Experiments with Improved Approximate Mean Value Analysis Algorithms

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1 Experiments with Improved Approximate Mean Value Analysis Algorithms Hai Wang and Kenneth C. Sevcik Department of Computer Science University of Toronto Toronto, Ontario, Canada M5S 3G4 Abstract. Approximate Mean Value Analysis (MVA) is a popular technique for analyzing queueing networks because of the efficiency and accuracy that it affords. In this paper, we present a new software package, called the Improved Approximate Mean Value Analysis Library (IAMVAL), which can be easily integrated into existing commercial and research queueing network analysis packages. The IAMVAL packages includes two new approximate MVA algorithms, the Queue Line (QL) algorithm and the Fraction Line (FL) algorithm, for analyzing multiple class separable queueing networks. The QL algorithm is always more accurate than, and yet has approximately the same computational efficiency as, the Bard- Schweitzer Proportional Estimation (PE) algorithm, which is currently the most widely used approximate MVA algorithm. The FL algorithm has the same computational cost and, in noncongested separable queueing networks where queue lengths are quite small, yields more accurate solutions than both the QL and PE algorithms. 1 Introduction Queueing network models have been widely adopted for the performance evaluation of computer systems and communication networks. While it is infeasible to compute the exact solution for the general class of queueing network models, solutions can be computed for an important subset, called separable (or product-form) queueing networks [2]. The solution for separable queueing networks can be obtained with modest computational effort using any of several comparably efficient exact algorithms. Mean Value Analysis (MVA) [9] is the most widely used of these algorithms. However, even for separable queueing networks, the computational cost of an exact solution becomes prohibitively expensive as the number of classes, customers, and centers grows. Numerous approximate MVA algorithms have been devised for separable queueing networks [1,10,4,3,6,14,17,15,11]. Among them, the Bard-Schweitzer Proportional Estimation (PE) algorithm [10] is a popular algorithm that has gained wide acceptance This research was supported by the Natural Science and Engineering Research Council of Canada (NSERC), and by Communications and Information Technology Ontario (CITO). R. Puigjaner et al. (Eds.): Tools 98, LNCS 1469, pp , c Springer-Verlag Berlin Heidelberg 1998

2 Experiments with Improved Approximate MVA Algorithms 281 among performance analysts [5,8], and is used in most commercial and research queueing network solution packages. In this paper, we present a new software package, called the Improved Approximate Mean Value Analysis Library (IAMVAL), which includes two new approximate MVA algorithms for analyzing multiple class separable queueing networks. Both algorithms have approximately the same computational cost as the PE algorithm. The Queue Line (QL) algorithm always yields more accurate solutions, and hence it dominates the PE algorithm in the spectrum of different algorithms that trade off accuracy and efficiency. The Fraction Line (FL) algorithm yields more accurate solutions than both the QL and PE algorithms, but only for noncongested separable queueing networks where queue lengths are quite small. These two new algorithms have the accuracy, speed, limited memory requirements, and simplicity to be appropriate for inclusion in existing commercial and research queueing network analysis packages, and the IAMVAL library can be integrated into these software packages to replace the PE algorithm library. The remainder of the paper is organized as follows. Section 2 reviews material relating to approximate MVA algorithms. Section 3 presents the QL and FL algorithms and their computational costs. Section 4 contains a comparison of the accuracy of the solutions of the PE, QL, and FL algorithms and discusses their relative merits. Finally, a summary and conclusions are provided in Section 5. 2 Background Consider a closed separable queueing network [2,9] with C customer classes, and K load independent service centers. The customer classes are indexed as classes 1 through C, and the centers are indexed as centers 1 through K. The customer population of the queueing network is denoted by the vector N = (N 1,N 2,..., N C ) where N c denotes the number of customers belonging to class c for c =1, 2,..., C. Also, the total number of customers in the network is denoted by N = N 1 + N N C. The mean service demand of a class c customer at center k is denoted by D c,k for c =1, 2,..., C, and k =1, 2,..., K. The think time of class c, Z c, is the sum of the delay center service demands of class c. For network population N, we consider the following performance measures of interest: R c,k ( N) = the average residence time of a class c customer at center k. R c ( N) = the average response time of a class c customer in the network. X c ( N) = the throughput of class c. Q c,k ( N) = the mean queue length of class c at center k. Q k ( N) = the mean total queue length at center k. Based on the Arrival Instant Distribution theorem [7,12], the exact MVA algorithm [9] involves the repeated applications of the following six equations, in which n =(n 1,n 2,..., n C ) is a population vector ranging from 0 to N; A (c) k ( n ) is the average number of customers a class c customers finds already at center k

