Constraints for PF|N Spin Locks

PF|N locks are a hybrid of the P|N locks and F|N locks considered previously:

they ensure that within each priority level requests are satisfied in FIFO order, and each request can be delayed at most once by a lower-priority request.

To begin with, similar to our analysis for P|N locks above, we establish a wait-time bound that provides a bound on the maximum delay encountered as part of single request for a resource `q issued with priority π. This wait-time bound is then used in turn to bound the maximum interference due to higher-priority requests. To this end, for a global resource `q, a task Ti, and a priority π, let Wq^PF|N(T_i, π) denote the smallest positive value that satisfies the following recurrence:

W_q^PF|N(Ti, π) , HP(`q, π) + SP (`q, π) + LP (`q, π) + 1. (6)

Here, HP (`q, π) denotes the maximum delay remote requests with a priority higher than π can contribute to the wait time of J_i’s request, which can be bounded based on the maximum number of jobs that exist during any interval of length Wq^PF|N(T_i, π):

HP(`_q, π) = X

Tx∈τ^R π>πx,q



njobs T_x, W_q^PF|N(T_i, π)
· Nx,q· L_x,q .

SP(`_q, π) accounts for the delay J_i’s request can incur due to remote requests with priority π, which are served in FIFO order:

SP(`q, π) =

m

X

P_k,P_k6=P (Ti)

Txmax∈τ (P_k){Lx,q|πx,q = π}.

Finally, LP (`q, π) accounts for the delay Ji’s request can incur due to remote

lower-priority requests, which in a PF|N lock (similar to a P|N lock) is limited to at most one critical section:

LP(`q, π) = max

Tx∈τ^R{Lx,q|πx,q > π}.

The fixed-point iteration can be aborted if no fixed-point with Wq^PF|N(T_i, π) ≤ di is found. If the recurrence for Wq^PF|N(Ti, π) converges, then it bounds the delay of a single request for `_q issued with priority π.

Lemma 11. Let t0 denote the time a jobJi of task Ti attempts to lock a resource`q with locking priorityπ, and let t1 denote the time that Ji subse-quently acquires the lock for `q. With PF|N locks,t1− t0 ≤ W_q^PF|N(Ti, π).

Proof. Let R denote Ji’s request for `q. In a PF|N lock, at any point in time t ∈ [t<sub>0</sub>, t<sub>1</sub>), J<sub>i</sub> is spinning non-preemptably because either (i) `_q is being used by a job with locking priority (with respect to `q) lower than π, (ii) `q is being used by a job with locking priority equal to π, or (iii) `_q is being used by a job with a locking priority greater than π. We bound the maximum duration for which each of these conditions can hold during an interval of length Wq^PF|N(Ti, π).

Case (i): Since requests are satisfied in priority order when using PF|N locks, R can be delayed by at most one lower-priority request for `q, which is accounted for by LP (`_q, π).

Case (ii): Since requests of equal priority are satisfied in FIFO order when using PF|N locks, with respect to each other processor, R can be delayed by at most one remote request for `q with priority π, for a total of at most SP(`_q, π) time units.

Case (iii): When using PF|N locks, any number of higher-priority requests can delay R. However, analogous to the response-time analysis of non-preemptive fixed-priority scheduling, the maximum number of higher-priority

requests for `<sub>q</sub>that exist during [t<sub>0</sub>, t<sub>1</sub>) bounds the length of the interval since Jiceases spinning and acquires `qas soon as `q is no longer contended. In any interval of length Wq<sup>PF|N</sup>(T<sub>i</sub>, π), at most njobs T<sub>x</sub>, Wq<sup>PF|N</sup>(T<sub>i</sub>, π)
jobs of each remote task Tx with a locking priority πx,q higher than π exist. Each such job issues at most N<sub>x,q</sub> requests for `_q, and holds `<sub>q</sub>for at most L<sub>x,q</sub> time units as part of each request. Each remote task Txwith a higher locking priority (with respect to `_q) hence holds `<sub>q</sub> for at most njobs T<sub>x</sub>, Wq<sup>PF|N</sup>(T<sub>i</sub>, π)
· Nx,q· L<sub>x,q</sub> during any interval of length Wq<sup>PF|N</sup>(Ti, π). The term HP (`q, π) thus bounds the cumulative length that J_i is spinning while a job with a higher locking priority uses `q during any interval of length Wq^PF|N(Ti, π).

