=Paper=
{{Paper
|id=Vol-2577/paper17
|storemode=property
|title=Adaptation of cloud computing as optimization of the process of rendering services to users in the conditions of limited computing resources
|pdfUrl=https://ceur-ws.org/Vol-2577/paper17.pdf
|volume=Vol-2577
|authors=Aleksandr Matov
|dblpUrl=https://dblp.org/rec/conf/its2/Matov19
}}
==Adaptation of cloud computing as optimization of the process of rendering services to users in the conditions of limited computing resources==
<pdf width="1500px">https://ceur-ws.org/Vol-2577/paper17.pdf</pdf>
<pre>
210


 Adaptation of cloud computing as optimization of the
process of rendering services to users in the conditions of
              limited computing resources

                                     © Aleksandr Matov

            Institute for Information Recording of NAS of Ukraine, Kyiv, Ukraine

                                  matov@ipri.kiev.ua



       Abstract. We consider the cloud computing (СС) infrastructure as an object of
       adaptation and the process of cloud computing adaptation as an optimization. The
       general formulation of the problem of adaptation of the discipline of providing
       computing resources to the users of СС is outlined. The technology of dynamic
       adaptive mixed discipline of providing computing resources to users of СС is
       offered. The direction of solving the problem of optimization of dynamic adap-
       tive mixed discipline is given.The well-known optimization functionality is pro-
       posed, based on the assumption that the results of the use of computing resources
       by the user (solving user problems) are depreciated in proportion to their time in
       the queue for the solution and the solution itself in the СС system. Other func-
       tionalities with time constraints are also possible. This is relevant for today's
       global real-time information and analytics systems using cloud computing tech-
       nology and can be critical with limited computing resources. It is stated that the
       optimization problem is solved by an iterative method using the appropriate an-
       alytical models of the operation of СС.The description of such models is
       given.The stochastic nature of the main factors and the need to quantify mass
       processes based on probability theory determines the use of queuing theory. It is
       proposed to develop analytical models of cloud computing as a queuing system
       with mixed service discipline. Models should consider failures and different fea-
       tures of operation and, where possible, have arbitrary distribution laws for certain
       probable processes. Then it is possible and appropriate to use the technology of
       dynamic adaptive mixed discipline of providing computational resource to the
       users of CC as a mechanism of adaptation of CC. The mathematical formulation
       and method of solving such tasks are given.

       Keywords: cloud computing, discipline of providing computing resources, ab-
       solute and relative priorities, adaptation and optimization of service disciplines,
       adaptation efficiency, mixed service discipline, mathematical model.




Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
                                                                                        211


Introduction

Creating adaptive cloud computing infrastructures that are able to adapt dynamically to
constantly changing conditions of operation, and developing appropriate computing or-
ganization methods is an important area of development of modern global information
and analytical systems using cloud computing technologies.
    Adaptation as control is secondary to the main control loop. If the management ful-
fills the basic goals, the realization of which ensures the functioning of the object, then
the adaptation ensures the quality of this functioning. Therefore, when there is a need
to improve (or maintain at the required level) the quality of the facility, there is always
a need for adaptation.
    The peculiarity of the adaptation system is that it is necessary to work in the condi-
tions of considerable uncertainty of the environment and the handling of the object.
    The ambiguity of the environment and the object is a feature that allows you to con-
sider adaptation as a specific type of control. In this case, the degree of uncertainty
determines the importance of solving the adaptation problem: the greater the uncer-
tainty, the greater the need for adaptation.


Cloud computing as an object of adaptation
Cloud computing is an object with a high degree of uncertainty in the operation process.
Here, the external uncertainty of the flow of computational resource (CR) requests (en-
vironment) is complemented by the internal uncertainty of the CC (object) associated
with the presence or absence of the required CR, the random failures of the CC system,
and the need to provide certain temporal characteristics for the many customers. This
is what determines the need for introducing adaptation into the process of functioning
of CC.
   In addition, the introduction of adaptation to the process of functioning of the CC is
associated with the need to maintain the system in an optimal and sometimes simply
operational state, regardless of the numerous external and internal factors that bring the
CC to the desired target state.
   All of the above can equally be attributed to the computational process as an object
of adaptation, because it develops in CC and is an integral attribute of it.
   The notion of adaptation as an active action (control) is usually embedded in two
meanings: adapting an object to a fixed environment (passive adaptation) and finding
an environment appropriate to that object (active adaptation) [1]. In the first case, the
adaptable entity functions to fulfill its goal in the best possible environment, that is, to
maximize its effectiveness in that environment. Active adaptation, on the contrary, im-
plies a change of environment in order to maximize the performance of the object.
   With respect to CC, as a queuing system), active adaptation can be seen as a change
in the intensity or quantity of incoming application flows, as well as the laws of the
distribution of the application process.
   Passive adaptation is most commonly used in the operation of CC, in which adaptive
influence may have different character. It may change either the parameters of the ad-
212


