A Stochastic Calculus for Network Systems with Renewable Energy Sources

A Stochastic Calculus for Network Systems with Rene wa ble Ener gy Sources Kui W u Dept. of Computer Science University of V ictoria V ictoria, British Columb ia, Canada Y uming Jiang Q2S Center of Excellence Norwegian Uni versity o f Science and T echnology T rondheim , Norway Dimitri Marinakis Dept. of Compu ter Science University of V ictoria V ictoria, British Columb ia, Canada Abstract —W e consider the perf ormance modeling and e va lua- tion of network systems power ed with renewable energy sources such as solar and wind energy . Such en ergy sources largely depend on envir onmental conditions, which are hard t o predict accurately . As such , it may only make sense to require the network systems to su pport a soft qu ality of service (QoS) guarantee, i.e., to guarantee a service requirement with a certain high probability . In this paper , we intend to build a solid mathematical foundation t o help better understand the stochastic energy constraint and the inherent correlation between QoS and the u ncertain energy sup ply . W e util ize a calculus approach to model the cumulative amount of charged energy and t he cumulative amount of consumed energy . W e d eriv e upp er and lower bound s on the remaining energy lev el based on a stochastic energy charging rate and a stochastic energy discharging rate. By buildin g the bridge between energy consumption and task execution (i.e., service), we stud y the QoS guarantee under the constraint of uncertain energy sources. W e furth er show how perfo rmance bound s can be improv ed if so me strong assumptions can be made. Index T erms —Stochastic Networ k Calculus, P erfo rmance Eval- uation, Renewable En ergy , Energy Schedulin g I . I N T RO D U C T I O N In the last deca des, there ha ve been in creasing dema nds on computin g, comm unication and storag e systems. The strong demand s drive mo dern I T infr astructures to extend an d scale at an unpre cedented s peed, raising seriou s “green” related con- cerns a bout hig h energy con sumption and greenh ouse emis- sion. Addre ssing this pro blem, th e use of ren ew able en ergy sources, such as solar and wind energy , plays an important role in the su pport o f sustaina ble co mputing [5]. V arious energy harvesting devices have b een developed and b roadly used in real-world applicatio ns. For example, to sup port perpetu al en vironm ental mon itoring, solar energy has be en u sed to power tiny sensor nod es de ployed in the wilderness [8]. While the ben efit of using renewable energy is clear, its application can po se g reat challenges in terms of assuring a satisfactory quality of services (QoS) for com puting systems. First, ren ew able energy sources are g enerally unr eliable and hard to predict. T he uncertainty in the energy supp ly makes it extremely hard to support strict QoS r equiremen ts. As a result, it only makes sense to demand the system to supp ort a soft QoS guarante e, i.e., to guara ntee the QoS requir ement (e.g., delay , throug hput) with a cer tain prob ability . Second, although modern compu ting and communication de vices could be eq uipped with r ate-adaptive capab ilities [2 3], allowing the devices to “smartly ” sched ule the execution o f various tasks, it is n ev ertheless non-trivial to design go od scheduling algorithm s that can effecti vely u tilize a limited and variable energy resou rce. Many aspects come into play , includin g fo r example, the predicted energy charging ra te, the tolerab le range of a QoS guaran tee, the rate-adap tiv e features o f the device, an d so o n. Th ird, QoS suppor t and task sched uling become ev en harder when the network includes multiple nodes powered by rene wable energy , because the task sched uling and energy managem ent of one node may have direct impact on the other node s. Such depend ency makes QoS supp ort extremely hard. Giv en the se c hallenges, we per ceiv e the strong need for a generic analy tical fr amew ork f or perfor mance mod eling and evaluation of a network system using ren ew able energy sources. Such a th eoretical mo del should (1) captu re th e stochastic f eatures in energy rep lenishment and energy con - sumption, (2) provide go od guidelines on the schedulab ility of given tasks un der uncertain en ergy constrain ts, (3 ) be applicable to a large g roup o f systems where the power -rate function [11], [23] may b e of different for ms, and (4) be ab le to analyze a single node as well as a ne twork system. V ariou s meth ods h av e been dev eloped for task sched ul- ing and p erform ance an alysis of e nergy constrained system, including for examp le Markov chain b ased methods [16], [20], calculu s appr oaches [1 1], [23], pre diction-b ased ap - proach es [ 7], [21], and so on. Ne vertheless, no analytica l model so far is sufficient to meet all the above requireme nts. W e are thu s motivated to develop a more ge neric analytical framework to fill the vacancy . I n this paper, we make the following co ntributions: • W e build a theo retical mod el for perform ance e valuation of network sy stems with renewable energy sources. Our model is based on recen t prog ress in stoch astic network calculus, but it much extends the concep ts of th e tra- ditional sto chastic network calculus b y intr oducing en- ergy charging/discharging models. Such extension is non- trivial since the new con cepts r equire special treatment in the deriv ation o f perform ance boun ds. • W e der iv e the stochastic up per and lower bo unds on a no de’ s residual energy level. These boun ds p rovide fundam ental guidance in the energy manag ement of a system with renewable energy supply . • By u sing a gen eric p ower -rate fun ction to brid ge the task execution and its energy consump tion, we derive the stochastic perf ormanc