CLOUD computing is quickly becoming an effective and

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 6, YEAR 203 Otial Multiserver Configuration for Profit Maxiization in Cloud Couting Junwei Cao, Senior Meber, IEEE, Kai Hwang, Fellow, IEEE, Keqin Li, Senior Meber, IEEE, and Albert Y. Zoaya, Fellow, IEEE Abstract As cloud couting becoes ore and ore oular, understanding the econoics of cloud couting becoes critically iortant. To axiize the rofit, a service rovider should understand both service charges and business costs, and how they are deterined by the characteristics of the alications and the configuration of a ultiserver syste. The roble of otial ultiserver configuration for rofit axiization in a cloud couting environent is studied. Our ricing odel takes such factors into considerations as the aount of a service, the workload of an alication environent, the configuration of a ultiserver syste, the service-level agreeent, the satisfaction of a consuer, the quality of a service, the enalty of a low-quality service, the cost of renting, the cost of energy consution, and a service rovider s argin and rofit. Our aroach is to treat a ultiserver syste as an M/M/ queuing odel, such that our otiization roble can be forulated and solved analytically. Two server seed and ower consution odels are considered, naely, the idle-seed odel and the constant-seed odel. The robability density function of the waiting tie of a newly arrived service request is derived. The exected service charge to a service request is calculated. The exected net business gain in one unit of tie is obtained. Nuerical calculations of the otial server size and the otial server seed are deonstrated. Index Ters Cloud couting, ultiserver syste, ricing odel, rofit, queuing odel, resonse tie, server configuration, service charge, service-level agreeent, waiting tie Ç INTRODUCTION CLOUD couting is quickly becoing an effective and efficient way of couting resources and couting services consolidation [0]. By centralized anageent of resources and services, cloud couting delivers hosted services over the Internet, such that accesses to shared hardware, software, databases, inforation, and all resources are rovided to consuers on-deand. Cloud couting is able to rovide the ost cost-effective and energy-efficient way of couting resources anageent and couting services rovision. Cloud couting turns inforation technology into ordinary coodities and utilities by using the ay-er-use ricing odel [3], [5], [8]. However, cloud couting will never be free [8], and understanding the econoics of cloud couting becoes critically iortant. One attractive cloud couting environent is a threetier structure [5], which consists of infrastructure vendors, service roviders, and consuers. The three arties are also called cluster nodes, cluster anagers, and consuers. J. Cao is with the Research Institute of Inforation Technology, Tsinghua National Laboratory for Inforation Science and Technology, Tsinghua University, Beijing 00084, China. E-ail: jcao@tsinghua.edu.cn.. K. Hwang is with the Deartent of Electrical Engineering, University of Southern California, Los Angeles, CA 90089. E-ail: kaihwang@usc.edu.. K. Li is with the Deartent of Couter Science, State University of New York, New Paltz, New York 256. E-ail: lik@newaltz.edu.. A.Y. Zoaya is with the School of Inforation Technologies, University of Sydney, Sydney, NSW 2006, Australia. E-ail: albert.zoaya@sydney.edu.au. Manuscrit received 23 Feb. 202; revised 8 June 202; acceted 28 June 202; ublished online 25 June 202. Recoended for accetance by V.B. Misic, R. Buyya, D. Milojicic, and Y. Cui. For inforation on obtaining rerints of this article, lease send e-ail to: tds@couter.org, and reference IEEECS Log Nuber TPDSSI-202-02-037. Digital Object Identifier no. 0.09/TPDS.202.203. in cluster couting systes [2], and resource roviders, service roviders, and clients in grid couting systes [9]. An infrastructure vendor aintains basic hardware and software facilities. A service rovider rents resources fro the infrastructure vendors, builds aroriate ultiserver systes, and rovides various services to users. A consuer subits a service request to a service rovider, receives the desired result fro the service rovider with certain service-level agreeent, and ays for the service based on the aount of the service and the quality of the service. A service rovider can build different ultiserver systes for different alication doains, such that service requests of different nature are sent to different ultiserver systes. Each ultiserver syste contains ultile servers, and such a ultiserver syste can be devoted to serve one tye of service requests and alications. An alication doain is characterized by two basic features, i.e., the workload of an alication environent and the exected aount of a service. The configuration of a ultiserver syste is characterized by two basic features, i.e., the size of the ultiserver syste (the nuber of servers) and the seed of the ultiserver syste (execution seed of the servers). Like all business, the ricing odel of a service rovider in cloud couting is based on two coonents, naely, the incoe and the cost. For a service rovider, the incoe (i.e., the revenue) is the service charge to users, and the cost is the renting cost lus the utility cost aid to infrastructure vendors. A ricing odel in cloud couting includes any considerations, such as the aount of a service (the requireent of a service), the workload of an alication environent, the configuration (the size and the seed) of a ultiserver syste, the service-level agreeent, the satisfaction of a consuer (the exected service tie), the quality of a service (the task waiting tie and the task resonse tie),

