On Adaptive Bandwidth Sharing with Rate Guarantees

Transcription

1 On Adaptive Bandwidth Shaing with Rate Guaantees N.G. Duffield y T. V. Lakshman D. Stiliadis y AT&T Laboatoies Bell Labs Rm A175, 180 Pak Avenue Lucent Technologies Floham Pak, 101 Cawfods Cone Road NJ USA Holmdel, NJ 07733, USA Abstact The objective of ecent eseach in fai queueing schemes has been to efficiently emulate a fluid-flow genealized (weighted) pocesso shaing (GPS) system, as closely as possible. A pimay motivation fo the use of fai-queueing has been its use as a means of poviding bandwidth guaantees and as a consequence end-to-end bounds fo taffic with bounded bustiness. The ate guaantees tanslate to scheduling weights which ae set when admission contol is done. A consequence of fai queueing systems closely emulating GPS is that when one o moe connections ae not backlogged, any excess bandwidth is distibuted to backlogged connections in popotion to thei weights. Howeve, weights ae set based on the long-tem equiements of taffic flows and not in any state-dependent manne that eflects instantaneous needs. In this pape, we question the notion that queueing systems should closely emulate a GPS system. Instead of emulating GPS, we popose thee modified scheduling schemes which peseve the ate guaantees of fai queueing (and hence peseve deteministic bounds) but adaptively edistibute the excess bandwidth such that eithe losses ae educed o s equalized. We compae the pefomance of the poposed schemes to that of fai queueing using diffeent taffic souces such as voice and video, as well as souces which have aggegate long-ange dependent behavio. We find that the poposed schemes, in compaison to packet GPS (), educe packet losses and cutail the tails of distibutions fo eal-time taffic and hence pemit the use of significantly smalle playout buffes fo the same netwok load. Keywods fai queueing, scheduling, jitte, bounds, eal-time taffic, taffic management I. INTRODUCTION Much eseach attention has ecently been focused on the development of fai-queueing systems [3], [10], [19], [20], [21], [24] that, within the nonpeemption constaints of a packet system, closely emulate fluid flow genealized pocesso shaing systems (GPS) and opeate at high speeds. Demes, Keshav, and Shenke poposed a fai queueing scheme that emulated GPS by using a simulated fluid flow GPS system as a efeence and basing packet scheduling decisions on the ode of depatues in the simulated GPS system. The computational buden of simulating GPS was educed in the self-clocked fai queueing scheme poposed by Golestani [10] which showed how fai queueing could emulate GPS without simulating GPS fo efeence. Subsequent wok has led to both impoved emulation bounds, futhe eductions in computational complexity and efficient methods fo implementation [11], [20], [21], [24]. Howeve, thee is no detailed examination of whethe the exclusive emulation of GPS is always appopiate fo all netwoking applications. In this pape, we question the notion that queueing systems must emulate GPS closely. A pimay motivation fo the expected use of fai queueinn outes and switches is its ability to guaantee wost case end-to-end bounds fo leaky bucket contolled souces [17]. Howeve, to guaantee wost case bounds the schedule meely needs to isolate flows so that each flow eceives its guaanteed shae of the link bandwidth [1], [19]. The guaantees do not depend on the popety of GPS systems that the unused shaes of non-backlogged flows be edistibuted to backlogged flows in popotion to thei guaanteed factional shae of link bandwidth (o equivalently scheduling weights). A fai queueing system need act as one only when all connections ae backlogged. When excess bandwidth (i.e. bandwidth beyond what has aleady been guaanteed to backlogged connections) is available, the fai queueing system, instead of emulating GPS that distibutes excess bandwidth accoding to long tem needs, should edistibute this excess bandwidth in a manne which eflects the cuent o instantaneous needs of backlogged flows (i.e. a flow state dependent edistibution). The edistibution policy can be picked to optimize measues such as packet loss pobabilities o to contol the tails of distibutions. It has been shown in [1], [23] that even when a FIFO schedule is used and all souces ae leaky-bucket shaped, the mean and 99th pecentile of s ae much lowe than the wost-case bounds guaanteed by a GPS seve. Howeve, thee ae sevice disciplines that can distibute the bandwidth in a moe elegant manne. Panwa et.al. [16] showed, that the shotest time to extinction policy is actually optimal fo scheduling customes with deadlines. Futhemoe, a policy that simply seves the longest queue has been shown to equie less buffe space to pevent buffe losses than FIFO o othe pe-flow disciplines [6], [13]. Similaly, policies that seve the queue that is moe likely to violate a bound ae poved to povide optimal buffe utilization [9]. Howeve, such disciplines by themselves cannot povide any bandwidth guaantees. Actually, low bandwidth constant bit ate connections ae likely to stave fo lonntevals of time, until thei queue becomes at least as lage as the queues of heavily busty, high bandwidth connections. Similaly, uses that can accept a significant loss ate, using some fom of Fowad Eo Coection, can incease thei thoughput by simply sending moe packets. Thee is no method to contol the bandwidth of individual uses. We popose thee modified fai queueing schemes that adaptively edistibute excess bandwidth while, like fai queueing, guaanteeing each flow its specified shae of the link bandwidth. This peseves fai queueing's ability to povide wost case end to end bounds and the schemes wok like fai queueing when thee is no excess bandwidth. In addition, the schemes povide wostcase fainess, bounding the time that a connection may not see any sevice. The excess bandwidth, howeve, is edistibuted as follows: 1. Longest fist (LDF) that seves the flow with cuent longest. 2. Least time to oveflow () that seves the flow with minimum diffeence between maximum allowed and cuent. 3. Least time to oveflow with leaky buckets ( LB) that seves the flow which would cause buffe oveflow fist if wost case aivals happen. We compae the pefomance of these policies to using both tace diven simulations as well as simulations with taffic models fo vaious types of souces. The LDF policy uses excess bandwidth to educe the vaiance of the distibution. This has the benefit of educing the playout buffe fo voice and video souces. Simulations with video taces and with voice taffic show that indeed this policy pefoms bette than without any sacifice of wost case guaantees. Since the deviation fom the maximum allowed is not taken into account, flows with small bounds (like voice) get almost no excess bandwidth in the pesence of flows with lage bounds. Long tem congestion o small eos in assigning weights can esult in these flows expeiencing losses much moe than flows with lage bounds. The policy ties to minimize packet losses by assigning excess bandwidth unde the assumption that the flow which is likely to oveflow the quickest has the most instantaneous bandwidth need. In doing so, it takes into account the cuent deviation of each flow fom its maximum allowed. Simulations with a mix of CBR, voice, and video souces with vey diffeent bounds show that this policy educes losses fo all classes as well as educes the vaiance of fo each class. The LB policy goes one step futhe by detemining the connection that is most likely to oveflow its buffe unde the assumption that aivals ae leaky-bucket shaped. A scheduling and buffe shaing scheme that ties to guaantee specific s while ensuing loss-fee behavio is pesented in [7]. Ou poposal is fundamentally diffeent fom the scheme pesented in [7]. We assume that ou system does not have enough buffes to guaantee zeo losses. So ou goal is not to define a system that will guaantee some specific s and at the same time zeo losses. Instead, we investigate the effect of state-dependent scheduling mechanisms in minimizing the tails of the distibutions while povidinsolation, and also investigate the effect of such mechanisms on the numbe of buffes that need to be eseved if limited losses ae acceptable.

