129 lines
6.5 KiB
Plaintext
129 lines
6.5 KiB
Plaintext
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Lemma 1:
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If ps_tq is scheduled, ps_tq_active is 1. ps_tq_int() can be called
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only when ps_tq_active is 1.
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Proof: All assignments to ps_tq_active and all scheduling of ps_tq happen
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under ps_spinlock. There are three places where that can happen:
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one in ps_set_intr() (A) and two in ps_tq_int() (B and C).
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Consider the sequnce of these events. A can not be preceded by
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anything except B, since it is under if (!ps_tq_active) under
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ps_spinlock. C is always preceded by B, since we can't reach it
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other than through B and we don't drop ps_spinlock between them.
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IOW, the sequence is A?(BA|BC|B)*. OTOH, number of B can not exceed
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the sum of numbers of A and C, since each call of ps_tq_int() is
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the result of ps_tq execution. Therefore, the sequence starts with
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A and each B is preceded by either A or C. Moments when we enter
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ps_tq_int() are sandwiched between {A,C} and B in that sequence,
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since at any time number of B can not exceed the number of these
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moments which, in turn, can not exceed the number of A and C.
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In other words, the sequence of events is (A or C set ps_tq_active to
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1 and schedule ps_tq, ps_tq is executed, ps_tq_int() is entered,
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B resets ps_tq_active)*.
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consider the following area:
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* in do_pd_request1(): to calls of pi_do_claimed() and return in
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case when pd_req is NULL.
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* in next_request(): to call of do_pd_request1()
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* in do_pd_read(): to call of ps_set_intr()
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* in do_pd_read_start(): to calls of pi_do_claimed(), next_request()
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and ps_set_intr()
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* in do_pd_read_drq(): to calls of pi_do_claimed() and next_request()
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* in do_pd_write(): to call of ps_set_intr()
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* in do_pd_write_start(): to calls of pi_do_claimed(), next_request()
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and ps_set_intr()
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* in do_pd_write_done(): to calls of pi_do_claimed() and next_request()
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* in ps_set_intr(): to check for ps_tq_active and to scheduling
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ps_tq if ps_tq_active was 0.
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* in ps_tq_int(): from the moment when we get ps_spinlock() to the
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return, call of con() or scheduling ps_tq.
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* in pi_schedule_claimed() when called from pi_do_claimed() called from
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pd.c, everything until returning 1 or setting or setting ->claim_cont
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on the path that returns 0
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* in pi_do_claimed() when called from pd.c, everything until the call
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of pi_do_claimed() plus the everything until the call of cont() if
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pi_do_claimed() has returned 1.
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* in pi_wake_up() called for PIA that belongs to pd.c, everything from
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the moment when pi_spinlock has been acquired.
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Lemma 2:
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1) at any time at most one thread of execution can be in that area or
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be preempted there.
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2) When there is such a thread, pd_busy is set or pd_lock is held by
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that thread.
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3) When there is such a thread, ps_tq_active is 0 or ps_spinlock is
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held by that thread.
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4) When there is such a thread, all PIA belonging to pd.c have NULL
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->claim_cont or pi_spinlock is held by thread in question.
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Proof: consider the first moment when the above is not true.
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(1) can become not true if some thread enters that area while another is there.
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a) do_pd_request1() can be called from next_request() or do_pd_request()
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In the first case the thread was already in the area. In the second,
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the thread was holding pd_lock and found pd_busy not set, which would
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mean that (2) was already not true.
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b) ps_set_intr() and pi_schedule_claimed() can be called only from the
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area.
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c) pi_do_claimed() is called by pd.c only from the area.
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d) ps_tq_int() can enter the area only when the thread is holding
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ps_spinlock and ps_tq_active is 1 (due to Lemma 1). It means that
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(3) was already not true.
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e) do_pd_{read,write}* could be called only from the area. The only
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case that needs consideration is call from pi_wake_up() and there
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we would have to be called for the PIA that got ->claimed_cont
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from pd.c. That could happen only if pi_do_claimed() had been
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called from pd.c for that PIA, which happens only for PIA belonging
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to pd.c.
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f) pi_wake_up() can enter the area only when the thread is holding
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pi_spinlock and ->claimed_cont is non-NULL for PIA belonging to
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pd.c. It means that (4) was already not true.
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(2) can become not true only when pd_lock is released by the thread in question.
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Indeed, pd_busy is reset only in the area and thread that resets
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it is holding pd_lock. The only place within the area where we
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release pd_lock is in pd_next_buf() (called from within the area).
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But that code does not reset pd_busy, so pd_busy would have to be
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0 when pd_next_buf() had acquired pd_lock. If it become 0 while
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we were acquiring the lock, (1) would be already false, since
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the thread that had reset it would be in the area simulateously.
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If it was 0 before we tried to acquire pd_lock, (2) would be
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already false.
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For similar reasons, (3) can become not true only when ps_spinlock is released
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by the thread in question. However, all such places within the area are right
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after resetting ps_tq_active to 0.
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(4) is done the same way - all places where we release pi_spinlock within
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the area are either after resetting ->claimed_cont to NULL while holding
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pi_spinlock, or after not tocuhing ->claimed_cont since acquiring pi_spinlock
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also in the area. The only place where ->claimed_cont is made non-NULL is
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in the area, under pi_spinlock and we do not release it until after leaving
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the area.
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QED.
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Corollary 1: ps_tq_active can be killed. Indeed, the only place where we
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check its value is in ps_set_intr() and if it had been non-zero at that
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point, we would have violated either (2.1) (if it was set while ps_set_intr()
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was acquiring ps_spinlock) or (2.3) (if it was set when we started to
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acquire ps_spinlock).
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Corollary 2: ps_spinlock can be killed. Indeed, Lemma 1 and Lemma 2 show
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that the only possible contention is between scheduling ps_tq followed by
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immediate release of spinlock and beginning of execution of ps_tq on
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another CPU.
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Corollary 3: assignment to pd_busy in do_pd_read_start() and do_pd_write_start()
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can be killed. Indeed, we are not holding pd_lock and thus pd_busy is already
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1 here.
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Corollary 4: in ps_tq_int() uses of con can be replaced with uses of
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ps_continuation, since the latter is changed only from the area.
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We don't need to reset it to NULL, since we are guaranteed that there
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will be a call of ps_set_intr() before we look at ps_continuation again.
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We can remove the check for ps_continuation being NULL for the same
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reason - the value is guaranteed to be set by the last ps_set_intr() and
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we never pass it NULL. Assignements in the beginning of ps_set_intr()
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can be taken to callers as long as they remain within the area.
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