OVERFLOW PROBABILITY IN AN ATM QUEUE WITH SELF-SIMILAR INPUT TRAFFIC

Copyright by IEEE OVERFLOW PROBABILITY IN AN ATM QUEUE WITH SELF-SIMILAR INPUT TRAFFIC Bori Tybakov Intitute for Problem in Information Tranmiion Ruian Academy of Science Mocow, Ruia e-mail: bt@ippi ac mk u and Nicola D Georgana Multimedia Communication Reearch Laboratory(MCRLab) Dept of Electrical Engineering Univerity of Ottawa Ottawa, Ontario, Canada KN 6N5 Tel: (63) 56-5800 x 65; Fax: (63) 56-575 e-mail: georgana@mcrlab uottawa ca Abtract Real meaurement in high-peed communication network have recently hown that traffic may demontrate propertie of long-range dependency peculiar to elf-imilar tochatic procee Meaurement have alo hown that, with increaing buffer capacity, the reulting cell lo i not reduced exponentially fat a it i predicted by Markov-model-baed queueing theory but, in contrat, decreae very lowly Preenting a theoretical undertanding to thoe experimental reult i till a problem The paper preent mathematical model for elf-imilar cell traffic and analyze the overflow behavior of a finite-ize ATM buffer fed by uch a traffic An aymptotical upper bound to the overflow probability, which decreae hyperbollically, h a, with buffer-ize-h i obtained A lower bound i alo decribed, which demontrate the ame h a aymptotical behavior, thu howing an actual hyperbolical decay of overflow probability for a elf-imilar-traffic model ---- Thi work wa upported in part by a grant from Nortel (BNR) and in part by NSERC operating grant A-8450 and by NWO project 73-9 --

INTRODUCTION Recent traffic meaurement in corporate LAN, Variable-bit-rate video, ISDN controlchannel and other communication ytem have indicated traffic behavior of elf-imilar nature [Leland 94], [Beran] With actual traffic meaurement, it wa hown that the overall cell lo decreae very lowly with increaing buffer capacity, in harp contrat to Poion-baed model where loe decreae exponentially fat with increaing buffer ize The problem i how to get a theoretical explanation of uch empirically oberved behavior An extenive bibliographical guide with 40 reference to previouly publihed paper on elf-imilar traffic and analyi i given in [Willinger] In thi paper, four model for elf-imilar cell input traffic are conidered Aymptotical upper bound to overflow probability in an ATM buffer queue fed by three type of traffic are preented Thee bound are expreed in term of the rate of input traffic, capacity of ATM channel, buffer ize, and the Hurt parameter (the lat how the level of traffic elf-imilarity) The derived bound decreae much lower (hyperbolically) than exponentially with buffer-ize growth For a model with contant ource rate and a channel of capacity, we are able to compare the obtained upper bound againt the lower bound preented here and proved by u in {TG] MODELS FOR SELF-SIMILAR TRAFFIC We introduce here mathematical model of elf-imilar input cell traffic to an ATM buffer Self-imilar procee Firt we recall the definition of a elf-imilar proce Conider a econd-order tationary real-number tochatic proce X = (, X, X, X,) of dicrete 0 argument (time) t I = {,, 0,,} Denote by µ = EX t < and σ = var X t <, the mean and the varience of X t repectively Denote by r(k) the autocorrelation function of proce X The mean µ, the varience σ, and the autocorrelation function r( k) do not depend on time t, and r( k) = r( k) Now we give the definition and ome propertie of exactly and aymptotically elf-imilar procee ( ee [Cox], [ST], [Leland 94], [TG]) Definition A proce X i called exactly econd-order elf-imilar (e) with parameter β H =, 0 < β <, if it autocorrelation function i r( k) = [( k + ) β k β + ( k ) β ] = ˆ g( k), k I = {,,} --

