J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, September 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the mean or distribution of the number of customers at a station, the customer throughput at a station, or the waiting time. The success of queueing networks stems mainly from the fact that for the class of product form networks [5] very efficient analysis algorithms, such as Buzen s algorithm [24] or mean value analysis [86] are known, and that software tools for the specification and analysis of QN models were available at In this paper, we do not consider the line of research on non Markovian models such as described, for example, in [47] an early stage [89, 100] Although QN have ....

J.P. Buzen. Computational Algorithms for Closed Queueing Networks with Exponential Servers. Communications of the ACM, 16:527--531, 1973.

....in [8] The importance of product form networks has led to many techniques for analysing them [9] The primary performance measure for circuit switched models is the blocking probability. This may be calculated from the normalising constant (G of (1) G may be calculated by convolutional methods [10,11], numerical inversion of generating functions [12] or by Monte Carlo integration [3, 13] This paper investigates the performance of Markov chain Monte Carlo simulation as an alternative means of estimating blocking probabilities in product form networks [14,15] see also [16] In addition to ....

....numerical problem for realistic sized networks. Moreover, in many cases, it is not sufficient to know the blocking probability, and it is desirable to sample from the distribution itself (see for example [17] Monte Carlo techniques, such as the FGS, bridge the gap between exact algorithms [10 12] and approximations [8,19,20] They allow a quantifiable tradeoff between computational time and accuracy, while being conceptually simple. This section presents the construction of a surrogate Markov chain k : k = 1, 2, with state space whose steady state probabilities are given ....

J. P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Comm. ACM, 16, 1973, 527--531.

....size constraints [1) the time and space computational requirements of the normalization constant may be very large due to the large number of feasible network states present in any non trivial model. The convolution algorithm for product form queueing networks was first dis covered by Buzen [4] for single chain networks and extended by Chandy, Herzog and Woo [5] and by Reiser and Kobayashi 2] to multi chain networks. Consider a network. of M service centers with K closed routing chains. Let N k be the popula tion size of chain k. The convolution algorithm encounters difficulties ....

....performance measures, such as chain throughputs and mean queue lengths, requires the computation of various other normalization constants. For a tutorial treatment of this topic, see [3] or [24] The throughput of chain k at center m for a network of closed chains with population vector is [4, 7, 22] Tmk(N) mk = k G(N) for k = 1,2, K, m 1,2, M and N (11) where G( is the normalization constant of the same network with population vector J k and mk is the relative arrival rate of chain k customers to center m. 11) is applicable for both fixed rate and queue dependent ....

Buzen, J.P., "Computational Algorithms for Closed Queueing Networks with Exponential Servers," Communications of the ACM, Sept. 1973, pp. 527-531.

....has recently been successfully solved by the discovery of a simple dynamic scaling technique [27] The second problem is the extremely large computational time and space re quirements to evaluate G(N) even for moderate values of K and Nk . For example, if the convolution algorithm is used [3,28], an array of G values indexed from 0 to N is necessary7 and the storage requirement for the array alone is proportional to the product N1N2.o.N K. The time requirement of the convolution al O(MKN1N2o.NK) o Below we first illustrate how one can model a virtual chan nel by a closed chain, ....

....to be given by [3,29] oC(w) 4. 14) This is identical to a network model with a single closed chain and with the service rate of channel i reduced to uCi 1 Pi2) Finally, the throughput seen by the virtual channel is given by: Y1 = Prob[source server is busy] C 1 45 Using Buzens result [28] 1 can be conveniently expressed as: GC(w 1) The mean number of chain 1 packets at channel i can be ob tained from: E[nil] nilP(n l) 416) n lEV n l: njl= W The mean number of chain 2 (open chain) packets at channel i has the following simple form [29] i2 E[ni2] Ci i 2 ....

