AKAR,N.--SOHRABY, K., Finite and Infinite QBD Chains: A Simple and Unifying Algorithmic Approach, Proceedings of IEEE INFOCOM, (1997), pp. 1105--1113.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[17] If the level transitions are only possible between adjacent ones, such queueing systems are called QBD (Quasi Birth Death) processes. For the steady state solution of this class of two dimensional Markov chains, several efficient methods have been developed and improved over recent years [2, 4, 6, 10, 11, 14]. However, there are only few works dealing with such twodimensional queueing systems in which upper bounded arrival and or departure batches occur, i.e. multiple jumps in level dimension are possible. The involvement of this kind of queueing systems is quite reasonable and useful during modeling ....

....or indirect way. The most well known indirect solution way is to perform re blocking (as shown in Fig. 2) in the transition probability matrix of the QBD M process to get a standard QBD process, then solving this transformed QBD process by one of the efficient methods available in literature [2, 4, 6, 11]. Along this line, the method of NAOUMOV et al. 11] and the spectral expansion method [6] are most appealing to use due to their fast and accurate performance. We will refer to these two methods as NA after BL and SE after BL method, respectively. The direct solution way is represented by the ....

AKAR, Nail -- SOHRABY, Khosrow, Finite and Infinite QBD Chains: A Simple and Unifying Algorithmic Approach, In Proceedings of IEEE INFOCOM, (1997), pp. 1105--1113.

....by not directly computing G, but iterating over a sequence of matrices which can be considered a factorization of A 0 G, and which can be computed faster. Note that Naoumov s method is equivalent to the cyclic reduction approach [8] applied to QBD systems. The Invariant Subspace (IS) approach [5, 6]. In this approach, the matrix R is obtained by computing the invariant subspace associated to a matrix derived from the coe#cient matrices A 0 , A 1 and A 2 . While there are several approaches to perform this task, the authors propose to evaluate the so called matrix sign function of this ....

N. Akar and Khosrow Sohraby. Finite and infinite QBD chains: A simple and unifying algorithmic approach. In Proc. IEEE Infocom 97. IEEE Computer Society Press, 1997.

....(25) x i = x 0 p i Gamma1 i Gamma1 X r=1 x r p i Gammar ]p Gamma1 0 ; for i 1; 26) where p i = P M r=i 1 p r . 4) In recent years new methods have been proposed to solve models of M=G=1 type and QBD (Quasi Birth and Death) models, see, e.g. Bini and Meini [4] Akar and Sohraby [2] and Li and Sheng [11] Due to the simple nature of the D BMAP D 1 queue (infinite or finite) we did not investigate the use of the newly developed methods in our models. 2.5 Computation of the Steady state Probabilities for the Transition Probability Matrix P Let uK (j) 1 X i=K 1 x(i; j) ....

Nail Akar and Khosrow Sohraby. Finite and infinite QBD chains: a simple and unifying algorithmic approach. In IEEE Infocom '97, pages 1105--1113.

.... Delta bn 0 1 C C C A G n;n 1 = an . bn 1 an 1 0 B B B 0 Delta Delta Delta Delta Delta Delta bn 1 1 0 U 0 n Delta Delta Delta Delta Delta Delta bn 2 0 1 C C C A : Note that if an = b n 1 for all n, then the generator Q in (1) is called level dependent QBD process [1], 2] 3] 16] 17] We assume that the generators Q, Gn (0 n N ) in (1) are irreducible and that Q is conservative and D 0 n e 6= 0 and U 0 n 1 e 6= 0 (0 n N Gamma 1) Then Dn An Un (0 n N ) are conservative generators and all Gn are transient. Let n (0 n N ) be the 1 Theta m ....

....on simple block matrix operations including addition, multiplication and solving systems of linear equations except the first step. Most of the computation is devoted to the first step where rate matrices R n;1 and R n;2 (0 n N ) are obtained. We can use efficient algorithm proposed recently in [1], 2] to obtain the rate matrices. For numerical example, we consider an overload control based on thresholds where N = 2. See Example and Fig.1) We consider a finite buffer system with a single exponential server and an MMPP arrival process with representation ( Q; We assume that when ....

N. Akar and K. Sohraby, Finite and infinite QBD chains: A simple and unifying algorithmic approach, in: Proc. of IEEE Infocom'97, (1997).

....a numerical example to demonstrate its computational features. Keywords: Finite QBD process, matrix geometric approach, invariant subspaces, matrixsign function, real Schur decomposition, Sylvester matrix equation, numerical linear algebra. This submission is a revised and extended version of [2]. y This work was supported by DARPA ITO under grant A0 F316 and by NSF under grant NCR 950814. Akar, Oguz, and Sohraby, A Novel Computational Method for Finite QBD Processes 1 1 Introduction In this paper, we examine a finite Quasi Birth and Death (QBD) process which is a Markov chain with ....

N. Akar and K. Sohraby. Finite and infinite QBD chains: a simple and unifying algorithmic approach. In Proc. IEEE INFOCOM, 1997.

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AKAR,N.--SOHRABY, K., Finite and Infinite QBD Chains: A Simple and Unifying Algorithmic Approach, Proceedings of IEEE INFOCOM, (1997), pp. 1105--1113.

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