## 4 Problem Statement and Algorithm Framework  
### Problem Formulation  
We propose the resource allocation problem to minimize the backhaul transport latency by mapping all service chains of predefined slices into the appropriate substrate network resources. The problem can be formulated as an ILP with linear constraints, subject to the specific service requirements and network capacity and bandwidth. The inputs to the resource allocation phase are slice traffic, slice latency thresholds, network capacity, and substrate link bandwidth and length. The output is the optimal deployment path for concurrent slice requests that minimizes the transport network latency while meeting the structural characteristics. In this regard, the optimization of latency has two considerations. Network capacity is firstly considered for describing the maximum transmission rate that can be provided for forwarding traffic, which also plays a critical role for network load-balancing. In addition, we also take into account the propagation distance of forwarded traffic in terms of the location of substrate nodes and substrate link bandwidth. The maximization of total backhaul throughput is equivalent to the minimization of the total number of transmission time [3]. After summarizing the transmission and propagation latency, we can find a latency-optimal mapping path for each communication service instance. The mathematical formulation of the NS resource allocation problem is as follows. \[\text{(P1)}:\min_{x^{(s,k)}_{u,v},s_{s}}\sum_{s\in S}\sum_{u,v\in N ^{p}}\sum_{k,l\in N^{v}}\alpha^{(s,k),(s,l)}_{(u,v)}x^{(s,k),(s,l)}_{(u,v)} \lambda^{(s,k),(s,l)},\] \[\text{s.t. (\ref{eq:1})}-(\ref{eq:2}),\] \[\sum_{u,v\in N^{p}}\sum_{k,l\in N^{v}}x^{(s,k),(s,l)}_{(u,v)} \lambda^{(s,k),(s,l)}=W^{s}_{vnu}-W^{s}_{csi},\forall s\in S, \tag{8}\] \[\sum_{u,v\in N^{p}}\sum_{k,l\in N^{v}}\alpha^{(s,k),(s,l)}_{(u,v)} x^{(s,k),(s,l)}_{(u,v)}\lambda^{(s,k),(s,l)}\leq T^{s}_{th},\forall s\in S,\] (9) \[\sum_{s\in S}x^{(s,k),(s,l)}_{(u,v)}\lambda^{(s,k),(s,l)}\leq\beta ^{(s,k),(s,l)}_{(u,v)},\forall u,v\in N^{p},k,l\in N^{v},\] (10) \[\sum_{s\in S}\sum_{k,l\in N^{v}}x^{(s,k),(s,l)}_{(u,v)}f^{(s,k),(s, l)}\leq C^{(s,k),(s,l)}_{(u,v)},\forall u,v\in N^{p},\] (11) \[x^{(s,k),(s,l)}_{(u,v)}f^{(s,k),(s,l)}=x^{(s,l),(s,n)}_{(v,t)}f^{( s,l),(s,n)},\forall u,v,t\in N^{p},k,l,n\in N^{v},\] (12) \[NI^{(s,k)}_{u},NI^{(s,l)}_{v}\geq NI_{th},\forall s\in S,u,v\in N ^{p},k,l\in N^{v},\] (13) \[m^{(s,k)}_{u},m^{(s,l)}_{v},x^{(s,k),(s,l)}_{(u,v)},\lambda^{(s,k),(s,l)} \in\{0,1\},\forall s\in S,u,v\in N^{p},k,l\in N^{v}. \tag{14}\]  
Here, \(W^{s}_{vnu}\) and \(W^{s}_{csi}\) denote the number of VNUs and communication service instances in the slice \(s\), respectively. Constraint (8) ensures that the hop count for mapping substrate network resources is equal to the number of virtual links in a slice, avoiding the closed-loop path or unnecessary extension of the physical mapping path during link mapping. In order to guarantee the user experience, constraint (9) assures the transport network latency of each NSI will not exceed the latency threshold specified by the service requirement. The threshold might be variable depending on the forwarded traffic \(f^{s}\) or other performance requirements. For example, for an industrial control NSI, the latency threshold will be lower as the service is latency-sensitive, whereas a higher one is set for serving latency-tolerant applications. The amount of required VNUs can not exceed the available substrate resources, so we indicate the corresponding VNU capacity that the substrate processing link \((u,v)\)accommodates in (10). Constraint (11) ensures that the resource capacity of substrate link \((u,v)\) can afford the forwarded traffic \(f^{s}\) of slice \(s\) to deploy and constraint (12) ensures flow conservation. The amount of flow goes into the substrate node \(v\) equals the amount of flow going out of it.