Analysis of BMAP/PH/c Retrial Queue with Threshold- Controlled Retrials and Synchronous Working Vacation

This work analyzes a multi-server retrial queueing system regulated by a batch Markovian arrival process and phase-type service times, incorporating a control mechanism for retrial customers. Retrial attempts from the orbit succeed only when the number of busy servers is at or below specific thresholds, which vary depending on the state of the arrival process. Customers waiting to retry may abandon the system independently due to impatience. When all servers become idle, they first remain on standby for a limited time, serving any arriving primary or retrial customers immediately; if no arrival occurs before this period ends, the servers take a synchronous working vacation, providing service at a reduced rate. Upon batch arrival, if insufficient servers are free, the batch is rejected or served partially, with the rest going into the orbit–a policy applied in both normal and working vacation modes. While in normal service mode, a disaster may occur at any time, causing all main servers to fail and forcing all customers in service to leave the system, causing no impact on the orbit. Repair begins immediately, during which no service is provided, and arriving batches may leave the system or join the orbit. The system state evolves as a multidimensional Markov chain in the asymptotically quasi-Toeplitz class, which facilitates obtaining the ergodicity conditions, enabling the derivation of steady-state probabilities and system performance measures. A case study involving the resolution of an optimization challenge is shown. Results highlight the influence of batch arrivals and system parameters on performance measures.