Session Dynamic Analysis of Communication Networks

Analysis of Burstiness and Jitter in Real-TimeCommunicationsZheng WangJon CrowcroftDepartment of Computer Science,University College LondonGower Street,London WC1E6BT,United Kingdomemail:z.wang@,j.crowcroft@tel:+440713877050ext7214,fax:+440713871397Session:Dynamic Analysis of Communication NetworksSponsoring Committee:the Communications Switching CommitteeABSTRACTThis paper examines burstiness and jitter in real-time communications.In this paper,we make so assumptions about the arrival patterns of the incoming traffic but characterize trafficwith two parameters.We assume the synchronization process is adaptive so theflow can bedivided into smaller synchronization units,and defined jitter using the delay experienced by thefirst packet in a synchronization unit as the target delay.We presente the results on the relation-ship between burstiness and jitter,and on the upper bounds of burstiness and jitter.1.IntroductionOne of the most interesting parameters to consider when supporting real-time communications is the delay jitter.When a stream of packets traverse the network,each packet may experience different delay;this variation in delay is often called delay jitter or just jitter.For traditional non-real-time applications,such asfile transfer and telnet,incoming packets can usually be consumed immediately by the applications when they are received correctly and in sequence.The variation in delay,or jitter,often has little real impact on these applications as the performance is largely dominated by the average delay that the packets have experienced.Real-time applications such as audio and video,however,have to faithfully recreate the original data stream at the receiver by playing back the data after afixed delay offset from the original departure time.The packets arriving with shorter delay may have to wait in the receiver’s buffer in order for the packets with longer delay to arrive before their playout time.Thus,the synchronization process is usually determined by the packets with worst delay.The main objective of this paper is to examine the relationship between burstiness and jitter.Jitter is a natural result of buffering in packet-switched networks.Whenever packets are buffered,the information about their inter-packet timing is lost.The regenerated packet stream is often smoothed as the result of buffering.Buffering can be caused by the contention for resources among differentflows but fair queuing algorithms such as FairQueuing,Virtual Clock,HRR and PGPS can reduce such buffering to the minimum for well-behaved sources[1-4].However,when the bandwidth allocated to a particularflow is less than the peak rate of thatflow,the packets at the peak of theflowfluctuation are still buffered and smoothed.This type of buffering is determined by the burstiness in the trafficflow itself.Generally speaking,the jitter increases when the traffic becomes more bursty.Exact analysis of such systems based on realistic stochastic traffic models is often intractable.Motivated by the fact that the synchronization process is dominated by the packets with worst case delay,the approach we take is to use non-probabilistic worst case analysis.Such approaches have been used in the performance analysis of various queuing algorithms and congestion schemes[5-8],and have yielded considerable insight.The rest of the paper is organized as follows.In Section2and3,we present the synchronization model and a characterization of traffic.In Section4and5,we present the definitions of burstiness and jitter.In Section6,we present the main results.Section7concludes the paper.2.Synchronization ModelTo recreate the original data stream at the receiver,the application has to play back the data after a delay offset from the original