Kronecker Modeling and Analysis of Multidimensional Markovian Systems by Tuǧrul Dayar

Author:Tuǧrul Dayar
Language: eng
Format: epub
ISBN: 9783319971292
Publisher: Springer International Publishing

and for , the drift is [112]

When ∑ c = 1 C(λ (c)∕μ (c)) < S, coefficients of all infinite variables are negative; hence, for all and condition (ii) at the beginning of this section holds. Besides, is finite for any arbitrarily chosen γ > 0. Therefore, condition (i) at the beginning of this section also holds.

Example 6.

(cont’d) In our model,

The Lyapunov function is given by

The drift for is

and the drift for is

.

The global maximum drift is χ = 4. 9359375 and attained when i 6 + i 8 = 2 at state (0, 0, 0, 1, 0, 1, 0, 1). With a lower bound of 90% on the steady-state probability mass (i.e., ɛ = 0. 1), we compute γ = 44. 4234 and .

Now, we provide the truncated multidimensional CTMC and a Kronecker representation of its generator matrix. A block tridiagonal generator matrix as in the previous two models of this section will not be used since the order of diagonal blocks increases considerably with a large number of dimensions as the partition index increases, and such a block partitioning does not yield itself to a scalable steady-state solver.

Traditionally, truncation of a countably infinite reachable state space for exponentially distributed retrial times is achieved by imposing an upper bound, say M, on the total number of customers in the orbit [131, 301]. This implies blocking of an arriving customer if the total number of retrial customers is already M. We consider a similar truncated model in which the size of retrial phase o of orbit c is limited by

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