Selection of Best Queuing Model among Five Classes of Models for Reduction of Waiting Times, Queue Lengths, and Enhanced Resource Allocation

Long customer waiting times and congestion at service points pose significant challenges to operational efficiency and customer satisfaction within the Nigerian banking sector. This study evaluates and compares five distinct classes of queuing models to identify the optimal framework for minimizing waiting times, reducing queue lengths, and enhancing resource allocation. Primary operational data, encompassing arrival times, service initiation times, and transaction completion times, were collected using observational methods over five working days across two Automated Teller Machine (ATM) galleries (Stands I and II) at Fidelity Bank at Federal Polytechnic, Nekede, Imo State, Nigeria. Five standard queuing configurations were evaluated: single-channel infinite capacity under First-Come, First-Served, multi-channel infinite capacity, single-channel Random Order service, single-channel finite system capacity, and single-channel limited source population. Empirical metrics revealed that single-server models operates under severe congestion, with utilization factors ranging from \(\rho \approx 0.9245\) to \(\rho \approx 0.9825\), culminating in extreme instability \(\rho \approx 1.1111\) during peak periods where customer arrival rates \(\lambda=0.8444\) /min exceeded service capacity \(\mu=0.7600 / \min (\$ \mathrm{mu}=0.7600 / \operatorname{text}\{\min \} \$)\). The multi-server model (M/M/2) consistently outperformed all competing configurations across all datasets, effectively halving system utilization \(\rho=0.4623-0.5556\), | stabilizing the system under heavy arrival traffic, and drastically reducing average customer waiting times in queue \(W_q\) from up to 68.17 minutes down to under 0.60 minutes. Consequently, the M/M/s multi-server framework is identified as the best-fitting queuing model, offering bank administrators a mathematically grounded decision tool for dynamic capacity allocation and service delivery optimization.