Online Mechanism Design for VMs allocation inPrivate Cloud

Xiaohong Wu1, Yonggen Gu1, Guoqiang Li2,∗, Jie Tao1, Jingyu Chen3, andXiaolong Ma4

1 School of Information and Engineering, Huzhou University, Zhejiang, 313000, China2 School of Software, Shanghai Jiao Tong University, Shanghai, China

3 Institute of Computer Application Technology, Hangzhou Dianzi University,Zhejiang, China

4 School of Information Management and Engineering, Shanghai University ofFinance and Economics, Shanghai, China

Abstract. Resource allocation mechanism plays a critical role towardsthe success of cloud computing. Existing allocation mechanisms in pub-lic cloud is unsuitable to private IaaS cloud because they either can-not maximize the sum of users value, or provide no service guarantee.For overcome these shortcomings, we propose a novel online, model-freemechanism that makes different allocations for flexible jobs and inflex-ible jobs. Users presenting job are incentivized to be truthful not onlyabout their valuations for VM units, but also about their arrival, de-parture and the character of jobs(flexible or inflexible). We simulate theproposed online mechanism using data from RICC, showing that usingour mechanism leads to high social welfare and percentage of served usercompared with the mechanism which adopts same allocation method forall jobs.

Keywords: mechanism design, incentive compatible, resource reserva-tion, greedy allocation

1 Introduction

With the rapid development of cloud computing technology, Infrastructure-as-a-Service has gained popularity in recent years due to the flexibility, scalabilityand reliability. For a private IaaS cloud, the objective of resource allocation isto maximize the efficiency of resources. That is, the IaaS private cloud providerneeds to find an optimum resource allocation for all users in order to maximizethe social welfare which is the sum of users value.

Existing allocation mechanisms in public IaaS cloud are pay-as-you-go andbid-based allocation. Pay-as-you-go is a first-come first-serve allocation mecha-nism where the user is not concerned about the value of an allocation. In fact,the efficiency of an allocation can be improved if the cloud allocates VMs tousers with higher valuation by knowing user-centric valuation. Amazon [1] hasused bid-based mechanism in spot instance market to make up for this short-coming, where users periodically submit bids to the provider, who in turn postsa series of spot prices. Users gain resource access, until the spot price rises above

their bids, at which time they are rejected. So, due to the dynamic changingof the spot price, it provide no service guarantee to those jobs which should becompleted by VMs during multiple time units period.

To overcome above shortcomings in existing mechanisms, we are motivated todesign a new auction mechanism for VM allocation in private IaaS cloud which ispresented as an online greedy allocation with reservation(OGAWR) mechanism.The OGAWR mechanism has three characteristics compared with the auction inspot instance. First, the auction in our online mechanism is carried out in eachtime unit as long as user comes, while the auction in spot instance is carriedout in each period that includes multiple time units. Second, OGAWR mecha-nism will provide service guarantee, that is, each job which should be processedduring multiple time units will not be terminated before it is completed. Third,especially, we use different allocation methods of VMs for the flexible jobs andinflexible jobs. Flexible job refers to those jobs that the users only care aboutwhether they could be completed before their deadline, and the process detailsare ignored. For example, a finance firm has to process the daily stock exchangedata for guaranteeing the trading in next day. Obviously, the finance firm onlycares about whether the job can be finished before the deadline of an hour beforethe next trading day, and does not care about how the job is processed. In con-trast with flexible job, inflexible job refers to those jobs that must be processedcontinuously when they start to be processed.

The rest of paper is organized as follows. In section 2 we discuss relatedwork. After formalizing the problem model in Section 3, in section 4 we designan online greedy allocation with reservation(OGAWR) mechanism, and analyzethe properties of OGAWR mechanism in section 5. The experiment evaluationare showed in section 6. Finally, conclusions appear in section 7.