e boun ds on delay an d system backlog . W e study b oth the single-no de case and the network case. 2 • W e extend th e model to analyze systems that utilize multiple energy so urces. • W e poin t ou t a m ethod to improve the p erform ance bound s if some strong assumptions, such as independence between multiple energy sources, can be made. The rest o f the paper is organiz ed as follows. W e introduce the basic notation s in Section II. In Section III, we present new energy models to captu re th e stochastic fe atures in the energy charging an d discharging processes. Based on th e new energy models, we derive the stochastic bound s on a node’ s remaining energy . W e analyze the p erform ance boun ds of a single node with resp ect to delay and system backlo g in Section IV. The perfor mance of a network system is provid ed in Sectio n V. Methods of hand ling m ultiple energy sources an d improvin g perfor mance bounds ar e intro duced in Section s VI and VII, respectively . Related work is discussed in Sectio n VIII. Th e paper is conclu ded in Section IX. I I . B A S I C N O TA T I O N S W e first introduce the basic n otations fo llowing the con ven- tion of stochastic network calculu s [9], [ 10], [13]. W e deno te by F the set of n on-negative, wide-sense incr easing fun ctions, i.e., F = { f ( · ) : ∀ 0 ≤ x ≤ y , 0 ≤ f ( x ) ≤ f ( y ) } , and b y ¯ F the set of no n-negative, wide-sense d ecreasing function s, i.e., ¯ F = { f ( · ) : ∀ 0 ≤ x ≤ y , 0 ≤ f ( y ) ≤ f ( x ) } . For any random variable X , its d istribution fun ction, deno ted by F X ( x ) ≡ P rob { X ≤ x } , belongs to F , and its com plementary distribution f unction, denoted by ¯ F X ( x ) ≡ P rob { X > x } , belongs to ¯ F . For any function f ( t ) , we u se f ′ ( t ) to d enote its de riv a ti ve, if it exists. The following o perations will be used in this pap er: • The (min , +) con volution of functions f and g under th e (min , +) algebra [1], [4], [12] is defin ed a s: ( f ⊗ g )( t ) ≡ inf 0 ≤ s ≤ t { f ( s ) + g ( t − s ) } . (1) • The (min , +) deconvolution of fu nctions f and g is defined as: ( f ⊘ g )( t ) ≡ sup s ≥ 0 { f ( t + s ) − g ( s ) } . (2) • The (max , +) convolution of func tions f and g is de fined as: ( f ¯ ⊗ g )( t ) ≡ sup 0 ≤ s ≤ t { f ( s ) + g ( t − s ) } . (3) • The pointwise infimum or pointwise minimum of functions f and g is defined as ( f ∧ g )( t ) ≡ min[ f ( t ) , g ( t )] . (4) In addition, we adop t: • [ x ] + ≡ max { x, 0 } , • [ x ] 1 ≡ min { x, 1 } . Throu ghout this pa per , we assume that the energy charging curve and the en er gy dischar ging curve , both of which will be defined later, are no n-negative and wide- sense increasing function s. In this paper, C ( t ) and C ∗ ( t ) are u sed to denote the cumulative e nergy a mount that h as been charged and d epleted in the tim e interval (0 , t ] , respectively . For any 0 ≤ s ≤ t , let C ( s, t ) ≡ C ( t ) − C ( s ) and C ∗ ( s, t ) ≡ C ∗ ( t ) − C ∗ ( s ) . Similarly , A ( t ) and A ∗ ( t ) are used to denote the cumulative amount of data traffic th at has arrived and dep arted in time interval (0 , t ] , respec ti vely , and S ( t ) is used to deno te the cumulative amo unt of serv ice pr ovided by the system in time interval (0 , t ] . For any 0 ≤ s ≤ t , let A ( s, t ) ≡ A ( t ) − A ( s ) , A ∗ ( s, t ) ≡ A ∗ ( t ) − A ∗ ( s ) , and S ( s, t ) ≡ S ( t ) − S ( s ) . By default, A (0) = A ∗ (0) = S (0) = 0 , and C (0) = C ∗ (0) = 0 . I I I . S T O C H A S T I C E N E R G Y M O D E L S A N D R E S I D UA L E N E R G Y A. Stochastic Ener g y Char ging a nd Dischar ging Models Definition 1: The stochastic energy charging model: Th e cumulative e nergy amo unt C ( t ) is said to follow s tochastic e ner gy c har ging (s.e.c.) curves α 1 ∈ F and α 2 ∈ F with bound ing fu nctions f 1 ∈ ¯ F and f 2 ∈ ¯ F , respecti vely , den oted by C ∼ sec < f 1 , α 1 , f 2 , α 2 >, if for all t ≥ s ≥ 0 and all x ≥ 0 , P r ob { inf 0 ≤ s ≤ t [ C ( s, t ) − α 1 ( t − s ) > x ] } ≥ f 1 ( x ) , (5) and P r ob { s up 0 ≤ s ≤ t [ C ( s, t ) − α 2 ( t − s ) > x ] } ≤ f 2 ( x ) . (6) W e call th e cu rves α 1 and α 2 the lower curve and the upper curve, respectively , an d the fun ctions f 1 and f 2 the lower bo unding fu nction and the upper bou nding fun ction, respectively . Remark 1: The practical meaning of (5) is to lo wer bound the cumu lativ e amoun t of charged en ergy , i.e., the energy harvester should provide a m inimal lev el of energy with a high probab ility . The m eaning of (6) is that the cumulative am ount of charged en ergy m ay b e up per boun ded. For instan ce, in the case of solar-po wered sensor nodes, the total c harged energy should no t beyond the best case scen ario, e.g. , sunny a ll the time. Remark 2: The ( ρ, σ 1 , σ 2 )-source in the harvesting theory [11] is a special case o f our stochastic energy charging model, by de fining α 1 ( t ) = ρt − σ 1 , α 2 ( t ) = ρt + σ 2 , an d setting the bound ing functio ns to 0 . Definition 2: The stochastic energy discharging model: Under th e constraint of ch arged energy am ount C ( t ) , the cumulative d ischarged energy amou nt C ∗ ( t ) is said to have s tochastic energy dischar ging (s.e.d.) curves β 1 ∈ F and β 2 ∈ F with bou nding fun ctions g 1 ∈ ¯ F and g 2 ∈ ¯ F , respectively , denoted by C ∗ ∼ sed < g 1 , β 1 , g 2 , β 2 >, 3 if for all t ≥ 0 an d all x ≥ 0 , P r ob { C ⊗ β 1 ( t ) − C ∗ ( t ) > x } ≥ g 1 ( x ) , (7) and P r ob { C ⊗ β 2 ( t ) − C ∗ ( t ) > x } ≤ g 2 ( x ) . (8) W e call cur ves β 1 and β 2 the upp er curve and the lower curve, respectively , and the function s g 1 , g 2 the u pper bou nding function and the lower b oundin g fun ction, r espectiv ely . Remark 3: The pr actical mea ning of the above d efinition is as follows. If we u se β 1 ( t ) to deno te the cumu lati ve virtual energy disch arging amoun t (i.e., the energy b udget