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positional analysis

We first introduce well-regulated VCPUs before we discuss how to remove abstraction overhead in compositional analysis for the vC2M systems.

Well-regulated VCPUs

Definition 6.1. A VCPU has well-regulated resource supply patterniff the resource

supply patterns in each of its periods are the same. We call such a VCPU as a well-regulated VCPU.

In other words, if a well-regulated VCPU V Pi does (or does not) execute at t1, the VCPU will (or will not) execute at t1 +k·Πi, where Πi is the VCPU’s period

and k _∈N.

Theorem 6.2. VCPUs are well-regulated VCPUs if they satisfy the following con-

ditions: (1) they use periodic server mechanism to manage their budgets; (2) their periods are harmonic and their release offsets are the same; (3) they are scheduled under EDF scheduling and are schedulable; (4) the scheduler uses a deterministic priority-tie breaking policy: for two VCPUs with the same deadline, the VCPU with a smaller period has higher priority; if they still have the same priority, the VCPU with a smaller index has higher priority.

Proof. We will prove each VCPU satisfying the conditions has a well-regulated re-

source supply.

Because of the condition (1) and (2), the EDF scheduling becomes the fixed- priority scheduling: a VCPU with a smaller period and a smaller index always has higher priority.

For the highest-priority VCPU V P1, it always executes immediately when it starts a new period. So it has a well-regulated resource supply pattern.

For the second-priority VCPU V P2, we will prove it has a well-regulated resource supply pattern by contradiction. Supposet1is the first time when the VCPU supplies differently at t1 and t2 = t1 +k ·Π2, where k ≥ 1. It has two cases: the VCPU executes at t1 but does not executes at t2 and vice versa. In the first case, because the VCPU still has budget at t1 and t1 is the first time when the VCPU’s supply pattern changes, the VCPU should also have budget att2. So the fact that the VCPU

V P2 does not execute at t2 is because the highest-priority VCPU V P1 executes at

t2. The fact that the VCPU V P2 executes at t1 indicates that the highest-priority

VCPU V P1 does not execute at t1. Because VCPUs are harmonic and V P2 has

larger period than V P1, we have Π2 = m·Π1, where m ∈ N and m ≥ 1. Since

t2 =t1+k·Π2 =t1+k·m·Π1, the VCPUV P1 should have the same resource supply at t1 and t2, which contradicts to the observation that the V P1 supplies differently at these two time points. So the first case does not occur. Similarly, we can prove that the second case does not happen either. So it is proved.

By induction, we can prove each VCPU has a well-regulated resource supply.

VCPUs in vC2_M can be configured to well-regulated VCPUs with minor modifi-

cations to Xen RTDS scheduler.

The Xen RTDS scheduler, which is used in vC2_M for scheduling VCPUs, does

not satisfy the condition (4) in Theorem 6.2: the scheduler breaks the priority-tie in an arbitrary order. We add the deterministic priority-tie breaking policy into the Xen RTDS scheduler with 20 lines of change. With the modified RTDS scheduler, the vC2_M system satisfies the condition (4).

The Xen RTDS scheduler uses EDF scheduling. By ensuring the total utilization of VCPUs on each core is no larger than 1,vC2_M satisfies the condition (3).

The release offsets of VCPUs under the Xen RTDS scheduler are always 0. By configuring VCPUs’ periods to be harmonic, vC2_M satisfies the condition (2).

Although VCPUs are implemented as deferrable server under the Xen RTDS

scheduler, vC2_M can run a background CPU-intensive task on each VCPU to turn

them to periodic servers as in [69]. The condition (1) is satisfied. Abstraction overhead-free analysis

We use properties of well-regulated VCPUs to remove the abstraction overhead for the analysis of vC2M.

Lemma 6.3. A well-regulated VCPU V Pi with a PRM model Ωi = (Πi,Θi) always

provides Θi time in any time interval with length Πi.

Proof. We consider a time interval [t1, t1+ Πi] for the VCPUV Pi. If t1 is the start

of a period of the VCPU V Pi, the VCPU provides Θi time in the time interval,

according to the VCPU’s definition.

