Stochastic Service Guarantee Analysis Based on Time-Domain Models

Stochastic Service Guarantee Analysis Based on T ime-Domain Models Jing Xie Department of T elematics Norwegian Univ ersity of Science an d T echnology Email: ji ngxie@item.ntnu.no Y uming Jiang Department of T e lematics & Q2S Center Norwegian Univ ersity of Science an d T echnology Email: ymjiang@ieee.org Abstract —Stochastic network calculus is a theory f or stochastic service guarantee analysis o f computer com- munication netw orks. In the current stocha stic network calculus literature, its traffic and serv er models are typ- ically defined based on the cumulative amo unt of t raffic and cumulative amount of service respectively . However , there are network scenarios where the applicability of such models is limited, and hence n ew ways of modeling traffic and service ar e needed t o address this l imitation. This pa- per presents time-d omain models a nd results for stochastic network calculus. Particularly , we define traffic models, which are defined based on pr obabilistic lo wer -bounds on cumulative packet inter-arrival time , a nd server models, which are defined based on probabilistic upper-bounds on cumula tive pa ck et service tim e . In additio n, examples demonstrating the use of the pr oposed time-domain models are provided. On the basis of the proposed models, the five basic properties o f stochastic network calculus a re a lso proved, which impli es broad applicability of the proposed time-dom ain approach. I . I N T RO D U C T I O N Stochastic network calculus is a theory dea ling with queueing sy stems foun d in c omputer co mmunication networks [1][2] [3][4]. It is p articularly useful for analyz ing net- works where se rvice guarantees are provided stochas- tically . Su ch networks include wireless networks, multi- acces s networks a nd multimedia networks whe re appli- cations ca n tolerate so me certain violation o f the desired performance [5]. Stochastic network calculus is ba sed on properly de- fined traf fic models [6][3][4][7][8][9] and server mo dels [3][4]. In the existing mod els o f stoch astic network calculus, an a rri v al process an d a service p rocess are typ- ically modeled by some stocha stic arri val curve, which probabilistically uppe r -bounds the cumulative amount of a rrival , and res pectively by some stochas tic ser- vice curve, which probabilistically lo wer- bounds the cumulative amount of se rvice . In this paper , we c all such models spa ce-domain mode ls. Ba sed on the space- domain traffic and server models, a lot of resu lts have been deri ved for stochas tic network c alculus. Amo ng the others, the most fundamental on es are the fiv e basic properties [3] [4]: (P .1) Service Guarantees including delay bou nd and backlog boun d; (P .2) Ou tput Charac- terization ; (P .3) Concatenation Pr operty ; (P .4) Le ftover Service ; (P .5) Superposition Pr operty . Examples demon- strating the n ecess ity of having the se basic prope rties and t heir use can be found [3] [4]. Nev ertheless, there are still many open resea rch chal- lenges for stocha stic network c alculus, an d a critical one is time-domain modeling a nd analysis [4]. T ime- domain modeling for service guaran tee a nalysis has its root from the deterministic Guaranteed Rate (GR) server model [10], where service g uarantee is captured by comparing with a (dete rministic) v irtual time func- tion in the time-doma in. This time-doma in model ha s been extended t o design aggregate-sche duling networks to s upport pe r -flow (deterministic) service gu arantees [11][12], while f ew such results are a vailable from space- domain mo dels. Othe r network s cenarios where ti me- domain mo deling may be p referable include wireless networks an d multi-acces s networks. In wireless networks, the v arying l ink condition may cause failed tr ansmission when the link is in ‘bad’ con- dition. T he s ender may hold u ntil the link condition becomes ‘good’ or re-transmit. For su ch c ases, it is dif ficult to directly find the stocha stic service c urve in the spac e-domain bec ause we need to chara cterize the stochas tic nature of the impaired se rvice caused by the ‘bad’ link cond ition. A pos sible way is that we use an impairment process [3] to ch aracterize the impaired service. Howe ver , how to defin e a nd find the impairment process a rises an other dif ficulty . Even though we c an define an impairment p rocess, we may first co n vert the impairment process into s ome existing stochas tic network calculus models, and then further analyze the performance bounds. The obtained performance bou nds may become loos e because of su ch c on version. If we characterize the seri vce proce ss i n the t ime-domain, we can us e random variables to represent the time intervals when the link is in ‘bad’ cond ition. Analyzing the stochas tic nature of suc h random variables would be easier . In add ition, this way can av oid the dif ference introduced by the intermediate con version. In contention-ba sed mu lti-access networks, backoff scheme s are often emplo yed t o reduce collision occuring. Becaus e the backof f process is characterized by backoff windows which may v ary with the dif ferent backof f stages, it is quite c umbersome for a spa ce-domain server model to characterize the service proces s with the con- sideration o f the backof f process. This a lso p rompts the possibility of characterzing the service process in the time-domain. Having sa id this, however , how to defin e a stochas tic version of the virtual time function and how to perform the corresponding analysis are yet open [4]. The objec ti ve of this p aper is to define traffic mode ls and s erver models in the time-domain an d de ri ve the correspond ing five basic p roperties for stoch astic ne t- work c alculus. Particularly , we de fine traffic mod els that are base d on probabilistic lower bo unds on cu mulative packet inter-arrival time . Also, we defin e server models that are b ased on s ome virtual time function and proba- bilistic upper bounds on c umulative pack et service time . In addition, we es tablish relationships among the pro- posed time-domain models, and the mapp ings b etween the proposed t ime-domain models and the e xisting spac e- domain models. F urthermore, we prove the fi ve basic properties b ased o n the proposed ti me-domain models. The remainder is structured a s follo ws. Sec. II in- troduces the mathematical background and fundamental space -domain mod els and relev ant results of stochastic network calculus. In Sec. III, we first introduce the time-domain de terministic traf fic an d server models, an d then extend them to stoch astic versions. In addition, the relationships among them as well as wit h so me existing space -domain models are es tablished. Se c. IV explores the fi ve basic prope rties. Sec . V summarizes the work. I I . N OTA T I O N A N D B AC K G RO U N D T o ea se exp ression, we as sume networks with fixe d unit length 1 packets. By con vention, we assume that 1 The results can also be exten ded to netw orks with variable-length packe ts while the expression and results wil l be more complicated. a packet is considered to be recei ved by a network element wh en and only wh en its last bit has a