the enalty of a low-quality service, the cost of renting, the cost of energy consution, and a service rovider s argin and rofit. The rofit (i.e., the net business gain) is the incoe inus the cost. To axiize the rofit, a service rovider should understand both service charges and business costs, and in articular, how they are deterined by the characteristics of the alications and the configuration of a ultiserver syste. The service charge to a service request is deterined by two factors, i.e., the exected length of the service and the actual length of the service. The exected length of a service (i.e., the exected service tie) is the execution tie of an alication on a standard server with a baseline or reference seed. Once the baseline seed is set, the exected length of a service is deterined by a service request itself, i.e., the service requireent (aount of service) easured by the nuber of instructions to be executed. The longer (shorter, resectively) the exected length of a service is, the ore (less, resectively) the service charge is. The actual length of a service (i.e., the actual service tie) is the actual execution tie of an alication. The actual length of a service deends on the size of a ultiserver syste, the seed of the servers (which ay be faster or slower than the baseline seed), and the workload of the ultiserver syste. Notice that the actual service tie is a rando variable, which is deterined by the task waiting tie once a ultiserver syste is established. There are any different service erforance etrics in service-level agreeents [2]. Our erforance etric in this aer is the task resonse tie (or the turn around tie), i.e., the tie taken to colete a task, which includes task waiting tie and task execution tie. The service-level agreeent is the roised tie to colete a service, which is a constant ties the exected length of a service. If the actual length of a service is (or, a service request is coleted) within the service-level agreeent, the service will be fully charged. However, if the actual length of a service exceeds the service-level agreeent, the service charge will be reduced. The longer (shorter, resectively) the actual length of a service is, the ore (less, resectively) the reduction of the service charge is. In other words, there is enalty for a service rovider to break a service-level agreeent. If the actual service tie exceeds certain liit (which is service request deendent), a service will be entirely free with no charge. Notice that the service charge of a service request is a rando variable, and we are interested in its exectation. The cost of a service rovider includes two coonents, i.e., the renting cost and the utility cost. The renting cost is roortional to the size of a ultiserver syste, i.e., the nuber of servers. The utility cost is essentially the cost of energy consution and is deterined by both the size and the seed of a ultiserver syste. The faster (slower, resectively) the seed is, the ore (less, resectively) the utility cost is. To calculate the cost of energy consution, we need to establish certain server seed and ower consution odels. To increase the revenue of business, a service rovider can construct and configure a ultiserver syste with any servers of high seed. Since the actual service tie (i.e., the task resonse tie) contains task waiting tie and task execution tie, ore servers reduce the waiting tie and faster servers reduce both waiting tie and execution tie. Hence, a owerful ultiserver syste reduces the enalty of breaking a service-level agreeent and increases the revenue. However, ore servers (i.e., a larger ultiserver syste) increase the cost of facility renting fro the infrastructure vendors and the cost of base ower consution. Furtherore, faster servers increase the cost of energy consution. Such increased cost ay counterweight the gain fro enalty reduction. Therefore, for an alication environent with secific workload which includes the task arrival rate and the average task execution requireent, a service rovider needs to decide an otial ultiserver configuration (i.e., the size and the seed of a ultiserver syste), such that the exected rofit is axiized. In this aer, we study the roble of otial ultiserver configuration for rofit axiization in a cloud couting environent. Our aroach is to treat a ultiserver syste as an M/M/ queuing odel, such that our otiization roble can be forulated and solved analytically. We consider two server seed and ower consution odels, naely, the idle-seed odel and the constant-seed odel. Our ain contributions are as follows. We derive the robability density function (df) of the waiting tie of a newly arrived service request. This result is significant in its own right and is the base of our discussion. We calculate the exected service charge to a service request. Based on these results, we get the exected net business gain in one unit of tie, and obtain the otial server size and the otial server seed nuerically. To the best of our knowledge, there has been no siilar investigation in the literature, although the ethod of otial ulticore server rocessor configuration has been eloyed for other uroses, such as anaging the ower and erforance tradeoff [7]. One related research is user-centric and arket-based and utility-driven resource anageent and task scheduling, which have been considered for cluster couting systes [7], [20], [2] and grid couting systes [4], [2], [9]. To coete and bid for shared couting resources through the use of econoic echaniss such as auctions, a user can secify the value (utility, yield) of a task, i.e., the reward (rice, rofit) of coleting the task. A utility function, which easures the value and iortance of a task as well as a user s tolerance to delay and sensitivity to quality of service, suorts arket-based bidding, negotiation, and adission control. By taking an econoic aroach to roviding service-oriented and utility couting, a service rovider allocates resources and schedules tasks in such a way that the total rofit earned is axiized. Instead of traditional syste-centric erforance otiization such as iniizing the average task resonse tie, the ain concern in such coutational econoy is user-centric erforance otiization, i.e., axiizing the total utility delivered to the users (i.e., the total user-erceived value). 2 AMULTISERVER MODEL Throughout the aer, we use P½eŠ to denote the robability of an event e. For a rando variable x, we use f x ðtþ to reresent the robability density function of x, and F x ðtþ to

