Uniform In Time Bounds For “No-Wait” Probability In Queues Of Mt/Mt/S Type

Abstract

In this paper we present new analytical results concerning long-term staffing problem in high-level telecommunication service systems. We assume that a service system can be modelled either by a classic M t /M t /S queue, or M t /M t /S queue with batch service or M t /M t /S with catastrophes and batch arrivals when empty. The question under consideration is: how many servers guarantee that in the long run the probability of zero delay in a queue is higher than the target probability at all times? Here the methodology is presented, which allows one to construct uniform in time upper bound for the value of S in each of the three cases and does not require the calculation of the limiting distribution. These upper bounds can be easily computed and are accurate enough whenever the arrival intensity is low, but become rougher as the arrival intensity is further increased. In the numerical section one compares the accuracy of the obtained bounds with the exact vales of S , obtained by direct numerical computation of the limiting distribution. 1 That is a uniform in time bound.

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

“…One of the appealing candidates are queueing systems with balking, in which arrival intensities decrease with the growth of the total number of customers in the system. Yet another direction of research follows from the work of Zeifman et al (2016b) and is related to optimization of "no-wait" probabilities. Fig.…”

Bounds on the rate of convergence for inhomogeneous $M/M/S$ systems with either state-dependent transitions, or batch arrivals and service, or both

“…One of the appealing candidates are queueing systems with balking, in which the arrival intensities decrease with the growth of the total number of customers in the system. Another one direction of research follows from [36] and is related to the optimization of no-wait probabilities in such queueing systems.…”

Evaluation of the complexity of modeling the dynamics of complex systems

“…Например, если имеется всего лишь m = 7 различных элементов (СМО), для каждого из которых известны n = 2 состояния, то общая модель должна включать в себя N = 2 [4,5]. Различные оценки для систем массового обслуживания с учетом современных реалий формирования потоков данных получены в циклах статей А. Зейфмана и других авторов, представленных в трудах ежегодной конференции Европейского Совета по моделированию [6][7][8][9][10][11][12][13][14][15][16][17]. Подходы на основе имитационного моделирования рассмотрены в [18,19].…”

“…Очевидно, что с учетом сложности некоторых функций распределения вероятностей решение (5) и (10) ( )…”