Optimality of Threshold Admission and Greedy Scheduling

1 Optimality of Threshold Admission and Greedy Scheduling Zhe Yu, Shiyao Chen, and Lang Tong, Fellow, IEEE, I. M AIN R ESULT We present the proof of...
Author: Shauna Skinner
1 downloads 0 Views 147KB Size
1

Optimality of Threshold Admission and Greedy Scheduling Zhe Yu, Shiyao Chen, and Lang Tong, Fellow, IEEE,

I. M AIN R ESULT We present the proof of the optimality of Threshold Admission and Greedy Scheduling (TAGS) policy under both average and deterministic settings. We first show that EDF-LMO always maintains low cost sources charging plan feasible. Then under the stochastic setting, the admission problem is formulated as a stochastic dynamic programming problem and the optimality of Threshold Admission policy is shown for identical charging demand case. For deterministic setting, we first establish an upper bound of the optimal competitive ratio by construct an adversary game. Then we show that TAGS achieves the upper bound. II. EDF-LMO The greedy scheduling and outsourcing policy used is simply two conditions to activate chargers. If EV Ji is 1) one of EVs with the earliest M deadlines, or 2) has a 0 laxity, activate its charger and purchase energy from grid if necessary. To analyze the priority of EVs and amount of outsourcing energy, it is intuitive to view the system as there are in total M chargers powered by local renewable chargers and infinitely many chargers powered by expensive outsourcing energy. The scheduler assigns M EVs to low cost chargers following Earliest Deadline First (EDF) principle and other EVs to outsourcing chargers at the last minute when they are about to miss their deadlines (LMO). Proposition 1 shows that EDF-LMO scheduling policy maintains low cost sources charging plan feasible. Proposition 1. Under EDF-LMO, 1) low cost chargers always finish planned low cost charging workload for all EVs accepted; 2) the amount of outsourcing energy is the minimum for all charger activation policies with EDF rule to guarantee feasibility in low cost charging time. Proof: See Appendix.

arrival. Here, the Threshold Admission policy is proved to be optimal for the fixed charging demand scenario. Assume the initial charging demands are identical, i.e., ji = j. The initial laxity and the inter-arrival time are random and I.I.D.. The admission problem can be viewed as a multi-stage stochastic dynamic programming. At the beginning of the ith stage, EV Ji is released and the initial laxity is revealed. The online scheduler determines whether to decline Ji or accept it and use EDF-LMO to schedule. The objective is to maximize the expected total profit from the EVs. The state of the system is defined by the real laxity Xi (ri ) since it is the maximum low cost charging time can be assigned to Ji without affecting other admitted EVs. Note that, if the real laxity, Xi (ri ), of the newly arrived EV, Ji , is so small that accepting it will cause negative profit even at its arrival, it is intuitive to reject Ji . On the other hand, if Xi (ri ) is so large that Ji can be fully charged by the low cost charger (Xi (ri ) > j), since all EVs have the same charging demand, rejecting Ji and accepting some other EV will do no better. Intuitively, we have: Lemma 1. If Xi (ri ) < c−1 c j, the optimal admission policy rejects EV Ji ; if Xi (ri ) ≥ j, the optimal admission policy accepts EV Ji . Proof: See Appendix. The difficulty of determining whether to accept EV Ji at ri is that the decision not only affects the profit collected from Ji but also affects the possible low cost charging time of future EVs. Consider EV Ji and Ji+1 at the arrival ri of Ji . Denote 1 0 Xi+1 (Xi+1 ) the real laxity of Ji+1 if Ji is admitted (declined) at ri , which is a function of the real laxity Xi , the initial laxity li+1 , and the inter-arrival time σi+1 between Ji and Ji+1 . If 1 0 Xi increases, Xi+1 (Xi+1 ) will not decrease. If Xi increases 1 0 by ∆Xi , the increase of Xi+1 (Xi+1 ) should be less or equal to ∆Xi because of possible idleness and the fact that the real laxity of Ji+1 will not exceed di+1 − ri+1 . To summarize, the following lemma holds for any li+1 and σi+1 : Lemma 2.

III. AVERAGE C ASE When the outsourcing energy is costly (c > 1), the scheduler needs to determine whether to accept particular EV at its Z. Yu, S. Chen, and L. Tong are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA. Email: {zy73,sc933,lt35}@cornell.edu. Part of this work was presented at 49th Annual Allerton Conference 2011, Monticello, IL. USA, and the IEEE SmartGridComm 2012. This work is supported in part by the National Science Foundation under Grant CNS1248079 and CNS-1135844.

1 1 Xi+1 (Xi , li+1 , σi+1 ) ≤ Xi+1 (Xi + ∆Xi , li+1 , σi+1 ) 1 ≤ Xi+1 (Xi , li+1 , σi+1 ) + ∆Xi ; 0 0 Xi+1 (Xi , li+1 , σi+1 ) ≤ Xi+1 (Xi + ∆Xi , li+1 , σi+1 ) 0 ≤ Xi+1 (Xi , li+1 , σi+1 ) + ∆Xi ; 1 0 Xi+1 (Xi , li+1 , σi+1 ) ≤ Xi+1 (Xi , li+1 , σi+1 ).

