Baltzer Journals Received 7 January 1997; revised 16 April 1997 An Interpolation Approximation for the GI/G/1 Queue Based on Multipoint Pade Approxima

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1 Baltzer Journals Received 7 January 1997; revised 16 April 1997 An Interpolation Approximation for the GI/G/1 Queue Based on Multipoint Pade Approximation Muckai K Girish 1 and Jian-Qiang Hu 1 Telesis Technologies Laboratory 5000 Executive Parkway, Suite 333 San Ramon, CA mgirish@ttl.pactel.com Dept. of Manufacturing Engineering Boston University 15 St. Mary's Street Boston, MA hu@enga.bu.edu The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy trac conditions. In this paper, we propose a new approximation technique, which combines the light and heavy trac characteristics. This interpolation approximation is based on the theory of multipoint Pade approximation which is applied at two points: light and heavy trac. We show how this can be applied for estimating the waiting time moments of the GI/G/1 queue. The light trac derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy trac limits of the GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of trac intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. Keywords: GI/G/1 queue, heavy trac limits, MacLaurin series, multipoint Pade approximation. 1 Introduction Queueing systems are used extensively to model various manufacturing, communication and computer systems. The estimation of the congestion measures in these systems is one of the main issues in queueing theory and has been studied widely. The lack of Queueing Systems: Theory and Applications, Vol. 6, 69-84, 1997.

2 M. K. Girish and J-Q. Hu / An Interpolation Approximation analytical solutions for many systems have led to the development of approximations in the literature. These approximations can be very accurate, but, can also be inaccurate under certain conditions. Many approximations have been built upon the behavior of queues in light or heavy trac conditions which make them work well in and around the respective trac intensities. It has, thus, become clear that a robust approximation should takeinto account both the light trac and heavy trac characteristics. A few such approximations have appeared in the literature which use various interpolation methods. The contribution of this paper is the development of a novel approach ofinterpolation based on the light and heavy trac characteristics. In particular, we apply the approachto estimate the moments of the waiting time in the GI/G/1 queue. We use the heavy trac limits and our method provides the exibility tochoose as many light trac derivatives as needed. The performance measures that we are interested in estimating in this paper are the moments of the steady state waiting time. We consider a parameterized queue in which the service time is S, where is a scale parameter and S is the service time random variable in the original queue. Therefore, when = 0, the parameterized queue is in light trac and when =1= this queue is in heavy trac, where is the trac intensity of the original queue. Note that some authors prefer to set the arrival rate equal to zero in light trac and equal to the service rate in heavy trac. Then, we calculate the light trac derivatives and the heavy trac limits of all the waiting time moments. Using this information, we develop an approximation procedure based on the technique of multipoint Pade approximation to estimate the performance measures. The estimates of the waiting time moments are obtained by setting =1. Analysis of queues in light trac have been carried out by many authors. Methods are developed in [18] to calculate the derivatives of the performance measures of arbitrary order in light trac for single server queues with Poisson arrival processes. Likelihood ratio methods were used in [0] toevaluate the light trac derivatives. Extension of the light trac behavior to multi-server queues was initiated by []. A new way to calculate the light trac derivatives of the GI/G/1 queue was proposed in [7]. They showed that the moments of the waiting and system time can be expanded as MacLaurin series in terms of a scale paramenter in the service distribution. These MacLaurin series coecients can be calculated using simple recursive equations. We use this procedure to evaluate the light trac derivatives in this paper. A procedure was developed in [5] to analyze tandem queueing networks using the light trac derivatives. Their numerical results show that this MacLaurin series approximation technique works well in most cases. It was also realized that the heavy trac limits play a critical role in the accuracy of the performance measures especially when the trac intensity gets closer to one. The property of single server queues in heavy trac is well known. In heavy trac, the departure process approaches the service distribution and these heavy trac limits have been successfully used in many approximations. Simple approximations based on the heavy trac limits were prosed in [15] and [16]. This was extended to open queueing networks with general external arrival and service distributions by [1]. Diusion approximations and reected Brownian Motions are very popular heavy trac approximation schemes (see [11], [14] and the references therein). Atwo-moment approximation using this approach for open queueing networks was developed in [10]. An interpolation approximation method was studied by [19]. They consider a queueing system with Poisson arrival process and evaluate the light trac derivatives and the