If t1 is in the middle of a period, we define t0 as the start of the period where ti

resides and let t1 = t0+η. We split the time interval [t1, t1 + Πi] to two intervals [t0+η, t0+ Πi]and(t0+ Πi, t0+η+ Πi]. Because the VCPU’s resource supply repeats

in each period, the resource supply in the second interval (t0 + Πi, t0 +η+ Πi] is

the same with the interval (t0, t0 +η]. By combining the interval (t0, t0 +η] and the interval[t0+η, t0+ Πi], we get the interval[t0, t0+ Πi], which provides Θi time.

The VCPU provides the same amount of time in the time interval [t0, t0+ Πi] and [t1, t1+ Πi]. It is proved.

Lemma 6.4. The resource supply bound function of a well-regulated VCPUV Pi with

a PRM model Ωi = (Πi,Θi) is sbfV Pi(t) =      x_·Θi+ max{t−x·Πi−y,0}, t≥Πi−Θi 0, otherwise where x=_bΠt ic and y= Πi−Θi.

Proof. According to Lemma 6.3, a well-regulated VCPU V Pi always provides Θi

time for any time interval with length Πi. Let the start time of t as the start time

of a hypothetical resource supply period of V Pi, which has length Πi but does not

necessarily overlap with the period of V Pi. The worst-case resource supply occurs

when the VCPU services its resource as late as possible in its hypothetical resource supply period.

When t < Πi − Θi, the VCPU provides 0 time, according to the worst-case

When t_≥Πi−Θi, there areb_Πt_icfull hypothetical periods of V Pi, each of which

provides Θi time. In the last partial hypothetical period oft, the VCPU provides no

resource for Πi−Θi time before it provides continuous resource for the rest of the

period. Therefore, the VCPU provides max_{t_{− b} t

Πic ·Πi−(Πi−Θi),0} in the last

hypothetical period. In total, the VCPU provides b t

Πic ·Θi + max{t− b

t

Πic ·Πi−

(Πi−Θi),0} resource when t >Πi−Θi.

With the tighter resource supply bound function for well-regulated VCPUs, we can remove the abstraction overhead in computing a VCPU’s parameters from its tasks’ resource demand.

Theorem 6.5. A harmonic task sets τi ={τi1, ...τin}, where τ

j i = (p j i, e j i), is schedu-

lable under EDF on a well-regulated VCPU V Pi with a PRM modelΩi = (Πi,Θi) if

Πi = min_τj i∈τip j i and Θi Πi = P τ_ij∈τi ej_i pj_i.

Proof. We first prove a widget to be used later: for any value aand k _≥1, k_{· b}a_{kc ≤}

ba_c. Let a = k _· m + r, where m is the largest possible integer for which r is

nonnegative. We have ba_{c ≥} k _·m. Because _ka = m+ _kr, we get m = _ba_kc, and

k_·m=k_{· b}a_kc. So _ba_{c ≥}k_{· b}a_kc.

Recall that the task setτi is schedulable if it satisfies Theorem 6.1. We now prove

that the dbf of τi is always no larger than the sbf of the VCPU V Pi with the PRM

model Ωi.

Because the task set is harmonic and Πi = min_τj i∈τip j i, each task τ j i’s period is pj_i =k_ij_·Πi, where k j i ∈N, and WCET is e j i =u j i ·p j i =u j i ·k j i ·Πi, where u j i = ej_i pj_i.

The dbf of the task set τi isdbfτi(t) = P τ_ij∈τib t pj_ic ·e j i = P τ_ij∈τib t kj_i·Πic · uj_i·k_ij_·Πj_i.

According to the proved widget above,dbfτi(t)≤ P τ_ij∈τib t Πic·u j i·Πi =b_Πt_ic· P τ_ij∈τiu j i· Πi =b_Πt_ic ·Θ_Πi_i ·Πi =b_Πt_ic ·Θi.

The sbf of the well-regulated VCPU V Pi with the PRM model Ωi satisfies

sbfV Pi(t)≥ b

t

Πic ·Θi ≥dbfτi(t). It is proved.

VCPUs, we can compute each VCPU’s parameters with Theorem 6.5. We remove the abstraction overhead because each VCPU’s bandwidth is equal to its tasks’ total utilization.