2 II. ARCHITECTURE Fai Queueing systems attempt to offe the same nomalized sevice to any two connections that ae continuously backlogged duing an inteval of time (t 1 ; t 2 ], whee by nomalized sevice we mean the atio of the offeed sevice and the allocated ate. The basic featue of such a system is that it teats connections in the same way, iespective of thei bustiness chaacteistics. Ou claim is that end-uses ae not necessaily inteested in a fai distibution of fee bandwidth. Instead, fo many applications, end-uses ae meely inteested in minimizing thei packet losses o seeing low tails in thei distibutions. Of couse, the queueing scheme must be able to povide some guaantees egading wost-case sevice offeed to a connection when the system is fully loaded. Isolation and potection equiements such as these wee defined in [1] as necessay to potect the system fom misbehaving uses that will ty to monopolize esouces. A. Peliminaies The definition of ou state-dependent queueing system uses the methodology pesented in [20] fo designing fai queueing systems. The method is based on the class of seves called Rate Popotional Seves (RPS). The key concept is a vitual time function v i (t) o system potential associated with each connection in the system that epesents the total nomalized sevice offeed to a connection duints backlogged peiods. The vitual time of a connection is defined as a non-deceasing function of time duing a system-busy peiod. When connection i is backlogged, its vitual time inceases exactly by the nomalized sevice it eceives. That is, fo any inteval of time (; t] that connection i is continuously backlogged, v i (t) v i ( ) = W i(; t) : whee, W i (; t) denotes the amount of sevice eceived by session i duing the inteval (; t] and is the scheduling weight allocated to connection i. When the connection is idle, howeve, the vitual time of the connection is updated though the system vitual time. The system vitual time is a non-deceasing function of time that keeps tack of the pogess of the total wok done by the schedule. When an idle session i becomes backlogged at time t, its vitual time v i (t) is set to v(t) to account fo the sevice it missed. Schedules use diffeent functions to maintain the system vitual time, giving ise to widely diffeent and fainess-behavios. Exactly this flexibility of the ate-popotional seves allows a simple implementation of the system without the equiement of simulatinn paallel a fluid system. At any instant t, the schedule sevices only the subset of connections with the minimum vitual time, and each connection in this subset eceives sevice in popotion to its eseved ate. Thus, the schedule can be seen to incease the vitual times of the connections in this subset at the same ate. A packet seve appoximates this behavio by soting the packets in the ode of thei finishing vitual times unde the fluid seve, and tansmitting them in inceasing ode of the finishing vitual times. The finishing vitual time of the kth packet of a session i, aiving at the schedule at time t, is computed as F k i = min(fi k 1 ; v(t)) + Lk i ; (1) whee L k i is the length of the packet and the allocated ate of session i. That is, the finishing vitual time is computed by adding the sevice time of the packet at the allocated ate to its stating vitual time. The latte, in tun, is the minimum of the finishing vitual time of the pevious packet eceived fom session i, and the cuent system vitual time. At the time that a connection becomes backlogged, its vitual time is updated based on the system vitual time function that keeps tack of the pogess of the total wok done by the schedule. When an idle session i becomes backlogged at time t, its vitual time v i (t) is set as v i (t) = max(v i (t ); v(t)); to account fo the sevice it missed. Rate-Popotional Seves may use a wide ange of functions to maintain the system vitual time, but the function chosen must satisfy two fundamental popeties: Fist, duing any inteval (t 1 ; t 2 ] within a system-busy peiod, the system vitual time function must be inceased with a ate of at least one, that is, v(t 2 ) v(t 1 ) t 2 t 1 : (2) Second, the system vitual time function must neve exceed the vitual time of any backlogged connection. These two conditions ae sufficient to achieve a bound equal to that of. Shape Di(t1,t2) Si(t1,t2) B. State Dependent Fai Queueing State Dependent Schedule Rate Popotional Schedule Fig. 1. Scheduling System Zi(t1,t2) Ri(ti,t2) + Wi(t1,t2) An illustation of the logical stuctue of ou system is pesented in Figue 1. Initially, all aiving packets ente pe-connection queues in a shape device. A packet is consideed as having aived to the system only when the last bit of the packet aives to the system. Similaly, packet sevice is consideed to be completed only when the last bit of the packet has been seved. The shape eleases packets to the schedule with a ate exactly equal to the allocated. We denote by S i (; t) the sevice offeed by the shape to connection i duing an inteval of time (; t]. Packets ae tansfeed fom the shape to the the schedule with infinite capacity. Let R i (; t) denote the sevice offeed by the RPS schedule. Packets that have not become eligible fo sevice emain in the coesponding connection queue in the shape, while all the eligible packets wait fo sevice in the RPS schedule queue. Sevice is always povided fom the RPS schedule queues, as long as packets ae available thee. When all schedule queues ae empty, a state dependent schedule (SDS) is invoked and selects a packet fom the shape queues fo tansmission. The sevice offeed to a connection does not change the state of the shape. Let us denote with D i (t 1 ; t 2 ) and Z i (t 1 ; t 2 ) the amount of taffic that is fowaded and seviced by the SDS schedule espectively. If the state dependent schedule is