rri ved to the network element, an d a packet is cons idered out of a network element when and only whe n its last bit has been transmitted by the network element. A packet can be served only when it s last bit has arriv e d. All queues are a ssumed to be empty at time 0 . Packets within a flow are served in the first-in-fir st-out (FIFO) orde r . A. Notati on Let p n , r ( n ) , a ( n ) and d ( n ) ( n = 0 , 1 , 2 , ... ) den ote the n th packet of a flow , its allocate d service rate, its arri val time and its depa rture time, res pectively . Let A ( t ) and A ∗ ( t ) res pectively de note the number of cumulati ve arri val packets and the number of cumu lati ve departure packets by time t . By con vention, we assu me a (0) = 0 , d (0) = 0 , A (0) = 0 and A ∗ (0) = 0 . For an y 0 ≤ s ≤ t , we denote A ( s, t ) ≡ A ( t ) − A ( s ) and A ∗ ( s, t ) ≡ A ∗ ( t ) − A ∗ ( s ) . In this paper , a ( n ) an d A ( t ) will be used to rep resent an a rri val proce ss interchan geably . A depa rture process will b e rep resented b y d ( n ) and A ∗ ( t ) interch angeab ly . The followi ng function sets are often use d in this paper . S pecifically , we use G to deno te the set of non- negati ve wide-sense increa sing functions as follows: G = { g ( · ) : ∀ 0 ≤ x ≤ y , 0 ≤ g ( x ) ≤ g ( y ) } W e de note b y ¯ G the set of non-negati ve wide-se nse decreas ing functions: ¯ G = { g ( · ) : ∀ 0 ≤ x ≤ y , 0 ≤ g ( y ) ≤ g ( x ) } Let ¯ F denote the set of functions in ¯ G , where for each function f ( · ) ∈ ¯ F , its n th-fold integration, den oted by f ( n ) ( x ) ≡  R ∞ x dy  n f ( y ) , is bou nded for ∀ x ≥ 0 and still b elongs to ¯ F for ∀ n ≥ 0 , or ¯ F =  f ( · ) : ∀ n ≥ 0 ,  Z ∞ x dy  n f ( y ) ∈ ¯ F  . For ease of expos ition, we ad opt [ x ] + ≡ max[0 , x ] and [ x ] 1 ≡ min[1 , x ] , and as sume that for any b ounding function f ( x ) , f ( x ) = 1 for ∀ x < 0 . B. Max-plus and Min-plus Algebra Basics An es sential idea of (stochastic) network ca lculus is to use alternate algebra s particularly the min-plus algeb ra and max-plus algebra [13] to transform c omplex non- linear network systems into analytica lly tr actable linear systems [4]. T o the best of our k nowledge, the existing models a nd results o f stoc hastic network ca lculus are mainly under the space-do main and ba sed on min-plus algebra that has b asic o perations pa rticularly suitable for c haracterizing cumulative a rri val an d cumulative ser- vice. For charac terizing arriv al and service processes in the time-domain , interestingly , the max-plus algebra has basic operations that well suit t he need. In this pape r , t he foll owing max -plus an d min-pl us operations are often used: • Max-Plus Convolution of g 1 and g 2 is ( g 1 ¯ ⊗ g 2 )( x ) = sup 0 ≤ y ≤ x { g 1 ( y ) + g 2 ( x − y ) } • Max-Plus De con v olution of g 1 and g 2 is ( g 1 ¯ ⊘ g 2 )( x ) = inf y ≥ 0 { g 1 ( x + y ) − g 2 ( y ) } • Min-Plus Co n volution of g 1 and g 2 is ( g 1 ⊗ g 2 )( x ) = inf 0 ≤ y ≤ x { g 1 ( y ) + g 2 ( x − y ) } • Min-Plus De con v olution of g 1 and g 2 is ( g 1 ⊘ g 2 )( x ) = sup y ≥ 0 { g 1 ( x + y ) − g 2 ( y ) } In this pape r , when applying supr emu m and infi mum , they may be interpreted as max imum and minimum whenever appropriate, respec ti vely . C. Pr eliminaries The follo w ing lemma is often us ed for later an alysis and t hus li s ted: Lemma 1. F or the s um of a collection of random variables Z = P n i =1 X i , no matter whether they are independ ent or not, ther e ho lds for the compleme ntary cumulative distribution function (CCDF) of Z : (See Lemma 1.5 in [4]) ¯ F Z ( z ) ≤ ¯ F X 1 ⊗ · · · ⊗ ¯ F X n ( z ) (1) where ¯ F Z = P { Z > z } , −∞ < z ≤ ∞ . For later ana lysis, we need some transformation be - tween the nu mber of cumulative arr i val pa ckets by time t , i.e., A ( t ) , and the time of a packet a rri ving to the system, i.e., a ( n ) . If A ( t ) is upper-bounded with respect to some func- tion α ( t ) ∈ G , we have the following lemma. Lemma 2. F or f unction α ( t ) ∈ G , ther e ho lds: 1) the following s tatements ar e equ ivalent: a) ∀ 0 ≤ s ≤ t , A ( s, t ) ≤ α ( t − s ) + x for ∀ x ≥ 0 ; b) ∀ t ≥ 0 , A ( t ) ≤ A ⊗ α ( t ) + x for ∀ x ≥ 0 ; 2) if ∀ t, x ≥ 0 , A ( t ) ≤ A ⊗ α ( t ) + x holds, then we have a ( n ) ≥ a ¯ ⊗ λ ( n ) − y , w here λ ( n ) ∈ G is the in verse fun ction of α ( t ) and define d as foll ows λ ( n ) = inf { τ : α ( τ ) ≥ n } (2) and y = sup k ≥ 0 [ λ ( k ) − λ ( k − x )] . (3) Pr oof: (1) For ( a ) → ( b ) , from the condition, we obtain A ( s, t ) − α ( t − s ) − x ≤ 0 for ∀ 0 ≤ s ≤ t . Then, there holds sup 0 ≤ s ≤ t [ A ( s, t ) − α ( t − s ) − x ] ≤ 0 which implies A ( t ) − i nf 0 ≤ s ≤ t [ A ( s ) + α ( t − s )] − x ≤ 0 . Thus, we conclud e A ( t ) ≤ A ⊗ α ( t ) + x for ∀ t, x ≥ 0 . For ( b ) → ( a ) , from the condition, we ha ve A ( t ) − i nf 0 ≤ s ≤ t [ A ( s ) + α ( t − s )] − x ≤ 0 which implies sup 0 ≤ s ≤ t [ A ( s, t ) − α ( t − s ) − x ] ≤ 0 . Then t here must be A ( s, t ) − α ( t − s ) − x ≤ 0 for ∀ 0 ≤ s ≤ t . Thus, A ( s, t ) ≤ α ( t − s ) + x holds for ∀ 0 ≤ s ≤ t and ∀ x ≥ 0 . (2) From (1), we know that A ( t ) ≤ A ⊗ α ( t ) + x is equiv ale nt t o A ( s, t ) ≤ α ( t − s ) + x for ∀ 0 ≤ s ≤ t and ∀ x ≥ 0 . The n for ∀ 0 ≤ m ≤ n , we have A  a ( m ) , a + ( n )  ≤ α  a + ( n ) − a ( m )  + x where a + ( n ) = a ( n ) + ǫ with ǫ → 0 . W e also kn ow n − m ≤ A  a ( m ) , a + ( n )  ≤ α  a + ( n ) − a ( m )  + x T aking the in verse func tion of α  a + ( n ) − a ( m )  yields a + ( n ) − a ( m ) ≥ λ ( n − m − x ) = λ ( n − m ) − [ λ ( n − m ) − λ ( n − m − x )] ≥ λ ( n − m ) − sup n − m ≥ 0 [ λ ( n − m ) − λ ( n − m − x )] (4) Let k = n − m . Eq.(4) ca n be written a s a + ( n ) − a ( m ) ≥ λ ( n − m ) − sup k ≥ 0 [ λ ( k ) − λ ( k − x )] from which we obtain a ( n ) ≥ a ( m ) + λ ( n − m ) − y (5) becaus e ǫ → 0 and y = sup k ≥ 0 [ λ ( k ) − λ ( k − x )] . Since Eq.(5) holds for ∀ 0 ≤ m ≤ n , we hav e a ( n ) ≥ sup 0 ≤ m ≤ n [ a ( m ) + λ ( n − m ) − y ] = a ¯ ⊗ λ ( n ) − y . Example 1 . Suppose the numbe r of cumulative arri val packets of a flow , A ( t ) , is upper-bounded b y α ( t ) + x for t ≥ 0 , where α ( t ) = ρ · t + σ . Let n ≡ α ( t ) . W e can get the i n verse func tion of α ( t ) , λ ( n ) = ( n − σ ) + ρ . W e c an use Eq.