We show that the fluid loss ratio in a fluid queue with finite buffer $b$ and constant link capacity $c$ is always a jointly convex function of $b$ and $c$. This generalizes prior work [6] which shows convexity of the $(b,c)$ trade-off for large number of i.i.d. multiplexed sources, using the large deviations rate function as approximation for fluid loss. Our approach also leads to a simpler proof of the prior result, and provides a stronger basis for optimal measurement-based control of resource allocation in shared resource systems

Kumaran, K.

Mandjes, M.R.H. (Michel)

Stolyar, A.

English

CWI's Institutional Repository

C e n t r u m  v o o r  W i s k u n d e  e n  I n f o r m a t i c aPNAProbability, Networks and Algorithms Probability, Networks and AlgorithmsConvexity properties of loss and overflow functionsK. Kumaran, M.R.H. Mandjes, A. StolyarREPORT PNA-R0207 FEBRUARY 28, 2002CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO).CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.Probability, Networks and Algorithms (PNA)Software Engineering (SEN)Modelling, Analysis and Simulation (MAS)Information Systems (INS)Copyright © 2001, Stichting Centrum voor Wiskunde en InformaticaP.O. Box 94079, 1090 GB Amsterdam (NL)Kruislaan 413, 1098 SJ Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199ISSN 1386-3711Convexity Properties of Loss and Overflow FunctionsKrishnan Kumaran, Michel Mandjes, and Alexander Stolyaremail: kumaran@lucent.com, michel@cwi.nl, stolyar@lucent.com Bell Labs/Lucent Technologies,600 Mountain Ave., Murray Hill, NJ 07974, USACWIP.O. Box 94079, 1090 GB Amsterdam, The Netherlands;Faculty of Mathematical Sciences, University of TwenteP.O. Box 217, 7500 AE Enschede, The NetherlandsABSTRACTWe show that the fluid loss ratio in a fluid queue with finite buffer  and constant link capacity  isalways a jointly convex function of  and . This generalizes prior work [6] which shows convexityof the   trade-off for large number of i.i.d. multiplexed sources, using the large deviations ratefunction as approximation for fluid loss. Our approach also leads to a simpler proof of the prior result,and provides a stronger basis for optimal measurement-based control of resource allocation in sharedresource systems.2000 Mathematics Subject Classification: 60K25 (primary).Keywords and phrases: queueing theory, trade-off between network resources, large deviations.Note: Part of this work was done while the second author was at Bell Laboratories/Lucent Technologies,Murray Hill NJ, US.11 IntroductionA queueing system can be described as a set of resources, typically service capacities (or bandwidth incommunication networks), and buffers (where excess traffic can be stored temporarily). The traffic offeredto the queues is determined by the routing and scheduling disciplines, which in turn determines systemperformance. In general, the resources at the queues have to be designed such that a performance criterionis satisfied, for instance the buffer overflow probability has to be below some small fraction . For a fixedperformance level, the resources involved often trade off. As each of these resources has its specific price, itis important to have more detailed knowledge of this trade-off. Hence, efficient (i.e., economical) networkdesign is enabled through insight into the shape of the resulting ‘iso-performance curve’.IIn this note we focus on one of the simplest queueing networks: a single queue, with finite buffer size and constant service rate , fed by a stationary fluid (arbitrarily divisible) source. We show that the fluidloss ratio function   is jointly convex on  and . (This in particular implies that for a given   , thetrade-off curve between  and , defined by     is convex.) A crucial observation in our analysis isthe following general principle:Sharing resources is always better than partitioning. More precisely: compare (1) a ‘parti-tioned’ system in which random input processes feeding queues with resources   (with   ), with (2) a ‘shared’ system in which  feeds a queue with buffer  and linkrate  . The amount of traffic lost in the shared system is dominated by the (total) amountof traffic lost in the partitioned system.Our analysis shows that this general principle easily implies the joint convexity of  . In the largedeviations regime, which involves scaling up the resources as    and   , and replacing the inputprocess by  i.i.d. copies, previous work established that the overflow probability decays exponentially in [1, 2, 9], with decay rate function 	 . It is proved in [6] that the trade-off curve defined by 	   Æis convex. Based on our observations about  , we also give a simpler proof of that result.However, the main advance of this work is the generalization of [6] by proving the joint convexity ofthe loss ratio  , even away from the asymptotic large  regime. This generalization can offer severalpractical advantages. Firstly, it permits more optimal resource partitioning. As in [6], this can be done op-timally using our results for inhomogeneous traffic and differential QoS requirements, and unlike [6], evenwhen each traffic class has only a few connections. Secondly, joint convexity permits use of more complexQoS metrics without losing the computational tractability of the multi-class resource optimization problem.Finally, when measurement-based control is feasible, the loss ratio   is a more appropriate, and moredirectly measurable, parameter in comparison to the rate function 	 . These considerations make itdesirable to directly prove the universal joint convexity