reresent the cuulative distribution function (cdf) of x, and x to reresent the exectation of x. A cloud couting service rovider serves users service requests by using a ultiserver syste, which is constructed and aintained by an infrastructure vendor and rented by the service rovider. The architecture detail of the ultiserver syste can be quite flexible. Exales are blade servers and blade centers where each server is a server blade [6], clusters of traditional servers where each server is an ordinary rocessor [7], [20], [2], and ulticore server rocessors where each server is a single core [7]. We will sily call these blades/rocessors/cores as servers. Users (i.e., custoers of a service rovider) subit service requests (i.e., alications and tasks) to a service rovider, and the service rovider serves the requests (i.e., run the alications and erfor the tasks) on a ultiserver syste. Assue that a ultiserver syste S has identical servers. In this aer, a ultiserver syste is treated as an M/M/ queuing syste which is elaborated as follows. There is a Poisson strea of service requests with arrival rate, i.e., the interarrival ties are indeendent and identically distributed (i.i.d.) exonential rando variables with ean =. A ultiserver syste S aintains a queue with infinite caacity for waiting tasks when all the servers are busy. The first-coe-first-served (FCFS) queuing disciline is adoted. The task execution requireents (easured by the nuber of instructions to be executed) are i.i.d. exonential rando variables r with ean r. The servers (i.e., blades/rocessors/cores) of S have identical execution seed s (easured by the nuber of instructions that can be executed in one unit of tie). Hence, the task execution ties on the servers of S are i.i.d. exonential rando variables x ¼ r=s with ean x ¼ r=s. Notice that although an M/G/ queuing syste has been considered (see, e.g., [3]), the M/M/ queuing odel is the only odel that accoodates an analytical and closedfor exression of the robability density function of the waiting tie of a newly arrived service request. Let ¼ =x ¼ s=r be the average service rate, i.e., the average nuber of service requests that can be finished by a server of S in one unit of tie. The server utilization is ¼ = ¼ x= ¼ = r=s, which is the average ercentage of tie that a server of S is busy. Let k denote the robability that there are k service requests (waiting or being rocessed) in the M/M/ queuing syste for S. Then, we have ([4,. 02]) 8 >< k ¼ ðþ k 0 ; k ; k! >: k 0 ; k ;! where 0 ¼ X k¼0 ðþ k k! þ ðþ!! : The robability of queuing (i.e., the robability that a newly subitted service request ust wait because all servers are busy) is P q ¼ X k¼ k ¼ ¼ ðþ 0! : The average nuber of service requests (in waiting or in execution) in S is N ¼ X k¼0 k k ¼ þ P q: Alying Little s result, we get the average task resonse tie as T ¼ N! ¼ x þ P q ¼ x þ ð Þ ð Þ 2 : The average waiting tie of a service request is W ¼ T x ¼ ð Þ 2 x: The waiting tie is the source of custoer dissatisfaction. A service rovider should kee the waiting tie to a low level by roviding enough servers and/or increasing server seed, and be willing to ay back to a custoer in case the waiting tie exceeds certain liit. 3 POWER CONSUMPTION MODELS Power dissiation and circuit delay in digital CMOS circuits can be accurately odeled by sile equations, even for colex icrorocessor circuits. CMOS circuits have dynaic, static, and short-circuit ower dissiation; however, the doinant coonent in a well-designed circuit is dynaic ower consution P (i.e., the switching coonent of ower), which is aroxiately P ¼ acv 2 f, where a is an activity factor, C is the loading caacitance, V is the suly voltage, and f is the clock frequency [6]. In the ideal case, the suly voltage and the clock frequency are related in such a way that V / f for soe constant >0 [22]. The rocessor execution seed s is usually linearly roortional to the clock frequency, naely, s / f. For ease of discussion, we will assue that V ¼ bf and s ¼ cf, where b and c are soe constants. Hence, we know that ower consution is P ¼ acv 2 f ¼ ab 2 Cf 2þ ¼ðab 2 C=c 2þ Þs 2þ ¼ s, where ¼ ab 2 C=c 2þ and ¼ 2 þ. For instance, by setting b ¼ :6, ac ¼ 7:0, c ¼ :0, ¼ 0:5, ¼ 2 þ ¼ 2:0, and ¼ ab 2 C=c ¼ 9:492, the value of P calculated by the equation P ¼ acv 2 f ¼ s is reasonably close to that in [] for the Intel Pentiu M rocessor. We will consider two tyes of server seed and ower consution odels. In the idle-seed odel, a server runs at zero seed when there is no task to erfor. Since the ower for seed s is s, the average aount of energy consued by a server in one unit of tie is s ¼ rs, where we notice that the seed of a server is zero when it is idle. The average aount of energy consued by an -server syste S in one unit of tie, i.e., the ower suly to the ultiserver syste S, isp ¼ s ¼ rs, where ¼ x is the average nuber of busy servers in S. Since a server still consues soe aount of ower P even when it is idle (assue that an idle server consues certain base ower P, which includes static ower dissiation, short-circuit ower dissiation, and other leakage and wasted ower []), we will include P in P, i.e., P ¼ ðs þ P Þ¼rs þ P. Notice that when P ¼ 0, the above P is indeendent of.