(3) to get y . For ∀ k ≥ 0 , we have y = sup k ≥ 0 n ( k − σ ) + ρ − ( k − σ − x ) + ρ o =        x ρ k ≥ σ + x, < x ρ σ ≤ k < σ + x, 0 k < σ. from which we get y = x ρ . Then, we know that for any packet, its arri val time satisfies a ( n ) ≥ sup 0 ≤ m ≤ n  a ( m ) + ( n − m − σ ) + ρ  − x ρ . If a ( n ) is lo we r -bounded with res pect to some func- tion λ ( n ) ∈ G , we have the following lemma. Lemma 3. F or f unction λ ( n ) ∈ G , the r e holds: 1) the following s tatements ar e equ ivalent: a) ∀ 0 ≤ m ≤ n , a ( n ) − a ( m ) ≥ λ ( n − m ) − y for ∀ y ≥ 0 ; b) ∀ n ≥ 0 , a ( n ) ≥ a ¯ ⊗ λ ( n ) − y for ∀ y ≥ 0 ; 2) if ∀ n, y ≥ 0 , a ( n ) ≥ a ¯ ⊗ λ ( n ) − y holds , then we have A ( t ) ≤ A ⊗ α ( t ) + x , wh ere α ( t ) ∈ G is the in verse function of λ ( n ) and defined as follows α ( t ) = sup { k : λ ( k ) ≤ t } (6) and x = su p τ ≥ 0 [ α ( τ + y ) − α ( τ ) + 1] . (7) Pr oof: (1) The ( a ) → ( b ) p art has been p roved i n Lemma 2(2). W e only prove the ( b ) → ( a ) pa rt. From the condition, we ha ve a ( n ) − sup 0 ≤ m ≤ n { a ( m ) + λ ( n − m ) } + y ≥ 0 which implies inf 0 ≤ m ≤ n { a ( n ) − a ( m ) − λ ( n − m ) } + y ≥ 0 . Thus t here holds a ( n ) − a ( m ) ≥ λ ( n − m ) − y for ∀ 0 ≤ m ≤ n and ∀ y ≥ 0 . (2) For ∀ 0 ≤ s ≤ t , we can find m, n ≥ 0 a ccording to t he follo wing f unctions A ( t ) = n = sup { k : a ( k ) ≤ t } A ( s ) = m = sup { k : a ( k ) ≤ s } . Thus, we hav e A ( s, t ) = n − m and a ( n ) − a ( m + 1) ≤ t − s . From (1), we know that a ( n ) ≥ a ¯ ⊗ λ ( n ) − y is equiv ale nt to a ( n ) − a ( m ) ≥ λ ( n − m ) − y . Th en we have t − s ≥ a ( n ) − a ( m + 1) ≥ λ ( n − m − 1) − y . T aking the in verse func tion of λ ( n − m − 1) y ields n − m − 1 ≤ α ( t − s + y ) Becaus e A ( s, t ) = n − m , we have A ( s, t ) ≤ α ( t − s + y ) + 1 = α ( t − s ) + [ α ( t − s + y ) − α ( t − s ) + 1] ≤ α ( t − s ) + sup t − s ≥ 0 [ α ( t − s + y ) − α ( t − s ) + 1] Let τ = t − s . Th e a bove inequality is written as A ( s, t ) ≤ α ( t − s ) + su p τ ≥ 0 [ α ( τ + y ) − α ( τ ) + 1] = α ( t − s ) + x Since A ( s, t ) − α ( t − s ) − x ≤ 0 holds for ∀ 0 ≤ s ≤ t , we have sup 0 ≤ s ≤ t [ A ( s, t ) − α ( t − s ) − x ] ≤ 0 from which we f urther obtain A ( t ) − i nf 0 ≤ s ≤ t [ A ( s ) + α ( t − s )] − x ≤ 0 . W e then conclude A ( t ) ≤ A ⊗ α ( t ) + x . D. R elated S pace-dom ain Results This s ub-section re views s ome related sp ace-domain results und er min-plus algeb ra [4]. It is worth high- lighting that the follo wing resu lts are for d iscrete time systems with unit discretization step. The virtual-backlog-centric (v .b .c) stoc hastic arri val curve mod el [14] is define d b ased on a proba bilis- tic uppe r- b ound on cumulativ e arri val. T o ease later analysis, the de finition of v .b.c stocha stic arri val curve model presented in this paper is bas ed on the nu mber of cumu lati ve arriv al pac kets while not the a mount of cumulativ e arriv a l (in bits) which has been w idely used in t he network calculus literature. The v .b .c s tochastic arriv al curve model explores the virtual backlog pr o perty of deterministic arri val curve, which is that t he q ueue len gth of a virtual single s erver queue (SSQ) fed with the sa me flow with a deterministic arri val curve is uppe r -bounded. For a flow having arri val curve α ( t ) , we construct a virtual SS Q system fed with the sa me flow . The SSQ system has infi nite buf fer space and the buf fer is initially empty . Suppose the virtual SSQ system provides s ervice α ( t ) to the flow for all t ≥ 0 . Then the unfinished wor k or b acklog in the virtual SSQ s ystem by time t is B ( t ) = A ( t ) − A ∗ ( t ) . The Lindely equation can be use d to deri ve B ( t ) , which is B ( t ) = max { 0 , B ( t − 1) + A ( t − 1 , t ) − α ( t − t + 1) } (8) Eq.(8) means that the a mount of traffic backlogged in the system by time t equa ls the a mount o f traffic backlogg ed by time t − 1 p lus the amount of traf fi c having arri ved between t − 1 and t minus the amount of traf fic having been s erved between t − 1 a nd t . By applying Eq.(8) iterati vely to its right-hand s ide, it becomes B ( t ) = sup 0 ≤ s ≤ t [ A ( s, t ) − α ( t − s )] . (9) If the flow is constrained by arriv al curve α ( t ) + x for all t ≥ 0 , it follows f rom Eq.(9) that the system b acklog is also upper -bou nded b y x . Th e v .b .c stochastic arri val curve is defined base d on the virtual backlog pr operty . Definition 1. (v .b .c Stochastic Arrival Curve). A flo w is said to have a virtual-backlog-centric (v .b .c) sto- chastic arrival curve α ( t ) ∈ G with bounding function f ( x ) ∈ ¯ G , denote d by A ∼ vb h α, f i , if for all t ≥ 0 an d all x ≥ 0 , t here holds P  sup 0 ≤ s ≤ t [ A ( s, t ) − α ( t − s )] > x  ≤ f ( x ) . (10) Eq.(10) c an a lso b e written as follows: P  A ( t ) > A ⊗ α ( t ) + x  ≤ f ( x ) . (11) Based on the existing space-domain traf fic a nd server models, a lot of results hav e been deriv e d for stocha stic network calculus wh ich include the fi ve basic p roperties [4] as introduc ed in Sec . I. In this pap er , the following result is specifically made use of in later analysis and hence l isted: Lemma 4. (Superp osition Property). Consider N flows with arriva l pr ocess es A i ( t ) , i=1,...,N, r espec tively . Let A ( t ) denote the aggre ga te arrival pr ocess . If ∀ i , A i ∽ vb h α i , f i i , t hen A ∽ vb h α, f i with α ( t ) = P N i =1 α i ( t ) , and f ( x ) = f 1 ⊗ · · · ⊗ f N ( x ) . I I I . T I M E - D O M A I N M O D E L S This section re views the deterministic arri val curv e and the d eterministic service curve models defin ed in the time-domain. W e generalize the d eterministic models and define time-domain stochastic arriv al curve and stochas tic service curve models. A. Deterministic Arrival Curve Consider a flow of which packets arri ve to a system at time a ( n ) . In order to deterministically guarantee a certain level of qua lity of s ervice (QoS) to this flow , the traf fic se nt by this flow mus t be constrained. The deterministic netw ork calculus traf fic model in the ti me- domain characterizes pac ket inter- a rri val time us ing a lower -bound function, c alled arri val curve in this paper and defined as foll ows [15]: Definition 2. (Arrival Curve). A flow is said to have a (deter ministic) arriva l curve λ ( n ) ∈ G , if its arr ival pr oc ess a ( n ) satisfies, for all 0 ≤ m ≤ n , a ( n ) − a ( m ) ≥ λ ( n − m ) . (12) The arri val curve model has the follo wing triplicity principle which will be used as the basis in defining the stochas tic arri val curve mo dels in the subs equen t subsec tions. Lemma 5. The following statements are equivalent: 1) ∀ 0 ≤ m ≤ n , a ( n ) − a ( m ) ≥  λ ( n − m ) − x  + for ∀ x ≥ 0 ; 2) ∀ n ≥ 0 , su p 0 ≤ m ≤ n n  λ ( n − m ) − x  + − [ a ( n ) − a ( m )] o ≤ 0 for ∀ x ≥ 0 ; 3) ∀ n ≥ 0 , sup 0 ≤ m ≤ n sup 0 ≤ q ≤ m n  λ ( m − q ) − x  + − [ a ( m ) − a ( q )] o ≤ 0 for ∀ x ≥ 0 , where λ ∈ G . Pr oof: I t is tri vially true that λ ( n − m ) − [ a ( n ) − a ( m )] ≤ sup 0 ≤ m ≤ n { λ ( n − m ) − [ a ( n ) − a ( m )] } from which, (2) i mplies (1). In addition sup 0 ≤ m ≤ n { λ ( n − m ) − [ a ( n ) − a ( m )] } ≤ sup 0 ≤ m ≤ n sup m ≤ k ≤ n  λ ( k − m ) − [ a ( k ) − a ( m )]  = sup 0 ≤ k ≤ n sup 0 ≤ m ≤ k  λ ( k − m ) − [ a ( k ) − a ( m )]  = su p 0 ≤ m ≤ n sup 0 ≤ q ≤ m  λ ( m − q ) − [ a ( m ) − a ( q )]  with which, (3) impli e s (2). For (1) → (2), it h olds since a ( n ) − a ( m ) ≥ λ ( n − m ) − x f or ∀ 0 ≤ m ≤ n . For (2) → (3), sup 0 ≤ m ≤ n sup 0 ≤ q ≤ m { λ ( m − q ) − [ a ( m ) − a ( q )] } ≤ sup 0 ≤ m ≤ n [ x ] = x. Thus (1), (2) and (3) are equiv a lent. From Definition 2, the right-hand side of a ( n ) − a ( m ) ≥ λ ( n − m ) − x in Lemma 5.