On the Relationship Between Packet Size and Router Performance for Heavy-Tailed Traffic 1

Transcription

1 On the Relationshi Between Packet Size and Router Performance for Heavy-Tailed Traffic 1 Imad Antonios antoniosi1@southernct.edu CS Deartment MO117 Southern Connecticut State University 501 Crescent St. New Haven, CT Lester Lisky lester@engr.uconn.edu CSE Deartment University of Connecticut 371 Fairfield Road, Unit 2155 Storrs, CT Abstract The roblem of characterizing the relationshi between acket size and network delay has received little attention in the field. Research in that area has been limited to either simulation studies or emirical observations that are detached from analytic traffic modeling. From a queueing viewoint, it is simle to show that these three variables are inter-related, which necessitates a more careful study. We resent a traffic model of a router fed by ON/OFF-tye sources with heavy-tailed burst sizes. The traffic model considered is consistent with the evidence that Web traffic is heavytailed. The analysis cases that are considered establish a quantitative characterization of the comlex relationshi among acket ayload and header sizes, traffic burstiness, and router queueing delay. 1 Introduction From the advent of acket-switched networks, it has been recognized that the choice of acket size bears some significance with resect to erformance metrics such as user resonse time, and acket delay. The literature includes many simulation studies that attemt to determine the otimal MTU size in an Ethernet or a wireless network (e.g. [Lett98]), while other studies have aroached the roblem more indirectly by looking at how acket fragmentation across a wide area network affects erformance (e.g. [Cárc91, Roma95]). Although queueing delay contributes to a significant ortion of end-to-end delay on the Internet, there have been no analytic studies that quantify the relationshi between acket size and erformance. Considering the size difference between an Ethernet acket and an ATM cell, where the first is roughly 28 times larger than the second, it seems sensible to tout 1 This work was artially suorted by the Deutsche Telekom for a roblem suggested by Prof. Eike Jessen of The Technische Universität München.

2 where otimality lies when it comes to acket size, and more generally how acket size affects router delay. In this aer, we resent a queueing model that allows us to exlore the relationshi between acket size and router delay. The queue at the router is fed by a set of ON/OFF-tye sources transmitting data that comes in bursts that are heavy-tailed in size. The traffic generated by the model exhibits the long-range deendence roerty of actual Web traffic, which is generally attributed to heavy-tailed transmissions [Crov96]. The remainder of the aer is structured as follows. In the next section, we lay out our modeling assumtions and discuss how the roblem can be framed as a queueing roblem. In Section 3, we resent our traffic model, and set u closedform solutions for acket delay in two secial cases. In Section 4, we resent some calculations of acket delay in terms of acket ayload and header size, as well as another arameter describing the degree of traffic burstiness. Section 5 rovides a conclusion. 2 Packet Size and Queueing Theory Since we are only interested in erformance from a queueing modeling oint of view, we emloy the terms acket and cell interchangeably to refer to an atomic information entity that is transmitted through a routing device modeled as a queueing system. At the host level, an alication submits a burst of data for transmission, which is then divided into ackets by the lowerlevel network rotocol levels. A host cycles between a eriod of time during which it transmits a burst (termed the ON eriod), and one where it s idle (the OFF eriod). This behavior is that of an ON/OFF rocess, which is the basis of the N-Burst model described later. We will make the assumtion that ackets are exonentially distributed in size. This modeling assumtion is reasonable since the erformance roerties of a queueing system with exonential service time are only marginally different from one with deterministic service time. Inherent in the design of a acket is a header, which is tyically a fixed number of bytes that carries network control

3 information. The size of the header is related to the comlexity of the associated network rotocol. The remaining art of the acket is the ayload, which contains the data intended for transmission by the alication. The size of the acket is thus the sum of the size of the header and the ayload. It can be easily inferred that if the header-to-ayload size ratio is too large, then the network becomes inefficient because it sends a significant amount of time rocessing header data. On the other hand, as the acket size gets larger, resonse times of interactive network-based alications might suffer as data transmission is delayed until a acket is filled with data. Although these intuitions are correct, they do not aint a comlete icture of how acket size influences erformance, and certainly do not allow for a quantitative analysis. We revert to queueing theory to reframe the roblem of acket size and erformance. Let us first consider an arrival stream where acket inter-arrival time is exonential, with rate λ. Assume that ackets are of size n. If this stream is fed into a router with exonential holding time, then we have the equivalent of an M/M/1 queue, whose associated steady-state system time, T n is described by: tsn T n =, (1) 1 ρ where t is the mean service time for a acket of size n, and s ρ = t λ n s is the router utilization. Now n consider the same model, excet in this case the ackets are half the size, n 2 acket arrivals is still a Poisson rocess but the arrival rate on a er-acket basis doubles ( λ 2λ ). If we assume that the service time of acket is roortional to its size, then the service time is halved ( ts t 2 n s n ), ensuring that the router utilization remains the same ( ρ = 2 λ t s 2 = λ t n s ). The n result of this change is that the system time, T n, decreases roortionally to the factor increase in acket size since the utilization rate remains the same. As acket size aroaches 0, this urorts that system time also aroaches 0. This conclusion suddenly changes if we attach a fixed-size

