Table 1 Cloud Information Notation

F

# of clouds

S

#of job types

M

# of VM types

m_s

VM type of type s job

g_s

# of VMs required by type s job

l_s

Execution time of type s job

L

Max. # of execution time

d_s

Max. waiting time of type s job

v_s

Data volume processed by type s job

\(r_{i}^{s}(t)\)

# of type s jobs accepted at cloud i, slot t

\(R_{i}^{s}\)

Quota of accepting type s jobs at cloud i per slot

\(p_{i}^{s}(t)\)

Service price for each type s job at cloud i, slot t

δ

Unit cost for transmitting data

\(D_{i}^{s(max)}\)

Max. # of type s job drops per slot

\(\xi _{i}^{s}\)

Penalty for dropping a type s job at cloud i

\(W_{i}^{s}(t)\)

Length of FIFO queue of type s job at cloud i, slot t

\(Q_{i}^{s}(t)\)

Length of importance queue of type s job at cloud i, slot t

\(Z_{i}^{s}(t)\)

Length of virtual queue of type s job at cloud i, slot t

ε

Constant positive parameter for \(Z_{i}^{s}(t), \forall i \in \{1,...,F\}\)

V

Penalty coefficient for Lyapunov framework

β_i(t)

Operational cost for each server at cloud i, slot t

\(N_{i}^{m}\)

# of servers hosting type m VMs owned by cloud i

\(C_{i}^{m}\)

Max. # of type m VMs a server can host