3 282 H. Wang and K.C. Sevcik when it arrives there, given the network population n ; 1 c is a C-dimensional vector whose c th element is one and whose other elements are zeros; and ( n 1 c ) denotes the population vector n with one class c customer removed: A (c) k ( n )=Q k ( n 1 c ), (1) R c,k ( ( ) n )=D c,k 1+A (c) k ( n ), (2) R c ( n )= K R c,k ( n ), (3) k=1 X c ( n c n )= Z c + R c ( n ), (4) Q c,k ( n )=R c,k ( n ) X c ( n ), (5) Q k ( n )= C Q c,k ( n ), (6) c=1 with initial conditions Q k ( 0 )=0fork =1, 2,..., K. The key to the exact MVA algorithm is the recursive expression in equation (1) which relates performance with population vector n to that with population vector ( n 1 c ). This recursive dependence of the performance measures for one population on lower population levels causes both space and time complexities of the exact MVA algorithm to be Θ(KC C c=1 (N c + 1)). Thus, for large numbers of classes (more than ten) or large populations per class, it is not practical to solve networks with the exact MVA algorithm. The approximate MVA algorithms for separable queueing networks improve the time and space complexities by substituting approximations for A (c) k ( N) that are not recursive. Among all approximate MVA algorithms, the Bard-Schweitzer Proportional Estimation (PE) algorithm [10] is a popular algorithm that is currently in wide use. The PE algorithm is based on the approximation { Q j,k ( N Nc 1 1 c ) N c Q c,k ( N) for c = j, Q j,k ( N) for c j. Hence, the approximation equation that replaces equation (1) in the PE algorithm is A (c) k ( N)=Q k ( N 1 c )= C Q j,k ( N 1 c ) Q k ( N) 1 Q c,k ( N). (7) N c j=1 The system of nonlinear equations (2) through (7) of the PE algorithm can be solved iteratively by any general purpose numerical techniques, such as the successive substitution method or Newton s method [10,3,17,8]. Zahorjan et al. [17] found that no single implementation of the PE algorithm is guaranteed to always work well. When the PE algorithm is solved by Newton-like methods, the

