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1 HOW FAST IS THE STOCHASTIC EVALUATION OT THE EXPECTED PERFORMANCE AND ITS PARAMETRIC SENSITIVITY Michael Shalmon Universite du Quebec INRS-Telecommunications 16 Place du Commerce, Verdun, Quebec, Canada, H3E-1H6 Abstract. Telecommunications systems are typically modelled by queueing networks. While a crude apriori indication of the network performance can be obtained in special cases by mathematical analysis, the precise performance evaluation by o line stochastic simulation and by on line measurements is a central issue in systems analysis, design and operation. O line and especially on line evaluation of the gradient of the expected performance with respect to the various parameters (such as arrival rate, service rate or routing probabilities) is also very important as it not only measures the sensitivity to parameter change but is also needed for optimizing the network conguration (ow and/or capacity assignement). For ressource allocation and for determining the speed with which optimization algorithms adapt to changing networks conditions, it is important to know the necessary computing or measurement time for performance evaluation and its optimization, given a needed accuracy. This paper provides analytical formulas for this purpose. 1. Introduction. For the complex queueing networks arising in modelling telecommunication systems, an analytical solution is typically not available unless the model is drastically simplied. The precise evaluation by o line stochastic simulation and/or by on line measurements of the expected performance (such as the mean or the variance of the steady state waiting time) and of its gradient with respect to the various parameters (such as arival rate, service rate and routing parameters) is a central issue in systems analysis, design and operation. In this paper we examine and give an analytical answer (in a restricted setting) to the folowing fundamental question. Given a needed accuracy, what is the necessary computing or measurement time for performance eval- Supported by NSERC grant OGP

2 uation and its optimization? Note that answering this question is important not only in determining ressource allocation but also in determining the speed with which optimization algorithms adapt to changing networks conditions such as for example an increase in the incoming ow of packets. 1.1 Stochastic solutions to network optimization problems. Consider the following network optimization problems, Kleinrock [6]: Problem 1. Capacity assignement. Given link ows f i g (i.e. given routing), choose link service rates f i g constrained by given overall capacity i = M to minimize overall mean delay i ED i. Problem 2. Flow assignement. Given link service rates f i g, choose link ows f i g constrained by given origin-destination pair trac (i.e. choose routing) to minimize overall mean delay. To solve the rst problem, one needs to i, the gradient with respect to the service rate, where as to solve the second problem one needs to i, the gradient with respect to the arrival rate. As already mentioned, the only way to precisely evaluate these gradients for real systems is to stochastically simulate the system o line and/or to rene and update the evaluation from on line measurements. Once one has an ecient estimation method for the gradient it is possible to use the estimated values in a stochastic optimization algorithm, see Rubinstein [8]. While the expected performance can be estimated by simply calculating the sample mean, obtaining the gradient of the expected performance is trickier. Estimating at two slightly dierent parameter values and approximating the gradientby the nite - 2 -

3 dierence of expected performances, while an obvious possibility for o line simulation (but not for on line evaluation), produces bias which isinversely proportional to the variance of the error. However, other methods recently advocated in the queueing litterature produce estimators which are (asymptotically) unbiased, and also circumvent the need to simulate at two dierent parameter values. Because only a single sample path is required, these methods can be used during operation without disturbing the system, the stochastic optimization procedures (of the Robbins-Monroe type) being as in the deterministic case, with the estimated gradient replacing the true one. There are basically two such methods. The rst is obtained by computing the (in- nitesimal) change in the sample path likelihood due to an (innitesimal) change in parameter, Rubinstein [7],(1989), Reiman and Weiss [6],(1989), and is called innitesimal likelihoodratio (LR) or score fuction. The second is obtained by computing the (innitesimal) change in the sample path, Ho and Cao [3],(1983), Suri and Zazanis [13],(1988), Suri [14],(1989), and is called innitesimal perturbation analysis (IPA). While the variance of the estimators can be estimated during the evaluation by using batches or regenerative cycles, and the condence interval of the estimator thus computed, in order to design the procedure for performance evaluation and its optimization, it is important to have an a priori estimate of the time required to achieve the needed accuracy. As already mentioned, the importance lies not only in determining ressource allocation but also in determining the speed with which optimization algorithms adapt to changing networks conditions such as for example an increase in the incoming ow of packets