(1) defines an arri val curve λ ( n − m ) − x . In addition, we can construct a virtual single server q ueue (SSQ) sys tem that is initially empty , fed wit h the same traffic fl ow , and has a service curve λ which makes d ( n ) ≤ a ¯ ⊗ λ ( n ) (see Definition 6). The n, the delay in t he virtual SSQ system is upper -bo unded b y d ( n ) − a ( n ) ≤ sup 0 ≤ m ≤ n [ λ ( n − m ) − ( a ( n ) − a ( m ))] ≤ x , and t he maximum sys tem delay for the first n packets is upper-bounded by sup 0 ≤ m ≤ n { d ( m ) − a ( m ) } ≤ su p 0 ≤ m ≤ n sup 0 ≤ q ≤ m { λ ( m − q ) − [ a ( m ) − a ( q )] } ≤ x. Example 2 . The Generic Cell Rate Algorithm (GCRA) [16] with pa rameter ( T , τ ) is a pa rallel algo - rithm to t he L eaky Bucket algorithm and ha s been us ed in fixed-length p acket ne tworks su ch a s Asynchron ous T ran sfer Mode (A TM) networks. The GCRA measures cell rate at a specified time sc ale and assume s that cells will h ave a minimum interval betwee n them. Here, T denotes the assume d mini mum interval between cells and τ de notes the maximum acceptab le excursion that quantifies how early cells may arr i ve with respect to T . It can be verified that if a flow is GCRA ( T , τ ) -con strained, it has an arri val curve λ ( n ) =  T · n − τ  + . B. Inter -arrival-time Stochas tic Arrival Curve Lemma 5.(1) define s a deterministic arriv al c urve λ ( n ) − x which lower -bound s the inter- a rri val time between any two pac kets. Bas ed on this, we define its probabilistic counterpart as foll ows: Definition 3. (i.a.t S tochastic Arrival Curve). A flow is said to have an inter-arrival-time (i.a.t) s tochastic arrival c urve λ ∈ G with bound ing function h ∈ ¯ G , denoted by a ( n ) ∼ it h λ, h i , if for all 0 ≤ m ≤ n and all x ≥ 0 , t here holds P n λ ( n − m ) − [ a ( n ) − a ( m )] > x o ≤ h ( x ) . (13 ) Example 3 . Cons ider a flo w with fixed unit packet size. S uppose its pa cket inter-arri val times follow an exponential distrib u tion with mean 1 ρ . The n, the packet arri val time ha s an Erlang distribution with para meter ( n, ρ ) [17 ]. And, for any two packets p m and p n , their inter -arri val tim e a ( n ) − a ( m ) satisfies, for ∀ x ≥ 0 , P n 1 ρ ( n − m ) − [ a ( n ) − a ( m )] > x o ≤ 1 − n − m − 1 X k =0 e − ρy ( ρy ) k k ! − ρ e − ρy ( ρy ) n − m − 1 ( n − m − 1)! where y = 1 ρ ( n − m ) − x . The i.a.t stoc hastic a rri val curv e is intuiti vely simple, but it has limited use if n o a dditional c onstraint is enforced. Let us consider a simple example to understand this problem. Consider a single no de with constant pe r packet service time T and its input flo w F satisfying a ( n ) ∼ it h τ · n, h i whe re τ ≥ T . Suppose we a re inter - ested in the delay D ( n ) , wh ere, by definition, D ( n ) = d ( n ) − a ( n ) . Becau se t he node has constant per packet service time T , it h as a (de terministic) se rvice curve T · n which implies d ( n ) = sup 0 ≤ m ≤ n [ a ( m ) + T · ( n − m )] . Then we have D ( n ) = sup 0 ≤ m ≤ n  a ( m ) + T · ( n − m )  − a ( n ) = sup 0 ≤ m ≤ n  a ( m ) + T · ( n − m ) − a ( n )  ≤ sup 0 ≤ m ≤ n  τ · ( n − m ) − [ a ( n ) − a ( m )]  (14) From Eq.(14), we have difficulty in further deri ving more results if no ad ditional constraint is added bec ause we only know P { τ · ( n − m ) − [ a ( n ) − a ( m )] > x } ≤ h ( x ) . When in vestigating t he performance metrics such as d elay bou nd and backlog bound in Section IV -A, we meet the similar dif ficulty . C. V irtua l-system-delay Stochastic Arrival Curv e The p revious su bsection stated the dif ficu lty of ap- plying i.a.t s tochastic arri val curve to service guarantee analysis. This subsection introduc es another stoch astic arri val curve model that can he lp a void such dif ficu lty . This model is called virtual-system-delay ( v .s.d ) stochas- tic arri val curve. The model explores the virtual system delay pr op erty of deterministic arri val curve a s implied by Lemma 5.(2), which is that the amount of t ime a packet spen ds in a virtual SSQ fed with the sa me flow with a deterministic arri val curve is lo we r -bounded. For a flow having de terministic a rri val curve, we construct a virtual SSQ system fed wi th the flow , which has infinite buf fer space a nd the buf fer is initi ally empty . Suppose the virtual SSQ sys tem provides a deterministic service curve λ to the flow o r d ( n ) = a ¯ ⊗ λ ( n ) for all n ≥ 0 . The amount of time packet n spe nds in the virtual SSQ s ystem is W s ( n ) = d ( n ) − a ( n ) = sup 0 ≤ m ≤ n { λ ( n − m ) − [ a ( n ) − a ( m )] } . If the flow is constrained by a rri val curve λ ( n ) − x for all n ≥ 0 , W s is also lower -bounde d by x . Based on the virtual system time p roperty , we define virtual-system-delay (v .s .d) stoch astic a rri val cu rve to characterize the arri val proces s. Definition 4. (v . s.d Stochastic Arrival Curve). A flo w is said to have a virtual-system-delay (v .s.d) stochastic arrival c urve λ ∈ G with bound ing function h ∈ ¯ G , denoted b y a ( n ) ∼ vd h λ, h i , if for all 0 ≤ m ≤ n and all x ≥ 0 , t here holds P n sup 0 ≤ m ≤ n  λ ( n − m ) − [ a ( n ) − a ( m )]  > x o ≤ h ( x ) . (15) Eq.(15) c an a lso b e written as P  a ¯ ⊗ λ ( n ) − a ( n ) > x  ≤ h ( x ) . (16) a ¯ ⊗ λ ( n ) can be considered as the expected ti me that the pa cket would arri ve to the sys tem if the flo w had passe d through the virtual SSQ with service cu rve λ ( n ) . x deno tes the diff erence be tween the expec ted arri val time and the actua l arriv al time. Eq.(16) c haracterizes this d if feren ce x b y introducing a bou nding function h ( x ) . Example 4. Consider a flow with the same fixed packet size. Suppose all pac ket inter -arriv al times are exponentially distrib u ted with me an 1 µ . Base d on the steady-state prob ability mas s func tion (PMF) of the queue-waiting time for an M/D/1 queue [18], we say that the flow has a v .s.d s tochastic arri val curve a ( n ) ∼ vd h D · n, h exp i for ∀ D < 1 µ , wit h ρ = µ · D and h exp ( x ) = 1 − (1 − ρ ) ⌊ x/D ⌋ +1 X i =0 e − µ ( − x ) [ µ ( − x )] i i ! where, ⌊ x/D ⌋ de notes the greatest integer less than or equal t o x/D . The follo wing theorem es tablishes relationsh ips be- tween i.a.t stoc hastic arri val curve and v .s.d stoc hastic arri val curve. Theorem 1. 1) If a flo w ha s a v .s. d stochastic arrival curve λ ∈ G with boun ding function h ∈ ¯ G , then the flow has a n i.a.t stochastic arrival curve λ ∈ G with the same bounding function h ∈ ¯ G . 2) In versely , if a fl ow has an i.a.t stochastic a rrival curve λ ∈ G with bounding function h ∈ ¯ F , it a lso has a v .s.d stochastic arr ival curve λ − η ∈ G with bounding f unction h η ∈ ¯ G where λ − η ( n ) = λ ( n ) − η · n h η ( x ) = h h ( x ) + 1 η Z ∞ x h ( y ) dy i 1 for ∀ η > 0 . Pr oof: The first part follo ws from that λ ( n − m ) − [ a ( n ) − a ( m )] ≤ sup 0 ≤ m ≤ n { λ ( n − m ) − [ a ( n ) − a ( m )] } holds f or ∀ 0 ≤ m ≤ n . For the seco nd part, t here holds sup 0 ≤ m ≤ n  λ − η ( n − m ) − [ a ( n ) − a ( m )]  ≤ st sup 0 ≤ m ≤ n  λ − η ( n − m ) − [ a ( n ) − a ( m )]  + Since f or ∀ x ≥ 0 , P  { λ ( n − m ) − η · ( n − m ) − [ a ( n ) − a ( m )] } + > x  = P  { λ ( n − m ) − η · ( n − m ) − [ a ( n ) − a ( m )] } > x  ≤ h  x + η · ( n − m )  , we have P n sup 0 ≤ m ≤ n { λ − η ( n − m ) − [ a ( n ) − a ( m )] } > x o ≤ n X m =0 P n { λ − η ( n − m ) − [ a ( n ) − a ( m )] } + > x o ≤ n X m =0 h ( x + η · ( n − m )) = n X k =0 h ( x + η · k ) ≤ ∞ X k =0 h ( x + η · k ) = h ( x ) + ∞ X k =1 h ( x + η · k ) ≤ h ( x ) + 1 η Z ∞ x h ( y ) dy . (17) which is mea ningful only when Eq.(17) is upper - bounde d by one. The 1-fold integrati on of h ( x ) is bounde d by one beca use the condition as sumes h ∈ ¯ F as for the [8]. Then t he second part follo ws from Eq.