3 M. K. Girish and J-Q. Hu / An Interpolation Approximation 3 heavy trac limit. The performance measures are then approximated by rational functions. But, the selection of the function is quite arbitrary and there is no systematic way of selecting the correct function to be used, except for some special cases. This interpolation approximation has been found to work well for those cases, for which the denominator of the rational functions can be chosen correctly. Our approach considers queues with general independent arrival and service distributions and we give avery easy procedure to calculate the coecients of the rational functions. Another interpolation approximation was proposed by [] for the average workload in the GI/G/1 queue. This method uses the rst three moments and the density function of the interarrival time and the mean and variance of the service distribution. This approximation also works well, but, is restricted in terms of the number of terms to use and the evaluation of the approximation parameters is not striaghtforward. An ecient technique to approximate the performance measures which have derivatives at any point is the so called Pade approximation (see [1] and [17] for a comprehensive mathematical treatment). [9] showed that Pade approximation can be applied to a wide class of these systems including the GI/G/1 queue. This method was used by [5] to approximate the moments of the waiting and the interdeparture time in tandem queueing networks. If a performance measure function has information (such as values or derivatives) at several points, then we can use multipoint Pade approximation. The mathematical theory of multipoint Pade approximation is described in detail in [1] which forms the basis of our approximation. Multipoint Pade approximation has been used by [8] for estimating the waiting time and is applied at many points where the derivatives could be found by derivative estimation techniques or simulation. But, they do not use the heavy trac limits. The performance measures of interest in this paper are the moments of the steady state waiting time. Our approximation scheme essentially considers the heavy trac limit and an arbitrary number of the light trac derivatives. We then apply the theory of multipoint Pade approximation to estimate the performance measures. We consider a number of numerical examples in this paper which show that the waiting time moments can be estimated with extremely high accuracy. A signicant feature of this method is that even low orders of the rational approximant produce excellent estimates. We also compare this method with an example of the interpolation approximation of [] for the mean workload in a GI/G/1 queue. The rest of this paper is organized as follows. In Section, we introduce some notation and review the results pertaining to light and heavy trac behavior of queues. We present the multipoint Pade approximation technique in Section 3. A framework for the development of simple approximations is presented in Section 4. In Section 5, we consider some numerical examples and this paper is concluded in Section 6. Light and Heavy Trac Analysis In this section, we review the results in queueing theory that treat a queue in light and heavy trac conditions in order to lay the groundwork for our approximation technique. The relevant theory is taken from [5], [7] and [14]. First we introduce the following

4 M. K. Girish and J-Q. Hu / An Interpolation Approximation 4 notation: W = steady-state waiting time of a job T = steady-state system time of a job A = interarrival time between two adjacent jobs S = service time of a job, where S is independent of = a scale parameter in the service time w nm = The coecient of m in the MacLaurin series of E[W n ]=n! t nm = The coecient of m in the MacLaurin series of E[T n ]=n! f() = probability density function (p.d.f.) of A k = f (k) (0 + )=k-th right hand derivative off()at0 k = E[S k ]=k! k = E[A k ]=k! = E[S]=E[A] The interarrival and service distributions that we consider in numerical examples in this paper are mostly mixed Erlang distributions. It has been established that mixtures of Erlang distributions are dense and can approximate any distribution. A mixed Erlang distribution is characterized by the following parameters: b = number of branches p i = probability of branch i, for i =1;:::;b n i = number of stages in branch i, for i =1;:::;b i = rate in branch i, for i =1;:::;b It is well known that E[W n ] is inversely proportional to (1, ) n and so as a rst step, we normalize E[W n ] by a factor of (1, ) n in our estimation. Let n = minimum(n 1 ;n ;:::;n b ). It was observed by [5] that for the GI/G/1 queue with mixed Erlang interarrival distribution, the MacLaurin series coecients of E[W n ], w nm = 0 for m =0;1;:::;n+n,1. Hence, we estimate (1, ) n E[W n ]= n+n in this paper. Note that we can set n = 1 without loss of generality, in case we use distributions other than mixed Erlang (for example, uniform or deterministic). The only reason for using n is that the approximation produces the same estimates using lower orders of the approximant..1 Light Trac Derivatives Queues in light trac have been studied in detail by many authors. It was shown by [7] that the light trac derivatives, w nm and t nm can be calculated based on the following recursive equations t nm = w nm = 8 < : P n ; m = n n i=1 n,iw i(m,n+i) ; m>n 0; m<n P m,n,1 i=0 i t (n+1+i)m ; m>n 0; mn (1) ()