In the constant-seed odel, all servers run at the seed s even if there is no task to erfor. Again, we use P to reresent the ower allocated to ultiserver syste S. Since the ower for seed s is s, the ower allocated to ultiserver syste S is P ¼ ðs þ P Þ. 4 WAITING TIME DISTRIBUTION Let W denote the waiting tie of a new service request that arrives to a ultiserver syste. In this section, we find the df f W ðtþ of W. To this end, we consider W in different situations, deending on the nuber of tasks in the queuing syste when a new service request arrives. Let W k denote the waiting tie of a new task that arrives to an M/M/ queuing syste under the condition that there are k tasks in the queuing syste when the task arrives. We define a unit iulse function u z ðtþ as follows: 8 >< u z ðtþ ¼ z; 0 t z ; >: 0; t > z : The function u z ðtþ has the following roerty: Z 0 u z ðtþdt ¼ ; naely, u z ðtþ can be treated as a df of a rando variable with exectation Z 0 tu z ðtþdt ¼ z Z =z 0 tdt ¼ 2z : Let z!and define uðtþ ¼li z! u z ðtþ.it is clear that any rando variable whose df is uðtþ has exectation 0. The following theore gives the df of the waiting tie of a newly arrived service request: Theore. The df of the waiting tie W of a newly arrived service request is f W ðtþ ¼ð P q ÞuðtÞþ e ð Þt ; where P q ¼ =ð Þ and ¼ 0 ðþ =!. Sketch of the Proof. Let W k be the waiting tie of a new service request if there are k tasks in the queuing syste when the service request arrives. We find the df of W k for all k 0. Then, we have f W ðtþ ¼ X k¼0 k f Wk ðtþ: Actually, W k can be found for two cases, i.e., when k< and when k. A colete roof of the theore is given in Section 9. tu Notice that a ultiserver syste with ultile identical servers has been configured to serve requests fro certain alication doain. Therefore, we will only focus on task waiting tie in a waiting queue and do not consider other sources of delay, such as resource allocation and rovision, virtual achine instantiation and deloyent, and other overhead in a colex cloud couting environent. 5 SERVICE CHARGE If all the servers have a fixed seed s, the execution tie of a service request with execution requireent r is known as x ¼ r=s. The resonse tie to the service request is T ¼ W þ x ¼ W þ r=s. The resonse tie T is related to the service charge to a custoer of a service rovider in cloud couting. To study the exected service charge to a custoer, we need a colete secification of a service charge based on the aount of a service, the service-level agreeent, the satisfaction of a consuer, the quality of a service, the enalty of a low-quality service, and a service rovider s argin and rofit. Let s 0 be the baseline seed of a server. We define the service charge function for a service request with execution requireent r and resonse tie T to be 8 ar; if 0 T c r; s 0 ar d T c >< r ; s 0 Cðr; TÞ ¼ if c r<t a s 0 d þ c r; s 0 0; if T> a d þ c >: r: s 0 The above function is defined with the following rationals:. If the resonse tie T to rocess a service request is no longer than ðc=s 0 Þr ¼ cðr=s 0 Þ(i.e., a constant c ties the task execution tie with seed s 0 ), where the constant c is a araeter indicating the servicelevel agreeent, and the constant s 0 is a araeter indicating the exectation and satisfaction of a consuer, then a service rovider considers that the service request is rocessed successfully with high quality of service and charges a custoer ar, which is linearly roortional to the task execution requireent r(i.e., the aount of service), where a is the service charge er unit aount of service (i.e., a service rovider s argin and rofit).. If the resonse tie T to rocess a service request is longer than ðc=s 0 Þr but no longer than ða=d þ c=s 0 Þr, then a service rovider considers that the service request is rocessed with low quality of service and the charge to a custoer should decrease linearly as T increases. The araeter d indicates the degree of enalty of breaking the service-level agreeent.. If the resonse tie T to rocess a service request is longer than ða=d þ c=s 0 Þr, then a service rovider considers that the service request has been waiting too long, so there is no charge and the service is free. Notice that the task resonse tie T is coared with the task execution tie on a server with seed s 0 (i.e., the baseline or reference seed). The actual seed s of a server can be decided by a service rovider, which can be either lower or higher than s 0, deending on the workload (i.e., and r) and syste araeters (e.g.,,, and P ) and the service charge function (i.e., a, c, and d), such that the net business gain to be defined below is axiized.

