# Time-Dependent Performance of a Multi-Hop Software Defined Network
## 4 Transient Analysis of Multiple SDN Switches  
Consider a network of \(M\) stations with an arbitrary topology with routing probabilities \(r_{ij}(t)\). We follow the approach of [69] developed for the steady-state network model then adapted to transient analysis in [70]. Additionally, we introduce time-dependent routing to model an SDN network. The first step to solve the network model is to decompose the network--i.e., to determine the input traffic parameters \(\lambda_{i}\), \(C_{Ai}^{2}\) at every station \(i\) and then apply the single server model of the previous section to each station separately. In the transient state, we should distinguish at any station \(i\) the input traffic \(\lambda_{i-in}(t)\) and the output traffic intensities \(\lambda_{i-out}(t)\):  
\[\lambda_{i-out}(t)=[1-p_{0i}(t)]\mu_{i}\]  
which are different; \(p_{0i}(t)\) denotes the probability that the station \(i\) is idle at time \(t\), i.e., the diffusion process related to this station is inside the barrier at \(x=0\). The term \(1-p_{0i}(t)=\varrho_{i}(t)\) presents probability that the station \(i\) is busy and customers are leaving it with the rate \(\mu_{i}\). Figure 8: \(f(x,t;0)\), \(t=0.750\) s.  
Figure 7: Mean queue for different \(C_{A}^{2}\). The traffic equations balancing the flows of stations are:  
\[\lambda_{i-in}(t)=\lambda_{0i}(t)+\sum_{j=1}^{M}\lambda_{j-out}(t)r_{ji}(t)\,\ \ \ \ \ \ \ \ \ i=1,\ldots,M, \tag{21}\]  
where the first term \(\lambda_{0i}\) represents the traffic flows coming from the outside of the network directly to station \(i\). The routing probabilities \(r_{ji}(t)\) change each interval \(\Delta\) following the decisions of the controller, remaining constant inside the interval, and the flow parameters may change every interval \(\delta<\Delta\); we assume \(\Delta=n\delta\), in numerical examples below \(n=10\). This way all model parameters are constant witin intervals \(\delta\) when the solution (14) is computed. Denote by \(f_{Aj}(x,t)\) and \(f_{Bj}(x,t)\) the density functions of the interarrival and service times distributions at station \(j\) at time \(t\). The pdf \(f_{Dj}(x,t)\) of the interdeparture times from this node at time \(t\) may be expressed as:  
\[f_{Dj}(x,t)=\varrho_{j}(t)f_{Bj}(x,t)+[1-\varrho_{j}(t)]f_{Aj}(x,t)*f_{Bj}(x,t) \,\ \ \ \ \ j=1,\ldots,M, \tag{22}\]  
where \(*\) denotes the convolution with respect to \(x\). The first term of the right side in (22) represents the interdeparture times of packets when the node \(j\) is working, and the second term gives the interdeparture times when it is idle. The formula (22), known as [PERSON]'s theorem [71], is exact for [PERSON] input (the pdf of the idle period distribution that should be used in the second term of (22) is the same as \(f_{Aj}(x,t)\)) and approximate in other cases. From (22), we receive:  
\[C_{Dj}^{2}(t)=\varrho_{j}^{2}(t)C_{Bj}^{2}(t)+C_{Aj}^{2}(t)(1-\varrho_{j}(t))+ \varrho_{j}(t)[1-\varrho_{j}(t)]. \tag{23}\]  
where \(C_{Dj}^{2}(t)\), \(C_{Bj}^{2}(t)\), and \(C_{Aj}^{2}(t)\) are time-dependent square coefficients of the variation in interdeparture, service, and interarrival times, respectively.