th at a service sh ould fo llow), and if we want to use β 1 ( t ) to up per bound th e actual discharged energy C ∗ ( t ) , then unde r the constraint of cu mulative charged en ergy a mount C ( t ) , at any time instance t , we have C ∗ ( t ) ≤ β 1 ( s ) + C ( t − s ) . (9) Note that we sho uld not r eplace β 1 ( s ) with C ∗ ( s ) since otherwise we ob tain C ∗ ( t − s ) ≤ C ( t − s ) f or any time interval ( s, t ] , which is a co nstraint too restrictiv e. Because the in equality (9) holds for any s ≤ t , we hav e C ∗ ( t ) ≤ inf 0 ≤ s ≤ t { β 1 ( s )+ C ( t − s ) } , which is C ∗ ( t ) ≤ C ⊗ β 1 ( t ) by th e definition of (min,+) con volution. Inequality (7) in Definition 2 represents the stocha stic version o f the above relationship. In practice, we may also want to lower bound the discharged energy (with a cur ve β 2 ) to effectively utilize the h arvested energy , since overcharged energy beyond the energy stor age capacity will be wasted. In this case, the stochastic lower bound is represen ted b y the inequality (8). Since C ( t ) and C ∗ ( t ) den ote the cumu lativ e amo unt of charged energy and the cumulative amount of discharged energy , respectively , the remaining energy amoun t, d enoted by E ( t ) , in the system at time t can be c alculated as: E ( t ) = C ( t ) − C ∗ ( t ) . (10) B. Stochastic Analysis on Remaining Ener gy For any energy -aware scheduling , we first need to under- stand the feasible ener gy depletion r e gion . Particularly , we need to answer the following question : what ar e the lower and upp er bo unds of P r ob { E ( t ) > x } ? The practical m eaning o f the lower boun d c omes from reliability consider ation: the r emaining energy at any time instance shou ld be with a high p robability larger than a minimal thresho ld value (called safety thres hold ) to guaran tee the reliable operation of the system. The upper bound is related to the effecti ve use of the energy sou rce: sin ce the cap acity o f energy storage is limited (in this case, x could be con sidered as the upp er limit of the ene rgy capacity), tasks should be scheduled to consume energy ef fectiv ely so that the probability that the system is overcharged is small. T o an swer the above question , we start with the following lemma. Lemma 1: Assume that th e complementa ry cumu lative dis- tribution fun ctions (CCDF) o f rando m variables X , Y are ¯ F X ∈ ¯ F an d ¯ F Y ∈ ¯ F , re spectiv ely . Assume that f 1 ( x ) ≤ ¯ F X ( x ) ≤ f 2 ( x ) an d g 1 ( x ) ≤ ¯ F Y ( x ) ≤ g 2 ( x ) . De note Z = X + Y . Then no matter whether X and Y a re independ ent or not, there holds for ∀ x ≥ 0 , ¯ F Z ( x ) ≤ f 2 ⊗ g 2 ( x ) , (11) and ¯ F Z ( x ) ≥ f 1 ¯ ⊗ g 1 ( x ) − 1 . (12) The p roofs o f all lemmas are referr ed to App endix of this paper . Theor em 1: Assume that the system has an en ergy charging source with cumulative ch arged en ergy am ount C ( t ) ∼ sec < f 1 , α 1 , f 2 , α 2 > an d it provides serv ice with cu mulative depleted energy amoun t C ∗ ( t ) ∼ sed < g 1 , β 1 , g 2 , β 2 > . Th e remaining energy amoun t of the system at a ny time instant t , E ( t ) , is lower b ounde d by P r ob { E ( t ) > x } ≥ f 1 ¯ ⊗ g 1 ( x − α 1 ⊘ β 1 (0)) − 1 , (13) and is upper boun ded by P r ob { E ( t ) > x } ≤ f 2 ⊗ g 2 ( x − α 2 ⊘ β 2 (0)) , (14) Pr oo f: W e only prove the lower bou nd, since the upper bound can be easily proved follo wing the s ame argument using the curves α 2 and β 2 , the bound ing fu nctions f 2 and g 2 , and Inequa lity (11). For any t ≥ s > 0 , we have E ( t ) = C ( t ) − C ∗ ( t ) = C ( t ) − C ⊗ β 1 ( t ) + C ⊗ β 1 ( t ) − C ∗ ( t ) = s up 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) + α 1 ( t − s ) − β 1 ( t − s ) } + C ⊗ β 1 ( t ) − C ∗ ( t ) ≥ s up 0 ≤ s ≤ t { inf 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) } + α 1 ( t − s ) − β 1 ( t − s ) } + C ⊗ β 1 ( t ) − C ∗ ( t ) = inf 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) } + sup 0 ≤ s ≤ t { α 1 ( t − s ) − β 1 ( t − s ) } + C ⊗ β 1 ( t ) − C ∗ ( t ) = inf 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) } + s up t> 0 { α 1 ( t ) − β 1 ( t ) } + C ⊗ β 1 ( t ) − C ∗ ( t ) = inf 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) } + C ⊗ β 1 ( t ) − C ∗ ( t ) + α 1 ⊘ β 1 (0) (15) If we consider inf 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) } and C ⊗ β 1 ( t ) − C ∗ ( t ) as two ran dom variables depending on t , Ine quality (13) follows accordin g to the definition s of s.e.c. and s .e.d. cur ves and Inequ ality (12) in Lemm a 1. Remark 4: Theore m 1 provides the following fundamen tal guidanc e in the energy manag ement of systems powered with renewable energy: • Given the statistical featu re of a ren ew able en ergy sou rce and a safety thresh old on remain ing e nergy lev el, we can find a suitable en ergy co nsumption budget to ser vice requirem ents (i.e., β 1 ( t ) in Theorem 1) b ased on ( 13), so that the system ha s sufficient energy with a certain probab ility . 4 • Given the statistical feature of a renewable energy sou rce and the maximum capacity of energy stor age in th e system, we can o btain a minima l energy expenditure rate based o n (14), such tha t th e pro bability that th e sy stem is overcharged is lower than a cer tain probab ility . • Overall, given a renew able energy source, a safety th resh- old energy level, and a n up per limit for energy stor age, the two in equalities in Theorem 1 d escribe the stochasti- cally feasible region of energy budget for services. I V . S T O C H A S T I C S E RV I C E G UA R A N T E E O F A S I N G L E N O D E A. System Model W ith the above energy mo dels, we next study the problem of pr oviding a stochastic serv ice guar antee g iv en an uncertain energy sup ply . W e start with the single-nod e case. For the pap er to be self-co ntained, we n eed to b riefly introdu ce the core