4 header to every acket, which can result in a very inefficient network; the network now sends a significant amount of time transmitting header information. Researchers have for long recognized that the effective utilization of the network is an imortant measure of its success, and have sought to determine a acket size that does not incur significant overhead. Now what haens when we instead grou ackets into larger ackets while maintaining the Poisson roerty? Without any header data, and alying the same analysis to (1), the conclusion here is that larger ackets yield worse erformance. The intuition of this finding is that delay statistics of the larger acket only account for the time of the last sub-acket that is rocessed. The tradeoff in this case is that as we increase the size of the acket, keeing the useful data to be transmitted constant, the cost associated with transmitting header data goes down. 3 The Traffic Model The above discussion offers a baseline analysis of the tradeoffs in acket and header sizes and router erformance, but its modeling assumtions are too simlistic to cature the behavior of realworld traffic. We therefore base our formulation on a queueing model that rovides for the generation of heavy-tailed traffic, the N-Burst model [Schw00b]. The arrival rocess of this model allows for the secification of sources that transmit heavy-tailed bursts and is reresented as an MMPP, or more generally a semi-markov (SM) rocess. The erformance characteristics of the modeled router can be obtained by solving the corresonding SM/M/1 queue using techniques develoed in [Neut81], which is described in detail in [Schw00b]. 3.1 Truncated Power-Tail Distribution At the heart of our traffic modeling is a Matrix Exonential reresentation of a ower-tail (PT) distribution. This is motivated by the discovery that if the ON time (or burst size) distribution is heavy-tailed then the long-range deendence of the measured traffic can be accounted for, as shown

5 in [Crov96, Fior98, Schw00b]). There is a good reason to believe that such functions are involved in network traffic. It has been shown in numerous laces that the file size distribution in many facilities, including those transmitted over the Internet, tend to have PT behavior [Fior98, Crov96]. In our modeling, we make use of a family of heavy-tailed distributions, called Truncated Power- Tail distributions (for a full treatment of TPT see [Grei99]), which are hase distributions [Lis92, Neut81], and thus can be used in analytic models of Markovian-like systems. The TPT distribution is governed by several arameters, one of which is the truncation arameter T that determines the range of the tail. In Figure 1, we lot on a logarithmic scale the reliability function of a TPT with different values of T. As the value of T increases, it can be seen that the exonential dro-off of the function occurs for larger values of the burst length. Figure 1: Reliability function of the TPT for different values of T on a logarithmic scale. 3.2 Overview of the N-Burst Model The N-Burst model [Schw99a-b, Schw00a-b] serves as a owerful descritor of the behavior of data traffic sources, and rovides the means for carrying out detailed erformance analysis using a linear algebraic aroach to queueing theory. The model mimics the behavior of N statistically identical

6 and indeendent hosts that intermittently ut data onto a telecommunications line leading to a router. Each of the hosts is an ON/OFF rocess reresented as an MMPP with a ower-tail ON time and an exonential OFF time. Each source transmits ackets at a rate of κ, contributing to the total acket generation rate of λ, which is simly N κ. During an ON eriod, the source transmits ackets at a rate of λ, the eak rate. The burst arameter, b, is defined to be the fraction of the time that a source is OFF, namely OFF ( ON + OFF). This arameter is instrumental in characterizing the degree of burstiness of a traffic source, in that as b aroaches 0 traffic resembles a Poisson stream, whereas when b aroaches the other limit of 1 traffic is resented for transmission in a very short eriod of time and the source is idle the majority of time (a bulk arrival). Keeing κ constant, as b goes from 0 to 1, we can observe that the acket arrival rate during an ON eriod grows unbounded. One of the most significant results of this work is the discovery of the conditions that cause blowus in acket delay at the router. Performance blowu turned out to be a function of the router utilization and the burst arameter, while its magnitude is deendent on the range of the ower-tail function and the tail exonent. In the model, the router is reresented as an exonential server with an average service rate of ν. With the arrival rocess being an MMPP, the system becomes an SM/M/1 queue, which can be solved analytically. The utilization ratio of the router, ρ, is simly N κ / ν. For a rocess with one ower-tailed source, or the 1-Burst model, a blowu oint is located where b = 1 ρ or equivalently where λ = ν. This indicates that it is sufficient for the eak rate of the source to exceed the router rate in order for a blowu to occur, irresective of the average rate at which a source is submitting ackets for transmission.