4 1.2 Previous work. While such an estimate is of clear practical interest, it is also dicult to obtain, and there are few analytical results known. Formulas for the variance of the mean queue length estimator for the M=G=1 queue are reviewed by W. Whitt [15],(1989) in his prize winning paper, where he develops heavy trac approximations based on the Brownian motion approximation of the queueing process. For queueing moments of higher order, such heavy trac approximations to the variance are computed by Asmussen [1],(1992), while heavy trac approximations to the variance of the LR based "score function" gradient estimator are computed by Asmussen and Rubinstein [2],(1992), (the formulas there are given in terms of recursively computed polynomials). There do not seem to be any computations for IPA estimators. 1.3 The paper. In this paper and in its companions [10],[11] we derive and study formulas for the variance of the estimators for the M=G=1 waiting time moments of arbitrary order and for their gradient with respect to either the service rate or the arrival rate. The formulas are valid at all trac intensities. The queueing analysis technique is taken from Shalmon [9],(1988), and relies on a busy period decomposition which has a triple interpretation: in terms of the ladder variable representation of the waiting time process, in terms of a birth-death branching process representation of the busy period, and in terms of the LCFS preemptive resume discipline. While [11] focusses on the queueing derivations, and [10] focusses on the (mostly heavy trac) study of the formulas for the estimators of the waiting time moments and their gradientbythe LR based score function method, this paper focusses on the study of the formulas for - 4 -

5 the estimators of the mean waiting time and gradient by the IPA method at all traf- c intensities. Note that the IPA method for computing the gradient is the fastest. While the formulas of this paper and its companions are strictly true only for the M=G=1 queue, note that the very simple heavy trac and (to a lesser extent) the low trac limit formulas are valid for much more general arrival processes. For example, it is well known that for quite general arrival processes the single server queueing process tends in the heavy trac limit (and after suitable scaling) to reected Brownian motion. Numerical calculations (in section 3) show that interpolating from the heavy and the light trac limit formulas is accurate within 25% in medium trac. This fact, together with the wide applicability of the heavy and ligth trac limits is particularly signicant for practitioners, as the formulas are to be used in the design of stochastic performance evaluation and optimization of more complex queueing models, particularly those with Poisson arrivals. In section 2 we rst prove a very useful 'Simulation exchange Lemma', valid for any single server queue, and which relates the eect of scaling changes in the service times to the eect of scaling changes in the interarrival times. For Poisson arrivals, the Lemma implies that changes with respect to the service rate are simply related to changes with respect to the arrival rate, and therefore one can easily translate an evaluation prole for varying service rate into an evaluation prole for varying arrival rate. This is also true for the gradients, and so, the variance formulas in the paper and which correspond to the gradient with respect to the arrival rate apply also with minor modications to the gradient with respect to the service rate. We then introduce the (known) various estimators for the M=G=1 queue, including a short - 5 -

6 original derivation of the IPA gradient estimator. In section 3 we present and study the formulas for the estimators performance. We relate the asymptotic bias and variance of the estimators to the variance and covariance of busy period statistics, and check numerically the heavy and light trac approximations at several trac intensities. We compare the performance of the two gradient estimators (nite dierence and IPA), relate it to the performance of the estimator for the mean, and point out their interesting behaviour in the light and heavy trac limits. In Section 4 we review the results, and suggest ways to improve the estimators performance

7 2. The queueing estimators Assume that we have a single server queue with with mean interarrival time 1=. Let W represent the steady state waiting time random variable, b k the k ; th moment of the service time, and = b 1 the trac intensity. For the M=G=1 queue, the Pollaczek-Khinchin formula for the mean waiting time is EW = b 2 2b 1 1 ; (2:1) the gradient of EW with respect to is = b (1 ; ) 2 (2:2) and there is a similar formula for the gradient withrespect to the service rate. In this paper we are interested in the time needed to obtain these quantities by stochastic evaluation. We rst show that one can easily translate an evaluation prole for varying service rate into an evaluation prole for varying arrival rate, and in particular one can obtain the gradient with respect to the service rate from the derivative with respect to the arrival rate. We then introduce the estimators for expectation, for the gradient by nite dierence and by IPA. 2.1 Simulation exchangeability. We denote the FCFS waiting time of the n ; th task in a single server queue by W (n). If all the service times are changed by a factor (say), and if all the interrarival intervals are changed by a factor (say), then we denote the waiting time of the n ; th task by W (n )