5 M. K. Girish and J-Q. Hu / An Interpolation Approximation 5 The moments of the waiting time was shown to be analytic at = 0 under some mild conditions by [1]. By analyticity, we mean that the MacLaurin series exists and converges to the right value around =0. Since we are interested in estimating (1, ) n E[W n ]= n+n, n =1;;:::,we need to obtain the MacLaurin series for these which, derived with suitable modications is given below: (1, ) 1X n X n E[W n ]=n! w n+n n(m+n+n) (,) i n m+i (3) i m=0 i=0 The above MacLaurin series expansion is based on the derivatives at = 0 and the light trac derivatives for our performance function can be obtained from this.. Heavy Trac Limits It is very easy to determine the heavy trac limits of the GI/G/1 queue, which can be obtained by taking the limits as! 1. The heavy trac limit for the rst moment of the waiting time is well known in the literature and is given below: lim (1, )E[W a+c s) 1!1= ]=(c where c a and c s represent the squared coecients of variation (variance divided by the square of the mean) of the interarrival and service time distributions, respectively. For the GI/G/1 queue, as! 1, the moments of the interdeparture time approach the moments of the service time. i.e., E[D n ]! n! n n ; for n =1;;::: (5) where D is the interdeparture time random variable. Then we use the following relationship between the waiting and interdeparture time presented in [5]: where E[ W n X n n! ] = (,1) n+j n+1,j E[ W j,1 (j, 1)! ]n+1,j j=1 3 + (,1)n+1 4 D n+1 n+1 X n+1, E[ 1, 1 (n + 1)! ]+ (,1) j n+1,j n+1,j j 5 (6) j = " j, X i=0 ( j,i, 1 j,i,1 )E[ W i j=1 i! ]j,i,1 +(,1) j,i j,i ix l=0 i,l E[ W l l! ]i,l # Iteratively, starting with n =, we set = 1 = 1! 1 in Equation (6) and use Equations (4) and (5) to obtain the heavy trac limits of the moments of the waiting time in the GI/G/1 queue as: lim (1,!1= )n E[W n (c ]=n! a +c n s) 1 (7) In the next section, we will show how the light trac derivatives and the heavy trac limits can be combined using the idea of multipoint Pade approximation. (4)

6 M. K. Girish and J-Q. Hu / An Interpolation Approximation 6 3 The Interpolation Approximation In this section, we show how the Multipoint Pade approximation technique given in [1] can be applied to estimate the moments of the waiting time. Let us call the function to be approximated as g () =(1,) n E[W n ()]= n+n, for n =1;;:::. The derivatives of g () are known at 1 = 0, and its limit is known at = 1=. Let g n 1 () be the MacLaurin expansion of g n () up to order m 1, 1 and let g n () be the Taylor expansion of g n () expanded around = up to order 0. These can then be expressed as: g 1 () = mx 1,1 j=0 u 1j j + O[ m1 ] (8) g () = u 0 + O[, ]; (9) where, from Equations (3) and (7), we get for n =1;;:::, (c u 0 = n! n+n a + c s) 1 u 1m = n! n X i=0(,) i n i The rational approximant that we are interested in is g () P L () Q M n (10) w n(m+n+n,i): (11) () : (1) We call Equation (1) a [L/M] multipoint Pade approximant, where L + M = m 1. Note that we use m 1,1 light trac derivatives and the heavy trac limit. The approximating rational function has polynomials of order L and M in the numerator and denominator, respectively. Next we dene an appropriate auxiliary function and by substituting it in the general multipoint Pade approximation scheme derived by [1] and dierentiating and taking limits (we omit the details here since the procedure is quite straightforward), we get: mx 1,1 i = u 0 i,m1,1, q () = u 0 q,m1 m1 + k=0 X u 1(m 1,i,k) k+1 m 1,q,1 m 1,1,k,q k=0 X i=0 u 1(m 1,1,q,k,i) (, ) m1,k,1 It is now easy to evaluate the rational approximant which is given in Equation (15). i+1 (13) (14)

