Definitions and Cost Model

Chapter 1 Introduction

4.2 MILP Framework

4.2.2 Definitions and Cost Model

In this section, we define the various parameters used in GCACP problem formulation and also define the cost models considered in the objective function.

The notation used in this chapter is listed in Table 4.1.

Demand: Given that there is reasonably accurate information on workload across different client regions in a time window, letL^ah_u denote the total demand generated from a client region u for an application typea during the hour h∈T.

Delay: LetD_su be the propagation delay between user location u and data center location s. Let D_max be the maximum latency allowed for a client based on the SLA with the cloud provider. We also define a binary variable, y_su to represent the ability of data center s to serve requests from client region u.

Mean service rate: Let the mean service rate of each server be B bps and each application is characterized by a request generation rate and job size. To model, change in service rate before and after failure, we assume the mean file size for an application to vary between J_a⁰ to J_a¹. The mean service rate of application type a

Input Symbol Description

Data center

p Percentage of total servers failed at any data center M^min Minimum number of servers at any data center M^max Maximum number of servers at any data center α Server acquisition cost

Client

L^ah_u Total number of requests generated for application a from user location u during hourh

Dsu Propagation delay between client regionuand data centers D_max The maximum tolerable latency

Server Utilization

B Service rate of server in bits per second J_a^o Mean file size of application ain kB

γs^{f h} Average server utilization at data center s during hour h and failed data center f

γ^max Maximum value of γs^{f h} to avoid waiting

Energy

θ^h_s Electricity price per kWh at data centersat hour h G^h_s Total available green energy at data center locationsduring

hourh

δ_s^h Utility sell-back price at data center sduring hourh E_max Maximum capacity of the battery

ZD Maximum energy that can be withdrawn from the battery ( during hourh)

ZC Maximum energy that can be supplied to the battery ( during hourh)

Table 4.1: Summary of notation used in the chapter

can be defined as _J^Bo

a jobs per second, where o denotes the occurrence of failure (0 without failure, 1 with failure).

Average data center utilization: We have considered that the data center failure

4.2 MILP Framework

at a site can be partial or complete, with p percentage of servers failing during an event of failure and therefore the available compute capacity reduces to (1−p)ms. Assuming that requests are first placed in a common queue, before being served by any of the available servers, we define the average utilization, γ_s^{f h} to be

γ_s^{f h} =















P

u,aλ^{af h}su J_a^o

msB ∀s, h, f 6=s

P

u,aλ^{af h}su J_a¹

(1−p)msB f =s, p <1

0 f =s, p= 1

(4.1)

i.e., data center offers degraded service to the applications after the failure at a site.

Power consumption: Let P_idle be the average power drawn in idle condition and Ppeak be the power consumed when server is running at peak utilization. Then total power consumed at a data center locations∈S, at hour h∈H is given by [16]

P_s^{f h} = m_s(P_idle + (E_s − 1)P_peak) + m_s(P_peak − P_idle)γ_s^{f h} + , ∀s, h, f (4.2) where E_s is the PUE of a data center at s and is an empirical constant.

Power model: The power from various renewable energy generators is assumed to be known based on the meteorological data (may be obtained from [31]). G^h_s is the actual renewable energy generated (both wind and solar energy) at a data center s during hourh. For each data center s, hourh and failed data center f, we denoteGU_s^{f h}as the renewable energy used, GS_s^{f h} as the renewable energy sold (net metering) and Z_s^{f h} as the energy supplied to the battery (when Z_s^{f h} > 0) or that consumed from the battery (when Z_s^{f h} < 0). Thus, the available green energy can be expressed as

G^h_s =GU_s^{f h}+Z_s^{f h}+GS_s^{f h} ∀s, h, f

We also defineP B_s^{f h} as the total brown energy drawn from utility at a data centers during hourhand failed data centerf. P B_s^{f h}is calculated as the difference between

the power consumed at a data center (P_s^{f h}) and the amount of renewable energy used (GU_s^{f h}), given by

P B_s^{f h}=P_s^{f h}−GU_s^{f h} ∀s, h, f (4.3) Battery model: For each data center s, during hour h, we denote E_max as the maximum battery capacity¹, and Z_D and Z_C as the maximum power that can be supplied and withdrawn, respectively. The battery level at any data centers, during hour h, after f^th data center failed is given by

E_s^f(h+1) =E_s^{f h}+Z_s^{f h} ∀s, h, f (4.4)

Cost models: The TCO consists of the following components:

• Server cost: Let α be the server acquisition cost. The total cost of servers across the data centers is

Φ = αX

s

m_s (4.5)

• Brown energy cost: To account for the spatio-temporal variation in the electricity price, we define θ^h_s as the electricity price at location s during hour hof the day. We also express the cumulative brown energy cost across all data centers throughout the time horizon as

Θ = X

s,h,f

θ_s^hP B_s^{f h} (4.6)

• Renewable energy sell-back revenue: We denote the utility sell-back price by δ_s^h at a data center sduring hour h. Let R be the cumulative on-site sell back revenue generated across all the data centers throughout the time horizon, defined as

R =X

s,h,f

δ_s^hGS_s^{f h} (4.7)

1Energy storage devices are known to have enough capacity to power a data center at its maximum load between 5−30 minutes [78].

4.2 MILP Framework