Fig.. Service charge CðrÞ versus r and. Fig. 2. Noralized service charge CðrÞ=ar versus r and. To build our discussion uon our earlier result on task waiting tie, we notice that the service charge function can be rewritten equivalently in ters of r and W as 8 ar; if 0 W c r; s 0 s a þ cd d >< r dw; s Cðr; WÞ ¼ 0 s c if r<w a s 0 s d þ c r; s 0 s 0; if W> a d þ c >: r: s 0 s The following theore gives the exected charge to a service request: Theore 2. The exected charge to a service request is P q C ¼ ar ððs rþðc=s 0 =sþþþ ; ððs rþða=d þ c=s 0 =sþþþ where P q ¼ =ð Þ and ¼ 0 ðþ =!. Sketch of the Proof. The roof is actually a detailed calculation of C, which contains two stes. In the first ste, we calculate CðrÞ, i.e., the exected charge to a service request with execution requireent r, based on the df of W obtained fro Theore. In the second ste, we calculate C based on the df of r. A colete roof of the theore is given in Section 9. tu In Fig., we consider the exected charge to a service request with execution requireent r, i.e., dp q CðrÞ ¼ar ð Þ e ð Þðc=s 0 =sþr e ð Þða=dþc=s0 =sþr : We assue that r ¼ billion instructions, ¼ 7 servers, s 0 ¼ billion instructions er second, s ¼ billion instructions er second, a ¼ 0 cents er one billion instructions (Note: The onetary unit cent in this aer ay not be identical but should be linearly roortional to the real cent in US dollars.), c ¼ 3, and d ¼ cents er second. (Note: These araeters are chosen only for illustration and should be scaled to any values.) For ¼ 6:5; 6:35; 6:55; 6:75; 6:95 service requests er second, we show CðrÞ for 0 r 3. It can be seen that the service charge is a decreasing function of, since the waiting tie and lateness enalty increase as increases. It can also be seen that the service charge is an increasing function of r, i.e., large service requests generate ore revenue than sall service requests. In Fig. 2, we further dislay CðrÞ=ar using the sae araeters in Fig.. Since ar is the ideal (axiu) charge to a service request with execution requireent r, CðrÞ=ar is considered as the noralized service charge. For ¼ 6:5; 6:35; 6:55; 6:75; 6:95 service requests er second, we show CðrÞ=ar for 0 r 3. It can be seen that the noralized service charge is a decreasing function of, since the waiting tie and lateness enalty increase as increases. It can also be seen that the noralized service charge is an increasing function of r, i.e., the ercentage of lost service charge due to waiting tie decreases as service requireent r increases. In other words, it is ore likely to ake rofit fro large service requests and it is ore likely to give free services to sall service requests. It can be verified that as r aroaches 0, the noralized service CðrÞ charge is li r!0 ar ¼ P q, where P q increases (and P q decreases) as increases. It can also be verified that as r aroaches infinity, the noralized service charge is CðrÞ li r! ar ¼, for all. 6 NET BUSINESS GAIN Since the nuber of service requests rocessed in one unit of tie is in a stable M/M/ queuing syste, the exected service charge in one unit of tie is C, which is actually the exected revenue of a service rovider. Assue that the rental cost of one server for unit of tie is. Also, assue that the cost of energy is er Watt. The cost of a service rovider is the su of the cost of infrastructure renting and the cost of energy consution, i.e., þ P. Then, the exected net business gain (i.e., the net rofit) of a service rovider in one unit of tie is G ¼ C ð þ PÞ, which is defined as the revenue inus the cost. The above equation is G ¼ C ð þ ðrs þ P ÞÞ, for the idle-seed odel, and G ¼ C ð þ ðs þ P ÞÞ, for the constant-seed odel. In Figs. 3 and 4, we deonstrate the revenue C and the net business gain G in one unit of tie as a function of for the two ower consution odels, resectively, using the sae araeters in Figs. and 2. Furtherore, we assue

Fig. 3. Revenue and net business gain versus (idle-seed odel). that P ¼ 2 Watts, ¼ 2:0, ¼ 9:492, ¼ :5 cents er second, and ¼ 0: cents er Watt. For 0 7, we show C and G. The cost of infrastructure renting is ¼ 4 cents er second, and the cost of energy consution is 0:5 þ 7 cents er second for the idle-seed odel and 0.5 cents er second for the constant-seed odel. We observe that both C and G increase with alost linearly and dro sharly after certain oint. In other words, ore service requests bring ore revenue and net business gain; however, after the nuber of service requests er unit of tie reaches certain oint, the excessive waiting tie causes increased lateness enalty, so that there is no revenue and negative business gain. There are two situations that cause negative business gain. In the first case, there is no enough business (i.e., service requests). In this case, a service rovider should consider reducing the nuber of servers and/or server seed s, so that the cost of infrastructure renting and the cost of energy consution can be reduced. In the second case, there is too uch business (i.e., service requests). In this case, a service rovider should consider increasing the nuber of servers and/or server seed, so that the waiting tie can be reduced and the revenue can be increased. However, increasing the nuber of servers and/or server seed also increases the cost of infrastructure renting and the cost of energy consution. Therefore, we have the roble of selecting the otial server size and/or server seed so that the rofit is axiized. 