J.P. Buzen, Computational Algorithms for Closed Queueing Networks with Exponential Servers CACM 16 (1973) 527-531

.... size constraints [1) the time and space computational requirements of the normalization constant may be very large due to the large number of feasible network states present in any non trivial model The convolution algorithm for product form queueing networks was first dis covered bv Buzen [4] for single chain networks and extended by Chandy, Herzog and Woo [5] and by Reiser and Kobayashi 2] to multi chain networks Consider a network of M service centers with K closed routing chains Let N k be the popula tion size of chain k. The convolution algorithm encounters difficulties when ....

....performance measures, such as chain throughputs and mean queue lengths, requires the computation of various other normalization constants. For a tutorial treatment of this topic, see [3] or [24] The throughput of chain k at center m for a network of closed chains with population vector is [4, 7 22] Tmk( mk G(N) for k = 1,2, K, m = 1,2, M and N 1 k (tl) where G( is the normalization constant of the same network with population vector 1 k and mk is the relative arrival rate of chain k customers to center m (tt) is applicable for both fixed rate and queue dependent ....

Buzen, J.P., "Computational Algorithms for Closed Queueing Networks with Exponential Servers," Communications of the ACM, Sept. 1973, pp. 527-531.

.... 1, m j the ratio xk = vk k replaced by v k v k for 1 #=j m, i.e. j l) i=1 n i = l 0; n j =0 =1 vk v j k j # n k (j l) is just a normalising constant that may be computed efficiently, along with Gm (n c 1) and G(n 1) by Buzen s algorithm [6]. Thus we define the recursive function , for real vector y = y1, y a ) and integers a, b (0 M, 0 N 1) by: y,a,b) y,a 1,b) ya (y,a,b 1) a, b 0) y,a,0) 1 (a 0) y, 0,b) 0 (b Thenwehave Gm (l) xm,M m, l) 0# l G0 (n 1) x,M,n 1) j l) w j ,m ....

J. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16:527--531, 1973.

.... performance analysis of computer systems, communication networks and other complex systems [5,14,19,28,38] The success of these models has largely been due to the excellent algorithms for computing the steady state performance measures that have been developed, such as the convolution algorithm [6,35], the mean value analysis (MVA) algorithm [36] the tree convolution algorithm [27] the recursion by chain algorithm (RECAL) 12,13] the mean value analysis by chain (MVAC) algorithm [15] and the distribution analysis by chain (DAC) algorithm [37] see [5,14,28] for an overview. While these ....

BUZEN, J. P. Computational algorithms for the closed queueing networks with exponential servers. Commun. ACM 16 (1973), 527-531.

....V. COMPUTING SUMS OF MONOMIALS While the monomial sums S#c; # defined in (27) could be computed by the convolution algorithm [19, 20, 1] we develop more efficient recursion formulas for these specific forms of sums. The derivation is a novel extension of Buzen s ideas in his treatment [22] of the convolution algorithm. Before we treat the general monomial sums (27) let us consider simpler sums of the form s#c; k## # ### # # # # ### ## # for c ##; C and k ##; K; 29) where b ### # #b # b # ### b # # # is a truncated vector of traffic class trunk requirements. ....

J. P. Buzen, "Computational algorithms for closed queueing networks with exponential servers," Communications of the ACM, vol. 16, pp. 527--531, Sept. 1973.

....V. COMPUTING SUMS OF MONOMIALS While the monomial sums S(c, #) defined in (27) could be computed by the convolution algorithm [13, 14, 15] we develop more efficient recursion formulas for these specific forms of sums. The derivation is a novel extension of Buzen s ideas in his treatment [17] of the convolution algorithm. Before we treat the general monomial sums (27) let us consider simpler sums of the form s(c, k) # i# k i T b (k) c 1 for c = 0, C and k = 1, K, 29) where b (k) b 1 b 2 b k ) T is a truncated vector of traffic class trunk ....

J. P. Buzen, "Computational algorithms for closed queueing networks with exponential servers," Communications of the ACM, vol. 16, pp. 527--531, Sept. 1973.

