**1. Introduction**

The world economy suffered a lot of losses after the coronavirus pandemic and it will take a long time for its rehabilitation. An important role in the development of the world economy will have the transportation and communication systems because it is necessary to renovate the economic communications among countries. For the investigation of the transportation and communication systems, mathematical models of queuing systems with moving servers are widely used. Typical examples of queues with moving servers are the lifts'systems. Lifts and communication systems, the traffic, the airport and the shipping facilities have a lot of similarities. All of them are united by the same principle – servers are moving in these systems. Hence, mathematical models of lifts'systems can be applied for the investigation of other systems with moving servers. As investigation of such systems by analytical approaches faces troubles, the use of the modern computers can allow to simulate their behavior. The simulation of such systems, various systems of programming (*Wolfram Mathematica* and others) allow to get close to reality, the numerical data of the desired parameters and give some advice for applications. The simulation also can also be a hint for the continuation of the analytical research. Today it is difficult to imagine the development of modern cities, such as New York, Moscow, Shanghai, Istanbul and others, without skyscrapers. The process of designing skyscrapers needs an effective planning and operating of the lifts'systems, which allows to improve various characteristics (customer's waiting and service times) and to reduce energy expenses. An important problem is also to introduce various control rules for the lifts'systems. At a first view, it seems that the lifts'systems have simple structures. In fact, from a scientific point of view, these systems have complicated structures and the construction of mathematical models needs some non-standard approaches. Moreover, although these problems are formulated in the frame of standard queues models with moving servers, for their investigations it is necessary to develop new methods and approaches, using different fields of mathematics. The lifts'systems, the traffic problems, the airport and the shipping facilities have a lot of similarities. The investigation of these systems can allow to estimate the main operating parameters (customers' waiting and service times, energy expenses and others) and to make the necessary recommendations for constructors and engineers. There are many publications in this field e.g. [1–3] and even special scientific journals are published *(International J. Transportation Science and Technology, Research in Transportation Business & Management* and many others*).* Unfortunately, the complicated lifts'systems, with various control rules, are not yet investigated widely. In [4–7], various mathematical models of lifts'systems with, different control rules, were introduced. The construction of mathematical models of lifts'systems and their research by analytical approaches, face some difficulties, because as it was mentioned above, these models have complicated structures. Hence, one of the effective methods are the simulation and the collection of simulated data, which can be used for estimating the various parameters of such lift systems, by comparing different control rules, finding optimal regimes for their operation. As the customers' arrival process into the lifts'systems has a stochastic structure, hence it leads to constructing and investigating the new stochastic models, approaches and programs for their simulation.

In this paper, the authors consider various lift systems with different parameters and different control rules. This paper can be regarded as a continuation of the authors' investigations presented in [4, 5]. Hence, we follow the notations introduced in these papers.
