Browsing by Subject "Matrix exponential distribution"

Now showing 1 - 3 of 3

Open Access
Numerical methods for the transient analysis of multi-regime Markov fluid queues
(2019-01) Gürsoy, Ömer
Markov fluid queue models have served as one of the main tools for the performance analysis of computer and communication systems and networks. These models have also been used in other disciplines such as insurance risk, finance, inventory control, etc. This thesis focuses on the time dependent (transient) analysis of Markov uid queue models. In particular, a numerical method is proposed to obtain both the transient and first passage time distributions of a Multi-Regime Markov Fluid Queue (MRMFQ). The proposed method is based on obtaining the steady-state solution of an auxiliary MRMFQ that is to be constructed from the original MRMFQ which then leads to the related transient measures of interest. First, in order to model the deterministic time horizon, the Erlangization method is used. Then, as an alternative to Erlangization, ME-fication technique which efficiently replaces the Erlang distribution with a Concentrated Matrix Exponential (CME), is used. As an application of the proposed method, an M/M/S+G queue with generally distributed impatience times is modeled by using MRMFQs and our transient analysis method is subsequently applied to obtain the time dependent distributions. Numerical examples are given to show the effectiveness of the proposed transient analysis method while employing ME-fication.
Open Access
Solving the ME/ME/1 queue with state–space methods and the matrix sign function
(Elsevier BV * North-Holland, 2006) Akar, N.
Matrix exponential (ME) distributions not only include the well-known class of phase-type distributions but also can be used to approximate more general distributions (e.g., deterministic, heavy-tailed, etc.). In this paper, a novel mathematical framework and a numerical algorithm are proposed to calculate the matrix exponential representation for the steady-state waiting time in an ME/ME/1 queue. Using state-space algebra, the waiting time calculation problem is shown to reduce to finding the solution of an ordinary differential equation in state-space form with order being the sum of the dimensionalities of the inter-arrival and service time distribution representations. A numerically efficient algorithm with quadratic convergence rates based on the matrix sign function iterations is proposed to find the boundary conditions of the differential equation. The overall algorithm does not involve any transform domain calculations such as root finding or polynomial factorization, which are known to have potential numerical stability problems. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.
Open Access
Solving the single server semi-Markov queue with matrix exponential kernel matrices for interarrivals and services
(ACM, 2006) Akar, Nail; Sohraby, K.
Markov renewal processes with semi-Markov kernel matrices that have matrix-exponential representations form a superset of the well-known phase-type renewal process, Markovian arrival process, and the recently introduced rational arrival process. In this paper, we study the steady-state waiting time distribution in an infinite capacity single server queue with the auto-correlation in interarrival and service times modeled with this general Markov renewal process. Our method relies on the algebraic equivalence between this waiting time distribution and the output of a feedback control system certain parameters of which are to be determined through the solution of a well known numerical linear algebra problem, namely the SDC (Spectral-Divide-and- Conquer) problem. We provide an algorithmic solution to the SDC problem and in turn obtain a simple matrix exponential representation for the waiting time distribution using the ordered Schur decomposition that is known to have numerically stable and efficient implementations in various computing platforms.