Performance Analysis of Priority Queueing Schemes in Internet Routers

Conference on Information Sciences and Systems, The Johns Hopkins University, March 8, Performance Analysis of Priority Queueing Schemes in Internet Routers Ashvin Lakshmikantha Coordinated Science Lab and Department of Electrical and Computer Engineering University of Illinois Urbana-Champaign email : lkshmknt@ifpuiucedu C L Beck Coordinated Science Lab and Department of General Engineering University of Illinois Urbana-Champaign email: beck@uiucedu R Srikant Coordinated Science Lab and Department of Electrical and Computer Engineering University of Illinois Urbana-Champaign email: rsrikant@uiucedu Abstract It is well-known that file sizes in the Internet have a very large variance One implication of this fact is that most of the files are small and contribute to a small amount of the total traffic, while a few large files account for most of the traffic in the Internet If all the files share the available capacity in a fair manner to approximate weighted processor sharing, then short files may experience delays that are large in comparison to their file sizes In this paper, we study the performance of a simple priority scheme at the router which preferentially treats shortflows Using simple fluid models, we show that such a priority scheme improves the performance of shortflows significantly while the performance of long-flows does not deteriorate much I Introduction Many studies have shown that the file size distribution in the Internet is heavy tailed (see, for example, []) This fact implies that most of the files transferred over the Internet are small in size (of the order of a few tens of kilobytes) and only a small fraction of the files are large It is roughly estimated that -% of the flows (that correspond to large files) carry about 8-9% of the Internet traffic During the transmission of the small files, which we will refer to as short-flows, the underlying TCP congestion control mechanism does not leave the slow start phase For a discussion of TCP dynamics, we refer the reader to [] Any packet loss incurred in the slow-start phase of TCP causes restarts, ie, the window size is reset to one, thus resulting in a significant degradation of throughput The degradation may be dramatic, especially when the router is severely congested A natural solution to the above problem is to provide priorities to the short-flows Queues with class based priorities have been extensively studied in the past (see, for example, []) The main conclusion of these studies has been that one can achieve a reduction in the waiting times for higher priority queues, but this comes at the expense of the customers in the lower priority queues This conclusion led many to raise objections to providing higher priority to short-flows (commonly referred to as short jobs in queueing-theoretic literature) [] Specifically, it has been argued that providing priorities to smaller jobs can lead to large waiting times for the larger jobs These claims were investigated by Bansal and Harchol-Balter [] in a more recent work on SRPT (Shortest Remaining Processing Time) Under this policy, after a customer departs from the system, the server processes that customer who would take the least amount of time to complete service Their findings suggest that, if the file-size distribution is heavy tailed, then the additional delays suffered by the long-flows are insignificant Implementing SRPT in the Internet is however, a very difficult task; each router must be made aware of the remaining file sizes of all the incoming flows This requires per-flow information and given the current technology, such policies cannot be implemented without significantly affecting router speeds Therefore, one needs to look for alternative schemes that can achieve similar performance without compromising on the router abilities One such scheme is to divide files into two classes and enforce strict priority for one class over the other In other words, all files that are below some threshold will be given higher priority The priority can be enforced either at the edges of the network by setting a bit in each of the packet headers that belong to short-flows (for example, using the DiffServ architecture [, ]), or by identifying long flows at the router Using a sampling technique proposed by Estan and Varghese [] for traffic measurements, an algorithm called SIFT has been developed by Prabhakar et al [] where long-flows are identified and moved to a low-priority buffer at the router Assuming that long and short-flows are identified using some unspecified algorithm, the effect of such priority based policies is examined via ns- simulations by Chen and Heidemann [] The results in [, ] indicate that it is possible to achieve a significant improvement in the bandwidth of short-flows, while keeping the losses of long-flows to a minimum In this paper, we evaluate the performance of such prioritybased schemes in the Internet via a fluid model Specifically, we first show that, without priorities, the throughput attained by short-flows is small due to the nature of bandwidth sharing that occurs in the current Internet We further show, via analysis and simulations, that by providing priorities one can significantly reduce the delays faced by the short-flows, while the additional delays experienced by the long-flows are very small We note that we do not address the problem of identifying short and long-flows We assume that such mechanisms