8 Simulation exchange Lemma: W (n 1) = W (n 1 ;1 ) (2:3a) Proof: If the n rst (service time minus interarrival time) variables are all scaled by the same factor, W (n) is also scaled by the same factor (including the case when it equals zero). The Lemma follows by simple arithmetic Let be the service rate. The service times are (inversely) scaled by and (for Poisson arrivals) the interarrival time distribution is (inversely) scaled by. The Lemma shows how to translate a simulation prole for varying into a simulation prole for varying. Let W denote the stationary FCFS waiting time with parameters. We can restate the Lemma in statistical terms: Corollary 1. For Poisson arrivals: W ;1 W (2:3b) By taking expectations in (2.3b), setting =1+ and letting go to zero, we obtain the following corollary (rst given in Shalmon and Rubinstein [10]): Corollary 2. For = ;kew (2:4) Thus the estimators for the gradient with respect to the arrival rate (which we exclusively examine in this paper), together with the estimators for expectation also give the gradient with respect to the arrival rate

9 2.2 The estimator for the expectation. Within a busy period rank the tasks in order of arrival the task initiating the busy period has rank 0, the next arriving task has rank 1, and so on. Let W (r) represent the waiting time of the task with rank r. By the regenerative theorem, EW k = E P ;1 r=0 W k (r) E = lim m!1 P 1 ;1 r 1 =0 W k (r 1 )++ P m ;1 r m =0 W k (r m ) m (2:5) where represents the number of customers in a busy period. So the expectation EW k can be estimated by the ratio of two sample means taken over a number (to be determined) of i.i.d. busy-idle cycles. We denote by hw k i m the numerator in right hand side of (2.5), i.e. m times the sample mean taken over m cycles, and by [MW k ] m the m cycle estimator for the expectation EW k. If m = 1 the indexing will be omitted. With this notation we write EW k = EhW k i EhW 0 i [MW k ] m = hw k i m hw 0 i m (2:6a) (2:6b) where hw 0 i is alternative for, and [MW k ] m is the m cycle estimator for EW k. Note that the numerator and denominator in (2.6b) are correlated. We turn next to the gradient estimators. 2.3 The nite dierence gradient estimator. The nite dierence 2m-cycle gradient estimator is derived from the dierence of two m-cycle expectation estimators obtained from evaluating at two arrival rates diering by a (small) quantity. Thus the centered nite dierence gradient estimator of the k ; th waiting time moment is given by [G c W k ] 2m = ;1 [[MW k +=2 ] m ; [MW k ;=2 ] m] (2:7a) - 9 -

10 while the forward one sided nite dierence estimator is given by [G f W k ] 2m = ;1 [[MW k + ] m ; [MW k ] m] (2:7b) It is important to distinguish between two situations. In the rst, the evaluations at the two slightly diering paramweters are independent. In the second the evaluation at the perturbed parameter is derived from that at the nominal parameter. This is the case with nite perturbation analysis (PA). 2.4 The IPA gradient estimator. Next, we give a short and original derivation of the formula for the innitesimal perturbation analysis (IPA) gradient estimator of the k ; th waiting time moment. Consider the n ; th -realization busy period. Increasing the arrival rate by a factor (1 + =) is (for Poisson arrivals) equivalent in distribution to decreasing the interarrival intervals by the same factor, while the service times do not change. A rst increase in the waiting time of the r ; th ranked task is due only to the preceding tasks inside the n;th -realization busy period and it amounts to [=(+)] (n r), where (n r) denotes the time elapsed since the begining of that busy period and till the arrival of the r ; th ranked task. A second (potential) increase in the waiting time is due to the preceding tasks outside the n ; th -realization busy period. It is independent of the rst increase, it is the same for all the tasks belonging to the n ; th -realization busy period, and we denote it by (n). We can write W +! (n r) =W! (n r)+ +!(n r)+! (n) (2:8) where! represents the realization of the interrarrival times assuming unity mean, thus emphasizing the common origin of the sample path with parameter and of the

11 sample path with parameter +. Both queueing processes (with parameters and + ) are regenerative (the busy periods with parameter + being formed by the concatenation of a random number of busy periods with parameter ) and therefore asymptotically stationary and ergodic. We want to show now that the stationary moments of (n) are o(). The key observation, borrowed from Kaplan [4],(1980) is that (n) is unchanged if we replace each preceding -realization busy period by a point arrival with service time equal to the increase in the unnished work across that busy period. The variable we seek becomes the waiting time in an M=G=1 queue with arrival rate + and service time distributed as [=( + )] where is the length of the -realization busy period. Note now that from the well known Pollaczek-Khinchin recursive formula for the waiting time moments (see for example equation A.1 of [10]) it is immediately inferred that the k ; th moment of the waiting time in an M=G=1 queue is a linear combination of the second, third (k +1); th moments of the service time. As the service time in the equivalent M=G=1 queue is [=( + )], we have that E k = o() for all k 1. From this conclusion and (2.8) it follows by applying the binomial formula that W k +! (n r) ; W k! (n r) = k W k;1! (n r)!(n r)+! (n) (2:9) where the stationary random variable has mean o(). By taking empirical averages in (2.9) (i.e. by summing over r and n and dividing by the number of tasks), and replacing them in the limit n!1with the stationary expectations, we obtain EW k + ; EWk = k EWk;1 + E (2:10)

