Transient Analysis of the M/M/k/N/N Queue using a Continuous Time Homogeneous Markov System with Finite State Size Capacity

In the present, we study the problem of strong ergodicity in nonhomogeneous Markov systems. In the first basic theorem, we relax the fundamental assumption present in all studies of asymptotic behavior. That is, the assumption that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix. In addition, we study the practically important problem of the rate of convergence to strong ergodicity for a nonhomogeneous Markov system (NHMS). In a second basic theorem, we provide conditions under which the rate of convergence to strong ergodicity is geometric. With these conditions, we in fact relax the basic assumption present in all previous studies, that is, that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix geometrically fast. Finally, we provide an illustrative application from the area of manpower planning.

“…due to condition (17). Since N 2 (t, ν) < ∞ for every r and ν and it is a sum of positive numbers, we have that its tail goes to zero.…”

Section: Definitionmentioning

confidence: 94%

Section: Introductory Notesmentioning

confidence: 99%

Strong Ergodicity in Nonhomogeneous Markov Systems with Chronological Order

Vassiliou

2024

“…By building upon the models of Kim [24] and Vasiliadis [25], we propose the following performance indicators.…”

Section: System Operation Indicatorsmentioning

confidence: 99%

Analysis of a Pre-Emptive Two-Priority Queuing System with Impatient Customers and Heterogeneous Servers

Yin,

Yan,

Guo

et al. 2023

This paper presents a queuing system model that incorporates multiple priorities, multiple abandonments, and heterogeneous servers. Waiting for service easily leads to impatient behaviors. The impact of two kinds of impatient behaviors, balking and reneging, on queueing system performance is examined. The problem is formulated as continuous-time Markov chains. It also introduces a special state called the non-sojourn state to record the number of customers who abandon the system. The state transition rate matrix is transformed into a block tridiagonal matrix by appropriately setting the state numbers. A novel indicator called interstate transition frequency is proposed, which aids in distinguishing state transitions during the system evaluation process. Based on the interstate transition frequency, a set of indicators is derived to offer additional analytical perspectives for the queuing system. Finally, the proposed model is applied to an automobile repair shop to validate its effectiveness in practical scenarios.

“…In [30], the author discussed the case of closed HMS with finite capacities of the states. In [31], the close relation between M/M/k/T/T queues and close HMS in continuous time is presented. More recent results on NHMS in continuous time could be found in [32], while a more recent review on the subject was given by [33].…”

Section: Introductory Notesmentioning

confidence: 98%

Limiting Distributions of a Non-Homogeneous Markov System in a Stochastic Environment in Continuous Time

Vassiliou

2022

The stochastic process non-homogeneous Markov system in a stochastic environment in continuous time (S-NHMSC) is introduced in the present paper. The ordinary non-homogeneous Markov process is a very special case of an S-NHMSC. I studied the expected population structure of the S-NHMSC, the first central classical problem of finding the conditions under which the asymptotic behavior of the expected population structure exists and the second central problem of finding which expected relative population structures are possible limiting ones, provided that the limiting vector of input probabilities into the population is controlled. Finally, the rate of convergence was studied.