of   without using 	  as approximation,which is what we accomplish here using arguments that are simpler than those of [6].The rest of this note is organized as follows. Section 2 gives a number of key sample path relations, andSection 3 establishes the convexity of  . In Section 4 the large deviations regime is examined, and theconvexity of the trade-off curve for buffer overflow probability is proved before conclusion in Section 5.22 Key sample path relationsConsider a single server with (constant) processing capacity , and a buffer size . The server is fed by afluid input flow. We denote by  the cumulative amount of fluid arrived by time   . In this sectionwe assume that  is a fixed function (interpreted later as a sample path of a random process), which isnon-decreasing continuous. Also, we consider the evolution of such (fluid) system in the interval 	, so      and, by convention,   .Suppose the buffer size is finite,   , and the initial queue length (buffer content) is , where    . Then, the queue length at every time    is determined as follows:      	     	    (1)where the lower and upper regulations  and  are unique non-decreasing continuous functions such that           ~~	  	  ~~   ~~ and	  	  ~~   ~~see Harrison [5], Proposition 2.4.6.The functions  and  we will also refer to as the (cumulative) idleness and (cumulative) loss function,respectively.In case of an infinite buffer,  , the queue length is defined similarly:      	   (2)with     ~~  	  	     In addition, for infinite buffer the expression for the idleness  is explicit (see [5], Proposition 2.2.3):  	       where  denotes the minimum of two numbers. Note that, for any   ,  is monotone non-decreasingon the initial condition .The following two simple lemmas are needed to prove the key Lemma 2.3 below.Lemma 2.1 Let   , be the queue length functions in three infinite buffer systems, defined by the triples     , and   , respectively, where     ,    ,   . Then,         (3)3Proof. If      (4)then	                  	    	   which implies (3). The case when condition (4) does not hold is trivial, because then 	   .Lemma 2.2 Consider a 4-tuple    ,     , and suppose non-decreasing contuinuousfunctions (of )  and  are such that   	    	  with           	  	     (The condition is absent that  can only increase when  is equal to .) Then,  	       (5)Proof. This result can be easily obtained for example by slightly extending the proof of Proposition2.4.6 in [5]. Namely, the pair 	    	    is the unique minimal fixed point of a mono-tone operator mapping a pair   into a pair  . It is easy to see that   is a fixed pointof that operator, and therefore the majorization (5) does hold. We omit details.The following lemma is the key in establishing the convexity of  , where   is the fraction oflost fluid in a finite buffer system with buffer  and server rate .Lemma 2.3 Consider three finite buffer systems defined by 4-tuples    ,    ,and    , respectively, where     ,    ,    , and    .Then,	     	     	       Proof. Using Lemma 2.1, it is easy to verify directly that ,  ,   	    	   , and        satisfy the conditions of Lemma 2.2. Therefore,    for all .43 Convexity of the fluid loss ratioIn this section we prove the convexity result for the fluid loss ratio. The importance and universality ofthis result is in the fact that the convexity holds for a system with any fixed parameters – not only in anasymptotic regime.Let       now be a random process with continuous non-decreasing paths and stationaryincrements. We can use normalization    without loss of generality. The mean (fluid) arrival rate  is well defined. The steady state fraction of lost fluid in a finite buffer system defined by    is definedas follows:  	   	assuming that the right-hand side is well defined and does not depend on . (It is easy to see that thisassumption holds in virtually all cases of interest, and so the above definition matches most of the commondefinitions of steady state ‘loss probability’ or ‘fluid loss ratio’ in systems with finite buffers.)Theorem 3.1 Suppose the random input flow process  with stationary increments is fixed. Then the func-tion   is convex jointly on  and .Proof. Consider a fixed sample path  of the input process, and a fixed initial buffer content . Letus fix positive constants and , such that   , and positive constants      , suchthat    and    . We have for any   :	     	     	    	     	   where the inequality follows directly from Lemma 2.3 and the equality follows from the fact that an (aux-illary) system defined by    ,    , is the system defined by     ‘scaleddown’ by a factor .4 Convexity of the trade-off curve for buffer overflow probability in thelarge deviations regimeIn this section we consider a system with infinite buffer space (i.e., no traffic loss). We consider traffic from independent, statistically identical, sources feeding into a buffered resource. This resource is modeled asa queue with constant depletion rate . As in the previous section, a single traffic source is described by arandom process       continuous non-decreasing paths and stationary increments. (It will beclear from our development that the continuity assumption is not important.) We remind that   represents the amount of traffic arrived by time , so we assume that   .5Since we consider an infinite buffer system we assume that the mean arrival rate (for a single source)therefore, the mean (fluid) arrival rate     so that the system is stable.We are interested in the steady-state overflow probability  , which is the probability of the queuelength