12 Letting now go to zero, we obtain an expectation formula for the gradient ofew = k EWk;1 (2:11) The m-cycle IPA estimator for the gradient ofthek ; th waiting time moment, to be denoted by [G IPA W k ] m, is therefore [G IPA W k ] m = k hw k;1 i m hw 0 (2:12) i m As an aside, note that formulas for EW k are independently derived as an intermediate step of the queueing analysis in [12], and could have been used to establish (2.11) in an indirect way

13 3. The accuracy of the M=G=1 estimators as a function of time We use in this section the variance and covariance formulas derived in [12] to determine the number of busy-idle cycles, the number of tasks (customers), and the time required for a given accuracy of of the estimate of EW and of its gradient with respect to. Let V : : be the (asymptotic) coecient of variation per cycle for a particular estimator. For precise denitions see (3.1),(3.5),(3.9). The asymptotic gaussianity of the estimators (see below) implies that to an asymptotic coecient of variation of q% corresponds (with probability.95) a relative accuracy of 2q%. Then, the number of cycles needed to attain a coecient of variation q% equals 10 4 V 2 : :=q, the number of tasks is obtained upon multiplication by (1 ; ) ;1 (the mean number of tasks per cycle), while the needed measurement time is obtained upon multiplication by b 1 (1 ; ) ;1 (the mean duration of a cycle). 3.1 The accuracy of the expectation estimator. The estimator for EW k is given in (2.6). The square of the coecient ofvariation (standard deviation relative to the mean) of the numerator hw k i is by denition C 2 E k = VAR hw k i [EhW k i] 2 while the square of the coecient of variation of the denominator is given by the same expression by setting k = 0. From (2.6) we can write: [MW k ] m EW k = 1+ k(m) 1+ 0 (m) =1+[ k(m) ; 0 (m)] 1 ; 0 (m)+ 2 0 (m) 1+ 0 (m) where k (m) =(EhW k i;m ;1 hw k i m )=EhW k i is the relative error in the estimation of EhW k i over m cycles. k (m) has zero mean, variance m ;1 C 2 E k and is asymptotically gaussian. The covariance E 0 (m) k (m) equals m ;1 [(EhW k i=eehw k i) ; 1]

14 Therefore, as m grows large and the relative error becomes small (the desired situation), we approach asymptotically the equalities m VAR[MWk ] m [EW k ] 2 C 2 E k + C2 E 0 ; 2 EhW k i EEhW k i ; 1 4= V 2 M k m E[MWk ] m ; EW k EW k ; C 2 E 0 + EhW k i EEhW k i ; 1 (3:1a) (3:1b) The right hand side of (3.1a) is the (asymptotic) squared coecient ofvariation of the k-th moment of the waiting time per cycle which we seek to compute. From (3.1a) it is sucient to compute the coecients of variation of hw k i, and their covariance. The right hand side of (3.1b) is the asymptotic relative bias. Note that the relative bias decreases as m ;1 while the coecient of variation decreases as m ; 1 2, and so, for small coecients of variation the relative bias is negligible. Remark. We will use the expression "relative deviation" p to mean that the coecient of variation equals p. C E k is given at all trac intensities by the recursive formulas (A.1) and (A.4) in the companion papers [10],[11]. C 2 E 1 explicitely equals: CE 1 2 =5b 2 b 2 (1 ; ) ;1 + 1 (2+8 b 3 b 1 b 2 ; b 1 b 3 b 2 ;1 b 1 b 3 b 2 2 ; b 4 b 2 2 ; 5 b 2 b 2 ) +(3+ b 4 1 b 2 ; 5 b 2 b 2 )+ 1 (3:2a) Thus C E 1 reduces to 5(1 ; ) ;1 +(8=3) ; 1+(4=3) ;1 for the M/D/1 queue and to 10(1 ; ) ;1 +8 ; 1+2 ;1 for the M=M=1 queue,while C 2 E 0 equals (1 ; );1 for the M/D/1 queue and (1 + )(1 ; ) ;1 for the M/M/1 queue. Note that the ratio C 2 E 1 =C2 E 0 decreases from innity at = 0 to 5 as! 1, and so the inuence of the correlation between hw i and is greatest in heavy trac

