Oe2423112320

N.D.Sheba kumari, B.Bharathi / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-august 2012, pp.2311-2320
Pre-Eminent Pricing For Cloud Cache
N.D.Sheba kumari, M.Tech(CS), B.Bharathi,Asst.Prof.,
ADAMS Engineering College Kammam,INDIA
ADAMS Engineering College Kammam,INDIA

Abstract
Caching on the edge of the Internet is permanently stored in Internet servers and probably
becoming a popular technique to improve the cached temporarily on the user side. Current
scalability and efficiency of delivering dynamic businesses on cloud computing such as Amazon Web
web content and it is rising. These clouds inturn Services and Microsoft Azure have begun to offer
helps in providing good quality query services.In data management services: the cloud enables the
order to query the data users pay for the users to manage the data of back-end databases in a
infrastructure they use required for the query. transparent manner. Applications that collect and
The goal of cloud economy is to optimize user query massive data, like those supported by CERN,
satisfaction and cloud profit.To increase the cloud need a caching service, which can be provided by the
profit an oppropriate price demand model should cloud. The goal of such a cloud is to provide efficient
guarantee the user satisfaction that enables the querying on the back-end data at a low cost, while
best possible pricing of query services. The model being economically viable, and furthermore,
should be reliable in that it reflects the correlation profitable. Fig. 1 depicts the architecture of a cloud
of cache structures involved in the queries. Best cache. Users pose queries to the cloud through a
possible pricing is achieved based on a dynamic coordinator module, and are charged on the- go in
pricing scheme that adapts to time changes. This order to be served. The cloud caches data and builds
paper proposes a new price-demand model data structures in order to accelerate query execution.
designed for a cloud cache and a dynamic pricing Service of queries is performed by executing them
scheme for queries executed in the cloud cache. either in the cloud cache (if necessary data are
The pricing solution employs a new method that already cached) or in a back-end database. Each
estimates the correlations of the cache services in cache structure (data or data structures) has an
an time-efficient manner. The experimental study operating (i.e., a building and a maintenance) cost. A
shows the efficiency of the solution. price over the operating cost for each structure can
ensure profit for the cloud. In this work, we propose a
Index terms -- Cloud data management, data novel scheme that achieves optimal pricing for the
services, cloud service pricing,price demand services of a cloud cache.
modeling,Dynamic Pricing
. 1.1 Price Setting for Cloud Cache Services
INTRODUCTION The cloud makes profit from selling its
The leading trend for service infrastructures services at a price that is higher than the actual cost.
in the IT domain is called cloud computing, a style of Setting the right price for a service is a nontrivial
computing that allows users to access information problem, because when there is competition the
services. Cloud providers trade their services on demand for services grows inversely but not
cloud resources for money.The quality of services proportionally to the price. There are two major
that the users receive depends on the utilization of the challenges when trying to define an optimal pricing
resources. The operation cost of used resources is scheme for the cloud caching service. The first is to
amortized through user payments. Cloud resources define a simplified enough model of the price
can be anything, from infrastructure (CPU,memory, demand dependency, to achieve a feasible pricing
bandwidth, network), to platforms and applications solution, but not oversimplified model that is not
deployed on the infrastructure. Cloud management representative. For example, a static pricing scheme
necessitates an economy, and, therefore, cannot be optimal if the demand for services has
incorporation of economic concepts in the provision deterministic seasonal fluctuations. The second
of cloud services. The goal of cloud economy is to challenge is to define a pricing scheme that is
optimize:1) user satisfaction and 2) cloud profit. adaptable to 1) modeling errors, 2) time-dependent
While the success of the cloud service depends on the model changes, and 3) stochastic behavior of the
optimization of both objectives, businesses typically application.
prioritize profit. To maximize cloud profit we need a
pricing scheme that guarantees user satisfaction while
adapting to demand changes. Recently, cloud
computing has found its way into the provision of
web services Information, as well as software is

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the cloud to schedule ahead associative actions for
the maintenance of the cloud infrastructure and the
cloud data. Moreover, the cloud can schedule the
service availability according to the guarantees for
the overall revenue estimated by the longterm
optimization. Nevertheless, it is important that the
long-term optimization process is flexible enough to
receive corrections while it is still in progress. The
corrections may refer to the difference between the
estimated and the actual price influence on the
Fig. 1. A cloud cache. demand of services.