2 Related Work

The resources allocation in cloud computing is an important topic because it isclosely related to the revenue of both cloud users and providers. Many literatureshave conducted the studies focusing on this topic, and there are two main linesof research for this problem. one of these investigates the VMs allocation bysolving an optimization problem. These works focus on the optimization of objectfunctions, but generally without considering any strategic behaviours amongusers (e.g., the VM allocation approach for spot markets in paper [9]). The otheris game theory based approach to analyze and design a reasonable mechanism.For instance, a cloud resource allocation approach based on game theory [7] isproposed, and assume that the allocation would start after all users submit theirrequest. Combinatorial auctions are supposed to apply in VMs allocation in someliteratures [8] [10] [11], and all these works only consider resource allocations inone time unit and restrict their discussions in a single offline auction period.However, in cloud computing, the cloud users arrive and leave randomly, so thestatistic analysis and design based game theory is not suitable for it.

Online mechanism is an important expansion of mechanism theory in themulti-agent and economics literature, generally applied in dynamic environment,which is consistent with the environment characters in cloud computing. Accord-ing to the research on online mechanism, there are two frameworks of research

in this field. One of these is model-based approach which aims for developing on-line variants of Vickrey- Clarke-Groves (VCG) mechanisms [3] [4]. These worksrely on a model of future availability, as well as future supply (e.g., Parkes andSingh [3] use an MDP-type framework for predicting future arrivals). The otheris model-free approach which requires fewer assumptions, and makes comput-ing allocations more tractable than the first one (e.g., the online mechanism inelectric vehicle charging in [5]).

The online mechanisms have been used in cloud computing [12] [13] [14]. [14]only introduces an online mechanism framework for cloud resources allocationwithout detail allocation algorithm. The online mechanism in [12] is a resourceallocation approach for batch jobs, and the value functions for users is contin-uous. Zaman et al [13] design a truthful mechanism that allocates the VMs tousers by greedy allocation, and allows those allocated users continuously usethose VMs for the entire period requested.

Based on those works in [12] [13] [14], we also aims to design an online truthfulmechanism for VMs allocation, but pay attention to the following points:

1. In our model, a user requests one VM for multiple time units to finish the jobduring the arrival-departure interval. According to the demand in processtime, all jobs are classified into two classes: flexible jobs and inflexible jobs.

2. We choose different allocation methods for the two classes of jobs, especiallydesign a discontinuous resource allocation based on reservation-ratio for theflexible jobs, by which the distribution of workload of users can be adjustedat their arrival-departure interval and the total workloads processed in cloudwill be improved.

3. We focus on all the users with single-valued preference. That is, each usercould get a non-zero constant value brought by the job only if it could befinished completely.

3 Modeling and Notations

We consider a private cloud provider who provides only one type of VM instances,and the total number of VM instances is denoted by C. Consider discrete timeperiods T = 1, 2, ..., indexed by t and possible infinite.

An agent presents a user i who submit their jobs to the cloud randomly,which can be characterized by the ’type’ θi = (ai, di, li, ei, Vi) ∈ Θi, where Θi isits type space. Here, ai and di present the arrival and departure time of agenti, and li is its total computation workload, i.e, the job size. Assume that eachagent requires at most one VM in one time unit. The workload li is the numberof time units that agent i must be allocated one VM. The last component of θiis Vi, the value agent i obtains if its job is completed, and Vi ≥ 0.

As described in section 1, the jobs are classified into flexible jobs and inflexiblejobs. In order to distinguish the job classes, a character parameter ei points outthe agent is flexible (ei = 1) or inflexible (ei = 0).

We define πi = (πaii , π

ai+1i , ..., πdi

i ) as the allocation for agent i. πti = 1 if

agent i is allocated one VM at time t ∈ [ai, di], otherwise πti = 0.

The allocation result for agent i is denoted by Ai.

Ai(πi) =

{1 if Σt∈[ai,di]π

ti ≥ li and πt

i ≤ 1

0 otherwise(1)

Each agent i is characterized by a valuation function vi defined as follows:

vi =

{Vi if Ai = 1

0 otherwise(2)

The challenge of the cloud provider is to make allocation decisions πt dy-namically while trying to maximize the sum of agents value. The problem isdescribed as follows:

maxΣvi

s.t

πti ≤ 1

Σni=1π

ti ≤ C, t ∈ T

(3)

Table I lists main symbols defined in this paper.