7 If there are 2 or more sources, the situation becomes more comlicated. An N-Burst rocess gives rise to N blowu oints, each with increasing delay severity as b increases. The value of λ for each blowu is given by ν ( N i) κ λ ( i N) : =. i Since κ = ( 1 b) λ and ρ = Nκ / ν, the b values of the blowu oints can be given as ν Nκ 1 ρ b( i N) : = =. (2) ν ( N i) κ 1 ( N i) ρ / N An examle of this behavior is given in Figure 2, which shows the mean acket delay for a 3-Burst rocess with different truncation arameters. It can be seen that as T grows the blowu in delay grows in severity (Note that the y-axis is on a logarithmic scale). Figure 2: Mean Packet Delay for a 3-Burst rocess. T = 1 corresonds to exonential ON times. The T = 9 curve shows 2 of the three blowu oints: b ( i 3) = {0.5,0.6,0.75}.

8 3.3 Parameter Sace Reformulation The N-Burst model considers the acket as the most rimitive modeling unit, which further needs to be reformulated in order to deal with a acket at a finer level of detail. We now reresent a acket as having the following two comonents: h = Packet header in bytes = Mean acket ayload in bytes This refinement necessitates the introduction of the random variable Y, which denotes the size of a burst in bytes. E(Y ) is then the exected number of bytes in a burst, or in other terms the mean file size submitted for transmission by the host. Two other indeendent arameters are introduced to deal with the distinction between useful work and overhead as follows: ν = Router service rate in bytes er second λ = Maximal (eak) source transmission rate in bytes er second Let L be the random variable denoting the number of ackets in a burst. Then the mean number of ackets in a burst can now be exressed as follows: n = E( L) = E( Y ). The rate at which ackets can be rocessed by the router is: ν ν = + h The arameters N, OFF, λ, and λ can be understood using their earlier interretation in the N- Burst model. The burst arameter as well as the router utilization can be exressed in terms of the new arameters as follows: b = 1 E( Y ) λb λ

9 ρ λ λb E( Y )( + h) = [0 ν ν = ] The above assumes that the burst arameter and the intra-burst acket rate are indeendent of h, which means that each source sends ayload data at the maximal rate λ 0]. One of the imortant characteristics of this reformulation is that the new model reduces to the N-Burst model for h = 0, which rovides a very useful mechanism to validate our calculations relative to earlier results. We can observe that when h is zero, the router utilization becomes indeendent of acket size. For other values of h, it becomes useful to further refine our notion of router utilization. First, let then we can define: δ = h, [ ρ( δ ) = ρ(0)(1 + δ ), where ρ(0) = λ b E( Y ) ν. (3) We interret ρ(0) as being the useful work utilization, which is the fraction of time sent by the router rocessing ayload information, while ρ(δ ) is the fraction of time the router is busy. It can be seen that if the header-to-ayload size ratio increases, that is, if ackets are made smaller or the header made bigger, ρ(δ ) will be bigger for the same ayload. Since ρ(δ ) should not exceed 1, δ must never be allowed to exceed 1 ρ(0) 1in our calculations. It is not immediately clear how this affects router erformance metrics over the range of values of the burst arameter as this requires intensive calculations. However, for values of b near the endoints, we can draw qualitative conclusions about the model behavior based on our understanding of the base model. When the burst arameter b is near zero, the system behaves like an M M / 1 queue as the λ / ν OFF eriods vanish and the overall inter-acket arrival rocess becomes governed by the intraburst acket arrival rocess, which is Poisson. Since the merging of several Poisson streams yields a Poisson stream, the model behavior is indeendent of the number of sources (see [Lis01] for

10 details). For the other endoint, namely when b is close to one, the system behaves like an / [ ] M ON λ M ν b /1 queue, which is a bulk arrival queue with Poisson inter-burst arrivals, where the size of each burst is described by the ON-time distribution. This model is once again indeendent of the number of sources. Our distribution of interest for the ON times is a TPT, which as shown in Figure 2 causes blowu oints for sufficiently large values of T. As the blowu oints are a function of the router utilization, we next investigate how δ affects erformance in the corresonding blowu regions. 3.4 Shift in Blowu Points In the 1-Burst rocess, blowu occurs when the acket arrival rate during an ON eriod exceeds the rate at which the router can rocess them. In terms of the reformulated arameters, a blowu occurs when λ ν + h or λ ν 1 + δ. This can be rewritten in a form that is more at for interretation, namely ν λ 1 + δ. The numerator describes the rate at which the router could rocess ackets of size in the absence of overhead. Therefore, if it is necessary to stay below the blowu oint in the resence of acket overhead, the intra-burst acket rate must be reduced by the factor 1 (1 + δ ). Of course, every acket must have a header of finite size, so an alternate comarison can be made between two different acket sizes with the same h. Suose we wish to comare erformance between a system with acket size (system 1), and one with 2 (system 2). The latter sends twice as many ackets as the former, but in the same amount of time. This leads to the two blowu oints occurring at:

11 ν λ 1 = and 1 + δ λ ν = 1+ 2δ 2. So, halving the acket size will cause a reduction in the accetable maximal data flow rate by the factor ( 1+ δ ) (1 + 2δ ) TPT30, α = 1.4, θ = 0.5, T = 30, ρ(0) = 0.25, h = 40, E(Y) = = 48 = 100 = 500 = 1000 = mcd Burst arameter b Figure 3: Mean cell delay calculations for various ayload sizes. The blowu oints can also be exressed in terms of the burst arameter. From (2), we have b ( 1 1) = 1 ρ ( δ ) = (1 + δ ). Figure 3 shows how the blowu oints shift with δ, which takes on the values, { 0.833, 0.4, 0.08, 0.04, }. This corresonds to blowu oints in the range b < The figure also reveals the comlexity in characterizing the relationshi between introducing overhead and the degree of traffic burstiness. The value of δ for which the mean Cell Delay (mcd) is a minimum is a very comlicated function of b. For instance, if we consider the lots with the lowest and highest values of, we observe that when b equals 0, the mcd for the first lot exhibits the best erformance, and when b is one, erformance is reversed. In

12 a similar fashion, one can find the blowu oints for the N-Burst rocess by using (2), relacing ρ with ρ ( 0)(1 + δ ) as follows: 1 ρ(0)(1 + δ ) b( i N) = for 1 i N. 1 ( N i) ρ(0)(1 + δ ) / N In this form, it is clear that the blowu oints deend on and h only in their ratio, δ. 3.5 Otimality for b = 0 At b = 0, the mean acket delay for the base model satisfies (1), yielding: 1 1 mcd ( b = 0) = = ν λ ν + h λ b E( Y ). This equation is indeendent of N, as long as λ b is interreted as the total burst rate from all sources. In order to find for what value of is this exression a minimum, we hold h constant and solve from which we obtain: mpd = 0, 2 ν = λ E( Y )(1 + δ ). b We know that ν = λ E( Y ) / ρ(0), which means that otimality occurs when ρ (0)(1 δ ) 2 = 1, which can also be exressed as: b 1 ρot ( δ ) =. 1 + δ + ot In other words, as δ is increased from 0, the router becomes busier according to (3), but the mean cell delay decreases until = 1 ρ(0) 1, after which oint it begins to increase. It s imortant δ ot to recognize that δ ot is only otimal for b = 0.

13 3.6 Behavior at b = 1 The analysis of acket delay in the case where b = 1is similar to that of the revious section excet that it is more comlex. As b aroaches 1, a host transmits ackets at a faster rate, becoming a bulk arrival rocess in the limit. The governing equation for mcd given a bulk arrival rocess can be exressed as follows (see [Coo81] for details): 1 ν mcd( b = 1) = D, where 1 ρ E( L( L 1)) D =. ) + 2 E( L With L being the random variable denoting the number of ackets generated in one burst, which is roortional to the ON-time distribution, we can derive the acket size distribution from the MMPP of our model. It is imortant to note that D above grows linearly with the variance of the burst size, which for TPT distributions can be unboundedly large, even if the router utilization is small (see [Anto04] for details). 4 Analysis Results We aim to cature the relationshi between the relevant arameters of our traffic model and delay at the router. From the N-Burst model, we know that router erformance is deendent on the burst arameter b and the ON-time distribution. Having augmented the N-Burst model to define a acket in terms of a ayload and a header, we are interested in revealing the erformance interlay between these last arameters and the ones from the N-Burst model. We first look at mcd as a function of ayload and acket size for a 1-Burst rocess with exonential ON time, which is shown in Figure 4. The mean number of bytes in a burst, E( Y ), is , and the burst arameter is set to 0.5. It can be observed that as the header-to-ayload ratio increases (for a given header size), mcd grows