concep ts in traditional stoch astic net- work calculus– stochastic traffic ar riv al cur ves and service curves [10]. For traffic arrivals , we have the following model. Definition 3: The v .b.c. mo del: A flow A ( t ) is said to have a v irtual-backlog-c entric (v .b .c.) stocha stic arriv al cur ve α ∈ F with boun ding fun ction f ∈ ¯ F , den oted by A ∼ vb < f , α >, if for all t ≥ 0 an d all x ≥ 0 , it hold s [9], [ 10] P r ob { sup 0 ≤ s ≤ t [ A ( s, t ) − α ( t − s )] > x } ≤ f ( x ) . (16) For service models, we hav e the followings. Definition 4: The s.c. model: A server is said to provide a flo w A ( t ) with a s tochastic service curve (s.c.) β ∈ F with bound ing fu nction g ∈ ¯ F , denoted by S ∼ sc < g , β >, if for all t ≥ 0 an d all x ≥ 0 , it hold s [9], [ 10] P r ob { s up 0 ≤ s ≤ t [ A ⊗ β ( s ) − A ∗ ( s )] > x } ≤ g ( x ) . (17) Remark 5: The s.c. model is adopted in th is p aper fo r ease of expressing the results, particu larly the concatenatio n proper ty fo r the network case analysis in Section V. Howev er , the s.c . mo del may be too r estrictiv e. In the liter ature, a variation of the mo del is av ailable, wh ich is called weak stochastic service curve (w .s) in [9]. The w.s. m odel is mu ch less restrictive and has been wid ely adop ted in the sto chastic network calculus literatu re. W e would like to stress that the w.s. m odel can also be u sed here, an d it can be verified (e.g. see [9]) that the delay a nd backlog bound results in this paper remain unch anged with the w.s. model. The affected one is the analysis on the network case, which would have much complicated expression. Definition 5: The s.s.c. model: A server is said to p rovide a flow A ( t ) with a s trict stochastic service curve (s.s.c.) β ∈ F with boun ding fun ction g ∈ ¯ F , den oted by S ∼ ssc < g , β >, if du ring any period ( s, t ] and f or any x ≥ 0 , it hold s [ 9], [10] P r ob { S ( s, t ) < β ( t − s ) − x } ≤ g ( x ) . (18) Remark 6: The s.s.c. model is to deco uple the serv ice and traffic arrivals. It is useful in energy o utage analysis as shown later in Section IV -C. The fo llowing measures are of interest in service guaran tee analysis under network calculus: • The back log B ( t ) in the sy stem at time t is defined as: B ( t ) = A ( t ) − A ∗ ( t ) . (19) • The delay D ( t ) at time t is defined as: D ( t ) = inf { τ ≥ 0 : A ( t ) ≤ A ∗ ( t + τ ) } . (20) B. Stochastic Service Gu arantee T o ease descrip tion, we define the following terms: Definition 6: Power -rat e function [23] is a function th at translates the amount of service to the amount of co nsumed energy . W e use a generic no tation P to denote the power -rate function . Definition 7: Energy-oblivious service curve of a system is the service cu rve that th e system would hav e if there were no energy co nstraint. Note that most moder n devices, such as compu ter servers and wireless transceivers, have th e capability of adjusting processing/tr ansmission rate. Associated with a r ate, there is a correspon ding power expen diture th at is governed by the p ower r ate fu nction. Gener ally speaking , a low proc ess- ing/transmission rate requires low energy consump tion. The power -rate f unction is system depend ent. For example, fo r most encod ing sch emes in wireless communica tion, the re- quired power is a convex function of the rate [23]. In ad dition, there are some devices, e.g., Atmel micropr ocessors, who se energy con sumption is domin ated by its on state. Th at is, as long as the device is powered on, it co nsumes e nergy at a rough ly constant speed, n o m atter whe ther o r n ot its CPU remains idle or executes tasks. T o av oid those different details, we use a generic notation P to build a b roadly -applicable model. Giv en a cumulative energy amount C ( t ) , the serv ice amount that C ( t ) can suppo rt, deno ted as S C ( t ) , can thus be calculated as: S C ( t ) = Z t 0 P − 1 ( C ′ ( x )) dx, (21) Similarly , g iv en a serv ice amoun t S ( t ) , the corre sponding energy consumption amoun t, denoted by C ∗ S , can be calculated as: C ∗ S ( t ) = Z t 0 P ( S ′ ( x )) dx. (22) Remark 7: W e assume th at the inverse o f p ower rate func- tion P , the derivati ve of the energy c harging curve C , an d th e deriv ati ve of the service curve S , all exist. This as sumption has been used and justified in [23]. This assumption is r easonable, because ( 1) th ere is u sually a one -to-one mapp ing between the served data amou nt and th e used energy amoun t, and (2) cumulative energy chargin g amount and cumulative service amount are m ostly co ntinuou s. Ne vertheless, even if the above 5 assumption m ay n ot be tr ue in so me specific situatio ns, numerical appro ximation can be used to estimate S C and C ∗ S . In the rest o f the pap er , we will use Eq uations (21) and (2 2) without giving f urther explanation. W ith all n otations being in troduce d, we are ready to answer the following qu estion regarding a QoS g uarantee: Assume that a d ata flow A ( t ) with the traffic arr i val curve A ∼ vb < f , α > is input into a node , which has an ene rgy-oblivious serv ice curve S ∼ sc < g , β > . Assume that the system is powered by an energy source following C ( t ) ∼ sec < f 1 , α 1 , f 2 , α 2 > . For any time t , wh at ar e the stochastic bo unds on the flow’ s delay D ( t ) and backlog B ( t ) ? The main difficulty in solving the above question is that th e data flow may not be able to get a service following the service curve β due to th e en ergy constrain t. In other words, a service is possible on ly if the system has eno ugh energy . By “enough energy”, we mea n that the system’ s energy level is above the safety threshold. T o simp lify presentation, we assume the safety thresho ld value is 0 in this section. Otherwise, trivial modification is req uired fo r th e following p