wok-conseving, then the system is wok conseving as well. Note also, that fo evey time t, D i (t 1 ; t 2 ) = Z i (t 1 ; t 2 ). We will efe to these systems as simply FQ-SDS. An agument that can be made against the above adaptive policies, is that a geedy use will always eceive moe bandwidth. In such a case, uses may equest less bandwidth and ty to eceive moe by sending moe busty taffic. Limiting such behavio can be explicity done in many ways. The simplest is policing. In most netwoks that povide some type of guaantees, it is easonable to assume that thee is some kind of policing. So misbehaving uses will be easily isolated. If policins not a viable option, the adaptive policies can themselves be tailoed to counte geedy behavio. Instead of picking the use with maximum queue length o the minimum time to oveflow, we can, fo example, pick the second such candidate. In this case, uses that consistently oveflow the buffes cannot depend on eceiving any excess bandwidth; it will not be in thei best inteests to be excessively busty. C. Redistibuting Excess Bandwidth This section descibes the schemes that we use fo the state dependent component of ou schedules. Note that this component is used only to edistibute the excess bandwidth and does not affect the guaanteed ates. The basic idea is that the schedule, in picking connections to seve, should use infomation about the state of connections so as to minimize packet losses, educe aveage s fo some taffic classes, o shape distibutions. Simple Longest Queue Fist has been shown to minimize the oveall packet losses in a system with finite buffes [16]. Hee, the schedule always picks a packet fom the longest queue fo tansmission. Howeve, if the aival ates and guaanteed shaes ae such that the expected s ae ae not the same fo all connections then the longest queue fist scheme will always favo high (and geneally high bandwidth connections). Also, if buffe allocations ae not the same, seving the longest queue fist may lead to buffe oveflow fo connections with a low buffe o low elative bandwidth guaantee. To account fo these diffeences in buffe sizes and bandwidth equiements, we can simply weight the queue size by the allocated bandwidth to make the schedule a Longest Delay Fist schedule. The schedule then uses the excess bandwidth to ty and equalize the s of all connections in the system and minimize the packet losses if buffe sizes have been allocated popotional to bandwidth equiements. We will efe to this algoithm as FQ-LDF. Note that this still does not take into account the maximum allowed fo each connection. An enhancement to FQ-LDF is to explicitly take into account the diffeent equiements that applications may have. We would like to be able to allow low maximum s to voice souces, somewhat highe maximum s to video souces, and much highe maximum s fo data souces. Also, constant bit ate souces may equie extemely low s, even when thei bandwidth allocation is vey low. To account fo these diffeences, a

3 maximum, that is usually a function of the buffe allocated to a connection, is associated with each connection. The schedule seves connections that ae most likely to exceed thei allocated maximum. Let D i denote the maximum allocated to connection i. Let us also denote by d i = Q i (t)= the that the last packet of the queue associated with that connection at time t may see. The connection that should be seved is the one that is moe likely to oveflow and exceed its bound. Thus, connections should be seved in inceasing ode of the (D i d i ) values. The above schemes have not taken into account othe chaacteistics of the souce that may be known duing admission contol. Chaacteistics could be the maximum bust that can be tansmitted by a connection o its peak aival ate. Fo example, if souces ae leaky bucket shaped, as is the case fo souces in the Contolled Load taffic class, we could use this infomation to pedict futue aivals fom souces and calculate moe pecisely the connectionsthat ae likely to oveflow o see s highe than the allocated. In systems, whee a dual, o multiple leaky bucket is used to shape the input taffic, a simila appoach can be used. In fact, we do not equie that a connection is shaped by a leaky bucket. All we need is that thee is a function specified that povides infomation as to the maximum bust that can aive fo a connection, as well as the wost-case time that this bust may aive. The main idea consists of tacking the state of the connection based on its packet aivals. This tackins simila to the policing function that is used to dop packets fom connections when they exceed thei taffic specification. Based on the state of the connection we can estimate the wost-case aivals fo a given connection. In ode to minimize the effect of wost-case aivals fo connections, we can seve connections in a way that will minimize the pobability that they will be penalized fo such an aival patten. Note, that since the system is only distibuting the instantaneous available bandwidth in a state dependent manne and since connections ae unlikely to send thei wost case taffic at the same time, such an appoach may lead to significant impovements in maximum s o packet losses. Let B i (; t) be the function that descibes the maximum amount of taffic that can aive fo connection i duing the inteval (; t] and let A i (; t) denote the actual aivals fom session i duing the same peiod. That is, fo evey inteval (; t]: A i (; t) B i (;t) (3) Then at time t, a maximum bust equal to B i B i (; t) A i (; t) (4) may aive with the peak ate. If the queue size of connection i is equal to Q i (t) and the maximum buffes allocated to that connection ae i, then a connection may oveflow if i Q i (t) B (5) Among the connections that may oveflow, we will select to tansmit a packet fom the connection that may