4 Experiments with Improved Approximate MVA Algorithms 283 algorithm involves solving a nonlinear system of at least C equations [3,8], and the algorithm may fail to converge, or may converge to infeasible solutions [17]. When the PE algorithm is solved by the successive substitution method, the algorithm may converge very slowly in some cases, although it always yields feasible solutions [5] and converges very quickly for most networks [17]. When the PE algorithm is solved by the successive substitution method, the space complexity of the algorithm is O(KC), and the time complexity is O(KC) per iteration [10]. Thus, application of the PE algorithm is practical even for networks of up to about 100 classes and about 1000 service centers. The number of customers per class does not directly affect the amount of computation (although it may influence the speed of convergence). 3 The Queue Line and Fraction Line Approximate MVA Algorithms In this section, two new iterative approximate MVA algorithms, the Queue Line (QL) algorithm and the Fraction Line (FL) algorithm, for multiple class separable queueing networks are presented. Both algorithms improve on the accuracy of the PE algorithm while maintaining approximately the same computational efficiency as the PE algorithm. The improvement with the FL algorithm occurs only in lightly loaded separable queueing networks, but the QL algorithm is more accurate than the PE algorithm in all cases. 3.1 The Queue Line Algorithm The QL approximation is motivated by observing the graph of Q c,k ( n ) versus n (as shown in Figure 1). The PE approximation is equivalent to interpolating the value between the values of Q c,k ( n ) for 0 and n c. The QL approximation interpolates instead between the values at 1 and n c. As can be seen in Figure 1, the QL approximation is more accurate than the PE approximation for both bottleneck and non-bottleneck centers. The QL algorithm is based on the following approximations: Approximation 1 (Approximations of the QL Algorithm). (1) when N c =1and c = j, (2) when N c > 1 and c = j, Q j,k ( N 1 c )=0, Q j,k ( N) Q j,k ( N 1 c) 1 = Q j,k( N) Q j,k ( N (N c 1) 1 c) (3) when N c 1 and c j, Q j,k ( N 1 c )=Q j,k ( N), N c 1, where N 1 c denotes the population N with one class c customer removed, and N (N c 1) 1 c denotes the population N with N c 1 class c customers removed.

5 284 H. Wang and K.C. Sevcik Like the PE algorithm, the QL algorithm assumes that removing a class c customer does not affect the proportion of time spent by customers of any other classes at each service center. However, as shown in Figure 1, while the PE algorithm assumes the linear relationship between Q c,k ( N) and N c, the QL algorithm estimates Q c,k ( N 1 c ) by linear interpolation between the points (1,Q c,k ( N (N c 1) 1 c )) and (N c,q c,k ( N)). PE Approximation QL Approximation 45 line Bottleneck center(s) Q c,k(n) Non-bottleneck centers 0 1 n c - 1 n c Fig. 1. Approximations of the PE and QL algorithms Approximation 1 leads to the following approximation equation that replaces equation (1) in the QL algorithm [13,16]. QL Approximation Equation: Under Approximation 1, the approximation equation of the QL algorithm is (1) when N c =1, A (c) k ( N)=Q k ( N) Q c,k ( N), (2) when N c > 1, A (c) k ( N)= Q k ( [ N) 1 N c 1 Q c,k ( N) Q c,k ( N (N c 1) ] 1 c ) ] = Q k ( N) 1 N c 1 Q c,k( N) Z c+ K l=1 D c,k [1+Q k ( N) Q c,k ( N) [ {D ]} c,l 1+Q l ( N) Q c,l ( N). (8) Note that when there is a single customer in class c, its residence time at a center k is given by R c,k ( N (N c 1) 1 c )=D c,k [1+Q k ( N) Q c,k ( ] N), because it is expected that the arrival instant queue length is simply the equilibrium mean queue length of all other classes, excluding class c. This fact is used in the final substitution in equation (8) (and also equation (9)).

6 Experiments with Improved Approximate MVA Algorithms The Fraction Line Algorithm The FL approximation is motivated by observing the graph of Q c,k( n ) n c versus n c (as shown in Figure 2). In this case, the PE approximation takes the value of this ratio to be the same at (n c 1) as at n c. The FL approximation estimates the value at (n c 1) by interpolating between the values at 1 and n c. As can be seen in Figure 2, when n c is sufficiently small, the FL approximation becomes more accurate than that of PE approximation for both bottleneck and non-bottleneck centers. The FL algorithm is based on the following approximations: Approximation 2 (Approximations of the FL Algorithm). (1) when N c =1and c = j, (2) when N c > 1 and c = j, Q j,k ( N) N c Q j,k( N 1 c) N c 1 = 1 N c 1 (3) when N c 1 and c j, Q j,k ( N 1 c )=0, [ ] Q j,k ( N) N c Q j,k( N (N c 1) 1 c) 1, Q j,k ( N 1 c )=Q j,k ( N), where N 1 c denotes the population N with one class c customer removed, and N (N c 1) 1 c denotes the population N with N c 1 class c customers removed. Like the PE and QL algorithms, the FL algorithm assumes that removing a class c customer does not affect the proportion of time spent by customers of any other classes at each service center. Furthermore, as shown in Figure 2, the FL algorithm estimates Q c,k ( N 1 c ) by linear interpolation between the points (1,Q c,k ( N (N c 1) 1 c )) and (N c, Q c,k( N) N c ). PE Approximation FL Approximation 1 Bottleneck center(s) Q c,k(n) n c Non-bottleneck centers 0 1 n c - 1 n c Fig. 2. Approximations of the PE and FL algorithms