exceeding level . (Here we use notation  for a ‘threshold value’ rather than the buffer size.) Undernon-restrictive conditions, this probability decays exponentially in the scaling parameter . We define thecorresponding exponential decay rate:	      The key result on 	 , based on large deviations arguments, is given below in Theorem 4.3. A majorcontribution to the development of these results was given by Botvich and Duffield [1], whereas relatedresults were derived in [2, 9]. Recently, a significant improvement was made by Likhanov and Mazumdar[7].Assumption 4.1 See [7] – Assume 	  is larger than   , for  large enough, and a positive , where	   	    (6)An additional assumption has to be made on the regularity of the traffic. For this purpose we here useHypothesis 1(iv) from [1], which essentially implies that the decay rate for discrete time carries over tocontinuous time. This is proven analogously to the proof of Theorem 1 of [1, p. 302].Hypothesis 1(iv) is stated as follows. Define   where ’s are i.i.d. copies of a single source process. (To be precise, the above notation also assumes thatthe process  is extended to be defined for all real .) Then it is required that     It is easily verified that, due to the stationarity and i.i.d. assumptions, this requirement reduces to thefollowing.Assumption 4.2 See [1] – Assume that for all   ,     Theorem 4.3 Under Assumptions 4.1 and 4.2,	   		   (7)6Applicability. The striking feature of the above theorem is that it is applicable for almost all typesof fluid traffic sources of practical interest. Short-range dependent sources satisfy the conditions (Markovfluid sources, on-off sources with light-tailed on-times), see [1], but also long-range dependent processes(fractional Brownian motion, on-off sources with heavy tailed on-times), see [4, 7, 8].Now, let us recall the meaning of 	 . By the classical Crame´r theorem (cf. Theorem 2.2.3 in [3]),	  is nothing else but the value of the large deviations rate function for the sequence of distributions ofat point   . This in particular means that, for a given , 	  is essentially a function of a singlevariable  . Using standard properties of rate functions, we can rewrite (7) as	   	     ~~   ~~         	 (8)Theorem 4.4 Defined implicitly by 	   Æ with Æ  , buffer  as a function of service capacity , isconvex.Proof. Let us fix two (positive) pairs   and   such that 	   	   Æ. Let us fixpositive and , such that   . Denote    and    . We need toshow that 	   Æ.Let    be the value of  that optimizes (7) for the pair  , and therefore    optimizes theright-hand side of (8). We obviously have the following property:Either      or      (9)Without loss of generality, let us assume that the former inequality holds.Property (9) is of course a matter of simple arithmetic. However, it is key here, and can be interpretedthe same way as Lemma 2.1 (and in fact can be obtained as a simple corollary from it):If the amount of fluid    arrives in the interval 	  into the system with parameters  , then  . Therefore, if we partition the system into two, with parameters  ,    , and feed theamounts of fluid   in 	  into them, then the queue length must be at or above threshold inat least one of them.By (8) and (9), we haveÆ  	   	   	  5 ConclusionIn this work, we have proved that the loss ratio is always a jointly convex function of buffer and band-width sizes in buffer-bandwidth queuing systems, which also implies convexity of buffer-bandwidth trade-off curves. This result generalizes previous work [6] that proves convexity of the loss function, approximated7by the large deviations rate function, for large numbers of multiplexed flows. Our results hence establishthat the same practical advantages mentioned in [6] regarding optimal design of partitioned buffer-bandwidthsystems continue to apply if loss ratio were the criterion instead of the rate function, and without the needfor large numbers of multiplexed flows. Apart from theoretical generality, this may offer some practicaladvantages in terms of more efficient resource allocation, better use of loss measurements, and the ability toadmit more complex QoS requirements.Acknowledgements. The authors would like to thank Iraj Saniee for insightful comments.References[1] D. Botvich and N. Duffield. Large deviations, the shape of the loss curve, and economies of scale inlarge multiplexers. Queueing Systems, 20: 293 – 320, 1995.[2] C. Courcoubetis and R. Weber. Buffer overflow asymptotics for a buffer handling many traffic sources.Journal of Applied Probability, 33: 886 – 903, 1996.[3] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. 2nd ed. Springer, 1998.[4] N. Duffield. Queueing at large resources driven by long-tailed M/G/-modulated processes. QueueingSystems, 28: 245 – 266, 1998.[5] J.M. Harrison. Brownian Motion and Stochastic Flow Systems. Wiley, 1985.[6] K. Kumaran and M. Mandjes. The buffer-bandwidth trade-off curve is convex. Queueing Systems, 38:471 – 483, 2001[7] N. Likhanov and R. Mazumdar. Cell loss asymptotics in buffers fed with a large number of independentstationary sources. Journal of Applied Probability, 36: 86 – 96, 1999.[8] M. Mandjes and S. Borst. Overflow behavior in queues with many long-tailed inputs. Advances inApplied Probability, 32: 1150 – 1167, 2000.[9] A. Simonian and J. Guibert. Large deviations approximation for fluid queues fed by a large number ofon/off sources. IEEE Journal of Selected Areas in Communications, 13: 1017 – 1027, 1995.8

Convexity properties of loss and overflow functions

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