Since Wq^PF|N(T_i, π) is by definition the smallest value that satisfies Equa-tion (6) (if one exists), after at most Wq^PF|N(Ti, π) time units after Ji started spinning, `_q is no longer unavailable due to a lower-priority (with respect to

`q) request (case (i)), `q is no longer unavailable due to earlier-issued equal-priority requests (case (ii)), and `<sub>q</sub> is no longer contended by jobs of tasks with higher locking priority (case (iii)). Hence, throughout an interval of length Wq<sup>PF|N</sup>(T<sub>i</sub>, π), resource `_q is unavailable for at most Wq^PF|N(T_i, π) − 1 time units. Thus, Ji’s request is served after at most Wq^PF|N(Ti, π) time

units after it was issued. 

If W_q^PF|N(T_i, π) does not exist, that is, if the recurrence Equation (6) does not converge, then starvation cannot be ruled out and Constraints 14 and 15 do not apply.

Similar to the monotonicity properties of Wq^P|N(Ti, π) (Equations (4) and (5)), W_q^PF|N(Ti, π) is monotonic with respect to scheduling priority and locking

priority:

∀T_h∈ τ^lh: W_q^PF|N(T_i, π) ≥ W_q^PF|N(T_h, π) and (7)

π⁰ < π → W_q^PF|N(T_i, π) ≥ W_q^PF|N(T_i, π⁰). (8)

Based on the wait-time bound Wq^PF|N(Ti, π), we next present constraints on the maximum spin delay incurred by any Ji when using PF|N locks. Recall from Section 6.3.3 that π_q^minLP and π_q^minHP denote the minimum locking priority of any lower-priority and higher-priority task, respectively, on Ti’s processor that accesses the global resource `q. For convenience, we repeat the definitions here:

π_q^minLP , max

Tx∈τ^ll{π_x,q|`_q∈ Q ∧ N_x,q > 0}, π_q^minHP , max

Tx∈(τ^lh∪{Ti}){πx,q|`q∈ Q ∧ Nx,q > 0}.

Similar to Constraint 10 for P|N locks, we can impose a simple constraint on the maximum spin delay due to higher-priority requests.

Constraint 14. In any schedule ofτ with PF|N locks:

∀P_k, P_k6= P (T_i) : ∀`_q ∈ Q^g : ∀T_x ∈ τ (P_k), π_x,q < π_q^minHP :

N_x,qⁱ

X

v=1

X_x,q,v^S ≤ njobs Tx, W_q^PF|N(Ti, π_q^minHP)
· Nx,q· ncs(Ti, q).

Proof. Analogous to Constraint 10. Each request R for `q issued by Ji

remains incomplete for at most Wq^PF|N(T_i, π) time units. Due to the mono-tonicity property stated in Equation (7), this also holds true for any request issued for `<sub>q</sub> by a job of a higher-priority task that preempted J<sub>i</sub>. At most ncs(Ti, q) requests are issued for `q by Ti and local higher-priority tasks while J_i is pending. Hence at most ncs(T_i, q) · njobs T_x, Wq^PF|N(T_i, π)
· Nx,q

requests of each remote task Tx with higher locking priority delay Ji. 

Next, we establish a constraint on arrival blocking due to the non-preemptable spinning of lower-priority jobs that are delayed by remote requests with higher locking priority.

Constraint 15. In any schedule ofτ with PF|N locks:

∀`q ∈ Q^g : ∀Tx∈ τ^R, πx,q < π_q^minLP :

N_x,qⁱ

X

v=1

X_x,q,v^A ≤ njobs Tx, W_q^PF|N(Ti, π_q^minLP)
· Nx,q· Aq.

