Table 1 Notations

S

Cloud service providers (sellers)

B

Cloud service users (buyers)

t

Time-slot

t^′

Time (within each time-slot)

α

Number of VMs (\(\alpha \in \mathbb {N}\))

θ_i

Type of agent i

\(\hat \theta \)

Collection of all agents’ types

\(\hat \theta ^{t}\)

Collection of all agents’ types in time-slot t

\(\hat \theta ^{\leqslant t}\)

Collection of all agents’ types from the beginning

till the end of time-slot t

v_i

The single unit valuation by agent i

q_i

The total VMs’ quantity that agent i wants to trade

a_i

The agent’s arrival time

d_i

The agent’s departure time

p

Agent’s presence period in a time-slot

\(U_{i}^{\hat \theta ^{t}}\)

Seller’s i utility at time-slot t

\(U_{j}^{\hat \theta ^{t}}\)

Buyer’s j utility at time-slot t

P

Payment rule

s_i,j

Seller’s obtained value from the auctioneer

b_i,j

Buyer’s payment value to the auctioneer

Q

Allocation rule

M^t

Matchable pairs of asks and bids at time-slot t

\(\sigma _{i}(\hat \theta ^{t})\)

Seller’s ask satisfiability in time-slot t

\(c^{p}_{i}(\hat \theta ^{t})\)

Seller’s ask criticality in time-slot t and time t^′

\(\sigma _{j}(\hat \theta ^{t})\)

Buyer’s bid satisfiability in time-slot t

\(c^{p}_{j}(\hat \theta ^{t})\)

Buyer’s bid criticality in time-slot t and time t^′

τ

Satisfiability threshold

ϕ(t)

Market price

\(e^{p}_{i}(t)\)

Agent i’s defer rate

ε

Defer rate threshold