oveflow soone, if the maximum bust aives with the peak ate allocated to that connection. O, j = min k: k Q k(t)bk ( B k P k ) (6) whee P i is the peak ate allocated to connection i. A. Peliminaies III. ANALYSIS Definition 1: A system busy peiod is a maximal inteval of time duing which the seve is neve idle. Duing a system busy peiod the seve is always tansmitting packets. Definition 2: A backlogged peiod fo session i is any peiod of time duing which packets belonging to that session ae continuously queued in the system. Let Q i (t) epesent the amount of session i taffic queued in the seve at time t, that is, Q i (t) = A i (0; t) W i (0; t): A connection is backlogged at time t if Q i (t) > 0. Definition 3: A session i busy peiod is a maximal inteval of time ( 1 ; 2 ] such that fo any time t 2 ( 1 ; 2 ]; packets of connection i aive with ate geate than o equal to, o, A i ( 1 ; t) (t 1 ): A session busy peiod is the maximal inteval of time duing which if the session wee seviced with exactly the guaanteed ate, it would emain continuously backlogged. It is impotant to ealize the basic distinction between a session backlogged peiod and a session busy peiod. The latte is defined only in tems of the aival function and the allocated ate. Thus, the busy peiod seves as an invaiant fo evaluating the wost-case behavio of diffeent scheduling algoithms unde the same aival patten. B. Wost-Case Pefomance In [19] it was shown that any RPS schedule has the same wost-case pefomance as a Weighted Fai Queueing schedule, and it can thus guaantee the same wost-case bounds. Howeve, the addition of the shaping mechanism and the state dependent mechanism affects the aival patten at the schedule. The intuition is that the SDS schedule is only invoked when thee is a fee bandwidth available. Wost-case pefomance usually assumes a fully utilized system. We will now show that the wost-case sevice offeed by the system as descibed above is not affected by the method by which fee bandwidth is distibuted. Ou main goal is to povide a lowe bound fo the sevice offeed by the system to a session i duing a session busy peiod. Note, that accoding to ou definition a session busy peiod is defined as the maximal inteval of time duing which the taffic aiving fo that connection is at least equal to the allocated bandwidth. Notice, howeve, that the taffic that aives to the RPS schedule can be lowe than the taffic aiving to the system. The eason is, that the connection eceives excess bandwidth fom the SDS seve. That means, that the busy peiods that the RPS schedule sees fo a connection i, may be diffeent than the connection busy peiods as seen by the system. It is easy to veify, howeve, that a busy peiod in the RPS system can continue afte the end of a busy peiod as seen by the whole system. We will fist conside the fluid modeled RPS system. Let us denote with Wi F (; t) the sevice offeed by the fluid RPS seve and with Wi P (; t) the sevice offeed by the coesponding packet-by-packet seve. Lemma 1: Let be the beginning of a busy peiod fo connection i. The wost case offeed sevice to connection i fo any time t duing the same busy peiod is bounded by Wi F (; t) (t ) The poof is in the Appendix. The basic idea is that since the shape is offeing sevice at least equal to that eseved fo connection i, the schedule will always have enough packets to sevice connection i with a ate equal to the eseved. Note, that we have made no assumption fo the SDS seve. We can thus assume that it is a packet system. In addition, the packet and fluid RPS systems see exactly the same aivals, and they ae both wok conseving systems. Thus, in both systems, the SDS seve is seving packets duing the same intevals of time. Thus, if the packet seve offes diffeent sevice than the fluid seve, this diffeence is due to the discepancy between the packet-by-packet and fluid RPS seve. But we know fom [19] that this diffeence is bounded by L i = + L max= whee L i is the maximum packet size fo connection i, L max is the maximum packet size of any connection in the seve and is the link capacity. We can thus wite: Coollay 1: The wost case offeed sevice to the packets of the j-th busy peiod of connection i that stated at time is Wi;j P (; t) (t L i Lmax ): This Coollay is a sufficient condition to pove that the system offes the same wost-case end-to-end s as unde the ate-popotional assignments [19]. C. Wost-Case Fainess Wost-case fainess was initially defined by Paekh in [17] and late expanded by Bennett and Zhang [24]. The measue is based on the wost-case fo cleaing the backlog of a session's queue. Accoding to this, fo evey session i that is continuously backlogged duing the inteval (t 1 ; t 2 ], W i (t 1 ; t 2 ) (t 2 t 1 ) C i ; (7) whee W i (t 1 ; t 2 ) epesents the sevice offeed to session i duing the inteval (t 1 ; t 2 ], and C i is a constant. We will call the smallest value of C i satisfying this inequality as the wost-case fainess index (WFI). Bennett and Zhang [24] defined a fainess paamete based on the smallest value of C i satisfying the inequality, and nomalizint to the allocated ate of the session. The wost-case fainess index measues the deviation between the sevice eceived by a session in the packet-level schedule and the sevice eceived by the same session in a coesponding fluid seve, whee the session is seviced at a ate of at each instant. Minimizing this paamete impoves the taffic mix at the output of the schedule [24]. Howeve, it should be noted that this paamete does not specify how the fee bandwidth left ove by idle sessions is distibuted acoss the active ones. Specifically, even fo the algoithms that distibute the fee bandwidth in a state dependent manne, we can show that they offe the same wost-case fainess as Wost-Case Weighted Fai Queueing [24]. The poof is in the Appendix.