The reaon the proce X with r( k) = g( k) i called elf-imilar i that it doe not change it correlational tructure with averaging It mean that the averaged (over block of m ) proce (, X ( m), X ( m), X ( m),) where X ( m) 0 t = ( m X X tm m tm ), m I, t I + + + ha the autocorrelation function r ( m ) ( k ) uch that r ( m) ( k) = r( k) = g( k) H i called the Hurt parameter of proce X Definition A proce X i called aymptotically econd-order elf-imilar (ao) with β parameter H =, 0 < β <, if for all k I, lim r ( ( k) = g( k) ( ) m The above mean that X i ao if (after averaging over block of ize m and with m ) it correlational tructure tend to that of the e proce The important property of X, which guarantee that X i ao with parameter β r( k) H = i lim k k β = c, where 0 < c < i a contant and 0 < β < [TG] It i intereting to notice that thi limiting equation give a definition of long-range dependence (lrd) proce (ee [Beran], Sect ) It mean a lrd proce i alway an ao proce Source proce We conider a traffic a a tream of cell For convenience, we aume that a cell ha the length when it i tranmitted over the ATM channel The channel can tranmit C cell at a time unit lot; the time interval [ t, t + ) i called lot t, t I {,, 0,,} and C i called the channel capacity The cae of C = i conidered in Section 3 and 4; the general cae of C I i conidered in Sect 5 The dicrete-time ource proce Y = (, Y, Y0, Y,) i a random proce where Y t repreent the number of new cell arrived at time t, Yt I0 { 0,,,} The ource proce Y i contructed in the following way Let θ ( t ω + ) be a random equence over time t, aociated with a moment of arrival ω of ource, t I, ω I, θ ( ) I0 The ource are numbered by according to their arrival, ω + ω The θ ( ) repreent the number of cell generated by ource upon it arrival at ω and θ ( i) give the analogou number for the moment which i i time unit later than ω The ource generate it cell within time interval of length τ I {,, 3,} and the equence θ ( i) in thi interval ω i < ω + τ i called the ource active period We denote by -3-

γ = ˆ θ ( ) + + θ ( τ ) ( ) the total number of cell generated by ource during it active period and γ i called the volume of ource Let ξ t denote the number of new ource arrived at t We aume in thi paper that ξ t are independent and Poionian with parameter λ It i aumed that the random equencie (θ (),,θ (τ )) are i i d for different and they are independent of the equence ξ t and ω The ource proce Y = (, Y, Y, Y,) Yt = θ ( t ω + ), 0 i defined a ( 3) i e Y i a uperpoition of ource active period The Y t i the total number of cell generated by all active ource at time t 3 Four model for elf-imilar traffic In the next ection, the mot detailed reult on buffer overflow probability hall be obtained for elf-imilar input traffic which are the further narrowing of the ource proce Y We introduce them now Let u conider the four pecial cae of Y In the mot intereting cae of Y ( ), θ ( i) = R where R i a given contant independent of and i, R I The rate R which are le than can not be interpreted in the framework of proce Y ( ) (the ame hold for Y ( ) and Y ( ) 3 ), ince θ ( i) ha a meaning of the number of cell arrived at i However, thee rate can be interpreted in the framwork of proce Y ( 4 ) The proce Y ( ) with R= wa conidered in [Cox], while Y ( ) with arbitrary R I wa conidered in [LTG] and [TG] In the cae of Y ( ), θ ( i) = R( l) given τ = l and the function R( l) i monotonic and not random (given τ = l ) R( l) doe not depend on and i, R( l) I, l I However, for the ource without the condition τ = l, the rate R( τ ) i a random variable In the cae of Y ( 3 ), θ ( i) = R ( l) given τ = l where R ( l) i a random variable (even for given τ = l ) and the ditribution of R ( l) doe not depend on and i, R ( l ) I According to aumption in, the random variable R ( l ) (given τ = l ) are independent for different In the cae of Y ( 4 ), θ (i), ω i < ω + τ with different are the egment of i i d tationary dicrete-time procee The proce Y ( 4) wa introduced in [TG] A condition of aymptotical elf-imilarity of Y ( 4) i given by: Statement If t α Pr{τ > t} c, t, < α < (where c i a contant, 0 < c < ) and Eθ <, Eθ <, then Y ( 4) i aymptotically econd-order elf-imilar proce with parameter H = (3 α ) / > 05-4-

In what follow, we give the condition for elf-imilarity of procee Y (i), i =,, 3, and thu preent finally our remaining three elf-imilar cell traffic model Y ( i), i =,, 3 Statement The proce Y ( i) i aymptotically econd-order elf-imilar (ao) with β parameter H =, 0 < β <, if aymptotically with l Y () : Pr{τ = l} Y () : R (l)pr{τ = l} Y (3) : (ER (l))pr{τ = l} where c > 0 i a contant ~ cl ( β + ) ( 4) For Y ( ) with R =, ufficiency of the condition ( 4) for ao wa given by Cox [Cox] Statement for the cae of proce Y ( ) wa given in [TG] 3 ATM - BUFFER QUEUE TO CHANNEL OF CAPACITY : LOWER BOUND TO OVERFLOW PROBABILITY In thi ection, we conider a finite-buffer queueing ytem with input cell traffic Y () having ource of contant rate R = and denoted by Y We will obtain the lower bound to the tationary buffer-overflow probability, expreed in term of λ > 0, Pr{τ = n} and h, where h i the ize of the buffer The bound decay hyperbollically rather then exponentially fat with increaing h when τ ha Pareto-type ditribution and Y i elf-imilar Let y t be the number of new cell arrived at t and z t be the number of cell that were in the buffer at time t Definition A ervice dicipline d i in cla D(h), if it atifie the following two condition: (i) If y t > 0, then ome cell (out of the y t total) goe definitely into ervice at t ; (ii) If y t h +, then no cell are dicarded at time t If y t > h +, intead, then exactly y t h cell are dicarded at t The event { y t > h + } i called the buffer overflow at time t We are intereted in the tationary probability of overflow, P over Our reult (the detail are in [TG]) for the lower bound to P over i expreed by the following: Statement 3 The queueing ytem Y/D// h with any ervice dicipline from D(h) ha the overflow probability P over lowerbounded by Pover Pr{ τ n} (3 ) ( Eτ + Eκ ) n= n -5-

