Eigenvalue expression for a batch markovian arrival process
Shoichi Nishimura,Hajime Sato +1 more
TLDR
The main result gives the time-dependent form of the first and second factorial moments of the counting process, which is represented by eigenvalues and eigenvectors of the matrix generating function of the batch size.Abstract:
Consider a batch Markovian arrival process (BMAP) as the counting process of an underlying Markov process representing the state of environment. Such a process is useful for representing correlated inputs for example. They are used both as a modeling tool and as a theoretical device to represent and approximate superposition of input processes and complex large systems. Our objective is to consider the first and second moments of the counting process depending on time and state. Assuming that the probability generating functions of batch size are analytic, and that eigenvalues of the infinitesimal generator are simple, we derive an analytic diagonalization for the matrix generating function of the counting process. Our main result gives the time-dependent form of the first and second factorial moments of the counting process, which is represented by eigenvalues and eigenvectors of the matrix generating function of the batch size.read more
Citations
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BookDOI
Constructive Computation in Stochastic Models with Applications
TL;DR: The first € price and the £ and $ price are net prices, subject to local VAT, and the €(D) includes 7% for Germany, the€(A) includes 10% for Austria.
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A BMAP/SM/1 queueing system with Markovian arrival input of disasters
TL;DR: In this article, the authors obtained a queue length and virtual waiting time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters, where the arrival of negative arrivals causes all customers to leave the system instantaneously.
Journal ArticleDOI
Markovian bulk-arrival and bulk-service queues with state-dependent control
TL;DR: A modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control is studied, with important questions regarding hitting time and busy period distributions answered in detail and the Laplace transforms of these distributions presented.
Journal ArticleDOI
Markovian bulk-arriving queues with state-dependent control at idle time
Anyue Chen,Eric Renshaw +1 more
TL;DR: In this article, a Markovian bulk-arriving queue modified to allow both mass arrivals when the queue is idle and mass departures which allow for the possibility of removing the entire workload is considered.
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Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues
TL;DR: It is shown that this stopped Markovian bulk-arrival and bulk-service queueing model is always λC-transient and that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ∈[0,λC].
References
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Book
Finite Markov chains
John G. Kemeny,J. Laurie Snell +1 more
TL;DR: This lecture reviews the theory of Markov chains and introduces some of the high quality routines for working with Markov Chains available in QuantEcon.jl.
Journal ArticleDOI
Finite Markov Chains.
TL;DR: This lecture reviews the theory of Markov chains and introduces some of the high quality routines for working with Markov Chains available in QuantEcon.jl.
Journal ArticleDOI
New results on the single server queue with a batch markovian arrival process
TL;DR: This work generalizes results to the single server queue with the batch arrival process and emphasizes the resulting simplifications of the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process.