Proof. Analogous to Constraint 13. A request R issued by a local lower-priority task (with lower-priority at least π_q^minLP) can be delayed by all remote requests for `_qwith higher locking priorities. Exploiting Equations (7) and (8), R remains incomplete for at most Wq^PF|N(Ti, π_q^minLP) time units, which limits the maximum number of jobs of each remote task T_x with a (potentially) higher locking priority to njobs Tx, Wq^PF|N(Ti, π_q^minLP)
. The stated bound on the maximum number of transitively blocking remote requests with higher

locking priorities follows. 

Requests issued with the same locking priority are satisfied in FIFO order.

Hence, the spin delay due to remote equal-priority requests can be constrained similarly to how it is constrained in the analysis of F|N locks.

Constraint 16. In any schedule ofτ with PF|N locks:

∀P_k, P_k6= P (Ti) : ∀`q∈ Q^g : X

Tx∈τ (P_k) πx,q=π^minHP_q

N_x,qⁱ

X

v=1

X_x,q,v^S ≤ ncs(Ti, q).

Proof. Analogous to Constraint 8. Due to the FIFO-ordering of equal-priority requests, it follows that, with respect to each remote processor, at most one

earlier-issued, equal-priority request can delay each of the ncs(T_i, q) requests for `q issued by Ji and local higher-priority jobs. 

As mentioned before, assuming that all requests for `q issued by Ji and local higher-priority jobs are issued with locking priority π^minHP_q is safe due to the monotonicity property stated in Equation (8); the blocking incurred by any J_i does not exceed the bound implied by Constraint 16 if in the actual schedule some requests of Ji or local higher-priority jobs are issued with a locking priority higher than π_q^minHP .

Next, we constrain the maximum transitive delay due to the non-preemptable spinning of lower-priority jobs that are delayed by earlier-issued remote requests with equal locking priority.

Constraint 17. In any schedule ofτ when using PF|N locks:

∀`_q ∈ Q^g : ∀P_k, P_k6= P (T_i) : X

Tx∈τ (P_k) πx,q=π_q^minLP

N_x,qⁱ

X

v=1

X_x,q,v^A ≤ A_q.

Proof. Analogous to Constraint 9. Since requests with the same priority are served in FIFO-order, at most one request per processor for a resource `_q is-sued with the same locking priority can contribute to Ti’s arrival blocking. 

Finally, we constrain the maximum spin delay due to remote requests with lower locking priority.

Constraint 18. In any schedule ofτ with PF|N locks:

∀`q ∈ Q^g : X

Tx∈τ^R πx,q>π^minHP_q

N_x,qⁱ

X

v=1

X_x,q,v^S ≤ ncs(Ti, q).

Proof. Analogous to Constraint 11. Each request for `_q issued by T_i or a local higher-priority job can be delayed at most once by a remote request for

`_q issued with a lower priority. 

Similar reasoning applies to the maximum transitive delay due to the non-preemptable spinning of a lower-priority job that is delayed by a remote request with a lower locking priority.

Constraint 19. In any schedule ofτ with PF|N locks:

∀`q∈ Q^g : X

Tx∈τ^R πx,q>π^minLP_q

N_x,qⁱ

X

v=1

X_x,q,v^A ≤ Aq.

Proof. Analogous to Constraint 12. If J_i is transitively blocked due to a request for `q (i.e., if Aq= 1), then at most one remote request for `q issued with a priority less than π_q^minLP can contribute to T_i’s arrival blocking. 

This concludes our analysis of PF|N locks. Together with the generic Con-straints 1–7, the PF|N-specific ConCon-straints 14–19 define a MILP that bounds the maximum blocking incurred by any J_i. In the unlikely case that the re-currence given in Equation (6) does not converge for some `q, the constraints that depend on the wait-time bound Wq<sup>PF|N</sup>(T<sub>i</sub>, π), namely Constraints 14 and 15, must be omitted from the MILP for this resource `q.

Next, we present constraints for the analysis of preemptable spin locks. We start with generic constraints applicable to all preemptable types considered in this work.