7 PROFIT MAXIMIZATION To forulate and solve our otiization robles analytically, we need a closed-for exression of C. To this end, let P us use the following closed-for aroxiation, ðþ k k¼0 k! e, which is very accurate when is not too sall and is not too large [7]. We also need Stirling s aroxiation of!, i.e.,! ffiffiffiffiffiffiffiffiffi 2ð e Þ. Therefore, we get the following closed-for aroxiation of : ffiffiffiffiffiffiffiffiffi 2ð Þðe =eþ þ ; and the following closed-for aroxiation of P q : P q ffiffiffiffiffiffiffiffiffi 2ð Þðe =eþ þ : Fig. 4. Revenue and net business gain versus (constant-seed odel). By using the above-closed-for exression of P q, we get a closed-for aroxiation of the exected service charge to a service request as C ar ffiffiffiffiffiffiffiffiffi ð 2ð Þðe =eþ þ Þ ððs rþðc=s 0 =sþþþ! : ððs rþða=d þ c=s 0 =sþþþ For convenience, we rewrite C as C ¼ arð D D 2 D 3 Þ, where D ¼ ffiffiffiffiffiffiffiffiffi 2ð Þðe =eþ þ ; D 2 ¼ðs rþðc=s 0 =sþþ; D 3 ¼ðs rþða=d þ c=s 0 =sþþ: Our discussion in this section is based on the above-closedfor exression of C. 7. Otial Size Given, r, s, P,,,, a, c, and d, our first roble is to find such that G is axiized. To axiize G, we need to find such that @G @ ¼ @C @ ð þ P Þ¼0; for the idle-seed odel, and @G @ ¼ @C @ ð þ ðs þ P ÞÞ ¼ 0; for the constant-seed odel, where @C @ ¼ ar ðd D 2 D 3 Þ 2 @D D 2 D 3 @ þ D @D 2 D 3 @ þ D @D 3 D 2 : @ To continue the calculation, we rewrite D as D ¼ ffiffiffiffiffiffiffiffiffi 2ð ÞR þ, where R ¼ðe =eþ. Notice that ln R ¼ lnðe =eþ ¼ð ln Þ. Since @ r ¼ @ 2 s ¼ ;

Fig. 5. Net business gain G versus and (idle-seed odel). we get @R R @ ¼ð ln Þþ @ ¼ ln ; @ and @R ¼ R ln : @ Now, we have @D @ ¼ ffiffiffiffiffi 2 2 ffiffiffiffi ð ÞR þ ffiffiffiffi @ R @ þ ffiffiffiffi @R ð Þ @ ¼ ffiffiffiffiffi 2 2 ffiffiffiffi ð ÞR þ ffiffiffiffi R ffiffiffiffi ð ÞR ln ¼ ffiffiffiffiffi 2 2 ffiffiffiffi ð ÞR þ ffiffiffiffi R ffiffiffiffi ð Þðln ÞR ¼ ffiffiffiffiffi 2 2 ffiffiffiffi ð þ ÞR ffiffiffiffi ð Þðln ÞR : Furtherore, we have Fig. 6. Net business gain G versus and (constant-seed odel). Such server size otiization has clear hysical interretation. When is sall such that is close to, the waiting ties of service requests are excessively long, and the service charges and the net business gain are low. As increases, the waiting ties are significantly reduced, and the service charges and the net business gain are increased. However, as further increases, there will be no ore increase in the exected services charge which has an uer bound ar; on the other hand, the cost of a service rovider (i.e., the rental cost and base ower consution) increases, so that the net business gain is actually reduced. Hence, there is an otial choice of which axiizes the rofit. 7.2 Otial Seed Given, r,, P,,,, a, c, and d, our second roble is to find s such that G is axiized. To axiize G, we need to find s such that @G @s ¼ @C @s rð Þs 2 ¼ 0; for the idle-seed odel, and and @D 2 @ ¼ cs=s 0 ; @G @s ¼ @C @s s ¼ 0; for the constant-seed odel, where @D 3 @ ¼ as=d þ cs=s 0 : Although there is no closed-for solution to, we notice that @G=@ is a decreasing function of. Therefore, can be found nuerically by using the standard bisection ethod. In Figs. 5 and 6, we deonstrate the net business gain G in one unit of tie as a function of and for the two ower consution odels, resectively, using the sae araeters in Figs., 2, 3, and 4. For ¼ 2:9; 3:9; 4:9; 5:9; 6:9, we dislay G for large enough such that <. We notice that there is an otial choice of such that G is axiized. Using our analytical results, we can find such that @G=@ ¼ 0. The otial value of is 3.67479, 4.7928, 5.89396, 6.98457, 8.06655, resectively, for the idleseed odel, and 3.54842, 4.64834, 5.73478, 6.860, 7.8804, resectively, for the constant-seed odel. @C @s ¼ ar ðd D 2 D 3 Þ 2 @D D 2 D 3 @s þ D @D 2 D 3 @s þ D @D 3 D 2 : @s Siilar to the calculation in the last section, we have @ r ¼ @s s ¼ 2 s ; and @R R @s ¼ @ @s ¼ ð Þ; s and @R @s ¼ ð ÞR: s

Fig. 7. Net business gain G versus s and (idle-seed odel). Now, we have @D @s ¼ ffiffiffiffiffiffiffiffiffi 2 @ R þð Þ @R @s @s ¼ ffiffiffiffiffiffiffiffiffi 2 s R þ s ð Þ2 R ¼ ffiffiffiffiffiffiffiffiffi 2ð þ ð Þ 2 Þ R s : Furtherore, we have @D 2 @s ¼ c þðs rþ s 0 s s 2 ¼ c r s 0 s 2 ; and @D 3 @s ¼ a d þ c þðs rþ s 0 s s 2 ¼ a d þ c r s 0 s : 2 Although there is no closed-for solution to s, we notice that @G=@s is a decreasing function of s. Therefore, s can be found nuerically by using the standard bisection ethod. In Figs. 7 and 8, we deonstrate the net business gain G in one unit of tie as a function of s and for the two ower consution odels, resectively, using the sae araeters in Figs., 2, 3, 4, 5, and 6. For ¼ 2:9; 3:9; 4:9; 5:9; 6:9, we dislay G for s large enough such that <. We