aptation object (parametric adaptation) or its structure (structural adaptation).The in-
tensity and laws of the distribution of the application service process, restrictions on the
waiting time (stay) of applications in the queue (system), the order of service of appli-
cations (service discipline), etc. can be considered as managed parameters of CC. An
example of a structural adaptation that changes the number of servicing devices and the
relationship between them is the reconfiguration of multi-server CC. Structural adapta-
tion is more radical and is usually accompanied by parametric adaptation, because each
structure has its own parameters.
    Depending on whether the model is an object of adaptation or not, there are two very
important types of adaptation: adaptation with and without model (search adaptation),
which differ significantly from each other [1].
    In the presence of an adequate object model, it is sufficient to measure the state of
the environment for the synthesis of the adaptive impact, and using the model to deter-
mine the impact that should put the object in the desired state.
    However, very often, the object of adaptation is so complex that it is impossible to
build a model of it, and an adequate model is all the more so.
    At the same time, it is probably not possible to use the adaptation method with the
model, which forces to resort to search adaptation. This type of adaptation is distin-
guished by the presence of search, a specially organized process that allows you to
determine the necessary adaptive impact without having an object model. Search en-
gine adaptation is characterized by experiments with an object, in the process of which
they obtain information about its properties. This information determines the adaptive
impact of the object's performance.
    The search engine adaptation process itself is a consistent, multi-stage process - steps
are taken at each stage to improve the performance of the facility (as opposed to adapt-
ing to a one-stage adaptation model).
    If, when adapting to a model, the state of the object is to be measured only to adjust
its model and not required for the adaptation itself, then in the search for adaptation the
state of the object carries the basic information for forming the influencing adaptation.
    The difficulty of adapting to a model lies in the synthesis of the model of the object,
and the adaptation itself is the solution of the optimization problem of selecting such
an impact formation that would satisfy the adaptation goals. Search engine adaptation
has other difficulties - you need to experiment with the object at the same time and
adapt it.
    In all cases, when it is possible to build an adequate model of the object, the question
of choosing the type of adaptation weighs unequivocally in favor of adaptation with the
model, because only the presence of the model allows you to quickly adapt the object.


Cloud computing adaptation as optimization
The solution to the problem of adaptation is to determine the kind of control (adaptive)
impact that maximizes the performance of the object in the current situation.
   The situation is characterized by two factors: the state of the environment in which
the object is located and the state of the adaptation object itself.
                                                                                     213


   For CC, as a queuing system, the state of the environment can be understood, for
example, the intensity of incoming requests, and the state of the object (system) - the
number or time of waiting (stay) of requests in the queue (system), or the malfunction
of the serving device, system boot level, etc.
   Depending on the current situation, an adaptive effect should be formed that mini-
mizes the average number or average waiting time (stay) of applications in the queue
(system), or the time of entering the system in a steady state, or the total cost for the
system operation, or the probability of losing applications, etc. e. The purpose of the
adaptation may be to maximize revenue from service requests, eliminate system over-
load, and maintain it in a stationary mode.
   Thus, the adaptation of CC can be considered as a process of optimizing work in the
current situation.


General statement of the task of adaptation of the discipline of service
The task of adapting the discipline of service in the CC is due to unforeseen and uncon-
trolled changes in the environment and system, which inevitably alter the optimal set-
ting of the discipline of service, if one was implemented in the system. Therefore, the
systematic adjustment (adaptation) of the discipline of service is inevitable if you wish
to maintain the system in optimal mode, regardless of changes occurring in the envi-
ronment and system.
   We formulate in general terms the task of adapting the discipline of service [2].
   Let X and E be the controlled and uncontrolled states of the medium. The {X, E}
pair uniquely describes the environment in which the CC is located. For example, X is
the passport data of service requests and E is the intensity of their receipt.
   Similarly, the pair {Y, H} describes the state of the system. Here, Y and H are re-
spectively controlled and uncontrollable factors. For example, Y is the length of the
application queues, and H is the service intensity or system failure rate.
   The performance of the system is extreme. It is defined on the controlled states of
the environment and system:

                                Ý = Ý ( X ,Y ) .                                     (1)
The system performance indicators may be the average time (waiting) of applications
in the system (queues), the average length of the application queue, the average total
cost of waiting (staying) applications in the queue (system), etc.
   The status of system Y depends on X, E and H, as well as on the discipline of ser-
vicing S:

                              Y = F ( X , E, H , S ) ,                               (2)
where F is the system operator.
   Service discipline refers to the rule of selecting service requests depending on the
state of the environment and system:
214


                                    S = S ( X ,Y ) .                                   (3)
In most cases, the optimality of discipline S is related to the extremisation of the per-
formance of the system (1).This means that for the synthesis of optimal discipline, the
following optimization problem must be solved:
                    Ý [ X , F ( X , E , H , S )] → extr∗  S 0
                                                   S∈S          ,                     (4)

where - restrictions imposed on the choice of discipline of service S.
   These restrictions may be related, for example, with a certain set of predefined ser-
vice disciplines, etc.
   Obviously, it is impossible to solve problem (4) at the stage of designing a computer
system , because a priori unknown factors E and H. Averaging over these factors cannot
be entered because they can be non-stationary.
                                                                0
   Therefore, the problem of synthesis of optimal discipline S should be solved by the
adaptation of CC, that is, in the mode of their operation. Then adaptation is reduced to
solving the problem
                                Ý ( S ) → extr∗  S 0
                                             S ∈S



in different disciplines:
                             ^      ^           ^      ^
                             Ý 1 = Ý ( S1 ),..., Ý ξ = Ý ( Sξ ).


The adaptation algorithm must specify the sequence of transition from one discipline
to another: which leads to a solution that is optimal in the current situation.


Use for dynamic adaptive technology adaptationthe mixed discipline of
      providing computing resourcesusers of CC
Currently, a large number of different service disciplines are known. Of these, the dis-
ciplines of service with relative and absolute priorities are widely used in CC. However,
these disciplines are static and therefore have a number of significant disadvantages
that reduce the efficiency of computing systems (processes) in the uncertainty of the
environment and the behavior of the systems themselves.
   When using discipline with relative priority, the selection of a regular service request
can only be made after the completion of the current service, even if the service request
has a lower priority. As a result, the length of your stay at the CC may be unacceptably
long for some of your most important applications. Reducing the delay in servicing
important applications is achieved by interrupting, that is, introducing absolute priority
for these applications. However, the duration of low priority applications is increased
by the CC, and in some cases, with the intensive receipt of important applications, the
process of servicing low priority applications may be blocked, which also reduces the
effectiveness of the CC as a whole.
                                                                                        215


         In order to compensate for the disadvantages inherent in the disciplines of ser-
vice with relative and absolute priorities and taking into account their advantages, it is
advisable to implement in the mixed disciplines of service that use both relative and
absolute priorities.
   Consider one of the mixed service disciplines. Let the N input streams of applications
according to their importance and urgency in service be divided into M groups, between
which there is an absolute priority, and inside a relative one. This means that requests
from any stream from a group m( m = 1, M ) interrupt the service of requests belonging
                                                               Nm
to streams from groups with numbers. Each group contains  M       threads whose requests
                                                             Nm = N
do not interrupt each other m + 1, M . It is obvious that m=1        . The priority of any

application in a system with such a service discipline can be described by a pair of
                   n = 1, N m
numbers m and n,              , where it determines the number of the request flow in the
group with the number m.
    The described mixed discipline of service allows to adapt more flexibly to various
situations arising during the functioning of the CC due to its adaptation. In this case,
the adaptation of the discipline consists in changing the number and position of the
boundaries that separate the flow of applications into groups of absolute priority, that
is, in changing the number of groups and the number of flows in groups. Grouping
options will be called breakdowns. The total number of breakdowns of F is determined
by the number of requests streams N : Ô = 2 . Each breakdown ϕ (ϕ ∈ Ô ) is given by
                                            N −1

                  { N , N , …, N M }
a set of numbers 1 2                 .
   Such a discipline is rightly called the dynamic adaptive mixed discipline of provid-
ing computing resources to users of CC
   Introduced mixed discipline of service is a known practical interest, because in the
optimal selection of the breakdown of flows into groups, in principle, provides no worse
service compared to "pure" disciplines (with relative and absolute priorities). Thus
 M = N, Nm = 1
               , for all m = 1, M , there is a discipline of service with absolute priority,
           M = 1, N1 = N
and when                    - with relative.


