**4. Methodology**

In this study, the amortized potential function analysis of the objective is used to examine the performance of the proposed algorithm. Amortized analysis is a worst-case analysis of a sequence of operations—to obtain a tighter bound on the overall or average cost per operation in the sequence than is obtained by separately analyzing each operation in the sequence. The amortized potential method, in which we derive a potential function characterizing the amount of extra work we can do in each step. This potential either increases or decreases with each successive operation, but cannot be negative. The objective of study is to minimize the total IbFt+E, denoted by *G* = *F* + *E*. It reflects that the target is to minimize the quality of service and energy consumed. The input to the problem is the set of jobs *I*. A scheduler generates the schedule *S* of jobs in *I*. The total energy consumption *E* for the scheduling is ∞0 *s*(*t*)<sup>α</sup> *dt*. Let Opt be an optimal offline algorithm such that for any job sequence *I*, IbFt+E *FOpt*(*I*) + *EOpt*(*I*) of Opt is minimized among all schedule of *I*. The notations used in MBS are mentioned in the Table 1. Any online algorithm ALG is said to be *c*-competitive for *c* ≥ 1, if for all job

 are as

sequences *I* and any input the cost incurred is never greater than *c* times the cost of optimal o ffline algorithm Opt, and the following inequality is satisfied:

$$\left(F\_{ALG(I)} + E\_{ALG(I)}\right) \le c \cdot \left(F\_{Opt(I)} + E\_{Opt(I)}\right),$$

The traditional power function is utilized to simulate the working of the proposed algorithm and compare the e ffectiveness by comparing with the available best known algorithm. The jobs are taken of di fferent sizes and the arrival of jobs is considered in di fferent scenario to critically examine the performance of the proposed algorithm. Di fferent parameters (such as IbFt, IbFt+E, speed of processor and speed growth) are considered to evaluate the algorithm.

#### **5. An** *O***(1)-Competitive Algorithm**

An ON-C multiprocessor scheduling algorithm multiprocessor with bounded speed (MBS) is explained in this section. The performance of MBS is observed by using potential function analysis, i.e., the worst-case comparison of MBS with an o ffline adversary Opt. The competitiveness of MBS is *O*(1) with an objective to minimize the IbFt+E for *m* processors with the highest speed (1 + <sup>Δ</sup>/3*m*)η.

#### *5.1. Multiprocessor with Bounded Speed Algorithm: MBS*

At time *t*, the processing speed of *u* adjusts to *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*min impua*(*t*) Ϯ 1/α, η , where 0 < Δ ≤ 1 3α , Ϯ ≥ 1 and α ≥ 2 are constants. The importance *imp*(*j*) of a job is uninformed and acknowledged only at release time *<sup>r</sup>*(*j*). The policies considered for the multiprocessor scheduling MBSfollows:

Job selection policy: The importance-based/weighted round robin is used on every processor.

Job assignment policy: a newly arrived job is allotted to an idle processor (if available) or to a processor having the minimum sum of the ratio of importance to the executed size for all jobs on that processor (i.e., *min nua f* = 1 *impu*(*jf*) *exsu*(*jf*)).

Speed scaling policy: The speed of every processor is scaled on the bases of the total importance of active jobs on that processor. Every active job *ji* on *u* obtains the fraction of speed:

*processors speed importance o f ji total importance o f all active jobs on that processor* 

i.e., *sua*· *impu*(*ji*) *nua k* = 1 *impu*(*jk*) or *sua*· *impu*(*ji*) *impua* . The speed of any processor gets adjusted (re-evaluated) on alteration in total importance of active jobs on that processor. MBS is compared against an optimal offline algorithm Opt, using potential function analysis. The principal result of this study is stated in Theorem 1. The Algorithm 1 of MBS is given next and the flow chart for MBS is given in Figure 2.

**Theorem 1.** *When using more than two processors* (*<sup>i</sup>*.*e*., *m* ≥ 2) *and each processor has the permitted maximum speed* (1 + <sup>Δ</sup>/3*m*)η*, MBS is c-competitive for the objective of minimizing the IbFt*+*E, where c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α = *O*(1) *and* 0 < Δ ≤ 1 3α .

#### **Algorithm 1: MBS (Multiprocessor with Bounded Speed)**

Input: total m number of processors {*<sup>u</sup>*1, ... , *uk*, ... , *um*}, *na* NoAJ *j*1, ... , *ji*, ... , *jna* and the importance of all *na*active jobs *imp*(*j*1), ... , *imp*(*ji*), ... , *imp jna*.

 Output: number of jobs allocated to every processor, the speed of all processors, at any time and execution speed share of each active job.

Repeat until all processors become idle:



5. allocate job *ji* to a processor u with *min nua f*= 1*impu*(*jf*) *exsu*(*jf*)

$$|\mathfrak{G}, imp\_{\mathfrak{u}\mathfrak{a}}| = imp\_{\mathfrak{u}\mathfrak{a}} + imp\_{\mathfrak{u}}(j\_{\mathfrak{l}})$$

$$7. \text{ s}\_{\text{val}} = (1 + \text{A/year}) \cdot \min \left( \left( \frac{\text{imp}\_{\text{us}}}{\text{\text{\textdegree}}} \right)^{1/a}, \eta \right) \text{ where } 0 < \Delta \le \left( \frac{1}{3a} \right) \text{and } \textsuperscript{\text{\textdegree}}{\text{\textdegree}} > 1 \text{ is a constant value.}$$

8. Otherwise, if any job *ji* completes on any processor u and other active jobs are available for execution on that processor then

$$\begin{aligned} \text{9. } imp\_{\text{nat}} &= imp\_{\text{nat}} - imp\_{\text{u}}(j\_l) \\ \text{10. } s\_{\text{int}} &= (1 + \Delta/s\_{\text{f}}) \cdot \min\left(\frac{imp\_{\text{u}}}{\text{T}}\right)^{1/s}, \eta \end{aligned} \text{ where } 0 < \Delta \le \left(\frac{1}{3a}\right) \text{ and } \mathring{\Upsilon} \ge 1 \text{ is a constant value.} $$

11. the speed received by any job *ji*, which is executing on a processor u, is *sua*·*impu*(*ji*) *impua* 

12. otherwise, processors continue to execute remaining jobs

#### *5.2. Necessary Conditions to be Fulfilled*

A potential function is needed to calculate the *c*-competitiveness of an algorithm. An algorithm is called *c*-competitive if at any time *t*, the sum of augmentation in the objective cost of algorithm and the modification in the value of potential is at the most *c* times the augmentation in the objective cost of the optimal adversary algorithm. A potential function Φ(*t*) is required to demonstrate that MBS is *c*-competitive. A *c*-competitive algorithm should satisfy the conditions:

Boundary Condition*:* The value of potential function is zero before the release of any job and after the completion of all jobs.

Job Arrival and Completion Condition: The value of potential function remains same on arrival or completion of a job.

Running Condition: At time when the above condition do not exist, the sum of the (rate of change) RoC of *Ga* and the RoC of Φ is at the most *c* times the RoC of *Go*.

$$\frac{dG\_v(t)}{dt} + \gamma \cdot \frac{d\Phi}{dt} \le c \cdot \frac{dG\_v(t)}{dt}, \text{ where } \gamma > 0. \tag{1}$$

**Figure 2.** Flow chart of the MBS scheduling algorithm.

#### *5.3. Potential Function* Φ(*t*)

An active job *j* is lagging, if (*pwka*(*j*, *t*) − *pwko*(*j*, *t*)) > 0. Since *t* is the instantaneous time, this factor is dropped from the rest of the analysis. For any processor *u*, let *LGu* = *j*1, *j*2, ... , *jlgu* be a group of lagging jobs using MBS and these jobs are managed in the ascending order of latest time (when any job gets changed into lagging job). *LG* = *mu* = 1 *LGu* is a set of all lagging jobs on all *m* processors. Further, *implgu* = *lgu i* = 1 *impu*(*ji*) is the sum of the importance of lagging jobs on a processor *u*. Following this, *implg* = *mu* = 1 *implgu* is the sum of the importance of lagging jobs on all *m* processors. Our potential function Φ(*t*) for IbFt+E is the addition of all potential values of *m* processors.

$$\Phi(t) = \sum\_{u=-1}^{m} \Phi\_u(t) \tag{2}$$

$$\begin{array}{rcl} \clubsuit\_{u} \spadesuit\_{u} (t) &=& \begin{cases} \sum\_{i=1}^{lg\_{u}} \Big( \Big( \sum\_{k=1}^{i} \text{imp}\_{\mu}(j\_{k}) \Big)^{1-2\delta} \Big) \cdot \boldsymbol{\omega}\_{i} & \text{if } \sum\_{k=1}^{i} \text{imp}\_{\mu}(j\_{k}) \le \eta^{1/2\delta} \\\ \sum\_{i=1}^{lg\_{u}} \Big( \frac{1}{1-\delta} \Big) \cdot \Big( \sum\_{k=1}^{i} \text{imp}\_{\mu}(j\_{k}) \cdot \eta^{-1} \Big) \cdot \boldsymbol{\omega}\_{i} & \text{otherwise} \end{cases} \end{array} \tag{3}$$

$$\begin{array}{rcl}\text{Where } \boldsymbol{\omega}\_{i} = \max\{0, \left(pwk\_{a}(j\_{i\cdot}, t) - pwk\_{a}(j\_{i\cdot}, t)\right)\} \\ \boldsymbol{\delta} = \frac{1}{2a} \text{ and} \end{array} \tag{4}$$

$$\left(\sum\_{k=-1}^{i} \operatorname{imp}\_{\mu}(j\_k)\right)^{1-2\delta} \text{ and } \left(\frac{1}{1-\delta}\right)\left(\sum\_{k=-1}^{i} \operatorname{imp}\_{\mu}(j\_k) \cdot \eta^{-1}\right) \tag{5}$$

are the coefficients *ci* of *ji* on processor *u*

MBS is analyzed per machine basis. Firstly, the verification of boundary condition: the value of Φ is zero after finishing of all jobs and prior to release of any job on any processor. There will be no active job on any processor in both situations. Therefore, the boundary condition is true. Secondly, the verification of arrival and completion condition: at time *t*, on release of a new job *ji* in *I*, *ji* without execution is appended at end of *I*. ω*i* is zero as *pwka*(*ji*, *t*) − *pwko*(*ji*, *t*) = 0. The coefficient of all other jobs does not change and Φ remains unchanged. At the time of completion of a job *ji*, ω*i* becomes zero and other coefficients of lagging jobs either remains unchanged or decreases, so, Φ does not increase. Thus the arrival and completion criteria holds true. The third and last criterion to confirm is running condition, with no job arrival or completion.

