Steady state analysis of level dependent quasi-birth-and-death processes with catastrophes

Baumann, Hendrik; Sandmann, Werner

doi:10.1016/j.cor.2011.05.003

Cited by 36 publications

(28 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In the literature, the algorithm and its theory are usually presented for continuous-time Markov chains with generator Q, where π Q = 0 is to be solved, but since for any stochastic matrix P, the matrix P − I has the properties of a generator matrix, the above algorithm can be interpreted as a special case. If p i j just depends on j − i, it is due to Neuts [11], for the more general form presented above (and even with non-constant matrix dimensions), we refer to [4,9,8,10,2]. The existence of the inverses as well as the convergence of the R n is proved by giving a physical interpretation of the entries of the matrix R n (conditional expected sojourn times) and those of I − p nn − R n p n+1,n , see [4].…”

Section: Resultsmentioning

confidence: 99%

Two-sided continued fractions in Banach algebras— A Śleszyński–Pringsheim-type convergence criterion and applications

Baumann

2015

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

We consider two-sided continued fractions in Banach algebras, that iswhere the coefficients b n , a n , c n are elements of some Banach algebra. We prove convergence criteria which are exact generalizations of the well-known Pringsheim-type convergence criteria for complex continued fractions, and state a result concerning the rate of convergence. Both convergence criteria and error bounds improve all known results.

show abstract

Section: Resultsmentioning

confidence: 99%

Two-sided continued fractions in Banach algebras— A Śleszyński–Pringsheim-type convergence criterion and applications

Baumann

2015

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, extensions of the algorithm beyond LDQBD processes are considered, for example, along the lines of [3], in which matrixanalytic methods were applied to compute stationary distributions of LDQBD processes with catastrophes where in each state the level component may drop to zero such that the generator matrix deviates from the block-tridiagonal form in its first block column.…”

Section: Resultsmentioning

confidence: 99%

“…. , R N using recursion (3). Of course, this method delivers a better approximation for R N too, since, for N * → ∞, we have R * N → R N .…”

Section: Getting Started In the Case Of Infinite State Spacesmentioning

confidence: 93%

See 1 more Smart Citation

Computing Stationary Expectations in Level-Dependent QBD Processes

Baumann¹,

Sandmann²

2013

J. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

show abstract

“…Sudhesh [23] gave an explicit transient solution for the state probabilities of the same model studied in [27]. Baumann and Sandmann [2] studied a model about level dependent quasi-birth-and-death processes with catastrophes, and gave a matrix analytic algorithm to analyze M/M/c queues in a random environment with catastrophes and state dependent rates. Kim and Lee [16] investigated an M/G/1 queue with disasters in which the system is equipped with a substitute server to provide service during the repair period.…”

Section: Introductionmentioning

confidence: 98%

Analysis of the M/G/1 queue in multi-phase random environment with disasters

Jiang

Liu

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

This paper studies an M/G/1 queue in a multi-phase random environment. When in operative phase i, i = 1, 2, . . . , n, the system is subject to disastrous interruptions, causing all present customers (waiting and served) to leave the system. At an exponential failure instant, the server abandons the service and the system goes directly to repair phase. After an exponential repair time, the system moves to operative phase i with probability q i , i = 1, 2, . . . , n. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive results of the cycle analysis, the sojourn time distribution and the length of the server's working time in a service cycle. In addition, some special cases and numerical examples are presented.

show abstract

Steady state analysis of level dependent quasi-birth-and-death processes with catastrophes

Cited by 36 publications

References 54 publications

Two-sided continued fractions in Banach algebras— A Śleszyński–Pringsheim-type convergence criterion and applications

Two-sided continued fractions in Banach algebras— A Śleszyński–Pringsheim-type convergence criterion and applications

Computing Stationary Expectations in Level-Dependent QBD Processes

Analysis of the M/G/1 queue in multi-phase random environment with disasters

Contact Info

Product

Resources

About