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The problem of virtual machine placement on physical servers in cloud data centers is considered. The resource management system has a two-level architecture consisting of global and local controllers. Local controllers analyze the state of the physical servers on which they are located and determine possible underloading, overloading, and overheating states based on the forecast for the next observation window. If one of the listed states is detected, the local controller notifies the global controller, which selects the destination servers to host the virtual machines via migration. It is proposed to place virtual machines based on the criteria of minimum remaining unused resources and violation of SLA agreements. A mathematical formulation of the optimization problem is given, which is equivalent to the known main assignment problem in terms of structure, necessary conditions, and the nature of variables. Reducing the assignment problem to a closed transport problem allowed us to solve the problem of hosting virtual machines under many criteria in real time and significantly increase its dimension in comparison with heuristic algorithms, which makes it possible to maintain the quality of modern cloud services in the conditions of rapid growth of physical and virtual resources of data centers. The developed mathematical formulation of the problem and the results of computational experiments can be included in the mathematical software of virtual machine live migration.

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Cloud applications and services such as social networks, file sharing services, and file storage have become increasingly popular among users in recent years. This leads to the enlargement of data centers, and an increase in the number of servers and virtual machines. In such systems, live migration is used to move virtual machines from one server to another, which affects the quality of service. Therefore, the problem of finding the total migration time is relevant. This article proposes analytical approach to obtaining analytical expression of the probability density of the total migration time based on the use of the apparatus of characteristic functions. The obtained expression is used to calculate characteristics of migration, taking into account the applications contributing the most randomness to the total migration time. To simplify the calculation of migration characteristics, the use of the Laguerre series can be recommended as giving more reliable results compared to Gram-Charlier series.