System Model

Chapter 1 Introduction

7.2 Problem Formulation

7.2.1 System Model

We first describe the assumptions used in the formulation and define various components used in the formulation. Table 7.1 summarises the notation used in this chapter.

Assumptions: The data centers are over-provisioned with sufficient compute capacity across various locations to accommodate single data center failure. Every data center is aware of the volume of workload being served across all the other data centers.

We now define the input parameters used in the formulation.

Demand: We define λ^uf_s as the number of requests mapped from client region u to data center s. Here f = 0 represents no data center failed and f ∈ 1,2..|S|

represents the failure of the f^th data center.

Number of active servers: Let the processing rate of the server be µ requests per second. We assume that among m_s servers at a data center, m^f_s are activated after f^th data center has failed. A data center is modeled as an M/M/n queue.

The number of servers required to satisfy workload l within the maximum tolerable

7.2 Problem Formulation

Input Symbol Description

Data center

S Set of data center locations s Index of data center

M_s Total number of servers available at data centers

m^f_s Number of active servers at data centerswhen data centerfhas failed p Percentage of total servers failed at any data center

Client

U Set of client regions u Index of client region

Lu Total number of requests generated from user locationu P_idle Power consumed by server when idle

Ppeak Power consumed by server when running at its peak Server µ Service rate of server (in number of requests per second)

γ_s^f Average utilization of active servers at data centerswhen data centerfhas failed L⁰ Total workload served by failed data center before failure

L Total workload required to be distributed

L_g Workload required to be served by green energy sources L_r Workload which can be served by any energy source

λ^uf_s The number of requests from client regionuserved by data centerswhen data center fhas failed

Demand and

λ^f Vector formed usingλ^uf_s , which denotes the load balancing strategy

workload distri- bution

Φ^uf_si The number of requests from client regionubeing served at data centersbeing served by consuming energy of typeiwhen data centerfhas failed

Φ^f_si The cumulative workload at data centersby consuming energy of typeiwhen data centerfhas failed

Φ^f_s Vector of Φ^f_sifor data centerswhen data centerfhas failed.

Φ^f Vector of Φ^f_s, which signifies the total workload distribution among different data centers and energy sources

i Index of energy type

θ_si Electricity price per kWh at data centersfor energy of typei θ Vector of different electricity prices across all data centers

P_si^f Total power of typeiconsumed at data centerswhen data centerfhas failed Energy Θ^f_s The total cost of energy consumed at data centerswhen data centerfhas failed

ρ The minimum green energy usage bound E_s PUE at data centers

P_si^max Maximum available power of typeiat data centers

Delay

W_su Propagation delay between data centersand client regionu T Maximum tolerable queuing delay

Table 7.1: Summary of notation used in the chapter

queuing delay T is given by the equation 1

m^fsµ−l =T ∀s, f (7.1)

l =f(m^f_s) = m^f_sµ− 1

T ∀s, f (7.2)

In Eq. (7.2), the function f(m^f_s) gives the workload that can be served using m^f_s servers while meeting the queuing delay constraint. Similarly, we define m^f_s by

m^f_s = 1 µ(1

T +l) ∀s, f (7.3)

Server utilization: We define γ_s^f to be the average utilization of active servers at the s^th data center after f^th data center failed, given by

γ_s^f = P

uλ^uf_s µm^fs

(7.4) Power consumption model: Let P_idle be the average power drawn in idle condition and P_peak be the power consumed when server is running at peak utilization. We express the power consumed at a data center s, given workload l, after f^th data center failed, as [16]:

P_s^f = m^f_s(P_idle + (E_s−1)P_peak) + m^f_s(P_peak − P_idle)γ_s^f (7.5)

=c₁m^f_s +c₂l (7.6)

where c₁ = (P_idle + (E_s−1)P_peak), c₂ = ^(Ppeak _µ^−Pidle), andE_s is the PUE of a data center at s. Substituting m^f_s and γ_s^f in Eq. (7.6) from Eq. (7.3) and Eq. (7.4) gives

P_s^f = c₁ µ(1

T +l) +c₂l

= (c₁

µ +c₂)l+ c₁

T µ (7.7)

Surplus workload: LetL⁰ be the workload served by a failed data centerf, before the failure. After the failure, letm^f_f be the number of servers available. The surplus

7.2 Problem Formulation

workload violating the queuing delay constraint at failed data center can be obtained as

L=L⁰−f(m^f_f) ∀f (7.8)

Workload partitioning: We partition the workload into two components: (i) green workload, L_g which is served by servers powered with green energy and (ii) remaining workload, L_r served by other servers. Due to workload conservation rule, the following equation has to be satisfied.

L_g+L_r =L (7.9)

Let ρ denote the minimum fraction of workload to be served by servers powered with green energy. The green energy usage constraint is represented by

Lg ≥ρL (7.10)

In our model, we consider 4 types of energy sources namely, solar, wind, PPA and brown (indexed in the same order). Φ^uf_si denotes the workload from client region u at data center s being served by consuming energy of type i, after f^th data center had failed.

The following equation makes sure that all the client demand is served.

4

X

i=1

Φ^uf_si =λ^uf_s ∀s, u (7.11) Φ^f_si denotes the cumulative workload at data center s being served by consuming energy of typei(i.e., Φ^f_si =P

uΦ^uf_si). Therefore, in order to satisfy the green energy usage we have

X

s 3

X

i=1

Φ^f_si ≥L_g ∀s (7.12)

Energy cost: Let θsi be the cost of energy of type i consumed at a data center s and θ be the vector of different energy prices across all data centers. The cost of

energy consumed at data center s after f^th data center has failed can be expressed as

Θ^f_s =X

i

θsiP_si^f ∀s, f (7.13) where P_si^f is the energy of type i is consumed at data center s when data center f has failed.