Table 1. Symbols definition

Symbol Definitionai arrival time of agent idi departure time of agent ili job size of agent iei character parameter of agent iθi type of agent, θi = (ai, di, li, ei, Vi)θ type set of all agents, θi ∈ θθt type set of agents which participate the allocation at t

πi allocation of agent i, πi = (πaii , π

ai+1i , ..., πdi

i )πt allocation decision at time t, i ∈ πt if i is allocated at tπt−i allocation decision at time t if agent i were not present

before time t(including time t)A allocation results, A = (A1, A2, ..., An)Ai allocation result of agent i. If i is allocated, Ai = 1St supply about future period, St = (s(t), s(t+ 1), ...)St−i supply about future period if agent i were not present

before time t(including time t)s(t) supply at time tr(t) reservation-ratio at time t, r(t) = s(t)/Cpi payment of agent i

4 The Online Greedy Allocation with ReservationMechanism

4.1 Description of Mechanism

In this section we design a model-free online mechanism for the above setting.Denote the number of idle VMs at time t by s(t). The definition of greedy

allocation is as follows.

Definition 41 (Greedy allocation) At each step t allocate the s(t) VMs to theactive agents with the highest valuations.

If all agents request one VM only for one time unit, greedy allocation withappropriate payment could constitute a truthful mechanism [2]. However, in thecase of multiple time units demands, according to equation (2), whether agenti could get the value Vi is decided by all of its allocation in period [ai, di]. Thatis, the value brought by one VM at some time unit cannot be decided at first, sogreedy allocation for each time unit can not be performed. In order to maintainincentive compatibility, we extend the greedy allocation policy by allowing thesystem to reserve VMs for agents. By such allocation approach, the agent is notonly allocated one VM at current time t but also reserved one VM for multipletime units in future period. Define unit valuation as the valuation of one VMper unit time, and it is expressed as Vi/li to each agent i.

Definition 42 (Online greedy allocation with reservation(OGAWR)) At eachstep t allocate the s(t) VMs to the active agents with the highest unit valuations,at the same time, make the VM reservation for allocated agent i during period[t+ 1, di] if li > 1.

Fig. 1. An example for multi-time unit demand(C = 1)

Consider an example with 3 time units and 3 agents in Fig. 1, where θ1 =(1, 3, 2, 1, 10), θ2 = (1, 3, 2, 1, 8), θ3 = (2, 3, 1, 1, 3) showing the agents arrival,departure, job size, job class and valuation. Suppose furthermore that C = 1.Sort the agents by their unit valuation Vi/li. Because agent 1 has the highest

unit valuation at time 1, OGAWR method would allocate the VM to agent 1,and reserve one VM for it. Since it is a flexible job (e1 = 1), there are two choicesfor reserving: reserving at time 2 or at time 3. In Fig. 1, the VM in time unit 2is chosen to reserve for agent 1, so there is no idle VM to be allocated at time2. At time 3, although agent 2 has higher unit valuation than agent 3, the VMis still allocated to agent 3, because agent 2 has no sufficient time to finish thejob at that time.

It is worth to note that OGAWR might not be performed in some cases.That is, an agent with highest unit valuation cannot be allocated although thereis sufficient time from departure for process. In the above example, if the VM intime unit 3 is reserved for agent 1 at time 1 . At time 2, although agent 2 hashigher unit valuation than agent 3 and there is sufficient time to process, agent2 still cannot be allocated. The reason for this result is that the supply in futureis less than that of current time. Therefore, the OGAWR can be realized only ifit makes ’non-increasing reserving’.

Definition 43 (non-increasing reserving) Non-increasing reserving refers to aclass of reserving schemes which always satisfies s(t) ≤ s(t+ 1) ≤ s(t+ 2) ≤ ...after allocation at each time t.

In OGAWR mechanism, the agent participating the allocation at time t sat-isfies three conditions:(1)It arrives before time t. (2)Its departure time is longerthan t + li − 1. (3)It is still unallocated. The OGAWR mechanism consists ofallocation rule and payment rule described as follows.