4 IV. PERFORMANCE STUDIES We simulated the system shown in Figue 1 using a vaiety of souces to geneate taffic and using the diffeent schedules (FQ-LDF, FQ-, and ) that wee descibed in the pevious section. In the simulated system, each souce is connected to a leaky bucket shape. The shape is connected via an access line to the multiplexing system. Fo each connection, the schedule in the multiplexe maintains a sepaate buffe whose dain time at the guaanteed ate is equal to the connection's maximum allowed D i. The souces used ae: 1. Two-state exponential On-Off souces: We used these to emulate compessed voice souces and so used small maximum allowed s fo these souces. The mean load geneated pe connection is 1/32 of the link capacity. 2. Video teleconfeence souces: We used the DAR(1) model which is a Makov chain detemined by thee paametes: the mean, vaiance, and one fame-lag coelation. The tansition matix is computed as: P = I + (1 )Q (8) whee I is the identity matix, and each ow of Q consists of the negative binomial (o gamma) pobabilities (f 0 ; :::; f K ; F K ) whee F K = Pk>K f k and K is the peak ate. The DAR(1) model matches the autocoelation of the data ove appoximately hunded fame lags and has been shown to be a model of video teleconfeences accuate enough fo taffic studies [5]. We used video souces with a mean ate of 1.5 Mbps, a peak-to-mean atio of 5, and = : CBR souces. 4. Data souces. These wee modeled as On-Off souces whee the on-time is Paeto distibuted, and the off-time is exponential. The density of the Paeto distibution is given by f (x) = a k 1 (k 1)=x k ; x >= a; a = (k 2)=(k 1) (9) We set k = 5=2, esultinn infinite vaiance of the on-times and longange dependence in the aggegate taffic [12], [22]. The mean on-time was set to 6 packet tansmission times and the mean off-time to 200. The leaky bucket shapes wee tuned off by setting thei token ates equal to the access line ate. We fist compaed the pefomance of FQ-LDF with (which is simulated exactly) using only one type of souce. Table I shows packet loss ates fo diffeent small buffe sizes fom a simulation with 32 voice souces. All connections wee seved with equal weight and the utilization is about.98. At this high load, typically has twice the losses of FQ-LDF and edistibution of excess bandwidth to the longest queue educes the aggegate loss ate. We find simila esults of lowe loss ates fo FQ-LDF with video souces as well. With Paeto souces, the loss ates ae mostly simila fo FQ-LDF and. This can be accounted fo by the heavy-tailed distibutions of the on peiods. Fom this popety it follows that the most likely manne in which a lage bust in the aggegate ove all such souces occus, is that the bust of one souce is fa lage than the est. (Such consequences of undelyinnfinite vaiance models have been temed the Joseph and Noah effects; see [14], [15]). Most of the othe souces have non-backlogged queues and both schedules give most of the available bandwidth to the one vey long queue. Hence, and FQ-LDF tend to behave almost identically. Apat fom packet losses, we also compaed the distibution esulting fom FQ-LDF to that of. Fo the 32 voice souce simulation with lage enough buffes to avoid packet losses, Figue 2 shows the tail fequencies (the numbe of times a given is exceeded) on a log scale. The plots show that FQ-LDF needs significantly smalle playout buffes than. We see the same effect fo video souces fom the boxplots in Figue 3. The utilization is.88 and the buffes ae lage enough so that thee ae no losses. The distibution of mean s expeienced by diffeent connections with FQ-LDF and is shown in the boxplots by showing the median (the line in the middle of the box) of the means, the quatiles (the uppe and lowe edges of the box enclosing the median), and by showing the ange of values outside the quatiles by the lines extending out fom the box. The cutailment of the tail of the distibution due to the use of FQ-LDF is evident. Thus, fo simila loads FQ-LDF has a lowe playout buffe equiement than. Next, we compae the pefomance of FQ- with. FQ-LDF uses the excess bandwidth to equalize s and it woks vey well when all souces have the same allowed s. When diffeent souces have diffeent maximum allowed s, the FQ- schedule which takes s into account is expected to pefom bette than FQ-LDF and. We use 32 souces with 8 souces fom each of the types descibed befoe. CBR souces ae guaanteed thei mean ate. They geneate at most 2 back-to-back packets and have a small buffe allocation of 5 packets. Voice souces ae also guaanteed thei mean ate and have a maximum buffe allocation of 1200 packets (about 10 ms at thei guaanteed ates). Video souces ae guaanteed two times thei mean ates and have a buffe of packets. Data souces ae given a lage buffe of packets. Since they ae not sensitive, they ae guaanteed a ate below thei mean ate. Data souces ae expected to use some the excess bandwidth fom the othe classes to make up fo thei allocation below thei means. Figue 4 shows the box plots fo mean s and 99 th pecentile s fo the voice (exponential on-off), video, CBR and data classes. The schedule gives excess bandwidths to the data class in popotion to the weights. Since the data class has a vey lage allowance, the FQ- schedule gives pioity to the low classes in the distibution of excess bandwidth. Consequently, the lowe classes have vey compact distibutions wheeas the vey high data class (which also has undeallocated guaanteed bandwidth) has bette tail behavio with. The FQ- schedule achieves what one would want in this mixed scenaio. Wost case bounds ae always met even fo data (if it is given appopiate guaanteed ates). The lowe classes have vey small tails and hence need dastically lowe playout buffes than when using. Finally, we study the situation whee we have the same mix of souces as befoe but thei guaanteed ates ae diffeent even within a souce class. We delibeately set D i s to be below what they would be if allocation was stictly based on deteministic bounds, i.e, we