departure time.The delay offset represents the estimated upper bound of delay a packet may experience.There are two basic models for determining this delay offset:static and adaptive.In the static model, a delay offset is estimated before theflow is established and isfixed for the lifetime of a session.The application plays back the data after thisfixed delay offset from the original departure time,regardless of the actual delay dur-ing the session.In the adaptive model,the delay offset is adjustable by the application.The receiver measures the current delay experienced by the packets and modifies the delay offset accordingly.Thus,the actual delay offset used by the application depends on the network conditions.The recent experience with the VT and Vat packet voice protocols has shown that adjusting the delay offset is feasible and the impacts on the performance are often imperceptible if the adjustment is carried out during silent periods[9].Delay OffsetsSynchronization UnitsSenderReceiverFigure1:Adaptive SynchronizationIn this paper,we assume the application can adjust the offset delay during the session and the synchronization is required only between packets within a synchronization unit.A synchronization unit is defined as a group of packets that share a common fixed delay offset [9].The delay offset may be adjusted at the beginning of a syn-chronization unit but should remain unchanged until the end of this synchronization unit.The boundary of a syn-chronization unit is application-dependent,for example,the most common synchronization units are talkspurts for voice and frames for video.The assumption of adaptive synchronization allows us to divide the packet stream into smaller manageable units,and to consider the synchronization process only within a single synchronization unit.3.A Characterization of TrafficWe assume that all packets are of the same size and the packet header carries an one-bit indicator,such as the s bit in the audio/video transport protocol [10],which marks the boundary of the synchronization units.We make no assumptions about the arrival pattern of the incoming traffic.The traffic in a synchronization unit is character-ized by a pair of parameters (ρmax ,ρmin ).Suppose S is a synchronization unit with M packets and a i is the arrival time of the i th packet.Define A (t 1,t 2)to be the number of packets arrived in the interval [t 1,t 2),then A (a 1,a m )is the number of packets arrived before t =a m :A (a 1,a m )=m −1(1)Let X m =a m −a 1A (a 1,a m ) and Y m =a m −a m −11 (m =2,3,...,M ).We define ρmin and ρmax as follows:ρmin =min [X 2,X 3,...,X M ]ρmax =max [Y 2,Y 3,...,Y M ]We say synchronization unit S conforms to (ρmax ,ρmin ),or S ∼(ρmax ,ρmin ),if X m ≥ρmin and Y m ≤ρmax ,for m =2,3,...,M .A (a 1,a m )slope of ρmaxslope ρmina M a m a 1t Figure 2:The Interpretation of ρmax and ρminFigure 2shows how the traffic in a synchronization unit S is characterized by the two parameters.Note that traffic arrival curve is confined between the two lines with slope ρmin and ρmax .We can interpret ρmax to be the peak rate of the incoming traffic and ρmin to be the highest rate at which a server is never idle.The (ρmax ,ρmin )pair is a generalization of characteristics in variable rate audio and video traffic.In digital audio with silence suppression,there is a clear pattern of busy period and silent period [13].A synchronization unit starts with a busy period followed by a silent period.In such cases,ρmax equals to the sampling rate during the busy period and ρmin equals to the average rate of the whole unit.Similar patterns are also observed in variable rate video [14].Let us now examine some properties of synchronization units which conform (ρmax ,ρmin ).Lemma 1If S ∼(ρmax ,ρmin ),for m =2,3,...,M,a m ≤a 1+ρminA (a 1,a m ) Proof.By definition,a m −a 1A (a 1,a m ) ≥ρmin .Thus,we have a m ≤a 1+ρminA (a 1,a m ) Lemma 2If S ∼(ρmax ,ρmin ),for m =2,3,...,M,a m ≥a 1+ρmaxA (a 1,a m ) Proof.By definition,a m −a m −11 ≤ρmax .Thus,we have a m ≥a m −1+ρmax1 (2)From (2),we have a m −1≥a m −2+ρmax 1 .Applying (2)recursively,we have a m ≥a 1+ρmaxA (a 1,a m ) .Note that Lemma 1and 2can be combined asa 1+ρmax m −1 ≤a m ≤a 1+ρmin m −1 (3)Equation (3)implies that for each packet,there is an interval in which the packet has to arrive.We refer to theinterval a 1+ρmax m −1 ,a 1+ρmin m −1as the m th packet’s arrival interval .The length of this interval (ρmin m −1 −ρmaxm −1 ),as we will show later,is the upper bound of the burstiness.4.BurstinessThe concept of burstiness originates from the difference in the way in which servers and traffic sources operate.The servers usually process packets one by one at a constant rate while traffic sources often generate pockets in bursts or at a variable rate.This mismatch results in queuing at the servers.A number of measures,such as the peak to average rate ratio,token bucket size and busy to idle period ratio,have been used in the litera-ture to characterize the burstiness of the traffic for different purposes [4,5,12].In this paper,we present a more fine graded definition of burstiness which can reflect the burstiness of individual packets.We define the burstiness of the m th packet asb m =a 1+ρminA (a 1,a m ) −a m From Lemma 1we have b m ≥0.Note that b m is the distance between the actual arrival time and the right edge of the m th packet’s arrival interval.Another useful interpretation is as follows:If S is served at a rate ρmin ,a 1+ρminA (a 1,a m ) is then the time when the server starts to process m th packet.Thus,b m is also the amount of time that m th packet has to wait in a queue before it can be served when the traffic is served at a fixed rate ρmin .Wecan view a 1+ρminA (a 1,a m ) as the expected arrival time and b m as the difference between the actual arrival time and the expected arrival time.In this sense,the expected arrival time is similar to the Virtual Clock [3].A packet arrived at its expected arrival time experiences no queuing delay.Lemma 3If S ∼(ρmax ,ρmin ),for m =2,3,...,M ,b m ≤ρmin (m −1) (1−ρmaxρmin )Proof.By definition b m =a 1+ρmin m −1 −a m .From Lemma 1,a m ≥a 1+ρmaxA (a 1,a m ) .Thus,b m ≤(m −1)(ρmin 1 −ρmax 1 )=ρmin (m −1) (1−ρmaxρmin ) Lemma 3shows that the burstiness of each packet is bounded by the length of its arrival interval.Figure 3illustrates the upper bound of b m in terms of ρmaxρmin .Note that the upper bound decreases when ρmin and ρmax becomes closer to each other.This implies the minimum and maximum rate ratio can be used as a measure of burstiness.ρminm −1 ρmaxρmin 10Figure 3:Upper Bound of Burstiness5.JitterJitter is usually defined as the difference between the delay a packet has experienced and the target delay for the flow [11].Since the synchronization is assumed to be adaptive,the target delay can be adjusted at the begin-ning of a synchronization unit.In this paper,we use the delay experienced by the first packet of a synchronization unit as the target delay.Let d m be the time when the m th packet departs from the network,then jitter is defined as follows:j m =(d m −a m )−(d 1−a 1)The definition can also be written as:j m =(d m −d 1)−(a m −a 1)Thus,the jitter can also be viewed as the difference between a packet’s relative distance to the first packet at the source and at the destination.The synchronization process is determined by j m .A positive j m implies that m th packet has experienced longer delay than the first packet,and j m represents the amount of time the first packet has to wait in the buffer in order for the m th packet to arrive before the synchronization deadline.To