The subject of discussion in this chapter is the stochastic approach to model investigation of queuing systems (QS) with waiting buffer (queue) and server (service unit). Two types of QS are presented: single-channel and multi-channel QS. Single-channel QS is discussed in the first part of the chapter with presentation of main characteristics such as workload of the resource, queuing length, total number of requests in system, waiting time and time for presence in the system. The second part deals with organization of analytical investigation of QS and in particular presentation of discrete and continuous time Markov chains. The two possibilities for investigation were considered – during the development of a transient regime and the conditions for reaching a steady state regime. Three versions of QS are presented in QS – with an infinite buffer, with a limited buffer, and QS with request rejections in the absence of a buffer. Some examples for stochastic model investigation are presented in the last part.
Top1. Main Characteristics Of Single-Channel Queuing System
Mathematical modeling based on networks of queuing systems (QSs) is widespread in various fields such as optimization of various technical, physical, economic, industrial, and administrative systems. Confirming this, the paper (Dudin et al., 2020) discusses investigation by using stochastic approach various complex multi-server QS or single-server queues with random distribution of service time or semi-Markov service. A similar confirmation of the applicability of the QSs was made in (Atlasov et al., 2020) for minimizing the total time for searching and processing information throughout the system, with remote servers divided into groups of equal capacity.
One of the most frequently studied QS is single channel with different types of synchronous or asynchronous workflow (Gortsev & Nezhelskaya, 2022). The research conducted is of different orientation, and the basis of each is QS. One example is the improvement of services provided by a government institution, discussed in (Darmayunata et al., 2023). The purpose of the research is to establish effective organization of the queue for ordering customers based on registration through a website with previously expected service time (Ts). To model the process, Single Channel Multi Steps QS was used with the requirement to minimize waiting time (Tw) and the number of waiting requests (L).
The application of QT and, in particular, the model researched on the basis of single-channel QS, is relevant to processes of different nature, including in the investigation of the traffic characteristics of multi-service communication networks. An example of such an application is the random variable-based traffic analysis of incoming requests with constant time intervals, which is presented in (Likhttsinder & Bakai, 2022). Based on the generalized Hinchin-Pollachek formula for average queue values, queue parameters such as dispersion, correlation properties and load factor are defined. To represent the workflow of requests, a group Poisson flow is chosen, which has no correlation component, and the average value of the queue is determined entirely by the number of incoming requests and their service intervals. As a result of the research, it is shown that the requests are processed cyclically and in the last interval of each cycle the queue is empty.
The main components of a classical single-channel QS and the related parameters of the request service process were briefly discussed in Chapter 5 (part 5.3.2, Figure 28). A presentation of the parameters of single-channel QS related to a stochastic model investigation of computer processes is presented below.
Workload of the resource (R) – for single-channel QS it is the ratio of the real busy time of a server unit (SU) to the total busy time T, while for multi-channel QS it is the average number of SUs busy serving the requests for an interval T. The parameter is commensurate with utilization factor ρ and allows to determine a general utilization factor. A case study of low intensity of the input flow of requests λ and high speed of service time is discussed in (Zyulkov et al., 2022). The object of a model study is QS type M/M/1, and the study was conducted for a very short observation time and approximate dependencies for the system utilization factor were obtained. A simple probabilistic model in the form of a probabilistic mixture was developed to conduct the experiments. An analogous study of QS with high loading and small tail is done in (Tsitsiashvili, 2021), where two alternatives are proposed. The first one is uniting many single channel QSs in a new multi-channel QS. The second variant is based on the model of single-channel QS in which random fluctuations are determined depending on the load factor. The exponent of this degree has a critical value above which the tail tends to zero and below which it tends to infinity.