Tasks of dynamic adaptive mixed discipline providing computing resources with
      the model
Let's consider two practical problems of dynamic adaptive mixed discipline of provid-
ing computing resources (mixed discipline of service) with the model.
   One of the main indicators of the effectiveness of CC is indicators based on the tem-
poral characteristics of these systems. Such metrics can be set by the contract between
the supplier and the user of the CR CC and are of particular importance for real-time
systems.
   Due to the random nature of the computing process, there are additional delays in
the processing of information, violating the permissible restrictions on its time in the
CC, which adversely affects the effectiveness of solving targeted user tasks.
216


   In such situations, it is necessary to maintain the time characteristics of the system
at a predetermined level in order to ensure the necessary efficiency of the CC. In con-
ditions of scarcity of computing resources, this is possible only by improving the effi-
ciency of the computing process, in particular, by adapting the discipline of servicing.
At the same time, there is a problem of the most efficient use of available computing
resources at each moment of time of operation of the control CC. This task can also be
addressed by adapting the discipline of service.
   In view of the foregoing, we will choose the average total cost of provisioning time
(waiting in queues and time of use, ie staying in CCs as in queuing system) CR as a
measure of the performance of the CC according to the requests (requirements) of users.
To do this, we use the well-known functional [1] - the average total cost of time to
provide the CR:
                                                n
                                     C ( S ) =  α i λi vi( s )
                                               i =1               ,
  what do we have
                              M     N
                    C ( ϕ ) =   α ( m , n ) λ ( m , n ) v (ϕ ) ( m , n )
                              m =1 n =1                                      ,            (5)

  where
   α i - is the cost per unit of time CR for the i-th type of user requests;
   λ i - the intensity of the i-th flow of applications;
   v i( S )
         - the average time for submitting the CR applications to the i-th stream;
  n - is the number of application types;
  s - parameter characterizing the method of organization of the computing process;
  v (ϕ ) ( m, n)( m = 1, M , n = 1, N m ) - the average time of the provision of the CR in the
CC application (m, n)-th flow;
  α (m, n) - is the unit cost of the time the CR is submitted to the CC (m, n)-th request;
   λ (m, n) is the intensity of the (m, n) flow of the CR in the CC.

   The performance indicator is based on the assumption that the results of the user's
use of the CR are depreciated in proportion to their time in the CC system. Then the
goals of adaptation of the mixed discipline of service will be either to satisfy the re-
quirements of the timely stay (m, n) of applications in the system, which are set by
                              v (m, n)
acceptable values of this time Ä       , or to minimize the functional (5). This goal is
achieved by finding the appropriate optimal breakdowns ϕ , that is, the tasks of adapt-
                                                           0


ing a mixed service discipline with relative absolute priority are optimization problems,
the general formulation of which is discussed above.
                                                                                       217


   Since the above goals of adapting mixed discipline to service can be achieved with
several different breakdowns of requests flow into groups of absolute priority, there is
a need to introduce additional restrictions on the choice of breakdown ϕ .
   The presence of an absolute priority in CC requires some technological losses of the
CR, which are proportional to the number of groups (levels) of the absolute priority. In
this regard, it is optimal to consider such a breakdown that ensures that the adaptation
goals are attained with a minimum number of absolute M priority groups.
   Then the considered problems of adaptation of mixed discipline of service can be
formally set as follows:

                             v (ϕ ) ( m , n ) ≤ v Ä ( m , n )  ϕ 0 ,
                                           ϕ ∈Φ ,
                                         M = min .                                     (6)

                                 C (ϕ ) → min  ϕ 0 ,
                                           ϕ ∈Φ ,
                                         M = min .                                     (7)
   It is not possible to solve the problems of finding the optimal partition (6) and (7)
using known analytical optimization methods. The only way to solve these problems is
a heuristic approach, which has no formal justification, but relies solely on the specifics
of the mathematical models [3…11] and related understanding.
   It follows from expressions (5) - (7) that the achievement of the goals of adaptation
of the mixed discipline of service is combined with the need to estimate the value of
the average residence time in the application system (m, n) -type - v(m, n). Therefore,
there is a need to synthesize a mathematical model of CC with a mixed discipline of
maintenance [3].



