R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math., 51 (1987), 143--180.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Numerical examples of SANs are also given to illustrate the fast convergence rate of the method. Key Words: Stochastic Automata Networks, Circulant Preconditioners, Conjugate Gradient Method. 1 Introduction Stochastic Automata Networks (SANs) are widely used in modeling queueing systems [6, 7, 8], communication systems [17, 24] manufacturing systems and inventory control systems [9] The SANs approach has more compact and efficient representations when compared with the generalized stochastic Petri nets [1, 3, 20] Moreover the matrix vector multiplication of the generator matrix and a ....

....simple example of SAN with 2 automata, namely a 2 queue overflow network. The SAN consists of two queues with exogenous Poisson arrivals and exponential servers. Whenever Queue 2 is full, the arrival customer will overflow to Queue 1 if it is not yet full. Otherwise the customer will be lost, see [6, 8]. In order to write down the generator for the network, let us first define the following queueing parameters: a1) i , the exogenous input rate of queue i(i = 1; 2) a2) i , the service rate of queue i(i = 1; 2) 3 (a3) l i Gamma s i Gamma 1, buffer sizes for queue i(i = 1; 2) a4) s i ....

R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math., 51 (1987), pp. 143--180.

....for the SANs. The preconditioners considered here are easy to construct, can be inverted efficiently and give very fast convergence rate. Key Words: Stochastic Automata Networks, Preconditioners. 1 Introduction Stochastic Automata Networks (SANs) are widely used in modeling queueing systems [2, 3, 4], communication systems [12, 18] manufacturing systems and inventory control [5] and also computer networks [13] The SANs approach has more compact and efficient representation when compare to the generalized stochastic Petri nets. Moreover the matrix vector multiplication of the generator ....

R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math., 51 (1987), pp. 143--180.

No context found.

R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math., 51 (1987), 143--180.

No context found.

Chan, R., Iterative Methods for Overflow Queueing Models I. Numer. Math., V51, 1987, pp. 143-180.

No context found.

Chan, R., Iterative Methods for Overflow Queueing Models. NYU Comp. Sci. Dept. Tech. Rep., # 171, 1985.

No context found.

R. Chan, Iterative Methods for Overflow Queueing Models I , Numerische Mathematik 51, (1987) 143--180.

....as R 1 is a bidiagonal matrix, the matrix vector multiplication By thus requires approximately 2n 2 O(n 2 ) operations. Hence solving the normal equations requires 4n 2 O(n 2 ) operations per iteration. The storage requirement is n 1 n 2 O(n i ) since we need to store the Q i s. In Chan [4], we mention two alternates that require the storage of only one of the Q i s. We note that there is no need to compute the last entry of Phi. In fact, Lemma 2.2.2 Phi n2 , the last diagonal entry of Phi, can be defined arbitrarily. Proof: Since fl 2;n2 = 0, it follows from (2.2.16) 2.1.14) ....

....the imaginary part. The numbers z j j e x j ; j = 1; Delta Delta Delta ; n l can be computed by using n l complex multiplications. The expression P n l j=1 e z j can be evaluated by the Fast Fourier Transform. This requires only O(n l log n l ) operations for arbitrary n l , see Chan [4]. From (2.2.17) we see that the work and storage required for computing the matrix vector multiplication By are thus reduced to O(n 2 log n 2 ) and O(n 2 ) respectively. The work of generating the whole null vector p can also be reduced to O(n 2 log n 2 ) We remark that when using the Fast ....

[Article contains additional citation context not shown here]

Chan, R., Iterative Methods for Overflow Queueing Models. NYU Comp. Sci. Dept. Tech. Rep., # 171, 1985.

....preconditioned conjugate gradient methods. We note that the MILU [31, 38] and MINV [25, 26] based preconditioners are not appropriate due to their expensive construction costs. One of the early applications of preconditioned conjugate gradient methods in solving queueing networks was done by Chan [9, 10]. For Markovian overflow networks with traffic density close to 1, the generator matrices are close to the discretization matrices of elliptic equations. Using techniques from elliptic equations, such as the fast Poisson solvers and domain decomposition methods [11] Chan has constructed efficient ....

....the number of nonzero columns in A. A matrix A is said to be nonnegative, denoted by A O, if all the entries of A are nonnegative. To introduce the terminologies and notations of SANs, let us consider a simple example of SANs with 2 automata. It is the 2 queue overflow network considered in [30, 9]. The network consists of two queues (automata) with exogenous Poisson arrivals and exponential servers. Whenever queue 2 is full, the arriving customers will overflow to queue 1 if it is not yet full. Otherwise the customers will be blocked and lost, see Figure 1. Overflow Figure 1. ....