assume that some statistical multiplexing will happen. The question then is how well, which ties to pevent buffe oveflows and minimize s, compaes with. The esults ae shown in Figue 5. We get vey low aveage s fo eal-time souces, in compaison to the s expeienced unde, even when thei bandwidth allocations ae vey low. This is because the D i s ae set low fo eal-time souces, and some of the excess bandwidth which would have gone to data souces unde the scheme is now used to futhe educe the s of eal time souces. The diffeences in s between and can be vey high as is seen by compaing s fo the voice class. Note that if data taffic losses must be pevented, this can be done by giving those connections an appopiate weight. V. DISCUSSION AND CONCLUDING REMARKS The main contibution of this pape is the modified fai queueing poposal that edistibutes excess bandwidth in a state dependent manne. The dawback of GPS that all fai queueing systems inheit in thei close emulation of GPS is that GPS seveely esticts state-dependent bandwidth shaing. The only statedependency in GPS is in the numbe of backlogged connections. Thee is no futhe latitude and shains detemined by the guaanteed ates which ae set based on long tem needs of the connections. This estiction on bandwidth shains moe stingent than that necessay to peseve a key popety of fai queueing, the ability to guaantee bounds fo leaky bucket contolled taffic souces. Consequently, fo many applications, thee is no need fo fai queueing systems to emulate the potentially suboptimal excess bandwidth shaing of GPS. With this in mind, we popose modified fai queueing schemes that emulate fai queueing only in that they povide ate guaantees and behave like fai queueing when all connections ae backlogged. This is sufficient to guaantee wost case bounds using ou scheme. Futhemoe, the schemes we popose ae wost case fai. Ou schemes do not emulate GPS's method fo excess bandwidth edistibution. Instead, we poposed thee methods (the longest fist, longest time to oveflow, and longest time to oveflow takinnto account leaky bucket states) fo edistibuting the excess bandwidth. Simulations show that the FQ-LDF and FQ- policies pefom vey well in compaison to. The adaptive bandwidth edistibution educes packet losses and makes the distibutions less skewed. The contol of s, in compaison to, is vey significant when vey sensitive taffic, such as CBR, is mixed with taffic with moe laxity in s. The low class has vey shot tails in compaison to and hence would need much lowe playout buffes. Analytical explanation of ou simulation esults emains a challenging poblem. Unless the fainess equiement of GPS is equied fo policy easons, fai queueing systems can impove thei pefomance by adaptive edistibution of excess bandwidth without losing thei wost case fainess popety o thei ability to guaantee wost case bounds. Simila extensions of ou scheme can be easily applied to othe types of schedules based eithe on Ealiest Deadline Fist [8], o on Sevice Cuves [2]. Note, that both of these appoaches define how packets must be seved in the wost-case so as not to violate some specific bounds. Howeve, thee is no equiement as to how packets can be seved if thee is available fee bandwidth in ode to minimize the tails of the distibutions. Actually, fo both of the above appoaches thee is no ham if packets ae tansmitted ealie than expected. Ou adaptive bandwidth edistibution techniques only define the method fo selecting packets when excess bandwidth is available.

5 Buffe size in pkts/conn Cell Loss Ratio: 13.18e e e e e e e e-4 Cell Loss Ratio:FQ-LDF 8.3e e e e e e-4 < 10 6 < 10 6 TABLE I C ELL L OSS R ATIOS FOR 32 E XPONENTIAL O N -O FF S OURCES WITH S HORT B UFFERS -1.5 log feq(>d) Log tail feq. fo 32 exponential On-Off souces, log feq(>d) Log tail feq. fo 32 exponential On-Off souces, LDF d d Fig. 2. Log tail fequencies fo 32 exponential On-Off souces, accoding to sevice discipline (FQ-LDF o ). 28 simulated video souces: 99% simulated video souces: mean LDF LDF Queueing Discipline Queueing Discipline Fig. 3. Boxplots of mean (left plot) and 99th pecentile (ight plot) accoding to excess sevice discipline (in each plot: FQ-LDF on left, on ight) fo 28 simulated video souces. Key: shaded between fist and thid quatiles; clea ba is median; boundaies at minimum and maximum. R EFERENCES [1] [2] [3] D. Clak, S. Shenke, L. Zhang, Suppoting Real-Time Applications in an Integated Sevices Packet Netwok: Achitectue and Mechanism, Poceedings of ACM SIGCOMM 1992, pp R. Cuz, Quality of sevice guaantees in vitual cicuit switched netwoks, IEEE Jounal on Selected Aeas In Communications, vol. 13, pp , August A. Demes, S. Keshav, S. Shenke, Analysis and Simulation of a Fai Queueing Algoithm Intenetwoking: Reseach and Expeience, pp, 3- [4] [5] [6] 26, vol. 1, N. Duffield, T.V. Lakshman, D. Stiliadis On Adaptive Bandwidth Shaing with Rate Guaantees, Bell Laboatoies Technical Memoandum TM. A. Elwalid, D. Heyman, T. V. Lakshman, D. Mita, A. Weiss, Fundamental Bounds and Appoximations fo ATM Multiplexes with Applications to Video Teleconfeencing, IEEE Jounal on Selected Aeas in Communications: Special Issue on Fundamental Advances in Netwoking, pp , August H. Gail, G. Gove, R. Guein, S. Hantle, Z. Rosbeg, M. Sidi, Buffe

6 8 Exp. On-Off in 32 mixed souces: mean 8 Exponential On-Off in 32 mixed souces: 99% Sim. video in 32 mixed souces: mean 8 Sim. video in 32 mixed souces: 99% CBR in 32 mixed souces: mean 8 CBR in 32 mixed souces: 99% Paeto OnOff in 32 mixed souces: mean 8 Paeto On-Off in 32 mixed souces: 99% Fig. 4. Boxplots of mean (left column) and 99 th pecentile (ight column) accoding to excess sevice discipline (in each plot: FQ- on left, on ight), displayed by ow fo each of following 2 classes of souces out of 4 classes being multiplexed: (top to bottom) Exponential On-Off, Simulated Video, CBR, Paeto On-Off. 32 souces total. Key: shaded between fist and thid quatiles; clea ba is median; boundaies at minimum and maximum.