15 Remark. In the companion paper [10], the explicit formulas for CE 1 2 in the particular M=D=1 and M=M=1 cases are incorrectly given, although the general formulas for the computation of C 2 E k given in that paper are correct. Note also that the high trac limit is unafected by the mistake, and so all the conclusions of [10] remain valid. EhW k i is given at all trac intensities by a recursiveformula derived in Shalmon [11]. For k =1,the explicit result is: EhW i EEhW i ; 1=2b 2 b 2 (1 ; ) ;1 + 1 (1 + b 3 ; 2 b 2 b 1 b 2 b 2 ) +(1; 2 b 2 1 b 2 ) 1 (3:2b) Thus [EhW i=eehw i;1] = CORRf hw igc E 1 C E 0 equals (1 + )(1 ; ) ;1 for the M=D=1 queue and (1 + 3)(1 ; ) ;1 for the M=M=1 queue. We omit the expressions for the correlation coecient but note, as an aside, that the correlation coecient between and hw i in the case of an M=M=1 queue tends to 1= p 2 :7 as! 0, equals 5 p 2=9 :78 at.5 trac intensity, and tends to 2= p 5 :9 as! 1. By carrying out the calculations in (3.2) for k =1we obtain VM 1 2, the asymptotic squared coecient of variation per cycle for the mean waiting time estimator: VM 1 2 =2b 2 b 2 (1 ; ) ;1 + 1 (1+6 b 3 b 1 b 2 ; b 1 b 3 b 2 ;1 b 1 b 3 b 2 2 ; b 4 b 2 2 ; 2 b 2 b 2 ) +(1+ b 4 1 b 2 ; 2 b 2 b 2 )+ 1 (3:3) V 2 M 1 reduces to 2(1 ; );1 +(8=3) +(4=3) ;1 for the M=D=1 queue and to 4(1 ; ) ; ;1 for the M=M=1 queue. The (1 ; ) ;1 term corresponds to the

16 heavy trac approximation! 1, while the ;1 term corresponds to the light trac approximation! 0. The asymptotic behaviour is highlighted below: Comment 1: In very light trac (where EW is almost proportional to, while E 1) the number of simulation tasks for a given "relative (standard) deviation" is proportional to ;1 (). In very heavy trac (where E and EW are each proportional to (1;) ;1 ) the number of simulation tasks for a given "relative (standard) deviation" is proportional to (1 ; ) ;2 ((1 ; ) ;4 ). For a given "relative deviation" one needs less tasks for medium trac than for either very light trac or very heavy trac. The "very light" trac behaviour can also be obtained by a single measurement at "light" trac followed by linear interpolation backwards to zero. As EW is almost proportional to (1 + ) in light trac, such a procedure results in a relative bias equal to. For an allowable bias of 1% (5%) one must interpolate backwards from = :01 ( = :05). For the M=M=1 queue, the estimator "relative deviation" per task at = :01 ( = :05) still exceeds that at = :5by a factor of 6.7 (1.6). See the concluding section for further discussion of this point. Numerical values computed both from the exact formulas and for the combined heavy and light tracapproximations are presented in Table 1 for the squared coef- cient of variation of the M=M=1 mean waiting time estimator (asymptotically) per cycle, and in Table 2 for the squared coecient ofvariation (asymptotically) per task. Multiplication of the coecients by 10 4 gives the number of cycles or tasks needed to obtain 1% "relative deviation"

17 Trac intensity VM (207) 28 (24) 15.5 (12) 51.5 (44) 412 (402) Table 1. The squared coecient of variation (asymptotically) per cycle for the estimator of the M=M=1 mean waiting time. The combined high and low trac approximation is indicated in paranthesis. Trac intensity VM 1 2 =(1 ; ) 209 (209) 31 (27) 31 (24) 515 (440) (40200) Table 2. The squared coecient of variation (asymptotically) per task for the estimator of the M=M=1 mean waiting time. The combined high and low trac approximation is indicated in paranthesis. Comment 2. For the estimator of the M=M=1 (and also M=D=1) mean waiting time, the combined light and heavy trac approximation is lower than the exact value, and is within 25% of it in medium trac. 3.2 The accuracy of the nite dierence gradient estimator. We turn now to the nite dierence gradient estimators [G W ] 2m dened in (2.7). For small enough, and from the Taylor expansion, the bias of the centered nite dierence estimator approximately equals ( 2 =24)(@ 3 EW =@ 3 )whilethebias of the forward one sided nite dierence estimator equals (=2)(@ 2 EW =@ 2 ) From (2.1) it follows that to ensure small bias b 1 =2 must be small compared to (1 ; ), and that the centered gradient estimator has relative bias given by b B c 1 2 2(1 ; ) (3:4a) while for the forward one sided gradient estimator the relative bias is B f b 1 1 ; (3:4b) From (3.4) one can determine for a given relative bias. Note that for a relative bias constant with one must take proportional to (1 ; ). Note also that for the