The last relationship is intuitive, accepting Ji will not the low cost charging time of Ji+1 .

2

Denote by Vi∗ (Xi ) the optimal expected to-go reward knowing that the real laxity of the newly released Ji is Xi . The Bellman equation of the ith stage is stated as Vi∗ (Xi ) = max VN∗ (XN ) = max

∗ (X 1 (X , l [j + (Xi − j)− c + EVi+1 i i+1 , σi+1 )), i+1 ∗ (X 0 (X , l EVi+1 , i i+1 σi+1 ))] i+1 [j + (XN − j)− c, 0]

The former term in the Bellman equation indicates accepting EV Ji . The online scheduler will collect j income from EV Ji and pay −(Xi − j)− c for the outsourcing energy. The latter term indicates declining Jn . Clearly, if vN = j + (XN − j)− c ≥ 0, JN will be accepted. The threshold of JN is simply 0. For i < N , introduce function f (Xi ) as the profit difference of acceptance and decline. f (Xi ) ,

∗ 1 j + c(Xi − j)− + EVi+1 (Xi+1 (Xi , li+1 , σi+1 )) ∗ 0 −EVi+1 (Xi+1 (Xi , li+1 , σi+1 )).

If f (Xi ) ≥ 0, accepting Ji is more profitable. Otherwise, the optimal admission policy should reject Ji . According to Lemma 1, f (Xi = j) ≥ 0 and f (Xi = c−1 c j) ≤ 0. Because f is clearly continuous in Xi , there is a zero point between j and c−1 c j. Note for any i, 0 ≤ Vi∗ (Xi + ∆Xi ) − Vi∗ (Xi ) ≤ ∆Xi c. If the available low cost charging time increases by ∆Xi , the scheduler can at most get ∆Xi c more profit by shifting ∆Xi work from outsourcing charger to the low cost charger. Combining this observation and Lemma 2, we have Lemma 3.

Theorem 2. The optimal competitive ratio for deadline scheduling with admission control for single local charger case is given by  1 if c ≤ 1; ∗ √ √ (1) C (1, c) = ( c − c − 1)2 if c > 1, Proof: The case c ≤ 1 is when the unit cost of expensive energy is less than the revenue of unit length charging. In this case, all EVs should be accepted because every EV generates profit even if it is entirely charged by expensive chargers. Therefore, the admission policy is Admit-All policy, i.e., to accept all EVs upon arrival. To show that C ∗ (1, c) = 1 when c ≤ 1, we need to show that the scheduling policy Earliest Deadline First-Last Minute Expensive charger activation with Admit-All admission policy minimizes the expensive charging cost among all online and offline scheduling policies. This is equivalent to show that the amount of work unable to finish by its deadline is minimized, which has been proven in [1] (Theorem 5.1). The case when c > 1 is more complicated. The proof is ∗ deferred √ √ to the 2rest of section where we show that C (1, c) ≤ ( c − c − 1) using an adversary game argument in Section IV-B. In Section IV-C, we analyze Threshold Admission Greedy Scheduling policy and show it achieves the optimal competitive ratio given in (1). B. An Upper Bound of Competitive Ratio

Lemma 3. The function f (Xi ) ,



∗ 1 EVi+1 (Xi+1 (Xi , li+1 , σi+1 ))

j + c(Xi − j) + ∗ 0 −EVi+1 (Xi+1 (Xi , li+1 , σi+1 ))

is monotonely increasing in Xi when Xi ≤ j. Proof: See Appendix. Lemma 3 gives the threshold structure of the optimal admission policy. If Xi is large enough such that f (Xi ) > 0, it is optimal to accept EV Ji . Otherwise, it is optimal to reject EV Ji . The optimal admission policy is stated as follows: Theorem 1. The Threshold Admission policy is optimal for the identical charging demand case. For any EV Ji , there is a threshold τi such that if and only if Xi ≥ τi , Ji is admitted. Remark 1. Within the identical charging demand case, the threshold τi on Xi (ri ) is equivalent to a threshold νi on the EV value vi = j + c(Xi − j)− where νi = j + c(τi − j)− . The optimal threshold τi depends not only on the cost of outsourcing energy but also on the distribution of the initial laxity li and inter-arrival time σi . IV. D ETERMINISTIC C ASE A. Optimal Competitive Ratio The optimal competitive ratio for single charger case (denoted by C ∗ (1, c), where the first argument indicates the number of local chargers) depends on the (normalized) marginal outsourcing cost c as follows.

