How can you apply basic queuing models to your work?

1 What is a queuing system?

A queuing system consists of three main components: the arrival process, the service process, and the queue discipline. The arrival process describes how customers or items arrive at the system, such as the rate, the pattern, and the distribution of arrivals. The service process describes how customers or items are served by the system, such as the number of servers, the capacity, the duration, and the distribution of service times. The queue discipline describes how customers or items are selected for service, such as first-come-first-served (FCFS), last-come-first-served (LCFS), or priority-based.

2 How to classify queuing models?

Queuing models can be classified using the Kendall notation, which summarizes the characteristics of the arrival process, the service process, and the queue discipline. This notation is composed of symbols such as A for the distribution of interarrival times (e.g., M for exponential, D for deterministic, G for general); B for the distribution of service times (e.g., M for exponential, D for deterministic, G for general); C for the number of servers (e.g., 1 for single server, N for N servers, ∞ for infinite servers); D for the queue discipline (e.g., FCFS, LCFS, PR for priority); E for the system capacity (e.g., K for finite capacity, ∞ for infinite capacity); and F for the population size (e.g., N for finite population, ∞ for infinite population). For instance, a queuing model with exponential interarrival times, exponential service times, a single server, a FCFS queue discipline, an infinite system capacity, and an infinite population size can be denoted as M/M/1/FCFS/∞/∞.

3 How to measure queuing performance?

Queuing performance can be measured by various metrics such as the average number of customers or items in the system (L), the average number of customers or items in the queue (Lq), the average time spent by a customer or item in the system (W), and the probability that there are n customers or items in the system (Pn). These metrics can be calculated with different formulas depending on the type of queuing model. For example, the M/M/1/FCFS/∞/∞ model requires formulas for L = ρ/(1-ρ), Lq = ρ^2/(1-ρ), W = 1/(μ-λ), Wq = ρ/(μ-λ), ρ = λ/μ, Pn = (1-ρ)ρ^n, P0 = 1-ρ, and Pw = ρ. Additionally, for an M/M/C/FCFS/∞/∞ model, formulas for L, Lq, W, Wq, ρ, Pn, P0, and Pw must be used. Lastly, for an M/M/∞/FCFS/∞/∞ model, formulas for L, Lq, W, Wq, ρ, Pn, P0 and Pw are used where λ is the arrival rate and μ is the service rate. Calculating these metrics can help you understand and improve your queuing system.