The demand for services, for instance, may 1.2 Related Work
depend in a nonpredictable way on factors that are Existing clouds focus on the provision of
external to the cloud application, such as web services targeted to developers, such as Amazon
socioeconomic situations. A representative model for Elastic Compute Cloud (EC2) , or the deployment of
the cloud cache should take into account that the servers, such as GoGrid . Emerging clouds such as
cache structures (table columns or indexes) may the Amazon SimpleDB and Simple Storage Service
compete or collaborate during query execution. The offer data management services. Optimal pricing of
demand for a structure depends not only on its price, cached structures is central to maximizing profit for a
but also on the price of other structures. For example, cloud that offers data services. Cloud businesses may
consider the query select A from T where B = 5 and offer their services for free, such as Google Apps
C = 10. Out of the set of candidate indexes to run the and Microsoft Azure or based on a pricing scheme.
query efficiently, indexes Ib =T(B), Ic = T(C), and Ibc Amazon Web Service (AWS) clouds include separate
= T(BC) are most important, since they can satisfy prices for infrastructure elements, i.e., disk space,
the conditions in the “where” clause. If the cache CPU, I/O, and bandwidth. Pricing schemes are static,
uses Ibc, then the indexes Ib and Ic, will never be used, and give the option for pay as-you-go. Static pricing
since Ibc can satisfy both conditions. Therefore, the cannot guarantee cloud profit maximization. In fact,
presence of Ibc has a negative impact on the demand as we show in our experimental study, static pricing
for Ib and Ic. Alternatively, if the cache uses Ib, then Ic results in an unpredictable and, therefore,
can also improve query performance via index uncontrollable behavior of profit.Closely related to
intersections, hence increasing the profit for the cloud computing is research on accounting in wide-
cloud. Therefore, indexes Ib and Ic have positive area networks that offer distributed services.
impact on each other’s demand. An appropriate discusses an economy for querying in distributed
estimation method is necessary to model price- databases. This economy is limited to offering budget
demand correlations among cached structures. The options to the users, and does not propose any pricing
peculiarity of the pricing problem for the application scheme. Other solutions for similar frameworks focus
of the cloud DBMS, in comparison with other on job scheduling and bid negotiation, issues
businesses, is that the selling good is not a orthogonal to optimal pricing. Pricing schemes were
consumable product, but a persistent service. A proposed recently for the optimal allocation of grid
consumable product diminishes with demand and has resources in order to increase revenue, or to achieve
to be ordered, whereas a cloud cache service can an equilibrium of grid and user satisfaction, assuming
satisfy infinite demand as long as it is maintained. knowledge of the demand for resources or the
Moreover, the demand for a cache service pauses if possibility to vary the price of a resource for different
this service is not available. A consumable product users. These works are orthogonal to ours, as we do
may cost to maintain depending on the stored not assume that service demand is known a priori and
amount, whereas the maintenance cost of a cache all users are charged the same for the consumption of
service depends only on time. Moreover, a cache the same service. Similarly, dynamic pricing for web
service may have a setup cost each time it is loaded services focuses on scheduling user requests. This
in the cloud. A big challenge for the cloud is to work is orthogonal to ours, as we require that users’
optimize the set of offered services, i.e., decide which requests for service are satisfied right away.
services to offer and when, depending on their Moreover, dynamic pricing for the provision of
demand while they are available. Roughly, the cloud network services, aims at achieving a game-theoretic
has to schedule online and offline periods of the equilibrium through price control among competitive
offered services, which affects the maintenance and Internet Service Providers. This work is orthogonal to
the setup cost. Furthermore, the optimization of the ours, as we focus on the maximization of cloud profit
cloud profit has to be scheduled for a long period in in the presence of competitive services within the
time while it is flexible during this period to adjust to same cloud provider. The problem of revenue
the real evolution of the service consumption. The management through dynamic pricing is well
long-term profit optimization is necessary in order for studied.Based on the rationale that price and demand