(17). Note that in the secon d part of the a bove theorem, h ( x ) ∈ ¯ F while not ∈ ¯ G . If the requirement o n the bounding function is relaxed to h ( x ) ∈ ¯ G , the above relationship ma y not hold in gene ral. The v .s.d stochastic arriv al curve has a co unterpart defined in the space-do main, the v .b .c stoch astic arri val curve as defined in Definition 1. The following theorem establishes relati onships between these tw o models. Theorem 2. 1) If a flo w has a v .b .c stochas tic arrival curve α ( t ) ∈ G with b ounding fun ction f ( x ) ∈ ¯ G , the flo w has a v .s. d stochastic arrival curve λ ( n ) ∈ G with bounding function h ( y ) ∈ ¯ G , whe r e λ ( n ) = inf { τ : α ( τ ) ≥ n } a nd h ( y ) = f  sup τ ≥ 0 [ α ( τ + y ) − α ( τ ) + 1]  . 2) If a flow has a v .s.d s tochastic arr ival curve λ ( n ) ∈ G with b ounding fun ction h ( y ) ∈ ¯ G , the flow has a v .b .c stochastic ar rival cu rve α ( t ) ∈ G with bound ing function f ( x ) ∈ ¯ G , where α ( t ) = sup { k : λ ( k ) ≤ t } and f ( x ) = h  sup k ≥ 0 [ λ ( k ) − λ ( k − x )]  . Pr oof: (1) From Lemma 2, we know t hat for ∀ x, t ≥ 0 , event {A ( t ) ≤ A ⊗ α ( t ) + x } impli es e vent { a ( n ) ≥ a ¯ ⊗ λ ( n ) − y } where y is obtained from Eq.(3). Thu s, there holds P {A ( t ) ≤ A ⊗ α ( t ) + x } ≤ P { a ( n ) ≥ a ¯ ⊗ λ ( n ) − y } . W e f urther h ave P {A ( t ) > A ⊗ α ( t ) + x } ≥ P { a ( n ) < a ¯ ⊗ λ ( n ) − y } . From the co ndition that the flow has a v .b .c stocha stic arri val cu rve α ( t ) , we know P {A ( t ) > A ⊗ α ( t ) + x } ≤ f ( x ) . According to Eq.(7), we obtain P { a ( n ) < a ¯ ⊗ λ ( n ) − y } ≤ f  sup τ ≥ 0 [ α ( τ + y ) − α ( τ ) + 1]  . (2) From Lemma 3, we know that for ∀ y ≥ 0 , e vent { a ( n ) ≥ a ¯ ⊗ λ ( n ) − y } implies event {A ( t ) ≤ A ⊗ α ( t ) + x } where x is obtaine d from Eq.(7). Thus, t here holds P { a ( n ) ≥ a ¯ ⊗ λ ( n ) − y } ≤ P {A ( t ) ≤ A ⊗ α ( t ) + x } W e f urther h ave P { a ( n ) < a ¯ ⊗ λ ( n ) − y } ≥ P {A ( t ) > A ⊗ α ( t ) + x } From the co ndition that the flow has a v .s.d stocha stic arri val curve λ ( n ) , we kno w P { a ( n ) < a ¯ ⊗ λ ( n ) − y } ≤ h ( y ) . According to Eq.(3), we hav e P {A ( t ) > A ⊗ α ( t ) + x } ≤ h  sup k ≥ 0 [ λ ( k ) − λ ( k − x )]  and complete the p roof. D. Ma ximum-(virtual)-system-delay Stochastic Arrival Curve The maximum-(virtual)-system-delay (m.s.d) stochas- tic arri val cu rve explores the maximum virtual sy stem delay pr o perty of deterministic a rri val curve implied by Lemma 5.(3), which is tha t the maximum system delay of a virtual SSQ fed with the same flow with a deterministic arr i val curve is lo wer-bounded. Similar to the discussion for v .s.d stochas tic arriv al curve, for a flow having arri val curve, we construct a virtual SSQ system fed with the flo w , which has infinite buf fer space and the buf fer is initially empty . Suppose the virtual SSQ s ystem provides a deterministic service curve λ to the flow or d ( n ) = a ¯ ⊗ λ ( n ) for all n ≥ 0 . The ma ximum s ystem delay in the virtual SSQ s ystem for the first n a rri val p ackets as su p 0 ≤ m ≤ n W s ( m ) = sup 0 ≤ m ≤ n sup 0 ≤ q ≤ m { λ ( m − q ) − [ a ( m ) − a ( q )] } . If the flow is constrained by arri val cu rve λ ( n ) − x for all n ≥ 0 , the maximum s ystem delay in the vir tua l SSQ is also u pper-bounded by x . Based on the maximum virtual system de lay prope rty , we de fine m.s.d stocha stic arri val curve model. Definition 5. (m.s.d Stochastic Arrival Curve). A flow is said to have a maximum-(virtual)-system - delay (m.s.d) stochastic arrival curve λ ( n ) ∈ G with bounding function h ( x ) ∈ ¯ G , de noted by a ( n ) ∼ md h λ, h i , i f for all 0 ≤ m ≤ n a nd a ll x ≥ 0 , t here holds P n sup 0 ≤ m ≤ n sup 0 ≤ q ≤ m  λ ( m − q ) − [ a ( m ) − a ( q )]  > x o ≤ h ( x ) . (18) E. Deterministic Service Curve T o provide service guarantees to an arri val-constrained flow F , the system usually needs to allocate a minimum service rate to F . A guaran teed minimum service rate is equ i valent to a guaranteed maximum service time f or each packet of the flow , and acc ordingly the packet’ s departure time from the sys tem is b ounded . Because packets of the same fl ow are se rved in FIFO ma nner , any packet p n from this flow will depa rt by ˆ d ( n ) which is i terati vely defined by ˆ d ( n ) = max [ a ( n ) , ˆ d ( n − 1)] + δ ( n ) (19) with ˆ d (0) = 0 , where δ ( n ) is the service time g uaranteed to p n . By applying Eq.(19) iterativ ely to its right-hand side, i t becomes ˆ d ( n ) = sup 0 ≤ m ≤ n [ a ( m ) + n X i = m δ ( i )] (20) where P n i = m δ ( i ) is the guaranteed c umulati ve s ervice time for packet p m to p n . Suppose we c an use a function γ ( n − m ) to denote P n i = m δ ( i ) , i .e. γ ( n − m ) = P n i = m δ ( i ) . Then, Eq uation(20) be comes ˆ d ( n ) = su p 0 ≤ m ≤ n [ a ( m ) + γ ( n − m )] = a ¯ ⊗ γ ( n ) which provides a ba sis for t he follo wing time-doma in (deterministic) se rver mod el that c haraterizes the service using an upper bound on the cumulativ e packet se rvice time [15] : Definition 6. (Service C urve). Consider a system S with input pr oces s a ( n ) and output pr ocess d ( n ) . The s ystem is said to pr ovide to the input a (deter ministic) s ervice curve γ ( n ) ∈ G , if for ∀ n ≥ 0 , d ( n ) ≤ a ¯ ⊗ γ ( n ) . (21) The (deterministic) service curve model ha s the fol- lowi ng duality principle: Lemma 6. F or ∀ x ≥ 0 , d ( n ) − a ¯ ⊗ γ ( n ) ≤ x for all n ≥ 0 , if and only if su p 0 ≤ m ≤ n [ d ( n ) − a ¯ ⊗ γ ( n )] ≤ x for ∀ n ≥ 0 , wh ere γ ∈ G . Pr oof: For the ”if” part, it holds beca use d ( n ) − a ¯ ⊗ γ ( n ) ≤ sup 0 ≤ m ≤ n [ d ( n ) − a ¯ ⊗ γ ( n )] . For t he ”only if” part, from d ( n ) − a ¯ ⊗ γ ( n ) ≤ x for ∀ n ≥ 0 , we have sup 0 ≤ m ≤ n [ d ( n ) − a ¯ ⊗ γ ( n )] ≤ sup 0 ≤ m ≤ n [ x ] = x . By the definition of s ervice curve, the fi rst pa rt of Lemma 6 define s a service c urve γ ( n ) + x . Lemma 6 states that if a server p rovides se rvice cu rve γ ( n ) + x , then su p 0 ≤ m ≤ n [ d ( m ) − a ¯ ⊗ γ ( m )] ≤ x holds, a nd vice versa. In this sense , we call Lemma 6 the du ality principle of service curve. F . Stochastic S ervice C urve For networks providing stochastic service g uarantees , follo wing the principle o f Eq.(20), we have the follo wing expression for the expected departure time of pa cket p n ˆ d ( n ) = sup 0 ≤ m ≤ n [ a ( m ) + n X i = m ( δ ( i ) + ǫ ( i ))] where we assume δ ( i ) is the d eterministic part while ǫ ( i ) the random part in the total s ervice time δ ( i ) + ǫ ( i ) guar- anteed to pa cket p i . W e call ǫ ( i ) s tochastic err or term assoc iated to δ ( i ) . Here, ǫ ( n ) is introduced to represen t the additional delay of p n due to some randomness . For example, an error-prone wireless link is o ften considered to ope rate in two states. If the link is in ‘good’ c ondition, it can se nd and receive data correctly; if the link is in ‘bad’ co ndition due to e rrors, the da ta that s hould be se nt immediately has to be queue d long er until the channel change s to ‘good’ condition. Then , ǫ ( n ) in this c ase represents the time period in which the channel is in ‘bad’ condition be tween the time when p n − 1 has bee n sent c orrectly and the t ime when p n can be sent. W ith the cons ideration of the stochastic error term, the (deterministic) service curve can b e extend ed to a stochas tic version as follo ws: Definition 7. (i .d Stochastic Service Cur ve). A system i s said to pr ovide an inter -dep arture time ( i.d) stochas tic s ervice curve γ ∈ G with bounding function j ∈ ¯ G , denoted by S ∼ id h γ , j i , if for all n ≥ 0 and all x ≥ 0 , ther e holds P n d ( n ) − a ¯ ⊗ γ ( n ) > x o ≤ j ( x ) . (22) Example 5. Co nsider two nodes, the se nder and the receiv er , commu nicate through an error- prone wireless link. Pac kets h ave fixed-length. Packets arri ving to the sender node are s erved in F IFO ma nner . Ass ume the guaranteed per-packet s ervice time is δ without any error . T o simplify the analysis , assume the time slot length e quals δ . The sender se nds packets c orrectly on ly when the link is in ‘go od’ condition. If the link is in ‘ba d’ con dition, no packets can be sent correctly . In addition, the sende r can se nd the head-of-queu e packet only at the beginning of a ti me s lot, i.e ., the time period during which the link is in ‘bad’condition should be an integer times of δ . The prob ability that a packet can be se nt correctly is determined b y p acket error rate (PER). P ER is determined by the packet length and the bit error rate (BER). Here, we assu me pa cket errors happen independ ently and the same PER denoted by P e is applied to all packets. The succe ssful transmiss ion probability of one packet is he nce 1 − P e . Suppose P { ∆( n ) = i } = P i − 1 e (1 − P e ) , i ≥ 1 , where ∆( n ) represents the numb er of time slots neces sary to succe ssfully se nd the n th packet with respect to the succe ssful transmission p robability 1 − P e . The number of time s lots necessary to succe ssfully send n packets is P n k =1 ∆( k ) which has the negative binomial d istrib u tion P n n X k =1 ∆( k ) = i o =      i − 1 n − 1 ! (1 − P e ) n P i − n e , i ≥ n 0 , i < n Then the se nder provides to its input a stochas tic s ervice curve γ which has the follo wing distrib ution P { γ ( n ) = ⌈ τ δ ⌉} = ⌈ τ δ ⌉ − 1 n − 1 ! (1 − P e ) n P ⌈ τ δ ⌉− n e where τ is the gu aranteed service time to s ucces sfully send n pa ckets and ⌈ x ⌉ de notes the smallest integer greater than or equal to x . W e c an find n 0 ≤ n suc h that a ¯ ⊗ γ ( n ) takes its maximum value, i .e., a ¯ ⊗ γ ( n ) = a ( n 0 ) + γ ( n − n 0 + 1) . From Eq.(22) , we have P { γ ( n − n 0 + 1) < d ( n ) − a ( n 0 ) − x } ≤ j ( x ) where j ( x ) = ⌈ d ( n ) − a ( n 0 ) − x δ ⌉− 1 X i = n i − 1 n − 1 ! (1 − P e ) n P i − n e In Sec. IV, we sho w that many resu lts can be de ri ved from the i.d stoch astic service curve model. Howe ver , without additional constraints, we hav e difficulty in prov- ing the c oncaten ation property for i.d stochas tic service curve. T o address thi s dif ficulty , we introduce a s tronger definition in the foll owing subs ection. G. Constrained Stochastic Servic e Curve The co nstrained stoc hastic s ervice curve mod el is generalized from the (deterministic) se rvice curve mode l based on its duality principle. From Lemma 6, we know that a sys tem with input a ( n ) and o utput d ( n ) has a service curve γ ( n ) if a nd only if f or ∀ n ≥ 0 , sup 0 ≤ m ≤ n { d ( m ) − a ¯ ⊗ γ ( m ) } ≤ x. (23) Inequality (23) p rovides the bas is to generalize the (deterministic) service curve model to the co nstrained stochas tic service curve defined as follo ws: Definition 8. (Constra ined Stochastic Service Curve). A system is said to pr ovide a c onstrained stochas tic service curve (c.s) γ ∈ G wit h boun ding func tion j ∈ ¯ G , denoted by S ∼ cs h γ , j i , if for ∀ n, x ≥ 0 , there holds P n sup 0 ≤ m ≤ n [ d ( m ) − a ¯ ⊗ γ ( m )] > x o ≤ j ( x ) . (24) The following theorem establishes relationships be- tween i.d stocha stic service c urve a nd c.s stochastic service curve. Theorem 3. 1) If a server S pr ovides to its input a ( n ) a c .s stochastic service cur ve γ ( n ) with bounding function j ( x ) ∈ ¯ G , it pr ovides t o the input a ( n ) a n i .d stochastic ser vice cur ve γ ( n ) with the same bounding function j ( x ) ∈ ¯ G , i.e., S ∼ id h γ , j i ; 2) If a serv er S pr ovides to it s input a ( n ) an i.d stochastic service curve γ ( n ) with boun ding func- tion j ( x ) ∈ ¯ F , i t pr ovides to the input a ( n ) a c.s stochastic service cu rve γ + η ( n ) = γ ( n ) + η · n with bounding f unction j η ( x ) ∈ ¯ F for ∀ η > 0 , wher e j η ( x ) = h 1 η Z n x − η · n j ( y ) dy i 1 . Pr oof: Th e first p art follows sinc e there always holds d ( n ) − a ¯ ⊗ γ ( n ) ≤ sup 0 ≤ m ≤ n { d ( m ) − a ¯ ⊗ γ ( m ) } . For the second pa rt, there holds for ∀ 0 ≤ m ≤ n , a ¯ ⊗ γ + η ( m ) ≥ a ¯ ⊗ γ ( m ) + η · m − η · n and t hen d ( m ) − a ¯ ⊗ γ + η ( m ) ≤ d ( m ) − a ¯ ⊗ γ ( m ) − η · m + η · n. Thus, we obtain P  sup 0 ≤ m ≤ n { d ( m ) − a ¯ ⊗ γ + η ( m ) } > x  ≤ P  sup 1 ≤ m ≤ n  d ( m ) − a ¯ ⊗ γ ( m ) − η ( m )  + > x − η · n  for which when x − η · n < 0 , the right hand side is equal to 1. In the follo wing, we assume x − η · n ≤ 0 under which, there h olds P  sup 0 ≤ m ≤ n { d ( m ) − a ¯ ⊗ γ + η ( m ) } > x  ≤ n X m =1 P  d ( m ) − a ¯ ⊗ γ ( m ) − η ( m )  > x − η · n  ≤ n X m =1 j ( x − η · n + η · m ) ≤ 1 η Z n x − η · n j ( y ) dy . Becaus e the probability is always not greater tha n 1, the second part follows from t he abov e inequality . In the second p art o f the ab ove theorem, j ( x ) ∈ ¯ F while n ot ∈ ¯ G . If the req uirement on the bou nding function is relaxed to j ( x ) ∈ ¯ G , the a bove relationship may n ot hold in g eneral. I V . B A S I C P RO P E RT I E S This s ection prese nts results deri ved from the time- domain traf fic mo dels and server mode ls introduc ed in Sec. III. Particularly , we in vestigate the five ba sic properties introduced in Sec. I, which are s ervice guar- antees including delay bo und and b acklog boun d, output characterization, c oncaten ation property and superposi- tion p roperty . Ho wever , some properties can directly be proved only for the comb ination of a specific traffic model and a specific s erver mod el. This explains why we need to establish the various relationsh ips be tween models in Se c. III. W ith these relationships, we can obtain the correspond ing results for models wh ich we are interested in. A. Service Guarantees This subsection in vestigates p robabilistic bounds on delay and bac klog u nder the combination of v .s .d stochas tic arri val curv e a nd i.d stochastic service curve. W e start with deriving the bound on delay t hat a packet would expe rience in a system. Theorem 4. (Delay B ound). Conside r a sys tem S pr oviding an i.d stochas tic service curve γ ∈ G with bounding function j ∈ ¯ G to the input which has a v .s.d arrival curv e λ ∈ G with bounding function h ∈ ¯ G . Let D ( n ) = d ( n ) − a ( n ) be the delay in the system o f the n th ( ≥ 0) p acket. F or ∀ x ≥ 0 , D ( n ) is b ounded by P { D ( n ) > x } ≤ j ⊗ h ( x − γ ⊘ λ (0)) . (25) Pr oof: F o r ∀ n ≥ 0 , there h olds d ( n ) − a ( n ) =  d ( n ) − a ¯ ⊗ γ ( n )  +  a ¯ ⊗ γ ( n ) − a ( n )  =  d ( n ) − a ¯ ⊗ γ ( n )  + sup 0 ≤ m ≤ n  λ ( n − m ) −  a ( n ) − a ( m )  + γ ( n − m ) − λ ( n − m )  ≤  d ( n ) − a ¯ ⊗ γ ( n )  + sup 0 ≤ m ≤ n  λ ( n − m ) −  a ( n ) − a ( m )  + sup 0 ≤ m ≤ n  γ ( n − m ) − λ ( n − m )  ≤  d ( n ) − a ¯ ⊗ γ ( n )  + sup 0 ≤ m ≤ n  λ ( n − m ) −  a ( n ) − a ( m )  + sup k ≥ 0  γ ( k ) − λ ( k )  . (26) The right-hand side of E q.