.... constants are discussed in Bertozzi and McKenna [2] but it has long been known that these generating functions can be useful; see Reiser and Kobayashi [10] For numerical inversion, we rely on the LATTICE POISSON algorithm in Abate and Whitt [1] as extended in Choudhury, Lucantoni and Whitt [6]. In the present paper we give a concise account of our algorithm for closed queueing networks; an expanded discussion appears in [3] The algorithm also applies to other product form models. We treat the special cases of circuit switched communication network models and resourcesharing models in ....

....process goes on until at step p the function on the righthand side becomes the p dimensional generating function and is explicitly computable. In each step we use the LATTICE POISSON inversion algorithm in [1] with modifications to improve precision and allow for complex inverse function as in [6]. We show below the inversion formula at the j th step. For simplicity, we suppress those arguments which remain constant during this inversion, letting g j (K j ) g ( j 1) z j 1 , K j ) and G j (z j ) g ( j) z j , K j 1 ) With this notation, the inversion formula (4.2) is g ....

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BUZEN, J. P. Computational algorithms for the closed queueing networks with exponential servers. Commun. ACM 16 (1973), 527-531.

....to calculating G, for example, by summing over all possible states, soon run into practical problems because of the number of states involved, not to say numerical difficulties such as round off. Algorithms to numerically evaluate these networks have been the subject of much interest. Buzen [4] developed the first algorithm, known as the convolution algorithm. This algorithm finds the normalisation constant G for a network of M centres and N customers using a simple recurrence relating G(M, N) to G(M 1, N) and G(M, N 1) Other performance metrics, such as mean queue lengths, ....

Buzen, J.P. "Computational Algorithms for Closed Queueing Networks with Exponential Servers." Communications of the ACM, Vol. 16, No. 9, pp 527-531, September 1973.

....a. That is y i = aD i . Thus we have G N y y y n n M n n M ( 1 2 1 2 # or G N y i n M n i ( 1 The problem with this formula is that to compute G(N) we must enumerate all the permutations of possible states. The convolution algorithm is thus developed by Buzen [Buzen73] to solve this problem. The idea is based on the following fact. If g n k y i n i k n ( 1 then we can rewrite it as a recursive form: g n k g n k y g n k k ( 1 1 34 where the initial values are g n g k ( 0 0 0 1 = 1, 2, ....

Buzen, J. P., "Computational algorithms for closed queueing networks with exponential servers," Communications of the ACM, vol. 16, no. 9, pp. 527-531, September 1973.

....no closed form is known. Thus an important problem in the area is the development of efficient algorithms (like MVA [22] and, in particular, the computation of those normalization constants, which are sums over all the possible states of the network. These calculation methods (MVA, convolution [5] . are quickly irrelevant, when the number of states increases. In these cases, we can use approximation methods. In Bell Laboratories, McKenna, Mitra and Ramakrishnan used in the eighties the asymptotic expansion of the normalization constants [12] 11] 13] 21] to derive efficient approximation ....

J.P. Buzen. -- Computational algorithms for closed queueing networks with exponential servers. -- Communications of ACM, 16:527--531, 1973.

....and finding all subpopulations having at least one customer of class one. Allen and Hynes did not explain the insight that led to this nonobvious approach, but the reasoning is similar to Buzen s convolution recurrence for computing the normalization constant of multiple class queueing networks [Buzen 1973]. Given the Partitions function, the rest of the code for multiclass MVA is not much more complicated than for the singleclass case; interested readers should refer to the function MultiCentralServer in [Allen and Hynes 1991] Multi class MVA is an excellent example of the trickiness of the ....

Buzen, J.P. 1973. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM 16(9): 527--531.

....contains as many terms as there are places in the net. Starting from this factorization two efficient algorithms have been derived [49, 50] for the computation of the PFS. Both have polynomial time and space complexities. They recall, respectively, the convolution algorithm derived by Buzen [51] for PFS Queueing Networks and the Mean Value Analysis algorithm of Reiser and Lavenberg [52] Basic to the derivation of the convolution algorithm is a recursive expression of the normalization constant that is a generalization of that derived by Buzen for multiple class product form queueing ....