h b where n = + a +, Eκ i given by λ Eκ = ( e ) (3 ) and a, b are given by a =, Eτ + Eκ b = a + (3 3) We apply the bound (3) to the mot intereting cae of Pareto-type ditribution, Pr{τ = n} = cn α, < α <, and elf-imilar traffic Y We get c Pover α( α )( Eτ + Eκ ) where α h + b a + α α + (3 4) Eτ = c n, c = ( n ), (3 5) n= n= and a and b are given by (3 3), wherea Eκ i given by (3) Aymptotically, when h i large, (34) give the bound P over c h α+ = c h β = c h ( H), where c i a contant independent of h but dependent on λ and α and where we ued the equation H = β = 3 α from Statement With thi bound, we come to the important concluion that for the conidered elf-imilar traffic Y, the overflow probability P over cannot decreae fater than hyperbolically with the growth of buffer ize h To get a numerical reult, let u conider, for example the cae of α= 5 (the Hurt parameter i H=(3-α)/=0 75 ) and λ = 0 In thi cae, c=0745, Eτ= 95, Eκ=4 5, and Pover 0 038 (3 6) h + 5 0 5 + According to (3 6) or (34), P over 79 0 3 for h = 0, P over 8 0 3, for h = 0, P over 93 0 4, for h = 00, P over 3 0 4, for h = 000, P over 94 0 5, for h = 0,000 4 ATM - BUFFER QUEUE TO CHANNEL OF CAPACITY : -6-

UPPER BOUNDS TO OVERFLOW PROBABILITY Here we conider the procee Y ( ) and Y () a the input cell traffic to ATM finite-buffer dicrete-time queue We are intereted in the buffer-overflow probability We adopt the uual ymbol to denote the queueing ytem Y/D//h/d mean that input traffic i Y (ie at time t, the input traffic provide Y t new requet or, in other word, it provide Y t new cell), ervice-time i contant and equal to, there i a ingle erver, the buffer ha ize h, and the evice dicipline i d In thi Section, we retrict our analyi to the cae of ATM channel having capacity C = The cae of greater channel capacity are the ubject of the next Section We begin with conideration of input traffic Y () with R =, denoted by Y 4 Upper bound to overflow probability for elf-imilar traffic Y A way to get our upper bound i baed on the large-deviation reult for a um of independent random variable with long-tailed ditribution In the cae of long-tailed ditribution, the large deviation of the um occur mainly at the expene of jut one (maximal) ummand Thee reult originated with Chityakov [Chityakov] and Chover et al [Chover] The more deep mathematical background for the problem i given in the book of Bingham, Goldie, and Teugle [Bingham] in context of regular variation Our upper bound i preented by: Statement 4 If for the aymptotically elf-imilar proce Y with parameter H = 3 α, the condition λe τ < i atified and the ource-active-period length τ ha the Pareto-type ditribution α Pr{ τ = } ~ l c l, < α <, l, (4 ) 0 then for the dicrete-time Y/D// h/ d ytem with any ervice dicipline d probability for the ize h buffer i upperbounded aymptotically a P over D( h), the overflow c 0 λ α(α )( λeτ) h α+, h (4 ) If for Y we alo conider the other ub-exponential ditribution Pr{ τ > l } with long tail (but not only the Pareto-type ditribution), for example, the lognormal ditribution or the Weibull ditribution adopted for the dicrete cae, and we ue the reult of Pake, Cline, Teugel,Willeken (ee [Bingham]), then under the condition of Statement 4, we get the aymptotical upper bound P over λ Pr{τ n} (4 3) ( λeτ) n=h -7-