Characterization of analytical models and mathematical formulation of tasks in
     queuing theory


   One of the main indicators of the effectiveness of CC is the indicators based on the
assessment of the time characteristics of these systems. Violation of permissible time
constraints, for example, the response time of the CC, affects the effectiveness of the
solution of user targets, which is of particular importance for real-time systems. First
of all it concerns special information systems, which are built using private CC.
   The stochastic nature of the main factors and the necessity of quantification of mass
processes on the basis of the theory of probability determines the use of the theory of
mass service. Then it is possible and appropriate to use the technology of the dynamic
adaptive mixed discipline of providing CR (maintenance) to users of the CC as mech-
anisms of adaptation of the CC [1].
218


   Analytical models for calculation of time characteristics are offered in the conditions
of the features of the functioning of the CC using a mixed discipline of service with
absolutely relative priorities and taking into account failures [3].
   It is proposed to develop analytical models of cloud computing as a queuing system
with mixed resource delivery discipline. Models should consider failures and different
features of operation and have arbitrary distribution laws for some probable processes.
   The general description of the models is as follows. Let the input of the CC system,
in which the discipline of service with a relatively absolute priority is implemented,
arrive N Poisson flows of applications of intensity
                                                         λ (m, n) (m = 1, M , n = 1, N m ) .
These flows are aligned with N priorities [1].
   The duration of the maintenance of applications of priority (m, n) is a random         vari-
                                   Bm, n (t )                                     b ( 2) (m, n)
able with a distribution function             , the first b (m, n) and the second
start point.
   An application of priority (m, n) whose service is interrupted by applications from
groups with 1, m − 1, numbers is returned to the queue. Updating its service is possible
either after servicing all interrupted applications (maintenance discipline A), or after
servicing all interrupted applications and all applications for accumulated flows, the m
group with ( m,1), ( m, n − 1) numbers (discipline of service upgrade B) .
   The serving device (CC) fails in accordance with the Poisson law with the 0 pa-
                                                                                        λ
rameter. The period of recovery of the device is a random       variable that has an arbitrary
                                        b0             b02
distribution law Во(t) with the first and second initial moments.
   During the restoration of the service device, requests of some streams in the queue
are accepted, while others are not accepted. This condition is given by the matrix-row
                n , i = 1, N                                      n =1
of coefficients i            , and in the case if requests of the i      stream are accepted
                               n =0
in the queue, and if requests i       are denied.
   Adaptation to bounce will be that in the period of recovery device incoming appli-
cations can either accumulate in the queue (discipline replenishment queue I), or re-
ceive a refusal and leave the system (discipline replenishment queue II).
   Failure of the servicing device can occur both during its free state and during service
of the application. In the latter case, the renewal of the service is carried out either from
the interrupted application, if there are no applications interrupting its service, (the dis-
cipline of the renewal of service C), or from applications of the senior relative priority
of the corresponding group, if any (discipline of renewal of service D).
   In case of repeated receipt of the servicing device, the interrupted application shall
be maintained from the place where it was interrupted. Within one priority, applications
are served in the order of receipt.
   The combination of service updating disciplines and queue replenishment allows
you to consider independent models of different types of systems that have the proper
designation. Different features of functioning consist of various combinations of disci-
plines A, B, C, D, I and II.
                                                                                                 219


   Let CC be in stationary mode, which RM <M KNr condition is for systems of type I, and
                                       RM =  ρ (m, n)
for systems of type II - RM < 1 . Here      m =1 n =1     - total loading of the device

applications ( ( ρ ( m , n ) = λ ( m , n ) b ( m , n ) - loading of the device (m, n) - applications),
     K = 1 /(1 + ρ 0 )                                                ( ρ = λ0b0
and r                  - the system readiness coefficient ( 0                    - loading the device
with refusals).
                                                         v(m , n)
   It is necessary to determine the average                          time spent in the system of ap-
plications of each (m, n) -priority, ie, the response time of the system CC. To determine
the average time of applications in the system (time response systems) use the known
direct method [1].


