Table 1 Key notations

M

The set of all connections

\(A_m(t)\)

A stochastic variable, connection m requires bandwidth \(A_m(t)\) on time slot t, which is i.i.d over slots

S

The set of all virtualized servers

\(P^s_{\text {NIC}}\)

All vswitches physical ports in server s, \(s \in S\)

\(P^s_{\text {VM}}\)

All vswitches virtual ports in server s, \(s \in S\)

\(P^s\)

All vswitches ports \(P^s_\text {VM} \bigcup P^s_\text {NIC}\) in server \(s \in S\)

\(P_{\text {VM}}\)

All vswitches virtual ports \(\bigcup _{s \in S} P^s_\text {VM}\)

\(P_{\text {NIC}}\)

All vswitches physical ports \(\bigcup _{s \in S} P^s_\text {NIC}\)

P

All vswitches ports \(P_{\text {VM}} \bigcup P_{\text {NIC}}\)

\(Q_u(t), Q_j(t)\)

Queue backlogs of vswitch port u and port j

\(z^{u}_m(t),z^{v}_m(t)\)

Connection m transmits through vswitch port \(u,v \in P\) on time slot t

\(\beta _{ij}\)

The bandwidth of network path (i, j), \(i,j \in P_{\text {NIC}}\)

\(r_{u}(t),r_{j}(t)\)

The vswitch resource allocation decision variables, which denotes the resource allocated to vswitch port \(u \in P^s_{\text {NIC}}\), \(j \in P^s_{\text {VM}}\) on time slot t, \(s \in S\)

\(y^{ij}_m(t)\)

The packet scheduling decision variable, which denotes the connection m transverses through the network path (i, j) on time slot t

\(f(\cdot )\)

A nondecrease, concave function \(f(\cdot )\) describes the relationship between the resource allocation on a vswitch port and the forwarding rate of the port

V

The tradeoff factor indicates the relation between the system stability and the object of the problem, \(V > 0\)