18 centered gradient estimator b 1 =2 cannot exceed, and so for the centered gradient estimator the relative bias cannot be chosen to exceed [=(1 ; )] 2 which is 1 for :5 and equals approximately 2 in light trac. On the other hand, for the one sided nite dierence gradient estimator in light trac one can chose b 1 to exceed provided the relative bias b 1 =(1 ; ) meets the requirement. We turn now to the variance. If the two evaluations are obtained by slightly perturbing the sample path (PA), then the variance is going to be very close to that of the IPA gradient estimator (see next subsection). Assume for the remaining of this subsection that the two evaluations are independent. The condition for small relative bias also ensures that the change in the mean estimator variance from ; =2 to + =2 is small. So, by independence: 2mV AR[G W ] 2m 4 2 mv AR[MW ] m (3:5a) Normalizing the standard deviation =@, which (from (2.1),(2.2)) equals ;1 (1 ; ) ;1 EW,we obtain a simple relationship between the squared coecients of variation (asymptotically) per cycle of the mean waiting time and of its independent replications nite dierence gradient: V 1 2 2mVAR[G W ] 2m 2(1 ; ) 2 VAR[MW m ] )2 b 1 (EW) 2 (3:5b) where b 1 (1 ; ) and for the centered estimator b 1 =2. We can now use (3.4),(3.5) together with (3.3) to determine the number of tasks needed to reach a desired relative precision in the estimation of the gradient of the mean waiting time with respect to the arrival rate. In all trac except the very

19 light, the requirement of small bias b 1 =2 (1 ; ) also ensures automatically that b 1 =2 (1 ; ). In very light trac, where (1 ; ), the situation is dierent for the centered and for the forward one sided estimators, as the limitation B c (b 1 =2) 2 2 applies to the centered estimator but not to the one sided estimator. From (3.4),(3.5) we have that: Comment 3. Given an allowable relative bias B, and for the same coecient of variation, the number of tasks needed by the independent replications centered gradient estimator is about equal to that needed by the mean estimator if 2 B, and otherwise exceeds it by a factor equal to 2 =B the number of tasks needed by the one sided estimator is 4 2 =B 2 times the number needed for the mean estimator. Thus, in all trac except the very light the centered estimator is superior to the forward one sided estimator, the breakpoint occuring at = B=2. However, as remarked before, the relative bias incurred by linearly interpolating backwards from = B=2 is B=2 and therefore the very light trac advantage of the one sided estimator is illusory. The denition of precision for a biased estimator and the choice of bias is somewhat arbitrary. A reasonable balance between bias and random error is obtained by allowing a relative bias about equal to (or somewhat smaller than) the coecient of variation, the overall error being somewhat greater than for an unbiased estimator. In Table 3 below, we give the squared coecient of variation (asymptotically) per task of the centered nite dierence gradient estimator for the M=M=1 queue assuming q% relative bias (and q>:01)

20 Trac intensity V 1 2 =(1 ; ) /q 41700/q /q Table 3. The squared coecient of variation (asymptotically) per task for the independent replications centered nite dierence estimator of the gradient of the M=M=1 mean waiting time. The relative biasisq%. 3.3 The accuracy of the IPA gradient estimator. We turnnow to the IPA gradient estimator [G IPA W k ] m as dened in (2.12). The squared coecient of variation for the numerator hw k;1 i is by denition C 2 IPA k = VARhW k;1 i [EhW k;1 i] 2 As m grows large and the relative error becomes small, we have by repeating the argument leading to (3.1) that the coecient of variation of [G IPA W k ] m asymptotically satises: m VAR [G IPAW k] m C 2 EhW ]2 IP A k + C2 E 0 ; 2 i 4= EEhW k;1 i ; 1 V 2 ; 1 ; CE 0 2 EhW k;1 + i EEhW k;1 i ; 1 m E[GIPA W k ] IP A k (3:6a) (3:6b) So, to evaluate asymptotically the IPA estimator for the gradient of the k-th moment of the waiting time, it is necessary and sucient to compute the coecientofvariation of hw k;1 i and its covariance with. Also, for small coecients of variation the relative bias is negligible. C IPA k is given at all trac intensities by a recursive formula in Shalmon [11].For k = 1,the explicit result is: CIPA 1 2 =15b 2 b 2 (1 ; ) ;1 + 1 b 3 ( b 1 b 3 b 2 ;1 b 1 b 2 ; 8 3 b 1 b 3 b 2 2 ; b 4 b 2 2 ; 15 b 2 b 2 ) +(5+ b 4 1 b 2 ; 15 b 2 b b 1 b 3 b 2 )+ (3:7)