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are dependent qualities, numerous variations of the horizon. The horizon includes sequential
problem have been tackled, for instance businesses nonoverlapping intervals that allow for scheduling
that sell products to retailers, seasonal products, structure availability. At the beginning of each
stochastic demand. Electronic businesses, and interval, the cloud redefines availability by taking
therefore cloud businesses can benefit from dynamic offline some of the currently available structures and
pricing policies.Cache services are distinguished taking online some of the unavailable ones. Pricing
from consumable products in two major ways: 1) optimization proceeds in iterations on a sliding time
they are not exhausted while they are consumed, and window that allows online corrections on the
2) the demand for a specific service pauses while this predicted demand, via reinjection of the real demand
is not available. To the best of our knowledge, this is values at each sliding instant. Also, the iterative
the first work that tackles the problem of optimal optimization allows for redefinition of the parameters
pricing of competitive data services within the same in the price-demand model, if the demand deviates
cloud cache provider. substantially from the predicted.
Research on the identification of
noncorrelated indexes using the query structure does 1.3 Contributions
not determine the negative and positive correlations. This paper makes the following contributions:
Identification of index correlations by modeling  A novel demand-pricing model designed for
physical design as a submodular and super-modular cloud caching services and the problem
problem is restricted to one-column indexes and one formulation for the dynamic pricing scheme that
index per query. Identification of negative index maximizes profit and incorporates the objective
correlation [2] does not consider the positive and no for user satisfaction.
correlation case. A recent index interaction model  An efficient solution to the pricing problem,
attempts to find all index correlations. As we show in based on nonlinear programming, adaptable to
Section 4, it does not satisfy three critical time changes.
requirements for the pricing scheme: 1) sensitivity to  A correlation measure for cache structures that is
the range of all possible correlations, 2) production of suitable for the cloud cache pricing scheme and a
ormalized values, and 3) fast computation. method for its efficient computation.
 An experimental study which shows that the
1.3 Our Proposal dynamic pricing scheme out performs any static
The cloud caching service can maximize its one by achieving 2 orders of magnitude more
profit using an optimal pricing scheme. This work profit per time unit.
proposes a pricing scheme along the insight that it is The rest of the paper is structured as
sufficient to use a simplified price-demand model follows: Section 2 presents the query execution
which can be reevaluated in order to adapt to model model, Section 3 models the optimal pricing problem,
mismatches, external disturbances and errors, and Section 4 models the price demand correlations
employing feedback from the real system behavior for data structures in the cloud cache. Section 5
and performing refinement of the optimization describes the solution of the pricing optimization
procedure. Overall, optimal pricing necessitates an problem and Section 6 presents the experimental
appropriately simplified price-demand model that study. Section 7 concludes the paper.
incorporates the correlations of structures in the
cache services. The pricing scheme should be 2. QUERY EXECUTION MODEL
adaptable to time changes. The cloud cache is a full-fledged DBMS
along with a cache of data that reside permanently in
Simple but not simplistic price-demand modeling. back-end databases. The goal of the cloud cache is to
We model the price-demand dependency employing offer cheap efficient multiuser querying on the back-
second order differential equations with constant end data, while keeping the cloud provider profitable.
parameters. This modeling is flexible enough to Our motivation for the necessity of such a cloud data
represent a wide variety of demands as a function of service provider derives from the data management
price. The simplification of using constant parameters needs of huge analytical data, such as scientific data,
allows their easy estimation based on given price- for example physics data from CERN and astronomy
demand data sets. The model takes into account that data from SDSS . Furthermore, a viable, and
structures can be available in the cache or can be moreover, profitable data service provider can
discarded if there is not enough respective demand. achieve cost and time efficient management of
Optional structure availability allows for optimal smaller scientific collections or any type of analytical
scheduling of structure availability, such that the data, such as digital libraries, multimedia data, and a
cloud profit is maximized. The model of price- variety of archived data. Users pose queries to the
demand dependency for a set of structures cloud, which are charged in order to be served.
incorporates their correlation in query execution. Following the business example of Amazon and
Price adaptivity to time changes. Profit Google, we assume that data reside in the same data
maximization is pursued in a finite long-term center and that users pay on-the-go based on the

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infrastructure they use, therefore, they pay by the an input to the presented optimal pricing
query. Service of queries is performed by executing scheme.Periodically (on predefined time intervals
them either in the cloud cache or in the back-end 𝑡[𝑖]) the cloud performs the pricing scheme proposed
database. Query performance is measured in terms of in this work. The pricing scheme schedules the
execution time. The faster the execution, the more availability and sets the prices P of the structures S
data structures it employs, and therefore, the more for a time horizon T as described in the rest of the
expensive the service. We assume that the cloud paper. The goal is to maximize the provider’s profit
infrastructure provides sufficient amount of storage and at the same time ensure that the user is not
space for a large number of cache structures. Each overcharged.
cache structure has a building and a maintenance
cost.
3.MODELING DYNAMIC PRICING
Alogorithm: This section describes the problem
Global: cache structures S, prices P, availability Δ formulation of maximizing the cloud profit. The
Query Execution ( ) presentation of the pricing scheme is guided by
if q can be satisfied in the cache then propositions that state the main rationale of our
(result, cost)←runQueryInCache (q) approach.
else
(result, cost)←runQueryInBackend (q) 3.1 Problem Formulation
end if This section defines the objective and the
S←addNewStructures () constraints of the problem, and gives the
return result, cost mathematical problem definition.
optimalPricing (horizon T, intervals t[i], S)
(Δ,P)←determineAvailability&Prices (T,t,S) 3.1.1 Objective
return Δ,P The cloud cache offers to the users query
main () services on the cloud data. The user queries are
execute in parallel tasks T1 and T2: answered by query plans that use cache structures,
T1: i.e., cached columns and indexes. We assume that the
for every new i do set of possible cache structures is S = {S1, . . . , Sm}.
slide the optimization window Whenever a structure S is built in the cache, it has a
OptimalPricin (T, t[i], S) one time building cost Bs. While S is maintained in
end for the cache it has a maintenance cost which depends on
T2: time, Ms(t). We assume that each structure is built
While new query q do from scratch in the cloud cache, as the cloud may not
(Result, cost)←query Execution (q) have administration rights on existing back-end
end while structures. Nevertheless, cheap computing and
if q executed in cache then parallelism on cloud infrastructure may benefit the
Charge cost to user performance of structure creation. For a column, the
else building cost is the cost of transferring it from the
Calculate total price and charge price to user backend and combining it with the currently cached
end if columns. This cost may contain the cost of
fig 2. Qurey Execution Model for cloud cache integrating the column in the existing cache table. For
indexes, the building cost involves fetching the data
Fig. 2 presents at a high level the query across the Internet and then building the index in the
execution model of the cloud cache. The names of cache. Since sorting is the most important step in
variables and functions are self-explanatory. The user building an index, the cost of building an index is
query is executed in the cache iff all the columns it approximated to the cost of sorting the indexed
refers to are already cached. Otherwise it is executed columns. In case of multiple cloud databases, the cost
in the back-end databases. The result is returned to of data movement is incorporated in the building
the user and the cost is the query execution cost (the cost. The maintenance cost of a column or an index is
cost of operating the cloud cache or the cost of just the cost of using disk space in the cloud. Hence,
transferring the result via the network to the user). building a column or an index in the cache has a one-
The cloud cache determines which structures (cached time static cost, whereas their maintenance yields a
columns, views, indexes) S to build in order to storage cost that is linear with time.In any case, the
accelerate query execution and reduce the query cost of a structure S as soon as it is built at time tbuilt
execution cost. Initially S is empty and gradually it is in the cache and until it is discarded is
filled with structures that would have or have Cs(t) = Bs + Ms(t – tbuilt). (1)
benefitted past queries. How S is populated and how
the costs of building and maintaining cache structures
as well as the query execution cost are computed is