erforma nce results. W e h av e the following impor tant tech nical lemmas. Lemma 2: Assume that functio ns f 1 , f 2 , f 3 ∈ F , and as- sume that f 1 (0) = f 2 (0) = f 3 (0) = 0 . For any t ≥ 0 , it holds that f 1 ⊗ ( f 2 ∧ f 3 )( t ) ≤ ( f 1 ⊗ f 2 ( t )) ∧ f 3 ( t ) (23) Lemma 3: If X 1 ≥ X 2 ≥ 0 and X 3 ≥ X 4 ≥ 0 , it holds X 1 ∧ X 4 − X 2 ∧ X 3 ≤ X 1 − X 2 + X 3 − X 4 . (24) Lemma 4: [10] Consider a rand om v a riable X . For an y x ≥ 0 , P r ob { [ X ] + > x } = P rob { X > x } . Theor em 2: Assume that a node has a n energy-o blivious service curve S ∼ sc < g , β > . Assume th at th e node is powered with an energy sour ce fo llowing C ( t ) ∼ sec < f 1 , α 1 , f 2 , α 2 > . T he actual service av a ilable to the task, denoted by S e ( t ) , follows S e ( t ) ∼ sc < ˙ g , β ∧ ˙ α 2 > , where ˙ g ( x ) = g ⊗ f 2 ( x ) , (25) ˙ α 2 ( t ) = Z t 0 P − 1 ( α 2 ( x )) dx. (26) Pr oo f: It is clear that a t a ny time t ≥ 0 , the actual serv ice provided by the system, S e ( t ) =  S ( t ) if S ( t ) ≤ S C ( t ) S C ( t ) otherwise, (27) where S C is defined by (refeq:SC). Th is is because if S ( t ) ≤ S C ( t ) , th e en ergy constraint do es no t play a r ole, an d th e actual ser vice am ount is equ al to S ( t ) . Otherwise, the ac tual service is equal to the amou nt allowed by the energy con - straint, i.e., S C ( t ) . In other words, S e ( t ) = S ( t ) ∧ S C ( t ) . Next, we pr ove that S e ( t ) ∼ sc < ˙ g , β ∧ ˙ α 2 > . Denote the arriv al flow as A ( t ) an d the output flow as A ∗ ( t ) , which is equal to S e ( t ) . For any time 0 ≤ s ≤ t , b ased on Lemma 2 and Lemma 3, we have A ⊗ ( β ∧ ˙ α 2 )( s ) − A ∗ ( s ) ≤ ( A ⊗ β ( s )) ∧ ˙ α 2 ( s ) − A ∗ ( s ) = ( A ⊗ β ( s )) ∧ ˙ α 2 ( s ) − S ( s ) ∧ S C ( s ) ≤ [ A ⊗ β ( s ) − A ∗ ( s )] + + [ S C ( s ) − ˙ α 2 ( s )] + (28) Note that in th e above, the first inequality is due to Lemma 2, the secon d equality is becau se that A ∗ ( s ) = S e ( s ) , and the last inequ ality is b ased on Lemma 3. W e thus have sup 0 ≤ s ≤ t { A ⊗ ( β ∧ ˙ α 2 )( s ) − A ∗ ( s ) } ≤ sup 0 ≤ s ≤ t { [ A ⊗ β ( s ) − A ∗ ( s )] + + [ S C ( s ) − ˙ α 2 ( s )] + } ≤ sup 0 ≤ s ≤ t { [ A ⊗ β ( s ) − A ∗ ( s )] + } + sup 0 ≤ s ≤ t { [ S C ( s ) − ˙ α 2 ( s )] + } (29) Based on the definitions of the s.c. curve and the s.e .c. curve, and Lemma 4, we have for any x > 0 , 0 ≤ s ≤ t , P r ob { sup 0 ≤ s ≤ t [ A ⊗ β ( s ) − A ∗ ( s )] + > x } ≤ g ( x ) , (30) P r ob { s up 0 ≤ s ≤ t [ S C ( s ) − ˙ α 2 ( s )] + > x } ≤ f 2 ( x ) . (31) For ( 31), we r emind the r eaders that the calculations of S C and ˙ α 2 do n ot chang e the bound ing fun ction. The theorem follows based on ( 30), (31), Inequ ality (1 1), and the d efinition of s.c. service curve. W e have the following th eorem for service guarantee . Theor em 3: Assume that a d ata flow A ( t ) with the traffic arriv al curve A ∼ vb < f , α > is input into a nod e. Assume that the node has an energy -oblivious service curve S ∼ sc < g , β > . Assume that the n ode is powered with energy supply following C ( t ) ∼ sec < f 1 , α 1 , f 2 , α 2 > . • For any time t ≥ 0 , x ≥ 0 , P r ob { D ( t ) > h ( α ( t ) + x, β ∧ ˙ α 2 ( t )) } ≤ f ⊗ ˙ g ( x ) (3 2) where ˙ g is defined with (25), ˙ α 2 is defined with (26), and h ( α ( t ) + x, β ∧ ˙ α 2 ( t )) is th e maximum ho rizontal distance between functions α ( t ) + x and β ∧ ˙ α 2 ( t ) . • For a ny t > 0 and x ≥ 0 , th e backlog B ( t ) is bo unded by P r ob { B ( t ) > x } ≤ f ⊗ ˙ g ( x − α ⊘ ( β ∧ ˙ α 2 )(0)) . ( 33) Pr oo f: Theore m 3 follows dir ectly by applying Theorem 2 to existing r esults of service guarantee in tr aditional stoch astic network calculus (refer to Chapter 5 of [10]). C. The Danger of Ener gy-Obliviou s Service An argument f or using determ inistic energy manageme nt is that th e system would be safe if the av erage en ergy depletion rate equals th e predicted ene rgy re plenishment rate. Since n o absolute g uarantee ca n b e made on th e ac curacy of ene rgy prediction , there is a chance that an energy outage occurs, i.e., the remaining energy-level is lower th an th e safety threshold. In this section, we sho w that th e energy outag e p robability is non-n egligible. Theor em 4: Assume th at a n ode provides a n energy- oblivious serv ice cu rve S ∼ ssc < g , β > . Assume that th e node is powered with ene rgy supply following C ( t ) ∼ sec < f 1 , α 1 , f 2 , α 2 > . Deno te C ∗ S ( t ) = R t 0 P ( S ′ ( x )) dx , and E ( t ) = C ( t ) − C ∗ S ( t ) . For any time t ≥ 0 , given the en ergy safety threshold x ≥ 0 , P r ob { E ( t ) < x } ≥ 1 − f 2 ⊗ g ( x − α 2 ⊘ ˙ β (0)) , (34) 6 where ˙ β ( t ) = R t 0 P ( β ( x )) dx . Pr oo f: For any t ≥ s ≥ 0 , we have E ( t ) = C ( t ) − C ∗ S ( t ) = C ( t ) − α 2 ( t ) + ˙ β ( t ) − C ∗ S ( t ) + α 2 ( t ) − ˙ β ( t ) ≤ s up 0 ≤ s ≤ t { C ( s, t ) − α 2 ( t − s ) } + sup 0 ≤ s ≤ t { ˙ β ( s, t ) − C ∗ S ( s, t ) } + sup t ≥ 0 { α 2 ( t ) − ˙ β ( t ) } = s up 0 ≤ s ≤ t { C ( s, t ) − α 2 ( t − s ) } + sup 0 ≤ s ≤ t { ˙ β ( s, t ) − C ∗ S ( s, t ) } + α 2 ⊘ ˙ β (0) . (35) Based on In equality (11), the definition o f s.e.c. curve, and the definition of s.s .c . curve (we note again th at the ca lculations of ˙ β and C ∗ S do not ch ange the bo undin g fu nction), we ob tain for any x > 0 , P r ob { E ( t ) > x } ≤ f 2 ⊗ g ( x − α 2 ⊘ ˙ β (0)) . ( 36) The theorem follows. As a n examp le, a ssume that there exists an upp er curve on the en ergy ch arging rate of α 2 ( t ) = ρt + σ , with a bound ing functio n f 2 ( x ) = e − ( x +2) , where σ is a g iv en safety thresh old energy level an d ρ is the average en ergy charging rate. Assume that the nod e co uld p rovide en ergy- oblivious service S ∼ s.c. < β , g > , where g ( x ) = e − ( x +2) . Assume that ˙ β ( t ) = R t 0 P ( β ( x )) dx = ρt . Clearly , the average energy charging rate is equal to the av erage energy discharging rate and initially the ene rgy level at the nod e is safe. With Theorem 4, howe ver , it is e asy to calculate that the energy outage pro bability is no less th an 72 . 