(26) implies a suf fic ient condition to obtain P { D ( n ) > x } , which is that P  d ( n ) − a ¯ ⊗ γ ( n ) > x  and P n sup 0 ≤ m ≤ n  λ ( n − m ) −  a ( n ) − a ( m )  > x o are known. T o ens ure the system’ s stability , we sh ould also hav e lim k →∞ 1 k [ γ ( k ) − λ ( k )] ≤ 0 . (27) In the rest of the paper , without explicitly stating, we shall assume inequa lity (27) holds. F rom L emma 1 an d sup k ≥ 0  γ ( k ) − λ ( k )  = γ ⊘ λ (0) , we conclude P { D ( n ) > x } ≤ j ⊗ h ( x − γ ⊘ λ (0)) . Next, we consider backlog b ound of a system. By definition, the back log in the system at time t ≥ 0 is B ( t ) = A ( t ) − A ∗ ( t ) . If a ( n ) is the arri val time of the latest packet arri v ing to the system by time t , then B ( t ) is B ( t ) ≤ inf  k ≥ 0 : d ( n − k ) ≤ a ( n )  . (28) Eq.(28) implies that, for ∀ x ≥ 0 , if B ( t ) > x , there must be a ( n ) < d ( n − x ) . Thus e vent {B ( t ) > x } implies ev ent { a ( n ) < d ( n − x ) } and P {B ( t ) > x } ≤ P { a ( n ) < d ( n − x ) } . Then we ha ve the foll owi ng result for backlog. Theorem 5. (Backlog Bound). Consider a system S pr oviding an i.d stochas tic service curve γ ∈ G with bounding function j ∈ ¯ G to the input which has a v .s.d stochastic arrival curve λ ∈ G with bou nding function h ∈ ¯ G . The bac klog a t time t ( t ≥ 0 ), B ( t ) , is bounde d by P {B ( t ) > H ( λ, γ + x ) } ≤ j ⊗ h ( x ) (29) for any x ≥ 0 , where, H ( λ, γ + x ) = sup n ≥ 0  inf [ k ≥ 0 : γ ( n − k ) + x ≤ λ ( n )]  is the max imum horizontal distance between functions λ ( n ) and γ ( n ) + x for ∀ x ≥ 0 . Pr oof: Simi la r to prove the delay bound, we have d ( n − x ) − a ( n ) = [ d ( n − x ) − a ¯ ⊗ γ ( n − x )]+[ a ¯ ⊗ γ ( n − x ) − a ( n )] =  d ( n − x ) − a ¯ ⊗ γ ( n − x )  + sup 0 ≤ k ≤ n − x  λ ( n − k ) − [ a ( n ) − a ( k )] + γ ( n − x − k ) − λ ( n − k )  ≤  d ( n − x ) − a ¯ ⊗ γ ( n − x )  + sup 0 ≤ k ≤ n − x  λ ( n − k ) −  a ( n + x ) − a ( k )  + sup 0 ≤ k ≤ n − x  γ ( n − x − k ) − λ ( n − k )  Let v = n − k . The above i nequality i s writ ten as d ( n − x ) − a ( n ) ≤  d ( n − x ) − a ¯ ⊗ γ ( n − x )  + sup 0 ≤ k ≤ n  λ ( n − k ) −  a ( n ) − a ( k )  + sup x ≤ v ≤ n  γ ( v − x ) − λ ( v )  Let x = H ( λ, γ + y ) , we have d  n − h ( λ, γ + y )  − a ( n ) ≤  d  n − h ( λ, γ + y )  − a ¯ ⊗ γ  n − h ( λ, γ + y )  + sup 0 ≤ k ≤ n  λ ( n − k ) − [ a ( n ) − a ( k )]  − y (30) Under the same con ditions as an alyzing the delay , we obtain P {B ( t ) > H ( λ, γ + x ) } ≤ j ⊗ h ( x ) . B. Output Character ization This s ubsec tion presents the result for cha racterizing the departure proces s from a system. Theorem 6 . (Output Charac terization). Conside r a sys- tem S pro v ides an i.d stochastic serv ice curve γ ( n ) ∈ G with bound ing function j ( x ) ∈ ¯ G to its inpu t which has a v .s .d stocha stic a rrival c urve λ ( n ) ∈ G with bounding function h ( x ) ∈ ¯ G . The output has an i.a.t stochastic ar rival curv e λ ¯ ⊘ γ ( n − m ) with bounding function j ⊗ h ( x ) ∈ ¯ G . Pr oof: For a ny two departure packets m < n , there holds d ( n ) − d ( m ) ≥ a ( n ) − a ¯ ⊗ γ ( m )+ a ¯ ⊗ γ ( m ) − d ( m ) −  d ( n ) − d ( m )  ≤  d ( m ) − a ¯ ⊗ γ ( m )  + a ¯ ⊗ γ ( m ) − a ( n ) ≤  d ( m ) − a ¯ ⊗ γ ( m )  + sup 0 ≤ k ≤ n  λ ( n − k ) − [ a ( n ) − a ( k )]  + sup 0 ≤ v ≤ m  γ ( v ) − λ ( n − m + v )  =  d ( m ) − a ¯ ⊗ γ ( m )  + sup 0 ≤ k ≤ n  λ ( n − k ) − [ a ( n ) − a ( k )]  − inf 0 ≤ v ≤ m  λ ( n − m + v ) − γ ( v )  Adding inf 0 ≤ v ≤ m  λ ( n − m + v ) − γ ( v )  to both sides of the above inequality , we get inf 0 ≤ v ≤ m  λ ( n − m + v ) − γ ( v )  − [ d ( n ) − d ( m )] ≤  d ( m ) − a ¯ ⊗ γ ( m )  + sup 0 ≤ k ≤ n  λ ( n − k ) − [ a ( n ) − a ( k )]  W ith the s ame c onditions as analyzing delay , we con- clude P n λ ¯ ⊘ γ ( n − m ) − [ d ( n ) − d ( m )] > x o ≤ j ⊗ h ( x ) C. Concatenation Pr op erty The concatenation property uses an equiv alen t system to rep resent a s ystem o f multiple se rvers connec ted in tandem, each of which provides stoc hastic service curve to the input. Then the equiv a lent sy stem provides the input a s tochastic se rvice c urve, which is deri ved from the stochas tic service curve provided by all in volved indi vidual s ervers. Theorem 7. (Conc atenation Property). Consider a flow passing thr o ugh a network of N sys tems in tandem. If each s ystem k (= 1 , 2 , ..., N ) pr ovides a c.s stochastic service curve S k ∼ cs h γ k , j k i to its input, then the network guarantees to the flow a c. s stocha stic se rvice curve S ∼ cs h γ , j i with γ ( n ) = γ 1 ¯ ⊗ γ 2 ¯ ⊗ · · · ¯ ⊗ γ N ( n ) (31) j ( x ) = j 1 ⊗ j 2 ⊗ · · · ⊗ j N ( x ) . (32) Pr oof: W e sh all o nly prove the two-node case, from which, the proof can be easily extended to the N -node case. The de parture of the first node is the arriv al t o the second node, so d 1 ( n ) = a 2 ( n ) . In addition, the arri val t o the ne twork is the arr i val to the first node, i.e., a ( n ) = a 1 ( n ) , and the de parture from the network is the departure from the second nod e, i.e ., d ( n ) = d 2 ( n ) , where, a ( n ) and d ( n ) denote the arri val process to a nd departure proce ss from the network, respectively . W e then h av e, sup 0 ≤ m ≤ n { d ( m ) − a ¯ ⊗ γ 1 ¯ ⊗ γ 2 ( m ) } = sup 0 ≤ m ≤ n { d 2 ( m ) − ( a 1 ¯ ⊗ γ 1 ) ¯ ⊗ γ 2 ( m ) } (33) Now let us consider any m , ( 0 ≤ m ≤ n ), for which we get, d 2 ( m ) − ( a 1 ¯ ⊗ γ 1 ) ¯ ⊗ γ 2 ( m ) = d 2 ( m ) − sup 0 ≤ k ≤ m  a 1 ¯ ⊗ γ 1 ( k )+ γ 2 ( m − k ) − d 1 ( k )+ a 2 ( k )  = d 2 ( m )+ inf 0 ≤ k ≤ m  d 1 ( k ) − a 1 ¯ ⊗ γ 1 ( k ) − γ 2 ( m − k ) − a 2 ( k )  ≤ sup 0 ≤ k ≤ m { d 1 ( k ) − a 1 ¯ ⊗ γ 1 ( k ) } + d 2 ( m ) + inf 0 ≤ k ≤ m  − [ a 2 ( k ) + γ 2 ( m − k )]  ≤ su p 0 ≤ k ≤ m { d 1 ( k ) − a 1 ¯ ⊗ γ 1 ( k ) } + [ d 2 ( m ) − a 2 ¯ ⊗ γ 2 ( m )] (34) Applying Eq.(33) to Eq .(34), we obtain sup 0 ≤ m ≤ n { d 2 ( m ) − ( a 1 ¯ ⊗ γ 1 ) ¯ ⊗ γ 2 ( m ) } ≤ sup 0 ≤ k ≤ n { d 1 ( k ) − a 1 ¯ ⊗ γ 1 ( k ) } + sup 0 ≤ m ≤ n { d 2 ( m ) − a 2 ¯ ⊗ γ 2 ( m ) } (35) with wh ich, since both n odes p rovide c.s stochas tic service curve to their input, the the orem follows from Lemma 1 and the de finition of c.s stocha stic service curve. D. Superpos ition Pr operty The superposition prope rty mea ns that the superposi- tion of flows can be represented using the same traf fic model. W ith this property , the aggregate of multiple indi vidual fl ows may be viewed as a sing le agg regate flow . Then t he service guarantees for the aggregate flow can be deriv ed in the sa me way as for a single flo w . First, we o nly consider the aggregate of two flows, F 1 and F 2 . Let a 1 ( n ) , a 2 ( n ) and a ( n ) be the a rri val process of F 1 , F 2 and t he a ggregate flow F A , r espec ti vely . For any packet p n of the aggregate flo w F A , it is either the m th packet from flow F 1 or the ( n − m ) th packet from flo w F 2 , where m ∈ [0 , n ] , i.e. a ( n ) = max { a 1 ( m ) , a 2 ( n − m ) } For example, a (1) is either max [ a 1 (0) , a 2 (1)] or max[ a 1 (1) , a 2 (0)] and the minimum of these tw o poss i- bilities, a (1) = inf  max[ a 1 (0) , a 2 (1)] , max [ a 1 (1) , a 2 (0)]  . W e can see a nother e x ample a (2) = inf  max[ a 1 (0) , a 2 (2)] , max [ a 1 (1) , a 2 (1)] , max[ a 1 (2) , a 2 (0)]  . Essentially , we have for a ny pac ket n of the aggregate flow a ( n ) = inf 0 ≤ m ≤ n  max[ a 1 ( m ) , a 2 ( n − m )]  . (36) W e generalize the result to the superposition of N ( ≥ 2 ) flows a ( n ) = inf P m i = n  max[ a 1 ( m 1 ) , a 2 ( m 2 ) , ..., a N ( n − N − 1 X i =1 m i )]  . (37) From Eq.