Abstract
This paper focuses on new measures of performance in single‐server Markovian queueing system. These measures depend on the moments of order statistics. The expected value and the variance of the maximum (minimum) number of customers in the system as well as the expected value and the variance of the minimum (maximum) waiting time are presented. Application to an M/M/1 model is given to illustrate the idea and the applicability of the proposed measures.
Highlights
Queueing systems have found wide applications in modeling and analysis of computer and communication systems, and several other engineering systems in which single server is attached to one or more workstations
The study of queueing systems has often been concerned with the busy period and the waiting time, because they play a very significant role in the understanding of various queueing systems and their management
The length of a busy period of an M/M/1 queue with constrained workload is discussed by Kinateder and Lee 1 using Laplace transform
Summary
Queueing systems have found wide applications in modeling and analysis of computer and communication systems, and several other engineering systems in which single server is attached to one or more workstations. Draief and Mairesse 2 analyzed the service times of customers in an M/M/1 queue depending on their position in a busy period. Takagi and Tarabia 4 provided an explicit probability density function of the length of a busy period starting with i customers for more general model M/M/1/L, where L is the capacity of the system; see Tarabia 5. Limit theorems are proved by investigating the extreme values of the maximum queue length, the waiting time and virtual waiting time for different queue models in literature. Artalejo et al presented an efficient algorithm for computing the distribution for the maximum number of customers in orbit and in the system during a busy period for the M/M/c retrial queue.
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