9 % , r egardless o f the time and the values o f ρ and σ . Remark 8: Theore m 4 an d the above exam ple illustra te the danger of en ergy-oblivious service. Since a service is practically feasible only if it do es not cause an energy outag e, any service sch eduling algor ithm sh ould con sider the energy constraint. This is the main reason why in Section III the energy discharging mod el is inher ently c oupled with the energy charging mo del a nd why in our analysis fo r service guaran tee, we enfo rce an energy co nstraint on the energy- oblivious serv ice. V . S T O C H A S T I C S E RV I C E G UA R A N T E E F O R A N E T W O R K E D S Y S T E M In the previous section, we an alyzed the single node case. Here, we study the serv ice g uarantee question in a n etwork for which the network nod es are p owered with a renewable energy source. It is well-known that we canno t simp ly add the d elay bo unds for each individual nodes to obtain the end - to-end delay bound alon g a path [ 3], [12], since otherwise the network-wide delay bo unds would be too loose to be usef ul. As a solution, the concatena tion prop erty of serv ice c urves should be proved an d applied [10]. Based on Theorem 2 and the concatenation p roperty of s.c . service model, the following theorem holds immediately . Theor em 5: Consider a traffic flow passing through of a network of N nodes in tandem. Assume that each n ode i (= 1 , 2 , . . . , N ) would provide an energy -oblivious service curve S i ∼ sc < g i , β i > to its input. Assume th at each n ode i is powered with an ene rgy supply following C i ∼ sec < f i 1 , α i 1 , f i 2 , α i 2 > . The network guarante es to the fl ow a stochas- tic service curve S ∼ sc < g , β > with g = ˙ g 1 ⊗ ˙ g 2 . . . ⊗ ˙ g N , (37) β = ˙ β 1 ⊗ ˙ β 2 . . . ⊗ ˙ β N , (38) where ˙ g i ( x ) = g i ⊗ f i 2 ( x ) , (39) ˙ β i ( t ) = β i ( t ) ∧ Z t 0 P − 1 ( α i 2 ( x )) dx. (40) Remark 9: Theore m 5 indicates th at we can treat the co n- catenation of multiple no des as a single system. T o ob tain th e end-to- end performan ce, we just need to ap ply Theore m 5 into Theorem 3, th at is, replac ing ˙ g and β ∧ ˙ α 2 ( t ) in T heorem 3 with g and β in Theo rem 5, respectiv ely . Remark 10: In Sections IV an d V, we only used the upper curve of th e c harged energy amount to con strain a ser vice. Actually , another set of similar analysis could be don e, if we use the lower curve of the charged energy amo unt to constrain a service. T he p ractical meaning o f th is c onstraint is to av oid overcharging the system and to e ffecti vely use the e nergy , i.e. , the serv ice sho uld dep lete the energy at a rate n o less than th e minimal charging rate. V I . N O D E S P OW E R E D W I T H M U L T I P L E E N E R G Y S O U R C E S It is possibly that a n etwork node m ight be ch arged with multiple energy sour ces. For examp le, sev eral types of energy such as solar, wind, vibrational, and thermal amo ng others can be scav enged fro m the sur round ings of a sensor node to replenish its batter y [1 9]. In this case, we need to study th e statistical features of the in tegrated en ergy source, fo r which we have the following theorem. Theor em 6: Assume tha t a n ode has N energy sou rces, denoted by C 1 , C 2 , . . . , C N , with C i following C i ( t ) ∼ sec < f i 1 , α i 1 , f i 2 , α i 2 > . Assume that the hardware per mits the multi- ple energy sou rces to charge the batter y simultan eously . The overall en ergy supply to the no de, de noted by C , follows C ( t ) ∼ sec < f 1 , α 1 , f 2 , α 2 > , where f 1 ( x ) = f 1 1 ¯ ⊗ f 2 1 . . . ¯ ⊗ f N 1 ( x ) − ( N − 1) , (41) α 1 ( t ) = α 1 1 + α 2 1 . . . + α N 1 ( t ) , (42) f 2 ( x ) = f 1 2 ⊗ f 2 2 . . . ⊗ f N 2 ( x ) , (43) α 2 ( t ) = α 1 2 + α 2 2 . . . + α N 2 ( t ) , (44) Pr oo f: W e only prove Eq uations (4 1) and (42) by apply- ing I nequality (12), as th e pro of for Equation s (43) and ( 44) is similar by u sing Inequality (11). In ad dition, we on ly n eed to prove the case that N = 2 , b ecause th e theorem holds by recursively applying the result for N = 2 . 7 As we assume th at the multiple ene rgy sources ca n charge the node simu ltaneously , fo r a ny time t > 0 , C ( t ) = C 1 ( t ) + C 2 ( t ) . For any t ≥ s > 0 , we have inf 0 ≤ s ≤ t { C ( s, t ) − α 1 ( t − s ) } = inf 0 ≤ s ≤ t { C 1 ( s, t ) + C 2 ( s, t ) − ( α 1 1 ( t − s ) + α 2 1 ( t − s )) } = inf 0 ≤ s ≤ t { C 1 ( s, t ) − α 1 1 ( t − s ) + ( C 2 ( s, t ) − α 2 1 ( t − s )) } ≥ inf 0 ≤ s ≤ t { C 1 ( s, t ) − α 1 1 ( t − s ) + inf 0 ≤ s ≤ t [ C 2 ( s, t ) − α 2 1 ( t − s )] } = inf 0 ≤ s ≤ t { C 1 ( s, t ) − α 1 1 ( t − s ) } + inf 0 ≤ s ≤ t { C 2 ( s, t ) − α 2 1 ( t − s ) } (45) The lower cur ve and the lower bo unding functio n of C ( t ) , i.e., Equation s (41) an d (42), are thus p roved ba sed on the a bove inequality and Inequ ality ( 12). V I I . F U RT H E R I M P R OV E M E N T O N B O U N D S So far , we have d ev eloped an analytical framework based on stochastic ne twork calculus to ev aluate the per forman ce of network systems with nodes powered by r enew able en ergy sources. Th is an alytical framework is general eno ugh no matter wheth er or not the en ergy-oblivious service pro cess and the en ergy charging proce ss are indepen dent, a nd no matter whether or not multiple energy sources, if they exists, are indepen dent. If we assume that the energy- oblivious ser vice process and the en ergy charging proc ess are ind ependen t or that multiple energy sources are independent, which we believ e is true in m ost applications, b etter performa nce b ounds could be obtained, using the following lem ma. Lemma 5: Assume that th e complementa ry cumu lative