According to previous discussion, for any processor *u*, let *dGua dt* = *impua* + *sua*<sup>α</sup> and *dGuo dt* = *impuo* + *suo*<sup>α</sup> be the alteration of IbFt+E in an infinitesimal period of time [*t*, *t* + *dt*] by MBS and Opt, respectively. The alteration of Φ because of Opt and MBS in an infinitesimal period of time [*t*, *t* + *dt*] by *u* is *d*Φ*uo dt* and *d*Φ*ua dt* , respectively. The whole alteration in Φ because of Opt and MBS in infinitesimal period of time [*t*, *t* + *dt*] by *u* is *d*Φ*u dt* = *d*Φ*uo dt* + *d*Φ*ua dt* . As this is multiprocessor system therefore to bound the RoC of Φ by Opt and MBS, the analysis is divided in two cases based on *na* and *m*, and then every case is further divided in three sub cases depending on whether *impua* > ηα and *implu* > η<sup>α</sup>, afterwards each sub case is further divided in two sub cases depending on *implg* > *impa* − 3 3+Δ ·*impa* and *implg* ≤ *impa* − 3 3+Δ ·*impa*, where 0 < Δ < 1, μ = 3 3+Δ . The potential analysis is done on individual processor basis, the reason behind it is that all the processors will not face the same case at the same time; rather different processors may face same or different cases.

**Lemma 1.** *For the positive real numbers x, y, A and B, if x*<sup>−</sup><sup>1</sup> + *y*<sup>−</sup><sup>1</sup> = 1 *holds then [2]:*

$$\mathbf{x}^{-1} \cdot \mathbf{A}^{\mathbf{x}} + \mathbf{y}^{-1} \cdot \mathbf{B}^{\mathbf{y}} \ge \mathbf{A} \cdot \mathbf{B} \tag{6}$$

\*\*Lemma 2.\*\*  $\|f\|\_{\mathfrak{u}} \le m$   $and$   $imp\_{l\mathfrak{g}\_{u}} \le \eta^{\alpha}$   $(a) \frac{d\Phi\_{\mathfrak{u}\mathfrak{g}}}{dt} \le \frac{s\_{\mathfrak{u}\mathfrak{g}}}{\alpha} + (1 - 2\delta) \cdot imp\_{l\mathfrak{g}\_{u}};$   $(b) \frac{d\Phi\_{\mathfrak{u}\mathfrak{u}}}{dt} \le -\left(s\_{\mathfrak{u}u} \cdot imp\_{l\mathfrak{g}\_{u}}\right)^{1 - 2\delta}$ 

*Appl. Sci.* **2020**, *10*, 2459

**Proof.** If *na* ≤ *m* then every processor executes not more than one job, i.e., every job is processed on individual processor.

(a) It is required to upper-bound *d*Φ*uo dt* for a processor *u*. To calculate the upper-bound, the worst-case is considered which occurs if Opt executes a job on *u* with the largest coefficient *clgu* = *implgu* 1−2δ. At this time, ω*i* increases at the rate of *suo* (because of Opt on *u*). The count of lagging jobs on some *u* may be only one.

$$\frac{d\Phi\_{\rm no}}{dt} \le c\_{l\xi\_u} \cdot \mathbf{s}\_{\rm no} \le \impproj\_{l\xi\_u} \, ^{1-2\delta} \cdot \mathbf{s}\_{\rm no} \tag{7}$$

Using Young's inequality, Lemma 1 (Equation (6)) in (7) such that *A* = *suo*, *B* = *implgu* <sup>1</sup>−2δ, *x* = α and *y* = 1 1−2δwe have:

$$\frac{d\Phi\_{\rm uo}}{dt} \le \frac{s\_{\rm uo}}{\alpha} + (1 - 2\delta) \cdot imp\_{\mathbb{Q}\_{\rm u}}\tag{8}$$

(b) Next, it is required to upper-bound *d*Φ*ua dt* for a processor *u*. To compute the upper-bound, consider that a lagging job *ji* on *u* is executed at the rate of *sua*· *impu*(*ji*) *nua k* = 1 *impu*(*jk*) or *sua*· *impu*(*ji*) *impua* , therefore, the change in ω*i* is at the rate of −*sua*· *impu*(*ji*) *impua* .

$$\frac{d\varPhi\_{\mu\mu}}{dt} = \sum\_{i=-1}^{l\lg a} \left( \left( \sum\_{k=-1}^{i} \operatorname{imp}\_{\mu}(j\_k) \right)^{1-2\delta} \right) \cdot \left( -s\_{\operatorname{int}} \cdot \frac{\operatorname{imp}\_{\mu}(j\_i)}{\operatorname{imp}\_{\mu\mu}} \right)$$

As only one job executes on a processor, therefore *impu*(*ji*) *impua* = 1 and *lgu* = *i* = 1,

$$\begin{array}{lcl}\frac{d\Phi\_{\text{nu}}}{dt} &= \begin{pmatrix}imp\_{l\S u}^{-1-2\delta} \end{pmatrix} \cdot \begin{pmatrix} -s\_{\text{nu}} \end{pmatrix} \\ \frac{d\Phi\_{\text{nu}}}{dt} &= \begin{pmatrix} s\_{\text{nu}} \cdot imp\_{l\S u}^{-1-2\delta} \end{pmatrix} \end{array} \tag{9}$$


\*\*Lemma 3.\*\*  $If \ n\_{\boldsymbol{\alpha}} \le \boldsymbol{m} \ and \ imp\_{l\S\boldsymbol{w}} > \eta^{\boldsymbol{\alpha}}$  
$$\text{(a) } \frac{d\Phi\_{\text{uv}}}{dt} \le \left(\frac{1}{1-\delta}\right) \cdot imp\_{l\S\boldsymbol{w}} \text{ (b) } \frac{d\Phi\_{\text{uv}}}{dt} = -\frac{(1+\delta/\mathfrak{m})}{(1-\delta)} \cdot imp\_{l\S\boldsymbol{w}}$$

**Proof.** If *na* ≤ *m* then every processor executes not more than one job, i.e., every job is processed on individual processor.

(a) It is required to upper-bound *d*Φ*uo dt* for a processor *u*. To calculate the upper-bound, the worst-case is considered which occurs if Opt executes a job on *u* with the largest coefficient *clgu* = 11−δ ·*implgu* ·η<sup>−</sup>1. At this time, ω*i* increases at the rate of *suo* (because of Opt on *u*) where *suo* ≤ η. The count of lagging jobs on any *u* may be only one.

$$\begin{array}{ll} \frac{d\Phi\_{\text{uv}}}{dt} \le c\_{l\lg u} \cdot s\_{\text{u}\eta} \le c\_{l\lg u} \cdot \eta \ = \left(\frac{1}{1-\delta}\right) \cdot imp\_{l\lg u} \cdot \eta^{-1} \cdot \eta\\ \frac{d\Phi\_{\text{uv}}}{dt} \le \left(\frac{1}{1-\delta}\right) \cdot imp\_{l\lg u} \end{array} \tag{10}$$

(b) Next, it is required to upper-bound *d*Φ*ua dt* for a processor *u*. To compute the upper-bound, consider that a lagging job *ji* on *u* is executed at the rate of *sua*· *impu*(*ji*) *nua k* = 1 *impu*(*jk*) or *sua*· *impu*(*ji*) *impua* , therefore the change in ω*i* is at the rate of −*sua*· *impu*(*ji*) *impua* . *impua* ≥ *impglu* > η<sup>α</sup>, *sua* = (1 + <sup>Δ</sup>/3*m*)·η

$$\frac{d\mathcal{O}\_{\rm un}}{dt} = \sum\_{i=-1}^{l\mathcal{G}\_{\rm u}} \left( \frac{1}{1-\delta} \right) \cdot \left( \sum\_{k=-1}^{i} imp\_{\mathcal{U}}(j\_k) \cdot \eta^{-1} \right) \cdot \left( -s\_{\rm un} \cdot \frac{imp\_{\mathcal{U}}(j\_i)}{imp\_{\rm un}} \right)$$