21 Thus C 2 IPA 1 reduces to 15(1 ; );1 ; (4=3) ; (23=3) + (4=3) ;1 for the M=D=1 queue and to 30(1 ; ) ;1 ; 4 ; 17+2 ;1 for the M=M=1 queue. Note that the ratio C 2 IPA 1 =C2 E 0 decreases from innity for = 0 to 15 as! 1, and thus the inuence of the correlation between hi and is greatest in heavy trac. EhW k;1 i is computed in Shalmon [11]. For k =1the explicit result is: Ehi EEhi ; 1=3b 2 b 2 (1 ; ) ;1 + 1 (2 + b 3 ; 3 b 2 b 1 b 2 b 2 ) +(1; 3 b 2 1 b 2 ) 1 (3:8) Thus [Ehi=EEhi;1] = CORRf higc IPA 1 C E 0 equals (1 + 2)(1 ; ) ;1 for the M=D=1 queue and ( )(1 ; ) ;1 for the M=M=1 queue. We omit the expressions for the correlation coecient but note, as an aside, that the correlation coecient between and hi in the case of an M=M=1 queuetends to 1= p 2 :7 as! 0, equals approximately.79 at.5 trac intensity, and tends to 3= p 15 :78 as! 1. By carrying out the calculations in (3.6) for k =1we obtain VIPA 1 2, the squared coecient of variation (asymptotically) per cycle of the IPA estimator for the gradient of the mean waiting time: VIPA 1 2 =10b 2 b 2 (1 ; ) ;1 + 1 ( b 1 b 3 b 2 ;1 b 3 ; 8 b 1 b 3 b 1 b 2 3 b 2 ; b 4 b 2 ; 10 b 2 b 2 ) +(4+ 4 b 1 b b 2 + b 4 b 2 ; 10 b 2 b 2 )+ 1 (3:9) V 2 IPA 1 reduces to 10(1;);1 ;(4=3);11=3+(4=3) ;1 for the M=D=1 queue and to 20(1;) ;1 ;3;8+2 ;1 for the M=M=1 queue. The (1;) ;1 term corresponds to

22 the heavy trac approximation! 1, while the ;1 term corresponds to the light trac approximation! 0. The "relative deviation" qualitative dependence on trac is (up to a constant) exactly as in the estimation of the mean waiting time. The dependence of the standard deviation on trac is dierent according to the dierence between (2.1) and (2.2). The asymptotic behaviour is highlighted below. Comment 4. For a given "relative deviation", the number of tasks needed for the IPA gradient estimator equals in very light trac the number needed for the mean estimator (and is proportional to ;1 ), while in very heavy trac, it exceeds the number needed for the mean estimator by a factor of 5 (and is proportional to (1 ; ) ;2 ). In very light trac is almost constant) the number of tasks needed for a given standard deviation is proportional to ;1, while in very heavy trac is proportional to (1 ; ) ;2 ) it is proportional to (1 ; ) ;6. Again, the number of tasks is less in medium trac than in either very light or very heavy trac. Linearly interpolating backwards from a measurement at produces a relative bias equal to 2. For an allowable relative bias of 1% (5%), one must interpolate backwards at = :005 ( = :025). For the M=M=1 queue the coecient of variation per task at = :005 ( = :025) exceeds that at = :5 by a factor of 6 (1.33). For further discussion on this point seesection 5. Comparing this performance with that of the independent replications centered nite dierence gradient estimator, we note that: Comment 5. For the same coecient of variation, and allowing the independent replications nite dierencegradient estimator a relative bias of q%, the IPA gradient estimator saving factor in tasks increases from about 1 in very light trac to about

