Queueing Theory Simulation: Analyzing Queue Length, Wait Time, Service Rate, Arrival and Departure Timestamps

Core Metrics Analysis in Queueing Theory Simulation

As a crucial branch of operations research, queueing theory is widely applied in service system optimization. Through simulation modeling, we can quantitatively evaluate the following key performance indicators:

Queue Length Metrics System queue length refers to the total number of customers in the system (including those being served), while queue length specifically indicates the number of customers waiting for service. During simulation, by recording the queue status at each timestamp, we can calculate average queue length and maximum queue length. Implementation typically involves maintaining a customer counter that increments on arrivals and decrements on departures.

Time-Based Metrics Waiting time represents the duration from customer arrival to service initiation, while sojourn time includes service time. The simulation program needs to track each customer's arrival timestamp and service completion events, calculating differences through event processing. Code implementation requires storing customer objects with arrival_time and service_start_time attributes.

Service Rate and Utilization Service rate reflects the average number of services completed per unit time, while system utilization indicates the proportion of time service facilities are occupied. In simulation, this is obtained by calculating the ratio of server busy time to total simulation time. Algorithm implementation involves accumulating busy_time counters and dividing by total_runtime.

Event Sequence Management Accurate recording of each customer's arrival time, service start time, and departure time forms the foundation of simulation. When using discrete event simulation engines, maintaining a future events list and processing events in chronological order is essential. Key functions include event scheduling with priority queues and time-advance mechanisms.

By establishing simulation models that incorporate random arrival intervals and service times, we can observe system performance characteristics under different loads, providing data support for resource allocation. Typical application scenarios include customer service center agent configuration and hospital triage system optimization. Code implementation often uses Poisson processes for arrivals and exponential distributions for service times.