Conclusions

− To develop the basics of creating adaptive cloud computing infrastructures capable
 of dynamically adapting to current changes in operating conditions. Dynamic adap-
 tive discipline for providing cloud computing resources is proposed.
− Cloud computing is an object with a high degree of randomness of the process of
 operation, the main factors of which are: the probability of flow of requests for com-
 puting resources; availability of necessary resources and accidental timing of their
 use by consumers; accidental failure of the СС infrastructure and time of their elimi-
 nation.
− Due to the random nature of the computing process, there are additional delays in
 the processing of information, violated permissible restrictions on its time in the sys-
 tem (at the time of the СС response), which adversely affects the effectiveness of
 solving targeted user tasks. This is relevant for real-time systems and, above all, for
 special information systems built using private clouds, and can be critical with limited
 ССcomputing resources.
− It is possible to reduce or reduce the impact of beneficial phenomena on the func-
 tioning of the ССby introducing adaptation to the functioning of the СС infrastruc-
 ture. In addition, the introduction of adaptation is associated with the need to support
 СС in the optimal (efficient use of resources) and sometimes just in working order,
 regardless of the many factors that drive the СС infrastructure from the desired target
 state.
− Depending on the situation, some adaptive influence should be generated, for ex-
 ample, minimizing the average number or average waiting time (stay) of applications
 in a queue (system), or the time of entering the system in a steady state, or the total
 cost for system operation, or the likelihood of losing applications, etc. .. The purpose
 of adaptation may be to maximize revenue from the service of applications, eliminate
 overloading the system and maintain it in a stationary mode of operation. Thus, the
 adaptation of the СС infrastructure can be considered as a process of optimizing the
 work in the current situation.
− The adaptation problem can be solved by using the adaptive discipline (order) of
 providing computing resources to users. Unforeseen and uncontrolled changes in the
220


 environment and the system will inevitably alter the optimal setting of the discipline,
 if one has been implemented in the system. Therefore, systematic adjustments (adap-
 tation) of the discipline are inevitable if you wish to maintain the system in optimal
 mode, regardless of changes occurring in the environment and system.
− Currently, a large number of different disciplines are known. Of these, disciplines
 with relative and absolute priorities are widely used for SS infrastructure. However,
 these disciplines are static and, as a result, have a number of significant drawbacks
 that reduce the efficiency of СС processes in the context of environmental uncertainty
 and the behavior of the systems themselves.
− The discipline of providing computing resources to data center users as an object of
 adaptation and the process of adaptation as an optimization is considered. The general
 statement of the adaptation problem as optimization is outlined.
− Finding the best discipline is not always linked to the extremisation of the data cen-
 ter performance indicator. The purpose of adaptation may also be to satisfy the per-
 formance constraint given by equals or inequalities. In any case, this formulation of
 the adaptation problem implies the need to implement dynamically changing several
 different or one mixed discipline of providing computing resources to СС users.
− The technology of dynamic adaptive mixed discipline for providing computing re-
 sources to СС users is proposed. The direction of solving the problem of optimization
 of dynamic adaptive mixed discipline is given.
− The mixed discipline of providing computing resources allows it to respond more
 flexibly to the various situations that arise in the functioning of СС due to its adapta-
 tion. In this case, the adaptation of the discipline consists of an optimal change in the
 number and position of boundaries that divide the flows of user requests for resources
 into groups of absolute priority, within which the relative priority, ie in the change of
 the number of groups and the number of flows in groups, operates.
− A well-known optimization functional is proposed, based on the assumption that
 the results of the use of computing resources by the user (solving user problems) are
 depreciated in proportion to their time in the СС. Other functionalities with time con-
 straints are also possible. Such metrics can be specified by the agreement between the
 supplier and the user of computing resources.
− It is not possible to solve the problems of finding the optimal breakdown into groups
 of flows of providing computing resources to users using known analytical methods
 of optimization. The only way to solve these problems is a heuristic approach, which
 has no formal justification but relies on the specifics of the problems (mathematical
 models) that should be able to determine the temporal characteristics of the СС, such
 as response time.
− СС analytical models are proposed as multi-threaded and multi-priority queuing
 systems with mixed service discipline. Models take into account failures and different
 features and have arbitrary distribution laws for some probable processes. The math-
 ematical formulation and the proposed method of solving service problems for ob-
 taining analytical expressions of temporal characteristics, including the response time
 of СС, which allow to realize dynamic adaptive discipline of providing computing
 resources to users of cloud computing, are made.
                                                                                                221


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