,048,576^2 =
2^{40}}@nm{1,048,576**2 = 2**40}, greater than the maximum signed 32-bit
integer value @am{2^{31} - 1, 2**31 - 1}.  An easy solution is to do the
multiplication as a 64-bit operation.  The product of @code{x} and
@code{y} is then @code{((int64_t) x) * y / f}.

Dividing two fixed-point values has opposite issues.  The
decimal point will be too far to the right, which we fix by shifting the
dividend @m{q} bits to the left before the division.  The left shift
discards the top @m{q} bits of the dividend, which we can again fix by
doing the division in 64 bits.  Thus, the quotient when @code{x} is
divided by @code{y} is @code{((int64_t) x) * f / y}.

This section has consistently used multiplication or division by @m{f},
instead of @m{q}-bit shifts, for two reasons.  First, multiplication and
division do not have the surprising operator precedence of the C shift
operators.  Second, multiplication and division are well-defined on
negative operands, but the C shift operators are not.  Take care with
these issues in your implementation.

The following table summarizes how fixed-point arithmetic operations can
be implemented in C.  In the table, @code{x} and @code{y} are
fixed-point numbers, @code{n} is an integer, fixed-point numbers are in
signed @m{p.q} format where @m{p + q = 31}, and @code{f} is @code{1 <<
q}:

@html
<CENTER>
@end html
@multitable @columnfractions .5 .5
@item Convert @code{n} to fixed point:
@tab @code{n * f}

@item Convert @code{x} to integer (rounding toward zero):
@tab @code{x / f}

@item Convert @code{x} to integer (rounding to nearest):
@tab @code{(x + f / 2) / f} if @code{x >= 0}, @*
@code{(x - f / 2) / f} if @code{x <= 0}.

@item Add @code{x} and @code{y}:
@tab @code{x + y}

@item Subtract @code{y} from @code{x}:
@tab @code{x - y}

@item Add @code{x} and @code{n}:
@tab @code{x + n * f}

@item Subtract @code{n} from @code{x}:
@tab @code{x - n * f}

@item Multiply @code{x} by @code{y}:
@tab @code{((int64_t) x) * y / f}

@item Multiply @code{x} by @code{n}:
@tab @code{x * n}

@item Divide @code{x} by @code{y}:
@tab @code{((int64_t) x) * f / y}

@item Divide @code{x} by @code{n}:
@tab @code{x / n}
@end multitable
@html
</CENTER>
@end html