As only one job executes on a processor, therefore *impu*(*ji*) *impua* = 1 and *lgu* = *i* = 1,

$$\begin{array}{rcl} \frac{d\boldsymbol{\uprho}\_{\rm u}}{dt} &= \left(\frac{1}{1-\delta}\right) \cdot \left(\boldsymbol{im}\boldsymbol{p}\_{\mathcal{S}\_{\rm u}} \cdot \boldsymbol{\upeta}^{-1}\right) \cdot \left(-s\_{\rm ua}\right) \\ &= -\left(\frac{1}{1-\delta}\right) \cdot \left(s\_{\rm ua} \cdot \boldsymbol{im}\boldsymbol{p}\_{\mathcal{S}\_{\rm u}} \cdot \boldsymbol{\upeta}^{-1}\right) \\ &= -\left(\frac{1}{1-\delta}\right) \cdot \left((1+\Delta/3\boldsymbol{m}) \cdot \boldsymbol{im}\boldsymbol{p}\_{\mathcal{S}\_{\rm u}} \cdot \boldsymbol{\upeta}^{-1}\right) \end{array}$$

$$\frac{d\boldsymbol{\uprho}\_{\rm ua}}{dt} = -\frac{(1+\Delta/3\boldsymbol{m})}{(1-\delta)} \cdot \boldsymbol{im}\boldsymbol{p}\_{\mathcal{S}\_{\rm u}} \tag{11}$$


**Lemma 4.** *If na* > *m and implgu*≤ ηα

$$\text{(a) } \frac{d\Phi\_{\text{uv}}}{dt} \le \frac{s\_{\text{u}\text{u}}}{\alpha} + (1 - 2\delta) \cdot imp\_{l\S\text{u}} \text{ (b) } \frac{d\Phi\_{\text{uu}}}{dt} \le -\frac{s\_{\text{u}\text{u}}}{(2 - 2\delta)} \cdot \left(\frac{imp\_{l\S\text{u}}}{imp\_{\text{u}\text{u}}}\right)^{2 - 2\delta} \text{ (b) } \frac{d\Phi\_{\text{uv}}}{dt} \le \text{(c) } \frac{s\_{\text{u}\text{u}}}{\alpha} \cdot \left(\frac{\delta}{2}\right)^{2} \cdot \left(\frac{\delta}{2}\right)^{2} \text{ (c) } \frac{d\Phi\_{\text{uv}}}{dt}$$

(a) It is required to upper-bound *d*Φ*uo dt* for a processor *u*. To calculate the upper-bound, the worst-case is considered which occurs if Opt is executing a job on *u* with the largest coefficient *clgu* = *implgu* 1−2δ. At this time, ω*i* increases at the rate of *suo* (because of Opt on *u*).

$$\frac{d\Phi\_{\rm uo}}{dt} \le c\_{l\xi\_u} \cdot \mathbf{s}\_{\rm uo} \;=\; imp\_{l\xi\_u}^{1-2\delta} \cdot \mathbf{s}\_{\rm uo} \tag{12}$$

Using Young's inequality, Lemma 1 (Equation (6)) in (12) such that *A* = *suo* , *B* = *implgu* 1−2δ, *x* = α and *y* = 1 1−2δwe have:

$$\frac{d\Phi\_{\rm uo}}{dt} \le \frac{s\_{\rm uo}\alpha}{\alpha} + (1 - 2\delta) \cdot \operatorname{imp}\_{l\S\mu} \tag{13}$$

(b) Next, it is required to upper-bound *d*Φ*ua dt* for a processor *u*, to compute the upper-bound consider that a lagging job *ji* on *u* is executed at the rate of *sua*· *impu*(*ji*) *nua k* = 1 *impu*(*jk*) or *sua*· *impu*(*ji*) *impua* , therefore the change in ω*i* is at the rate of −*sua*· *impu*(*ji*) *impua* . To make the discussion straightforward, let *hui* = *ik* = 1 *impu*(*jk*), *hu*0 = 0, *hulgu* = *implgu* and *impu*(*ji*) = *hui* − *hui*−1. (by using Equation (3):

$$\begin{array}{rcl} \frac{d\Phi\_{\mathrm{uu}}}{dt} &= \; \- \, \_{i=1}^{lgs\_{\mathrm{u}}} \left( \left( \sum\_{k=1}^{i} imp\_{u}(j\_{k}) \right)^{1-2\delta} \right) \cdot \left( -s\_{\mathrm{u}u} \frac{imp\_{u}(j\_{k})}{imp\_{\mathrm{uu}}} \right) \\ &= -\frac{s\_{\mathrm{u}u}}{imp\_{\mathrm{u}u}} \; \- \, \_{i=1}^{lgs\_{\mathrm{u}}} \cdot \left( \left( l\_{\mathrm{u}u} \right)^{1-2\delta} \right) \cdot \left( l\_{\mathrm{u}u} - l\_{\mathrm{u}u-1} \right) \\ &\leq -\frac{s\_{\mathrm{u}u}}{mp\_{\mathrm{u}u}} \; \- \, \_{i=1}^{lgs\_{\mathrm{u}}} \cdot \, \int\_{0}^{l} \cdot f^{2-2\delta} \, \, df \\ &\leq -\frac{s\_{\mathrm{u}u}}{mp\_{\mathrm{u}u}} \; \int\_{0}^{l} \cdot f^{2-2\delta} \, \, df \\ &= -\frac{s\_{\mathrm{u}u}}{mp\_{\mathrm{u}u}} \cdot \frac{h\_{\mathrm{u}u}}{(2-2\delta)} \\ &= -\frac{s\_{\mathrm{u}u}}{mp\_{\mathrm{u}u}} \cdot \frac{imp\_{\mathrm{gu}}}{(2-2\delta)} \end{array}$$

$$\frac{d\Phi\_{\mathrm{uu}}}{dt} \leq -\frac{s\_{\mathrm{uu}}}{(2-2\delta)} \cdot \left( \frac{imp\_{\mathrm{gu}}}{imp\_{\mathrm{u}}} \right)^{2-2\delta} \right) \tag{14}$$


**Lemma 5.** *If na* > *m and implgu*> ηα

$$\text{(a) } \frac{d\Phi\_{\text{uv}}}{dt} \le \left(\frac{1}{1-\delta}\right) \cdot imp\_{l\S\text{uv}}; \text{ (b) } \frac{d\Phi\_{\text{uv}}}{dt} \le -\frac{\left(1+\Lambda/\gamma\_{\text{uv}}\right)}{\left(2-2\delta\right)} \cdot \left(\frac{imp\_{l\S\text{uv}}}{imp\_{\text{uv}}}\right)^2$$

**Proof.** If *na* > *m* then: (a) It is required to upper-bound *d*Φuo *dt* for a processor u. To calculate the upper-bound, the worst-case is considered which occurs if Opt executes a job on u with the largest coefficient *<sup>c</sup>*lg*u* = 11−δ ·*imp*lg*u* ·η<sup>−</sup><sup>1</sup> (as *impua* ≥ *imp*lg*u*> η<sup>α</sup>). At this time, ω*i* increases at the rate of *suo* (because of Opt on *u*).

$$\begin{array}{ll} \frac{d\Phi\_{\mathsf{u}\mathsf{u}}}{dt} & \leq c\_{l\mathsf{\cdot}\mathsf{u}\cdot\mathsf{s}\_{\mathsf{u}\mathsf{u}}} \\ & = \left(\frac{1}{1-\delta}\right) \cdot \mathsf{imp}\_{l\mathsf{\cdot}\mathsf{u}} \cdot \eta^{-1} \cdot \mathsf{s}\_{\mathsf{u}\mathsf{u}} \\ & \leq \left(\frac{1}{1-\delta}\right) \cdot \mathsf{imp}\_{l\mathsf{\cdot}\mathsf{u}} \cdot \eta^{-1} \cdot \eta \; \; \{\cdot \colon s\_{\mathsf{u}\mathsf{o}} \leq \eta\} \end{array}$$

$$\frac{d\mathsf{\cdot}\mathsf{p}\_{\mathsf{u}\mathsf{o}}}{dt} \leq \left(\frac{1}{1-\delta}\right) \cdot \mathsf{imp}\_{l\mathsf{\cdot}\mathsf{u}} \tag{15}$$

(b) Next, it is required to upper-bound *d*Φ*ua dt* for a processor *u*. To compute the upper-bound, consider that a lagging job *ji* on *u* is executed at the rate of *sua*· *impu*(*ji*) *nua k* = 1 *impu*(*jk*) or *sua*· *impu*(*ji*) *impua* , therefore the change in ω*i* is at the rate of −*sua*· *impu*(*ji*) *impua* . To make the discussion uncomplicated, let *hui* = *ik* = 1 *impu*(*jk*), *hu*0 = 0, *hulgu* = *implgu* > η<sup>α</sup>, *impua* ≥ *implgu* > ηα and *impu*(*ji*) = *hui* − *hui*−1. Let *z* < *lu* be the largest integer such that *huz* ≤ η<sup>α</sup>. (using Equation (3)):

*d*Φ*ua dt* = *lgu i* = 1 *ci*·−*sua*· *impu*(*ji*) *impua* = − *sua impua* ·*zi* = 1*impu*(*ji*)·(*hui*)<sup>1</sup>−2<sup>δ</sup> + *lgu i* = *z*+<sup>1</sup> 11−δ ·*impu*(*ji*)·*hui*·η<sup>−</sup><sup>1</sup> ≤ − *sua impua* · *huz* 0 *f* <sup>1</sup>−2δ*d f* + 11−δ ·η<sup>−</sup>1· *hulgu huz fdf* = − *sua impua* · *huz* 2−2δ (<sup>2</sup>−2<sup>δ</sup>) + *hulgu* 2−*huz* 2 (<sup>2</sup>−2<sup>δ</sup>)·η = − *sua impua* · *huz* 2 (<sup>2</sup>−2<sup>δ</sup>)*huz* 1/α + *hulgu* 2−*huz* 2 (<sup>2</sup>−2<sup>δ</sup>)·η ≤ − (<sup>1</sup>+<sup>Δ</sup>/3*m*)·η *impua* · *huz* 2 (<sup>2</sup>−2<sup>δ</sup>)η + *hulgu* 2−*huz* 2 (<sup>2</sup>−2<sup>δ</sup>)·η %∵ *huz* ≤ ηα & = − (<sup>1</sup>+<sup>Δ</sup>/3*m*) *impua* · *hulgu* 2 (<sup>2</sup>−2<sup>δ</sup>) = − (<sup>1</sup>+<sup>Δ</sup>/3*m*) (<sup>2</sup>−2<sup>δ</sup>) ·*implgu* 2 *impua d*Φ*ua dt* ≤ −(<sup>1</sup> + <sup>Δ</sup>/3*m*) (2 − 2δ) ·⎛⎜⎜⎜⎜⎝ *implgu* 2 *impua* ⎞⎟⎟⎟⎟⎠ (16)

**Lemma 6.** *At all time t, when* Φ *does not comprise discrete alteration dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt* , *where c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>. *Assume that* γ = 116 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>.

