Queueing Theory M/M/m Model: Single Queue with Multiple Servers

The M/M/m queueing model with a single queue and multiple servers is a fundamental tool for analyzing service system performance. This model describes a system where customers form a single queue and are processed by multiple servers following a first-come-first-served (FCFS) discipline. The notation M represents Markovian processes, indicating that both inter-arrival times and service times follow exponential distributions, while m denotes the number of parallel service channels.

Key performance metrics in this model include average queue length, average waiting time, and server utilization rate. Through programming simulations, one can visualize system behavior under varying loads using discrete-event simulation algorithms. When customer arrival rates approach service rates, queue backlogs tend to accumulate rapidly. Implementing load-balancing logic in code demonstrates how increasing the number of servers can significantly reduce waiting times by distributing workload across multiple channels.

This model applies to real-world scenarios like bank teller counters and call centers with multiple agents. Simulation code typically includes parameter adjustment functions for server count and service rates, helping managers optimize resource allocation. By computationally testing different configurations, organizations can find optimal trade-offs between operational costs and customer satisfaction through sensitivity analysis algorithms.