Aymptotically in the cae of Pareto-type ditribution of τ, the upper bound (4) i again hyperbolical over h with the ame exponent ( α + ) a in the lower bound (3 4) Thi how that both bound demontrate the right aymptotical behavior of P over apart from a factor which i independent of h in (3 4) and (4) Now we give a numerical example to compare the lower bound (3 4) and the upper bound α (4 ) for the cae of Pr{ τ = l} = c0l, < α <, and elf-imilar traffic Y Chooing α = 5, λ = 0, we get 35 a the upper bound/lower bound ratio 4 Upper bound to overflow probability for elf-imilar traffic Y () and Y () with R > Here, we extend Statement 4 on the elf-imilar input traffic Y ( ) atifying ( 4) Beginning with the cae of Pr{ τ = l } atifying (4 ), we obtain finally P over c λ α+β 3 α(α )( λeγ ) h α β+, h, λe γ < (44) α c where c3 c0c, c c α β +, c,and c i the contant from (4) c 0 Thu, (4 4) i an aymptotical (for large h ) upper bound to the overflow probability P over for the dicrete-time Y () /D//h/d ytem with any ervice dicipline d D( h) in the cae of aymptotically elf-imilar input cell traffic Y ( ) defined in Sec and atifying ( 4) and (4 ) The bound i expreed in term of λ, α, β, and h, where (we remind) λ i the intenity of ource arrival, α i the parameter of ditribution of ource active-period length τ, β = ( H ) i the parameter of elf-imilar traffic ( 0 < β <, β α ), and h i the buffer ize Since Y ( ) i more general proce than Y ( ), we can derive [from (4 4)] the upper bound to P over for YR/D//h/d ytem where the ymbol YR denote the aymptotically elf-imilar traffic Y ( ) with any given rate R I Putting the retriction λre τ <, we obtain c λr α P over 0 α(α )( λreτ) h α+, h, < α < (4 5) Thu, (4 5) i an aymptotical upper bound to the overflow probability P over for dicrete-time YR/D//h/d ytem Next, we conider the cae of ditribution Pr{τ = l} which decreae exponentially, -8-

Pr{τ = l}~ c 4 e ϕl, l (4 6) where c 4 > 0 and ϕ > 0 are contant We obtain, P over c 4 λeϕ h, ϕ( λeγ ) (lnh) β h (4 7) when λeγ < (4 7) give an aymptotical upper bound to the overflow probability for a Y () /D//h/d ytem with elf-imilar input traffic Y () atifying ( 4) and (4 6) The bound (4 7) obtained for the exponential-tail ditribution Pr{τ = l} goe to zero a little fater than the bound (4 ), (4 4), and (4 5) obtained for hyperbollical-tail ditribution Pr{τ = l} 5 ATM-BUFFER QUEUE TO CHANNEL OF CAPACITY C : UPPER BOUNDS TO OVERFLOW PROBABILITY So far we conidered an ATM channel which could tranmit one cell at it time unit called lot Now we conider a channel which can tranmit C cell in it lot, C I Again, the problem i to find an upper bound to the buffer-overflow probability 5 Queueing model We conider the dicrete-time queueing ytem Y/DC//h which ha the input cell traffic Y, contant ervice-time C (where C i a poitive integer), a ingle erver, and finite buffer of h-cell ize In our conideration, dicrete-time Y/DC//h i equivalent to the ytem Y/D/C/h which ha C erver, each with contant ervice-time Taking it into account, it i more convenient for u to analyze Y/D/C/h ince, in thi cae, we hould deal with only one channel-cale unit and not partition the channel lot into ublot of length C each We conider only the ervice dicipline d which atify the following two condition: (i) If y t > 0 (where y t and z t were defined in Sect 3), min( y t, C) cell go into ervice at t, (ii) If y t h + C, then no cell are dicarded at t If y t > h + C, intead, then y t h C cell are dicarded at t We denote by D ( h) the cla of the conidered dicipline C The event { y t > h + C} i called the buffer overflow at time t and Pt over ˆ= Pr{y t > h + C} denote it probability Again, we are intereted in getting an upper bound to the teady-tate overflow t probability denoted by P over, P over ˆ= limup P t over -9-