exist and they can be implemented without causing significant errors [, ] The reason for studying fluid models, as compared to stochastic models, is that they are simpler to analyze and can be generalized to both non-exponential distributions (which we consider in a longer version of the paper) and systems with access bandwidth constraints, although we only discuss access bandwidth constraints in this paper and do not deal with non-exponential file sizes here On the other hand, the stochastic model of the considered system is very hard to solve analytically and closed form solutions exist only when the file-size distributions are exponential[, 7, ] The rest of the paper is organized as follows In Section II we briefly describe the model and evaluate the performance when no priorities are enforced In Section III we analyze the performance of the system when priorities are enforced The simulation results are contained in Section IV and we provide the conclusions in Section V II Fluid Models with No Priorities We consider a single link accessed by many users Shortflows arrive at the system according to a Poisson process with mean rate λ s We assume that the short-flow file sizes are exponentially distributed with mean Similarly, the longflows arrive according to another independent Poisson process with mean rate λ l and their file sizes are exponentially distributed with mean Note that from a well-known property of Poisson processes, this arrival model can be viewed as a single Poisson process, with a hyper-exponential file-size distribution In fact, one can generalize this model to allow the file size to be a mixture of many exponential distributions, thus closely approximating any heavy-tailed distribution [8] We do not consider such generalizations here; however, it would be fairly straightforward to extend our analysis to the more general model It is assumed that / / Typically, around 8-9% of the flows are short-flows and therefore, λ s λ l However, most of the bytes in the Internet are carried by the long-flows Hence we further assume λ s < λ l To avoid trivial cases, we assume λ s, λ l > The rate at which any user can transmit is determined by the capacity of the bottleneck link and by the way in which the link capacity is shared among the competing users Let the number of long-flows in the system be denoted by n l and the number of short-flows be n s It is well known that the data rate of a user can be approximated by its window size divided by its RTT (round-trip time) Thus, at any given time, the data rate of a flow is proportional to window size As the short-flows never leave the slow-start phase of TCP, the window sizes of the short-flows will be very small Thus, the bandwidth received by short-flows will be small On the other hand, the long-flows will be in the congestion avoidance phase and thus, their window sizes will be much higher than that of short-flows To model this, we assume that the capacity of the bottleneck link is shared in a proportional manner with weights w s and w l (w l > w s) Specifically, the data rate of a short-flow is given by x s = w sc n sw s + n l w l, () where C denotes the capacity of the link Similarly, the data rate of a long-flow is given by x l = w l C n sw s + n l w l () Such a sharing policy tends to be unfair to short-flows It is easy to see that xs x l = ws w l As w s < w l, the bandwidth seen by short-flows can be much smaller than the bandwidth seen by the large-flows Without loss of generality, we can assume C = Then, under an appropriate scaling [9], we can describe the evolution of the flows in the system by the following differential equations: n l = λ l n l x l, n s = λ s n sx s Let us denote the total load on the system by () ρ = λ l + λs = ρ l + ρ s () We will now calculate the average bandwidth received by a short-flow and a long-flow at equilibrium, when the system is critically loaded, ie, ρ = IIA Convergence to Equilibrium The stationary point of the system of differential equations () is given by ρ l = n l x l, ρ s = n () sx s, where n s and n l denote the equilibrium values We now show that under critically-loaded conditions, the system will asymptotically converge to