In deriving an upper bound of competitive ratio when c > 1, we identify the worst case instance by introducing a game between the online scheduler and an adversary. Such an approach has been considered by Koren and Shasha in [2]. We first specify √ the√game followed by analysis that establishes C ∗ (1, c) ≤ ( c − c − 1)2 . The adversary plays the dual role of an offline scheduler and the one who selects instances in order to minimize the competitive ratio. The basic adversarial strategy is to feed the online scheduler with unfavorable future releases, depending on the online scheduler’s decision history, so that a tight upper bound on the optimal competitive ratio can be obtained. 1) The Adversary Game: The game between the online scheduler and the adversary has multiple rounds. The key component of the adversary strategy is the use of tight EVs. Herein, an EV is called tight if the initial laxity is zero, meaning that the EV has to be charged immediately in order to meet the deadline. Before starting the game, the adversary chooses an arbitrary integer n to generate an EV instance with at most n EVs, so that the game lasts at most n rounds. Note that n is unknown to the online scheduler. An illustration of the game is given in Figure 1 (for n = 4). Round 1: The first tight EV J1 is released at time 0 with length j1 = 1. If the online scheduler declines J1 , the offline adversary will not release more EVs thus end the game. The adversary, as the offline scheduler realizes the reward j1 while the online scheduler realizes 0 reward. The profit ratio for this instance is 0. If the online scheduler accepts J1 , the game enters Round 2.

3

accept

stop releasing ratio

accept

accept

release J3

release J2 decline

accept release J1 decline release J0 decline ratio decline

ratio

ratio

j3 −(c−1)(j0 +j1 +j2 ) j3

j2 −(c−1)(j0 +j1 ) j3

j1 −(c−1)j0 j2

j0 j1

ratio 0 j0

Fig. 1: Illustration of the adversary game for n = 4. Round 2: Upon the online scheduler accepting J1 , the adversary chooses to release another tight EV J2 at time ǫ (ǫ is chosen to be small enough so that J2 arrives immediately after J1 ). The charging demand of J2 is j2 > j1 so that the online scheduler is tempted with a more profitable EV. Similar to Round 1, if the online scheduler declines J2 , the adversary ends the game by not releasing anymore EVs. In this case, the online scheduler is rewarded with j1 whereas the adversary scheduler, having the foresight that the second EV to come is more profitable, rejects the first EV and accepts the second. The profit ratio is for this instance is j1 /j2 . If the online scheduler accepts J2 , knowing that expensive energy is necessary, the game enters Round 3. Round k: Upon the online scheduler accepting Jk , the adversary chooses to release another tight EV Jk at time (k −1)ǫ with charging demand jk+1 > jk . If the online scheduler declines Jk , the adversary ends the game P by stop releasing P k−1 j −c k−2 j additional EVs. The profit ratio realized is i=1 i jk i=1 i . Round n (Termination round): If the game is in this round, the online scheduler has accepted all n−1 EVs. The adversary releases the last EV Jn . P Ifn−1 the online scheduler declines Jn , the Pn−2 i=1 ji i=1 ji −c . On the other hand, profit ratio realized is jn ifPthe online scheduler accepts J , the profit ratio realized is n Pn−1 n i=1 ji −c i=1 ji . jn 2) Analysis of Adversary Game: The use of adversary and tight EVs allows us to identify specific EV sequences that upper bound the competitive ratio. The analysis of the game leads to the following Lemma: Lemma 4. For single local charger scenario and unit expensive energy cost c > 1, the optimal competitive ratio can be upper bounded by √ √ C ∗ (1, c) ≤ ( c − c − 1)2 .

(2)

The proof of Lemma 4 follows the three steps below. Detailed proof can be found in Appendix. 1) Enumerate all possible admission decisions by the online scheduler, and compute all possible realized profit ratios as a function of the charging demand ji ’s of the EVs released; 2) Take the maximum over the realized profit ratios, which serves as an upper bound on C ∗ (1, c) (since the maximum is taken over all possible actions of the online scheduler);

3) Design the charging demand ji ’s of the EVs released to equate the maximum over the realized profit ratios with √ √ ( c − c − 1)2 . C. Optimality of TAGS D. Analysis of Achievability for TAGS √ prove that TAGS achieves the competitive ratio √We now ( c − c − 1)2 in single local charger case, equal to the adversarial upper bound. We follow the steps below in Section IV-D1, IV-D2 and IV-D3. 1) Upper bounding the optimal offline profit in pressing busy intervals Bp with maximum deadline of pressing EVs in Bp , and the optimal offline profit in non-pressing busy intervals Bn with charging demand of non-pressing EVs in Bn 2) Analyzing the structure of pressing busy intervals induced by TAGS to bound the maximum deadline of pressing EVs 3) Optimize √ parameter β to show competitive ratio √ ( c − c − 1)2 . 1) Busy Interval Structure: We first investigate the structure of pressing and non-pressing busy intervals induced by TAGS on an arrival EV sequence. There may be more than one pressing and non-pressing busy intervals for an arrival EV sequence. The busy interval structure applies to every one of them. Without loss of generality, when we later refer to pressing or non-pressing busy intervals, assume the starting time of the interval is 0. 1) Start with non-pressing busy interval, since it has simple structure: there are no EVs ever declined or accepted via the threshold admission procedure in non-pressing busy interval, and all EVs are accepted as non-pressing EVs and finished without expensive energy. For non-pressing busy intervals Bn , the optimal offline profit, the profit TAGS obtains, and the busy interval length are all equal. (Interval [7, 8] in Table I can be used as an example.) 2) For pressing busy interval, the structure is more complicated: there may be EVs released and classified as pressing, but declined due to failed threshold admission; there may be EVs released and classified as pressing, and accepted via the threshold admission. Both EVs accepted by threshold admission and EVs failed the threshold admission may convert previously non-pressing pending