ensure that all packets in a synchroniza-tion unit arrive before their playout deadline,the first packet has to wait for the packet which experiences the larg-est jitter.Define J max =max [j 1,j 2,...,j M ].The first packet of a synchronization unit should be played back at t =d 1+J max .All packets that has a smaller jitter than J max have to be buffered.6.Main ResultsIn this section,we examine the relationship between burstiness and jitter.We assume that the server is work-conserving (i.e.the server is never idle when there is backlog)and the flow is served at rate µ≥ρmin .Let d m be the time when the m th packet departs from the server.We say a packet has arrived or departed only when its lastbit has arrived or departed.We haved m =max [d m −1,a m ]+µ1 (4)for m =2,3,...,M .The delay experienced by the first packet depends on the backlog of packets left over from the previous synchronization unit.Suppose D is the time that the first packet has to wait before being served.We haved 1=D +µ1 +a 1(5)When a m ≥d m −1,there is a gap between the departure time of the (m −1)th packet and the arrival time of m th packet.Define this gap as i m ,so we havei m =a m −d m −1a m >d m −1i m =0a m ≤d m −1When the flow is served at a fixed rate,i m can be interpreted as the length of the interval during which the server is idle.Let I (i ,j )to be the sum of i m between the i th packet and the j th packet.Thus,I (i ,j )=n =i Σji n .Lemma 4If the input traffic conforms to (ρmax ,ρmin )and µ≥ρmin ,then the output traffic conforms to (µ,ρmin ).Proof.1)We first show d m −d m −11 ≤µ.If a m ≥d m −1,from (4)we haved m =a m +µ1 ≥d m −1+µ1 If a m <d m −1,from (4)we haved m =d m −1+µ1 Thus,d m −d m −11 ≤µ.2)we now show ρmin ≤d m −d 1m −1 .We proceed by induction on the index of m .First step:We show ρmin ≤d 2−d 11 .If a 2≤d 1,d 2=d 1+µ1 ≤d 1+ρmin1 If a 2>d 1,d 2=a 2+µ1 From Lemma 1,we havea 2≤a 1+ρmin1 Thus,d 2≤a 1+ρmin 1 +µ1 From (5),we also haved 1=a 1+µ1 +D ≥a 1+µ1 Therefore,we have d 2≤d 1+ρmin1 .Inductive step:We show d m +1≤d 1+ρmin m .if d m ≤d 1+ρminm −1 .If a m +1≤d m ,d m +1=d m +µ1 ≤d 1+ρmin m −1 +µ1 ≤d 1+ρminm If a m +1>d m ,d m +1=a m +1+µ1 From Lemma 1,we havea m +1≤a 1+ρminm Thus,d m +1≤a 1+ρmin m +µ1 From (5),we also haved 1=a 1+µ1 +D ≥a 1+µ1 Therefore,we haved m +1≤d 1+ρminm So we have proven d m ≤d 1+ρmin m −1 .This implies ρmin ≤d m −d 1m −1 .A simple consequence of this Lemma is that the burstiness of the output traffic can be calculated in the sameway as the input traffic,i.e.b m out =d 1+ρmin m −1 −d m .We also have d m ≤d 1+ρminm −1 thus j m ≥0.Theorem 1Suppose the input traffic conforms (ρmax ,ρmin )and µ≥ρmin .Let b m in is the burstiness of the mth packet in the input traffic and b m out is the burstiness of the mth packet in the output traffic.For m =2,3,...,M,b m in =j m +b mout Proof.From Lemma 4,we know the output traffic conforms to (µ,ρmin ).By definition,we haveb m in =a 1+ρminm −1 −a m b m out =d 1+ρminm −1 −d m Thus,we haveb m in −b m out =(d m −d 1)−(a m −a 1)=j mSo the Theorem is proven.Theorem 1is a very general result.It reveals the simple but interesting relationship between burstiness and jitter.Theorem 1indicates that any change in burstiness is transformed into jitter.We now look at two simple cases where the calculation of b m is straightforward.Lemma 5For m =2,3,...,M,d m ≥a m +µ1 Proof.If a m ≥d m −1,d m =a m +µ1 .If a m <d m −1,d m =d m −1+µ1 ≥a m +µ1 .Theorem 2For m =2,3,...,M ,j m ≥−DProof.Following Lemma 5and (5),we havej m =(d m −a m )−(d 1−a 1)≥−DNote that j m ≥0when D =0.From Theorem 1,we have b m in ≥b m out .Thus,when there is no backlog left over from previous synchronization units,the burstiness never increases.Theorem 3If S ∼(ρmin ,ρmax )is served at a fixed rate µ=ρmin ,j m =b min Proof.From Lemma 1,a 2≤a 1+ρmin 