notice that there is an otial choice of s such that G is axiized. Using our analytical results, we can find s such that Fig. 9. Net business gain G versus s and (idle-seed odel). @G=@s ¼ 0. The otial value of s is 0.6325, 0.76982, 0.90888,.049,.90, resectively, for the idle-seed odel, and 0.5705, 0.7009, 0.8545, 0.99348,.3584, resectively, for the constant-seed odel. Such server seed otiization also has clear hysical interretation. When s is sall such that is close to, the waiting ties of service requests are excessively long, and the service charges and the net business gain are low. As s increases, the waiting ties are significantly reduced, and the service charges and the net business gain are increased. However, as s further increases, there will be no ore increase in the exected services charge which has an uer bound ar; on the other hand, the cost of a service rovider (i.e., the cost of energy consution) increases, so that the net business gain is actually reduced. Hence, there is an otial choice of s which axiizes the rofit. 7.3 Otial Size and Seed Given, r, P,,,, a, c, and d, our third roble is to find and s such that G is axiized. To axiize G, we need to find and s such that @G=@ ¼ 0 and @G=@s ¼ 0, where @G=@ and @G=@s have been derived in the last two sections. The two equations can be solved by a nested bisection search rocedure. In Figs. 9 and 0, we deonstrate the net business gain G in one unit of tie as a function of s and for the two ower consution odels, resectively, using the sae araeters in Figs., 2, 3, 4, 5, 6, 7, and 8, where ¼ 6:9. For ¼ 4; 5; 6; 7; 8, we dislay G for s large enough such that <. Using our analytical results, we can find and s such that @G=@ ¼ 0 and @G=@s ¼ 0. For the idle-seed Fig. 8. Net business gain G versus s and (constant-seed odel). Fig. 0. Net business gain G versus s and (constant-seed odel).

odel, the theoretically otial values are ¼ 5:56827 and s ¼ :4689, which result in the axiu G ¼ 49:2536 by using the closed-for aroxiation of C. Practically, can be either 5 or 6. When ¼ 5, the otial value of s is.62236, which results in the axiu G ¼ 49:650. When ¼ 6, the otial value of s is.37044, which results in the axiu G ¼ 49:8888. Hence, the ractically otial setting is ¼ 6 and s ¼ :37044, and the axiu net business gain in one unit of tie is G ¼ 49:29273 by using the exact exression of C. For the constant-seed odel, the theoretically otial values are ¼ 5:79074 and s ¼ :35667, which result in the axiu G ¼ 47:80769 by using the closed-for aroxiation of C. Practically, can be either 5 or 6. When ¼ 5, the otial value of s is.55839, which results in the axiu G ¼ 47:63979. When ¼ 6, the otial value of s is.323, which results in the axiu G ¼ 47:78640. Hence, the ractically otial setting is ¼ 6 and s ¼ :323, and the axiu net business gain in one unit of tie is G ¼ 47:9830 by using the exact exression of C. 8 SIMULATION SETTINGS AND RESULTS See Section of the suleentary aterial, which can be found on the Couter Society Digital Library at htt:// doi.ieeecoutersociety.org/0.09/tpds.202.203. 9 PROOFS OF THEOREMS AND 2 See Section 2 of the suleentary aterial, available online. 0 CONCLUDING REMARKS We have roosed a ricing odel for cloud couting which takes any factors into considerations, such as the requireent r of a service, the workload of an alication environent, the configuration ( and s) of a ultiserver syste, the service level agreeent c, the satisfaction (r and s 0 ) of a consuer, the quality (W and T) of a service, the enalty d of a low-quality service, the cost ( and ) of renting, the cost (,, P, and P ) of energy consution, and a service rovider s argin and rofit a. By using an M/M/ queuing odel, we forulated and solved the roble of otial ultiserver configuration for rofit axiization in a cloud couting environent. Our discussion can be easily extended to other service charge functions. Our ethodology can be alied to other ricing odels. ACKNOWLEDGMENTS The authors are grateful to three anonyous reviewers for their constructive coents. Part of the work were erfored while K. Hwang, K. Li, and A.Y. Zoaya were visiting Tsinghua National Laboratory for Inforation Science and Technology, Tsinghua University, in the winter of 20 and the suer of 202 as Intellectual Ventures endowed visiting chair rofessors. This work is artially suorted by Ministry of Science and Technology of China under National 973 Basic Research Grants No. 20CB302805 and No. 20CB302505, and National 863 High-tech Progra Grant No. 20AA04050; Ministry of Industry and Inforation Technology of China under the Internet of Things rogra; the Innovation R/D Tea Progra of Guangdong Province, China, under contract No. 2000D0047265; Australian Research Grant DP0970. REFERENCES [] htt://en.wikiedia.org/wiki/cmos, 202. [2] htt://en.wikiedia.org/wiki/service_level_agreeent, 202. [3] M. Arbrust et al., Above the Clouds: A Berkeley View of Cloud Couting, Technical Reort No. UCB/EECS-2009-28, Feb. 2009. [4] R. Buyya, D. Abrason, J. Giddy, and H. Stockinger, Econoic Models for Resource Manageent and Scheduling in Grid Couting, Concurrency and Coutation: Practice and Exerience, vol. 4,. 