4

EV index 0 1 2 3 4 5

rn 0 1 2 3 4 7

jn 2 3 1 2 1 1

dn 3 4 6 6 5 10

Admission decision admitted admitted admitted admitted declined admitted

TABLE I: Example of an input EV sequence EVs to pressing ones upon its release. (Still use the example in Table I: at time t = 0, EV 0 is non-pressing; at time t = 1, the release of EV 1 turns EV 0 to pressing.) For pressing busy intervals Bp , denote by P the set of pressing EVs released in Bp . The optimal offline profit ever achievable for P is no greater than the latest deadline in the EV set P minus the beginning of interval Bp , since no matter how many pressing EVs are released, they all contend for the time period from beginning of Bp to the latest deadline in the EV set P. (Interval [0, 6] in Table I can be used as an example.) 2) Upper Bounding Latest Deadline: Next we establish bound on the latest deadline for Bp in Lemma 5. To this end, we first investigate the detailed structure of Bp . In the pressing busy interval Bp , there could be multiple EV arrivals that raise the threshold admission procedure, some pass while others fail. The structure of Bp can be viewed as a repeated procedure of accepting EVs via threshold admission. Over the course of accepting EVs via threshold admission, some of the previously non-pressing EVs are converted to pressing EVs (as an example, in Table I EV 0 is converted from non-pressing to pressing when EV 1 is released, and EV 2 is converted to pressing when EV 3 is released). Specifically, we introduce the notations to represent this structure of pressing busy interval Bp . Denote the EVs that go through and pass the threshp,threshold old admission upon arrival in Bp by J1p,threshold , . . . , Jm p,threshold ’s are pressing by definition) in chronical order of (Ji p,threshold release (r1p,threshold ≤ . . . ≤ rm ). The planned expensive charging time of Jip,threshold is denoted by oi . Then the repeated structure begins: 1) When J1p,threshold arrives and gets accepted, denote by J0p,converted the set of EVs turned to pressing from nonpressing due to J1p,threshold (J0p,converted is explicitly used in Eq. (10) and (11) in the proof of Proposition 2). p,threshold 2) Similarly, when Ji+1 arrives and gets accepted, dep,converted the set of additional EVs turned to note by Ji pressing from non-pressing. p,converted 3) Denote by Jm the set of pressing EVs that are p,threshold turned pressing after the admission of Jm in Bp . p,threshold Define Bi to be the time period length from ri (release time of Jip,threshold ) to the end of the tentative pressing busy p,threshold , and interval evaluated in the declining option for EV Ji+1 gi to be the tentative profit value evaluated in the declining p,threshold . option for EV Ji+1 Example for Bi and gi : Still illustrate with the example in Table I for i = 1. EV 1 and EV 3 is the first (J1p,threshold ) and second (J2p,threshold) EV admitted via threshold admission in

the pressing busy interval. Thus, release time of J1p,threshold is rip,threshold = 1. The end of the tentative pressing busy interval evaluated in the declining option for J2p,threshold (EV 3) is 5 (EV 2 is converted to pressing upon arrival of EV 3). Thus, B1 = [1, 5]. The tentative profit g1 evaluated in the declining option for J2p,threshold(EV 3) is (2 + 3 + 1) − c × (0 + 1 + 0). We first explain with a qualitative argument why the latest deadline in Bp can be upper bounded: The deadline of both accepted pressing EVs (denoted by S p ) and declined pressing EVs (denoted by D) cannot be far reaching into the future relative to the length of the pressing busy interval, since otherwise they will instead be non-pressing EVs. Now we present Lemma 5, which formally establishes the upper bound of the deadlines of the pressing EVs either accepted or declined during Bp . Lemma 5. Suppose J p is either declined, or an accepted p,threshold pressing EV arrived between rip,threshold (inclusive) and ri+1 (non-inclusive) in the pressing busy interval Bp . Then the deadline of J p can be bounded with dp ≤ rip,threshold + cBi + βgi . Proof: See Appendix. Remark 1. When β increases, upper bound on dp provided by Lemma 5 becomes looser. This can be explained as follows: with a large threshold, TAGS has a higher standard in accepting EVs that incur expensive charging. Therefore TAGS chooses to decline more opportunity due to the fear of the expensive energy cost. More declination translates to longer time period in which TAGS is not collecting revenue while the offline schedule potentially can with a priori knowledge in admission. After bounding the deadlines of pressing EVs by rip,threshold + cBi + βgi , we still need an inequality bounding the quantity rip,threshold + cBi + βgi , provided in Proposition 2. We will use the short notation ri for rip,threshold from here on. Proposition 2. The following inequality holds 1 β ri + Bi ≤ gi . c 1+β−c

(3)

The ratio between profit gi obtained by TAGS and 1c ri + Bi indicates the efficiency of TAGS in turning local charging time into profit. The disadvantage for TAGS in admission is the unawareness of the future releases, and therefore TAGS may sometimes regret accepting a previous EV afterwards. These EVs that ideally should not have been accepted will incur expensive energy cost and erode the revenue TAGS collects. Proposition 2 lower bounds this ratio by 1+β−c , confirming β the efficiency of TAGS to turn busy time into profit. The proof can be found in Appendix. β decreases and ProposiRemark 2. When β increases, 1+β−c tion 2 provides a tighter upper bound on 1c rm + Bm . This can be explained as follows: with a large threshold, TAGS has a high standard in accepting EVs that incur expensive charging cost. Therefore less expensive energy is accumulated and this reduction in expensive energy boosts the efficiency in turning busy time into profit.