– Allocation rule At each time t, it makes allocation as follows.Stage 1 Greedy allocation: Allocate the s(t) VMs using greedy allocation,breaking ties at random.Stage 2 Non-increasing reservation: Make non-increasing reservation for a-gents who are allocated in stage 1 if necessary. If one VM is reserved foragent i at time k, πk

i = 1.Let θt = (θ1, θ2, ..., θn) denote the set of agent types participating the alloca-tion at time t, and πt denotes the decision policy at time t. The mechanismmakes a sequence of allocation decisions (π1, π2, ...), and πt includes all thoseagents allocated at time t. For example, in Fig. 1, π1 = (1, 1, 0), π1 = {1}and A1 = 1 after allocation at time 1, and it is not changed at time 2. Afterallocation at time 3, π3 = (0, 1), π3 = {3}, A3 = 1.

– Payment ruleWe design a critical payment which is equal to the critical value for allocatedagents, and the definition of critical value is as follows.Given type θi = (ai, di, li, ei, Vi), the critical value for agent i is defined as

V c(ai,di,li,ei)

(θ−i) =

minV ′i s.t. Ai(θ

′i, θ−i) = 1,

for θ′i = (ai, di, V′i )

∞ no such V ′i exists

(4)

where θ−i = (θ1, θ2, ..., θi−1, θi+1, ...).

We define payment policy pi(θ) as

pi(θ) =

{V c(ai,di,li,ei)

(θ−i) if Ai = 1

0 otherwise(5)

4.2 The Algorithm Design of OGAWR Mechanism

In this section, the algorithm based on the proposed rules for allocation andpayment is designed. First, we introduce two reserving methods for inflexibleagents and flexible agents respectively.

Continuous reserving : Continuous reserving is suitable for inflexible agents,which is similar to the allocation in MOVMPA mechanism proposed in paper[13]. If agent i wins the auction at time t, one VM will be reserved continuouslyfor it in next li − 1 units. That is, πk

i = 1, k = t+ 1, ..., t+ li − 1, if πti = 1.

Discontinuous reserving based on reservation-ratio(Discontinuous reserving):This reserving method reserves one VM for agent i in next li−1 time units withlowest reservation-ratio, and reserve the VM in earliest time unit if there aremultiple time units with same reservation-ratio.

Reservation ratio denoted by r(k) is the ratio of the number of reserved VMsand total capacity C at future time k expressed as r(k) = s(k)/C. Obviously,r(k) is changed with time.

Fig. 2. An example for continuous allocation(C = 2)

For inflexible agents, continuous reserving and discontinuous reserving bothcould be used. In the next example, we compare the allocation results by usingabove two reserving methods respectively. Consider the example with 3 time u-nits and 3 agents in Fig. 2 and 3, where θ1 = (1, 3, 2, 0, 10), θ2 = (1, 3, 2, 1, 8), θ3 =(2, 3, 2, 1, 3), and C = 2. Fig. 2 shows the results by continuous reserving for allagents. At time 1, agent 1 and agent 2 are allocated and reserved VMs con-tinuously. At time 2 there is no idle VMs because two VMs are both reservedat time 1, so agent 3 cannot be allocated. At time 3, agent 3 still cannot beallocated because it has no sufficient time from departure to complete. Fig. 3shows another results by discontinuous reserving. At time 1, agent 1 and agent 2all be allocated one VM. Then it reserves VM at time 2 for agent 1 and reservesVM at time 3 for agent 2 based on reservation-ratio. We can see all agents could

Fig. 3. An example for noncontinuous allocation by reservation-ratio(C = 2)

be allocated by this method, and the sum of users value is higher than that ofcontinuous reserving. In our mechanism, we choose discontinuous reserving forinflexible agents. The steps of the allocation algorithm at time t are as follows:

Step 1 Sort all agents which participate the allocation at time t in non-increasingorder of Vi/li.

Step 2 Allocate s(t) idle VMs to s(t) agents with highest valuation, breakingties at random.

Step 3 Choose a suitable reserving method for each agent allocated at step2. Continuous reserving is chosen if ei = 0, and discontinuous reserving ischosen if ei = 1.

Table 2. Allocation algorithm: Allocate

Input: θt, St, tOutput: St, πt, A

1: if s(t) = 0, goto end2: sort all θt ⊆ Θ in non-increasing order of Vi/li3: (πt, A) = greedyallocate(θt, St)4: sort all i ∈ πt in non-decreasing order of di5: for each i ∈ πt

6: if ei = 17: (πi, S

t) = DiscontinuousReserve(li − 1)8: else9: (πi, S

t) = ContinuousReserve(li − 1)10: end if11: end for12: St+1 ← St \ {s(t)}13: end

Define a status vector St = (s(t), s(t + 1), s(t + 2), ..s(t + m − 1)) as theVM supplies in period [t, t+m− 1] before allocation at time t, where s(t+ k) isdenoted as the supply at future time (t+k) ∈ T , and m satisfies s(t+m−1) < C

and s(t + m) = C. For computing critical value, we define v(m)−i,t to be the mth

highest of unit valuations Vj/lj from all agents j in θt, j 6= i. Then vs(t)−i,t , for

supply s(t), is the lowest value that is still allocated a unit, if agent i were not

present not only at current t but also before t. Henceforth, we refer to v(s(t))−i,t as

the marginal clearing value of the idle VM for agent i at time t.OGAWR Mechanism runs in each time unit t, the algorithm is described as

follows:

Step 1 According to the allocation rule of OGAWR, the allocation is performedbased on St, θt, which generates an allocation set πt and a new status setSt+1 , and updates allocation result A.

Step 2 For each agent i who got its first unit at step 1, the critical value for

agent i at time t is computed as equation vci,t = vs(t)−i,t. Then, we execute

a suppositional allocation in which i is not present, and get suppositionalresults πt

−i and St+1−i .

Step 3 For each i who got the allocation before time t and t ≤ di − li + 1,according to the suppositional result St

−i which suppose that i had been not

present before t, we compute the critical value for agent i as vci,t = vs−i(t)−i,t .

We also execute a suppositional allocation in which i is not present basedon the suppositional status St

−i, and get suppositional results πt−i and St+1

−i .Step 4 For each i who satisfies t = di − li, the payment pi is computed. If

Ai = 0, the payment pi is zero, otherwise the payment can be computed as

pi = ( mint∈[ai,di−li+1]

vci,t) · li

Lemma 41 The payment in above algorithm is a critical payment. That is,( mint∈(ai,di−li+1)

vci,t) · li = V c(ai,di,Vi,li,ei)

(θ−i) for each allocated agent i.

Proof. Follows directly from the definition of critical value, expressed by equation(4).

5 Analysis of OGAWR Mechanism

We assume no early-arrival no late-departure misreports with ai ≤ a′i ≤ d′i ≤ di,because generally agent i don’t know its type until ai and the value of agent willbe zero if it is finished after di. We also assume no less job size misreports withl′i ≥ li because the agent i will have no sufficient time to process if l′i < li.

Definition 51 (Monotonic with resource demand) An allocation policy is mono-tonic with resource demand li if for any arrival-departure interval [ai, di], anyvaluation Vi and any job size report l′i ≥ li , we have Ai(ai, di, Vi, l

′i) = 1 ⇒

Ai(ai, di, Vi, li) = 1.

Definition 52 (Monotonic with arrival-departure interval) An allocation policyis monotonic with arrival-departure time if for any job size li, any valuationVi and any any arrival-departure time report a′i ≥ ai and d′i ≤ di , we haveAi(a

′i, d′i, Vi, li) = 1⇒ Ai(ai, di, Vi, li) = 1.

Table 3. Mechanism algorithm: OGAWR

Input: t, θt = {θ1, θ2, ..., θn}, St

Output: St+1, πt, pi, A1: πt = ø2: if s(t) = 0, goto end3: (St+1, πt, A)=Allocate(θt, St, t)4: for each πi ∈ πt do

5: vci,t = vs(t)−i,t

6: (St+1−i , π

t−i, A−i)=Allocate(θ

t−i, S

t, t)7: end for8: for each i /∈ πtand Ai = 1 and t ≤ di − li do

9: vci,t = vs−i(t)

−i,t

10: (St+1−i , π

t−i, A−i) = Allocate(θt−i, S

t−1−i , t)

11: end for12: for each i: t = di − li + 113: if Ai = 0, pi = 014: else pi = min

t∈[ai,di−li+1]vci,t · li

15: end for16: end

Lemma 51 The allocation policy in OGAWR mechanism is monotonic withresource demand and arrival-departure interval.

Proof. First, we prove the monotonicity with resource demand li.Assume for contradiction that l′i ≥ li and Ai(ai, di, l

′i) = 1, but Ai(ai, di, li) =

0. Because Ai(ai, di, Vi, l′i) = 1, it must satisfy Vi ≥ V c

(ai,di,Vi,l′i,ei)(θ−i). Since

V c(ai,di,Vi,l′i,ei)

(θ−i) = mint∈[ai,di−l′i+1]

vci,t · l′i, it follows that Vi/l′i ≥ min

t∈[ai,di−l′i+1]vci,t.

On the other side, because Ai(ai, di, Vi, li) = 0, it must satisfy Vi < V c(ai,di,Vi,li,ei)

(θ−i). Thus it follows that Vi/li < mint∈[ai,di−li+1]

vci,t.

Since [ai, di − l′i + 1] ⊆ [ai, di − li + 1] for l′i ≥ li,

mint∈[ai,di−li+1]

vci,t ≤ mint∈[ai,di−l′i+1]

vci,t.

So,Vi/li < min

t∈[ai,di−li+1]vci,t ≤ min

t∈[ai,di−l′i+1]vci,t ≤ Vi/l′i.

Then it must satisfy l′i < li. That is a contradiction with assumption.Second, we prove the monotonicity on arrival-departure interval. we say that

an arrival-departure interval [ai, di] is tighter than [a′i, d′i] if a′i ≥ ai and d′i ≤ di.

Also assume for contradiction that [ai, di] is tighter than [a′i, d′i], andAi(a

′i, d′i,

Vi, li)= 1, butAi(ai, di, Vi, li)= 0. BecauseAi(a′i, d′i, Vi, l

′i)= 1, Vi ≥ V c

(a′i,d′i,Vi,li,ei)

(θ−i). According to the computation of critical value,

V c(ai,di,Vi,li,ei)

(θ−i) ≤ V c(a′i,d

′i,Vi,li,ei)

(θ−i),

so Ai(ai, di, Vi, li) = 1 and a contradiction with the assumption.

Next, we discuss about whether an agent would get more utility by misreportei. First, an inflexible agent would not misreport ei = 1 because discontinuousallocation for this class job will cause zero value. Second, we find there is nodifference in allocation and payment to flexible agent between reporting ei = 1and reporting ei = 0. For allocation, due to the greedy allocation and non-increasing reserving, whether an agent can be allocated is only decided by theorder of its valuation and not related with ei. For payment, according to thecritical value equation(4), the critical value of agent i would not changes whenei changes, and the payment of the agent is equal to the critical value which isalso not related to ei. We assume that each agent is rational, that is, the agentwill choose to report true type when misreport cannot improve its utility.

Theorem 51 The OGAWR mechanism is incentive compatible with no-earlyarrival, no-late departure misreports and no less job size misreports.

According to the lemma 51, given valuation report Vi, the allocation policyis monotonic with resource demand and arrival-departure time, so agents willreport true arrival-departure interval and job size with no-early arrival, no-late departure misreports and no less job size misreports. Assume agent i istruthful. Next, we prove by case analysis that the agent i would get more u-tility if it is truthful to report the valuation Vi. (a) If agent i is not allocated,V c(ai,di,Vi,li,ei)

(θ−i) > Vi and to be allocated, the agent must report some V ′i > Vi.

But since the critical value is greater than the true valuation Vi, it will cause tonegative utility if i wins for V ′i .

ui = Vi − V c(ai,di,Vi,li,ei)

(θ−i) < 0.

(b)If agent i is allocated, the utility is nonnegative since V c(ai,di,Vi,li,ei)

(θ−i) ≤ Vi,and agent i does not want to report a lower valuation V ′i for which it wouldnot be allocated. Consider any V ′i for which agent i continue to allocated. Thecritical value is not changed since it is independent of the reported valuation V ′i .Therefore, OGAWR is incentive compatible.

5.1 Competitive analysis

We define the competitive ratio on social welfare as follows. An auction mecha-nism M is c− competitive with respect to the social welfare if for every biddingsequence θ , EM (θ) ≥ Eopt(θ)/c. Accordingly, c is the competitive ratio of M.Where, EM is the sum of agents value in mechanism M, and Eopt denotes thesum of agents value by the optimal algorithm.

Assume that VM to all agents has a same maximal unit valuation vmax andsame minimal unit valuation vmin, i.e, vi ∈ [vmin, vmax]. Define N = vmax

vmin. At

the same time, we assume the maximal job size is L and L ≥ 2 .For example, consider the case C = 1, and three bids θ1 = (1, L+1, 2, 1, 2(vmin+

ε)) , θ2 = (2, L+ 1, L, 1, L · vmax) and θ3 = (1, 1, 1, 1, vmin) where ε is a positivequantity, and vmin − ε > 0. In OGAWR agent 1 would be allocated one VM at

time 1, and reserved one VM at time 2 because it has highest unit valuation atthat time. But this reserving makes agent 2 lose the allocation, although agent2 has higher unit valuation. Because li = L for θ2, agent 2 should get the allo-cation at time 2, otherwise it has no sufficient time to process. Therefore, VMsis only allocated to agent 1 and get the total value 2(vmin + ε) in OGAWR. Onthe other hand, an optimum offline algorithm will allocate resources to agent 2and agent 3, and get the total value (L · vmax + vmin)/(2(vmin + ε)).

Theorem 52 OGAWR mechanism has a competitive ratio on social welfareN ·(L+1)

2 if C = 1.

Proof. Consider a period of time [t, t+L] ∈ T . Obviously, the potential maximalvalue of allocation is vmax · L. Now, we consider the potential worst case withminimal value of allocation. The worst case is that the VM during L time units isonly reserved in one time unit, and due to the reserving, VMs during other L−1units could not be allocated. The potential minimal value of allocated agentsin [t, t + L] is vmin. Moreover, if the minimal valuation is vmin, it must exitsanother time unit t1 before time t that the agent gets allocation and reservationby bids vmin.

Now, we consider the efficiency during the time period: t1 ∪ [t, t + L]. Fromabove analysis, we can compute the minimal value by our allocation in t1∪ [t, t+L], i.e.,

minEOGAWR(θ) = 2 · vmin.

The potential maximal value of allocation in t1 ∪ [t, t+ L] is

maxEopt(θ) = vmax · (L+ 1).

So, EOGAWR(θ) ≥ Eopt(θ)/c, where c = N ·(L+1)2

Consider there are C VMs. We can easily deduce that the competitive ratio

is C · N ·(L+1)2 .

6 Evaluation and Simulation

As analysed above, the competitive ratio c of OGAWR mechanism might bevery large because it is decided by L and N . That is, it may cause to very lowsocial welfare at the worst case. In this section, we will present the simulationresults and compare the OGAWR mechanism with two allocation methods. oneis an offline optimal approach designed under the assumption that we knowall the agents valuation beforehand and completely ignore the allocation timeconstraint in [ai, di]. Although it is not reasonable that OGAWR mechanism iscompared with the offline allocation without time constraint, but we can under-stand the actual level of proposed mechanism on social welfare and percentageof served agents by comparing the curves. The other method compared is a goodonline mechanism (MOVMPA) designed in paper [13]. The main difference be-tween MOVMPA mechanism and OGAWR mechanism is that MOVMPA usescontinuous allocation for each agent.

