de Boer, Pieter-Tjerk

English

The performance of state-independent changes-of-measure for estimating the overflow probability of the total population of a two-node Markovian tandem queue using importance sampling is studied. It is shown that only the change of measure proposed by Parekh and Walrand (1989) can be asymptotically efficient for large overflow levels. The behaviour of this change of measure all over the parameter space is categorized, using both analysis and numerical calculation

University of Twente Research Information

Analysis of state-independent IS measures for the two-node tandem queue— extended abstract for RESIM2004 —Pieter-Tjerk de Boer AbstractThe performance of state-independent changes-of-measure for estimating the overflow probability of the totalpopulation of a two-node Markovian tandem queue using importance sampling is studied. It is shown that onlythe change of measure proposed by Parekh and Walrand (1989) can be asymptotically efficient for large overflowlevels. The behaviour of this change of measure all over the parameter space is categorized, using both analysisand numerical calculation.1 IntroductionSince the end of the 1980s, there has been a steady interest in the problem of estimating rare event probabilitiesin queueing models, particularly networks of queues. This interest derives mainly from applications in the fieldof telecommunications, where buffers in routers and switches should be dimensioned such that overflow is a rareevent.The present work considers the use of importance sampling (IS) simulation to estimate the probability of over-flow of the total population of two (M)/M/1 queues in tandem. Importance sampling simulation artificially makesthe overflow less rare by changing the probability distributions (also called “change of measure” or “tilting”) inthe model, and compensating for this by keeping track of the likelihood ratio. In the present work, only state-independent changes of measure are considered; this means that only a global change of the arrival and servicerates, without regard to the model’s present state, is made.In rare-event simulation of queueing models, one typically strives for asymptotic efficiency: the relative error(defined as the estimator’s standard deviation divided by its mean) increases at most polynomially in the overflowlevel (at a fixed number of simulation replications), while the overflow probability itself decreases exponentially.An even better estimator may have bounded relative error: the relative error stays essentially constant as theoverflow level increases. However, a worse change-of-measure may lead to an exponentially growing relativeerror (as does standard simulation), or even to the very undesirable situation where the estimator variance (andthus the relative error) becomes infinite.In [PW89], a state-independent change-of-measure was proposed for Markovian queueing models, based onlarge-deviations theory; according to this change of measure, in the tandem queue model the arrival rate andthe bottleneck service rate should be exchanged. This change of measure will henceforth be referred to as P&W.Experimental results in [PW89] showed that this method works well in some cases, and not so well in othercases. In [GK95] the P&W method was studied in detail for tandem networks; it was proven for some partsof the parameter space (i.e., ranges of arrival and service rates) that the resulting simulation is asymptoticallyefficient, and for some other parts that it is not. Experimental results in [dB00] and [dBN02] indicate that, atthe cost of a significant increase in complexity, asymptotically efficient simulation is possible everywhere using astate-dependent change of measure.So on the one hand we have P&W’s state-independent tilting, which in several cases does not work well, andon the other hand we have the state-dependent tilting which works well even in those cases, but is much morecomplex. A natural question then is whether other state-independent tiltings than P&W could be asymptoticallyefficient. The fact that P&W is derived on the basis of large-deviations theory, and the fact that an adaptive ap-proach to finding a state-independent tilting fails to result in asymptotic efficiency in some cases (see Section 7.4.4in [dB00]), suggest that this might not be the case, but of course do not constitute a formal proof.In the present work, the above question is answered for the two-node tandem queue, and extensions of[GK95]’s results to the entire parameter space are given.2 Analytical resultsThis section lists some theorems with brief descriptions of the proofs; the full proofs will appear in a future fullversion of this paper.Dept. of Electrical Eng., Mathematics and Computer Science, University of Twente, Enschede, The Netherlands; ptdeboer@cs.utwente.nl1All of these theorems use the following notation:= the (untilted) arrival rate at the first queue,  = the(untilted) service rate at the first server, and  = the (untilted) service rate at the second server. Furthermore,  ,  and   represent the tilted rates.We limit the study to cases where both queues are stable, i.e.  and 	  ; otherwise, the overflow eventwould not be rare. Furthermore, we limit ourselves to the case where the second server is the bottleneck, i.e., .Theorem 1 For the two-node tandem queue simulation problem, no state-independent change of measure isasymptotically efficient if  2  4  (1  2  )((1  )2   2  2) The proof starts by defining two sample paths. Their contributions to the first and second moments of the esti-mator are calculated; from this, two necessary conditions for asymptotic efficiency follow. The inequality in thetheorem is true in the region where these two conditions are contradictory.Theorem 2 For the two-node tandem queue simulation problem, every state-independent change of measure forwhich either  , or    , or    , is not asymptotically efficient. I.e., the P&W tilt is the only one that maybe asymptotically efficient.The proof considers starting the simulation from any initial state and proposing a “guess” for the second momentthat would be obtained (as a function of the initial state); next, it is shown that with any non-P&W tilting thisis actually a lower bound on the true second moment; finally it is established that this lower bound alreadyincreases so quickly with the buffer size that asymptotic efficiency is impossible.Theorem 3 For the two-node tandem queue simulation problem, a state-independent change of measure accord-ing to P&W (i.e., with   ,     and     ) is not asymptotically efficient if 3 2 12  3  2  2   5 The proof uses a similar approach to the one used for the previous theorem.Theorem 4 For the two-node tandem queue simulation problem, a state-independent change of measure accord-ing to P&W (i.e., with   ,     and     ) leads to infinite variance ifM2 4  1  M with M defined as M  (  )2 The proof uses a similar approach to the one used for the previous two theorems. The difference however is thatthe “guess” is such that multiplying it by any positive number does not invalidate the proof that it is a lowerbound on the true second moment; consequently, the second moment must be infinite. ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! !  ! ! ! ! ! ! ! ! ! ! ! ! ! ! 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By repeating this for several overflow levels and studying the dependence of the resultingvariance on the overflow level, one can classify the simulation as asymptotically efficient or not. Furthermore,cases of infinite variance can be identified. Doing this on a grid of points in the    parameterspace for the P&Wtilting allows us to produce a complete picture of the behaviour of P&W, also in the areas where neither of theTheorems 3, 4 and 5 apply (i.e., the white area in Figure 1).The result of a scan of the entire    parameter space for the two-node tandem queue with P&W tilting isshown in Figure 2. Note that, in contrast to other results in this paper, this scan also covers the area in whichthe first queue is the bottleneck (i.e,    ).Theorems 3 and 4 provide sufficient conditions for exponential growth and infinite variance; these results (i.e.,curves C’ and D from Figure 1) have also been drawn in the figure, for comparison with the numerical results.Clearly, the numerically found boundaries follow the sufficient conditions rather closely.Finally, note that although the proofs only imply bounded relative error above line B in Figure 1, the numericalresults suggest that wherever the simulation is asymptotically efficient, it also has a bounded relative error;asymptotic efficiency without bounded relative error was never observed in the numerical study.0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.0µνbounded REexponential growthinfinite varianceFigure 2: Behaviour of the two-node tandem queue simulation under P&W tilting34 ConclusionsIn this paper, a detailed analysis of simulating a Markovian two-node tandem queue using importance samplingwith a state-independent change of measure has been presented. The main conclusion is that the only suchchange of measure that may be asymptotically efficient, is the one heuristically derived on the basis of largedeviations theory in [PW89]. Furthermore, the behaviour of the simulation under this change of measure hasbeen characterized further, thus extending the results from [GK95].Obviously, simulating this particular two-node queueing network is not very interesting by itself; typically,the overflow probability can be calculated numerically. However, the results from this paper do have implicationsfor future research in the general area of efficient simulation of queueing models.First of all, the fact that P&W is on the one hand not asymptotically efficient in parts of the parameter space,and on the other hand is the only state-independent tilting that may be asymptotically efficient at all, means thatmore complicated changes of measure are needed. Formally, this has only been established for this particularmodel, but given this model’s simplicity, it is to be expected that this is also true for a large class of other models.This gives a strong motivation for research into more complicated methods, like the adaptive state-dependentmethod from [dBN02]. Also, it means that methods (like [dBKR02]) that adaptively try to find the best state-independent tilting (rather than just assuming P&W) at least for this model will never be asymptotically efficientwhen P&W is not. (Still, such methods should not be discarded; they are simple and can still be used in the manysituations where an asymptotically efficient state-independent tilting does exist; and when the overflow level ofinterest is not too high, even a non-asymptotically efficient simulation may be acceptable.)Secondly, the results regarding the limitations of P&W tilting are of use in testing new simulation methods. Itis particularly interesting to test new simulation methods in parameter regions where P&W is not asymptoticallyefficient or yields infinite variance. Previously, P&W was only known to be not asymptotically efficient in a ratherlimited region of the parameter space (above curve C in Figure 1). The analysis and numerical study in this paperhave extended this region. Furthermore, we now know that there is also a region in which the variance underP&W is actually infinite. (Infinite variance under P&W previously was known for a different system, namely atwo-node tandem queue with feedback, see [RJ02].)Future extensions of this research could focus on finding analytical results for the parameter regions thatare not covered by the present theorems; on generalizing the statement that only P&W can be asymptoticallyefficient (Theorem 2) to a larger class of models; and on non-Markovian queueing networks.References[dB00] Pieter-Tjerk de Boer. Analysis and efficient simulation of queueing models of telecommunication sys-tems. PhD thesis, University of Twente, 2000.[dBKR02] P. T. de Boer, D. P. Kroese, and R. Y. Rubinstein. A fast cross-entropy method for estimating bufferoverflows in queueing networks. In Fourth Workshop on Rare Event Simulation and Related Combi-natorial Optimization Problems, RESIM/COP’02, 2002.[dBN02] Pieter-Tjerk de Boer and Victor F. Nicola. Adaptive state-dependent importance sampling simula-tion of markovian queueing networks. European Transactions on Telecommunications, 13(4):303–315,2002.[GK95] Paul Glasserman and Shing-Gang Kou. Analysis of an importance sampling estimator for tandemqueues. ACM Transactions on Modeling and Computer Simulation, 5(1):22–42, January 1995.[PW89] S. Parekh and J. Walrand. A quick simulation method for excessive backlogs in networks of queues.IEEE Transactions on Automatic Control, 34:54–66, 1989.[RJ02] R. S. Randhawa and S. Juneja. Simulating rare events by combining importance sampling and tem-poral difference methods. Draft, 2002.[Sad91] J. S. Sadowsky. Large deviations theory and efficient simulation of excessive backlogs in a GI + GI + mqueue. IEEE Transactions on Automatic Control, 36:1383–1394, 1991.4

Analysis of state-independent IS measures for the two-node tandem queue (extended abstract)

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Analysis of state-independent IS measures for the two-node tandem queue (extended abstract)

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