7 M. K. Girish and J-Q. Hu / An Interpolation Approximation 7 For its derivation, refer to [1]. P L () Q M () = det det M M,1 ::: M.... M+1 M ::: 1 M M M,1 ::: 0 M,1 ::: M... M+1 M ::: 1 M M,1 ::: 1 (15) In the next section, a framework for the development of simple formulas for estimating the waiting time moments is developed. Numerical results are presented in Section 4. 4 Simple Approximations The interpolation approximation developed in the previous section can be used to come up with simple approximations, by choosing a few coecients at a time. For example, let us start with a [/1] approximant for (1, )E[W ]= (note that we set n = 1). We omit the superscript for convenience since we deal only with the rst moment in this section. w 10 = w 11 =0 w 1 = 0 w 13 = ( ) w 14 = ( )+ 3 0( ) u 1m = w 1(m+), w 1(m+1) for m 0 1 = 3 u 0, u 1, u 11, 3 u 10 = u 0, u 11, u 10 1 () = u 0 3 +(1,)[u 10 +(u 11 + u 10 ) ] () = 3 u 0 3 +(1,)[u 10 +(u 11 + u 10 ) +(u 1 + u 11 + u 10 ) ] From the above, we have E[W()] = ( 0 (), 1 1 ()) (1, )(, 1 ) (16) Next, we estimate (1, )E[W ] with a [/1] approximant. Then after appropriate modications to the interpolation approximation formulas, we get the following leading to Equation (17): u 0 = (c a + c s) 1

8 M. K. Girish and J-Q. Hu / An Interpolation Approximation 8 1 = 3 u 0, u 1 = u 0 0 () = 3 u 0 3 +(1,)u 1 1 () = u 0 3 E[W ]= 1(c a+c s)( 0 1 (c s +1)+(1, )(c a + c s)) (1, )( 0 1 (c s +1)+(1,)(c a + c s)) The above can be extended to the higher moments as well as for higher orders of the approximant. This approach can serve as a framework for the development of simple approximations for the GI/G/1 queue. In the next section, we present several numerical examples of the sigle server queue. (17) 5 Numerical Examples In this section, we present several numerical examples to compare the accuracy of the multipoint Pade approximation with the exact (simulation) values and also with other approximation procedures. In the rst example, we compare three approximations. The MacLaurin Series Approximation (MSA) is based on the alternate method developed by [5] which uses rational approximation based on the MacLaurin series to estimate the moments of the interdeparture time and then calculates the moments of the waiting time. The Queueing Network Analyzer (QNA) is a popular two-moment based approximation developed by [1]. The interpolation approximation developed in this paper is compared with the above two techniques. In the second example, we compare the waiting time estimates of our interpolation approximation with that of simulation. In the last example, we compare our approximation with an interpolation approximation for the mean workload developed by []. Example 1. In this example, a number of arrival and service distributions are considered, which are either Erlang (E 3 ), hyperexponential (H ) or three-branched mixed Erlang (ME). The rst two moments of the waiting time are compared at various trac intensities ( =.1,.3,.5,.7,.9). The arrival rate is set at 1 for all the distributions. For H arrival distribution, the parameters are: (p 1 ;p ) = (.6,.4) and ( 1 ; ) = (3.0, 0.5). For H service distributions, the parameters are (p 1 ;p )=(:6;:4) and ( 1 ; ) = 8 >< For ME arrival distribution, the parameters are: >: (1:0; 8:0); when = 0:1 (3:0; 4:0); when =0:3 (:4; 1:6); when =0:5 (1:7143; 1:149); when = 0:7 (1:3333; 0:8889); when = 0:9. (p 1 ;p ;p 3 ) = (:;:3;:5) (n 1 ;n ;n 3 ) = (; 4; 3) ( 1 ; ; 3 ) = (:0; :4; 5:0);