J. P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, September 1973.

....because they can model a system in quite a natural way, and can be analysed reasonably efficiently, provided that they are separable, i.e. they obey certain criteria that allow for a product form solution. Two efficient solution methods for product form queueing networks are the convolution method [9] and mean value analysis (MVA) 29] of which the latter is more generally applied. One of the reasons for this is that approximations exist for MVA solutions (e.g. the iterative Schweitzer Bard approximation [34, 7] that reduce the analytical complexity (in the single class case) from O(NK) to ....

....times can be used (e.g. constant, uniform distribution, negative exponential distribution) However, often restrictions are imposed to keep the models analytically tractable. For instance, algorithms that make use of the product form properties of queueing networks (MVA [29] convolution method [9]) require negative exponential service time distributions for queueing centres. Standard Petri nets do not include a concept of time, but various types of timed Petri nets have been proposed. In performance modelling the so called Generalised Stochastic Petri Nets (GPSN) 1] an extension of ....

J.P. Buzen, "Computational algorithms for closed queueing networks with exponential servers," Communications of the ACM, vol. 16, Sept. 1973, pp. 527--531.

....a test model for our experiments. Finally we identify model characteristics that may lead to excessive pseudo events. 4 2.1 Example The model consists of a number, P , of computing clusters. The model is a closed queueing network with J Theta P jobs. Each cluster is a central server model (see [1]) with a single CPU and K I O devices. All queueing disciplines are FCFS (an assumption that is not necessary for uniformization to work) The service times at the CPU are assumed to be exponentially distributed with rate c . When a job leaves the CPU it goes to one of the I O devices, which is ....

J.P. Buzen, "Computational Algorithms for Closed Queueing Networks with Exponential Servers," Commun. ACM, vol. 16, no. 9, pp. 527-531, September 1973.

....error analysis, multiclass queueing networks, product form solutions. 0. Introduction Product form queueing networks are introduced in [12,10,3] Their queues have the M)M property [17,33] There are two important techniques to analyze closed queueing networks: the convolution method (Buzen [1]) and mean value analysis (MVA) Reiser [22] Because of their computational complexity, both methods are not applicable to large systems having several job classes and a large number ( 100) of jobs. To overcome this difficulty, heuristic methods to approximate MVA are proposed by Chandy and ....

J.P.Buzen, Computational algorithms for closed queueing networks with exponential servers, Comm. ACM 16, 9 (1973) 527-531.

.... = X all n s:t: J(n;v) m;nN =0 N Y i=1 (1 Gamma p i ) n i (1 Gamma p N ) X all n s:t: J(n;v) m Gammav N ;n N =0 N Y i=1 (1 Gamma p i ) n i 1 This is the same technique used to define the basic relationship for the convolution algorithm used to evaluate closed queueing networks [Buz73] 97 Alternatively we can write h(v; m) h GammaN (v; m) 1 Gamma p N )h GammaN (v; m Gamma v N ) 8.3) where h GammaN (v; m) is the same as h(v; m) except that it is evaluated for a system with node N removed, i.e. h GammaN (v; m) X all n s:t: J GammaN (n;v) m N ....

J. Buzen. Computational algorithms for closed queueing networks with exponential servers. Commun. ACM, 16(9):527--531, 1973. 108

....compute ff(P ; V ) for each (V ,r) corresponding to read sets in Omega N . The (V ,r) that yields the highest value for ff(P ; V ) is the optimal vote and 3 This is the same technique used to define the basic relationship for the convolution algorithm used to evaluate closed queueing networks [18]. quorum assignment. However, if the nodes are labeled such that p 1 p 2 : p N , we need only consider the non decreasing vote assignments which correspond to the members of EN generated by Algorithm 2. In other words, the members in Omega N Gamma EN need not be considered for ....