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Cache services are offered through query execution Value constraints. It is straightforward that both the
that uses cache structures. The demand for cache demand and the price of a structure must be positive
structures is defined as follows: numbers. Furthermore, it is necessary to impose an
upper bound on the price. The reason is that the
Definition 1. The demand for a cache structure S, optimum solution is to instantaneously raise the price
denoted as λs(t), is the number of times that S is of at least one structure to infinity, if this is
employed in query plans selected for execution at allowed.These bounds can be formulated as follows:
time t.
Naturally, in realistic situations the demand 0 ≤ 𝜆𝑖, 𝑖 = 1, … , 𝑚, (5)
for a structure is measured in time intervals. If a
structure S is built in the cache then query plans that 0 ≤ 𝑝 𝑖 ≤ 𝑝 𝑚𝑎𝑥 , 𝑖 = 1, … , 𝑚. (6)
involve it can be selected, i.e., λs(t) ≥ 0. otherwise
not, i.e., λs(t) = 0. Intuitively, there is a trade-off Dynamics of the demand. Naturally, the demand
between 1) keeping a structure in the cache and and the price of a structure are connected variables:
paying the maintenance cost, and 2) building and intuitively, as the price for a structure increases the
discarding the structure occasionally. This trade-off is demand decreases and vice versa. In order to to solve
reinforced toward the one or the other direction by the optimization problem (4), a mathematical
the demand of the structure: if the demand is low, it relationship, which models the interaction between
is possible that it is cheaper to discard the structure demand and price, is necessary. However, this
from the cache and pay the building cost multiple mathematical relationship should have a structure as
times, than pay the maintenance cost; if the demand flexible as possible, so that, upon a proper
is high, then the opposite tactic may be more identification of its parameters, it is able to represent
profitable for the cloud. The cloud makes profit by as many as possible functions of demand and price.
charging the usage of structures in selected query We make the following assumption:
plans for a price. Let us assume that the price of a
structure S at time t is ps(t). Then the profit of the
cloud at a specific time is
𝑚
𝑟 𝑡 = 𝑖=1 𝛿 𝑖 . ( 𝜆𝑠 𝑖 𝑡 . 𝑝𝑠 𝑖 𝑡 −
𝑐𝑠 𝑖 (𝑡)), 𝛿 𝑖 = 0,1, (2)

where 𝛿 𝑖 represents the fact that the structure 𝑆 𝑖 is
present in the cloud cache. Specifically, a structure
may be present or not in the cache at any time point
in [0, T] and not present before the beginning of
optimization time, i.e.,
0 𝑜𝑟 1, 𝑖𝑓 𝑡 ∈ 0, 𝑇 ,
𝛿𝑖 𝑡 =
0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒.

Based on this, the cost of a structure w.r.t. time Proposition 1. The demand of a structure S has
becomes memory: the demand at time t depends on the
𝑐 𝑠 𝑡 = 1 − 𝛿 𝑖 𝑡0 𝐵 𝑠 + 𝑀 𝑠 𝑡 − 𝑡0 , (3) demand before t. Consequently, the relationship
between price and demand can be modeled as an
where t0 is the start time of cost observation.
ordinary differential equation, which can be written
Structures can be built and discarded at any
in the general case in its implicit form
time 𝑡 ∈ 0, 𝑇 and the total profit of the cloud is
𝑇
𝑅 𝑇 = 0 𝑟 𝑡 𝑑𝑡. The goal is to maximize the total
profit in 0, 𝑇 by choosing which structures to build
or discard and which price to assign to each built
structure at any time

(7)
where m ≤ n, to respect the causality
principle, as m > n would imply that demand could
(4) change (due to a change of price) before the price has
3.1.2 Problem Constraints changed. In particular, since there is no inertia in
It is necessary to constrain the optimization setting a price for a structure, m = 0 and (7) can be
of the objective 4, so that a reasonable and correct rewritten in its explicit form
solution can be found.