5

3) Optimize Threshold: Now we are ready for Step 3, expressing competitive ratio as a function of β and optimize over β. Lemma 6. The optimal √ competitive ratio for single local √ charger case C ∗ (1, c) ≥ ( c − c − 1)2 . Proof: See Appendix.

F. Proof of Lemma 1

A PPENDIX E. Proof of Proposition 1 •

Proof: Suppose on one charger at time t, there are EVs J1 , . . . , Jn (in increasing order in deadline) in the system with remaining low cost charging times ˆj1 , . . . , ˆjn . We first show that at time t, EDF rule can finish the remaining charging times ˆj1 , . . . , ˆjn of EVs J1 , . . . , Jn on a charger, if the laxity vector (L1 , . . . , Ln ) has Li ≥ 0 for all i. To show this statement, according to EDF rule, EV Ji Pi will be finished at time t + m=1 jm . Since Li = di − Pi (t + m=1 jm ) ≥ 0 for all i, it is shown that EDF rule meets every deadline di successfully. Now consider a new EV Jn+1 accepted to a system with EVs J1 , . . . , Jn , and non-negative laxity vector Li ≥ 0 for all i. After accepting Jn+1 and before outsourcing energy planning for Jn+1 , the laxity vector (L′1 , . . . , L′n , L′n+1 ) calculated for J1 , . . . , Jn , Jn+1 using charging demand jn+1 for the remaining low cost charging time of Jn+1 may contain negative components for the EVs with deadline no earlier than Jn+1 . With the outsourcing charging time for Jn+1 , on+1 = max{0, − min1≤i≤n+1 L′i }, the following increase in laxity vector takes place. – The EVs with deadlines before that of EV Jn+1 will always complete successfully with EDF rule, with or without outsourcing charging time planning for EV Jn+1 . The laxity components corresponding to these EVs will not change, and remain positive. – For the EVs with deadlines no earlier than that of EV Jn+1 , after charge the amount on+1 using expensive source, their laxity component will increase by on+1 due to the reduction in low cost charging time for EV Jn+1 . The choice of on+1 ensures non-negativity of the laxity components, i.e., L′i + on+1 ≥ 0 (shown in Eq. (4)). L′i + on+1

= L′i + max{0, − ≥ L′i −



Then the laxity vector still has negative entry in the component arg min1≤i≤n+1 L′i after planning outsourcing energy for EV Jn+1 . This EV with negative laxity component cannot finish under EDF rule.

min

1≤i≤n+1

min

1≤i≤n+1 L′i } ≥ 0.

L′i } (4)

Therefore after outsourcing charging time planning for Jn+1 , laxity vector is non-negative. Thus, we have shown that EDF-LMO can always finish low cost charging workload for all EVs accepted. To show the amount of outsourcing energy is the minimum for all policies with EDF rule, suppose the newly released EV J is planned for an outsourcing energy strictly less than o = max{0, − min1≤i≤n+1 L′i }, where (L′1 , . . . , L′n+1 ) is the laxity vector for EVs J1 , . . . , Jn+1 .

Proof: The first half is trivial. If Xi (ri ) < c−1 c j, accepting EV Ji will end up with negative income while rejection will give zero profit. For the second half, assume Xi (ri ) ≥ j and EV Ji is rejected by some admission policy π. We consider two cases as follow. • Case 1: Policy π accepts no more EV after Ji . Construct policy π ′ exactly follows π but accepts Ji . Since Xi (ri ) ≥ j, policy π ′ earns j more profit than π. • Case 2: Policy π accepts some EVs after Ji . Assume among those accepted EVs afterwards under policy π, EV Ji′ has the earliest deadline. Consider admission policy π ′ which does exactly the same as policy π except accepting EV Ji instead of Ji′ . If di′ ≤ di , the feasible region of EV Ji is larger than that of Ji′ since ri′ ≥ ri . Under admission policy π ′ , construct a schedule policy EDF-LMO-prime which schedules the same charging sequence as EDF-LMO does under admission policy π and earns the same profit. Since EDF-LMO is optimal under any admission policy, EDFLMO-prime is suboptimal under policy π ′ . So we have Vπ−EDF-LMO = Vπ′ −EDF-LMO-prime ≤ Vπ′ −EDF-LMO which indicates accepting Ji does no worse than rejecting it. If di′ > di , we will also construct a similar scheduling policy EDF-LMO-prime. As shown in Fig. 2, assume under policy π, the real laxity for EV Ji′ as defined in Definition ?? is from t′1 to t′2 at its arrival. The real laxity for EV Ji is assumed to start from t1 at its arrival. Under policy π, EDF-LMO schedules two possible sets of EVs between t1 to t′1 ∗ . One set is EVs which were accepted before Ji and Ji′ and have deadlines later than Ji but earlier than Ji′ (denoted by Ji−1 ). The other is EVs accepted after Ji but earlier than Ji′ (denoted by Ji+1 ). Ji+1 has later deadlines than Ji′ by definition of Ji′ but it may be charged before Ji′ because of possible idleness. Under policy π ′ which accepts Ji instead of Ji′ , we construct suboptimal scheduling policy EDF-LMO-prime as follows. Starting from t1 , charge Ji the same amount as EDF-LMO charges Ji′ under π. Then charges Ji−1 and Ji+1 sequentially the same amount to t′2 . After t′2 , EDF-LMO-prime schedules the same charging sequence as EDF-LMO. EDF-LMO-prime is feasible. First of all, Xi ≥ j ≥ ˆji′ , so we can charge Ji by ˆji′ starting from t1 . Secondly, denote ˆji−1 (ri ) the low cost charging amount of Ji−1 planned when calculating Xi . Since EDF-LMO does not change the low cost charging time after acceptance, ˆji−1 (ri ) ≥ ∗ Since