5 Upper bound to the overflow probability for elf-imilar traffic Y For the dicrete-time Y/D/C/h/d, d D (h) queueing ytem with elf-imilar input C traffic Y atifying (4 ), the overflow probability for the ize h buffer and channel capacity C i upperbounded by c λ P over 0 α(α )(C λeτ) h α+, h, λeτ < C (5 ) 5 3 Generalization on the input traffic Y (4) Our conideration in 5 (the detail are in [TGa]) remain valid for the elf-imilar input traffic Y (4) if intead of the ditribution of τ, we ue the ditribution of γ (the ource volume), τ γ = θ(i) (5 ) i= where θ(i) i the generic ymbol for θ (i), and if Pr{γ = l} i a Pareto-type or ub-exponential ditribution Thu, for Y (4) /D/C/h/d, d D C (h), we obtain λ P over Pr{γ > n}, h, λeγ < C (5 3) C λeγ n=h A a firt example of Y (4), let u conider the ingular cae of θ(i) = R, R I In thi cae, Y (4) i Y ( ) If Pr{τ = l} atifie (4 ), then uing Pr{γ > x}= Pr{τ > x }, we obtain R c λr α P over 0 α(α )(C λreτ) h α+, h, λreτ < C (5 4) than The relation (5 4) i the direct generalization of (4 5) in the cae of channel capacity greater A a econd example of Y (4), let u conider i i d θ(i) with Pr{θ(i) =}= Pr{θ(i) = 0}= p (5 5) If Pr{τ = l} i the Pareto-type ditribution (4 ), then c λp α P over 0 α(α )(C λpeτ) h α+, h, λpeτ < C (5 6) The bound (5 6) i a natural extention of the bound (4 5) in the cae of R < and C Thee conideration can be extended to the traffic Y (4) with ergodic equence θ(i) -0-

5 4 Generalization on the input traffic Y () The conideration in 5 (the detail are in [TGa]) remain alo valid for the input traffic Y (), if intead of the ditribution of τ, we ue the ditribution of γ = τr(τ) a in 4 Eventually for the C cae, if λeγ < C, we get the upper bound which are the ame a (4 4) [for Y () /D/C/h/d with the traffic Y () atifying ( 4) and (4 )] and (4 7) [for Y () /D/C/h/d with the traffic Y () atifying ( 4) and (4 6)] with the only change of the term ( λeγ ) for the term (C λeγ ) in the denominator 6 CONCLUSIONS Four model for ATM cell traffic were conidered in thi paper Then a finite buffer fed by elf-imilar traffic Y i ( ) wa treated a a queueing ytem The problem wa to find an aymptotical bound to the buffer overflow teady-tate probability and with thi bound to explain the low decay of cell-lo probability meaured in high-peed network Indeed, we got bound which decreae hyperbollically with increaing buffer ize and thi decay i much lower than regular exponential decay The bound have imple and explicit expreion in term of intenity and the Hurt parameter of input traffic, ATM channel capacity, and buffer ize Reference [Beran] J Beran, "Statitic for Long-Memory Procee", New York, Chapman and Hall, 994 [Bingham] N H Bingham, C M Goldie, and J L Teugel, "Regular variation", Cambridge, New York, Melburn, Cambridge Univerity Pre, 987 [Chityakov] V P Chityakov, "A theorem on um of independent poitive random variable and it application to branching procee", Theory Prob Appl, vol 9, No 4, pp 640-648, 964 [Chover] J Chover, P Ney, and S Wainger, "Function of probability meaure", J d'anal Math, vol 6, pp 55-30, 973 [Cox] D R Cox, "Long-range dependence:a review", In Statitic:An Appraial, H A David and H T David, ed Ame, IA:The Iowa State Univerity Pre, 984, pp 55-74 [Leland 94] W E Leland, M S Taqqu, W Willinger, and D V Wilon, "On the elf-imilar nature of Ethernet traffic (Extended verion)", IEEE/ACM Tran on Networking, vol, No, pp -5, 994 [LTG] N Likhanov, B Tybakov and N D Georgana, "Analyi of an ATM buffer with elf-imilar ("Fractal") input traffic", Proc IEEE INFOCOM'95, Boton, MA, pp 985-99, 995 --

[ST] G Samorodnitky and M S Taqqu, Stable non-gauian random procee, New York, London, Chapman and Hall, 995 [TG] B Tybakov and N D Georgana, "On elf-imilar traffic in ATM queue: Definition, overflow probability bound and cell delay ditribution", ubmitted to IEEE/ACM Tran on Networking [TGa] B Tybakov and N D Georgana, "Self-imilar traffic and upper bound to bufferoverflow probability in ATM queue", ubmitted to Performance Evaluation [TK] B Tybakov and P Papantoni-Kazako, "The bet and the wort packet tranmiion policie", Information Tranmiion Problem, to appear in 996 [Willinger] WWillinger,MSTaqqu,and AErramilli, "A bibliographical guide to elf-imilar traffic and performance modelling for modern high-peed network" in "Stochatic network:theory and application", EdFPKelly, SZachary, and IZiedin, Oxford Univerity Pre,Oxford,996 --