[Article contains additional citation context not shown here]

R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math., 51 (1987), pp. 143--180.

No context found.

R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math. 51 (1987), pp. 143--180.

No context found.

R. Chan, (1987) Iterative Methods for Overflow Queueing Models I, Numer. Math. 51, pp. 143--180.

....these matrices into circulant ones and then using FFT s. However, the convergence rate will be governed by the condition number of A and hence preconditioning will be essential for the feasibility of this approach. Some preconditioners based on circulant matrices have been proposed: ffl [33, 5]: It consists in using a circulant matrix C to approximate a Toeplitz matrix T by first copying the main diagonals of T to C and then completing C to form a circulant matrix. This is illustrated in the fashion outlined in the following picture. copy ignored complete to make C circulant T = C = ....

R. Chan, Iterative Methods for Overflow Queueing Models I , Stud. Appl. Math., 74 (1987), pp. 171--176.

....i i ) is close to 1, i.e. i s i i = 1 O(n Gammaff i ) 4. 9) for some ff 0, then the queueing problem resembles a second order elliptic equation on a rectangle with an oblique boundary condition on one side (the side with overflow) and Neumann boundary conditions on the others, see [26]. The SOR method is one of the standard methods for solving this problem, see [124] However, in [26] the preconditioned conjugate gradient method has also been considered, with the preconditioner being constructed by changing the oblique boundary condition to Neumann boundary condition. This ....

.... queueing problem resembles a second order elliptic equation on a rectangle with an oblique boundary condition on one side (the side with overflow) and Neumann boundary conditions on the others, see [26] The SOR method is one of the standard methods for solving this problem, see [124] However, in [26], the preconditioned conjugate gradient method has also been considered, with the preconditioner being constructed by changing the oblique boundary condition to Neumann boundary condition. This preconditioner will be referred to as Neumann preconditioner below. The convergence rate of the ....

[Article contains additional citation context not shown here]

R. Chan, Iterative Methods for Overflow Queueing Models I , Numer. Math., 51 (1987), pp. 143--180.

....preconditioned conjugate gradient methods. We note that the MILU [30, 37] and MINV [25, 26] based preconditioners are not appropriate due to their expensive construction costs. One of the early applications of preconditioned conjugate gradient methods in solving queueing networks was done by Chan [9, 10]. For Markovian overflow networks with traffic density close to 1, the generator matrices are close to the discretization matrices of elliptic equations. Using techniques from elliptic equations, such as the fast Poisson solvers and domain decomposition methods [11] Chan has constructed efficient ....

....the number of nonzero columns in A. A matrix A is said to be nonnegative, denoted by A O, if all the entries of A are nonnegative. To introduce the terminologies and notations of SANs, let us consider a simple example of SANs with 2 automata. It is the 2 queue overflow network considered in [29, 9]. The network consists of two queues (automata) with exogenous Poisson arrivals and exponential servers. Whenever queue 2 is full, the arriving customers will overflow to queue 1 if it is not yet full. Otherwise the customers will be blocked and lost, see Figure 1. oe 1 s 1 Queue 1 oe ....

[Article contains additional citation context not shown here]

R. Chan, Iterative Methods for Overflow Queueing Models I, Numer. Math., 51 (1987), pp. 143--180.

.... i i ) is close to 1, i.e. i s i i = 1 O(n Gammaff i ) 4:9) for some ff 0, then the queueing problem resembles a second order elliptic equation on a rectangle with an oblique boundary condition on one side (the side with overflow) and Neumann boundary conditions on the others, see [25]. The SOR method is one of the standard methods for solving this problem, see [123] However, in [25] the preconditioned conjugate gradient method has also been considered, with the preconditioner being constructed by changing the oblique boundary condition to Neumann boundary condition. This ....

.... problem resembles a second order elliptic equation on a rectangle with an oblique boundary condition on one side (the side with overflow) and Neumann boundary conditions on the others, see [25] The SOR method is one of the standard methods for solving this problem, see [123] However, in [25], the preconditioned conjugate gradient method has also been considered, with the preconditioner being constructed by changing the oblique boundary condition to Neumann boundary condition. This preconditioner will be referred to as Neumann preconditioner below. The convergence rate of the ....

[Article contains additional citation context not shown here]

R. Chan, Iterative Methods for Overflow Queueing Models I , Numer. Math., 51 (1987), pp. 143--180.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC