1
An Efficient Cloud Market Mechanism for
Computing Jobs with Soft Deadlines
Ruiting Zhou, Zongpeng Li, Chuan Wu, Zhiyi Huang
Abstract—This work studies the cloud market for computing
jobs with completion deadlines, and designs efficient online
auctions for cloud resource provisioning. A cloud user bids for
future cloud resources to execute its job. Each bid includes (a)
a utility, reflecting the amount that the user is willing to pay
for executing its job, and (b) a soft deadline, specifying the
preferred finish time of the job, as well as a penalty function
that characterizes the cost of violating the deadline. We target
cloud job auctions that executes in an online fashion, runs in
polynomial time, provides truthfulness guarantee, and achieves
optimal social welfare for the cloud ecosystem. Towards these
goals, we leverage the following classic and new auction design
techniques. First, we adapt the posted pricing auction framework
for eliciting truthful online bids. Second, we address the challenge
posed by soft deadline constraints through a new technique
of compact exponential-size LPs coupled with dual separation
oracles. Third, we develop efficient social welfare approximation
algorithms using the classic primal-dual framework based on
both LP duals and Fenchel duals. Empirical studies driven by
real-world traces verify the efficacy of our online auction design.
Index Terms—Cloud computing, Auction Mechanism, Online
Algorithm.
I. INTRODUCTION
Cloud computing has emerged as a new computing
paradigm that offers users rapid on-demand access to resources
such as CPU, RAM and disk storage, with minimal management overhead. In the past decade, two types of cloud
platforms blossomed on the Internet, including (i) largescale Internet data centers, exemplified by Amazon EC2 [1],
Microsoft Azure and Linode [2], [3], which organize a shared
resource pool for serving their users; and (ii) co-location
data centers, often found in metropolitan areas, where smaller
clouds from different users are physically co-located, jointly
managed and serviced by the co-location [4].
Virtualization technologies help cloud providers pack their
resources into different types of virtual machine (VM), for
allocation to cloud users. For example, Amazon EC2 [1]
currently offers 23 types of different VM types in 7 categories.
Each type of VM has its focus and forte, and a large computing
R. Zhou is with the University of Calgary, Calgary, AB, Canada (e-mail:
[email protected]).
Z. Li is with the University of Calgary, Calgary, AB, Canada, and
also with the School of Computer and Collaborative Innovation Center of
Geospatial Technology, Wuhan University, Wuhan, China (e-mail: zongpeng@
ucalgary.ca).
C. Wu is with the University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]).
Z. Huang is with the University of Hong Kong, Kowloon, Hong Kong
(e-mail: [email protected]).
This work was supported in part by NSERC, Wedge Networks, and HKRGC (grant #: HKU 17202115E, HKU 718513, HKU 17204715, HKU
17225516, C7036-15G (CRF)).
job often requires cooperation among multiple VM instances.
For example, social games [5] and enterprise applications [6]
are often composed of a front-end web server tier, a load
balancing tier and a back-end data storage tier, each suited
for execution on a VM that is abundant in a particular type of
resource: bandwidth, CPU, or storage.
Cloud computing jobs can be categorized into two types,
depending on whether their computing need is elastic or not.
Cloud jobs such as large scale web servers utilize cloud service
as a utility, and require the rented VMs to be always active,
with possible dynamic size scaling. These jobs are similar
to the power users in a power grid who demands always-on
power supply. Other jobs such as big data analytics and Google
crawling data processing often have a batch processing nature.
They require a certain computing job to be completed without
demanding always-on VM service, and may tolerate a certain
level of delay in the job completion. These users are similar
to the energy users in a power grid who needs to draw a fixed
quantity of energy for powering a given job, but in a flexible
time window.
Existing market mechanisms for cloud computing, particularly the auction type mechanisms, have been implicitly
targeting the first type of non-elastic cloud jobs. In such oneround [7] and online [8] cloud resource auctions, once a bid
is accepted, the service time window of the corresponding
VMs is fixed, i.e. either in the current round [8] or between
the start and finish times prescribed in the bid [9]. Such
auction algorithms do not need to consider the scheduling
of accepted jobs. In sharp contrast, a well designed market
mechanism for the second type of elastic jobs must pay close
attention to not only whether to accept a bid, but when to
schedule its execution based on its deadline information. For
example, consider a cloud user who bids for a VM bundle
tailored for human genome analysis. Its job can be processed
within 3 hours if the specified VM bundle is provisioned;
however, as long as the computing result is available within
the next 24 hours, the user is happy. This leaves ample
space for job scheduling in the temporal domain, which a
well-designed auction algorithm should judiciously exploit to
maximize resource utilization and social efficiency — for
example, scheduling a job within its tolerance window to time
slots with relatively low demand.
This work generalizes existing auction design in the cloud
market by proposing online auctions that explicitly handle
jobs with prescribed deadlines. We further allow a cloud user
to express soft deadlines, described by both a preferred job
completion time, coupled with a penalty function that encodes
how much penalty is associated with different degrees of dead-
2
line violation. Compared with simple market mechanisms such
as fixed pricing, a well-designed auction provides automatic
price discovery, promptly adapts prices with the fluctuation of
supply and demand, and allocates cloud resources to jobs who
value them the most, maximizing the overall “happiness” of
everyone in the cloud ecosystem.
We simultaneously target the following goals in our cloud
auction design. First, we require the cloud auction to be
computationally efficient and executes in polynomial time.
Second, the auction should be truthful, so that bidding true
job valuation is the dominant strategy for a cloud user. Third,
the auction should maximize the social welfare of everyone
in the system including both the cloud provider and the cloud
users. Such cloud auction design is faced with a number of
challenges. First, truthfulness is a rather strong property that
comes only with a pair of carefully prepared VM allocation
and payment algorithms that work in concert with each other.
Furthermore, even if the cloud users can be assumed to be
altruistic and truthful bids are given for free, the winner
determination problem for social welfare maximization is an
integer linear program (ILP) that is NP-hard to solve. A new
challenge unique to this work is the non-traditional type of soft
deadline constraints, which is hard to model and handle with
traditional LP formulation and algorithm design. Last but not
least, we require the auction to be online, immediately making
a decision upon the arrival of each bid, without knowing future
bids in the market, yet still guaranteeing near-optimal decision
making as compared to the offline optimum.
We first consider a basic setting where resources in the cloud
are free of cost up to a known capacity limit, and that the
soft deadline can be expressed by enumerating a few hard
deadline options and their corresponding bidding prices. We
first present a natural ILP formulation of the social welfare
maximization problem. While polynomial in size, this ILP
involves both conventional constraints (capacity limits) and unconventional constraints (job deadlines). The latter further lead
to unconventional dual variables that are hard to interpret and
update in a primal-dual algorithm framework we will leverage.
We convert the natural ILP into a compact-exponential ILP that
has a compact formulation of conventional constraints only, at
the price of involving an exponential number of variables.
We apply the posted pricing primal-dual framework to
the compact-exponential ILP for online social welfare maximization. Although the dual has an exponential number of
constraints, we show fast dual oracles that can quickly update
the dual variables, which are interpreted as unit cost of cloud
resources in different time slots. We maintain carefully estimated resource costs based on recently designed exponential
cost functions [10]. Upon receiving a bid, we compare the
bidding price with the estimated cost of the bid. If the bidding
price is higher, the bid is accepted and dual variables are
updated; otherwise the bid is rejected. The posted pricing
framework charges winning jobs an estimated cost that is
independent from the bidding price, and is truthful [11]. We
conduct theoretical analysis on the competitive ratio and prove
its upper-bound.
We proceed to generalize our cloud auction design by
addressing two practical concerns. First, we model the cost
of resource provisioning in data centers, using a convex cost
function that characterizes server cost with Dynamic Voltage
Frequency Scaling [12]. Second, we consider the general form
of a soft deadline, specified by (i) a preferred deadline and (ii)
a non-decreasing penalty function for deadline violation. The
new social welfare maximization problem is an integer convex
program. We resort to a new primal-dual solution framework
for well-structured convex programs based on Fenchel dual
[13], and adapt our posted pricing auction framework from
the previous scenario to this general setting.
In the rest of the paper, we discuss related work in Sec. II,
and introduce the system model in Sec. III. Design and
analysis of the online cloud auctions are presented in Sec. IV
and Sec. V. Sec. VI presents simulation studies, and Sec. VII
concludes the paper.
II. P REVIOUS R ESEARCH
Market mechanism design for cloud computing, particularly
auction mechanisms for cloud resource trading, has attracted
substantial interest from the research community, with a large
number of VM auctions spawned in the past few years [7]–[9],
[14]–[16].
The earliest VM auctions are simple in that they are oneround auctions, and assume that the cloud provisions a single
type of VM, or that VM configurations are equivalent up to
linear scaling [14]. They also assume the scenario of static
VM provisioning, where the number and type of VMs to be
sold are predetermined prior to the auction start [15].
Dynamic VM provisioning, in which the cloud provider
makes decision on which VMs to assemble and how many
based on demand learned from user bid during the auction, has
been studied in the past two years [7]–[9]. Zhang et al. design
a randomized auction for dynamic resource provisioning in
cloud computing based on a convex decomposition technique,
which is truthful and guarantees a small approximation ratio
in social welfare [7]. Shi et al. further study dynamic resource
provisioning where cloud users are subject to budget constraints, and design online auctions where decision making
is coupled in the time domain due to fixed user budgets [8].
Online cloud auctions appear later than their one-round
counterparts. Zhang et al. is among the first to study online
cloud auction design, but they assume all VMs are of a uniform
type [16]. The work of Shi et al. [8] designs online auctions,
but does not consider the temporal correlation in decision
making due to jobs spanning multiple time slots. A recent
work of Zhang et al [9] study online cloud auctions where a
user bids into a fixed time window for job execution; hence the
scheduling dimension is non-present in their solution space.
There have been recent studies on mechanism deign for
batch jobs with deadlines. Lucier et al. study two scheduling
algorithms for jobs with deadlines in cloud computing clusters
[17]. They analyze the competitive ratio for non-committed
scheduling, which does not require to finish executing a
job that has started execution. They do not provide any
performance guarantee on the competitive ratio for committed
scheduling. Navendu et al. design a truthful allocation and
pricing mechanism for computing jobs with deadlines, but
3
restrict attention to the offline setting [18]. Azar et al. construct an online mechanisms for preemptive scheduling with
deadlines [19]. Their mechanism is truthful and achieves a
constant competitive ratio. All of those work consider only
one fixed deadline for each job, and fail to model the server’s
operation cost.
Compared with existing literature on cloud auctions, this
work is the first to design cloud auctions that explicitly
consider job elasticity and job execution deadlines, which are
important for practical applications to batch processing jobs.
Accordingly, we propose the compact-exponential optimization technique that can effectively handle the new job deadline
constraints in social welfare maximization for the cloud.
The online primal dual method (see [20] for a detailed
survey) is a power algorithmic technique that that has witnessed broad applications, such as solving the ski rental
problem, maximizing revenue in ad-auctions, and solving the
general packing problem. The original primal dual framework
works on linear programs, and is not used to solve problems
modelled by convex programs in our work. Rather recently,
new techniques were introduced to help apply the primal dual
framework for algorithm design to convex programs. Blum et
al. study online combinatorial auctions with production costs
using the online primal dual framework [21]. They presents
algorithms for various cost functions. Huang et al. further
investigate the same problem and propose mechanisms with
improved competitive ratio [11]. All those work don’t consider
the scheduling of jobs, and they cannot handle VM departures
and resource recycling.
III. S YSTEM M ODEL AND P RELIMINARIES
We consider a cloud data center hosting a pool of K types
of resource, including CPU, RAM and disk storage that can
be dynamically assembled into different types of VMs. Let
[X] denote the integer set {1, 2, . . . , X}. There are a total
ck unit of type-k resource in this cloud. The cloud service
provider acts as the auctioneer to lease VMs to cloud users
through an auction. User’s bid arrives randomly in a large
time span 1, 2, . . . , T . Note that multiple bids can arrive
simultaneously, and would be ordered randomly. There are
I users participating in the auction, and each user requests
multiple types of VM, and specifies in its bid: (i) rik , the total
amount of type-k resource, and (ii) wi , the number of slots
required to finish the job by the designated VMs. Job execution
doesn’t need to be continuous. A user i’s job can be executed
at any time slot as long as the total execution time meets wi
before the deadline. We consider two soft deadline models
in this work: a basic model with alternative deadlines and a
general model with penalty function and server operation cost.
A. Jobs with Alternative Deadlines
We first consider a basic scenario where each user submits
J optional bids to express disjunctive deadline options. A bid
from user i consists of a list of desired types of resource rik ,
8k; the number of requested slots wi , and deadlines for job
completion dij , 8j, each with a corresponding bidding price
bij . We use Bi to denote the bidding language of user i’s bids
submitted at time ti :
Bi = {ti , {rik }k2[K] , wi , {dij , bij }j2[J] }.
We adopt the XOR bidding rule that assumes a user can
win at most one bid among its J optional bids [7]. Upon the
arrival of each bid, the cloud provider decides immediately
whether to accept it, and if so, which deadline to choose and
how to schedule the job. A binary xij equals 1 if user i’s jth
bid wins, and 0 otherwise. Let another binary variable yi (t)
encode the scheduling of user i’s job: yi (t) = 1 if user i’s
job is scheduled to run at time t, and 0 otherwise. The cloud
provider also calculates the payment pi for each winner i.
Let vij be the true valuation of user i’s jth bid, then the
utility of that bid with is uij (bij ) = vij pi if xij = 1, and is
0 if xij = 0. In practice, user are assumed to be selfish with
a natural goal to maximize their own utilities; they may lie
about their true valuations in the hope of a higher utility. The
cloud provider instead pursues highest social welfare possible
to make everyone in the cloud system “happy”. Thus, it is
important for the cloud provider to elicit truthful bids.
Definition 1. (Truthful Auction): A cloud auction is truthful
if the dominant strategy for each user is to report its true
valuation, which always maximizes its utility: for all bij 6= vij ,
uij (vij ) uij (bij ).
Definition 2. (Social Welfare): The social welfare in the cloud
market
with alternative deadlines
is the aggregate user utility
P
P
P
v
x
p
i2[I] P j2[J] ij ij
i2[I] i plus the cloud provider’s
utility i2[I] pi . P
Payments
P cancel themselves, and the social
welfare becomes i2[I] j2[J] vij xij .
B. Jobs with Penalty Function and Operation Cost
We further consider a more general model where each user
submits a single preferred deadline di , with a penalty function
gi (⌧i ) defined over deadline violation ⌧i :
gi (⌧i ) =
⇢
gci (⌧i ),
+ 1,
if ⌧ 2 [0, T
otherwise
di ]
(1)
where di + ⌧i is the job completion time; bi gi (⌧i ) is the
bidding price, decreasing with job completion time; gci (·) is a
nondecreasing function and gci (0) = 0. User i’s bid with this
model is: Bi = {ti , {rik }k2[K] , wi , di , bi , gi (⌧i )}.
Existing studies on cloud auction design often ignore the
server operation cost of the cloud provider. It is natural to
include server cost in the computation of social welfare, albeit
the fact that it makes social welfare optimization substantially
more challenging (from linear to non-linear integer programming). The operation cost in the cloud comprises mainly
of power consumption for provisioning the virtual machines,
increasing as the amount of resources used grows. Let zk (t)
be the amount of type-k resource used at time t in the cloud,
then the cost function of type-k resource is defined as:
fk (zk (t)) =
(
k zk (t)
+ 1,
1+ k
,
if zk (t) 2 [0, ck ]
otherwise
(2)
Parameter k is the coefficient determined by the power
consumption of each type of resource. Recent measurement
studies suggest that the power consumption of memory, disk
are significantly lower than that of CPU [22]. k
0
modulates the shape of the cost function, following the
the operational model of physical servers in the cloud. For
4
example, Dynamic Voltage Frequency Scaling (DVFS) is a
technique widely adopted in virtualization platforms, adjusting
the frequency or voltage of CPUs to save power consumption
[12]. k is roughly 2 if the voltage is proportional to the usage
of CPU when DVFS is enabled, and equals 0 when DVFS in
disabled [23]. The shape of RAM and disk cost function is
different from that of CPU, with k 2 [0.5, 1] [22].
Similar to the notations in Sec. III-A, let a binary xi be an
auction decision and pi be the payment. vi gi (⌧i ) is the true
valuation of user i’s bid. The cloud provider’s utility equals
the
minus the operation cost, i.e.,
P aggregate
P user payment
P
p
f
i2[I] i
k2[K]
t2[T ] k (zk (t)). The definitions of user
i’s utility, truthful auction and social welfare are omitted here
as similar ones can be found in Sec. III-A. Table I summaries
notation for ease of reference.
TABLE I: Summary of Notations
I
# of users
[X] integer set {1, . . . , X}
T
# of time slots
J
# of bids per user
f
cost function
f⇤
convex conjugate of f
g
penalty function
ti
user i’s arrival time
pi user i’s payment
ui
user i’s utlity
rik
demand of type-k resource by user i
wi
# slots requested by user i
⌧i
# slots that passes the deadline for user i
dij (di )
deadline of user i’s jth (user i’s) bid
bij (bi )
bidding price of user i’s jth (user i’s) bid
vij (vi )
true valuation of user i’s jth (user i’s) bid
xij (xi )
user i’s jth (user i’s) bid wins (1) or not (0)
yi (t)
whether or not to allocate user i’s job in slot t
ck
capacity of type-k resource
pk (t)
marginal price of type-k resource at time t
zk (t)
amount of allocated type-k resource at time t
Uk (Lk ) maximum (minimum) value per unit
of type-k resource per unit of time
1
max{2, (1 + k ) k }
max{ ✓ckk k , ck (✓✓kk 1) ln(
✓k
⇢k
IV. O NLINE AUCTION M ECHANISM FOR J OBS WITH
A LTERNATIVE D EADLINES
In this section, we focus on the scenario where each user’s
job has J alternative deadlines. Sec. IV-A presents the social
welfare maximization problem and the framework to handle
such deadline problems. We design an online auction in
Sec. IV-B and conduct theoretical analysis in Sec. IV-C.
maximize
X X
bij xij
(3)
i2[I] j2[J]
subject to: yi (t)t 
X
j2[J]
wi xij 
X
X
j2[J]
dij xij , 8t 2 [T ], 8i 2 [I] : ti  t, (3a)
X
yi (t), 8i 2 [I],
rik yi (t)  ck , 8k 2 [K[, 8t 2 [T ],
(3c)
X
xij  1, 8i 2 [I],
(3d)
t2[T ]:ti t
i2[I]:ti t
j2[J]
xij , yi (t) 2 {0, 1}, 8i 2 [I], 8t 2 [T ], 8j 2 [J].
(3b)
(3e)
simplified compact-exponential ILP with a packing structure,
at the price of involving an exponential number of variables:
subject to:
XX
XX
bil xil
(4)
i2[I] l2⇣i
rik xil  ck , 8k 2 [K], 8t 2 [T ],
(4a)
X
(4b)
i2[I] l:t2l
l2⇣i
xil  1, 8i 2 [I],
(4c)
xil 2 {0, 1}, 8i 2 [I], 8l 2 ⇣i .
Constraints (4a) and (4b) are equivalent to (3c) and (3d).
⇣i is the set of time schedules that satisfy constraints (3a)
and (3b) for user i. The value of bil is based on schedule
l, and equals the corresponding bij . We relax the integrality
constraints of xil to xil 0 and formulate the dual problem.
By introducing dual variables pk (t) and ui to constraints (4a)
and (4b) respectively, the dual LP of the relaxed (4) is:
minimize
X
ui +
i2[i]
subject to: ui
bil
X X
pk (t), ui
X X
ck pk (t)
(5)
t2[T ] k2[K]
k2[K] t2l
Note that the following constraint is redundant, and is not
P
explicitly included in the ILP above: yi (t)  j2[J] xij , 8i 2
[I], 8t 2 [T ]. Constraint (3a) ensures that a job is scheduled
to run between its arrival time and deadline. Constraint (3b)
guarantees that the number of allocated slots is sufficient for
serving a successful bid. The capacity limit of each type of
resource is expressed in constraint (3c), and the alternative
deadlines are modelled with the XOR bidding rule by (3d).
Even in the offline setting, ILP (3) without constraints
(3a) and (3b) is still a NP-hard combinatorial optimization
problem, equivalent to the classic knapsack problem. The
challenge further escalates when we involve the jobs’ deadlines and pursue online decision making. To address these
challenges, we resort to the primal-dual algorithm design
technique. In preparation, we first design a new framework to
handle the unconventional constraints for deadline modelling.
More specifically, we reformulate the original ILP (3) into a
)}
competitive ratio of Aonline1 (Aonline2 )
↵1 (↵2 )
maximize
A. Social Welfare Maximization Problem
Under the assumption of truthful bidding (bij = vij ), the social welfare maximization problem with alternative deadlines
can be formulated into the following ILP:
0
Uk
k
k (1+ k )ck
rik pk (t), 8i 2 [I], 8l 2 ⇣i ,
(5a)
0, 8i 2 [I], 8k 2 [K], 8t 2 [T ].
(5b)
As we can observe, a feasible solution to ILP (4) has a
corresponding feasible solution in ILP (3), and the optimal
objective value of (4) is equal to that of (3). The number of
variables in ILP (4) is exponential as the number of possible
time schedules for user i is exponential in size. We next design
an efficient primal-dual allocation scheme that only updates a
polynomial number of variables, and can simultaneously solve
optimization problems (3), (4) and (5).
B. Online Auction Design
In the auction algorithm, the cloud provider needs to deicide
whether to accept a user i’s job and if so, how to schedule
its job to meet its deadline. If user i’s jth bid with schedule
5
l is accepted, then let xij = 1, and update the variable yi (t)
according to schedule l. To solve ILP (3), we adopt the primaldual technique to the compact-exponential ILP (4) and its
dual (5). For each primal variable xil in (4), there is a dual
constraint associated to it. Complementary slackness indicates
the update of the primal variable is based on its dual constraint.
xil is zero unless its associated dual constraint (5a) is tight.
Because the dual variable ui 0, we let ui be the maximum
of 0 and the right hand side (RHS) of (5a),
ui = max{0, max{bil
l2⇣i
X X
rik pk (t)}}.
(6)
t2l k2[K]
Accordingly, the cloud provider accepts user i if ui > 0, and
serves user i’s job according to the schedule that maximizes
RHS of constraint (5a); if ui  0, the bid is rejected.
Algorithm 1 A Primal-dual Online Auction Aonline1
Input: bidding language {Bi }, {ck }
1: Define function pk (zk (t)) according to (7);
2: Initialize xij = 0, yi (t) = 0, zk (t) = 0, ui = 0, pk (t) =
0, 8i 2 [I], 8j 2 [J], 8k 2 [K], 8t 2 [T ]; Let xil = 0, 8i 2
[I], 8l 2 ⇣i , by default;
3: Upon the arrival of the ith user
4:
xij , {yi (t)}, pi , {pk (t)}, {zk (t)}
{pk (t)}, {zk (t)} ;
=
pk (zk (t)) = Lk
Acore1 Bi , {ck },
if 9j 2 [J], xij = 1 then
Accept user i’s jth bid and allocated resources according to yi (t); Charge pi for user i;
7: else
8:
Reject user i.
9: end if
5:
6:
nential, we design a dual oracle that selects only a polynomial number of dual constraints. We fix a set of schedules
Li with polynomial size through
P the
P dual koracle, and set
ui = max{0, maxl2Li {bil
t2l
k2[K] ri pk (t)}}. Then
xil is updated to 1 when ui > 0. The dual oracle works as
follow. For each deadline dij of user i’s job, we select wi
slots with the minimum price for t 2 [ti , dij ], and let lj be the
corresponding schedule, and add lj to set Li . The schedule
that maximizes user i’s utility is the one with the minimum
price in set Li .
We next discuss the update of the dual variable pk (t). Recall
that pk (t) represents the marginal price per unit of type-k
resource at time t. We define a new variable zk (t) as the
amount of allocated type-k resource at time t, and let the
marginal price be a function of zk (t). pk (t) is increasing
with the growth of zk . Let Uk and Lk be the maximum and
minimum values per unit of type-k resource per unit of time,
respectively. pk (t) starts at Lk and exponentially increases
when zk (t) is close to the capacity ck . It reaches Uk when
zk (t) = ck because in this case, the cloud provider will never
allocate any type-k resource to any user. In summary, pk (t) is
defined as a function on zk (t) as following:
Algorithm 2 A Scheduling Algorithm Acore1
Input: bidding language {Bi }, {ck }, {pk (t)}, {zk (t)}
Output: xil , pi , {pk (t)}, {zk (t)}
P
k
1: c(t) =
k2[K] ri pk (t), 8t 2 [T ]; // price per slot
2: for all j 2 [J] do
3:
Select wi slots with minimum (c(t)) and zk (t) + rik 
ck , 8k 2 [K]
P within [ti , dij ], save the schedule in lj ;
4:
pij = t2lj c(t); uij = bij pij ;
5: end for
6: j ⇤ = arg maxj2[J] {uij };
7: if uij ⇤ > 0 then
8:
xij ⇤ = 1; yi (t) = 1, 8t 2 lj ⇤ , pi = pij ⇤ ;
9:
xilj⇤ = 1;
10:
ui = uij ⇤ ; zk (t) = zk (t) + rik , 8k 2 [K], t 2 lj ⇤ ;
11:
pk (t) = pk (zk (t)), 8k 2 [K], t 2 lj ⇤ ;
12: end if
13: Return xij ⇤ , {yi (t)}, pi , {pk (t)}, {zk (t)}
If we interpret dual variable pk (t) as
marginal price per
Pthe P
unit of type-k resource at time t, then t2l k2[K] rik pk (t) is
the total charge that user i should pay when its job is assigned
according to schedule l. The RHS of (5a) becomes the utility
of bid i with schedule l. Thus, the assignment of ui in (6)
effectively maximizes user i’s utility. This is a key step towards
achieving social welfare maximization and truthfulness.
Note that although the calculation of ui seems to take
exponential time as the size of dual constraint (5a) is expo-
Where
Uk
mini2[I],j2]J]
P
✓
Uk
Lk
◆ zk (t)
ck
(7)
b
maxi2[I],j2]J] w ijrk
and Lk
i i
.
k
w
r
k2[K] i i
Aonline1 in Alg. 1 with the schedule algorithm Acore1
running for each user in Alg. 2 is the online auction. Aonline1
first defines the price function and initializes the primal and
dual variables in lines 1-2. Upon the arrival of each user i,
we select the bid j ⇤ with schedule lj ⇤ that maximizes user
i’s utility through the dual oracle (lines 2-5). If user i obtains
positive utility, primal variables xij ⇤ and yi (t) are updated
according to schedule j ⇤ (line 8). We then increase the usage
for different resources (zk (t)) and update the price (pk (t)) for
t 2 lj ⇤ (lines 10-11).

bij
Slot 1
Slot 2
Slot 3
Slot 4
Slot 5
Z1(1)=2
p1(1)=1.2
Z1(2)=2
p1(2)=1.2
Z1(3)=2
p1(3)=1.2
Z1(4)=0
p1(4)=0
Z1(5)=0
p1(5)=0
C1=10
User i arrives at time 2
2 units bid 1: deadline 3, $6
2 slots bid 2: deadline 4, $4
{
Accept bid 2. User i’s job
is processed at time slots 3 and 4
Slot 1
Slot 2
Slot 3
Slot 4
Slot 5
Z1(1)=2
p1(1)=1.2
Z1(2)=2
p1(2)=1.2
Z1(3)=4
p1(3)=1.32
Z1(4)=2
p1(4)=1.148
Z1(5)=0
p1(5)=0
Fig. 1: An Example of the process in Aonline1 .
We next use an example to illustrate the winner determination process in Aonline1 , as shown in Fig. 1. Suppose the
online system spans 5 time slots. A cloud data center hosts
only one type of resource and the capacity is 10, i.e., c1 = 10.
Assume L1 = 1 and U1 = 2. Before the arrival of user i,
assume the marginal price per unit of resource at time t is
p1 (1) = p1 (2) = p1 (3) = 1.2; p1 (4) = p1 (5) = 0. The
amount of allocated resource at time t is z1 (1) = z1 (2) =
z1 (3) = 2; z1 (4) = z1 (5) = 0. User i arrives at time 2,
6
requiring 2 units of resource and 2 time slots to execute its job.
It submits two optional bids: it is willing to pay $6 if its job
is completed before time 3 or $4 if its job is finished before
time 4. The bidding price of the user i can be expressed as
Bi = {2, 2, 2, {3, $6}, {4, $4}}. Upon the arrival of the user
i, Acore1 is executed to deicide whether to accept it and if so,
how to schedule the job. The price per slot is calculated at line
1 in Acore1 and c(1) = c(2) = c(3) = 2.4; c(4) = c(5) = 0.
For the first bid of user i, lines 3-4 in Acore1 compute the
schedule, payment and utility of it: l1 = [2, 3], p11 = 4.8
and u11 = 1.2. For the second bid of user 1, l2 = [3, 4],
p12 = 2.4 and u12 = 1.6. Now user i’s maximum utility is
larger than 0, i.e., u12 > 0, primal and dual variables are
updated accordingly at lines 8-11 in Acore1 . Here z1 (t) and
p1 (t) at slots 3 and 4 are updated, i.e., z1 (3) = 4, z1 (4) = 2
4
2
and p1 (3) = 1 · 2 10 ⇡ 1.32, p1 (4) = 1 · 2 10 ⇡ 1.148. User i’s
second bid is accepted and its job is processed at time slots 3
and 4. The cloud provider charges $2.4 for user i. This process
is repeated until the last user’s job is handled.
C. Theoretical Analysis
i) Correctness, Polynomial Time, and Truthfulness.
Theorem 1. Aonline1 computes a feasible solution to ILP (3),
ILP (4) and LP (5) in polynomial time.
Proof: (Correctness): Aonline1 outputs a feasible solution for
ILP (3) because line 3 in Acore1 guarantees that the schedule
lj for user i’s jth bid satisfies constraints (3a), (3b) and (3c).
Constraint (3d) holds as only one bid per user can be accepted
by Acore1 in line 6. Furthermore, the corresponding relation
between xij and xil implies xil is a feasible solution for ILP
(4).
the dual problem (5), Acore1
P For P
P assigns
P 0 tok ui if bil 
k
r
p
(t),
and
b
il
k2[K]
t2l i k
k2[K]
t2l ri pk (t) to ui
otherwise, ensuring the feasibility of Aonline1 .
(Polynomial running time): Lines 1-2 can be executed in
linear time for the initialization of the cost function, primal and
dual variables. Upon the arrival of user i, Algorithm Acore1
first takes T steps to calculate the price of each slot. The
for loop iterates J times to select the best slots for each bid.
Line 3 in Alg. 2 takes O(T K) time to schedule the job and
check the capacity limit. Line 4 can be done in O(1) steps.
Thus, the running time of the for loop in Alg. 2 is O(JKT ).
Then line 6 records the bid with the maximum utility in J
steps. The body of the if statement (line 8-11) takes O(KT )
time to update the primal and dual variables and compute the
payment. To sum up, the running time of Acore1 is O(JKT ).
The last step of Aonline1 (lines 5-9) is to announce the auction
decision, which can be done in constant time. In conclusion,
Aonline1 runs in polynomial time (O(IJKT )).
Theorem 2. The online auction Aonline1 is truthful.
Proof: Our auction Aonline1 belongs to the family of posted
pricing mechanisms [11]. Upon the arrival of user i, the
payment that user i needs to pay to the cloud provider if
it wins, depends only on the amount of resource that has
been sold, and user i’s demand. It is independent of user
i’s bidding price. Consequently, user i cannot improve its
utility by lying about its bidding price as its utility equals
its valuation minus the payment, i.e., uij = vij
pi .
Furthermore, Aonline1 always selects the schedule with the
maximum utility among all possible schedules for user i.
Hence, truthful bidding guarantees that each user obtains its
maximum utility in Aonline1 .
ii) Competitive Ratio.
We next examine the competitive ratio of our online auction.
The competitive ratio is the upper-bound ratio of the social
welfare achieved by the optimal solution of ILP (3) to the
social welfare achieved by our online auction Aonline1 . We
first introduce the primal-dual analysis framework in Lemma
1, which guides the final proof of the competitive ratio.
Let OP T1 and OP T2 denote the optimal objective values of
ILP (3) and (4), respectively. We know that OP T1 = OP T2 .
Let Pi and Di be the objective value of primal problem (4)
and that of dual problem (5) returned by an algorithm after
processing user i’s bids. Let P0 = D0 = 0 be the initial values.
Then PI and DI are the final primal and dual objective values
achieved by the algorithm.
Lemma 1. If there exists a constant ↵1
1 such that Pi
1
Pi 1
(D
D
)
for
all
i,
then
the
algorithm is ↵1 i
i
1
↵1
competitive in social welfare.
Proof: When we sum up the inequalities for each i, we
P
P
1
have PI =
Pi 1 )
Di 1 ) = ↵11 DI .
i (Pi
i (Di
↵1
According to weak duality [24], DI
OP T2 , therefore
1
1
PI
OP
T
=
OP
T
.
So
we
can
conclude
that the
2
1
↵1
↵1
algorithm is ↵1 competitive.
Aonline1 guarantees P0 =D0 =0. We next define an
Allocation-Price Relationship and show that if it holds for a
given ↵1 , then the primal and dual objective values achieved
by Aonline1 satisfy the inequality in Lemma 1. pik (t) denotes
the price of type-k resource after handling user i. zki (t) is the
amount of allocated type-k resource after processing i’s job.
Definition 3. The Allocation-Price Relationship for Aonline1
with ↵1 1 is pik 1 (t)(zki (t) zki 1 (t)) ↵11 ck (pik (t) pik 1 (t)),
8i 2 [I], 8k 2 [K], 8t 2 l.
Lemma 2. If the Allocation-Price Relationship holds for a
1
given ↵1 1, then Aonline1 guarantees Pi Pi 1
↵1 (Di
Di 1 ) for all i 2 [I].
Proof: If user i is rejected, then Pi Pi 1 = Di Di 1 = 0.
In the following analysis, we assume that user i’s jth bid
is accepted, and let l be the schedule of user i’s job. The
increment of the primal objective value is: Pi Pi 1 = bil .
Note that Aonline1 makes the constraint (5a) tight when
bid bij with schedule l is accepted. Thus, bil = ui +
P
k2[K]
P
t2l
pik 1 (t)(zki (t)
zki
1
(t)).
The increase of the dual objective value is: Di Di 1 =
P
P
ui + k2[K] t2l ck (pik (t)
pik 1 (t)). By summing up the
Allocation-Price Relationship over all k 2 [K] and t 2 l, we
can obtain: Pi Pi 1 ui + ↵11 (Di Di 1 ui ). Since ui 0
and ↵1 1, it is obvious that Pi Pi 1 ↵11 (Di Di 1 ).
The Allocation-Price Relationship involves only the variables for type-k resource, we next try to find the corresponding
↵1,k for each resource k that satisfies the Allocation-Price
Relationship. The value of the approximation ratio ↵1 is just
the maximum value among all ↵1,k . In order to compute ↵1,k ,
7
we assume that rik ⌧ ck , then zki (t) zki 1 (t) can be expressed
as dzk (t). The derivative of the Allocation-Price Relationship
under the above assumption is:
Definition 4. The Differential Allocation-Price Relationship
k
for Aonline1 with ↵1,k 1 is pk (t)dzk (t) ↵c1,k
dpk (t), 8i, k, t.
Lemma 3. ↵1,k = ln
and the marginal price defined in (7)
satisfies the Differential Allocation-Price Relationship.
Proof: The derivative of the marginal price function is:
zk (t)
1
dpk (t) = p0k (zk (t)) = Lk (Uk /Lk ) ck ln(Uk /Lk ) ck . The
Differential Allocation-Price Relationship is:
reformulate the above convex optimization to the following
compact-exponential convex problem:
subject to:
Lk
Uk
Lk
◆ zk (t)
ck
dzk (t)
) ↵1,k
ck
Lk
↵1,k
Uk
ln
.
Lk
✓
Uk
Lk
◆ zk (t)
ck
1
Uk
ln
dzk (t)
ck
Lk
i2[I] l:t2l
In this section, we present the online auction design for the
general model that includes a penalty function and operation
cost. We focus on the more challenging case of superlinear
cost function with k > 0. The auction design for linear cost
with k = 0 is similar and is omitted here.
A. Social Welfare Maximization Problem
Under the assumption of truthful bidding, the social welfare
maximization problem in the general model is:
i2[I]
gi (⌧i ))
X X
fk (zk (t))
yi (t)t  di + ⌧i , 8t 2 [T ], 8i 2 [I] : ti  t,
X
wi xi 
yi (t), 8i 2 [I],
i2[I]:ti t
(8)
t2[T ] k2[K]
subject to:
X
(9a)
X
xil 2 {0, 1}, zk (t)
(8a)
(8b)
t2[T ]:ti t
yi (t)rik  zk (t), 8k 2 [K[, 8t 2 [T ],
(8c)
⌧i , zk (t) 0, xi , yi (t) 2 {0, 1},
8i 2 [I],8t 2 [T ], 8k 2 [K].
(8d)
Again, constraint yi (t)  xi , 8i, t is redundant and is implied
by the other constraints. Recall the definition of the cost
function in (2) (fk (zk (t)) = +1 if zk (t) > ck ), constraint
(8c) guarantees that the amount of allocated resource never
exceeds its capacity.
Let ⇣i be the set of time schedules that satisfy constraints
(8a) and (8b) for user i, we adopt the same framework to
(9b)
xil  1, 8i 2 [I],
0, 8i 2 [I], 8l 2 ⇣i , 8k 2 [K], 8t 2 [T ].
(9c)
We introduce dual variables pk (t) and ui to (9a) and (9b).
The Fenchel dual [13] of the relaxed convex problem (9) is:
X
minimize
ui +
X X
X X
bil
fk⇤ (pk (t))
(10)
t2[T ] k2[K]
k2[K] t2l
V. O NLINE AUCTION D ESIGN FOR THE G ENERAL M ODEL
WITH P ENALTY F UNCTION AND O PERATION C OST
(bi xi
rik xil  zk (t), 8k 2 [K], 8t 2 [T ],
l2⇣i
subject to: ui
Theorem 3. The online auction Aonline1 in Alg. 1 is ↵1 Uk
competitive in social welfare with ↵1 = maxk2[K] ln L
.
k
Proof: According to the proof in Lemma 3, ↵1 =
Uk
maxk2[K] ln L
satisfies the Differential Allocation-Price Rek
lationship. Under the assumption that dzk (t) = zki (t)
zki 1 (t)) is much smaller than the capacity of type-k resource
(ck ), we have dpk (t) = p0k (zk (t))dzk (t) = pik (t) pik 1 (t).
As a result, we can obtain the Allocation-Price Relationship
Uk
holds for ↵1 = maxk2[K] ln L
. Then, combining Lemma 1
k
and Lemma 2 we finish the proof.
X
(9)
t2[T ] k2[K]
i2[i]
Uk
Therefore this lemma holds for ↵1,k = ln L
.
k
maximize
XX
X X
fk (zk (t))
bil xil
i2[I] l2⇣i
Uk
Lk
✓
XX
maximize
rik pk (t), 8i 2 [I], 8l 2 ⇣i , (10a)
0, 8i 2 [I], 8k 2 [K], 8t 2 [T ].
(10b)
pk (t), ui
Where fk⇤ (pk (t)) is the convex conjugate [25] of the cost
function fk (·), defined as: fk⇤ (pk (t)) = supzk (t) 0 {pk (t)zk (t)
fk (zk (t))}. The explicit expression of the conjugate is as
following:
8
⇣ p (t) ⌘ 1+ k
>
k
k
>
k
<
· 1 ,
⇤
1
+
k
k
fk (pk (t)) =
k
>
>
:
1+ k
ck pk (t)
,
k ck
where zk0 (t) = (
Proof:
fk⇤ (pk (t))
pk (t)
k (1+ k )
= sup
zk (t) 0
(
)
1
k
zk0 (t)  ck
zk0 (t)
(11)
> ck
.
pk (t)zk (t)
pk (t)zk (t)
k zk (t)
1+ k
,
1,
zk (t) 2 [0, ck ]
zk (t) > ck
We observe that pk (t)zk (t) 1 = 1 when zk (t) > ck , thus
we only need to obtain the conjugate of f when zk (t) 2 [0, ck ],
1+ k
Let k (zk (t)) = pk (t)zk (t)
. The derivative
k zk (t)
of k (zk (t)) with respect to zk (t) is :
k (zk (t))
0
= pk (t)
k (1
+
k )zk (t)
k
.
When we let k (zk0 (t))0 = 0, the local maximum happens at
1
k (t)
the point zk0 (t) and zk0 (t) = ( kp(1+
) k.
k)
Note that the domain of zk (t) is within the range [0, ck ],
therefore the supremum of k (zk (t)) is zk0 (t) only if zk0 (t) 2
[0, ck ]. Otherwise, when zk0 (t) > ck , we can obtain that
0
k (zk (t)) > 0, which means
k (zk (t)) monotonically increases with the increment of zk (t) and the supremum happens
at zk (t) = ck .
To sum up, we derive the conjugate of the cost function as
shown in (11).
B. Online Auction Design
We adopt the same posted pricing primal-dual framework
from Sec. IV to solve the convex problem (8). Similarly, the
primal-dual technique is applied to its compact-exponential
problem (9) and its dual (10): the primal variable xil remains
zero unless its dual constraint (10a) becomes tight. The
assignment of ui is the same as that in (6).
8
Although there are an exponential size of dual constraints
in the computation of ui , we design a dual oracle based
on dynamic programming to output a polynomial size of
schedules, then only dual constraints associated to this set of
schedules need to be considered.
The basic idea of the dual oracle is as follows. We fix the
completion time of user i’s job to be tc (tc 2 [ti + wi 1, T ]),
and construct the best schedule lj with the minimum price
in this case. The set that includes all such lj has polynomial
size, and is the output of the dual oracle. The construction
of lj is based on the dynamic programming method. The
base case is the schedule l0 with t 2 [ti , ti + wi 1]. We
move the
P completion time one slot forward each time. Let
c(t) = k2[K] rik pk (t) be the price of user i’s job running
at time t. If the completion time tc passes the deadline
di , thePcorresponding penalty is added to the price, i.e.,
c(t) = k2[K] rik pk (t)+g(tc di ). When the old competition
time is replaced with the new one, we only need to compare
the price of the old competition time and wi 1 slots before the
old competition time. For example, if user i arrives at time 1
with wi = 4, then the basic case is l0 = {1, 2, 3, 4}. Assume
that arg maxt2{1,2,3} c(t) = 2. We next fix the completion
time to 5, the best schedule is then {1, 4, 3, 5} if c(4) < c(2)
and {1, 2, 3, 5} otherwise. The process is repeated until the
completion time reaches T .
The marginal price pk (t) per unit of type-k resource at time
t can be defined as the derivative of the cost function, i.e.,
fk 0 (zˆk (t)) if the overall demand of resource k at t (zˆk (t)) is
known. But in the online setting, it is impossible for the cloud
provider to acquire the complete knowledge of the system.
The cloud provider predicts the final demand at future slots
as ✓k (✓k > 1) times of the current demand at those slots if
the predicted final demand is below the capacity, and set the
marginal price to fk 0 (✓k zk (t)) where zk (t) is the amount of
current allocated resource k at t. Let Uk0 be the maximum value
per unit of type-k resource per unit of time. The marginal price
grows exponentially when the predicted demand is larger than
the capacity, and reaches Uk0 if zk (t) = ck . More specifically,
the marginal price function is defined as:
pk (zk (t)) =
8
0
>
< fk (✓k zk (t)),
>
: fk0 (ck )e⇢k (zk (t)
ck
✓k
ck
ck
)
✓k
, zk (t) >
✓k
zk (t) 
(12)
1
with parameters ✓k = max{2, (1 + k ) k },
✓k
✓k
⇢k = max{
ln(
k,
ck
ck (✓k 1)
Uk0
)},
k
k (1 + k )ck
where Uk0  maxi2[I] wbirk .
i i
The online auction Aonline2 for the general model is presented in Alg. 3. Upon the arrival of the ith user, Aonline2
calls Acore2 in Alg. 4 to make decision. Acore2 computes the
best schedule for user i through the dual oracle (lines 1-10)
to maximize its utility ui . If ui > 0, the corresponding primal
and dual variables are updated in lines 14-17.
C. Theoretical Analysis
i) Correctness, Polynomial Time, and Truthfulness.
Lemma 4. The running time of Acore2 is O(KT + T 2 ).
Algorithm 3 A Primal-dual Online Auction Aonline2
Input: bidding language {Bi }, {ck }, { k , k }
1: Define cost function fk (zk (t)) according to (2);
2: Define function pk (zk (t)) according to (12);
3: Initialize xi = 0, yi (t) = 0, zk (t) = 0, ⌧i = 0, ui = 0, pk (t) =
0, 8i 2 [I], 8k 2 [K], 8t 2 [T ]; Let xil = 0, 8i 2 [I], 8l 2 ⇣i ,
by default;
4: Upon the arrival of the ith user
5:
xi , {yi (t)}, pi , {pk (t)}, {zk (t)}
{pk (t)}, {zk (t)} ;
=
Acore2 Bi , {ck },
if xi = 1 then
Accept user i’s bid and allocated resources according
to yi (t); Charge pi for user i;
8: else
9:
Reject user i.
10: end if
6:
7:
Algorithm 4 A Scheduling Algorithm Acore2 .
Input: Bi , {ck }, {pk (t)}, {zk (t)}
Output: xi , {yi (t)}, pi , {pk (t)}, {zk (t)}
1: Add slot t 2 [ti , T ] to set T if zk (t) + rik  ck , 8k 2 [K];
2: Let schedule l0 include the first wi slots (t1 , t2 , . . . , twi )
in T ; Define j = 1;
3: while wi + j  |T | do
4:
lj = lj 1 ;
5:
Let tc is
Pthe (wi + j)th slot in T ;
6:
c(t) = k2[K] rik pk (t), 8t 2 {t1 , t2 , . . . , twi , tc };
7:
If tc > di , c(tc ) = c(tc ) + g(tc di );
8:
tm = arg maxt2{t1 ,...,twi 1 } c(t);
9:
If c(twi ) < c(tm ), for schedule lj , replace the slot tm
with twi and
P save tc into twi ;
10:
Pj = t2lj c(t); j = j + 1;
11: end while
12: j ⇤ = arg minj {Pj };
13: if bi
Pj ⇤ > 0 then
14:
xi = 1; yi (t) = 1, 8t P
2 lj ⇤ ; xP
ilj ⇤ = 1;
15:
ui = bi Pj ⇤ ; pi = k2[K] t2lj⇤ rik pk (t);
16:
zk (t) = zk (t) + rik , 8k 2 [K], t 2 lj ⇤ ;
17:
pk (t) = pk (zk (t)), 8k 2 [K], t 2 lj ⇤ ;
18: end if
19: Return xi , {yi (t)}, pi , {pk (t)}, {zk (t)}
Proof: Line 1 initializes a feasible slot set T in O(KT ) steps.
Line 2 takes wi steps to define a schedule lo The while loop
(lines 3-11) is to compute the best schedule lj if the completion
time is fixed, will iterate at most T wi times. Within the
while loop body, lines 4-7 takes O(wi + 1) steps to update
c(t). The running time of finding the maximum price in line 8
is linear to wi . Lines 9-10 takes constant time for the comparison and addition. Thus, the while loop can be executed in
O((T wi )wi ) steps. Line 12 can be done in O(T wi ) steps
to find the schedule with the minimum price. The running time
to execute the if body is O(KT ). In summary, the running
time of Acore2 is O(KT + wi (T wi ))  O(KT + T 2 ).
Theorem 4. Aonline2 in Alg. 3 is a truthful auction that returns
9
feasible solutions for convex problems (8), (9) and (10) in
polynomial running time.
Proof: (Polynomial running time:) The running time of the
initialization process in lines 1-3 is linear. By Lemma 4, Acore2
in line 5 processes each user in O(KT + T 2 ) time. The If
statement in lines 6-10 can be done within constant time.
Therefore, after handling the last user, the overall running time
of Aonline2 is O(I(KT + T 2 )).
(Correctness and Truthfulness:) We omit the proof here as
similar proofs can be found in Theorem 1 and Theorem 2.
ii) Competitive Ratio.
The proof follows the same structure as that in Sec. IV-C.
Let Pi and Di be the primal (9) and dual (10) objective
values achieved by Aonline2 after handling user i’s request. By
Lemma 1, Aonline2 is ↵2 -competitive in social welfare if there
1
is a constant a2 1 such that Pi Pi 1
Di 1 )
↵2 (Di
for all i. We next define the Allocation-Price Relationship for
Aonline2 , and show that if the Allocation-Price Relationship
1
holds for a given ↵2 , then Pi Pi 1
Di 1 ) also
↵2 (Di
holds. The last step is to define the differential version of the
Allocation-Price Relationship and prove there exists a ↵2,k that
satisfies this relationship. By setting ↵2 = maxk2[K] {↵2,k },
we can obtain the competitive ratio of Aonline2 .
Definition 5. The Allocation-Price Relationship for Aonline2
with ↵2 1 is pik 1 (t) zki (t)
zki 1 (t)
fk (zki (t))
fk (zki 1 (t))
1
↵2
fk⇤ (pik (t))
fk⇤ (pik 1 (t))
Pi
Pi
1
= bil
fk0 (zk (t)) =
0
fk⇤ (zk (t))
fk (zki (t))
fk (zki
1
=
⇢
k (1
+
+ 1,
8⇣
<
:
ck ,
pk (t) ⌘
k (1 + k )
pk (t) =
k (1
X X
pik 1 (t) zki (t)
zki
1
(
k (1
+
1 ⇣
↵2,k
k )(✓k zk (t))
pk (t)
+ k)
k (1
k)
1
k
1
fk (zki (t))
= ui +
k ✓k
✓k
fk (zki
1
k
1
Since ui
0 and ↵1
1
(D
D
i
i
1 ).
↵2
k (1
1
k
k (1
k +1
+
k )ck
pk (t) >
k (1
+
k )ck
k
k
e
ck
✓k
⇢k (ck
)
Uk0 .
+
+
k )zk (t)
k )✓k
k
k
)dzk (t)
k zk (t)
1
k
dzk (t). (13)
k
2 · 2 k)
·2 k
k (4 · 2
=
k
2
1
2
1
4 k (2 k 1)

 4 k < ↵2,k
2 k 1
k2
=
1
k ✓k
= ✓k (1 +
✓k 1
X X
fk⇤ (pik (t))
Case 2: zk (t) >
ui +
1
(Di
↵2
Di
1
k)
1
k
< e, and
0, it is obvious that Pi
1
Definition 6. The Differential Allocation-Price Relationship
for Aonline2 with ↵2,k 1 is: pk (t)dzk (t) fk0 (zk (t))dzk (t)
1
f ⇤ 0 (pk (t))dpk (t), 8i, 8k, 8t
↵2,k k
2 l.
< e(1 +
k)
< ↵2,k .
In this case, the marginal price zk (t) is:
k (1
+
k
k )ck
e
⇢k (zk (t)
ck
✓k
)
.
fk0 (zk (t)))dzk (t)
1
ck ⇢k pk (t)dzk (t).
↵2,k
(14)
By Lemma 7, we can obtain pk (t) fk0 (zk (t))
pk (t)
1
k
p
(t)
p
(t),
thus
to
prove
(14),
it
is
sufficient
to
k
k
1+ k
1+ k
prove:
k
Pi
k)
Note that dpk (t) = ⇢k pk (t)dzk (t), then the Differential
Allocation-Price Relationship is:
(pk (t)
ui ).
ck
✓k :
pk (t) =
fk⇤ (pik 1 (t))
By summing up the Allocation-Price Relationship for
Aonline2 over all k 2 [K] and t 2 l, we can obtain:
Pi
k )ck
< 1, then ✓k = (1 +
(t)) .
t2l k2[K]
Pi
k (1
k +1
The second equality holds because Aonline2 update the value
of dual variables such that dual constraint becomes tight and
rik = zki (t) zki 1 (t). Then the increase of the dual objective
value is:
Di
, pk (t) 
} = 2, we can obtain
ii) If
t2l k2[K]
Di
, if zk (t) 2 [0, ck ]
otherwise
Cancelling the common term on both sides, (13) becomes
1
k +1
(✓kk 1)
. i) If k
1, ✓k = max{2, (1 +
↵2,k k ✓
k2[K] t2l
X X
k
+
k
⌘
(t))
(t)
1
k
k
Recall that Uk0 is the maximum value per unit of resource k
per unit of time. It is clear when the marginal price is larger
than Uk0 , no bids can win. Thus, we may assume zk (t)  ck
in the rest of the proof, and fk0 (zk (t)) = k (1 + k )zk (t) k .
Next, we divide our proof into two cases:
Case 1: zk (t)  ✓ckk : Because pk (t) = f 0 (✓k zk (t)) = k (1 +
k
 k (1 + k )ckk , the Differential Allocationk )(✓k zk (t))
Price Relationship can be rewritten as:
t2l k2[K]
= ui +
k )zk (t)
When the amount of allocated type-k resource reaches the
capacity, i.e., zk (t) = ck , according to the definition of
marginal price in (12),
, 8i, 8k, 8t 2 l.
Lemma 5. If the Allocation-Price Relationship for Aonline2
holds with a given ↵2
1, then Aonline2 guarantees Pi
1
Pi 1
(D
D
)
for
all
i 2 [I].
i
i 1
↵2
Proof: If user i is rejected, then Pi Pi 1 = Di Di 1 = 0.
In the next analysis, we assume that user i is accepted, and let
l be the schedule of user i’s job. The increment of the primal
objective value is:
X X
U0
Lemma 6. ↵2,k = max{4(1 + k ), 2(1+k k ) ln( (1+ k )c k )}
k
k k
and the marginal price function defined in (12) satisfy the
Differential Allocation-Price Relationship.
Proof: We first write down the explicit expressions for the
differentials of the cost function (2) and its convex conjugate
(11):
1+
1
k
ck ⇢k pk (t)dzk (t) ) ⇢k 
↵2,k
ck (1 +
pk (t)dzk (t)
k
By the value of ⇢k , either i)
⇢k =
=
✓k
ck
k

e
ck
k
ck (1 +
k)
k
e(1 +
k)

k
ck (1 +
k)
↵2,k .
k)
↵2,k .
10
normalize ck to 1, the competitive ratio of Aonline2 is close
to 6 with a small Uk0 and k , as demonstrated in the left
figure. The right figure shows that if ck is a large number,
the competitive ratio is determined by k and increases with
the increment of k .
✓k
Uk0
Or ii) ⇢k =
ln(
)
k
ck (✓k 1)
k (1 + k )ck
0
2
Uk

ln(
)
k
ck
k (1 + k )ck
2(1 + k )
Uk0
k
=
ln(
)
k
ck (1 + k )
k
k (1 + k )ck

k
ck (1 +
k)
VI. P ERFORMANCE E VALUATION
↵2,k .
In conclusion, we have finished the proof for both cases.
Lemma 7. When zk (t) > ✓ckk , the marginal price pk (t) is
larger than the marginal cost by a factor of at least 1 + k :
pk (t)
(1 +
0
k )fk (zk (t)).
Proof: When zk (t) > ✓ckk , pk (t) = k (1+
So Lemma 7 is equivalent to verify
e⇢k zk (t)
zk (t) k
(1 +
k )e
k )ck
k
e
ck
✓k
⇢k (zk (t)
⇢k ck
✓k
)
.
(15)
ckk
We first show that the inequality (15) holds when zk (t) = ✓ckk .
If zk (t) takes the value of ✓ckk , (15) becomes ✓kk
1+ k
which is obviously true.
Next, it suffices to show the left side of (15) is nondecreeing as zk (t) increases. Let L(zk (t)) denote the left hand
of (15). The derivative of L(zk (t)) is
L0 (zk (t)) =
e⇢k zk (t) (⇢k zk (t)
zk (t)1+ k
k)
.
✓k
ck
Because ⇢k
0
k
ck k and zk (t) > ✓k , then ⇢k zk (t)
0
and the derivative L (zk (t)) is nonnegative. Consequently, the
lemma follows.
Theorem 5. The online auction Aonline2 in Alg. 3 is ↵2 competitive in social welfare with ↵2 = maxk2[K] ↵2,k .
Proof: Because ↵2 is the maximum number among all ↵2,k ,
then Differential Allocation-Price Relationship also holds with
↵2 . We assume that dzk (t) = zki (t) zki 1 (t) is much smaller
than the capacity of type-k resource (ck ), then
fk (zki (t))
fk⇤ (pik (t))
fk (zki
1
(t)) = fk0 (zki
fk⇤ (pik 1 (t))
=
1
(t))(zki (t)
⇤0 i 1
fk (pk (t))(pik (t)
zki
1
(t)),
pik 1 (t)).
Therefore, the Allocation-Price Relationship holds with ↵2 .
Combinng Lemma 1 and Lemma 5, we finish the proof.
14
30
25
Competitive Ratio
Competitive Ratio
35
Uk’=3, ck =1
20
15
Uk’=50, ck =1000
10
8
6
0.6
10
5
0.6
12
0.4
0.2
βk
0
0.5
1
1.5
γ
k
2
2.5
0.4
0.2
βk
0
0.5
1.5
1
γ
2
2.5
k
Fig. 2: An illustration of the competitive ratio of Aonline2 (↵2 )
under different settings.
of
We plot the value of ↵2 in Fig. 2 when we vary the value
0
k , k , ck and Uk [22], [23]. We can observe that if we
We evaluate our online auctions Aonline1 and Aonline2
through trace-driven simulation studies. We exploit the trace
version 1 in Google Cluster Data [26], which contains the
information for each job including the start time, execution
duration, and resource demands (CPU and RAM). We translate
each job into a bid, arriving sequentially in 18 hours. We
assume that each user’s job consumes [1, 12] slots and each
time slot is 5 minutes [26]. User’s job deadline is generated
uniformly at random between its arrival time and the system
end time. The bidding price of each job equals its overall
resource demand times unit prices randomly picked in the
range [Lk , Uk ]. By default, Lk = 1 and Uk0 = Uk = 50. The
demand for CPU and RAM units is normalized so that the
maximum capacity is 1. For the cost function, k is set within
[0.4, 0.6] for CPU and within [0.005, 0.02] for RAM [22]. k
is set within [1.7, 2.2] for CPU and within [0.5, 1] for RAM
[23].
Performance of Aonline1 . We examine the performance of
Aonline1 in terms of the competitive ratio, social welfare and
user satisfaction.
Fig. 3 shows the competitive ratio of Aonline1 with different
number of users (I) and bids per user (J). The observed
competitive ratio is much better than the theoretical bound
and remains at a low level (< 2). It fluctuates with the
increase of the number of users and sightly decreases when
the number of bids per user grows. This is because when each
user provides a larger number of optimal bids, Aonline1 is
more likely to optimize the schedule of its job, leading to a
better performance. In Aonline1 , the marginal price function
is defined based on the real value of Uk and Lk . We vary
the value of Uk /Lk , and use the estimated values of Uk
as the input of Aonline1 , to examine the performance. As
shown in Fig. 4, there is a downward trend as the value of
Uk /Lk decreases, while there is no large difference with either
underestimation and overestimation. The observation confirms
the analysis in Theorem 3 that the value of Uk /Lk determines
the competitive ratio. Underestimation is more desirable than
the overestimation, as compared to that achieved by the real Uk
(labelled by 100%). Overestimation makes the price rise more
rapidly, filtering out users that are supposed to be accepted.
We next study the social welfare achieved by Aonline1 in
Fig. 5 and Fig. 6. The 3d figure in Fig. 5 plots the social
welfare under different number of users and bids per user. Our
online auction Aonline1 achieves a higher social welfare when
there is larger number of users participating the auction. The
change of bids per user doesn’t have major influence on the
social welfare. When the number of users grows, the number
of bids with larger bidding price also increases. As a result,
Aonline returns a larger social welfare. The social welfare
under different number of slots and Uk /Lk is illustrated in
2
1.6
1.61.6
1.6
8000
8000
80008000
8000
1.4
1.4
1.41.4
1.4
1.2
1.21.2 1.2
1.2
1
1 1 1
10 10 20 20 3030 4040 5050
60
1 10
30 30U40/L40 50 50 60 6060
101020 20 20
50
60
kU k /L
k k 40
U /L30
0.5
0.5
0.5
40
50
70 70
80 80
90 90
100 100
0.5
40 60
50
0.5
40
6060 60
7070 80
9090 100
40 50
50Number
80
of Users
of
40
50
60 Number
70
80Users
90 100
100
Number
of
Users
Number of Users
k
Number of Users
Ukk /L
Uk /L
k
k
6000
6000
60006000
6000
4000
4000
40004000
4000
2000
2000
20002000
2000
0
0
0 120
0
120
0
120 120
80
10 10
10 8 10
801208080
8 6 88
40
10
6 4 4 66
8
40 80 40
2
4
40
Number
ofUsers
Users40 00 20 0 2 4
J4J 6
Number
of
J
Number
of
Users
0
2
0
Number
of Users
0 0
2 J
J
Number
of Users
0
0
Provider
Revenue
Provider
Revenue
Provider Revenue
Competitive Ratio
1.6
Social Welfare
J=2
J=2
J=2
J=4
J=2
J=2J=4
J=4
J=6
J=4
J=4J=6
J=6
J=6
J=6
1
Competitive Ratio
1
111
Competitive Ratio
1.5
1.5
1.5
1.5
1.5
50%
50%
100%
50%
50%
50%
100%
150%
100%
100%
100%
150%
150%
150%
150%
Social Welfare
1.8
1.8
1.81.8
1.8
Social Welfare
SocialWelfare
Welfare
Social
2 2 22
2
Competitive
Ratio
Competitive
Ratio
Competitive Ratio
2
222
Competitive Ratio
Competitive
Ratio
Competitive
Ratio
Competitive Ratio
Ratio
Competitive
11
1.8
Fig.
1:
Competitive
ratioratio
of of
Aonline1
Fig.
Social
welfare
ofAA
with
Fig.
ratio
ofofonline1
AAonline1
Fig.
1: Competitive
Aonline1
3: Social
Social
welfare
with
Fig.
2:Competitive
Competitive
ratio
online1
online1
Fig.
5:
welfare
ofonline1
online1
Fig.
1:
Competitive
of
Fig.Fig.
3: Social
welfare
of A
withwith
Fig.
2:2:
ratio
of of
online1
Fig.
4:Competitive
Competitive
ratio
A
Fig.
3:
ratio
ofof A
AA
Fig. 1:
1: Competitive
Competitive ratio
ratio
online1
3:
Social
welfare
of
Aof
2:
Competitive
ratio
ofAof
Aonline1
online1 Fig.
online1
online1
Fig.
Competitive
ratio
of
A
Fig.
3:
Social
welfare
A
with
Fig.
2:
Competitive
ratio
A
online1
with
different
number
of
users
and
J.
with
different
number
of
users
and
J.
online1
number
of
different
number
of
users
and
J.
online1
with
different
U
/L
.
with
different
U
/L
.
different
number
of
users
and
J.
k
k
k
k
with
different
number
of
users
and
J.
different
number
of
users
and
J.
with
different
U
/L
.
with different
different kUk /L
k k ..
with
different
users and
J.
withdifferent
differentnumber
numberofof
different
number
of users
and J.
k .
with
number
of users
users and
and J.
J. with
different
number
of
users
and
J.
2
with differentUkU/L
/L
2
k
k
k
UUk /L/Lk =10
=10
U k /L k =10
k k
UU
/L/L=10
=30
=30
kUk k/L
U k /L k =30
k kU
k /L =10
UU
/L
=30
/L
=50
k k
kUk k/L
U k /L k =50
k =50
k U
k /L =30
U k /L k =50
k k
U k /L k150
=50
Provider Revenue
k
Social Welfare
Competitive Ratio
100
Social Welfare
Percentage of Winners
Percentage of Winners
k
PercentageofofWinners
Winners
Percentage
Percentage of Winners
k
U k/L
=10
U kk/L
UU
/L k=20
=20 k=20 9090
10090
k k/L
U k/Lkk=20
90
U /L
=30 80 90
80
/L
=20
U /LU=30
5
/L k=10
UU/L
=10
k =10
1.5 U k/L U
k k k
/L k/L
=10
=30
4
U k/L k=30
UkU/L
k =30
k
k Uk /L =10
U kU
/L
=30
/L
k =50
k
U k/L k=50
kk =50
k
6000
U /L
k U=50
k /L =30 6000 6000
U /L
3
k k
6000
Social Welfare
Social
SocialWelfare
Welfare
Competitive Ratio
110110
110
110
U=10
/L =10
UU
/L/L=10
k k
100
k kk k
110
100
100
U /L =10
Social Welfare
Social
Welfare
Social
Welfare
Social
Welfare
Social
Welfare
7000
7000 7000
7000
7000
6000
6000 6000
6000
5000 5000
5000
6000
5000
4000 4000
4000
5000
4000
3000 3000
3000
4000
3000
2000 2000
2000
3000
2000
1000 1000
1000
2000
1000
50%
1.6
150
150
1501.4
U k /L
U kk=10
/L k =10
U k /L k =10
100%
UUkk/L
/L
=30
Ukkk=10
/L k =30
U k /L k =30
150%
U k /L k =10
UUk /L
=30
/L
=50
U k /L k =50 k Ukk /L =50
k k
U k /L k =50U k /L k =30
U k /L k =50
Fig.
Competitive
Users801: 90
50Number
60 of70
100
ratio of A
Provider Revenue
1
J=2
U k/L k=50
k k
U
80
150
k k/L
k k =30
6000
4000
U k/Lkk=30
80
1.2
100 100
J=4
4000 4000
100
2
70
70
U
/L
=40
100
4000
U=40
/L kk=30
80
J=6
UU
/L
70
k
k
k /L
k =40
70
Ukk/Lkk=40
4000
0.5
50 1 100
2000 2000
50
1
60 7060
30
40
40
50
60
702000
90 U 100
’=3
Uk’=5050
Uk10
’=100 20
U k/L
=50
60
/L
=40
U k/LUk=50
2000 80
k
50
60
k k
U k/Lkk=50
Uk /L k
Number of Users
50
U k/L k=50
50
0
0
0
2000
0
50
0
60
40
50
60
70
80
90
100
0
50 40
50
60
70
80
90
100
0 120 120140 140160 160180 180200 200
12
14 0
16 0
18 70 20 90
50
1000
70
90
110
U k/L k=50
0
50 0 050
70 10
90
110
40
50
60
70 of 80
80 90
90 100
100
50 0 50
70
90
110 110
Users
120
160
180
40
50Number
60Number
of 70
Users
Number
of
Slots 200
90Number
110of Users
70 of90
90 110110
120 140
140
160
180
200
Number
of Slots
of
Number
Users
5050Number
70
90Users
110
50
Number
of70
Users
5050
Number of Users
Number
of70Users
0
Number of Slots
2:Number
0Number ofonline1
0 Competitive
Users Fig. Number
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Number
of
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50
70Users
90
110
50 of70
90
110
Number
of
Users
with
different
number
of
users
and
J.
with
different
Ukof/L
k.
ofwelfare
Slots of A
4: Number
Social
Aonline1
Fig.
5:
Percentage
ofwinners
winners
Fig. Fig.
6:
welfare
withwith
Number
of Users
Number
Users
Fig.
Percentage
AA
Fig.
8:
Competitive
ratio
Aonline2
with
online1
online1
4: Social
of Aofonline1
with
Fig.
5:7:Percentage
of of
winners
in in
Ain
online1
Fig.
6: Social
welfare
and of
cloud
provider’s
online1
Fig.
6:
Social
welfare
and
cloud
provider’s
Fig.
4:4:Social
ofof AAonline1
Fig.
5:
Percentage
of winners
winners
in AAonline1
online1Fig.
Fig.different
Social
welfare
with
Fig.
5:
Percentage
of
in
0
Fig.
6:Social
Social
welfare
andcloud
online1 with
T welfare
and
Ukkk/L
with
different
and
different
T and
U
6:in
welfare
and
with
different
I Iand
k
k /L
different
I A
and
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with
different
I and
U
/L
k /L
kU/L
k k
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Icloud
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kU
k
online2
k /Lk
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revenue
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I and
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110 Uk /Lk
different T and U /L
with different I and U /L7000
120
140Number
160of Slots
180
200
40
k
k
Percentage of Winners
Fig.
4: Social
Fig. different
5: Percentage
winners
in Aonline1
different
T andwelfare
Ukk /Lkk of Aonline1 withwith
I and of
Ukk/L
revenue
withI Iand
and
/L
kk
Fig. 6:ininSocial
welfare
cloud
revenue
AA
UkU/L
online2
kprovider’s
online2
k k
U /L
=10with and
6000
100
different T and Uk /Lk
with different I and Uk /L
k
revenue in Aonline2 with I and Uk /Lk
50
of
U
U
/L k=20UF.’=10
Li,U kand
Lau,90
k
k
k
k
k
Provider Revenue
Social Welfare
k
Percentage of Winners
Social Welfare
Percentage of Winners
Percentage
of Winners
Percentage
of Winners
Percentage
of Winners
Percentage
of Winners
Fig. 6. Both
’=10
U
Shi,Zhang,
L. Zhang,
C. Wu,
“An online
auction
100 number of slots and the value of Uk /Lk
[8] W. Shi, L.
C. U Wu,
Z. Li,Z. and F. Lau, “An online
auction
100 the
U F.
’=30 Lau,
4000
U resource
.=30
framework
for
dynamic
resource
provisioning
in80cloud
computing,”
in
[8]
W.
Shi,for
L.dynamic
Zhang,
C.Wu,
Wu,
Z.Li,Li,
and
“Anonline
onlineauction
framework
provisioning
in
cloud
computing,”
in auction
[8]
W.
Shi,
L.
Zhang,
C.
Z.
and
F.
Lau,
“An
100
influence the
welfare.
A
is
able
to
allocate
more
100social
U
/L
=30
online1
k k
U ’=50
80 80
U ’=50
Proc.
of ACM
SIGMETRICS,
2014.
3000
Proc.
of ACM
SIGMETRICS,
2014.
framework
fordynamic
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Ruiting Zhou received a B.E. degree in telecommunication engineering from Nanjing University of
Post and Telecommunication, China, in 2007, a
M.S. degree in telecommunications from Hong Kong
University of Science and Technology, Hong Kong,
in 2008 and a M.S. degree in computer science
from University of Calgary, Canada, in 2012. Since
March, 2016, she has been a PhD candidate at
the Department of Computer Science, University
of Calgary, Canada. Her research interests include
smart grids, cloud computing and mobile network
Zongpeng Li received his B.E. degree in Computer
Science and Technology from Tsinghua University
(Beijing) in 1999, his M.S. degree in Computer
Science from University of Toronto in 2001, and his
Ph.D. degree in ECE from University of Toronto in
2005. He has been with the University of Calgary
since 2005, where he is now Professor of Computer
Science. In 2011-2012, Zongpeng was a visitor at the
Institute of Network Coding, Chinese University of
Hong Kong. His research interests are in computer
networks, network coding, cloud computing, and
Chuan Wu received her B.Engr. and M.Engr. degrees in 2000 and 2002 from the Department of
Computer Science and Technology, Tsinghua University, and her Ph.D. degree in 2008 from the
Department of Electrical and Computer Engineering,
University of Toronto, Canada. Between 2002 and
2004, She worked in the Information Technology
industry in Singapore. Since September 2008, Chuan
Wu has been with the Department of Computer
Science at the University of Hong Kong, where she
is currently an Associate Professor. Her research is
in the areas of cloud computing, data center networking, online and mobile
social networks, and mobile wireless networks. She was the co-recipient of
the best paper awards of HotPOST 2012 and ACM e-Energy 2016.
Zhiyi Huang is an assistant professor of Computer
Science at the University of Hong Kong since 2014.
He got his PhD under Sampath Kannan and Aaron
Roth from Penn in 2013 and worked as a postdoc
with Tim Roughgarden from 2013-2014. His research interest is in the area of theoretical computer
science, including in particular algorithmic game
theory, differential privacy, and online algorithms.