Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations

This paper proposes a discrete‐time multi‐server queue with multiple synchronous vacations under balking and reneging. Arriving customers decide whether to join the system or balk on the basis of some state‐dependent joining/balking probabilities, and renege according to a geometric distribution when servers are busy. The servers take a vacation together if there are no customers in the system at a service completion instant. When the servers are on vacation, an arriving customer activates an impatience timer which is geometrically distributed. The inter‐arrival times, service times and vacation times are assumed to be independent and geometrically distributed. We obtain closed‐form expressions and develop a computational algorithm for calculating the steady‐state probabilities. Specifically, we establish the application of the proposed framework in analyzing a multi‐server queueing system with synchronous vacation under balking and reneging. Applications of such models can be found in a wide variety of real‐time systems including call centers, computer and communication systems, cloud computing, quality control and maintenance in industrial establishments. We develop a cost model to determine the optimal service rate. Various performance measures and numerical examples are sketched out to demonstrate the impact of the proposed method. Some special cases of the model have also been discussed. Finally, we show that in the limiting case the results converge to the corresponding continuous‐time counterparts.