ri ≤ ri′ , di ≤ di′ , we know t1 ≤ t′1 .

6

ˆji−1 . Xi ≥ j indicates that after charge Ji by j, the low cost charger can still charge Ji−1 by ˆji−1 (ri ) which makes shifting the scheduling order feasible. In the end, shifting ˆjn+1 is feasible because dn+1 ≥ dn′ > dn by assumption. The constructed scheduling policy EDF-LMO-prime has the same profit as EDF-LMO, which makes admission policy π ′ associated with EDF-LMO has a profit no worse than π associated with EDF-LMO. Ti Ji

j Ti′ j

Ji′

Ji−1 Ji+1 ˆ ji−1 ˆ ji+1

Low cost Charger 0

t1

Ji′ ˆ ji′

Ji−1 ˆ ji−1

Ji ˆ ji = ˆ ji′

Low cost Charger 0

t1

t′1

π di t′2

t′1

di′

Ji+1 ˆ ji+1 di t′2

time π′

di′

time

Fig. 2: Construction of EDF-LMO-prime.

G. Proof of Lemma 3 Proof: For simplicity, the functions 0 1 Xi+1 (Xi , li+1 , σi+1 ) and Xi+1 (Xi , li+1 , σi+1 ) will be 0 1 written as Xi+1 (Xi ) and Xi+1 (Xi ) in short. Note ∗ ∗ 0 ≤ Vi (Xi + ∆Xi ) − Vi (Xi ) ≤ ∆Xi c. That is, if the real laxity of EV Ji is increased by ∆Xi , the optimal reward is increased at most by ∆Xi c, which indicates shifting ∆Xi work from expensive sources to local renewables. And we have f (Xi + ∆Xi ) − f (Xi ) =

∆Xi c ∗ 1 +EVi+1 (Xi+1 (Xi + ∆Xi )) ∗ 1 −EVi+1 (Xi+1 (Xi )) ∗ 0 −EVi+1 (Xi+1 (Xi + ∆Xi )) ∗ 0 +EVi+1 (Xi+1 (Xi )).

Since 1 1 Xi+1 (Xi + ∆Xi , ) ≥ Xi+1 (Xi ),

we have 1 ∗ 1 ∗ (Xi+1 (Xi )) ≥ 0. EVi+1 (Xi+1 (Xi + ∆Xi )) − EVi+1

And by Lemma 2, 0 0 Xi+1 (Xi + ∆Xi ) ≤ Xi+1 (Xi ) + ∆Xi .

We have ∗ 0 EVi+1 (Xi+1 (Xi + ∆Xi ))

∗ 0 ≤ EVi+1 (Xi+1 (Xi ) + ∆Xi ) ∗ 0 ≤ EVi+1 (Xi+1 (Xi )) + ∆Xi c

So f (Xi + ∆Xi ) − f (Xi ) ≥ 0, f (Xi ) is monotone increasing in Xi .

H. Proof of Lemma 4 Proof: For the designed EV release sequence with at most n rounds, there are n + 1 possible actions of the online scheduler, the first n possible actions indexed by the first declined EV Ji , i = 1, . . . , n, and the last possible action being accepting J1 , . . . , Jn all the way and declining nothing, as illustrated in Fig. 1. This enumeration leads to the following adversarial bound of the competitive ratio ever achievable for the constructed arrival EV sequence Pn−1 Pi−1 ji − (c − 1) k=1 jk jn − (c − 1) k=0 jk , }. max{ max 1≤i≤n−1 ji+1 jn (5) Now we design the charging demand ji so that the value of Eq. (5) can be made small to provide a tight upper bound. To bound the maximum of the n terms by choosing ji ’s, choose ji ’s to make the terms all equal. Since we have n − 1 ji ’s to determine (j0 = 1 is already determined), we can set the first n − 1 terms (all terms but the last term) inside the maximum in Eq. (5) to be 1/α, where α > 1, as shown in Eq. (6). In addition, if we can find appropriate α such that the last term in Eq. (5) is no greater than 1/α, then the maximum in Eq. (5) is bounded by 1/α, and 1/α thus serves as an upper bound on the competitive ratio ever achievable. α(ji − (c − 1)

i−1 X

k=1

jk ) = ji+1 , for i = 1, . . . , n − 1.