14 until it blows u as indicated by the sikes. Aside from this blowu, the surface in the figure does not reveal any region where mcd is not accetable. When the ON-time distribution is a TPT, erformance looks radically different as shown in Figure 5. All other arameters values are the same as in Figure 4. There are three henomena that are imortant to extract from this figure. First, we see that for small header sizes when ayload size increases, mcd grows. This behavior is consistent with our analysis from Section 2 that as acket size increases, so does system time. However, for a realistic range of ayload sizes such as the ones in the figure, mcd is still in an accetable range. The second region of interest is defined by the blowu in mcd as the header size increases over the range of ayload sizes. This blowu corresonds to the one defined for the N-Burst model, excet there it was exressed in terms of the burst arameter. In Figure 5, we see that for a fixed value of b, the location of the blowu oint occurs earlier for small ayloads than large ayloads. In other words, for small ayload sizes, the range of values of header size that yield accetable erformance is smaller than it is for larger ayload sizes. This is exlained by (3), which effectively shifts the blowu oint as the router utilization increases, keeing the burst arameter constant. As such, we are able to define a region of accetable service with resect to mean cell delay at the router. There is a third blowu equivalent to the one seen with the exonential ON time, and that is due to high header-to-ayload size ratio, which is indicated by the sikes in the figure. We last look at mcd as a function of b and ayload size for a fixed header size of 5000, as shown in Figure 6. As before, an increase in mcd as ayload grows can be observed. A region of accetable service can also be exressed here, but in this case it is in terms of allowable burst arameter values over a range of ayload sizes for a fixed header. As ayload size increases, the blowu in mcd occurs for larger values of b. The increase in ayload size reduces δ, which lowers the router utilization, effectively shifting the value of b at which a blowu occurs.

15 mcd calculations for 1 Burst: T = 1, θ = 0.5, ρ(0) = 0.25, b = mcd x ayload header x 10 4 Figure 4: Mean Cell Delay as a function of ayload and header size for a 1-Burst rocess with exonential ON time. mcd calculations for 1 Burst: T = 30, θ = 0.5, ρ(0) = 0.25, b = mcd x ayload header x 10 4 Figure 5: Mean Cell Delay as a function of ayload and header size for a 1-Burst rocess with ower-tailed ON time.

16 mcd calculations: TPT30, θ = 0.5, ρ(0) = 0.5, h = mcd x ayload burst arameter Figure 6: Mean Cell Delay as a function of ayload size and the burst arameter for a 1-Burst rocess with ower-tailed ON-time and fixed header size. 5 Conclusion In this aer, we have resented an analytic model with a rich arameter set based on the N-Burst model. Our formulation rovides us with the means to study the intricate relationshi among ayload, header size and mean cell delay at the router in the resence of heavy-tailed traffic. For secial cases of the burst arameter, namely when the arrival rocess reduces to either a Poisson or a bulk arrival rocess, we rovide closed-form solutions. When the ON time distribution is exonential, we show that delay is tame as long as the header-to-ayload size ratio is within a realistic range. When burst durations are ower-tailed, however, we are able to characterize of region of accetable service, which is defined by an allowable range of header sizes for each ayload size, beyond which delay blows u according to the results of the N-Burst model.

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18 [Schw99b] [Schw00a] [Schw00b] Schwefel, H.-P., Fiorini, P., and Lisky, L., Analytic Model of Performance in Telecommunication Systems, Based on ON-OFF Traffic Sources With Self-Similar Behavior, 7th International Conference on Telecommunication Systems, Nashville, TN, March 18-21, Schwefel, H.-P., Imact of Aggregated, Self-Similar ON/OFF Traffic on Delay in Stationary Queueing Models (extended version), Performance Evaluation, Schwefel, H.-P., Performance Analysis of Intermediate Systems Serving Aggregated ON/OFF Traffic with Long-Range Deendent Proerties, PhD Dissertation, Technische Universität München, 2000.