7 286 H. Wang and K.C. Sevcik Approximation 2 leads to the following approximation equation that replaces equation (1) in the FL algorithm [13,16]. FL Approximation Equation: Under Approximation 2, the approximation equation of the FL algorithm is (1) when N c =1, A (c) k ( N)=Q k ( N) Q c,k ( N), (2) when N c > 1, A (c) k ( N)=Q k ( N) 2 N c Q c,k ( N)+Q c,k ( N (N c 1) 1 c ) D c,k [1+Q ] k ( N) Q c,k ( N) = Q k ( N) 2 N c Q c,k ( N)+ Z c+ K l=1 {D c,l [ ]}. 1+Q l ( N) Q c,l ( N) (9) The system of nonlinear equations of the FL algorithm consists of (9) and (2) through (6), while that of the QL algorithm consists of (8) and (2) through (6). As with the PE algorithm, the system of nonlinear equations of either the QL or the FL algorithm can be solved by any general purpose numerical techniques. When the QL and FL algorithms are solved by Newton-like methods, like the PE algorithm, each involves solving a nonlinear system of at least C equations, and the algorithm may fail to converge, or may converge to infeasible solutions [13,16]. When all three algorithms are solved by Newton-like methods, they involve solving the same number of nonlinear equations, and hence have approximately the same space and time complexities [13,16]. When the QL and FL algorithms are solved by the successive substitution method, like the PE algorithm, either algorithm may converge very slowly in some cases, although they converge quickly for most networks [13,16]. When all three algorithms are solved by the successive substitution method, their space complexities are all O(KC) because of their similar structures. Moreover, for the networks in which N c > 1 for all c, the QL algorithm requires (10KC K C) additions/subtractions and (6KC + C) multiplications/divisions per iteration, while the FL algorithm requires (10KC K C) additions/subtractions and (7KC + C) multiplications/divisions per iteration, as contrasted to (4KC K) additions/subtractions and (3KC + C) multiplications/divisions for the PE algorithm. For networks in which N c = 1 for all c, all three algorithms require the same number of operations. Hence, the time complexity of either the QL or FL algorithm is O(KC) per iteration and is identical to that of the PE algorithm when all three algorithms are solved by the successive substitution method. 4 Accuracy of the Solutions of the Algorithms We have experimentally evaluated the accuracy of the solutions of the QL and FL algorithms relative to that of the PE algorithm. We choose to compare the QL and FL algorithms against the PE algorithm because it is the most popular and most widely used of the many alternative approximate MVA algorithms. In each of these experiments, two thousand random networks were generated and solved by each of the three approximate algorithms and by the exact MVA