J. Buzen, "Computational algorithms for closed queueing networks with exponential servers," Commun. ACM, vol. 16, no. 9, pp. 527--531, 1973.

....if a brute force enumeration technique is to be used. For some specific situations like queueing networks and stochastic knapsack models where the f i (n i ) have the form ae n i i =n i or ae n i i , efficient recursive algorithms to sum over the state space are available. See for example (Buzen 1973) for queueing networks with constant population and (Ross 1995) for stochastic knapsacks. In this paper we describe a transform technique to sum functions of the form Q M i=1 f i (n i ) over a state space defined by a set of linear equality and or inequality constraints with integer coeeficients. ....

Buzen, J. P. (1973). Computational algorithms for closed queueing networks with exponential servers.

....to calculating G, for example by summing over all possible states, soon run into practical problems because of the number of states involved, not to say numerical difficulties such as round off. Algorithms to numerically evaluate these networks have been the subject of much interest. Buzen[2] developed the first algorithm, known as the convolution algorithm. This algorithm finds the normalisation constant G for a network of M centres and N customers using a simple recurrence relating G(M;N) to G(M Gamma 1; N) and G(M;N Gamma 1) Other performance metrics, such as mean queue ....

J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, September 1973.

....made in queueing network theory evident in (Baskett, Chandy, Muntz, and Palacios 1975) The classic Erlang Loss model is extended to consider multiple classes of customers each with its own service demands and arrival patterns. Efficient methods based on recursion similar to that presented in (Buzen 1973) have been developed to compute performance measures. Aein (Aein 1978) introduced a simple birth death model that analyzed access constraints imposed on user classes as a strategy to sharing transmission capacity. A product form solution to the birth death state equations is derived. The analysis ....

Buzen, J. P. (1973, September). Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM 16 (9), 527--531.

....example of models with a product form solution, and are therefore a popular formalism to model (traditional) computer systems. A queueing network is separable if it obeys certain criteria [6] The most widely used product form solution methods for queueing networks are the convolution algorithm [8] and mean value analysis (MVA) 23] At the expense of some accuracy faster results can be obtained using the iterative Bard Schweitzer approximation of MVA [5, 26] If a queueing network is separable, parts of the model can be aggregated into one flow equivalent queueing centre with a ....

J.P. Buzen, "Computational algorithms for closed queueing networks with exponential servers," Comm. ACM, vol. 16, Sept. 1973, pp. 527--531.

....their steady state distributions are known explicitly up to a normalization constant. There are three well known non randomized computational methods for throughput the most important performance measure in CMP networks: convolution algorithms and related recursions such as mean value analysis [4, 8, 38, 39], asymptotic expansions [34, 31, 32, 33, 37] and transform inversion methods [10, 11] This paper was motivated by the recent work of Ross, Tsang and Wang [41] on specialized Monte Carlo methods for throughput in these networks (see also [42, 43, 44] They developed two importance sampling ....

Buzen, J.P. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM 16, 527-531. 1973.

....The first model is designed to allow direct parametric control over critical parameters, in order to conduct simulator performance sensitivity studies and to directly compare the performance of the the different synchronization protocols. The model is a network of central server models [4], similar to the models considered in [13] The basic element in the model is a cluster of central server queuing network models, each comprised of one CPU and five IO devices. The CPU has exponential service with mean 2; the IO service time distribution uses a parameter ff 2 [0; 1] the ....

J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Commun. ACM, vol. 16, no. 9, 527-531, September 1973.

....system modification at the relatively higher expense of simulation. The key property of the separable networks is that their state distribution function has CHAPTER 2. OVERVIEW OF PREVIOUS STUDIES 17 a form of a product of factors, each of which is associated with a queuing center in the model [30]. Hence, the components of such a system are separated . The separability leads to simplifications in the mathematical description of the behavior of the system and can be outlined as follows [31, 32, 33] ffl Job Flow Balance: The number of arrivals at each center is equal to the number of ....