dis- tribution fu nctions (CCDF) of non-negative rando m v ariables X , Y are ¯ F X ∈ ¯ F and ¯ F Y ∈ ¯ F , respectively . Assume that f 1 ( x ) ≤ ¯ F X ( x ) ≤ f 2 ( x ) , g 1 ( x ) ≤ ¯ F Y ( x ) ≤ g 2 ( x ) , and f 1 , f 2 , g 1 , g 2 ∈ ¯ F . Den ote Z = X + Y . Assume tha t X and Y are inde penden t, there hold s f or ∀ x ≥ 0 , ¯ F Z ( x ) ≤ 1 − ¯ f 2 ⋆ ¯ g 2 ( x ) , (46) and ¯ F Z ( x ) ≥ 1 − ¯ f 1 ⋆ ¯ g 1 ( x ) , (47) where ¯ f 1 = 1 − [ f 1 ] 1 , ¯ f 2 = 1 − [ f 2 ] 1 , ¯ g 1 = 1 − [ g 1 ] 1 , ¯ g 2 = 1 − [ g 2 ] 1 , and ⋆ is the Stieltjes convolution operatio n [17]. W ith Lemma 5, if the indepe ndence assumption cou ld be made, the b ounds in the theor ems of this p aper could be revised acco rdingly . That is, we cou ld apply Lemma 5 instead of Lemma 1 in the p roofs of the theo rems to get better bound s. For examp le, if we assume that two energy sources are in depend ent, th e lower and up per boun ds in T heorem 6 should be revised, resp ectiv ely , to f 1 ( x ) = 1 − (1 − f 1 1 ) ⋆ (1 − f 2 1 ) , (48) f 2 ( x ) = 1 − (1 − f 1 2 ) ⋆ (1 − f 2 2 ) . (49) As an example, assume that f 1 1 ( x ) = f 2 1 ( x ) = e − 2 x and f 1 2 ( x ) = f 2 2 ( x ) = e − x . W ith Theorem 6, we have the lower bound f 1 ( x ) = e − 2 x and the upper boun d f 2 ( x ) = 2 e − x/ 2 . But, by applying Lemm a 5, we c an get new lower b ound f 1 ( x ) = (1 + 2 x ) e − 2 x and new up per bound f 2 ( x ) = 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Bounding Functions old upper bound new upper bound old lower bound new lower bound Fig. 1. Comparison of old and ne w bounds (1 + x ) e − x . It is easy to verif y that the new bo unds are tighter as shown in Fig. 1. V I I I . R E L A T E D W O R K W ith the increasing importance of greenin g compu ting, QoS and perfo rmance e v aluation of network systems with unc ertain energy supp ly have attracted much attention . Many p erfor- mance mo deling ap proache s have been developed, wh ich could be divided into two main categories, determ inistic and stochastic approach es. In the deterministic group, Zafer [23] et al. use deterministic network calculus to model traf fic arrival and traffic departur e. They use a power -rate function to link the traffic dep arture rate and energy consumption rate. By considering the special features of specific power -rate fun ctions, th ey form ulate an d solve the optimal transmission scheduling problem under the giv en energy constraints. Their work only focuses on single - node analy sis and assumes th at traffic ar riv a ls and serv ice rates are determ inistic. Kan sal et al. [11] pro pose a so-termed harvesting theory to h elp energy man agement of a sen sor node an d d etermine the p erforma nce lev els that the sensor node can supp ort. The basic idea o f the harvesting theory is to use a leaky bucket mod el to represent energy sup ply and ene rgy dep letion. Mo ser et al. [ 15] de scribe en ergy-aware scheduling and prove the conditions for a scheduling algorithm to b e optimal in a system whose en ergy stor age is reple nished pr edictably . In the stoch astic grou p, Markov chain models h av e been used extensi vely . Susu e t al. [20] use a discrete-time Markov chain in wh ich states rep resent d ifferent energy lev els. Some work [16] uses a Mar kov chain m odel to capture the influence of clouds and win d o n solar rad iation intensity . Rele vant to stochastic en ergy modeling, ther e are many efforts to predict a stoc hastic energy supp ly . L u et al. [14] assess thr ee p re- diction techniq ues: regression analy sis, moving a verage, an d exponential smooth ing. Recas et a l. [1 8] prop ose a weather- condition ed moving average ( W C M A ) model, which adapts to long-ter m seasonal ch anges and short-term sudden weather changes. Moser et al. [15] introdu ce energy variability cur ves to predict the power p rovided b y a harvesting unit. W e develop our analytical fram ew ork based on stochastic network calculu s [2], [6], [ 9], [ 10]. Un like determ inistic net- work calculus [1], [1 2], which searches for the worst-case per- forman ce boun ds, stochastic network c alculus tries to derive tighter p erforma nce boun ds, but with a small probab ility that the bo unds may not hold true. Since m ost ren ew able energy 8 sources, such as solar and wind energy , ar e not d eterministic, stochastic n etwork calculus is a go od fit for the perfo rmance ev aluation of system s u sing r enew able energy . Nevertheless, traditional stochastic network calculus was no t o riginally targeted at modeling such systems. Substantial work is thus required to extend this useful theory . Recent interesting work b y W ang et al. [22] uses stoch astic network calculu s to ev aluate the reliability of the power grid with re spect to r enewable energy . Their en ergy supply and demand mo dels are a subset of the mo dels we present in Section III. Th eir work shows a g ood examp le of how to tailor our mode ls for a specific a pplication. Another fun damental difference is that W ang et al. d efine the energy supply and energy dem and as two de-coup led r andom processes, wh ile in our work energy discharging is inh erently coup led with energy charging. Finally , related to an alytical frameworks for perfo rmance modeling , ther e is a large body of researc h on energy -aware scheduling algorith ms. For examp le, Niy ato et al. [16] in- vestigate the im pact of different sleep and wake-up strategies on data communication am ong solar-po wered wireless nod es. In [ 21], V igo rito et al. pro pose an ad aptive duty -cycling algorithm that ensu res operationa l power levels at wir eless sensor nodes regar dless of chan ging environmental conditions. In [7], G orlatova e t al. measure th e energy av ailability in indoor environment and based on the measu rement results they develop alg orithms to d etermine energy