(37), it is difficult to directly characterize the packet inter-arri val ti me of the aggregate flow . W e kno w that if a flow has a v .s.d stoc hastic arriv al curve, with Theorem 2(2), this fl ow has a v .b .c stochas tic a rri val curve, for which the superposition property h olds [4]. Thus, we can indirectly prove that the superposition property holds for the v .s.d stoch astic arriv al curve. If flow i has a v .s.d s tochastic a rri val curv e a i ( n ) ∼ vd h λ i , h i i i = 1 , 2 , ..., N , from Theorem 2(2), flow i has a v .b . c stochastic arri val curve α i ( t ) with bounding function f i ( x ) = h i  sup τ ≥ 0 [ α i ( τ + y ) − α i ( τ ) + 1]  , where α i ( t ) = sup { k : λ i ( k ) ≤ t } . According to Lemma 4, the aggregate flow has a v .b .c stoch astic arri val curve α ( t ) = P N i =1 α i ( t ) with bo unding function f ( x ) = f 1 ⊗ · · · ⊗ f N ( x ) . W e a pply Theorem 2(1) and obtain the follo wing result: Theorem 8. Conside r N fl ows wi th arrival pr o cesse s a i ( n ) ∼ vd h λ i , h i i , i = 1 , ..., N . F or the aggr egate of these flows, ther e holds a ( n ) ∼ vd h λ, h i with λ ( n ) = inf { τ : α ( τ ) ≥ n } a nd h ( y ) = f  sup k ≥ 0 [ λ ( k ) − λ ( k − x )]  , where α ( t ) = P N i =1 α i ( t ) a nd f ( x ) = f 1 ⊗ · · · ⊗ f N ( x ) with α i ( t ) = sup { k : λ i ( k ) ≤ t } f i ( x ) = h i  sup τ ≥ 0 [ α i ( τ + y ) − α i ( τ ) + 1]  . E. Lefto v er Service Characterization This subsection explores the leftover service cha r - acterization und er aggregate schedu ling. T o ease the discuss ion, we co nsider the simplest ca se wh en there are two flo ws co mpeting resource in a s ystem under FIFO aggregation. Suppos e tha t if packets arri ve to the system simultaneously , they a re inserted into the FIFO q ueue randomly . C onsider a system fed with a fl ow F A which is the aggregation of two constituent flows F 1 and F 2 . Suppose both the service charac terization fr om the server and traf fi c charac terization from F 2 are known. W e are interested in characterizing the se rvice time recei ved by F 1 , with which per-flo w bounds for F 1 can be then easily obtained using earlier res ults derived in the previous subsec tions. Theorem 9. Consider a system S with inpu t F A that is the aggr egati on o f two constituent flows F 1 and F 2 . Suppos e F 2 has a (deterministic) arrival cu rve λ 2 ( n ) ∈ G , and the sy stem pr ovides to the input an i.d s tochastic service curve γ ∈ G with bounding function j ( x ) ∈ ¯ G . Then if γ  n + su p[ q : λ 2 ( q ) ≤ a 1 ( n )]  ∈ G , F 1 r e ceives an i.d stochastic ser vice cur ve γ  n + sup[ q : λ 2 ( q ) ≤ a 1 ( n )]  with the same bounding function j ( x ) . Pr oof: Suppos e pa cket p n 1 is the ( n + m ) th packet of F A , i.e., a ( n + m ) = a 1 ( n ) , where m repres ents the number of packets from F 2 . As the system p rovides an i.d stoch astic service curve γ ( n ) to the agg regate flow F A , there holds P { d ( n + m ) − a ¯ ⊗ γ ( n + m ) > x } ≤ j ( x ) . a 1 ( n ) = a ( n + m ) indicates a 2 ( m ) ≤ a 1 ( n ) . Let ¯ m = sup[ q : λ 2 ( q ) ≤ a 1 ( n )] . As λ 2 is the (de terministic) arri val curve of F 2 , we ha ve ¯ m ≥ m because of a 2 ( m ) ≥ λ 2 ( m ) . Then γ ( n + ¯ m ) ≥ γ ( n + m ) . Let γ 1 ( n ) = γ ( n + ¯ m ) . From γ 1 ( n ) ≥ γ ( n + m ) , we have a 1 ¯ ⊗ γ 1 ( n ) ≥ a ¯ ⊗ γ ( n + m ) . As d ( n + m ) = d 1 ( n ) , t here holds d 1 ( n ) − a 1 ¯ ⊗ γ 1 ( n ) ≤ d ( n + m ) − a ¯ ⊗ γ ( n + m ) . Thus, we conclud e P { d 1 ( n ) − a 1 ¯ ⊗ γ 1 ( n ) > x } ≤ j ( x ) and complete the p roof. F . Discus sion In t his section, we have presented t he fi ve basic prop- erties of stocha stic network calculus und er various traf fic models and se rver mo dels defined in the time-domain and introduced some simple a pplications. For example, a GCRA-constrained flo w has a deterministic arri val curv e. If a flo w’ s packet inter-arri val times a re exponentially distrib uted, the n this flow has a v .s.d stochastic arriv al curve. The service process of an error -pron e wireless li nk can be modeled by an i.d stocha stic service curve. For each basic property , we in vestigated one com- bination of a specific traffic model and a specific server model. P articularly , we pro ved that the service guarantees and the output c haracterization hold for the combination of v .s.d stochas tic a rri val curve and i.d stochas tic s ervice curve. For the conc atenation property , we in vestigated the case that a ll se rvers provide the constrained service cu rve to their inpu t but did not specify the type of arri val curve. In order to prove the superpos ition property , we u sed the transformation be- tween v .s.d stochastic service curve and v .b .c stochastic service curve. The lefto ver service characterization was only p roved f or the combination of deterministic arri val curve an d i.d stochastic se rvice curve. W ith the relationships and trans formations among models establishe d in Sec. I II, these five prope rties may be directly o r indirectly proved for other combinations of traf fic mode ls an d server models. For example, it is easy to prove the service g uarantees and output charac- terization for the combination o f m.s.d stochastic arr i val curve and i.d stoch astic service curve. Considering space limitation, these results are not included. Howe ver , to prove the con catenation property and superposition prop- erty for other server models and traf fic mod els, it will require additional transformations among models. For the leftover s ervice characteriza tion, we may need more constraints or transformations whe n proving it for othe r combinations of traf fic mode ls and server models . W e leav e the se as our future work. V . C O N C L U S I O N For s tochastic service guaran tee ana lysis, we intro- duced several time-domain models for traf fic and se rvice modeling. The essential i dea of them is t o base the model on cu mulati ve packet inter-arr i val time for traffic and on cumula ti ve service time for service. Simple e xamples have been gi ven to demo nstrate the use of them. Based on the proposed time-domain models, the five basic properties for stocha stic network ca lculus we re de ri ved, with which, the results can be ea sily applied to b oth the single-node a nd t he network cases. As Example 5 showed, we can directly obtain the service curve in the time-domain. W ith the result o f service guarantee s, the p robabilistic delay boun d and probablistic b acklog boun d can b e rea dily obtaine d. W e believ e, the propos ed time-domain models and d eri ved results c an b e p articularly u seful for an alyzing stoc hastic service guarantee s in systems, where the behavior of a server in volves some stochas tic processe s which ca n be d irectly c haracterized in the time-domain, whil e it is dif fi cult to characterize such s tochastic processes in the space -domain. Such sy stems include wireless links an d multi-access ne tworks where backoff sche mes may be employed. 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