23 20=q in very heavy trac. As already remarked, it is reasonable to choose the relative bias about equal to (or somewhat smaller than) the standard deviation, in which case the q% in comment 5. represents the overall relative precision, in the sense that the error will lie within plus or minus 3q%. Numerical values computed both from the exact formulas and for the combined high and light trac approximations are presented in Table (4) for the squared coecient of variation (asymptotically) per cycle of IPA gradient estimator of the M=M=1 mean waiting time, and in Table (5) for the squared coecient of variation per task. Trac intensity VIPA (220) 32 (42) 34 (44) 191 (201) 1991 (2002) Table 4. The squared coecient of variation (asymptotically) per cycle for the IPA gradient estimator of the M=M=1 mean waiting time. The combined high and low trac approximation is indicated in paranthesis. Trac intensity VIPA 1 2 =(1 ; ) 214 (222) 36 (46) 69 (88) 1910 (2010) (200200) Table 5. The squared coecient of variation (asymptotically) per task for the IPA gradient estimator of the M=M=1 mean waiting time. The combined light and heavy trac approximation is indicated in paranthesis. Comment 6. For the IPA gradient estimator of the M=M=1 (and also M=D=1) mean waiting time, the combined light and high trac approximation is higher than the exact value, and is within 25% of it in medium trac

24 4. Discussion and Conclusions. We examined the estimators for the M=G=1 mean waiting time and its IPA and nite dierence gradient with respect to the arrival (or service) rate, and presented detailed bias and variance calculation as a function of the estimation time. Note that the distribution and the moments of the M=G=1 queue length are related to the distribution and moments of the M=G=1 waiting time, and so the estimators analyzed here can be used to infer the queue length moments as well. In particular, and for much more general settings, the mean queue length is related to the mean waiting time by Little formula. The variance of the estimators for the M=G=1 queue length can also be computed directly by an analogous busy period decomposition interpretable in terms of the LCFS nonpreemptive discipline [12]. Note also that the method of analysis also extends in principle to the GI=GI=1 queue, Shalmon [9], but the calculations are much more complex. We review next our main ndings. (1) For small "relative deviations", both the mean waiting time estimator and the IPA gradient estimator have negligible relative bias, while the nite dierence gradient estimator has an irreducible bias. (2) For a given "relative deviation", and allowing the independent replications centered nite dierence gradient estimator a relative bias of q%, the IPA gradient estimator saving factor in simulation tasks increases from 1 in very light trac to 20=q in very heavy trac. (3) For a given "relative deviation", the number of simulation tasks needed for the

25 IPA gradient estimator equals in very light trac the number needed for the mean estimator, while in very heavy trac it exceeds the number needed for the mean estimator by a factor of 5. The saving of the IPA gradient estimator in (2) is signicant for high precision note however as an aside that the same comparison for i.i.d. observations of exponentially distributed random variables shows a saving of 100=q, and that unlike in (3), without any increase with respect to the estimator of the mean. (4) Numerical calculations show that the combined low and heavy trac are accurate within 25% in medium trac. This is an encouraging observation as the low and trac approximations are valid for much more general settings than the M=G=1 queue. Recursive formulas for (arbitrarily) higher waiting time moments and their IPA gradient, and valid at all trac intensities appear in Shalmon [11]. From their explicit low and heavy trac limits we can generalize our ndings on the performance of the mean waiting time estimator and of its IPA gradient estimator as a function of trac intensity: (5) The "relative deviation" for the estimator of the k ; th moment ofthewaiting time has (up to a known constant) the same asymptotic behaviour for all k. In very light trac the number of simulation tasks for a given relative (standard) deviation is proportional to ;1 (), while in very heavy trac it is proportional to (1 ; ) ;2 ((1 ; ) ;(2k+2) ). (6) The "relative deviation" for the IPA gradient estimator of the k ; th moment of the waiting time has (up to a constant) the same asymptotic behaviour for all k. In very light trac the number of simulation tasks for a given relative

26 (standard) deviation is proportional to ;1 ( ;1 ), while in very heavy trac it is proportional to (1 ; ) ;2 ((1 ; ) ;(2k+4) ). Remark 1. The light trac result implying for the same relative deviation longer simulations than in medium trac might seem at rst sight puzzling. It is caused by the fact that in very light trac almost all tasks encounter an empty queue, and thereby are wasted for expectation calculations. We showed that by linearly interpolating backwards in very light trac we partially solve the problem. Note that linear interpolation in very light trac is nothing but importance sampling according to likelihood ratio, and this can be done at all trac intensities. The evaluation of the variance for such acombined LR-IPA is left for future work. Remark 2. The heavy trac result (both for relative deviation and in particular for standard deviation) is very disapointing. The main lesson to be drawn is that one must use analysis to reduce the amount of evaluation time, for example using the analytical fact that the heavy trac queueing process is approximated (after suitable scaling) by reected Brownian motion

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