**Proof.** The analysis is divided in two cases based on *na* > *m* or *na* ≤ *m*, and then each case is again alienated in three sub-cases depending on whether *impua* > ηα or *impua* ≤ ηα and *implgu* > ηα or *implgu* ≤ η<sup>α</sup>, afterwards each sub-case is again alienated in two sub-cases depending on whether *implgu* > *impua* − 3 3+Δ ·*impua* or *implgu* ≤ *impua* − 3 3+Δ ·*impua*, where 0 < μ = 3 3+Δ < 1 and Δ = (1/3α). As a job in MBS which is not lagging must be an active job in Opt,

$$\text{imp}\_{\text{uo}} \ge \text{imp}\_{\text{ua}} - \text{imp}\_{\text{lg}\_{\text{u}}} \ge \text{imp}\_{\text{ua}} - (\text{imp}\_{\text{ua}} - \mu \cdot \text{imp}\_{\text{ua}}) \ge \mu \cdot \text{imp}\_{\text{ua}} \Rightarrow \text{imp}\_{\text{ua}} \le \frac{\text{imp}\_{\text{uo}}}{\mu} \tag{17}$$

$$
\mu = \left(\frac{3}{3+\Delta}\right) \tag{18}
$$

$$\gamma = \frac{1}{16} \cdot \left( 1 + \left( 1 + \mathbb{A} / 3^n \right)^a \right) \tag{19}$$

$$\mathcal{L} = \left(\frac{9}{8} + \frac{3\Delta}{8}\right) \left(1 + \left(1 + \Delta/3n\right)^{a}\right) \tag{20}$$


**Case I:** When *na* ≤ *m* and *impua* ≤ η<sup>α</sup>, since *implgu* ≤ *impua* we have *implgu* ≤ η<sup>α</sup>, and *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*minimpua*1/<sup>α</sup>, η = (1 + <sup>Δ</sup>/3*m*)·*impua*1/<sup>α</sup>.

(a) If *implgu* > *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS is *d*Φ*u dt* = *d*Φ*uo dt* + *d*Φ*ua dt* .

(using Equations (8) and (9))

$$\frac{d\Phi\_u}{dt} \le \left(\frac{s\_{uo}}{\alpha} + (1 - 2\delta) \cdot imp\_{l\xi\_u}\right) - \left(s\_{ua} \cdot imp\_{l\xi\_u}\right)^{1 - 2\delta} \tag{21}$$

(by using Equations (1) and (21)) *dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + <sup>γ</sup>·*suo*<sup>α</sup> α + (1 − <sup>2</sup><sup>δ</sup>)·*implgu* − *sua*·*implgu* <sup>1</sup>−2<sup>δ</sup> = *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γα ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)·*impua*1/<sup>α</sup>·*implgu* <sup>1</sup>−2<sup>δ</sup> ≤ γα ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*impua* − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)·*impua*1/<sup>α</sup>·<sup>1</sup> − 3 3+Δ ·*impua*<sup>1</sup>−2<sup>δ</sup> = γα ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ·(<sup>1</sup> − 2δ) − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)· Δ3+Δ <sup>1</sup>−2<sup>δ</sup> ≤ γα ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ − γ· Δ3+Δ <sup>1</sup>−2<sup>δ</sup> ≤ γα ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ − γ· Δ3+Δ = γ α ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 3 3+Δ ≤ γα ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ = γ α ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 116 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> (by using Equation (19)) ≤ γα ·*suo*<sup>α</sup> + *impuo* μ · 1716 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> (by using Equation (17)) = γ α ·*suo*<sup>α</sup> + *impuo*· 1716 ·1 + Δ3 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> (by using Equation (18)) ≤ γα ·*suo*<sup>α</sup> + *impuo* · 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> (by using Equation (20))

$$\frac{dG\_{\rm uu}}{dt} + \gamma \cdot \frac{d\mathcal{O}\_{\rm u}}{dt} \le \frac{\mathcal{Y}}{\alpha} \cdot \mathbf{s}\_{\rm uo}{}^{\alpha} + imp\_{\rm uo} \cdot \mathbf{c} \tag{22}$$

Since *c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> and γ = 116 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>, we have

$$\Rightarrow c = \left(\frac{9}{8} + \frac{3\Delta}{8}\right) \cdot 16 \gamma \Rightarrow \frac{\mathcal{V}}{c} = \frac{1}{18 + 6\Delta} < 1 \Rightarrow \frac{\mathcal{V}}{c} < 1 \Rightarrow \mathcal{V} < c$$

$$\text{Since } \mathcal{V} < c \text{ and } a > 1 \Rightarrow 1 > \frac{1}{\alpha} \Rightarrow \frac{\mathcal{V}}{\alpha} < c \tag{23}$$

(by using Equation (23) in Equation (22))

$$\frac{d\mathbf{G}\_{\text{uur}}}{dt} + \mathbf{\dot{y}} \cdot \frac{d\boldsymbol{\Phi}\_{\text{u}}}{dt} \le \mathbf{c} \cdot \mathbf{s}\_{\text{no}}\,^{\alpha} + \mathbf{c} \cdot \mathbf{i} \text{imp}\_{\text{uo}}\, \quad = \mathbf{c} \cdot (\mathbf{s}\_{\text{no}}\,^{\alpha} + i \text{mp}\_{\text{uo}}\,) \, = \mathbf{c} \cdot \frac{d\mathbf{G}\_{\text{u}}}{dt}$$

Hence the running condition is fulfilled for *na* ≤ *m*, *impua* ≤ η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* > *impua* − 3 3+Δ ·*impua*, *c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>.

(b) If *implgu* ≤ *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS depends on *d*Φ*uo dt*since *d*Φ*ua dt*≤ 0.

(by using Equation (8))

$$\frac{d\Phi\_{\rm ul}}{dt} \le \left(\frac{s\_{\rm no}}{\alpha} + (1 - 2\delta) \cdot \operatorname{imp}\_{l\S\_{\rm ul}}\right) \tag{24}$$

(by using Equations (1) and (24)) *dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + γ· *suo*<sup>α</sup> α + (1 − <sup>2</sup><sup>δ</sup>)·*implgu* = *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ α ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* ≤ γ α ·*suo*<sup>α</sup> + *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*impua* = γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ·(<sup>1</sup> − 2δ) ≤ γ α ·*suo*<sup>α</sup> + *impuo* μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + γ·(<sup>1</sup> − 2δ) (by using Equation (17)) ≤ γ α ·*suo*<sup>α</sup> + 1 μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + γ ·*impuo* = γ α ·*suo*<sup>α</sup> + 1 μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + 116 · 1 + (1 + <sup>Δ</sup>/3*m*) <sup>α</sup>·*impuo* (by using Equation (19)) = γ α ·*suo*<sup>α</sup> + 1716 · 1 + Δ 3 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equation (18)) ≤ γ α ·*suo*<sup>α</sup> + 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* ≤ *<sup>c</sup>*·*suo*<sup>α</sup> + *<sup>c</sup>*·*impuo* (by using Equations (18) and (23)) *dGua dt* + γ· *d*Φ*u dt* ≤ *<sup>c</sup>*·(*suo*<sup>α</sup> + *wuo*) *dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is satisfied for *na* ≤ *m*, *impua* ≤ η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* ≤ *impua* − 3 3+Δ ·*impua* , *c* = 9 8+ 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

**Case II:** When *na* ≤ *m*, *impua* > η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, and *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*minimpua*1/<sup>α</sup>, η = (1 + <sup>Δ</sup>/3*m*)η.

(a) If *implgu* > *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS is *d*Φ*u dt* = *d*Φ*uo dt* + *d*Φ*ua dt* .