7 8 Exp. On-Off in 32 mixed souces: mean 8 Exponential On-Off in 32 mixed souces: maximum Sim. video in 32 mixed souces: mean 8 Sim. video in 32 mixed souces: max CBR in 32 mixed souces: mean 8 CBR in 32 mixed souces: max Paeto OnOff in 32 mixed souces: mean Paeto OnOff in 32 mixed souces: max Fig. 5. Boxplots of mean (left column) and maximum (ight column) accoding to excess sevice discipline (in each plot: on left, on ight), displayed by ow fo each of 4 classes of souces: (top to bottom) Exponential On-Off, Simulated Video, CBR, Paeto On-Off. 32 souces total. Key: shaded between fist and thid quatiles; clea ba is median; boundaies at minimum and maximum.

8 size equiements unde longest queue fist, Poceedings IFIP' 92, 1992 [7] L. Geogiadis, R. Guein and A. Paekh Optimal multiplexing on a single link: Delay and buffe equiements, in Pocessdings of IEEE INFO- COM' 94, pp [8] L. Geogiadis, R. Guein, V. Peis and K.N Sivaajan Efficient netwok QOS povisioning based on pe-node taffic shaping, in Poceedings of IEEE INFOCOM'96, pp [9] A. Biman, H.R. Gail, S.L. Hantle and Z. Rosbeg, An optimal sevice policy fo buffe systems, Jounal of the Association fo Computing Machiney, pp vol. 42, no. 3, May [10] S. J. Golestani, A Self-Clocked Fai Queueing Scheme fo Boadband Applications, Poceedings of INFOCOM' 94, pp , [11] P. Goyal, H.M. Vin and H. Chen, Stat-time Fai Queing: A Scheduling Algoithm fo Integated Sevices Packet Switching Netwoks, Poceedings ACM SIGCOMM' 96, pp [12] K. R. Kishnan, The Hust Paamete of Non-Makovian On-Off Taffic Souces, Intenal Bellcoe Repot, [13] D. S. Lee, Weighted Longest Queue Fist: An Adaptive Scheduling Discipline fo ATM Netwoks, Poceedings INFOCOM' 97, [14] B.B. Mandelbot and J.W. Van Ness (1968). Factional Bownian Motions, Factional Noises and Applications. SIAM Review, [15] B.B. Mandelbot and J.R. Wallis (1968). Noah, Joseph, and Opeational Hydology. Wate Resou. Res., [16] S. Panwa, D. Towsley, J. Wolf, Optimal scheduling policies fo a class of queues with custome deadlines to the beginning of sevices, Jounal of ACM, vol. 35, no. 4, pp , [17] A. K. Paekh R. G. Gallage, A Genealized Pocesso Shaing Appoach to Flow Contol in Integated Sevices Netwoks: The Multiple Node Case, Poceedings of INFOCOM' 93, pp , [18] J.L. Rexfod, A.G. Geenbeg, F.G Bonomi, Hadwae Efficient Fai Queueing Achitectues fo High-Speed Netwoks, Poceedings of IN- FOCOM' 96, pp , [19] D. Stiliadis, A. Vama, Latency-Rate Seves: A Geneal Model fo Analysis of Taffic Scheduling Algoithms, Poceedingsof INFOCOM' 96, pp , [20] D. Stiliadis and A. Vama, Design and analysis of Fame-based Fai Queueing: A New Taffic Scheduling Algoithm fo Packet-Switched Netwoks, Poceedings of ACM SIGMETRICS ' 96, pp , May [21] S. Sui, G. Vaghese and G. Chandamenon, Leap Fowad Vitual Clock: A New Fai Queuing Scheme with Guaanteed Delays and Thoughput Fainess Poceedings INFOCOM' 97, [22] W. Willinge, M.S. Taqqu, R. Sheman, D.V. Wilson, Self-Similaity Though High-Vaiability: Statistical Analysis of Ethenet LAN Taffic at the Souce Level, Poceedings of ACM SIGCOMM [23] D. Yates, J. Kuose, D. Towsley, M.G. Hluchyj, On pe-session end-toend distibutions and the call admission poblem fo eal-time applications with QoS equiements, Poceedings of ACM SIGCOMM 1993, pp.2-12, Septembe [24] J.C.R. Bennett, H. Zhang, WF 2 Q: Wost-case Fai Weighted Fai Queueing, Poceedings of INFOCOM' 96, pp , APPENDIX WORST-CASE FAIRNESS OF FQ-SDS Let us denote by t k, the times that a packet is eleased by the shape to the RPS schedule. In ode to be able to calculate an exact bound fo the wostcase fainess we need to account fo the fluid seve that the RPS system is simulating. Let us assume that Wi F (; t) is the sevice offeed by this fluid to seve to connection i duing an inteval (; t]. We will fist pove the following lemma: Lemma 2: Let Wi F (; t) be the sevice eceived by session i in the fluid RPS system duing an inteval (; t] in which it is continuously backlogged. If at time a packet was eleased fom the shape then, W F i (; t) (t ): Poof: We know that since the beginning of the system busy peiod the shape eleases packets with ate equal to. We also know that the connection vitual time is inceasing by the nomalized sevice offeed to the connection, when the connection is not idle. Othewise, it inceases based on the system vitual time. Let us denote with t 0 <, the last time that connection vitual time was updated by the use of the system vitual time. Thus, fo time t 0, v(t 0 ) = v i (t 0 ) (10) Obviously, this happened afte an idle peiod of connection, whee the connection vitual time has emained behind the system vitual time. This update was initiated by a packet aival. That means that at time t 0 a packet was eleased fom the shape. We know that the total aivals to the RPS schedule since time t 0 ae bounded by S i (t 0 ; ) ( t 0 ) (11) We also