9 M. K. Girish and J-Q. Hu / An Interpolation Approximation 9 and for ME service distributions, the parameters are: (p 1 ;p ;p 3 ) = (:3;:;:5) (n 1 ;n ;n 3 ) = (4; ; 3) ( 1 ; ; 3 ) = 8 >< >: (40:0; 10:0; 50:0); when = 0:1 (15:0; 4:0; 1:5); when = 0:3 (8:0; 4:0; 6:0); when =0:5 (5:0; :71; 4:8); when = 0:7 (4:0; :0; 3:75); when = 0:9. The approximations that we consider are the Multipoint Pade (M-Pade), Girish and Hu's MacLaurin Series Approximation (MSA) and Ward Whitt's Queueing Network Analyzer (QNA). The simulation runs consist of 40 replications of 50,000 customers with 10,000 warm-up customers. All the simulation results in this paper are presented as estimate standard deviation. Numerical results are given in Tables I-VI. We evaluate the average absolute percentage error for all the approximants and this serves as a summary comparison characteristic among the various approximation schemes. For the M E=G=1 queue (Tables I and II), the M-Pade results are clearly superior to MSA as well as QNA for both E[W ] and E[W ] estimates. Even the [/] M-Pade approximation is better than the others; and the [3/3] M-Pade approximant improves the accuracy even more. Note that the [6/5] MSA approximant leads to large errors and so we used a very high approximant, [16/16], which of course does well, but not as well as the interpolation approximation. For the E[W ] estimates of the H =G=1 queue (Tables III), the [/] M-Pade approximation again leads to the lowest average absolute % error. But note that the [16/16] MSA approximant's error is only 0.73%. It has been found in [5] that MSA works very well when c a and/or c s are greater than one. But, we need higher orders of MSA to get to this accuracy level. For the E[W ] estimates, the error of the [/] M-Pade approximation is only worse than the [16/16] MSA error. The reduction in the order of the approximant that gives estimates with reasonably high accuracy is quite prominent and supports the use of the interpolation approximation. For the E 3 =G=1 queue, for both E[W ] and E[W ](Tables V and VI), the [/] M- Pade approximation is better than QNA and [6/5] MSA. Similar to the E[W ] estimates of the H =G=1 queue, the [16/16] MSA is very accurate. But the extra computation required for this high order of the approximant doesnotoutweigh the reduction in the accuracy for the interpolation approximation which is only slightly worse and is well within the acceptable level. Example. The arrival process in this example is uniformly distributed between [0,] and the service distribution is Erlang with stages. We consider various trac intensities and compare the rst two moments of the waiting time evaluated using multipoint Pade approximation with that of simulation. We actually estimate (1,) n E[W n ], for n =1; in this example. It can be easily observed that i =0,fori1. Numerical results are given in Table VII. Our interpolation approximation's errors are.006% and 9.394% for the rst and second moment, respectively which are reasonable. It was proved by [1] that the moments of the waiting time are not analytic for this system since the uniform interarrival distribution function is not analytic. We believe that this might explain the relatively poorer performance of the interpolation approximation.