507-542, 2007. [5] R. Buyya, C.S. Yeo, S. Venugoal, J. Broberg, and I. 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Junwei Cao received the bachelor s and aster s degrees in control theories and engineering in 996 and 998, resectively, both fro Tsinghua University, Beijing, China. He received the PhD degree in couter science fro the University of Warwick, Coventry, United Kingdo, in 200. He is currently a rofessor and vice director, Research Institute of Inforation Technology, Tsinghua University, Beijing, China. He is also the director of Coon Platfor and Technology Division, Tsinghua National Laboratory for Inforation Science and Technology. Before joining Tsinghua University in 2006, he was a research scientist at MIT LIGO Laboratory and NEC Laboratories Euroe for about five years. He has ublished ore than 30 aers and cited by international scholars for over 3,000 ties. He is the book editor of Cyberinfrastructure Technologies and Alications, ublished by Nova Science in 2009. His research is focused on advanced couting technologies and alications. He is a senior eber of the IEEE and IEEE Couter Society, and a eber of the ACM and CCF. Kai Hwang received the PhD degree fro the University of California, Berkeley in 972. He is a rofessor of EE/CS at the University of Southern California. He also chairs the IV-endowed visiting chair rofessor grou at Tsinghua University in China. He has ublished eight books and ore than 28 scientific aers in couter architecture, arallel rocessing, distributed systes, cloud couting, network security, and Internet alications. His oular books have been adoted worldwide and translated into four foreign languages. His ublished aers have been cited ore than,000 ties by early 202. His latest book Distributed and Cloud Couting: fro Parallel Processing to the Internet of Things (with G. Fox and J. Dongarra) was just ublished by Kaufann in 20. He received the 2004 CFC Outstanding Achieveent Award, and the Founders Award for his ioneering work in arallel rocessing fro IEEE IPDPS in 20. He has served as a founding editor-in-chief of the Journal of Parallel and Distributed Couting for 28 years. He has delivered 34 keynote addresses on advanced couting systes and cutting-edge inforation technologies in ajor IEEE/ACM Conferences. He has erfored advisory, consulting and collaborative work for IBM, Intel, MIT Lincoln Lab, JPL at Caltech, ETL in Jaan, ITRI in Taiwan, GMD in Gerany, INRIA in France, and Chinese Acadey of Sciences. He is a fellow of the IEEE (986). Keqin Li is a SUNY distinguished rofessor in couter science and an Intellectual Ventures endowed visiting chair rofessor at Tsinghua University, China. His research interests are ainly in design and analysis of algoriths, arallel and distributed couting, and couter networking. He has contributed extensively to rocessor allocation and resource anageent; design and analysis of sequential/arallel, deterinistic/robabilistic, and aroxiation algoriths; arallel and distributed couting systes erforance analysis, rediction, and evaluation; job scheduling, task disatching, and load balancing in heterogeneous distributed systes; dynaic tree ebedding and randoized load distribution in static networks; arallel couting using otical interconnections; dynaic location anageent in wireless counication networks; routing and wavelength assignent in otical networks; energy-efficient ower anageent and erforance otiization. He has ublished ore than 240 research ublications and has received several Best Paer Awards for his highest quality work. He is currently on the editorial board of IEEE Transactions on Parallel and Distributed Systes, IEEE Transactions on Couters, International Journal of Parallel, Eergent and Distributed Systes, International Journal of High Perforance Couting and Networking, and Otiization Letters. He is a senior eber of the IEEE. Albert Y. Zoaya is currently the chair rofessor of high erforance couting and networking and Australian Research Council Professorial fellow in the School of Inforation Technologies, The University of Sydney. He is also the director of the Centre for Distributed and High Perforance Couting which was established in late 2009. He is the author/coauthor of seven books, ore than 400 aers, and the editor of nine books and conference roceedings. He is the editor-in-chief of the IEEE Transactions on Couters and serves as an associate editor for 9 leading journals, such as, the IEEE Transactions on Parallel and Distributed Systes and Journal of Parallel and Distributed Couting. He is the reciient of the Meritorious Service Award (in 2000) and the Golden Core Recognition (in 2006), both fro the IEEE Couter Society. Also, he received the IEEE Technical Coittee on Parallel Processing Outstanding Service Award and the IEEE Technical Coittee on Scalable Couting Medal for Excellence in Scalable Couting, both in 20. He is a chartered engineer, a fellow of AAAS, IEEE, and IET (United Kingdo).. For ore inforation on this or any other couting toic, lease visit our Digital Library at www.couter.org/ublications/dlib.