(6)

We can then obtain the recursion for ji ’s from Eq. (6), j1 = 1, j2 = α, α(ji+1 − cji ) = ji+2 − ji+1 ,

(7)

with the characteristic function j 2 − (α + 1)j + αc = 0. After designing j1 to jn according to Eq. (7), it is ensured that the first n− 1 terms (all terms but the last term) inside the maximum in Eq. (5) is equal to 1/α. We need, in addition, that the last term in Eq. (5) is no greater than 1/α, to guarantee that the maximum in Eq. (5) is bounded by 1/α. We will next search for the range of α such that 1/α is a valid upper bound on the competitive ratio ever achievable. To examine the condition associated with the last term in Eq. (5), we have due to Eq. (7) Pn−2 Pn−1 jn−1 − (c − 1) k=0 jk jn − (c − 1) k=0 jk 1 ≤ = . jn α jn (8) Eq. (8) can be reduced to jn ≤ cjn−1 , and further to jn+1 ≤ jn due to Eq. (7). Therefore we have so far fixed a value for α > 1 and chosen the value of ji ’s according to the recursion Eq. (7). The only condition to satisfy in order for the validity of the bound 1/α is the turnaround property of the sequence jn , i.e., there exists an integer N such that ji−1 ≤ ji for 2 ≤ i ≤ N , and jN +1 ≤ jN (jn+1 ≤ jn can be ensured by choosing n = N ). This turnaround property can be explained as follows: the increasing rate of the charging demand will eventually slow down, so that the online scheduler cannot obtain a higher profit ratio by accepting all the way up to Jm+1 . Therefore given any value of α that ensures the turnaround property, we can

7

design an adversary game with a finite m that guarantees that the best competitive ratio ever achievable cannot exceed 1/α. To find out the range of α ensuring the turnaround property, we examine the characteristic function h(j) = j 2 − (α + 1)j + αc that governs the growth of the sequence ji . When h(j) has two real roots, the sequence ji increases monotonically, while when h(j) has a pair of complex roots, the sequence ji oscillates and there exist integers N such that jN +1 ≤ jN . Picking the smallest such integer N will ensure the turnaround property. Now the issue turns to finding the value of α such that h(j) has a pair of complex roots. To this end, we simply need to find the value of α under which h(j) has a double root. To do this, we take the derivative of h(j) and set it to 0, h′ (j ∗ ) = 2j ∗ − (α + 1) = 0, which yields j ∗ = α+1 2 . ∗ into h(j ) = 0 implies that the value Substituting j ∗ = α+1 2 α under which h(j) has a double root satisfies the polynomial f (z, 1) = (z + 1)2 − 4cz. With some algebra (detailed algebra steps omitted here), it is shown that for any 1 ≤ α < z(c, 1) (where z(c, 1) is the larger real root of the polynomial f (z, 1) = (z + 1)2 − 4cz), h(j) has a pair of complex roots, and the turnaround property holds. Therefore we can use α arbitrarily close to z(c, 1) to construct the length ji ’s, for which the best competitive ratio ever achievable is no greater than 1/α. Taking the limit of α → z(c, 1) completes the proof of Lemma 4.

• •

If J p is turned pressing upon the release of an accepted pressing EVs, then dp ≤ rip,threshold + Bi . If J p is turned pressing upon the release of a declined EVJD , then if the deadline of J p satisfies dp ≤ dD , then by the analysis of the declined pressing EVcase, we can see that dp ≤ dD ≤ rip,threshold + cBi + βgi . Otherwise, if dp > dD , then we know that after fitting the EVJD in, the end time τt of the tentative pressing busy interval at time t can at most be extended by the local charging time planned for JD (i.e., xD − oD ), to τt + jD − oD . Since after fitting in the EV JD the EV J p is turned pressing, it must hold that the deadline of Ji satisfies

di

≤ τt + jD − oD ≤ τt + (jD − coD ) + (c − 1)oD ≤ (rip,threshold + Bi ) + βgi + (c − 1)Bi = rip,threshold + βgi + cBi ,

due to the fact that JD cannot pass the threshold admission procedure. We complete the proof that dp is necessarily less than rip,threshold + cBi + βgi .

I. Proof of Lemma 5 Proof: We treat the declined and accepted pressing EVs in Bp separately. 1) Declined pressing EVs: There is only one way a EV J p can be declined in the pressing busy interval Bp : J p is released, has to go through the threshold admission procedure, and has insufficient profit to get accepted. Hence if J p was declined between rip,threshold (inclusive) p,threshold (non-inclusive), then and ri+1 dp − (rip,threshold + Bi ) ≤ j p − op = j p − cop + (c − 1)op ≤ βgi + (c − 1)Bi , (9) where the first inequality follows from the fact that EDFLME packs the entire interval [rip,threshold , rip,threshold +Bi ] full, and in the final inequality, j p − cop ≤ βgi follows from the threshold admission rule of TAGS. Specifically, it holds that Profitaccept (t)

= j p − cop + Profitdecline (t) ≤ (1 + β)Profitdecline (t),

which leads to xp − cop ≤ βProfitdecline (t) ≤ βgi . The final piece we need, op ≤ Bi , is obvious since the expensive charging time planned for J p can at most be as large as the length of the time period from rip,threshold (rip,threshold is no later than rp ) to rip,threshold + Bi (the end point of the pressing busy interval immediately before p,threshold ). ri+1 2) Accepted pressing EVs: We further divide our discussion into the following two scenarios.

J. Proof of Proposition 2 Proof: We prove Eq. (3) by induction. β β The base case: 1c r0 + B0 = B0 ≤ 1+β−c B0 = 1+β−c g0 , since we assume r0 = 0 and there is no expensive charging planned during the time period B0 before the release of the first EV J1p,threshold accepted via threshold admission, and therefore it holds that B0 = g0 .

Now assume the statement holds for i ≤ m − 1. We will show the statement holds for m. Before proving m case, define o¯m = rm−1 + Bm−1 − rm , i.e., o¯m is the time period from rm to the end of the tentative pressing busy interval immediately before the mth accepted pressing EV. We prepare some equalities for later use.

p,converted Bm = jm − om + jm + o¯m

(10)

p,converted = gm − gm−1 jm − com + jm

(11)

8

Optimizing over β yields the best competitive ratio achiev-

Then we have 1 rm + Bm c



= = ≤ = =





≤ =

able by TAGS, maxβ [β + 1+β−c ]−1 . Some algebra shows 1 p,converted √ rm + jm − om + o¯m + jm (by Eq. (10)) √ cβ c ] = ( c + c − 1)2 , attained when that minβ [β + 1+β−c p 1 rm + gm − gm−1 + com − om + o¯m (by Eq. (11))β − (c − 1) = c(c − 1). Therefore the competitive ratio p c c(c − 1) is of√ TAGS with the choice of β = c − 1 + 1 √ rm + gm − gm−1 + p¯ om (12) ( c − c − 1)2 . c 1 ( − c)rm + gm − gm−1 + c(rm + o¯m ) c R EFERENCES 1 ( − c)rm + gm − gm−1 [1] K. R. L. Kruk, J. Lehoczky and S. Shreve, “Heavy traffic analysis for c EDF queues with reneging,” Annals of Applied Probability, vol. 21, no. 2, +c(rm−1 + Bm−1 ) (by definition of o¯m ) pp. 484–545, 2011. 1 [2] G. Koren and D. Shasha, “Dover: An optimal on-online scheduling ( − c)rm + gm − gm−1 c algorithm for overloaded uniprocessor real-time systems,” SIAM Journal 1 of Computing, vol. 24, pp. 318–339, 1995. +c( rm−1 + Bm−1 ) + (c − 1)rm−1 c 1 ( − 1)rm + (c − 1)(rm−1 − rm ) + gm − gm−1(13) c β +c gm−1 1+β−c β c 1+β−c − 1 gm (14) gm + 1+β β gm , 1+β−c

where Eq. (12) follows from the fact that om ≤ o¯m (the part p,threshold of Jm charged by the expensive charger can at most be as large as the length of the interval [rm , rm−1 + Bm−1 ], which is o¯m ), Eq. (13) is due to the induction hypothesis, and Eq. (13) follows from the fact that rm−1 ≤ rm (releases of p,threshold J1p,threshold , . . . , Jm follows chronical order), (1 + β)gm−1 ≤ gm (threshold admission structure) and unit outsourcing cost c > 1. This concludes the induction. K. Proof of Lemma 6 Proof: After the execution of TAGS the collection I of EVs ever released will be partitioned into I = S n ∪ S p ∪ D where S n (S p ) denotes the successful non-pressing(pressing) EVs and D the declined EVs (necessarily pressing) under TAGS. Each pressing(non-pressing) EV is accounted for in exactly one pressing(non-pressing) busy interval. Therefore, for all pressing and non-pressing busy intervals, the optimal offline profit is bounded in Eq. (18) ∗ Voffline (I)

≤ ≤

∗ ∗ Voffline (S n ) + Voffline (S p ∪ D) X X X ji + [ maxp di − minp ri ] Bn ri ∈Bn



X X



X X

Bn ri ∈Bn

Bp

ji +

ri ∈B

ri ∈B

(15)

X p,threshold [rm + cBm + βgm ] (16) Bp

X ji + [

Bn ri ∈Bn

cβ 1+β−c

cβ gm + βgm ] (17) 1 + β−c Bp X X X + β][ ji + gm ]



[

=

cβ ∗ [ + β]VTAGS (I), 1+β−c

Bn ri ∈Bn

Bp

(18)

where Eq. (15) follows from the bounding strategies for nonpressing and pressing EVs, Eq. (16) follows from Lemma 5 and Eq. (17) from Proposition 2. By Eq. (18) we conclude that the competitive ratio achieved cβ )−1 . by TAGS is bounded below by (β + 1+β−c

Suggest Documents