Simulation of M/M/1 Queueing System

The M/M/1 queueing system is one of the fundamental models in queueing theory, consisting of a single server, an infinite-capacity waiting queue, Poisson arrival process, and exponentially distributed service times. Through simulation of the M/M/1 system, we can better understand its operational mechanisms and evaluate performance metrics such as average queue length and average waiting time. In code implementation, this typically involves maintaining system state variables and event queues.

### Introduction to Event-Scheduling Method The event-scheduling method is a discrete-event simulation approach that simulates system dynamics by defining and scheduling key events. In M/M/1 queue simulation, two primary events are involved: customer arrival events and customer departure events. The simulation process maintains an event list, processes events in chronological order, and updates system states accordingly. From a programming perspective, this requires implementing an event calendar or priority queue to manage event timing efficiently.

### Simulation Logic Initialization: Set simulation duration, arrival rate (λ), and service rate (μ). Initialize the queue as empty and generate the first customer arrival event. In code, this involves setting initial parameters and creating the initial event object. Event Processing: Arrival Event: When a new customer arrives, if the server is idle, service begins immediately and a departure event is generated. If the server is busy, the customer joins the queue. Simultaneously, schedule the next arrival event using exponential distribution for inter-arrival times. Programmatically, this requires condition checks and random number generation for event scheduling. Departure Event: After service completion, if the queue is not empty, the first customer in queue begins service, generating a new departure event. This involves queue management operations and service time calculations. Statistical Metrics: Average Queue Length: Record instantaneous queue length values during simulation and calculate time-weighted average. Implementation requires maintaining cumulative statistics and time-stamped measurements. Average Waiting Time: Record each customer's duration from arrival to service start, then compute the average. This needs customer object tracking with arrival and service start timestamps.

### Performance Analysis Theoretical performance indicators for M/M/1 systems can be directly calculated using queueing theory formulas, but simulation methods allow verification under more complex scenarios (such as non-Poisson arrivals or non-exponential service times). By adjusting arrival and service rates, we can observe system performance under light and heavy loads, such as congestion formation and resolution. Simulation code can include comparative analysis between theoretical and simulated results.

### Extended Considerations The flexibility of the event-scheduling method makes it suitable for more complex queueing networks or priority queueing systems. By introducing additional event types (such as server breakdowns or customer balking), we can further enhance simulation practicality and accuracy. Programmatically, this would involve extending the event class hierarchy and modifying event handling logic.