Calculating steady-state probabilities of queueing systems using hyperexponential approximation

Abstract

This article proposes an analysis of the results of the application of hyperexponential approximations with parameters of the paradoxical and complex type for calculating the steady-state probabilities of the G/G/n/m queueing systems with the number of channels n = 1, 2 and 3. The steady-state probabilities are solutions of a system of linear algebraic equations obtained by the method of fictitious phases. Approximation of arbitrary distributions is carried out using the method of moments. We verified the obtained numerical results using potential method and simulation models, constructed by means of GPSS World.

Calculating steady-state probabilities of single-channel closed queueing systems using hyperexponential approximation

“…Works [5][6][7][8] show that the use of hyperexponential approximation (denoted by l H ) makes it possible to determine with high accuracy the steady-state probabilities of non-Markovian queuing systems. These probabilities are determined using solutions of a system of linear algebraic equations obtained by the method of fictitious phases.…”

“…x is the state, when there are k customers in the system (1 ) k m   , and , , , i j u v are the number of customers for which the generation time is in the first, second, third and fourth phase, respectively, and s is the phase number of service time. The proposed numbering of the states differs from that introduced in the works [6][7][8] and helps reduce the number of states of the H 4 /H l /1/m closed system. We denote by 0( , , , )…”

Calculating steady-state probabilities of single-channel queueing systems with changes of service times depending on the queue length

“…Papers [5][6][7] suggest the use of hyperexponential approximation (denoted by k H ) for calculating of steady-state probabilities of the non-Markovian queueing systems. If variation coefficients 1 V and 2 V of distributions of the interarrival time between two consecutive customers and the service times satisfy the conditions 1 2 0.6 V V   and 1 2 max{ , } 2 V V  then we are able to calculate steady-state probabilities with high accuracy (higher than in the case of using simulation models); see [7]. These probabilities are determined using solutions of a system of linear algebraic equations obtained by the method of fictitious phases.…”

Reliability assessment of a series system with redundancy and repair facilities

“…Works [2,[4][5][6][7][8] show that the use of hyperexponential approximation (denoted by l H ) makes it possible to determine with high accuracy the steady-state probabilities of non-Markovian queuing systems. These probabilities are determined using solutions of a system of linear algebraic equations obtained by the method of fictitious phases.…”