Probabilistic Performance Analysis of Networks using an Improved Network Service Envelope Approach

Probabilisti c P erformance Analysis of Net works using an Improv ed Net w ork Service En v elop e Approac h K. Angrishi, U. Killat Institute of Communic ation Networks Hambur g University of T e chnolo gy 21075 Hambur g, Germany { kishor e.angrishi,kil lat } @tu- harbur g.de Abstract Sto c hastic net w ork calculus is an ev olving theory whic h accoun ts for sta- tistical m ultiplexing and uses an en velope approac h f o r probabilistic dela y and bac klog analysis of netw orks. One of the key ideas of sto c hastic net- w ork calculus is the possibilit y to describ e the service offered at a net w ork no de as a stochastic servic e en ve lo p e, whic h in turn can b e used to describe the sto c hastic service a v ailable in a net w or k of no des and determine end-to- end probabilistic dela y and bac klog b o unds. This pap er in tro duces a new definition of sto c hastic service env elop es whic h yields a simple netw ork ser- vice en velope and tighter end-to-end p erformance b o unds. It is sho wn for ( σ ( θ ) , ρ ( θ )) - constrained traffic model that the end-to-end p erformance mea- sures computed using the new sto c ha stic netw ork service en velope are tigh t in comparison to the ones obtained using the existing start-of-the-art defini- tion of statistical net w ork service en ve lo p e and are b ounded b y O ( H log H ), where H is the n um b er of no des tra v ersed b y the arriv al traffic. Keywor ds: Sto c hastic Net work Calculus, Net w ork Service En v elop e, Qualit y of Service 1. In tro duction The conv ergence of data , voic e and video traffic ov er the In ternet has in- creased the significance of p erformance analysis of data netw orks. The critical asp ect in the p erformance analysis of data netw orks is the efficien t mo deling of arriv al traffic and service av ailable to t he a r r iv al traffic in the net work. Pr eprint s ubmitte d to Elsevier Scienc e Novemb er 20, 2018 The exactness of the p erformance measures dep ends on the accuracy of the mo dels describing the ar riv al traffic a nd service av ailable in the net w ork, but with increased mathematical complexit y . In most cases, b ounds on the p er- formance measure are suffi cient for net work analysis. Net w ork calculus is one of the p opular theories useful for computing w orst-case p erformance bo unds in data net works with the help of deterministic env elop es describing the ar- riv al traffic a nd servic e a v a ilable in a net w ork no de. The probabilistic ve r - sion of netw ork calculus is called sto c hastic net w ork calculus 1 whic h retains the en v elop e approach and man y fa vorable c ha r acteristics of (deterministic) net w ork calculus and deriv es probabilistic p erformance b ounds. The r a ison d’ ˆ etre of net work calculus is the p o ssibility to compute probabilistic b ounds on end-to-end perfo rmance measures using a net w ork service en velope whic h describes the service a v ailable in a netw ork a s a single abstract no de. It has b een sho wn in [1] that the end-to-end w orst case p erfo rmance b ounds o b- tained by summing the per- no de results scale in the order of O ( H 2 ) and the b ounds computed using net work serv ice env elop e scale in the o rder of O ( H ), where H is the num b er of no des tra vers ed b y the arriv al flow. There has b een man y attempts to ac hieve a similar linear scaling of end-to-end p erformance b ounds in statistical net w ork calculus, but with little success. W e direct the in terested readers t o [2] for a more elab o r ate discussion on what makes the probabilistic extension of netw ork calculus so difficult. In [3], authors pre- sen ted a sto c hastic net w ork service en velope whic h allows the computation of end-to-end probabilistic p erfo rmance measures that is sho wn to b e b ounded b y O ( H log H ) for exp o nen tial b ounded burstiness ( EBB) tra ffic mo del. In this pap er, w e presen t a differen t definition for statistical service en ve- lop e based on the sto c hastic service pro cess ch a racterizing the service offered at a net work no de. The new definition of statistical service env elop e allo ws to compute tigh t er end-to-end p erformance measures than the ones obtained using the exis ting definition of net w ork service env elop e fr om [3] while still main taining the O ( H log H ) scaling of the end-to-end b ounds for ( σ ( θ ) , ρ ( θ )) - constrained tra ffic mo del. La ter in the pap er, we will use Mark o v mo du- lated on-off traffic mo del as in [3] t o demonstrate t he tightnes s of the dela y measure computed using the new definition of statistical service en ve lop e. The rest of the pap er is structured as fo llo ws: In Section 2, we giv e a n 1 The terms statistical netw or k calculus, stochastic netw ork calc ulus and pr o babilistic net work calculus are used interch a ngeably in the literature 2 o ve rview of the statistical net work calculus and define our not io n of statistical service en v elop e. Then, w e use the en v elop e functions to deriv e p erformance b ounds on delay , bac klog and output burstiness. The scaling prop erties of the deriv ed end-to-end p erfor mance b ounds are sho wn in Section 3. In Section 4, a n umerical example using Marko v Mo dulated On-Off traffic is presen t ed for illustration. Brief conclusions are presen ted in Section 5. Throughout the pap er w e use discrete time mo del t ∈ N 0 = { 0 , 1 , 2 , . . . } and assume the random pro cesses to b e stationary , that is, t he random pro cesses dep ends only on t he length of the in t erv al ( s, t ] (∆ = t − s ) , but not on s or t itself. 