(by using Equations (8) and (9))

$$\frac{d\Phi\_u}{dt} \le \left(\frac{\text{s}\_{uo}}{\alpha} + (1 - 2\delta) \cdot \text{imp}\_{l\text{\text{\textdegree}u}}\right) - \left(\text{s}\_{ua} \cdot \text{imp}\_{l\text{\textdegree}u}\right)^{1 - 2\delta} \tag{25}$$

(by using Equations (1) and (25))  $\frac{dG\_m}{dt} + \gamma \cdot \frac{d\Phi\_h}{dt}$ 

 ≤ *impua* + *sua*<sup>α</sup> + <sup>γ</sup>·*suo*<sup>α</sup> α + (1 − <sup>2</sup><sup>δ</sup>)·*implgu* − *sua*·*implgu* <sup>1</sup>−2<sup>δ</sup> = *impua* + (1 + <sup>Δ</sup>/3*m*) α·η<sup>α</sup> + γ α ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)·η·*implgu* 1−2δ ≤ *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ α ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*impua* − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)·*implgu* 2δ·*implgu* 1−2δ = γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*) α + γ·(<sup>1</sup> − 2δ) ·*impua* − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)·*implgu* ≤ γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*) α + γ ·*impua* − γ·(<sup>1</sup> + <sup>Δ</sup>/3*m*)· 1 − 3 3+Δ ·*impua* ≤ γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*) α + γ − γ· Δ 3+Δ ·*impua* = γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*) α + γ· 3 3+Δ ·*impua* ≤ γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*) α + γ ·*impua* ≤ γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*) α + 116 · 1 + (1 + <sup>Δ</sup>/3*m*) α · *impuo* μ (by using Equations (17) and (19)) = γ α ·*suo*<sup>α</sup> + 1716 · 1 + Δ 3 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equation (18)) ≤ γ α ·*suo*<sup>α</sup> + 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equations (20) and (23)) ≤ *<sup>c</sup>*·*suo*<sup>α</sup> + *<sup>c</sup>*·*impuo* = *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* ) *dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is fulfilled for *na* ≤ *m*, *impua* > η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* > *impua* − 3 3+Δ ·*impua* , *c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

(b) If *implgu* ≤ *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS depends on *d*Φ*uo dt* since *d*Φ*ua dt* ≤ 0. (by using Equation (7))

$$\frac{d\Phi\_u}{dt} \le \left(\frac{s\_{uo}}{\alpha} + (1 - 2\delta) \cdot imp\_{l\lg u}\right) \tag{26}$$

(by using Equations (1) and (26)) *dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + <sup>γ</sup>·*suo*<sup>α</sup> α + (1 − <sup>2</sup><sup>δ</sup>)·*implgu* = *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·η<sup>α</sup> + γα ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γα ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* ≤ γα ·*suo*<sup>α</sup> + *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*impua* = γ α ·*suo*<sup>α</sup> + *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ·(<sup>1</sup> − 2δ) ≤ γα ·*suo*<sup>α</sup> + *impuo* μ ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ (by using Equation (17)) = γ α ·*suo*<sup>α</sup> + 1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 116 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>· *impuo* μ (by using Equation (19)) = γ α ·*suo*<sup>α</sup> + 1716 ·1 + Δ3 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impuo* (by using Equation (18)) ≤ γα ·*suo*<sup>α</sup> + 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impuo* ≤ *<sup>c</sup>*·*suo*<sup>α</sup> + *<sup>c</sup>*·*impuo* (by using Equations (20) and (23)) = *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo*) *dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is satisfied for *na* ≤ *m*, *impua* > η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* ≤ *impua* − 3 3+Δ ·*impua*, *c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>.

**Case III:** When *na* ≤ *m*, *impua* > η<sup>α</sup>, *implgu* > η<sup>α</sup>, and *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*minimpua*1/<sup>α</sup>, η = (1 + <sup>Δ</sup>/3*m*)η .

(a) If *implgu* > *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS is *d*Φ*u dt*= *d*Φ*uo dt*+ *d*Φ*ua dt*.

(by using Equations (10) and (11))

$$\frac{d\Phi\_{\rm u}}{dt} \le \left(\frac{1}{1-\delta}\right) \cdot imp\_{l\S\rm u} - \frac{\left(1+\Delta/3m\right)}{\left(1-\delta\right)} \cdot imp\_{l\S\rm u} \tag{27}$$

(by using Equations (1) and (27)) *dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + γ· 11−δ ·*implgu* − (<sup>1</sup>+<sup>Δ</sup>/3*m*) (<sup>1</sup>−<sup>δ</sup>) ·*implgu* = *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·η<sup>α</sup> + γ· 11−δ ·*implgu* − <sup>γ</sup>·(<sup>1</sup>+<sup>Δ</sup>/3*m*) (<sup>1</sup>−<sup>δ</sup>) ·*implgu* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γ· 11−δ ·*impua* − <sup>γ</sup>·(<sup>1</sup>+<sup>Δ</sup>/3*m*) (<sup>1</sup>−<sup>δ</sup>) ·*impua* − 3 3+Δ ·*impua* = *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 11−δ − γ· 11−δ ·(1 + <sup>Δ</sup>/3*m*)· Δ3+Δ ≤ *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 11−δ − γ· 11−δ · Δ3+Δ = *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 11−δ · 3 3+Δ ≤ *impuo* μ ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 116 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>· 11−δ (by using Equations (17) and (19))

$$\frac{dG\_{\rm un}}{dt} + \gamma \cdot \frac{d\phi\_{\rm u}}{dt} \le \frac{imp\_{\rm no}}{\mu} \cdot \left( \left( 1 + \left( 1 + \Lambda/\gamma\_{\rm u} \right)^{\alpha} \right) + \frac{1}{16} \cdot \left( 1 + \left( 1 + \Lambda/\gamma\_{\rm u} \right)^{\alpha} \right) \cdot \left( \frac{2\alpha}{2\alpha - 1} \right) \right) \tag{28}$$

$$\text{Since } a > 1 \Rightarrow \frac{2a}{2a - 1} = \frac{2a - 1 + 1}{2a - 1} = 1 + \frac{1}{2a - 1} < 2 \Rightarrow 1 < \left(\frac{2a}{2a - 1}\right) < 2\tag{29}$$

(by using Equations (29) and (28))

$$\begin{array}{rcl} \frac{d\mathbf{G\_{0x}}}{dt} + \boldsymbol{\gamma} \cdot \frac{d\boldsymbol{\phi\_{0}}}{dt} & \leq & \frac{imp\_{\rm{uo}}}{\mu} \cdot \left( \left( 1 + (1 + \mathbb{A}/3\boldsymbol{u})^{\boldsymbol{a}} \right) + \frac{2}{16} \cdot \left( 1 + (1 + \mathbb{A}/3\boldsymbol{u})^{\boldsymbol{a}} \right) \right) \\ & = & \left( \frac{\boldsymbol{\phi}}{8} + \frac{3\boldsymbol{\Lambda}}{8} \right) \cdot \left( 1 + (1 + \mathbb{A}/3\boldsymbol{u})^{\boldsymbol{a}} \right) \cdot imp\_{\rm{uo}} \text{ (by using Equation (18))} \\ & = & c \cdot imp\_{\rm{uo}} \text{ (by using Equation (20))} \\ & \leq c \cdot (s\_{\rm{uo}} \boldsymbol{\alpha} + imp\_{\rm{uo}}) \\ \frac{d\mathbf{G\_{0x}}}{dt} + \boldsymbol{\gamma} \cdot \frac{d\boldsymbol{\Phi\_{0}}}{dt} & \leq c \cdot \frac{d\mathbf{G\_{0x}}}{dt} \end{array}$$

Hence the running condition is fulfilled for *na* ≤ *m*, *impua* > η<sup>α</sup>, *implgu* > η<sup>α</sup>, *implgu* > *impua* − 3 3+Δ ·*impua*, *c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>.

(b) If *implgu* ≤ *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS depends on *d*Φ*uo dt*since *d*Φ*ua dt*≤ 0.

(by using Equation (10))

$$\frac{d\mathcal{O}\_{\rm u}}{dt} \le \left(\frac{1}{1-\delta}\right) \cdot imp\_{l\mathcal{G}u} \tag{30}$$

(by using Equations (1) and (30))

*dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + γ· 11−δ ·*implgu* = *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·η<sup>α</sup> + γ· 11−δ ·*implgu* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γ· 11−δ ·*impua* = *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 2α 2α−1 ≤ *impuo* μ ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 2γ (by using Equations (17) and (29)) = *impuo* μ ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 216 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> (by using Equation (19)) = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impuo* (by using Equation (18)) = *<sup>c</sup>*·*impuo* (by using Equation (20)) ≤ *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* ) *dGua dt* +γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is satisfied if *na* ≤ *m*, *impua* > η<sup>α</sup>, *implgu* > η<sup>α</sup>, *implgu* ≤ *impua* − 3 3+Δ ·*impua*, for *c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>.

**Case IV**: When *na* > *m* and *impua* ≤ η<sup>α</sup>, since *implgu* ≤ *impua* we have *implgu* ≤ ηα and *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*minimpua*1/<sup>α</sup>, η = (1 + <sup>Δ</sup>/3*m*)*impua*1/<sup>α</sup> .

If *implgu* > *impua* − 3 3+Δ ·*impua*then total RoC of Φ because of Opt and MBS is *d*Φ*u dt* = *d*Φ*uo dt* + *d*Φ*ua dt* . (by using Equations (13) and (14))

$$\frac{d\Phi\_{\rm u}}{dt} \le \left(\frac{s\_{\rm no}}{\alpha} + (1 - 2\delta) \cdot imp\_{l\_{\rm gu}}\right) - \left(\frac{s\_{\rm ua}}{(2 - 2\delta)} \cdot \left(\frac{imp\_{l\_{\rm gu}}2^{-2\delta}}{imp\_{\rm ua}}\right)\right) \tag{31}$$

(by using Equations (1) and (31))

$$\begin{split} &\frac{d\mathbf{g}\_{uu}}{dt} + \boldsymbol{\chi} \cdot \frac{d\boldsymbol{\Phi}\_{\mathbf{u}}}{dt} \\ &\leq \operatorname{imp}\_{\mathrm{u}u} + \boldsymbol{s}\_{\mathrm{u}u} \,^{a} + \boldsymbol{\chi} \cdot \left( \left( \frac{\boldsymbol{s}\_{\mathrm{u}u}}{a} + (1-2\delta) \cdot \operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}} \right) - \left( \frac{\boldsymbol{s}\_{\mathrm{u}u}}{(2-2\delta)} \cdot \left( \frac{\operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}}}{\operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}}} \right) \right) \right) \\ &= \operatorname{imp}\_{\mathrm{u}u} + (1+\Lambda/3\boldsymbol{u})^{a} \cdot \operatorname{imp}\_{\mathrm{u}u} + \frac{\gamma}{a} \cdot \boldsymbol{s}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}} + \boldsymbol{\chi} \cdot (1-2\delta) \cdot \operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}} - \boldsymbol{\chi} \cdot \frac{(1+\Lambda/u)\operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}}}{(2-2\delta)} \cdot \left( \frac{\operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}}}{\operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}}} \right)^{2-2\delta} \\ &\leq \frac{\gamma}{a} \cdot \boldsymbol{s}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}} a + \left( 1 + (1+\Lambda/3u)^{a} \right) \cdot \operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}} \, \boldsymbol{\chi} \cdot \operatorname{imp}\_{\mathrm{|\mathcal{S}\_{\mathbf{u}}}} - \boldsymbol{\chi} \cdot \frac{(1+$$