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are m × 1 matrices of parameters, then the constraint
in (9) becomes
𝑑2˄ 𝑑2˄
𝑉. 𝑃 = 𝐴 𝑇 2 + 𝐵 𝑇 𝛤 𝑇 ˄, (10)
𝑑𝑡 𝑑𝑡
is actually a set of constraints of the form:
Justification 1. Economies as well as societies tend 𝑗 =𝑚 𝑑2 𝜆 𝑠 𝑑𝜆 𝑠
𝑖 𝑖
𝑗 =1 𝑏 𝑖.𝑗 . 𝑝 𝑠 𝑡 = 𝛼 𝑖 + 𝛽𝑖 + 𝛾 𝑖 . 𝜆 𝑆 𝑖 (𝑡)
to behave in a way that reflects past experience. More 𝑑𝑡 𝑑𝑡
formally, an economic system, such as the cloud
cache and its users has inertia, which means that the Problem definition. The previous discussion leads to
current system behavior depends on past and the following problem formulation for optimal
influences future behavior. Concerning the cloud pricing: The maximization of the cloud DBMS profit
cache, this means that the demand for structures has a is achieved with
time-resistant effect. For example, assume that the the solution of the following optimization problem:
demand for a structure built in the cloud cache does
not drop as fast as expected in a memory-less system
w.r.t. price increase. Two intuitive exemplifying
reasons for this are: 1) the structure is already built
and remains available because the building cost is
already amortized, while the maintenance cost is not
very high; and 2) the structure, for example a cached
column is requested for the execution of numerous
queries, because it involves information that is
currently very popular to users.
Associating the price with the nth derivative
of demand for a structure, guarantees n degrees of
freedom for the shape of their relationship. Therefore,
the bigger the order n is, the more flexible the price-
demand relationship is. Yet, as the order n increases, 3.2 Generalization of Optimization Objective
the number of parameters of the price-demand The problem of dynamic pricing as
relationship increase and more information is needed formulated in Section 3.1 consists of a sole objective:
in order to identify their values. We choose to the maximization of the cloud profit, subject to some
consider a second order differential equation as it is constraints. From a mathematical point of view, we
versatile enough to represent a price-demand expect a solution that is on the boundaries of the
relationship, where the demand drops smoothly at the feasible area, meaning a solution along the
beginning of time,3 as depicted in Fig. 3, while constraints of the problem that satisfies the objective.
keeping the number of parameters to identify low. The constraints on the price-demand dependency in
Therefore, the constraint is: (10) do not actually constrain the sought solution, but
only the value of the optimal profit, if the solution is
applied; therefore, the sought solution is expected to
be on the boundaries of the allowed price, (6), and
We constrain f to be an ordinary differential relation demand values, (5), meaning maximum price
between price and demand selections as long as the demand for structures is
𝑑2 𝜆 𝑠 𝑑 𝜆𝑠 above zero. This is called a bang-bang solution and
𝑝𝑠 𝑡 = 𝛼 + 𝛽 + ϒ. 𝜆 𝑠 𝑡 . (9) the mathematical reason for this expectation is that
𝑑𝑡 2 𝑑𝑡
the objective of the problem is linear w.r.t. the
The parameters α,β,ϒ are constrained to be control variables: the price p and the structure
constants. This means that the price model considers availability _. Intuitively, the objective of
a static relation between demand and price. In order optimization is the purely egoistic and
to make the pricing model realistic, we have to straightforward maximization of cloud profit. The
consider the influence of the price of one structure to optimization procedure shall try to achieve this goal
the demand of the rest. Therefore, it is necessary to as soon as possible, resulting in charging the highest
extend (9) so that it captures correlations of demand possible prices as long as there is structure demand.
and prices between pairs of structures. Let us assume Of course, the freedom of choosing the availability of
that V is a m × m matrix where the row and the structures complicates the optimization goal, but does
column i corresponds to the structure Si i = 1, . . .,m. not change the decision for maximum charge
Each element vij,i, j = 1, . . .,m corresponds to the whenever availability for a structure is decided.
correlation of the price of Sj to the demand of Si. We Naturally, one would expect that the user
call V the correlation matrix of prices and demands. dissatisfaction from high service charge, which is the
If ˄ and P are the m × 1 matrices of demands and actual reason for the demand drop, should be taken
prices for the respective structures in S, and A, B, Γ into consideration in a real cloud business. Simply,
the cloud risks to permanently lose the dissatisfied

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users in an open-market world. The user satisfaction procedure. The augmented optimization objective
is an altruistic tend of the optimization that is (14) leads the optimization procedure to seek a
opposite to the egoistic tend of cloud profit. trajectory that balances the opposite egoistic and
altruistic tends.
Proposition 2. The altruistic tend of pricing
optimization is expressed as: 1) a guarantee for a low 4 SOLVING THE DYNAMIC PRICING
limit on user satisfaction, or 2) an additional PROBLEM
maximization objective. The problem of optimal pricing is an
optimal control problem with a finite horizon, i.e., the
Justification 2. There are two policies in order to maximum time of optimization T is a given finite
incorporate an altruistic tend in pricing optimization. value. The free variables are the prices of the cache
The first is to give a much lower priority to user structures, pis, called the control variables, and the
satisfaction than cloud profit, which results into a dependent variables, called state variables, is the
constraint (static or time dependent) that passively demand for the structures, λis and the availability of
restricts the maximization of profit, i.e., expression the structures δis. The problem is augmented with
(4). The second is to handle it as a secondary goal of bounds on the values of both the control and the state
the pricing optimization, which results into a new variables and by a constraint on the dependency type
objective that actively restricts profit maximization. of the state on the control variables.
“Passive” restriction means that the altruistic tend
turns down pricing solutions proposed by the 4.1 Design of the Solution
optimization procedure, while “active” restriction The objective function of the problem is the
means that the altruistic tend is involved in the 𝑇
maximization of an integral, i.e., 𝑚𝑎𝑥 0 𝑟 𝑡 −
proposition of pricing solutions. If the altruistic tend
is expressed as low-limit guarantee on 𝑤.𝑢𝑡𝑑𝑡. The optimality scope of the sought solution
user satisfaction, then it can be formulated as an depends on the convexity of the objective function.
additional constraint of the optimization problem of The latter is bilinear w.r.t. the demand and the price
Section 3.1 on the demand drop (this is the result of factor λs(t).ps(t) in (2) and ps(t) in
(12)). It is not possible to prove that the objective
function is convex and, therefore, there is no
guarantee of global optimality of the solution.
Due to: 1) the nonlinearity of the objective
where λmin is the selected minimum value of function, 2) the presence of both integer inputs (the
demand drop rate. Alternatively, the user satisfaction δis control binary variables) and continuous inputs
can be defined as the difference of the structure price and states (the pis and the δis, respectively), and 3)
and the actual cost the potentially large scale of the system (when m is
high), it is almost impossible to find an analytical
𝑢 𝑡 ≡ 𝑝 𝑠 𝑡 − 𝑐 𝑠 (𝑡) (12) solution to the optimization problem. This calls for
In this case, the problem can accommodate, either a numerical optimization techniques, such as
new constraint or a new optimization objective. In the mixedinteger nonlinear programming, which present
first case, the constraint can be the advantage of being implementable online. A way
to implement dynamic optimization tools on real
systems is to
𝑢 𝑡 ≤ 𝑟 𝑚𝑖𝑛 , (13)
proceed as follow:
where rmin is the selected minimum value of 1. Solve the MINLP problem along a fixed
prediction horizon to compute a sequence
cloud profit. Adding one of the constraints (11) or
of values for the control variables.
(12) to the optimization problem does not change the
2. Apply the first values to the system.
objective of the optimization, which attempts to
maximize the prices while satisfying the new 3. Slide the prediction horizon and go back
constraints, (see Fig. 4b). to 1.
If the altruistic tend is expressed as a new
This approach, referred to as Optimal
maximization goal, the optimization objective is a
Control with Receding Horizon or as Model Prective
combination of (4) and (12)
Control (for which a trajectory is tracked) in the
control literature, has been successfully applied to a
very large number of uncertain, complex, and
nonlinear systems, in simulation as well at lab or
industrial scales. This methodology has shown its
ability to improve the performances of a large class
of systems, despite the use of simplified models, the
where w is a weight that calibrates the
influence of the altruistic tend to the optimization

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presence of uncertainty on model parameters, model horizon, as well as the number of optimization
mismatch, and process disturbances. repetitions. Formally
We propose the division of the prediction
horizon [0,T] into time intervals: let us assume that
there are time points 𝑡 𝑗 𝜖 0, 𝑇 , 𝑗 = 0, … , 𝑘, such that
t0 = 0 and tk = T on which built structures can be built
or discarded. Therefore, the problem is to maximize
the total profit in [0,T] by choosing (16)
For simplification, we consider all the
control variables in a time interval to be static, which
means that prices and availability of structures are
constant. Application-wise, we assume that the
availability of structures and their prices are set at the
beginning time of each repetition of the optimization
procedure. Of course, we could refine this
simplification by considering prices to be functions
of time in each interval. Yet, this would augment the
number of variables dramatically, reducing the
Fig.4.optimization procedure divided into efficiency of the method. For example, even for
short time intervals and iterations on a sliding time linear dependency of price on time: p = a.t + b with
window static a, b, the number of variables in the problem is
doubled.
which structures to built or discard on each
𝑡 𝑗 𝜖 0, 𝑇 , 𝑗 = 0, … , 𝑘, and which price to assign to 4.2 Parameters Estimation
each built structure Concerning the constraints on the price-
demand dependency in (10), it is necessary to
estimate the parameters A, B, Γ. For this, the
nonhomogeneous m order system of second order
differential equations in (10), has to be solved. One
way to do is to transform the system into a 2 × m
(15) order system of first order differential equations, by
Fig. 4 depicts the proposed repeated optimization breaking each second order equation into a set of
over a sliding time prediction horizon of length T. two. The result in both cases is a set of equations that
For simplicity, we consider equal time intervals, show the dependency of demand on price involving
𝑡 𝑗 +1 − 𝑡 𝑗 = 𝑡 𝑗 +2 − 𝑡 𝑗 +1 , 𝑖 = 0, … . , 𝑘 − 2. The the parameters
ptimization is performed repeatedly for k prediction
horizons beginning at tstart and ending at tend, such
that: 𝑡 𝑠𝑡𝑎𝑟𝑡 , 𝑡 𝑒𝑛𝑑 , 𝑡 𝑠𝑡𝑎𝑟𝑡 = 0, 𝑡1 , … . , 𝑇 and
𝑡 𝑒𝑛𝑑 = 𝑇, 𝑇 + 𝑡1 , 2𝑇, respectively. In this way we
achieve, on one hand, to optimize by taking into (17)
account the inertia of the cloud behavior in a long where F is a m × m matrix of functions on
prediction horizon, and on the other, to improve the time and elements of the parameter matrices A, B, Γ,
optimization by tuning the initial values of both the as well as the initial values of the demand and the
control and the state variables at each time interval rate of demand at the beginning of time. The solution
[tj, tj+1] to the values predicted by the current of the system is possible, if the m constraints in (10)
optimization results. We can further improve the are independent, i.e., if the m differential equations
optimization procedure, by injecting the real values are independent.
of the state variables, if these are available.
Specifically, if the actual time is close to the starting Proposition 3. It is always possible to manage the
time tstart of an optimization phase, then the real cache structures in a way that the constraints in (10)
demand values of the structures are available; if the are independent differential equations.
real values are different than the values predicted by
the previous optimization phase, then the real values Justification 3. Independency of the constraints in
can substitute the predicted ones in the new (10) means that there are no pair of cache structures
optimization phase, calibrating the procedure toward for which the demand depends in the exact same way
an improved overall result. We transform the from the prices of all the cache structures. Intuitively,
problem into a MINLP one by substituting each this is not a problem: assume two structures S1 and
control and state variable into a of k-arity set of S2. If these are competitive, each one has a negative
variables, where k is the number of time intervals of dependency on its own price and a positive
control variable einitialization in the optimization dependency on the price of the other; therefore, it is