2. Statistical Net work Calculus In this section, we giv e a brief o v erview of the statistical net w ork calculus and o ur notion of stat istical service en ve lop e. Then, we deriv e p erfo rmance b ounds in using our notion of statistical service en velope. The elegan t theory of netw ork calculus [1, 4] prov ides useful insigh ts for the understanding of fundamen tal concepts of in tegrated and differen tiated services , flow control, resource (bandwidth or buffer) dimensioning in dat a net w orks. The tw o main adv antages of netw ork calculus are ( i ) the relativ e ease in its abilit y to mo del diff erent sc heduling a lgorithms used at a net w ork no de, and ( ii ) the p o ssibility to mo del a net w or k of no des as a single abstract no de, which substantially reduces the complexit y in volv ed in netw ork anal- ysis. The mathematical theory of min-plus algebra forms the basis for the theory of net work calculus. In the fo llowing w e recall the t w o most commonly used min-plus op erations in net w ork calculus, namely , min-plus conv olution and min-plus de-con v olution op erations [1]. Definition 2.1. L et f ( s, t ) and g ( s, t ) b e two non-de cr e asing, r e al value d, bivariate functions defin e d at t ≥ s ≥ 0 . Then the min-pl us c onv o lution ( ⊗ ) and de-c onvolution ( ⊘ ) op er ations ar e d efine d as fol lows: f ⊗ g ( s, t ) = inf s ≤ k ≤ t { f ( s, k ) + g ( k , t ) } (1) f ⊘ g ( s, t ) = sup 0 ≤ k ≤ s { f ( k , t ) − g ( k , s ) } (2) The statistical net w ork calculus is t he proba bilistic v ersion of net work calculus and aims to profit from the statistical m ultiplexing in data net- w orks. The fundamen tal difference b et ween the statistical net work calculus 3 and its deterministic coun terpart is t hat the p erformance b ounds are ex- pressed as probabilistic tail b ounds, i.e., the deriv ed b ounds are violated with some pr o babilit y . Arriv al and departure pro cesses are describ ed us- ing real-v alued, biv ariate functions A ( s, t ) and D ( s, t ), respectiv ely , whic h represen t the cum ulative amoun t of da t a seen in the in terv al ( s, t ] for any 0 ≤ s ≤ t . W e a ssume that there a re no arr iv als in t he in terv al ( −∞ , 0 ] and A ( t ) = A (0 , t ), D ( t ) = D (0 , t ) for an y t ≥ 0. Sinc e w e assume the random pro cess to b e stationary , random pro cess dep ends only on the length of the interv al ( s, t ], that is ∆ = t − s , but not on s or t it self (therefore, A ( s, t ) = A (∆) , D ( s , t ) = D (∆)). F or an arriv al pro cess A , a non- decreasing, real v alued function is called statistical arriv al en velope function or effec- tiv e env elop e G [3] if the function satisfies the follo wing condition, for an y t, s, σ ≥ 0 : P { A ( s, t ) > G ( t − s ) + σ } ≤ ε g ( σ ) (3) where ε g is called error function whic h is a non-negativ e, decreasing f unction of σ b ounding the violation probabilit y . The sufficien t condition for the deriv ed p erformance b ounds to b e finite is tha t the error function is required to satisfy the in t egr a bilit y condition [3] giv en b elo w: Z ∞ 0 ε ( u ) du < ∞ (4) Similarly , the service a v ailable to a flo w is c har a cterized using a non- decreasing, real v alued function called statistical service env elop e or effectiv e service env elop e S [5 ] suc h that for a giv en arriv al pro cess ( A ) and departure pro cess ( D ), the service en velope satisfies t he fo llo wing condition for all t, σ ≥ 0: P { A ⊗ S ( t ) > D ( t ) + σ } ≤ ε s ( σ ) (5) where ε s is a decreasing error function b ounding the violatio n probabilit y . This error function is also required to satisfy t he in tegra bilit y condition from equation (4) as the sufficien t condition f o r t he deriv ed p erfo rmance b ounds to b e finite. In this pap er, we define a differen t notio n of statistical service en v elop e ( S ) whic h is deriv ed fro m the sto chastic service pro cess ( S ) c har a c- terizing the service offered at a net work no de. F or a give n arriv al pro ces s ( A ) and departure pro cess ( D ) at a netw ork no de, the sto c hastic service pro cess ( S ) describing the service offered at the no de satisfies the follo wing condition for an y fixed sample path and all t ≥ 0: A ⊗ S ( t ) ≤ D ( t ) (6) 4 The k ey observ ation is that the sto c hastic service pro cess ( S ) for an y fixed sample path is a deterministic service en v elop e of the service offered to the giv en ar r iv al pro ces s tra jectory , and equation (6) follo ws from the definition of deterministic service env elop e [1]. An y random pro cess S satisfying the ab o v e relationship (equation (6)) b etw een arriv al pro cess and departure pro cess for an y fixed sample path is referred t o as “dynamic F -serv er” [4]. W e no w define a differen t notion of statistical service en v elop e ( S ) based on the sto c hastic service pro cess ( S ) at the node. Definition 2.2. L et S b e the sto chastic servic e pr o c ess cha r acterizing the servic