allocation in systems with pred ictable energy inp uts and in systems wher e energy inputs a re stoch astic. In the stochastic mode l, they assume th at e nergy inputs are i.i. d. ra ndom variables. Un like the a bove work, ou r analytical fram ew ork is generic and uses only abstract notatio ns on energy c harging/dischargin g amount, traffic arriv al amou nt, etc. As such, all the above work could be tre ated as a spec ial case o f ou r mo re gener al framework in which the abstract fu nctions are replaced with concrete ones for specific applicatio ns. I X . C O N C L U S I O N A N D F U T U R E W O R K The application of ren ew able en ergy po ses m any chal- lenging question s wh en it com es to a QoS gu arantee fo r IT infr astructure. Particular ly , the strong call fo r a generic analytical too l for the pe rforman ce modeling and evaluation of such systems rem ains unan swered. W e, in this p aper, d ev elop such a the ory using a stochastic network calculus [2], [6], [9], [10] appro ach. W e enrich the existing stochastic network calculus theo ry to make it useful for ev aluating systems with a stoch astic energy supp ly . W e intro duce new models to capture th e dy namics in the energy chargin g an d discharging processes, an d d erive new analytical results to provide funda - mental per forman ce bound s, wh ich could be used to guide the energy manag ement and the d esign of task scheduling algorithm s. This paper mainly fo cuses on the introdu ction of a theo- retical f ramework. Along the line, we en visage many futur e research topics, including (1) the ap plication of energy charg- ing/discharging models in different applications (e.g. , [20], [22]), (2) analyzing the service model o f different scheduling strategies (e.g. , [15]), and (3) n etwork reliability an alysis (e. g., network-wide energy o utage analysis). 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W e thus have { X + Y > y } ⊃ ( { X > x } ∩ { Y > y − x } ) (51) and hence P r ob { X + Y > y } ≥ P rob { { X > x } ∩ { Y > y − x }} ≥ 1 − P r ob { X ≤ x } − P r ob { Y ≤ y − x } = P rob { X > x } + P r ob { Y > y − x } − 1 . (52) Since the above ine quality holds for all y ≥ x ≥ 0 , we get P r ob { X + Y > y } ≥ sup y ≥ x ≥ 0 { P rob { X > x } + P rob { Y > y − x }} − 1 , (53 ) which is ¯ F Z ( y ) ≥ ¯ F X ¯ ⊗ ¯ F Y ( y ) − 1 ≥ f 1 ¯ ⊗ g 1 ( y ) − 1 . (54) B. Pr oo f of Lemma 2 Pr oo f: By definitio ns of the ⊗ and ∧ operation s, we need to prove that inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ∧ f 3 ( t − s ) } ≤ inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ( t − s ) }∧ f 3 ( t ) . (55) Since f 3 ∈ F , we have inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ∧ f 3 ( t − s ) } ≤ inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ( t − s ) ∧ f 3 ( t ) } . (56) W e th erefore only need to prove that inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ( t − s ) ∧ f 3 ( t ) } ≤ inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ( t − s ) }∧ f 3 ( t ) . (57) First, fo r any gi ven t ≥ 0 , th ere e xists a s 1 (0 ≤ s 1 ≤ t ) , such that inf 0 ≤ s ≤ t { f 1 ( s ) + f 2 ( t − s ) } = f 1 ( s 1 ) + f 2 ( t − s 1 ) . (58) In addition, when t is given, f 3 ( t ) is a constan t. W e n ext p rove th at Inequ ality (5 7) h olds tr ue for the following two exclusive cases: • Case 1 : f 1 ( s 1 ) + f 2 ( t − s 1 ) ≤ f 3 ( t ) . T he right hand side o f I nequality (57) equ als f 1 ( s 1 ) + f 2 ( t − s 1 ) . Since f 1 ∈ F and f 1 (0) = 0 , we have f 1 ( s 1 ) ≥ 0 . Theref ore, f 2 ( t − s 1 ) ≤ f 3 ( t ) . (59) W e need to sh ow that the left h and side of Ineq uality (57) is no larger th an f 1 ( s 1 ) + f 2 ( t − s 1 ) . For the given t ≥ 0 , there exists a s 2 (0 ≤ s 2 ≤ t ) such th at inf 0 ≤ s ≤ t { f 1 ( s )+ f 2 ( t − s ) ∧ f 3 ( t ) } = f 1 ( s 2 )+ f 2 ( t − s 2 ) ∧ f 3 ( t ) . (60) W e have f 1 ( s 2 ) + f 2 ( t − s 2 ) ∧ f 3 ( t ) ≤ f 1 ( s 1 ) + f 2 ( t − s 1 ) ∧ f 3 ( t ) = f 1 ( s 1 ) + f 2 ( t − s 1 ) . (61) In the above, the first inequality is due to (60), and th e second equality is due to (59). • Case 2: f 1 ( s 1 ) + f 2 ( t − s 1 ) > f 3 ( t ) . The right hand side of In equality (5 7) equals f 3 ( t ) . W e also hav e f 1 (0) + f 2 ( t − 0) ≥ f 1 ( s 1 ) + f 2 ( t − s 1 ) due to (58). Ther efore, f 2 ( t ) ≥ f 1 ( s 1 ) + f 2 ( t − s 1 ) > f 3 ( t ) , and inf 0 ≤ s ≤ t { f 1 ( s ) + f 2 ( t − s ) ∧ f 3 ( t ) } ≤ f 1 (0) + f 2 ( t ) ∧ f 3 ( t ) = f 3 ( t ) (62) In conclusion, Ine quality (5 7) is tru e fo r both cases. The lemma is proved. C. Pr oof o f Lemma 3 Pr oo f: T o e valuate the value of X 1 ∧ X 4 − X 2 ∧ X 3 , we have: • Case 1: X 1 ≤ X 4 and X 2 ≥ X 3 . W e h av e X 1 = X 2 = X 3 = X 4 , because X 1 ≥ X 2 ≥ 0 and X 3 ≥ X 4 ≥ 0 . Thus (24) holds. • Case 2: X 1 ≤ X 4 and X 2 ≤ X 3 . W e have X 1 ∧ X 4 − X 2 ∧ X 3 = X 1 − X 2 ≤ X 1 − X 2 + X 3 − X 4 . • Case 3: X 1 ≥ X 4 and X 2 ≥ X 3 . W e have X 1 ∧ X 4 − X 2 ∧ X 3 = X 4 − X 3 ≤ 0 . Inequa lity (2 4) ho lds since the righ t side is no less than 0 . • Case 4: X 1 ≥ X 4 and X 2 ≤ X 3 . W e have X 1 ∧ X 4 − X 2 ∧ X 3 ≤ X 1 − X 2 ≤ X 1 − X 2 + X 3 − X 4 . The le mma is pr oved sin ce the above list c overs all possible scenarios. 10 D. Pr oo f of Lemma 5 Pr oo f: The p roof of ( 46) co uld be f ound at [17]. W e on ly prove (4 7). For independ ent no n-negative rand om v ariables, X an d Y , we have ∀ x ≥ 0 , F Z ( x ) = Z x 0 F X ( x − y ) dF Y ( y ) . Note that F X , F Y , ¯ f 1 , ¯ g 1 are wide-sense incre asing, F X ≤ ¯ f 1 and F Y ≤ ¯ g 1 . Hence, we have F Z ( x ) = Z x 0 F X ( x − y ) dF Y ( y ) ≤ Z x 0 ¯ f 1 ( x − y ) dF Y ( y ) = Z x 0 F Y ( x − y ) d ¯ f 1 ( y ) ≤ Z x 0 ¯ g 1 ( x − y ) d ¯ f 1 ( y ) = ¯ f 1 ⋆ ¯ g 1 ( x ) . (63) Note that the third eq uality ho lds because Stieltjes convolution operation is commutative. I nequality (47) is thus proved since ¯ F Z ( x ) = 1 − F Z ( x ) .