this stationary point The amount of work remaining in the system is given by W (t) = ns + n l Since the system is critically loaded the total work remaining in the system, denoted by W (t), is invariant with time To see this, note that Ẇ (t) = d dt ( ns + n l ) = ρ s n sx s + ρ l n l x l = (ρ s + ρ l ) (n sx s + n l x l ) = So, if at any point in time, say t, ṅ s(t ) =, then ṅ l (t ) = Further, for all t > t, the system will be at equilibrium To prove that either ṅ l (t) or ṅ s(t) reaches zero, we note the following Suppose, n s >, then by (), n l < Also, for all values of n s > and n l >, the functions, n s and n l are continuous and further, n l Therefore, n l is a monotonically decreasing function, bounded below by zero Convergence to equilibrium immediately follows IIB Equilibrium analysis ()

Now we analyze the system at equilibrium As observed previously, the amount of work in the system is constant Hence, n l + n s = n l() + ns() = W () (7) Using () and () in (), we get From (8) and (7), we get ρ s ρ l = n sw s n l w l (8) x s = ρsw l + ρ l w s (w l )W () x l = ρsw l + ρ l w s (w s )W () III Fluid Models with Priorities From the analysis carried out in the preceding sections, it is clear that the long-flows on average get much higher bandwidths due to the fact that they can transmit more packets in one RTT We now show that by giving priority one can significantly enhance the bandwidth provided to short-flows while still maintaining high bandwidth to the long-flows This analysis shows that the benefits obtained by the short-flows can be arbitrarily large while the additional delay suffered by the long-flows is well bounded Throughout this analysis we assume that the arrival pattern and the file-size distributions are unchanged We analyze the performance of the system under prioritiesin other words, the long-flows are served only when the system is devoid of the short-flow The short-flows transmit data while the underlying TCP is in the slow-start phase In the slow-start phase, the window size is slowly increased from The window size doubles each RTT if all the acknowledgments are received Thus, it is possible that even though enforcing priority ensures that the short-flows are served first, the short-flows may not end up utilizing all the bandwidth as the TCP is not designed for data transmissions at such high rates in the slow-start phase We model this by imposing a limit r s on the maximum rate at which a short-flow can transmit the data This would ensure that the maximum rate achieved by the short-flows cannot exceed the inherent limit imposed by TCP The simulations conducted show that the bandwidth received by the shortflows is very close to this limit The bandwidths received by short-flows and long-flows are given by, x s = min r s, n s, nsr s x l = max, I ns= n l With x s and x l defined as above the fluid model now is given by () We analyze this system at equilibrium n l (9) If the short-flows are given priority, then at equilibrium, their transmission rates would be x s = r s The number of short-flows at equilibrium is given by, n s = ρs r s Using the fact that the amount of work left in the system is invariant with time, we can calculate the number of long-flows at equilibrium: n l = W () ρs r s The bandwidth received by long-flows at equilibrium is therefore equal to x l = ρ l n l = ρ l W () ρs r s Let x swop (x swp ) denote the bandwidth received by the shortflows at equilibrium when the priorities are not enforced (enforced) Then, using (9), we get the improvement in performance of the short-flows to be x swp ρ l w s W () = x swop ρ sw l + ρ l w s r s Similarly, the degradation in throughput of the long-flows is given by (with x lwop and (x lwp ) defined appropriately), x lwop = + ρs w l ρ s x lwp ρ l w s r sw () () The above analysis indicates that it is possible to achieve arbitrary improvement in the performance of the short-flows, depending on the value of W () Thus, the main question is the following: how do priorities affect the performance of long-flows? To get a bound on the throughput degradation of longflows, we now consider the situation where r s Clearly, r s is going to benefit short-flows as they now can transmit at arbitrary rates Thus, the long-flows will suffer a higher degradation Taking this limit, we get from (), x lwop lim r s x lwp = + ρs ρ l w l w s Let us now suppose that =, ρ s =, ρ l = 8 These values are consistent with our brief discussion regarding the nature of Internet traffic in the first section of the paper Further, our numerical simulations indicate that the short-flows on an average receive about a third of the bandwidth received by the long-flows Using these values, we get x lwop x lwp = 7 Thus, the long-flows see a reduction in bandwidth of only 7%, while the short-flow performance can improve dramatically, especially if the work in the system is very high This very clearly supports giving priority to short-flows IIIA Equilibrium Analysis IV Simulation results