2318 | P a g e

not possible that they create the same constraint. If S1 [0,Tsmall) and [Tsmall,Tbig) or a procedure with one
and S2 are collaborative, creating the same constraint long phase [0,Tbig). For simplicity, the demand is a
means that they depend on the exact same way on step function as shown in Fig. 6, i.e.,𝜆 𝑠 𝑇 = 𝜆1 , 𝑡 ∈
each other’s price and on the price of the rest of the 0, 𝑇 𝑠𝑚𝑎𝑙𝑙 corresponding to price p1 and 𝜆 𝑠 𝑇 =
structures; this fact implies that S1 and S2 are always 𝜆2 , 𝑡 ∈ [𝑇 𝑠𝑚𝑎𝑙𝑙 , 𝑇 𝑏𝑖𝑔 ) corresponding to price p2 (for
employed together in the cloud; therefore, they can simplicity we ignore structure correlations). Assume
be represented as a set of structures with a single that the building cost of S is Bs and the maintenance
price. The parameters A, B,Γ, can be estimated by cost is 𝑀 𝑠 𝑡 = 𝑎. 𝑡and S is built once at time t = 0.
performing curve fitting (e.g., the least square The cloud profit in [0,Tsmall) is 𝑟 𝑠𝑚𝑎𝑙𝑙 =
method), on (17). The fitting is performed based on a 𝜆1 . 𝑝1 − 𝐵 𝑠 − 𝑀 𝑠 𝑇 𝑠𝑚𝑎𝑙𝑙 . If 𝑟 𝑠𝑚𝑎𝑙𝑙 < 0,the cloud
sample data set of pricedemand values. Ideally, we decides to discard S and the second optimization
need a data set with the values for ˄ for all phase starts with S not available. Since the demand is
combinations of a set of price values 𝑝 𝑣 𝜖 0, 𝑚𝑎𝑥 𝑝 , significant in (Tsmall,Tbig) , th(e cloud may decide to
where maxp is a maximum value, for all price build S again, at t ≥Tsmall, resulting in profit
variables P. The fitting of (17) necessitates the initial 𝑟 𝑏𝑖𝑔 −𝑠𝑚𝑎𝑙𝑙 ≤ 𝜆2 . 𝑝2 − 𝐵 𝑠 − 𝑀 𝑠 (𝑇 𝑏𝑖𝑔 − 𝑇 𝑠𝑚𝑎𝑙𝑙 ). For
values of demand and demand rate at the beginning the long-term optimization the profit is: 𝑟 𝑏𝑖𝑔 =
of time. Since time is an orthogonal issue to the curve 𝜆1 . 𝑝1 + 𝜆2 . 𝑝2 − 𝐵 𝑠 − 𝑀 𝑠 𝑇 𝑏𝑖𝑔 Obviously, 𝑟 𝑏𝑖𝑔 >
fitting problem, we can order Pv and assume that for
𝑟 𝑠𝑚𝑎𝑙𝑙 + 𝑟 𝑏𝑖𝑔 −𝑠𝑚𝑎𝑙𝑙 Therefore, the result of the two-
the fitting of each pair of data that consists of price
values of all structures and the respective demand phase short-term optimization procedure is not as
optimal as that of the one-phase long-term procedure.
value 𝑝 𝑣1 ,…, 𝑝 𝑣 𝑚 , 𝜆 𝑣 𝑖 , 𝑖 = 1, … . . , 𝑚, 𝜆 𝑣 𝑖 is the
Naturally, the prediction of future behavior of a
initial value of demand w.r.t. time. In order to get the system is subject to unpredictable perturbations.
initial value of demand rate at t = 0, we need another Hence, the longer the horizon is, the more error-
measurement of demand for each structure 𝜆΄ 𝑣 𝑖 that prone the optimization procedure is, as the prediction
is really close to 𝜆 𝑣 𝑖 , i.e., 𝜆 𝑣 𝑖 − 𝜆΄ 𝑣 𝑖 < 𝑒 𝜆 𝑖 ≈ 0. This accuracy of the behavior of demand, tends to
can be achieved by slightly changing the values in 𝑝 𝑣 , decrease with time.
producing 𝑝΄ 𝑣 = ( 𝑝 𝑣 𝑖 + 𝑒1 , … . 𝑝 𝑣 𝑚 + 𝑒 𝑚 , 𝑒 𝑖 ≈
0, 𝑖 = 1, … . , 𝑚. We propose to estimate the demand 4.4 Simplicity of the Model
𝑑𝜆 We have assumed that the parameters of the
rate as 𝑖 = 𝑒 𝜆 𝑖 , assuming that the smallest price
𝑑𝑡 𝑡=0 constraints in (10) are constant. Yet, it is possible that
change in two consequent observation time points is in a real system the dependency of demand on the
ei. prices changes with time, because of any reasons.
This means that the parameters, A, B, Γ, should be
time varying. Even though the dynamics of (10)
would be more realistic, they would highly increase
the complexity of the problem, as there is no way,
without a priori knowledge to determine time-varying
parameters with more confidence than fixed
parameters contrary to what can happen for physical
systems where degradation, e.g., of physical
parameters can be modeled. Hence the problem falls
in the scope of optimization of uncertain systems
(potentially subject to model mismatch or parametric
Fig.5.The optimization procedure may give a higher uncertainty or disturbances), which is an active
profit if performed in a long time period research domain.In this context, it can be shown that
the use of measurements and of feedback is able to
4.3 Optimization Horizon reject a part of the detrimental impact of parametric
An important issue is to estimate the uncertainty on the optimal performances. In our case,
appropriate length of the time period, in which we real demand values are fed back as the optimization
seek to optimize the cloud profit. Specifically, we horizon slides, which increases the robustness of the
have to determine the value of T which represents the proposed approach. As mentioned, Model Predictive
optimization horizon of (4). Intuitively, a long Control has been widely used in Industry, where
horizon allows the optimization procedure to take accurate dynamic models are almost never available.
into account the inertia of the system, whereas a short In these situations using tendency models (i.e.,
horizon may preclude the procedure from taking into models that capture the main trends of a process) and
account important long-term effects of current measurements is generally sufficient to improve the
optimization decisions. process performances up to such a level that the
Example 2. Assume a structure S with demand 𝜆 𝑠 (𝑡) costly efforts for identifying a more accurate process
and an optimization procedure of two short phases model are not justified by the loss of optimality.

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Finally, as the optimization proceeds, new data are
collected and this data can clearly be used to
reidentify the price/demand model periodically.

5 CONCLUSION
We define an appropriate price-demand
model and we formulate the dynamic pricing
problem. The proposed solution allows: on one hand,
longterm profit maximization, and, on the other,
dynamic calibration to the actual behavior of the
cloud application, while the optimization process is
in progress. We discuss qualitative aspects of the
solution and a variation of the problem that allows
the consideration of user satisfaction together with
profit maximization. The viability of the pricing
solution is ensured with the proposal of a method
that estimates the correlations of the cache services in
an time-efficient manner. This paper proposes a
novel pricing scheme designed for a cloud cache that
offers querying services and aims at the
maximization of the cloud profit.

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