10 M. K. Girish and J-Q. Hu / An Interpolation Approximation 10 Example 3. In this example, we compare our approximation technique with the results in Table II of [], where he developed approximations for the normalized mean workload, c z. The mean waiting time is related exactly to the mean workload by the following expression (see [3]): c z E[W ]= (1, ), 1 1 (18) We use Equation (18) to convert the mean workload given in Table II of [] to the mean waiting time. The rate parameters for the E and exponential distributions are given in Tables VIII and IX. For the H service distribution, p 1 = and p = and the rates are given in Table X. We have omitted the cases in Table II of [] with = 0.0 and 1.0. Numerical results for E[W ] are shown in Table VIII-X. It can be seen that our method is clearly superior (the average absolute % errors for the three cases are 0.45%, 0.19% and 0.576% for M-Pade and 7.374%, 7.431% and 3.433% for Whitt, respectively). 6 Conclusions In this paper, we proposed a novel approach of interpolation for the GI/G/1 queue which eciently combined the light and heavy trac behavior of queues in arriving at very accurate estimates for the moments of the waiting time. The theory of multipoint Pade approximation formed the basis of the method. This technique requires the heavy traf- c limits which can be easily calculated, and any number of the light trac derivatives which can be evaluated using recursive formulas easily. This method generalizes some of the interpolation approximations proposed in the literature. A signicant feature of this method is that it is applicable to any order of the moments. We also show that very low orders of the approximant produces very accurate and robust estimates. We conducted several numerical experiments and the results show that the multipoint Pade approximation is superior to any known approximations under all trac intensities. We also developed a framework for simple approximations using this interpolation scheme. Though not mentioned before, the same approach developed in this paper for the waiting time moments can be extended to the moments of the system time as well as the interdeparture time. The light trac derivatives of system time and interdeparture time have been derived in [7] and [13], respectively. The heavy trac limits of the moments of the system time and the interdeparture time can also be easily evaluated. A direction for future research will be the study of the convergence properties of multipoint Pade approximation. Our numerical experience indicated that when the number of stages in the mixed Erlang arrival distribution is high (more than 8), this procedure does not work well in medium to heavy trac conditions. Another possible direction is to extend this methodology to estimate the performance measures of other queueing systems as well for which light trac derivatives and heavy trac limits can be evaluated. An example is the estimation of the waiting and interdeparture time moments of the G/G/1 queue with Markov-modulated arrivals which has been widely used to model queueing systems with autocorrelated arrivals. The light trac derivatives of the waiting time and the interdeparture time were derived by [3] and [6], respectively which together with the heavy trac limits (if they exist) facilitate the application of interpolation approximations.

11 M. K. Girish and J-Q. Hu / An Interpolation Approximation 11 Acknowledgments This research is supported in part by the National Science Foundation under grants DDM , and EID-911. The authors wish to thank the anonymous referee for the helpful comments. References [1] G. A. Baker, Essentials of Pade Approximants. Academic Press, [] D. Y. Burman and D. R. Smith, A Light-Trac Theorem for Multi-Server Queues, Math. Oper. Res., 8, 15-5, [3] K. W. Fendick and W. Whitt, Measurements and Approximations to Describe the Oered Trac and Predict the Average Workload in a Single-Server Queue, Proceedings of the IEEE, Vol. 77, No. 1, , [4] K. W. Fendick, V. R. Saksena AND W. Whitt, Dependence in Packet Queues, IEEE Transactions on Communications, 37, , [5] M. K. Girish and J. Q. Hu, Higher Order Approximations for Tandem Queueing Networks, Queueing Systems: Theory and Applications, Vol., 49-76, 1996a. [6] M. K. Girish and J. Q. Hu, The Departure Process of the MAP/G/1 Queue, Journal of DEDS, submitted for publication, 1996b. [7] W. B. Gong and J. Q. Hu, The MacLaurin Series for the GI/G/1 Queue, J. Applied Prob., 9, , 199. [8] W. B. Gong, A. Yan and D. Tang, Rational Representations for Performance Functions of Queueing Systems, Princeton Conference on System & Information Sciences, 199. [9] W. B. Gong, S. Nananukul and A. Yan, Pade Approximation for Stochastic Discrete Event Systems, IEEE Trans. on Automatic Control, 40, No. 8, , [10] J. M. Harrison, and V. Nguyen, The QNET Method for Two-Moment Analysis of Open Queueing Networks, Queueing Systems: Theory and Applications, 6, 1-3, [11] J. M. Harrison and M. I. Reiman, Reected Brownian Motion on an Orthant, Ann. Prob., 9, , [1] J. Q. Hu, Analyticity of Single-Server Queues in Light Trac, Queueing Systems: Theory and Applications, 19, 63-80, [13] J. Q. Hu, The Departure Process of the GI/G/1 Queue, Operations Research, 44, No. 5, , [14] L. Kleinrock, Queueing Systems, Vol., Addison-Wesley, [15] P. J. Kuehn, Approximate Analysis of General Queueing Networks by Decomposition, IEEE Transactions on Communications, COM-7, , [16] K. T. Marshall, Some Inequalities in Queueing, Operations Research, 16, , [17] P. P. Petrushev, and V. A. Popov, Rational Approximation of Real Functions. Cambridge University Press, [18] M. I. Reiman and B. Simon, Open Queueing Systems in Light Trac, Math. Operations Research, 1988a. [19] M. I. Reiman and B. Simon, An Interpolation Approximation for Queueing Systems with Poisson Input, Operations Research, Vol. 36, No. 3, , 1988b. [0] M. I. Reiman and A. Weiss, Light Trac Derivatives via Likelihood Ratios, IEEE Trans. Inf. Technol., Vol. 35, , [1] W. Whitt, The Queueing Network Analyzer, The Bell System Technical Journal, 6, No.9, , [] W. Whitt, An Interpolation Approximation for the Mean Workload in the GI/G/1 Queue,