e offer e d a t the no de, then the statistic al servic e envelop e S for al l t, s, σ ≥ 0 c an b e define d as : P { S ( s, t ) < S ( t − s ) − σ } ≤ ε s ( σ ) (7) wher e ε s is a de cr e asing err or function b ounding the violation pr ob ability and fulfil ling the inte gr ability c ond i tion f r om e quation (4). The new definition of statistical service en v elop e do es not imply equation (5). How ev er, the sample path b ound of the statistical service en velope f rom equation (7) is also a v alid b o und to its coun terpart from equation (5), i.e., for all t ≥ 0 P { A ⊗ S ( t ) > D ( t ) + σ } ≤ P { A ⊗ S ( t ) > A ⊗ S ( t ) + σ } ≤ P  sup 0 ≤ k ≤ t {S ( t − k ) − S ( k , t ) } > σ  and, P {S ( t ) − S ( t ) > σ } ≤ P  sup 0 ≤ k ≤ t {S ( t − k ) − S ( k , t ) } > σ  The main adv antage of t he new definition of statistical servic e en v elop e from equation (7) is that its sample path b o und is a low er b o und to sample b ound b ound o f statistical service en velope from equation ( 5), whic h is necessary information required ab out the servic e offered in a netw ork to compute end- to-end p erfo rmance measures, i.e., P  sup 0 ≤ k ≤ t {S ( t − k ) − S ( k , t ) } > σ  ≤ P  sup 0 ≤ k ≤ t { A ⊗ S ( t − k ) − D ( k, t ) } > σ  5 Figure 1: Net work of H co ncatenated no des The sample path b ound of the statistical service en velope fro m equation (7) is giv en b y the followin g lemma. Lemma 2.1. Consider the servic e offer e d at a network no de b eing describ e d using a sto chastic servic e pr o c e s s S and let S b e the statistic a l s e rvic e envelop e derive d fr om sto chastic servic e pr o c ess S satisfying e quation (7) with an err or function ε s satisfying in te gr ability c on d ition fr om e quation (4). Then for any δ > 0 an d al l t, σ ≥ 0 P  sup 0 ≤ k ≤ t {S ( t − k ) − S ( k , t ) − δ · ( t − k ) } > σ  ≤ ∞ X u =0 ε s ( σ + δ u ) (8) The term δ is used in the ab ov e lemma to mak e the violation probability function of the en v elop e dep enden t on time, so that w e can compute the sample path violatio n probability in terms of the giv en violation probabilit y function of service en v elop e from equation (7). In t uitively , the term δ can b e seen as rate correction factor that reduces the guarantee d service b y a rate δ . Pr o of: F or a giv en t, σ ≥ 0 a nd δ > 0 we ha v e P  sup 0 ≤ k ≤ t {S ( t − k ) − S ( k , t ) − δ · ( t − k ) } > σ  ≤ ∞ X u =0 P {S ( u ) − ( δ u + σ ) > S ( u ) } ≤ ∞ X u =0 ε s ( σ + δ u ) The first inequalit y is due to the application of Bo ole’s inequalit y and setting u = t − k . The second inequalit y is from the definition of statistical service en v elop e from equation (7).  The new definition of statistical service env elop e fro m equation (7) will b e sho wn to b e b enefic ial for the end-to-end net w ork analysis in Section 3 of this pap er. F rom no w on, unless sp ecified otherwise, the term statistical service en velope refers to its definition from equation (7). W e no w state our main results using the statistical service en velope de- riv ed from the sto c hastic service pro cess a t the net w o rk no de. Consider a 6 flo w trav ersing through a net work of H no des connected in series as sho wn in Fig. 1 . W e assume that the service av ailable fo r a flo w at each ho p ( h ) is c haracterized b y a sto c hastic service pro cess ( S h ). The follo wing theorem will pro vide a p ossibilit y to characterize the service offered by a net w o rk of no des as sho wn in Fig. 1 in-terms of p er-no de statistical service en velopes. Theorem 2.1. Consider a flow tr aversing a network of H no des c onne cte d in serie s with e ach hop ( h = 1 , 2 , . . . , H ) offering a s ervic e char acterize d by its c orr esp on ding sto ch a s tic serv i c e pr o c e s s ( S h = S 1 , S 2 , . . . , S H ). Then , the sto chastic network servic e pr o c ess S net for any fixe d sample p a th is given by S net = S 1 ⊗ S 2 ⊗ · · · ⊗ S H (9) and the c orr esp onding statistic al network serv i c e envelop e S net is g iven by S net = S 1 ⊗ S 2 ⊗ · · · ⊗ S H (10) with a de cr e asing err or function ε s net , fo r an y δ > 0 , g iven by ε s net ( σ ) = inf σ 1 + ··· + σ H = σ ( H X h =1 ∞ X u =0 ε s h ( σ h + δ u ) ) (11) such that the statistic al network servic e envelop e S net satisfies the fol lowing c ondition , for a ny t, σ ≥ 0 and δ > 0 , P  sup 0 ≤ k ≤ t {S net ( t − k ) − S net ( k , t ) − δ · ( t − k ) } > σ  ≤ ε s net ( σ ) (12) Pr o of: The pro of of the theorem has tw o parts; The first is to prov e the sto c hastic net work service pro cess and the second part is to prov e t he statis- tical netw ork service en ve lop e. Let A = A 1 b e the arriv al tr a ffic at the no de 1 or ingress o f the netw ork and D = D H = A H +1 represen t the departure traf- fic from the net w ork of H no des connected in series as show n in Fig . 