8 Experiments with Improved Approximate MVA Algorithms 287 algorithm. The mean absolute relative errors in estimating the quantities, X c, R c, and Q c,k, relative to the corresponding exact values, Xc, Rc, and Q c,k, were calculated in each case according to the following formulae: For throughput, Γ X = 1 C C c=1 Xc X c X ; c For response time, Γ R = 1 C C c=1 Rc R c ; For average queue length, Γ Q = 1 KC C c=1 K k=1 R c Q c,k Q c,k Q. c,k Also, the maximum absolute relative error in Q c,k was noted: Λ Q = max c,k Q c,k Q c,k. Q c,k For each of these error measures (M), we calculated the sample mean (M) and sample standard deviation (S M ) over the 2000 trials in each experiment. We also recorded the maximum value of Λ Q (max(λ Q )) over the 2000 trials in each experiment. 4.1 Experiments for Multiple Class Separable Queueing Networks In the first set of experiments, two thousand random networks were generated and solved by each of the three approximate algorithms and the exact MVA algorithm for each number of classes from one to four. The parameters used to generate the random networks are given in Table 1. The mean absolute relative errors in throughput, response time, queue length, and the maximum absolute relative errors in queue length are shown in Table 2. Additional statistics on error measures are presented elsewhere [16]. In these the experimental results, the QL Table 1. Parameters for generating multiple class separable queueing networks Server discipline: Load independent Population size (N c): Class 1: Uniform(1,10) Class 2: Uniform(1,10) Class 3: Uniform(1,10) Class 4: Uniform(1,10) Number of centers (K): Uniform(2,10) Loadings (D c,k ): Uniform(0.1,20.0) Think time of customers (Z c): Uniform(0.0,100.0) Number of trials (samples): 2000 algorithm always yielded smaller errors than the PE algorithm for all randomly generated networks. We found that the FL algorithm yields larger errors than the PE and QL algorithms for some networks, although it yielded smaller errors

9 288 H. Wang and K.C. Sevcik Table 2. Summary of statistical results of each algorithm for multiple class separable queueing networks whose parameters are specified in Table 1 Measure Algorithm 1 class 2 classes 3 classes 4 classes PE 0.40% 0.73% 0.82% 0.90% Γ X QL 0.31% 0.67% 0.79% 0.88% FL 0.05% 0.47% 0.67% 0.80% PE 0.59% 0.99% 1.04% 1.09% Γ R QL 0.45% 0.91% 0.99% 1.06% FL 0.06% 0.63% 0.84% 0.96% PE 0.69% 1.23% 1.45% 1.64% Γ Q QL 0.55% 1.15% 1.40% 1.61% FL 0.11% 0.83% 1.20% 1.47% PE 15.19% 14.82% 14.63% 16.20% max(λ Q) QL 13.13% 14.41% 14.49% 16.12% FL 10.82% 11.98% 13.64% 15.62% than the PE and QL algorithms for most of the networks among our test cases. We also found that the error of the FL algorithm rises faster than those of the QL and PE algorithms as the number of classes increases. By applying the statistical hypothesis testing procedure [16], we can further conclude that, on the average, the accuracy of all three approximate MVA algorithms decreases as the number of classes increases, and the FL algorithm is the most accurate while the PE algorithm is the least accurate algorithm among the three algorithms for multiple class separable queueing networks with sufficiently small population. However, these conclusions are only valid for small networks with a small number of classes, and small customer populations as governed by the parameters in Table 1. Although we would like to have experimented with larger networks with more classes and larger populations, the execution time of obtaining the exact solution of such networks prevented us from doing so. 4.2 Experiments for Single Class Separable Queueing Networks In order to gain some insight into the behavior of the three approximate MVA algorithms for larger networks than those whose parameters are specified in Table 1, the second set of experiments was performed. This involved five experiments for single class separable queueing networks. We chose single class separable queueing networks because it is feasible to obtain the exact solution of such networks. The parameters used to generate the random networks in each of these experiments are given in Table 3. The detailed experimental results for these five experiments are presented elsewhere [16]. In these experiments, we also found the QL algorithm was more accurate than the PE algorithm for all randomly generated single class separable queueing networks. Moreover, we found that both the PE and QL algorithms always yield a higher mean response time and a lower throughput relative to the exact