1 ≤d 1.Also,d 1=D +µ1 +a 1,Thus,d 2=d 1+ρmin 1 =a 1+ρmin 2 +D Applying Lemma 1again,a 3≤a 2+ρmin2 ≤d 2Similarly,we haved m =d 1+ρminm −1 b m out =d 1+ρminm −1 −d m =0By Theorem 1,j m =b m in .Theorem 3indicates that the traffic is completely smoothed when µ=ρmin and all burstiness is transformed into jitter.Theorem 4If S ∼(ρmin ,ρmax )is served at a rate µ≥ρmax and D =0,j m =0Proof.From (5),d 1=a 1+µ1 .Note a 2≥a 1+ρmax1 .Thus,we have a 2≥d 1.Similarly,by using (5)and (2),we get a m ≥d m −1and d m =a m +µ1 .Thus,j m =(d m −d 1)−(a m −a 1)=ρmax1 +a 1−d 1=0 This result is not surprising.When the traffic is served at its peak rate or higher,there is no buffering taking place at all.We now look at a general case in which the traffic is served at a rate µ(ρmax ≥µ≥ρmin ).We are interested in the upper bound of the jitter.Theorem 5If flow S ∼(ρmax ,ρmin )and ρmax ≥µ≥ρmin ,thenj m ≤(m −1)(µ1 −ρmax1 )Proof.1)We first showa m ≥a m −1+ρmax1 +i m (6)d m =d m −1+µ1 +i m (7)If a m >d m −1,we have i m =a m −d m −1.Thus,a m =d m −1+i m ≥a m −1+µ1 +i m ≥a m −1+ρmax1 +i m andd m =a m +µ1 =d m −1+µ1 +i m If a m <d m −1,we have i m =0.Thus,d m =d m −1+µ1 =d m −1+µ1 +i m a m ≥a m −1+ρmax 1 =a m −1+ρmax1 +i m 2)We now showa m ≥a 1+ρmaxm −1 +I (1,m )(8)d m =d 1+µm −1 +I (1,m )(9)From (6),we havea m −1≥a m −2+ρmax1 +i m −1Thus,a m ≥a m −1+ρmax 1 +i m ≥a m −2+ρmax2 +I (m −1,m )Applying (6)recursively,we havea m ≥a 1+ρmaxm −1 +I (1,m )Similarly,we can getd m ≤d 1+µm −1 +I (1,m )Note that (9)actually says that the time between the departure of the first packet and the m th packet equals to the sum of all busy period and idle period.This is consistent with the work conserving assumption.3)We now prove the Theorem.Since j m =(d m −a m )−(d 1−a 1),we havej m ≤(m −1)(µ1 −ρmax 1 )mρmaxρmin µj m Figure 4:Upper Bound of JitterFigure 4shows that the upper bound of jitter in terms of µ.When µincreases,the upper bound of jitter decreases,and it decreases more rapidly when µis close to ρmin .The upper bound of jitter is also proportional to m and converges when µincreases.It is also interesting to note that the upper bound is independent of D .Theorem 5allows us to calculate the minimum bandwidth for a given jitter constraint.Suppose J max is the maximum jitter that the receiver can deal with.The minimum bandwidth required µcan be derived fromJ max ≤(M −1)(µ1 −ρmax1 )We can getµ=J max ρmax +(M −1)(M −1)ρmax (10)Equation (10)can be used to calculate the amount of bandwidth that has to be reserved for the flow based on the size of the playout buffer available at the receiver.It may also be used by the receiver to adjust the size of playout buffer when the bottleneck bandwidth can be estimated [15,16].7.ConclusionIn this paper,we examined burstiness and jitter in real-time communications.We made so assumptions about the arrival patterns of the incoming traffic but characterized traffic with two parameters.We assumed the syn-chronization process is adaptive and defined jitter using the delay experienced by the first packet in a synchroniza-tion unit as the target delay.We presented the results on the relationship between burstiness and jitter,and on the upper bounds of burstiness and jitter.8.References[1] A.Demers,S.Keshav,and S.Shenker,"Analysis and Simulation of a Fair Queueing Algorithm",in Proc.ofACM SIGCOMM’89,1989.[2]C.Kalmanek,H.Kanakia,and S.Keshav,"Rate Controlled Servers for Very High-Speed Networks",in Proc.of IEEE GlobeCom’90,1990.[3]L.Zhang,"VirtualClock:A New Traffic Control Algorithm for Packet Switching Networks",in Proc.of ACMSIGCOMM’90,1990.[4] A.Parekh and R.Gallager,"A Generalized Processor Sharing Approach to Flow Control-The Single NodeCase",in Proc.of IEEE INFOCOM’92,1992.[5]R.Cruz,"A Calculus for Network 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