Signal transmission using counter-propagation is increasingly adopted to enhance resource utilization in spectrally-spatially elastic optical networks (SS-EONs). Crosstalk is a well-known drawback of SS-EONs that increases blocking probability. Evaluating blocking probability analytically in counter-propagation is challenging due to additional constraints. Current studies typically employ simulation-based techniques or do not consider dynamic scenarios in their analytical models to estimate blocking probability. This paper proposes AnalyticalCP, an exact analytical continuous-time Markov chain model (CTMC) for SS-EONs considering counter-propagation, which estimates blocking probabilities and enhances resource utilization. AnalyticalCP is constructed in two phases: (i) First, a bipartite graph is formed to generate two separate sets of cores and modes, using an optimization approach focused on minimizing interference among cores and modes. (ii) Second, the resultant sets of cores and modes serve as input parameters for the analytical model, streamlining the generation of all feasible states and transitions to estimate exact blocking probability. AnalyticalCP accommodates both single- and multi-class requests: single-class requests uniformly utilize spectrum slots, while multi-class requests adapt slot usage to match bandwidth requirements. Numerical results indicate that AnalyticalCP and the simulations yield comparable performances in terms of blocking probabilities and resource utilization, with the iterative approximate model also demonstrating acceptable accuracy. AnalyticalCP outperforms a benchmark model that considers vertex elimination during bi-partitioning. We extend our CTMC model with a Markov-modulated Poisson process (MMPP) to evaluate bursty-traffic behavior in SS-EONS.

Recently, multi-core and multi-mode fibers (MCMMFs) have been considered to overcome physical limitations and increase transport capacity. They are combined with elastic optical networks (EONs) to form spectrally-spatially elastic optical networks (SS-EONs), an emerging technology. Fragmentation and crosstalk (XT) are well-known drawbacks of SS-EONs that increase blocking probability; evaluating blocking probability analytically is difficult due to additional constraints. When calculating blocking probabilities in MCMMFs-based SS-EONs, it is observed that all current studies either employ simulation-based techniques or do not consider defragmentation of their analytical models. This paper proposes an exact analytical continuous-time Markov chain model for blocking probabilities, named AnalyticalDF, in SS-EONs, which considers defragmentation and the XT-avoided approach. AnalyticalDF generates all possible states and transitions while avoiding inter-core and inter-mode XTs for single-class and multi-class requests. Single-class requests utilize the same number of slots, whereas multi-class requests adopt varying numbers of slots to accommodate client needs. We introduce an iterative approximation model for a single-hop link when AnalyticalDF is not tractable due to scalability. We evaluate AnalyticalDF, the iterative approximate model, and simulation studies for a single-hop link. The numerical results indicate that AnalyticalDF outperforms a non-defragmentation-aware benchmark model.

A well-known drawback of spectrally-spatially elastic optical networks (SS-EONs) is crosstalk, which increases blocking probability; evaluating blocking probability analytically is challenging due to additional constraints. Not surprisingly, all the existing studies at this time mainly use simulation-based techniques to quantify blocking probabilities in multi-core and multi-mode fibers-based SS-EONs. This paper proposes AnalyticalBP, an exact analytical continuous-time Markov chain model for blocking probabilities considering the crosstalk-avoided approach in SS-EONs. AnalyticalBP generates all the feasible states and their transitions while avoiding inter-core and inter-mode crosstalks for both single and multi-class requests. Single-class requests use the same number of slots, whereas multi-class requests adopt different numbers of slots to satisfy clients’ requirements. When AnalyticalBP is not tractable due to scalability, we introduce an iterative approximate model for a single-hop link. We further extend the single-hop model for a multi-hop network. We compare AnalyticalBP, the iterative approximate model, and Monte Carlo simulation studies for a single-hop link. Numerical results indicate that the performances obtained by AnalyticalBP and the simulation studies, in terms of blocking probabilities and resource utilization, are comparable, and the accuracy of the iterative approximate model is also acceptable.