know that since time t 0 the connection vitual time has only inceased by the nomalized sevice offeed to it. Thus, v i ( ) v i (t 0 ) = W F i (t0 ; ) (12) S i(t 0 ; ) Fom Eq.11 (13) ( t 0 ) (14) We also know that the system vitual time is inceasing with a ate at least linea to the eal time, o v( ) v(t 0 ) + ( t 0 ): (15) Fom Eq. (10),(14) and (15) we can conclude that v i ( ) v( ) (16) Thus when a packet is eleased fom the shape at time, the connection vitual time v i ( ) will be updated to a value no lage to that of the system vitual time. Duing the inteval (; t], the connection vitual time is inceasing by the nomalized sevice offeed to that connection. Thus, Theefoe, v i (t) v i ( ) = W i F (; t) : (17) W F i (; t) = (v i (t) v i ( )): (18) Since v i (t) v(t) and v i ( ) v( ), we can e-wite this as W F i (; t) (v(t) v( )): (19) Duing the inteval (; t], the system potential is inceasing with a ate at least equal to that of eal time. That is, v(t) v( ) t : (20) Fom equations (19) and (20), we have W F i (; t) (t ): 2 We can now evaluate the wost-case fainess index of the packet-based system. Theoem 1: The queueing of a session-i packet aiving at time in a packet FQ-SDS system is uppe-bounded by d P i ( ) QP i ( ) + L i ( 1 1 ); (21) whee Q P i ( ) is the backlog of session i at time, L i the maximum packet size of session i, and L max the maximum packet size acoss all sessions. Poof: Let us denote with t k the time at which the k-th packet of session i is eleased fom the shape to the schedule. Note that this time is identical in both the fluid and packet vesions consideed. We will pove the theoem fo each instant t in the inteval (t k ; t k+1 ]. Let, Q F i (t) be the backlon the fluid seve at time t and Q P i (t) that in the packet-by-packet seve. By Lemma 2, the sevice offeed to session i afte time t k is with a ate at least equal to, and without any latency. Let us assume that the backlog of connection i is cleaed at some time t in the fluid seve. Then we can wite Q F i (t) = W F i (t; t ) = W F i (t; t k+1) + W F i (t k+1; t ) W F i (t; t k+1) + max( (t t k+1 ); 0) (22) We assume that if t < t k+1, then Wi F (t k+1; t ) = 0. We can e-wite Eq. (22) as t t k+1 + QF i (t) W F i (t; t k+1) (t k+1 t) + t + QF i (t) W F i (t; t k+1) : (23)

9 The above equation holds fo the fluid seve. The time to clea the backlog at time t in the fluid seve is thus given by d F i (t) = t t. The packet seve may lag the fluid seve at most by the tansmission time of one packet [19]. Theefoe, the time d v i (t) needed to clea the backlog of connection i in the packet-by-packet seve afte time t is bounded by d v i (t) df Lmax i (t) + Q F i (t) W i F (t; t k+1) + (t k+1 t) (24) To evaluate the above expession, we will conside two sepaate cases. Case 1: Q v i (t) QF i (t). Let Lk i denote the size of the kth packet of session i. Then, W F i (t; t k+1) = Fom Eq. (24) and (25) we can wite L k i W F i (t k; t) L k i min(l k i ; (t t k)): (25) d v i (t) (t k+1 t) + QF i (t) Lk i + min(l k i g ; (t t k)) : (26) i By using the hypothesis that Q P i (t) QF i (t), this becomes Thus, Eq. (31) becomes d P i (t) (t k+1 t) + QP i (t) (L i + W ) + min(l k i + W; (t t k)) : (34) The ight-hand side of the above equation is maximized when L i + W = (t t k ). Theefoe, d P i (t) t k+1 t k Lk i + W L k i Q P i (t) Q P i (t) Lk i + W + QP i (t) + QP i (t) + L k i ( 1 1 ) + L i ( 1 1 ): (35) This concludes the poof of Theoem 1. Note that the wost-case fainess index of (L max=) + L i = L i = given by the theoem is identical to that of WF2Q [24]. d P i (t) (t k+1 t) + QP i (t) Lk i + min(lk i ; (t t k)) : (27) The ight-hand side of the above equation is maximized when L k i = (t t k). Thus, d P i (t) (t k+1 t k Lk i ) + QP i (t) Lk i Q P i (t) Q P i (t) + L k i ( 1 1 ) + Lk i + L i ( 1 1 ) : (28) Case 2: Q F i (t) > QP i (t). In this case, let Q = QF i (t) QP i (t). We can e-wite Eq.(24) as d P i (t) (t k+1 t) + QP i (t) + Q W F i (t; t k+1) : (29) At time t, the packet-by-packet seve has offeed moe sevice to connection i than the fluid seve. Howeve, packet k + 1 has not yet been eleased by the shape. Thus, the additional sevice that fluid seve has to offe until time t k+1 is equal to the additional sevice that the packet-by-packet seve has offeed until time t, plus the sevice it will offe until time t k+1. That is, Fom Eq. (29) and (30), W F i (t; t k+1) = Q + W P i (t; t k+1): (30) d P i (t) (t k+1 t) + QP i (t) W P i (t; t k+1) : (31) Notice that at time t k, the packet-by-packet seve can only be behind the fluid seve. Let us assume without loss of geneality, that W F i (0; t k) = W P i (0; t k) + W: (32) Fo the sevice offeed to connection i by the packet-by-packet seve afte time t, we can wite W P i (t; t k+1) = W P i (t k; t k+1 ) W P i (t k; t) L k i + W W P i (t k; t) L k i + W min(l i + W; (t t k )): (33)