1Lemma 1: 2 If ps_tq is scheduled, ps_tq_active is 1. ps_tq_int() can be called 3 only when ps_tq_active is 1. 4Proof: All assignments to ps_tq_active and all scheduling of ps_tq happen 5 under ps_spinlock. There are three places where that can happen: 6 one in ps_set_intr() (A) and two in ps_tq_int() (B and C). 7 Consider the sequnce of these events. A can not be preceded by 8 anything except B, since it is under if (!ps_tq_active) under 9 ps_spinlock. C is always preceded by B, since we can't reach it 10 other than through B and we don't drop ps_spinlock between them. 11 IOW, the sequence is A?(BA|BC|B)*. OTOH, number of B can not exceed 12 the sum of numbers of A and C, since each call of ps_tq_int() is 13 the result of ps_tq execution. Therefore, the sequence starts with 14 A and each B is preceded by either A or C. Moments when we enter 15 ps_tq_int() are sandwiched between {A,C} and B in that sequence, 16 since at any time number of B can not exceed the number of these 17 moments which, in turn, can not exceed the number of A and C. 18 In other words, the sequence of events is (A or C set ps_tq_active to 19 1 and schedule ps_tq, ps_tq is executed, ps_tq_int() is entered, 20 B resets ps_tq_active)*. 21 22 23consider the following area: 24 * in do_pd_request1(): to calls of pi_do_claimed() and return in 25 case when pd_req is NULL. 26 * in next_request(): to call of do_pd_request1() 27 * in do_pd_read(): to call of ps_set_intr() 28 * in do_pd_read_start(): to calls of pi_do_claimed(), next_request() 29and ps_set_intr() 30 * in do_pd_read_drq(): to calls of pi_do_claimed() and next_request() 31 * in do_pd_write(): to call of ps_set_intr() 32 * in do_pd_write_start(): to calls of pi_do_claimed(), next_request() 33and ps_set_intr() 34 * in do_pd_write_done(): to calls of pi_do_claimed() and next_request() 35 * in ps_set_intr(): to check for ps_tq_active and to scheduling 36 ps_tq if ps_tq_active was 0. 37 * in ps_tq_int(): from the moment when we get ps_spinlock() to the 38 return, call of con() or scheduling ps_tq. 39 * in pi_schedule_claimed() when called from pi_do_claimed() called from 40 pd.c, everything until returning 1 or setting or setting ->claim_cont 41 on the path that returns 0 42 * in pi_do_claimed() when called from pd.c, everything until the call 43 of pi_do_claimed() plus the everything until the call of cont() if 44 pi_do_claimed() has returned 1. 45 * in pi_wake_up() called for PIA that belongs to pd.c, everything from 46 the moment when pi_spinlock has been acquired. 47 48Lemma 2: 49 1) at any time at most one thread of execution can be in that area or 50 be preempted there. 51 2) When there is such a thread, pd_busy is set or pd_lock is held by 52 that thread. 53 3) When there is such a thread, ps_tq_active is 0 or ps_spinlock is 54 held by that thread. 55 4) When there is such a thread, all PIA belonging to pd.c have NULL 56 ->claim_cont or pi_spinlock is held by thread in question. 57 58Proof: consider the first moment when the above is not true. 59 60(1) can become not true if some thread enters that area while another is there. 61 a) do_pd_request1() can be called from next_request() or do_pd_request() 62 In the first case the thread was already in the area. In the second, 63 the thread was holding pd_lock and found pd_busy not set, which would 64 mean that (2) was already not true. 65 b) ps_set_intr() and pi_schedule_claimed() can be called only from the 66 area. 67 c) pi_do_claimed() is called by pd.c only from the area. 68 d) ps_tq_int() can enter the area only when the thread is holding 69 ps_spinlock and ps_tq_active is 1 (due to Lemma 1). It means that 70 (3) was already not true. 71 e) do_pd_{read,write}* could be called only from the area. The only 72 case that needs consideration is call from pi_wake_up() and there 73 we would have to be called for the PIA that got ->claimed_cont 74 from pd.c. That could happen only if pi_do_claimed() had been 75 called from pd.c for that PIA, which happens only for PIA belonging 76 to pd.c. 77 f) pi_wake_up() can enter the area only when the thread is holding 78 pi_spinlock and ->claimed_cont is non-NULL for PIA belonging to 79 pd.c. It means that (4) was already not true. 80 81(2) can become not true only when pd_lock is released by the thread in question. 82 Indeed, pd_busy is reset only in the area and thread that resets 83 it is holding pd_lock. The only place within the area where we 84 release pd_lock is in pd_next_buf() (called from within the area). 85 But that code does not reset pd_busy, so pd_busy would have to be 86 0 when pd_next_buf() had acquired pd_lock. If it become 0 while 87 we were acquiring the lock, (1) would be already false, since 88 the thread that had reset it would be in the area simulateously. 89 If it was 0 before we tried to acquire pd_lock, (2) would be 90 already false. 91 92For similar reasons, (3) can become not true only when ps_spinlock is released 93by the thread in question. However, all such places within the area are right 94after resetting ps_tq_active to 0. 95 96(4) is done the same way - all places where we release pi_spinlock within 97the area are either after resetting ->claimed_cont to NULL while holding 98pi_spinlock, or after not tocuhing ->claimed_cont since acquiring pi_spinlock 99also in the area. The only place where ->claimed_cont is made non-NULL is 100in the area, under pi_spinlock and we do not release it until after leaving 101the area. 102 103QED. 104 105 106Corollary 1: ps_tq_active can be killed. Indeed, the only place where we 107check its value is in ps_set_intr() and if it had been non-zero at that 108point, we would have violated either (2.1) (if it was set while ps_set_intr() 109was acquiring ps_spinlock) or (2.3) (if it was set when we started to 110acquire ps_spinlock). 111 112Corollary 2: ps_spinlock can be killed. Indeed, Lemma 1 and Lemma 2 show 113that the only possible contention is between scheduling ps_tq followed by 114immediate release of spinlock and beginning of execution of ps_tq on 115another CPU. 116 117Corollary 3: assignment to pd_busy in do_pd_read_start() and do_pd_write_start() 118can be killed. Indeed, we are not holding pd_lock and thus pd_busy is already 1191 here. 120 121Corollary 4: in ps_tq_int() uses of con can be replaced with uses of 122ps_continuation, since the latter is changed only from the area. 123We don't need to reset it to NULL, since we are guaranteed that there 124will be a call of ps_set_intr() before we look at ps_continuation again. 125We can remove the check for ps_continuation being NULL for the same 126reason - the value is guaranteed to be set by the last ps_set_intr() and 127we never pass it NULL. Assignements in the beginning of ps_set_intr() 128can be taken to callers as long as they remain within the area. 129