12 M. K. Girish and J-Q. Hu / An Interpolation Approximation 1 Operations Research, Vol. 37, No. 6, , [3] Y. Zhu and H. Li, The MacLaurin Expansion for a G/G/1 Queue with Markov-Modulated Arrivals and Services, Queueing Systems: Theory and Applications, , 1993.

13 M. K. Girish and J-Q. Hu / An Interpolation Approximation 13 Table I E[W ] estimates for the ME=G=1 queue in Example 1 Service M-Pade M-Pade MSA MSA QNA Simulation Dist. [/] [3/3] [6/5] [16/16] ME ME ME ME ME H H H H H E E E E E Avg abs % err Table II E[W ] estimates for the ME=G=1 queue in Example 1 Service M-Pade M-Pade MSA MSA QNA Simulation Dist. [/] [3/3] [6/5] [16/16] ME ME ME ME ME H H H H H E E E E E Avg abs % err

14 M. K. Girish and J-Q. Hu / An Interpolation Approximation 14 Table III E[W ] estimates for the H =G=1 queue in Example 1 Service M-Pade MSA MSA QNA Simulation Dist. [/] [6/5] [16/16] ME ME ME ME ME H H H H H E E E E E Avg abs % err Table IV E[W ] estimates for the H =G=1 queue in Example 1 Service M-Pade MSA MSA QNA Simulation Dist. [/] [6/5] [16/16] ME ME ME ME ME H H H H H E E E E E Avg abs % err

15 M. K. Girish and J-Q. Hu / An Interpolation Approximation 15 Table V E[W ] estimates for the E 3=G=1 queue in Example 1 Service M-Pade MSA MSA QNA Simulation Dist. [/] [6/5] [16/16] ME ME ME ME ME H H H H H E E E E E Avg abs % err Table VI E[W ] estimates for the E 3=G=1 queue in Example 1 Service M-Pade MSA MSA QNA Simulation Dist. [/] [6/5] [16/16] ME ME ME ME ME H H H H H E E E E E Avg abs % err

16 M. K. Girish and J-Q. Hu / An Interpolation Approximation 16 Table VII E[W ] and E[W ] estimates for the U=E =1 queue in Example [4/] M-Pade Simulation [4/] M-Pade Simulation E[W ] E[W ] E[W ] E[W ] Avg abs % err Table VIII E[W ] estimates for the E =E =1 Queue in Example 3 [/] M-Pade Whitt Exact Avg abs % err Table IX E[W ] estimates for the E =M=1 Queue in Example 3 [/] M-Pade Whitt Exact Avg abs % err

17 M. K. Girish and J-Q. Hu / An Interpolation Approximation 17 Table X E[W ] estimates for the E =H =1 Queue in Example 3 1 [/] M-Pade Whitt Exact Avg abs % err