*Appl. Sci.* **2020**, *10*, 2459

$$\begin{aligned} \lambda &\leq \frac{\mathcal{V}}{\alpha} \cdot \mathsf{s}\_{\mathsf{u}\mathsf{u}} ^{\alpha} + \mathsf{imp}\_{\mathsf{u}\mathsf{u}} \Big( 1 + \left( 1 + \mathsf{A} / \mathsf{s}\_{\mathsf{u}} \right)^{\alpha} + \mathcal{V} \cdot \frac{\left( \frac{\mathsf{A}}{\mathsf{s} + \mathsf{A}} \right)^{2 - 2\delta}}{\left( 2 - 2\delta \right)} \Big) \\\\ &\frac{dG\_{\mathsf{u}\mathsf{u}}}{dt} + \mathcal{V} \cdot \frac{d\Phi\_{\mathsf{u}}}{dt} \le \frac{\mathcal{V}}{\alpha} \cdot \mathsf{s}\_{\mathsf{u}\mathsf{u}} ^{\alpha} + \mathsf{imp}\_{\mathsf{u}\mathsf{u}} \Big( 1 + \left( 1 + \mathsf{A} / \mathsf{s}\_{\mathsf{u}} \right)^{\alpha} + \mathcal{V} \cdot \mathcal{V} \cdot \frac{\left( \frac{\mathsf{A}}{\mathsf{s} + \mathsf{A}} \right)^{2}}{\left( 2 - 2\delta \right)} \Big) \end{aligned} \tag{32}$$

$$\text{Since } a > 1 \Rightarrow \frac{2a}{2a - 1} = \frac{2a - 1 + 1}{2a - 1} = 1 + \frac{1}{2a - 1} > 1\\ \Rightarrow \left(\frac{1}{2 - 2b}\right) = \frac{a}{2a - 1} = \frac{1}{2} \cdot \left(\frac{2a}{2a - 1}\right) > \frac{1}{2} \tag{33}$$

(by using Equations (32) and (33))

*dGua dt* + γ· *d*Φ*u dt* ≤ γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ − γ 2 · Δ 3+Δ 2 = γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ 1 − 1 2 · Δ 3+Δ 2 = γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ <sup>2</sup>Δ2+11Δ+18 2Δ2+12Δ+18 ≤ γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ (by using Equations (17) and (19)) ≤ γ α ·*suo*<sup>α</sup> + *impuo* μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + 116 · 1 + (1 + <sup>Δ</sup>/3*m*) α = γ α ·*suo*<sup>α</sup> + *impuo* μ · 1716 · 1 + (1 + <sup>Δ</sup>/3*m*) α = γ α ·*suo*<sup>α</sup> + 1716 · 1 + Δ 3 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equation (18)) ≤ γ α ·*suo*<sup>α</sup> + 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* ≤ *<sup>c</sup>*·*suo*<sup>α</sup> + *<sup>c</sup>*·*impuo* (by using Equations (20) and (23)) = *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* )

*dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is fulfilled for *na* > *m*, *impua* ≤ η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* > *impua* − 3 3+Δ ·*impua* , *c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

(a) If *implgu* ≤ *impua* − 3 3+Δ ·*impua* then total RoC of Φ because of Opt and MBS depends on *d*Φ*uo dt* since *d*Φ*ua dt*≤ 0.

 (by using Equation (13))

$$\frac{d\Phi\_u}{dt} \le \left(\frac{s\_{\rm no}}{\alpha} + (1 - 2\delta) \cdot \operatorname{imp}\_{l\S u}\right) \tag{34}$$

(by using Equations (1) and (34))

*dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + γ· *suo*<sup>α</sup> α + (1 − <sup>2</sup><sup>δ</sup>)·*implgu* = *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ α ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* ≤ γ α ·*suo*<sup>α</sup> + *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*impua* = γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ·(<sup>1</sup> − 2δ) ≤ γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ (by using Equations (17) and (19)) ≤ γ α ·*suo*<sup>α</sup> + *impuo* μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + 116 · 1 + (1 + <sup>Δ</sup>/3*m*) α = γ α ·*suo*<sup>α</sup> + *impuo* μ · 1716 · 1 + (1 + <sup>Δ</sup>/3*m*) α = γ α ·*suo*<sup>α</sup> + 1716 · 1 + Δ 3 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equation (18)) ≤ γ α ·*suo*<sup>α</sup> + 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* ≤ *<sup>c</sup>*·*suo*<sup>α</sup> + *<sup>c</sup>*·*impuo* (by using Equations (20) and (23)) = *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* ) *dGua dt*+ γ· *d*Φ*u dt*≤ *c*· *dGuo dt*

Hence the running condition is satisfied for *na* > *m*, *impua* ≤ η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* ≤ *impua* − 3 3+Δ ·*impua* , *c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

**Case V:** When *na* > *m* and *impua* > η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, and *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*minimpua*1/<sup>α</sup>, η = (1 + <sup>Δ</sup>/3*m*)η .

(a) If *implgu* > *impua* − 3 3+Δ ·*impua* then the total RoC of Φ because of Opt and MBS is *d*Φ*u dt* = *d*Φ*uo dt* + *d*Φ*ua dt* .

(by using Equations (13) and (14))

$$\frac{d\varPhi\_{\rm u}}{dt} \le \left(\frac{s\_{\rm ua}}{\alpha} + (1 - 2\delta) \cdot imp\_{l\varPsi\_{\rm u}}\right) - \left(\frac{s\_{\rm ua}}{(2 - 2\delta)} \cdot \left(\frac{imp\_{l\varPsi\_{\rm u}}}{imp\_{\rm ua}}\right)\right) \tag{35}$$

(by using Equations (1) and (35)) *dGuad*Φ*u*

*dt* + γ· *dt* ≤ *impua* + *sua*<sup>α</sup> + γ· *suo*<sup>α</sup> α + (1 − <sup>2</sup><sup>δ</sup>)·*implgu* − *sua* (<sup>2</sup>−2<sup>δ</sup>)· *implgu* 2−2δ *impua* = *impua* + (1 + <sup>Δ</sup>/3*m*) α·η<sup>α</sup> + γ α ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*implgu* − γ· (<sup>1</sup>+<sup>Δ</sup>/3*m*)η (<sup>2</sup>−2<sup>δ</sup>) · *implgu* 2−2δ *impua* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ α ·*suo*<sup>α</sup> + γ·(<sup>1</sup> − <sup>2</sup><sup>δ</sup>)·*impua* − γ· (<sup>1</sup>+<sup>Δ</sup>/3*m*)η (<sup>2</sup>−2<sup>δ</sup>) · *implgu* 2−2δ *impua* ≤ γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ − γ· (<sup>1</sup>+<sup>Δ</sup>/3*m*)*implgu* 1/α (<sup>2</sup>−2<sup>δ</sup>) · *implgu* 2−2δ *impua* ≤ γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ − γ· 1 (<sup>2</sup>−2<sup>δ</sup>)· (<sup>1</sup>−( 3 3+Δ ))2 ·*impua*<sup>2</sup> *impua* 

$$\frac{dG\_{\rm uu}}{dt} + \gamma \cdot \frac{d\Phi\_{\rm u}}{dt} \le \frac{\mathcal{Y}}{\alpha} \cdot s\_{\rm uo}{}^{\alpha} + imp\_{\rm uo} \cdot \left(1 + \left(1 + \mathbb{A}/\mathfrak{s}\mathfrak{u}\right)^{\alpha} + \gamma - \gamma \cdot \frac{1}{\left(2 - 2\delta\right)} \cdot \left(\frac{\Delta}{3 + \Delta}\right)^{2}\right) \tag{36}$$