....by extensive system measurement and benchmarking. A validated model can afterwards be used to predict the effect of system modification or upgrade with a great degree of confidence. The solution to a separable QNM has a standard form, that is easy to compute up to a normalization constant. Buzen [30] provided the first method for efficient model solution by evaluation of the normalization constant. His method (convolution algorithm) is based on a recursive definition of this constant. The computation can be done in time proportional to the number of centers and the number of customers (for a ....

Jeffrey P. Buzen. Computational Algorithms for Closed Queueing Networks with Exponential Servers. Communications of the ACM, 16(9):527--531, September 1973.

....become increasingly popular. The solution for separable queueing networks can be obtained with modest computational effort using several comparatively efficient exact algorithms. The first of these algorithms is the convolution algorithm which was developed by Chapter 1. Introduction 2 Buzen [7]. Buzen s convolution algorithm applied only to single class separable queueing networks with First Come First Serve (FCFS) exponential centers. Then Reiser and Kobayashi [29] developed a similar algorithm for multiple class separable queueing networks. Reiser and Lavenberg [31] later developed an ....

J. P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527-531, September 1973.

.... for certain types of queueing networks [10] Efficient algorithms for the computation of sensitivities for closed QNMs with exponential servers are proposed in [4] As opposed to the approach presented here, the techniques proposed in [10] and [4] are based on Buzen s convolution algorithm [3]. Partial derivatives of the MVA main equation have been used in the context of optimizing load balancing in distributed systems by de Souza e Silva and Gerla [6] Similar intermediate results are also used to prove certain monotonicity properties in [11] The potential of using MVA based ....

J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, September 1973.

....note on the use of Fast Fourier Transforms in Buzen s Algorithm Nigel Bean and Mark Stewart Teletraffic Research Centre, Department of Applied Mathematics, University of Adelaide. Abstract Buzen [4] presented an algorithm for the computation of the equilibrium distribution of customers in closed queueing networks with exponential servers. This paper presents a refinement of that algorithm which is computationly more efficient. Additionally this algorithm is more amenable to the computation ....

....of many marginal distributions. 1 Introduction A significant milestone in queueing theory was the development of product form equilibrium distributions, 11, 12, 3, 13, 14] Practical use of the product form technique often requires that the normalising constant be efficiently evaluated. Buzen [4] published a convolution algorithm to evaluate the normalising constant for closed networks that was a significant development in the study of large queueing networks. The evaluation of convolution forms is not unique to queueing theory and since Cooley and Tukey [8] presented the Fast Fourier ....

J. P. Buzen. Computational Algorithms for Closed Queueing Networks with Exponential Servers. Communications of the Association for Computing Machinery, Vol. 16, 1973, pp. 527--531.

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J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, September 1973.

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J. Buzen. Computational algorithms for closed queueing networks with exponential servers. Comm. ACM, 16(9):527--531, 1973.

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Buzen, J. P. (1973). Computational algorithms for closed queueing networks with exponential servers. Commun. ACM,16(9), 527--531.

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J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Comm. ACM, 16(9):527--531, 1973.

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Buzen, J. P. (1973). Computational algorithms for closed queueing networks with exponential servers. Commun. ACM,16(9), 527--531.

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J. P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, 1973.

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Jeffrey P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527--531, September 1973.

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J.P. Buzen. Computational Algorithms for Closed Queueing Networks with Exponential Servers. Communications of the ACM, 16:527--531, 1973.

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J. P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527531, September 1973.

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J.P. Buzen, #Computational algorithms for closed queueing networks with exponential servers," Comm. ACM,vol. 16, Sept. 1973, pp. 527#531.

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J.P. Buzen. Computational algorithms for the closed queueing networks with exponential servers. Commun. ACM, 16:527--531, 1973.

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JP Buzen. Computational algorithms for closed queueing networks with exponential servers. Commun. ACM 16: 527-531, 1973.

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