As same as [13], the input data of the experiments are collected from theParallel Workload Archive [15], where collect many workload logs from largescale parallel systems in various places around the world. We select 10 thousandscontinuous records from log RICC-2010-2. In the log, the minimal time unitrecorded is second. In our experiment, we choose 10 minutes as one time unit,and all records selected are distributed randomly from time 0 to time 8000.Each record corresponds to one task, and the information of a task includesarrival time , wait time, runtime, number of allocated processors, etc. Each taskis processed by at most 8 thousands processors. According to the number ofallocated processors k, a task can be divided into k subtasks each of whichmust be processed serially in one processor. That is, one subtask requests atmost one VM in each time unit which is consistent with the assumption in ourmodel. Let each agent present one subtask(also is one job). After the step of taskdecomposition, there are about 285 thousands agents in these records.

Next, we discuss how the type of agent θ = (ai, di, li, ei, Vi) can be got.First, ai and li can be got from the log, where the real arrival time of therecord is ai and the runtime can be converted to the size li. As described above,if k agents are generated from one same record, they will have same arrivaltime and job size. Second, we produce the other information di and Vi. Assumethat the deadline and the valuation are exponential distribution. Deadline diand valuation Vi/li are computed as di = ai + li + li · exp(davg) and Vi/li =exp(vavg). Finally, the parameter ei is generated randomly. The table II showsthe simulation parameters.

Table 4. Simulation Parameters

Type Notaion Value ParameterArrival time ai form workload archivedeparture time di ai + li + li · exp(davg) davg = 2job size li from workload archivevaluation Vi Vi/li = exp(vavg) vavg = 50job character (flexible) ei 1 or 0, generate randomly

Fig. 4 and 5 shows the distribution of all those records we selected. Fig. 4shows the number of arrival subtasks at each time unit, while Fig. 5 is the sizedistribution of all agents.

Before running of the mechanism, we initialize the supply, the total number ofVMs, which is closely related to the allocation results. Define an initial supplyC0 is equal to average requirement for each time unit, i.e., C0 = Σn

i=1li/|T |,where |T | = 8000 is total time units we select.

Fig. 6 shows the social welfare, the sum of agents value under different C,and C changes continuously from C = 0.5 · C0 to C = 1.5 · C0 . First, we notethat the trends for the two scenarios are different C when supply is low andclose to C = 0.5 · C0, the OGAWR mechanism results only in a small overallimprovement in social welfare, However, when it grow to more than C0, thereis a very marked improvement. Especially, when C = 1.5 ·C0, the social welfare

Fig. 4. Distribution of arrival time Fig. 5. Distribution of job size

Fig. 6. The number of competed jobs underthree mechanisms

Fig. 7. The sum of agents value under threemechanisms

under OGAWR mechanism is very close to it under offline allocation, while itstill keep a low level in MOVMPA mechanism.

With respect to the number of completed jobs of individual agents, the resultsare shown as Fig. 7. The percentage of completed jobs increases with the increaseof supply in all allocation approaches, and in OGAWR, it is obviously higherthan that in MOVMPA when they are in the same supply capacity.

7 Conclusion

In this paper, we propose a online VM allocation mechanism for private IaaScloud resources, where the goal is to improve the social welfare. We constructa online resource allocation model in which jobs are divided into two classes:inflexible jobs and flexible jobs. Then, an online greedy allocation with reser-vation(OGAWR) mechanism under the dynamic cloud environment is designedand proved truthfully. We also performed extensive experiments to observe theresults of the mechanism. The results show that, from the aspects of improvingsocial welfare and the number of completed jobs, OGAWR is better than themechanism which allocate the inflexible agents as well as flexible agents.

For future work we plan to consider several issues. First, in this paper weassumed all agents need only one VM per time unit, but in the future we plan toextend the allocation model to deal with multiple VMs demands per time unit.Second, it would be interesting to design model-based mechanism and compare

the performance with the model-free online mechanism proposed in this paper.Finally, we also plan to study online mechanism design for public IaaS cloud infuture work, where the goal of mechanism is to maximal the revenue of cloudprovider.

8 Acknowledgments

The workload log from the RICC cluster was graciously provided by MotoyoshiKurokawa. This Work was supported by the National Natural Science Founda-tion of China under Grant No. 61170029, Zhejiang Provincial Natural ScienceFoundation of China under Grant No. Y1111000, and Zhejiang Provincial Sci-ence and Technology Plan of China under Grant No. 2013C31097.

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