(by using Equations (36) and (33))

$$\begin{split} \frac{d\mathbf{G}\_{\mathrm{na}}}{dt} + \mathsf{y} \cdot \frac{d\boldsymbol{\Phi}\_{\mathrm{n}}}{dt} &\leq \frac{\mathsf{y}}{\alpha} \cdot \mathsf{s}\_{\mathrm{no}} \,^{\alpha} + \mathrm{imp}\_{\mathrm{na}} \cdot \left( 1 + (1 + \mathsf{A}/\mathsf{3}\mathsf{m})^{\alpha} + \mathsf{y} - \frac{\mathsf{y}}{2} \cdot \left( \frac{\mathsf{A}}{3 + \mathsf{A}} \right)^{2} \right) \\ &= \frac{\mathsf{y}}{\alpha} \cdot \mathsf{s}\_{\mathrm{no}} \,^{\alpha} + \mathrm{imp}\_{\mathrm{nr}} \cdot \left( 1 + (1 + \mathsf{A}/\mathsf{3}\mathsf{m})^{\alpha} + \mathsf{y} \left( 1 - \frac{1}{2} \cdot \left( \frac{\mathsf{A}}{3 + \mathsf{A}} \right)^{2} \right) \right) \\ &= \frac{\mathsf{y}}{\alpha} \cdot \mathsf{s}\_{\mathrm{no}} \,^{\alpha} + \mathrm{imp}\_{\mathrm{nr}} \cdot \left( 1 + (1 + \mathsf{A}/\mathsf{3}\mathsf{m})^{\alpha} + \mathsf{y} \left( \frac{2 \mathsf{A}^{2} + 11 \mathsf{A} + 18}{2 \mathsf{A}^{2} + 12 \mathsf{A} + 18} \right) \right) \\ &\leq \frac{\mathsf{y}}{\alpha} \cdot \mathsf{s}\_{\mathrm{no}} \,^{\alpha} + \mathrm{imp}\_{\mathrm{na}} \cdot \left( 1 + (1 + \mathsf{A}/\mathsf{3}\mathsf{m})^{\alpha} + \mathsf{y} \right) \\ \text{By using Equations (17) and (19)} \end{split}$$

$$\begin{array}{c} \text{(by using Equations (17) and (19))}\\ \leq \underset{\alpha}{\overset{\vee}{\alpha}} \cdot \underset{\alpha}{\overset{\alpha}{\cdot}} + \underset{\beta}{\overset{\text{imp}\_{\text{av}}}{\overset{\text{imp}\_{\text{av}}}}}{\right) \cdot \underset{\beta}{\left(1 + (1 + \Delta/\gamma\_{\text{a}})^{\alpha} + \frac{1}{16}\right)}{\left(1 + \Delta/\gamma\_{\text{a}}\right)} \end{array}$$

$$\begin{array}{l} = \frac{\mathsf{Y}}{\mathsf{a}} \cdot \mathsf{s}\_{\mathsf{uo}} \,^{a} + \frac{\mathsf{imp}\_{\mathsf{vo}}}{\mathsf{a}} \cdot \Big( \left[ \frac{\mathsf{Z}}{16} \Big( 1 + (1 + \mathsf{A}/\mathsf{a}\mathsf{u})^{a} \Big) \right) \\ = \frac{\mathsf{Y}}{\mathsf{a}} \cdot \mathsf{s}\_{\mathsf{uo}} \,^{a} + \left( \frac{\mathsf{Z}}{16} \cdot \Big( 1 + \frac{\mathsf{A}}{3} \Big) \Big( 1 + (1 + \mathsf{A}/\mathsf{a}\mathsf{u})^{a} \Big) \right) \cdot \mathsf{imp}\_{\mathsf{uo}} \text{ (by using Equation (18))} \\ \leq \frac{\mathsf{Y}}{\mathsf{a}} \cdot \mathsf{s}\_{\mathsf{uo}} \,^{a} + \left( \left( \frac{\mathsf{Y}}{8} + \frac{\mathsf{A}}{8} \right) \Big( 1 + (1 + \mathsf{A}/\mathsf{a}\mathsf{u})^{a} \Big) \right) \cdot \mathsf{imp}\_{\mathsf{uo}} \\ \leq c \cdot \mathsf{s}\_{\mathsf{uo}} \,^{a} + c \cdot \mathsf{imp}\_{\mathsf{no}} \text{ (by using Equations (20) and (23))} \\ = c \cdot \left( \mathsf{s}\_{\mathsf{uo}} \,^{a} + \mathsf{imp}\_{\mathsf{uo}} \right) \\ \leftarrow \mathsf{s}\_{\mathsf{vo}} \,^{a} \end{array}$$

 ·

1 + (1 + <sup>Δ</sup>/3*m*)

*dGua dt*+ γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is fulfilled for *na* > *m*, *impua* > η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* > *impua* − 3 3+Δ ·*impua* , *c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

(a) If *implgu* ≤ *impua* − 3 3+Δ ·*impua* then total RoC of Φ due to Opt and MBS depends on *d*Φ*uo dt* since *d*Φ*ua dt* ≤ 0.

(by using Equation (13))

$$\frac{d\varPhi\_{\mathsf{u}}}{dt} \le \left(\frac{s\_{\mathsf{u}\mathsf{o}}}{\alpha} + (1 - 2\delta) \cdot \mathsf{imp}\_{l\mathsf{g}u}\right) \tag{37}$$

α

$$\text{(by using Equations (1) and (37))}$$

$$\begin{split} \frac{d\mathbf{G}\_{\mathsf{u}\mathsf{u}}}{dt} + \boldsymbol{\gamma} \cdot \frac{d\boldsymbol{\Phi}\_{\mathsf{u}}}{dt} &\leq \mathsf{imp}\_{\mathsf{u}\mathsf{u}} + \mathsf{s}\_{\mathsf{u}\mathsf{u}}\,^{\mathsf{a}} + \boldsymbol{\gamma} \cdot \left(\frac{\mathsf{s}\_{\mathsf{u}\mathsf{u}}\,^{\mathsf{a}}}{\boldsymbol{\alpha}} + (1 - 2\delta) \cdot \mathsf{imp}\_{l\mathsf{\{\zeta\}}}\right) \\ &= \mathsf{imp}\_{\mathsf{u}\mathsf{u}} + (1 + \mathsf{A}/\mathsf{\{\boldsymbol{\alpha}\}})^{\mathsf{a}} \cdot \boldsymbol{\eta}^{\mathsf{a}} + \frac{\mathsf{y}}{\boldsymbol{\alpha}} \cdot \mathsf{s}\_{\mathsf{u}\mathsf{o}}{}^{\mathsf{a}} + \boldsymbol{\gamma} \cdot (1 - 2\delta) \cdot \mathsf{imp}\_{l\mathsf{\{\zeta\}}} \\ &\leq \frac{\mathsf{y}}{\boldsymbol{\alpha}} \cdot \mathsf{s}\_{\mathsf{u}\mathsf{o}}{}^{\mathsf{a}} + \mathsf{imp}\_{\mathsf{\{\boldsymbol{\alpha}\}}} + \left(1 + \mathsf{A}/\mathsf{\{\boldsymbol{\alpha}\}}\right)^{\mathsf{a}} \cdot \mathsf{imp}\_{\mathsf{\{\boldsymbol{\alpha}\}}} + \mathsf{y} \cdot \mathsf{imp}\_{\mathsf{\{\boldsymbol{\alpha}\}}} \end{split}$$

= γ α ·*suo*<sup>α</sup> + *impua*· 1 + (1 + <sup>Δ</sup>/3*m*) α + γ (by using Equations (17) and (19)) ≤ γ α ·*suo*<sup>α</sup> + *impuo* μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + 116 · 1 + (1 + <sup>Δ</sup>/3*m*) α = γ α ·*suo*<sup>α</sup> + *impuo* μ · 1716 · 1 + (1 + <sup>Δ</sup>/3*m*) α = γ α ·*suo*<sup>α</sup> + 1716 · 1 + Δ 3 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equation (18)) ≤ γ α ·*suo*<sup>α</sup> + 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* ≤ *<sup>c</sup>*·*suo*<sup>α</sup> + *<sup>c</sup>*·*impuo* (by using Equations (20) and (23)) = *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* ) *dGua dt*+ γ· *d*Φ*u dt*≤ *c*· *dGuo dt*

 Hence the running condition is satisfied for *na* > *m*, *impua* ≤ η<sup>α</sup>, *implgu* ≤ η<sup>α</sup>, *implgu* ≤ *impua* − 3 3+Δ ·*impua* , *c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

**Case VI:** When *na* > *m* and *impua* > η<sup>α</sup>, *implgu* > η<sup>α</sup>, and *sua*(*t*) = (1 + <sup>Δ</sup>/3*m*)·*minimpua*1/<sup>α</sup>, η = (1 + <sup>Δ</sup>/3*m*)η.

(a) If *implgu* > *impua* − 3 3+Δ ·*impua* then total RoC of Φ because of Opt and MBS is *d*Φ*u dt* = *d*Φ*uo dt* + *d*Φ*ua dt* .

(by using Equations (15) and (16))

$$\frac{d\varPhi\_{\rm u}}{dt} \le \left( \left( \frac{1}{1-\delta} \right) \cdot \operatorname{imp}\_{l\varPsi\_{\rm u}} \right) - \left( \frac{\left( 1 + \Lambda/3n \right)}{\left( 2 - 2\delta \right)} \cdot \left( \frac{\operatorname{imp}\_{l\varPsi}}{\operatorname{imp}\_{\rm ua}} \right) \right) \tag{38}$$

(by using Equations (1) and (38)) *dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + γ· 1 1−δ ·*implgu* − (<sup>1</sup>+<sup>Δ</sup>/3*m*) (<sup>2</sup>−2<sup>δ</sup>) · *implgu* 2 *impua* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*) α·η<sup>α</sup> + γ· 1 1−δ ·*impua* − γ· (<sup>1</sup>+<sup>Δ</sup>/3*m*) (<sup>2</sup>−2<sup>δ</sup>) · *implgu* 2 *impua* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ· 1 1−δ ·*impua* − γ· 1 (<sup>2</sup>−2<sup>δ</sup>)· ( *impua*−( 3 3+Δ )·*impua*) 2 *impua* = *impua* + (1 + <sup>Δ</sup>/3*m*) α·*impua* + γ· 1 1−δ ·*impua* − γ· ( Δ 3+Δ ) 2 (<sup>2</sup>−2<sup>δ</sup>) ·*impua* = 1 + (1 + <sup>Δ</sup>/3*m*) α + γ· 2α 2α−1 − γ· ( Δ 3+Δ ) 2 (<sup>2</sup>−2<sup>δ</sup>) ·*impua* ≤ 1 + (1 + <sup>Δ</sup>/3*m*) α + 2γ − γ 2 · Δ 3+Δ 2 ·*impua* (by using Equations (29) and (33)) = 1 + (1 + <sup>Δ</sup>/3*m*) α + γ· 2 − 1 2 · Δ 3+Δ 2 ·*impua* = 1 + (1 + <sup>Δ</sup>/3*m*) α + γ· 1 + <sup>2</sup>Δ2+11Δ+18 2Δ2+12Δ+18 ·*impua* ≤ 1 + (1 + <sup>Δ</sup>/3*m*) α + 2γ ·*impua* ≤ *impuo* μ · 1 + (1 + <sup>Δ</sup>/3*m*) α + 216 · 1 + (1 + <sup>Δ</sup>/3*m*) α (by using Equations (17) and (19)) = *impuo* μ · 1816 · 1 + (1 + <sup>Δ</sup>/3*m*) α = 1816 · 1 + Δ 3 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* (by using Equation (18)) = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α ·*impuo* = *<sup>c</sup>*·*impuo* (by using Equation (20)) ≤ *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* ) *dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt* Hence the running condition is fulfilled for *na* > *m*, *impua* > η<sup>α</sup>, *implgu* > η<sup>α</sup>, *implgu*

> *impua* − 3 3+Δ ·*impua* , *c* = 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*) α .