1. The departure traffic D h from the no de at hop h b ecomes the arriv al traffic A h +1 to the do wnstream no de at hop h + 1 , i.e., A h +1 = D h for all h = 1 , . . . , H . In o r der to prov e, for any sample path the stochastic net work service pro cess S net = S 1 ⊗ S 2 ⊗ . . . ⊗ S H represen ts the service offered b y the net work sho wn in Fig. 1, one needs to sho w that the departure traffic D from the net w ork satisfies for any sample path the condition D ≥ A ⊗ S net . This 7 can b e sho wn in a straigh tf o rw ard fashion. F rom the prop erty of sto c ha stic service pro ces s characterizing the service offered at a no de (equation ( 6)), the resp ectiv e departure tra ffic satisfies for any sample path the condition D h ≥ A h ⊗ S h for h = 1 , 2 , . . . , H , where S 1 , S 2 , . . . , S H represen ts the sto c hastic net w ork service pro ces s offered b y the resp ectiv e H nodes in the net w o r k. Applying the condition on departure traffic fro m eac h no de iterativ ely for the departure t r a ffic D = D H from the netw ork, one gets fo r an y sample path D ≥ A ⊗ S 1 ⊗ S 2 ⊗ . . . ⊗ S H = A ⊗ S net . This prov es our first claim. F or the giv en t, σ ≥ 0 and δ > 0, w e hav e P  sup 0 ≤ k 1 ≤ t {S net ( t − k 1 ) − S net ( k 1 , t ) − δ · ( t − k 1 ) } > σ  = P  sup 0 ≤ k 1 ≤ t {S 1 ⊗ S 2 ⊗ · · · ⊗ S H ( t − k 1 ) − S 1 ⊗ S 2 ⊗ · · · ⊗ S H ( k 1 , t ) − δ · ( t − k 1 ) } > σ } ≤ P  sup 0 ≤ k 1 ≤ k 2 ≤ k 3 ···≤ k H ≤ t {S 1 ( k 2 − k 1 ) − S 1 ( k 1 , k 2 ) − δ · ( k 2 − k 1 ) + S 2 ( k 3 − k 2 ) − S 2 ( k 2 , k 3 ) − δ · ( k 3 − k 2 ) + · · · + S H ( t − k H ) − S H ( k H , t ) − δ · ( t − k H ) } > σ } ≤ P  sup 0 ≤ k 1 ≤ k 2 ≤ t {S 1 ( k 2 − k 1 ) − S 1 ( k 1 , k 2 ) − δ · ( k 2 − k 1 ) } + + sup 0 ≤ k 2 ≤ k 3 ≤ t {S 2 ( k 3 − k 2 ) − S 2 ( k 2 , k 3 ) − δ · ( k 3 − k 2 ) } + · · · + sup 0 ≤ k H ≤ t {S H ( t − k H ) − S H ( k H , t ) − δ · ( t − k H ) } > σ 1 + · · · + σ H  ≤ P  sup 0 ≤ k 1 ≤ k 2 ≤ t {S 1 ( k 2 − k 1 ) − S 1 ( k 1 , k 2 ) − δ · ( k 2 − k 1 ) } > σ 1  + P  sup 0 ≤ k 2 ≤ k 3 ≤ t {S 2 ( k 3 − k 2 ) − S 2 ( k 2 , k 3 ) − δ · ( k 3 − k 2 ) } > σ 2  + · · · + P  sup 0 ≤ k H ≤ t {S H ( t − k H ) − S H ( k H , t ) − δ · ( t − k H ) } > σ H  ≤ inf σ 1 + ··· + σ H = σ ( H X h =1 ∞ X k =0 ε s h ( σ h + δ k ) ) = ε s net ( σ ) The inequalit y in the third step is due to the prop erty of suprem um op er- ation, i.e., sup 0 ≤ s ≤ t { X ( s ) + Y ( s ) } ≤ sup 0 ≤ s ≤ t { X ( s ) } + sup 0 ≤ s ≤ t { Y ( s ) } [6]. 8 The final inequality is from the definition of sample path statistical service en v elop e (equation (8)) and the stationar it y assumption of sto c hastic service pro cesses.  If the error function ε s h of the individual statistical service en v elop e S h at hop h for h = 1 , 2 , . . . , H satisfies the integrabilit y condition f r om equation (4), then the error function ε s net of the statistical net work service en velope will b e finite (i.e., ε s net < ∞ ) . W e next describ e the probabilistic perfo rmance b ounds on bac klog, dela y and output burstiness using the statistical arriv al env elop e (equation (3)) and netw ork servic e env elop e (Theorem 2.1). In the follo wing theorem, we use the notation S h, − δ ( t ) = S h ( t ) − δ t and G δ ( t ) = G ( t ) + δ t to simplify the presen t a tion. Theorem 2.2. L et A an d D b e the arrival and dep artur e tr affic, r esp e ctively, fr om a network of H no des c onne cte d in series and G b e the c orr esp onding statistic al arrival envelop e with an err or function ε g . Assume S net is the sto chastic network servic e pr o c ess that ch ar acterizes the servic e offer e d by the network a nd S net is the c orr esp o n ding statistic a l network servic e envelop e with an err o r function ε s net . Then we have the fol lowing b ounds. 1. B acklo g b o und : Th e pr ob abilistic b ound o n the b acklo g in a network, for any t, σ ≥ 0 a n d δ > 0 , is given by P { B ( t ) > G δ ⊘ S net, − δ (0) + σ } ≤ ε ( σ ) (13) 2. D elay b o und : T h e pr ob abilis tic b ound on the delay in a network, for any t, σ ≥ 0 and δ > 0 , is give n by P { W ( t ) > d ( σ ) } ≤ ε ( σ ) (14) wher e d ( σ ) = inf { x : G δ ( t ) + σ ≤ S net, − δ ( t + x ) for al l t ≥ 0 } 3. O utput Burstiness : G δ ⊘ S net, − δ is a statistic al arrival envelo p e of the dep artur e tr affic fr om the network, w h ich satisfies the fol lowing c ondi- tion for any t, s, σ ≥ 0 and δ > 0 : P { D ( s, t ) > G δ ⊘ S net, − δ ( t − s ) + σ } ≤ ε ( σ ) (15) wher e the err or function ε is given by ε ( σ ) = inf σ g + σ s net = σ ( ∞ X k =0 ε g ( σ g + δ k ) + ε s net ( σ s net ) ) (16) = inf σ g + σ s 1 + ··· + σ s H = σ ( ∞ X k =0 ε g ( σ g + δ k ) + H X h =1 ∞ X k =0 ε s h ( σ s h + δ k ) ) (17) 9 The pro of of the theorem relies o n the sample pat h b ound of statistical arriv al en v elop e (equation (3)) for all t, σ ≥ 0 and an y δ > 0 giv en b y [3 ]. P  sup 0 ≤ k ≤ t { A ( k , t ) − G ( t − k ) − δ ( t − k ) } > σ  ≤ ∞ X u =0 ε g ( σ + δ u ) (18) Pr o of: W e no w pro ve the probabilistic b ound on backlog B ( t ), for some t ≥ 0. The backlog B ( t ) at a net w ork no de is given as A ( t ) − D ( t ). Therefore, for all t, σ ≥ 0 and an y δ > 0, w e ha ve P { B ( t ) > G δ ⊘ S net, − δ (0) + σ } = P { A ( t ) − D ( t ) > G δ ⊘ S net, − δ (0) + σ } ≤ P { A ( t ) − A ⊗ S net ( t ) − G δ ⊘ S net, − δ (0) > σ } ≤ P  sup 0 ≤ k ≤ t { A ( t ) − A ( k ) − G δ ( t − k ) − S net ( k , t ) + S net, − δ ( t − k ) } > σ  ≤ P  sup 0 ≤ k ≤ t { A ( k , t ) − G ( t − k ) − δ ( t − k ) } + sup 0 ≤ k ≤ t {S net ( t − k ) − S net ( k , t ) − δ ( t − k ) } > σ g + σ s net  ≤ P  sup 0 ≤ k ≤ t { A ( t ) − A ( t − k ) − G ( k ) − δ k } > σ g  + P  sup 0 ≤ k ≤ t {S net ( k ) − S net ( k ) − δ k } > σ s net  ≤ inf σ g + σ s net = σ ( ∞ X u =0 ε g ( σ g + δ u ) + ε s net ( σ s net ) ) = inf σ g + σ s 1 + ··· + σ s H = σ ( ∞ X u =0 ε g ( σ g + δ u ) + H X h =1 ∞ X u =0 ε s h ( σ s h + δ u ) ) The third inequalit y is due to the property of suprem um op erat ion, i.e., sup 0 ≤ s ≤ t { X ( s ) + Y ( s ) } ≤ sup 0 ≤ s ≤ t { X ( s ) } + sup 0 ≤ s ≤ t { Y ( s ) } [6]. The final inequalit y is from the definition of statistical net w ork service en v elop e (The- orem 2.1) and sample path statistical arriv al en v elop e (equation 1 8). The pro ofs of the probabilistic b ounds on dela y and departure pro cess are the immediate v ariations of the pro of presen ted ab ov e and are omitted.  