10 Experiments with Improved Approximate MVA Algorithms 289 Table 3. Parameters for generating single class separable queueing networks Parameter Experiment Value Population size (N) 1 Uniform(2,10) 2 Uniform(20,100) 3,4,5 Uniform(2,100) Number of centers (K) 1,2,3 Uniform(2,10) 4 Uniform(50,100) 5 Uniform(100,200) Loadings (D k ) 1,2,3,4,5 Uniform(0.1,20.0) Think time of customers (Z) 1,2,3,4,5 Uniform(0.0,100.0) Server discipline 1,2,3,4,5 Load independent Number of trials (samples) 1,2,3,4, solution. These results are consistent with the known analytic results [5,13]. We also found that the FL algorithm tended to yield a lower mean system response time and a higher throughput relative to the exact solution. However, some networks for which the FL algorithm yields a higher mean system response time and a lower throughput have been observed. By applying the same statistical testing procedure as in the first set of experiments [16], we conclude that given a network, as the population increases, the accuracy of the FL algorithm degrades. Moreover, given the network population, when the number of centers increases, the accuracy of the FL algorithm increases and the FL algorithm yields more accurate solutions than the other two algorithms. When the network is congested, the average queue length at some centers is large, and the approximations of both the PE and the QL algorithms lead to better approximations than those of the FL algorithm. These results are also consistent with the results of the first set of experiments. 4.3 Examples We investigated some specific examples of single class separable queueing networks to illustrate how the QL algorithm dominates the PE algorithm, and the FL algorithm is the most accurate in low congestion networks, but the least accurate in high congestion networks among the three approximate MVA algorithms. The network parameters of four such examples are given in Table 4. For these cases, the absolute value of the difference between the approximate response time and the exact response time is used to measure the accuracy of an approximate MVA algorithm. Figure 3 shows the errors for the PE, QL, and FL algorithms as a function of the population for Example 1. The corresponding graphs for the other three examples all have exactly the same form. From this we conclude that the basic form of the curves shown in Figure 3 is robust across broad classes of single class separable queueing networks.

11 290 H. Wang and K.C. Sevcik Table 4. Network parameters of examples Example Parameters (The subscript c on network parameters is dropped since C = 1.) 1 C =1,K =2,D 1 =1.0,D 2 =2.0,Z =0.0,N [1, 50] 2 C =1,K =2,D 1 =10.0,D 2 =20.0,Z =10.0,N [1, 50] 3 C =1,K =3,D 1 =10.0,D 2 =20.0,D 3 =30.0,Z =0.0,N [1, 50] 4 C =1,K =3,D 1 =10.0,D 2 =20.0,D 3 =30.0,Z =50.0,N [1, 50] 0.6 PE 0.5 QL FL 0.4 Absolute Error in R N Fig. 3. Plot of the absolute error in R vs N for Example 1 5 Summary and Conclusions Two new approximate MVA algorithms for separable queueing networks, the QL and FL algorithms, are presented. Both the QL and FL algorithms have approximately the same computational costs as the PE algorithm. Based on the experimental results, the QL algorithm is always more accurate than the PE algorithm. Moreover, as with the PE algorithm, the solutions of the QL algorithm are always pessimistic relative to those of the exact MVA algorithm for single class separable queueing networks. Specifically, both the PE and QL algorithms always yield a higher mean system response time and a lower throughput for single class separable queueing networks. These properties, which we have observed in the experiments described here, have been formally proven to hold as well [13,16]. The FL algorithm is more accurate than the QL and PE algorithms for noncongested separable queueing networks where queue lengths are small, and is less accurate than the QL and PE algorithms for congested separable queueing networks where queue lengths are large. The FL algorithm must be used only with caution since its accuracy deteriorates as the average queue lengths increase. The QL algorithm always has higher accuracy than the PE algorithm. In particular, the QL algorithm dominates the PE algorithm in the spectrum of different approximate MVA algorithms that trade off accuracy and efficiency. The IAMVAL library which includes the QL and FL algorithms can be integrated into existing commercial and research queueing network analysis packages to replace the PE algorithm library.

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