CiscoSystems Workgroup Switch Catalyst CiscoSystems Workgroup Switch Catalyst In this section, we study the effect of priorities on the system via ns- simulations We consider a Mb bottleneck link, which is being accessed by many flows (Fig ) The users arrive into the network as a Poisson process at the rate of 9 users/sec An arriving user belongs to the short-flow class with probability 9 These users will bring files that are exponentially distributed with mean KB Users belonging to the long-flow class have files that are exponentially distributed with mean MB Obviously, any arriving user belongs to the long-flow class with probability The load on the system is therefore equal to 9 (by ()) The data rates obtained by various flows in the presence and absence of priorities are plotted in Figure The shortflows receive data rates close to 8 KB/sec when there are no priorities However, when priorities are enforced, they receive an average data rate of 9 KB/sec This is a increase of more than 9% On the other hand the data rates seen by long-flows drops only by about 8% This shows that giving priority helps improve the throughput of 9% of the flows by about 9% while the reduction in bandwidth seen by the longflows is comparatively small The performance of the priority based queue under 9% load and 98% load is given in Figure and respectively These results show similar trends though the actual improvement achieved depends on the load on the system job streams In Proceedings of the 9th Annual ACM- SIAM Symposium on Discrete Algorithms, 998 [] S Blake, D Black, M Carlson, E Davies, Z Wang, and W Weiss An architecture for Differentiated Services In IETF RFC7, 998 [] X Chen and J Heidemann Preferential treatment for short flows to reduce web latency Computer Networks, [] ME Crovella and A Bestavros Self-similarity in the World Wide Web traffic: Evidence and possible causes IEEE/ACM transactions on Networking, pages 8 8, 997 [] C Estan and G Varghese New directions in traffic measurement and accounting In Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, [7] G Fayolle, I Mitrani, and R Iasnogorodski Sharing a processor among many classes Journal of the ACM, pages 9, 98 Mbps [8] A Feldmann and W Whitt Fitting mixtures of exponentials to long tail distributions to analyze network performance analysis In Proceedings of INFOCOM, 997 Figure : Simulated Topology V Conclusions In this paper, we have introduced a simple fluid model to study the performance of long and short-flows in the Internet We have shown that when the router is severely congested, giving priority to short-flows will significantly enhance their throughput without significantly degrading the data rates seen by the long-flows Acknowledgments: This work was supported in part by an AFOSR URI Grant F9--- References [] C Psounis A Ghosh and B Prabhakar SIFT: a simple algorithm for identifying large flows, Presented at the Stochastic Networks Conference, Montreal, Canada [] M Bender, S Chakrabarti, and S Muthukrishnan Flow and stretch metrics for scheduling continuous [9] HC Gromoll, A L Puha, and R J Williams The fluid limit of a heavily loaded processor sharing queue Annals of Applied Probability, [] N Bansal M Harchol-Balter Analysis of SRPT scheduling: Investigating unfairness In Proceedings of the ACM SIGMETRICS international conference on Measurement and modeling of computer systems, [] L Kleinrock Time shared systems : A theoretical treatment Journal of ACM, pages, 97 [] L Kleinrock Queueing Systems: Vol Wiley- Interscience Publication, 97 [] K Nichols, S Blake, and F Baker Definition of the Differentiated Services field (DS field) in the IPv and IPv headers In IETF RFC7, 998 [] TM O Donovan Direct solutions of M/G/ processor sharing models Operations Research, pages, 97 [] L L Peterson and BS Davie Computer Networks: A Systems Approach Morgan-Kaufman, 999

x Avg Bandwith =8KB/sec x Avg Bandwith =9KB/sec x 8 x Long Flows Avg Bandwith =9KB/sec x 8 x Long Flows Avg Bandwith =KB/sec (a) No priorities (a) No priorities x Avg Bandwith =9KB/sec x Avg Bandwith =KB/sec x 8 x Long Flows Avg Bandwith =79KB/sec x 7 x Long Flows Avg Bandwith =KB/sec (b) Priority for short-flow (b) Priority for short-flow Figure : Data rates got by short and long-flows in presence and absence of priorities at load 9 Figure : Data rates got by short and long-flows in presence and absence of priorities at load 9

x Avg Bandwith =8 KB/sec Bandwidth (KB/sec) x 8 x Long Flows Avg Bandwith =7KB/sec Bandwidth (KB/sec) (a) No priorities x Avg Bandwith =9KB/sec x 8 x Long Flows Avg Bandwith =KB/sec (b) Priority for short-flow Figure : Data rates got by short and long-flows in presence and absence of priorities at load 98