The optimal v alue of δ is c hosen to minimize the viola tion proba bility of the 10 p erformance b ounds. The probabilistic p erformance b ounds f r o m Theorem 2.2 can b e further impro ve d if the arriv a l traffic pro cess A and the sto chastic service pro cess S h at each hop h for h = 1 , 2 , . . . , H are statistically inde- p enden t of one another. In the follow ing theorem, w e use the notatio n for con v entional conv olution ε g ∗ ε s ( σ ) = R σ 0 ε g ( σ − u ) dε s ( u ) (as error functions are non-negativ e functions) to simplify the presen tation. Theorem 2.3. L et A b e the arrival tr affic indep endent of the servi c e of- fer e d at the network of H no des c onne cte d in series and G is the c orr esp ond- ing statistic al arrival envel o p e with an err or func tion ε g . Assume S h is the sto chastic servic e pr o c ess char ac terizi ng the servic e offer e d at the hop h for h = 1 , 2 , . . . , H and the servic es off e r e d at e ach hop ar e statistic al ly indep en- dent o f one another. L et S h b e the c orr es p ondin g statistic a l servic e envelop e with an err or func tion ε s h at hop h for h = 1 , 2 , . . . , H and S net b e the sta- tistic al network s e rvic e envelop e. T hen we have the fol lowing b ounds. 1. B acklo g b o und : Th e pr ob abilistic b ound o n the b acklo g in a network, for any t, σ ≥ 0 a n d δ > 0 , is given by P { B ( t ) > G δ ⊘ S net, − δ (0) + σ } ≤ 1 − ( ˜ ε g ∗ ˜ ε s 1 ∗ ˜ ε s 2 ∗ · · · ∗ ˜ ε s H )( σ ) (19) 2. D elay b o und : T h e pr ob abilis tic b ound on the delay in a network, for any t, σ ≥ 0 and δ > 0 , is give n by P { W ( t ) > d ( σ ) } ≤ 1 − ( ˜ ε g ∗ ˜ ε s 1 ∗ ˜ ε s 2 ∗ · · · ∗ ˜ ε s H )( σ ) (20) wher e d ( σ ) = inf { x : G δ ( t ) + σ ≤ S net, − δ ( t + x ) for al l t ≥ 0 } 3. O utput Burstiness : G δ ⊘ S net, − δ is a statistic al arrival envelo p e of the dep artur e tr affic D fr om the network, which sa tisfie s the fol lowing c on- dition for any t, s, σ ≥ 0 and δ > 0 : P { D ( s, t ) > G δ ⊘ S net, − δ ( t − s ) + σ } ≤ 1 − ( ˜ ε g ∗ ˜ ε s 1 ∗ ˜ ε s 2 ∗ · · · ∗ ˜ ε s H )( σ ) (21) wher e ˜ ε g ( σ ) = 1 − P ∞ k =0 ε g ( σ + δ k ) and ˜ ε s h ( σ ) = 1 − P ∞ k =0 ε s h ( σ + δ k ) for h = 1 , 2 , . . . , H . The pro of of the theorem relies on the Lemma 4.1 from [6], whic h states that for an y t wo non-nega tiv e indep enden t ra ndo m v ariables F and G with 11 P ( F > σ ) ≤ f ( σ ) and P ( G > σ ) ≤ g ( σ ) where f ( σ ) and g ( σ ) are non- negativ e, decreasing function for an y σ ≥ 0, then P { F + G > σ } ≤ 1 − ( ˜ f ∗ ˜ g )( σ ) (22) where ˜ f ( σ ) = 1 − f ( σ ) and ˜ g ( σ ) = 1 − g ( σ ). Pr o of: W e no w prov e the probabilistic b o und on bac klog B ( t ), for some t ≥ 0. The backlog B ( t ) at a net w ork no de is given as A ( t ) − D ( t ). Therefore, for all t, σ ≥ 0 and an y δ > 0, w e ha ve P { B ( t ) > G δ ⊘ S net, − δ (0) + σ } = P { A ( t ) − D ( t ) > G δ ⊘ S net, − δ (0) + σ } ≤ P { A ( t ) − A ⊗ S net ( t ) − G δ ⊘ S net, − δ (0) > σ } ≤ P  sup 0 ≤ k ≤ t { A ( k , t ) − G ( t − k ) − δ ( t − k ) + S net ( t − k ) − S net ( k , t ) − δ ( t − k ) } > σ } = P  sup 0 ≤ k ≤ t { A ( k , t ) − G ( t − k ) − δ ( t − k ) − δ ( t − k )+ S 1 ⊗ S 2 ⊗ · · · ⊗ S H ( t − k ) − S 1 ⊗ S 2 ⊗ · · · ⊗ S H ( k , t ) } > σ } ≤ P  sup 0 ≤ k ≤ t { A ( k , t ) − G ( t − k ) − δ ( t − k ) } + sup 0 ≤ k ≤ k 2 ≤ t {S 1 ( k 2 − k ) − S 1 ( k , k 2 ) − δ ( k 2 − k ) } + sup 0 ≤ k 2 ≤ k 3 ≤ t {S 2 ( k 3 − k 2 ) − S 2 ( k 2 , k 3 ) − δ ( k 3 − k 2 ) } + · · · + sup 0 ≤ k H ≤ t {S H ( t − k H ) − S H ( k H , t ) − δ ( t − k H ) } > σ  ≤ 1 − ( ˜ ε g ∗ ˜ ε s 1 ∗ ˜ ε s 2 ∗ · · · ∗ ˜ ε s H )( σ ) The third inequalit y is due to the property of suprem um op erat ion, i.e., sup 0 ≤ s ≤ t { X ( s ) + Y ( s ) } ≤ sup 0 ≤ s ≤ t { X ( s ) } + sup 0 ≤ s ≤ t { Y ( s ) } [6]. The final inequalit y follo ws from equations (8), (18) and (22). The pro of s o f the prob- abilistic b ounds on delay and departure pro ces s are the immediate v a r ia tions of the pro o f presen ted ab o v e a nd are omitted.  