(a) If *implgu* ≤ *impua* − 3 3+Δ ·*impua* then total RoC of Φ due to Opt and MBS depends on *d*Φ*uo dt* since *d*Φ*ua dt* ≤ 0.

(by using Equations (15))

$$\frac{d\Phi\_{\rm U}}{dt} \le \left( \left( \frac{1}{1-\delta} \right) \cdot \operatorname{imp}\_{l\S u} \right) \tag{39}$$

(by using Equations (1) and (39)) *dGua dt* + γ· *d*Φ*u dt* ≤ *impua* + *sua*<sup>α</sup> + <sup>γ</sup>· 11−δ ·*implgu* = *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·η<sup>α</sup> + γ· 11−δ ·*implgu* ≤ *impua* + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impua* + γ· 11−δ ·*impua* ≤ *impua*1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 11−δ = *impua*·<sup>1</sup> + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + γ· 2α 2α−1 ≤ *impuo* μ ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 2γ (by using Equations (17) and (29)) = *impuo* μ ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> + 216 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup> (by using Equation (19)) = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>·*impuo* (by using Equation (18)) = *<sup>c</sup>*·*impuo* (by using Equation (20)) ≤ *<sup>c</sup>*·(*suo*<sup>α</sup> + *impuo* ) *dGua dt* + γ· *d*Φ*u dt* ≤ *c*· *dGuo dt*

Hence the running condition is satisfied for *na* > *m*, *impua* > η<sup>α</sup>, *implgu* > η<sup>α</sup>, *implgu* ≤ *impua* − 3 3+Δ ·*impua*, *c* = 98 + 3Δ8 ·1 + (1 + <sup>Δ</sup>/3*m*)<sup>α</sup>.

The analysis of all cases and sub cases in Lemma 6 prove that the first condition, running condition is fulfilled. Aggregating the discourse about all conditions job arrival and completion condition, boundary condition and Lemma 6, it is concluded that Theorem 1 follows. The competitive values of related algorithms and MBS on α = 2 and 3 are shown in Table 3. Among all online clairvoyant and ON-C scheduling algorithms, the competitiveness of MBS is least, which reflects that the MBS outperforms other algorithms.

#### **6. Illustrative Example**

To observe the performance of MBS, a group of four processors and a set of seven jobs are considered. The best known result in the online non-clairvoyant scheduling algorithms is provided by the Azar et al. [40] in NC-PAR. NC-PAR is a super-constant lower bound on the competitive ratio of any deterministic algorithm even for fractional flow-time in the case of uniform densities. The processing of jobs using algorithms MBS and NC-PAR [40] is simulated and the results are stated in Table 4 as well as in Figures 3–11. The jobs arrived along with their importance but the size of jobs was computed on the completion of jobs. The response time (Rt) is the time interval between the starting time of execution and arrival time of a job. The turnaround time is the time duration between completion time and arrival time of a job. Most of the jobs using MBS have lesser turnaround time than using NC-PAR. The Rt of the jobs using MBS is better than NC-PAR. In Figures 3 and 4, the allocation and execution sequence of jobs on four processors is depicted with the help of triangles and rectangles using NC-PAR and MBS, respectively. As per the Figures 3 and 4, the importance of the jobs in NC-PAR increased with time where as in MBS the importance remains constant during the life time of the jobs. It is clearly evident from the Figures 3 and 4 that on any processor using NC-PAR at a time only one job has been executed, whereas using MBS the processor has been shared by more than one job. The hardware specifications are mentioned in the Table 5.


**Table 4.** Job details and execution data using MBS and NC-PAR.

**Table 5.** Hardware specifications.


**Figure 3.** Scheduling of jobs using NC-PAR.

**Figure 4.** Scheduling of jobs using MBS.

**Figure 5.** Speed of processors using MBS and NC-PAR.

Figures 5 and 6 present the speed of different processors and combined speed of all processors with respect to time using MBS and NC-PAR, respectively. As per the graphs of Figure 5, the speed of a processor using MBS goes high initially but later it reduces and most of the time the speed of processors using MBS is constant, but when processors executes jobs using NC-PAR the speed of processors have heavy fluctuations, which shows that some extra energy may be needed for such frequent fluctuation in NC-PAR. The graphs of the Figure 6 shows that the combined speed of processors using NC-PAR increased and decreased linearly whereas using MBS it increased and decreased stepwise. The count of local maxima and minima in the speed growth graphs (Figure 7) of NC-PAR is more than MBS. Therefore, not only individual processor's speed but also the combined speed of all the processors is reflecting the heavy fluctuation in NC-PAR and varying-constant mixed behaviour of MBS.

**Figure 6.** Combined speed of all processors using MBS and NC-PAR.

**Figure 7.** Growth of combined speed of all processors using MBS and NC-PAR.

**Figure 8.** Total power consumed by all processors using MBS and NC-PAR.

**Figure 9.** Power consumed by processors using MBS and NC-PAR.

In this simulation analysis the traditional power function is used and the value of α is 2. The processors are having the maximum limit of speed which is considered 3.6. The value of Δ = (<sup>3</sup>α)−<sup>1</sup> is considered for the analysis. The power consumed is square of the speed, i.e., proportional to the speed this fact can be viewed by comparing the graphs of Figures 5 and 9. Figure 8, shows that initially MBS consumed more power but power consumption decreased with respect to increase in time, whereas in case of NC-PAR there is no fix pattern, but power consumption is higher most of the time than in MBS.

The graphs of Figure 10 demonstrate the objective of the algorithm (important based flow time plus energy). It reveals that except one processor P1, all other processor have lesser objective value, when these processors executed jobs by using MBS than NC-PAR. The combined objective of all processor is

given in the Figure 11, which strengthen the previous observation of Figure 10 (the objective values using MBS is lesser than using NC-PAR). It can be concluded from the different observations and the Figure 11, that the algorithm MBS performs better than NC-PAR.

**Figure 10.** Importance-based flow time + energy consumed using MBS and NC-PAR.

**Figure 11.** Total importance-based flow time + energy consumed using MBS and NC-PAR.

#### **7. Conclusions and Future Work**

To date, the problem of ON-C scheduling algorithms with an objective to minimize the IbFt+E for multiprocessor setting is studied less extensively. A scheduling algorithm multiprocessor with bounded speed (MBS) is proposed, which uses importance-based/weighted round robin (WRR) for job selection. MBS extends the theoretical study of an ON-C multiprocessor DSS scheduling problem with an objective to minimize the IbFt+E using the bounded speed model, where every processor's maximum speed using MBS is (1 + <sup>Δ</sup>/3*m*)η and using offline adversary Opt is η. The speed of any processor

changes if there is a variation in the total importance of jobs on that processor. The competitiveness of MBS is 9 8 + 3Δ 8 · 1 + (1 + <sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*<sup>Δ</sup>/3*m*) α = *O*(1) against an o ffline adversary, using the potential function analysis and traditional power function. The performance of MBS is compared with best known algorithm NC-PAR [40]. A set of jobs and processors are used to simulate the working of MBS and NC-PAR. The average turnaround and response time of jobs, when they are executed by using MBS is lesser than NC-PAR. The speed scaling strategy and power consumption in MBS is better than NC-PAR. For all processors at any time, MBS provides the lesser value of the sum of important-based flow time and energy consumed than NC-PAR. Competitiveness of NC-PAR is 3 for α = 2 and 3.5 for α = 3, whereas the value of competitive ratio *c* of MBS for Δ = (<sup>3</sup>α) −1 , *m* = 2 and α = 2 is 2.442; for Δ = (<sup>3</sup>α) −1 , *m* = 2 and α = 3 is 2.399; for Δ = (<sup>3</sup>α) −1 , *m* > 2, α = 2 is 2.375 < *c* < 2.442; for Δ = (<sup>3</sup>α) −1 , *m* > 2, α = 3 is 2.333 < *c* < 2.399. These results demonstrate that the scheduling algorithm MBS outperforms other algorithms. The competitive value of MBS is least to date. Before these outcomes, there were no results acknowledged for the multi-processor machines in the ON-C model with identified importance, even for unit importance jobs [40]. The further enhancement of this study will be to implement the MBS in real environment. One open problem is to achieve a reasonably less competitive algorithm than MBS. In this study, author considers non-migratory and sequential jobs and this work may be extended to find a scheduling for migratory and non-sequential jobs. Other factors (such as memory requirement) may also be considered for analysis in future extension.

**Author Contributions:** All authors have worked on this manuscript together. Writing—original draft, P.S.; writing—review and editing, B.K., O.P.M., H.H.A. and G.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflicts of interest.
