N. Akar and K. Sohraby. An invariant subspace approach in M/G/1 and G/M/1 type markov chains. Technical report, Computer Science Telecommunications, University of Missouri-Kansas City, MO,64110, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

..... 0 0 0 Delta Delta Delta a i Gamma1 a i 0 Delta Delta Delta 0 0 7 7 7 7 7 7 7 7 7 (11) where a i are given by (5) Note that the P matrix is a structured M=G=1 type of matrix [21] Efficient matrixanalytic methods have recently been developed in queueing theory [2] [3] 4] which can be employed to get the steady state probabilities vector of such structured matrices of M=G=1 type. Some of these methods have been 14 implemented in a software package called TELPACK [1] We have used TELPACK to compute the steady state probability vector of the P matrix. The ....

N. Akar and K. Sohraby. An invariant subspace approach in M/G/1 and G/M/1 type markov chains. Technical report, Computer Science Telecommunications, University of Missouri-Kansas City, MO,64110, 1995.

....for y 0 and x. The last equality comes from the fact that W (1) and the higher order derivatives of W (x) should vanish as x 1. We note that, in the above formulation one can replace SA by any matrix SA whose row space is equal to the former. The two companion papers by Akar and Sohraby [6, 7] include fast and numerically reliable algorithms to compute a basis for the row space of SA without the need for solving the eigenvalues and eigenvectors of A in the more general framework of M G 1 and G M 1 type Markov chains. The emphasis here is introducing a new mathematical framework for ....

N. Akar and K. Sohraby. An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. submitted to Commun. Stat.-Stochastic Models, 1995.

....outside the unit disk by (A1) 6] It then follows by (18) that the roots of Delta(z) and det F (z) within the unit disk coincide, therefore det F (z) has m Gamma 1 roots in the unit disk and one simple zero at z = 1. We now give the following result which can be proven by direct substitution [1]. Theorem 2 G satisfies the matrix polynomial equation F (G) 0: 19) We now define a new polynomial matrix H(s) with the indeterminate s H(s) 1 Gamma s) F ( 1 s 1 Gamma s ) 20) which has the following properties: ffl H f is nonsingular since H f = Gamma1) F ....

....G based on theorem 4 and the matrix sign function iterations. The theory part for the equation A(R) is omitted since it follows the same lines of that of the equation G = A(G) but we will include the algorithm for G M 1 type Markov chains in the next section as well. The reader is referred to [1] for details. 4 Algorithms We first give the algorithm to to compute G for M G 1 type Markov chains. 1. Obtain a left coprime polynomial fraction A(z) D (z)N(z) where D(z) D 0 D 1 z Delta Delta Delta D d z and N(z) N 0 N 1 z Delta Delta Delta Nn z ; and the ....

N. Akar and K. Sohraby. An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. submitted to Stochastic Models, 1995.

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N. Akar and K. Sohraby. An invariant subspace approach in M/G/1 and G/M/1 type markov chains. Technical report, Computer Science Telecommunications, University of Missouri-Kansas City, MO,64110, 1995.

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