The p erformance b o unds from Theorem 2.2 and Theorem 2.3 are deriv ed us- ing the statistical en v elop es and require an additional rate correction fa ctor 12 Figure 2: Netw ork no de with thro ugh and cr oss flows δ whose v alue is chos en to minimize the violation probability of the p erfor- mance bo unds. Before w e pro ceed to analyzing the scaling prop erties of the perfor ma nce b ounds deriv ed in Theorem 2.2, we need an importa n t result on lefto v er sta- tistical service en v elop e using a generalized sc heduling mo del. The leftov er service en ve lop e is a generic env elop e mo deling the service a v ailable t o a flow of in terest whic h is left un used by its neighbor ing flo ws sharing the resources at a no de. The concept of leftov er service en ve lo p e was first in tro duced in de- terministic setting [1, 4] and then extende d to sto chastic domain in [3, 2, 7]. It should be noted that lefto ve r servic e en v elop e accoun ts for a pessimistic estimate of the off ered service at a no de for a flo w of interes t, as it c haracter- izes offered service as the w orst case servic e av ailable to a lo w priorit y flo w in a queue with static priorit y sc heduling. The following theorem describ es the lefto ve r statistical servic e en v elop e deriv ed with statistical service en v elop e from equation (7). Theorem 2.4. L et S agg b e statistic al servic e envelop e with an err or function ε s ag g and S agg b e a sto chastic servic e pr o c ess char acterizi n g the aggr e g a te servic e offer e d at a queue with the flow of inter est A a n d the neighb oring flow A c char acterize d using statistic al arrival envelop es G and G c with err or functions ε g ( u ) and ε g c ( u ) , r esp e ctively. L et D and D c b e the dep artur e flows fr om the queue for the c orr esp on ding flow of inter est A and the neighb oring flow A c , r esp e ctively. Assumin g that the stabili ty c o n dition at the q ueue i s satisfie d, i.e., E [ S agg ] ≥ E [ A ] + E [ A c ] , then the leftover sto chastic s e rvic e pr o c e ss S , for a ny sam ple p ath and for al l t ≥ 0 , i s give n by S ( t ) = S agg ( t ) − A c ( t ) (23) and the leftover s tatistic al servic e en v e lop e S w ith an e rr or function ε s , for al l t ≥ 0 , is given by S ( t ) = S agg ( t ) − G c ( t ) (24) wher e ε s is g iven by ε s ( σ ) = in f σ g c + σ s ag g = σ  ε g c ( σ g c ) + ε s ag g ( σ s ag g )  (25) 13 Pr o of: F rom the prop erty of sto c hastic service pro cess (equation (6)) of the service offered at a queue sho wn in the F ig . 2, w e hav e, for any sample path and for a ll t ≥ 0 D ( t ) + D c ( t ) ≥ ( A + A c ) ⊗ S agg ( t ) D ( t ) ≥ inf 0 ≤ k ≤ t { A ( k ) + A c ( k ) + S agg ( k , t ) } − D c ( t ) ≥ inf 0 ≤ k ≤ t { A ( k ) − A c ( k , t ) + S agg ( k , t ) } = A ⊗ ( S agg − A c )( t ) = A ⊗ S ( t ) The inequalit y in the third step is due to the fact that f or any sample path the departure traffic is alw ays b ounded by the arriv al traffic, i.e., D c ( t ) ≤ A c ( t ). This pro v es o ur claim a b out the lefto ver sto c hastic service pro cess. F or a giv en t ≥ 0, w e hav e P {S ( t ) − S ( t ) > σ } = P {S ( t ) − S agg ( t ) + A c ( t ) > σ } = P { A c ( t ) + S ( t ) − S agg ( t ) + S agg ( t ) − S agg ( t ) > σ } ≤ P { A c ( t ) − G c ( t ) > σ g c } + P {S agg ( t ) − S agg ( t ) > σ agg } ≤ in f σ g c + σ s ag g = σ  ε g c ( σ g c ) + ε s ag g ( σ s ag g )  = ε s ( σ ) This prov es our claim a b out the leftov er non-random statistical service en- v elop e.  F or a work conserving queue sho wn in Fig. 2 which is serv ed at a con- stan t rate C , the ag g regate sto c hastic service pro cess and statistical service en v elop e o f the service offered at the queue is C , i.e., S agg = S agg = C with error function ε s ag g = 0. F rom Theorem 2 .4, for all t ≥ 0, we get the left- o ve r sto chas t ic service pro cess for a ny sample path is S ( t ) = C t − A c ( t ) and the lefto ve r statistical service env elop e is S ( t ) = C t − G c ( t ) with the error function ε s = ε g c . 3. Scaling of End-to-End P erformance Bounds In this section, w e analyze the scaling prop erties of the p erformance b ounds deriv ed in Theorem 2.2. F or this purp ose, a s in [7], w e use the 14 Figure 3: Net work of H concatenated nodes with cross tra ffic ( σ ( θ ) , ρ ( θ )) - traffic mo del from [4] in a net w o r k of H nodes connected in series with cross traffic sho wn in Fig.3. The flo w of intere st is the one which tra ve r ses through the net w ork of H no des connected in series and is termed through flow A . The flow whic h tra nsits the net work at eac h hop is termed cross flow A c . The netw ork no de at each hop ha s a w ork conserving sc heduler whic h op erates at a constan t rate C . The go a l is t o determine the end-to-end p erformance (dela y and bac klog) b ounds for the through flow in presence of the cross flo w at eac h hop and identify its order of scaling. An arriv al traffic A with the effectiv e bandwidth α is said to b e a ( σ ( θ ) , ρ ( θ ) ) - constrained arriv al traffic if for an y t ≥ 0 and θ > 0 it satisfies the condition tα ( θ , t ) ≤ ρ ( θ ) t + σ ( θ ) (26) where ρ ( θ ) is a b ound on the time indep enden t v ersion of effectiv e bandwidth [8] (i.e., ρ ( θ ) ≥ lim t →∞ α ( θ , t )). F or t ≥ 0, G ( t ) = ρ ( θ ) t + σ ( θ ) can b e used as the statistical a r riv al en v elop e G of the arriv al traffic A with the error function ε g ( x ) = e − θ x , i.e., for all t ≥ s ≥ 0 and an y θ > 0 the follow ing condition holds from Chernoff ’s b ound and equation (26): P { A ( s, t ) > ρ ( θ )( t − s ) + σ ( θ ) + γ } ≤ e θ ( t − s ) α ( θ ,t − s ) − θ ρ ( θ )( t − s ) − θσ ( θ ) − θγ ≤ e − θ γ (27) Let t he through flow A with effectiv e bandwidth α a t the ingress of the net w ork and the cross flow A c with effectiv e bandwidth α c at eac h hop b e the ( σ ( θ ) , ρ ( θ )) and ( σ ( θ ) , ρ c ( θ )) - arriv al traffic with statistical arriv al en velopes G ( t ) = ρ ( θ ) t + σ ( θ ) and G c ( t ) = ρ c ( θ ) t + σ ( θ ), resp ectiv ely . F or all t ≥ 0 and an y θ > 0, the condition C ≥ ρ ( θ ) + ρ c ( θ ) must b e satisfied for stabilit y . The stochastic arriv al pro cesses A and A c describe t he through flow and cross flo w, resp ectiv ely . T he service a v ailable to the through flow at eac h hop can b e characterize d using lefto v er statistical service en velopes from Theorem 2.4. F or all t ≥ 0, the leftov er sto chastic service pro cess S h for h = 1 , . . . , H is giv en as S h ( t ) = C t − A c ( t ), the lefto v er statistical service en v elop e S h for h = 1 , . . . , H is giv en as S h ( t ) = C t − G c ( t ) with the error 15 function ε s h ( γ ) = ε g c ( γ ) = e − θ γ . The statistical service env elop e S net ( t ) from Theorem 2.1, for all t ≥ 0, is g iv en as S net ( t ) = ( C − ρ c ( θ )) t − H σ ( θ ) with error function ε s net ( γ ) = P H h =1 P t k =0 ε s h ( γ + δ k ). Throughout this section, w e will ev aluate the larger in terv al [0 , ∞ ] inste a d of [0 , t ] to simplify the deriv ation of conserv ativ e, closed-form p erformance b ounds. W e first deriv e the end-to- end bac klog B ( t ) b o und, for all t ≥ 0 , using the stat istical env elop es. F or any θ > 0 and C − ρ ( θ ) − ρ c ( θ ) 2 ≥ δ ≥ 0, we get G δ ⊘ S net, − δ (0) = ( H + 1) σ ( θ ). The probabilistic bac klog B ( t ) b ound from Theorem 2.2, for all t ≥ 0, is giv en as P { B ( t ) > ( H + 1) σ ( θ ) + γ } ≤ inf γ g + γ s 1 + ··· + γ s H = γ ∞ X k =0 ε g ( γ g + δ k ) + H X h =1 ∞ X k =0 ε s h ( γ s h + δ k ) = inf γ g + γ s 1 + ··· + γ s H = γ ∞ X k =0 e − θ ( γ g + δk ) + H X h =1 ∞ X k =0 e − θ ( γ s h + δk ) = inf γ g + γ s 1 + ··· + γ s H = γ 1 1 − e − θ δ e − θ γ g + H X h =1 1 1 − e − θ δ e − θ γ s h = ( H + 1) 1 − e − θ δ e − θγ H + 1 (28) The final step is due to the conv exit y of e − x . Usually we determine a ba c klog b ound so that P { B ( t ) > ( H + 1) σ ( θ ) + γ } ≤ ε , where ε is the giv en viola tion probabilit y . Setting the rig h t-hand side o f equation (28) to ε , using the optimal v alue of δ = C − ρ ( θ ) − ρ c ( θ ) 2 and solving for γ giv es, for an y θ > 0 γ = H + 1 θ log ( H + 1) ε  1 − e − θ ( C − ρ ( θ ) − ρ c ( θ )) 2  (29) Therefore the bac klog x = ( H + 1) σ ( θ ) + γ can b e explicitly b ounded as follo ws: x = inf θ > 0 H + 1 θ log ( H + 1) ε  1 − e − θ ( C − ρ ( θ ) − ρ c ( θ )) 2  + ( H + 1) σ ( θ ) (30) It is apparen t from equation (30) that the end-to- end bac klog measure com- puted using The o rem 2.2 is b ounded b y O ( H log H ). 16 The same tec hnique can b e used to deriv e the end-to-end dela y b ound using Theorem 2.2. F or an y θ > 0 a nd C − ρ ( θ ) − ρ c ( θ ) 2 ≥ δ ≥ 0, d ( γ ) from equation (1 4) becomes γ +( H +1) σ ( θ ) ( C − ρ c ( θ ) − δ ) , then the probabilistic b ound on dela y W ( t ) from Theorem 2.2, fo r all t ≥ 0, is giv en as P { W ( t ) > d ( γ ) } ≤ ( H + 1) 1 − e − θ δ e − θ ( C − ρ c ( θ ) − δ ) d ( γ ) H + 1 + θ σ ( θ ) (31) Usually we determine a dela y b ound so that P { W ( t ) > d } ≤ ε , where ε is the given violation pr o babilit y and d = d ( γ ). Setting the righ t- hand side of equation (31) to ε , using the optima l v alue of δ as C − ρ ( θ ) − ρ c ( θ ) 2 and solving for d ( γ ) giv es d = inf θ > 0 2( H + 1) θ ( C + ρ ( θ ) − ρ c ( θ )) log ( H + 1) ε  1 − e − θ ( C − ρ c ( θ ) − ρ ( θ )) 2  + 2( H + 1) σ ( θ ) ( C + ρ ( θ ) − ρ c ( θ )) (32) It is apparen t from equation (32) that the end-to-end dela y measure com- puted using The o rem 2.2 is b ounded b y O ( H log H ). 4. Numerical E xample The goal of this section is to illustrate the benefits of using the new def- inition of statistical service en velope from equation (7) o v er its coun terpart from equation (5) o n the efficiency and scalabilit y of the computed p erfo r - mance measures using a n umerical example. F or the n umerical experimen t w e consider a netw ork of H concatenated no des as sho wn in Fig. 3. The queue at eac h hop h is serv ed at a constan t determinis tic service rate C . W e use the Mark ov mo dulated on-off (MMOO) pro ces s t o describ e the arriv als o f N indep enden t through flows at the ingress of the netw ork a nd the a rriv als of M indep enden t cross flo ws at eac h hop h inside the net w or k. Mark ov mo d- ulated on-off pro ces s is a ty pical example of ( σ ( θ ) , ρ ( θ )) - constrained traffic mo del with parameters (0 , α ( θ )) and is commonly used to mo del the v oice [9] and video tra ffic [10] in the In ternet. Mark ov mo dulated on-off pro cess can b e in “On” state or “Off” state for a random time in terv al whic h is negativ e exp o nen tially distributed with a verage E [ T on ] and E [ T of f ], resp ectiv ely . In “On” state, arriv al traffic transmits data at a constan t rate P and no data is transmitted in “Off” state. The effectiv e bandwidth o f Mark o v mo dula t ed 17 on-off pro cess has an in teresting prop erty that α ( θ , t ) ≤ α ( θ ) and for an y θ > 0 is giv en by α ( θ ) = 1 2 θ  P θ − r 10 − r 01 + q ( P θ − r 10 + r 01 ) 2 + 4 r 10 r 01  (33) where r 10 = 1 E [ T on ] and r 01 = 1 E [ T of f ] . In the example, we determine the n umerical end-to-end dela y b ound for N through flows in the net w o r k with a violation probabilit y ε = 1 0 − 9 . The capacit y of the serv er C at eac h hop is set to 100 M bps . W e use tw o t yp es of Mark ov mo dulated on-off traffic mo del as in [3]: ( i ) “high burstiness” v ariant has the parameters E [ T on ] = 10 ms and E [ T of f ] = 90 ms and ( ii ) “lo w burstiness” v ariant has the parameters E [ T on ] = 1 ms and E [ T of f ] = 9 ms . Both v ariants of the o n-off traffic pro duce data a t an a v erage rate m = 0 . 15 M bps and emit data at a p eak rate P = 1 . 5 M bps during the ”‘On”’ state. The sto c hastic service a v ailable for N through flo ws is determined using the generalized sche duling mo del ( Theorem 2.4). 1 3 5 7 9 11 13 15 17 19 21 23 25 0 200 400 600 800 1000 1200 1400 1600 No of hops (H) Delay [ms] using service envelope from equation (7) using service envelope from equation (5) (a) with “high burs tiness” tra ffic 1 3 5 7 9 11 13 15 17 19 21 23 25 0 20 40 60 80 100 120 140 160 No of hops (H) Delay [ms] using service envelope from equation (7) using service envelope from equation (5) (b) with “low burs tiness” tra ffic Figure 4 : End-to-end delay bo und with a viola tion probability ε = 10 − 9 for incr e asing nu mber o f ho ps H with N = 1 34 thr o ugh MMOO flows and M = 33 3 cr o ss MMO O flows at each hop W e compare the end-to-end dela y b ounds for through flo ws determined using the statistical service en v elop e from equation(7) and equation(5). F or a v alidation of the latter w e repro duced the results presen ted in [3] using the statistical service en v elop e from equation(5). Fig. 4 shows the probabilistic end-to-end delay b ounds with a violation probabilit y ( ε ) of 10 − 9 as a f unction of increasing num b er of hops H . A t eac h hop, M = 333 cross flo ws ar e m ultiplexed with N = 1 34 through flows . The plot illustrates the O ( H log H ) 18 b ounds of end-to- end dela ys with statistical net w or k service en v elop e from Theorem 2.1 and v alidates that the end-to -end dela y b o unds determined using the statistical service env elop e from equation (7) pro vide tigh ter b ounds than the o nes computed using the statistical service en velope from equation (5). 100 200 300 400 500 600 20 40 60 80 100 120 140 160 180 200 Total number of flows (N+M) End−to−end delay bound [ms] H=5 H=2 H=1 using service envelope from equation (7) using service envelope from equation (5) H=10 (a) with “high burs tiness” tra ffic 100 200 300 400 500 600 20 40 60 80 100 120 140 160 180 200 Total number of flows (N+M) End−to−end delay bound [ms] H=5 H=2 using service envelope from equation (5) using service envelope from equation (7) H=10 H=1 (b) with “low burstiness” traffic Figure 5: Comparison of End-to-end statistical delay b ounds with a v io lation pro bability ε = 10 − 9 for MMOO tra ffic computed using tw o differen t de finitio n of statistical net work env elop es In Fig. 5 w e plot the pro babilistic end-to-end dela y b ound for N through flo ws in a net w ork with H = 1 , 2 , 5 , 10 hops for increasing N + M n umber of flo ws at eac h hop while main taining N = M . It can b e observ ed that the new definition of statistical serv ice en v elop e from equation (7) t o gether with Theorem 2 .2 yield a tighter dela y b o und ev en for single hop case. The b enefit of the new definition of statistical netw ork service env elop e is more o b vious when the n um b er of no des H tra vers ed b y the through flo ws is increased. 5. Conclusion W e presen ted a new for mulation of statistical service en velope using the sto c hastic serv ice pro ces s desc ribing the service offered at the net work no de. W e show ed for Mark ov mo dulated on- o ff t r affic mo del t ha t the new formu- lation o f statistical service env elop e yields end-to-end probabilistic p erfor- mance measures are tigh ter than the ones computed using existing state-of- the-art definition of statistical net w ork service en velope and are b ounded b y ( H log H ) for more general ( σ ( θ ) , ρ ( θ )) - constrained traffic mo del, where H is the n umber of no des tra v ersed b y the arriv al traffic. 19 References [1] J.-Y. L . Boudec, P . Thiran, Net w o r k Calculus: A Theory of Determin- istic Queuing Systems for the In ternet, Springer-V erlag, 2 001. [2] C. Li, A. Burchard, J. Lieb eherr, A netw ork calculus with effectiv e ba nd- width, IEEE/A CM T ransactions on Net working 15(6) (2007 ) 1442–145 3. [3] F. Ciuc u, A. Burchard, J. Liebeherr, Scaling prop erties of statistical end-to-end b ounds in the netw ork calculus, Information Theory , IEEE T ra nsactions o n 52 (6) (2006) 230 0 – 2312. [4] C.-S. Chang, P erformance Guarante es in Comm unicatio n Net works, Springer-V erlag, 2000. [5] R. L. Cruz, Quality o f service managemen t in in tegrated services net- w orks, in: Pro ceedings of 1st Semi-Ann ual Researc h Review, June 19 96. [6] Y. Jiang, A basic sto c hastic net w ork calculus, in: Proceedings of A CM SIGCOMM, 2006, pp. 123–134. [7] M. Fidler, An end-to-end probabilistic net w ork calculus with momen t generating functions, in: Pro ceedings of IW QoS, 2006. [8] F. P . Kelly , Notes on effectiv e bandwidths, Sto c ha stic Netw orks: Theory and Applications Oxford, Roy al Stat istical So ciet y Lecture Notes Series, (1996) 141–168 . [9] ITU-T Recommendation P .59, Artificial conv ersational sp eec h (1993). [10] B. Maglaris, D. Anastassiou, P . Sen, G. Karlsson, J. D. Ro bbins, Pe r - formance mo dels of stat istical m ultiplexing in pa c k et video comm unica- tions, IEEE T ransactions on Comm unicatio ns 36 (1988) 834–843. 20