21.1 Introduction to queuing theory

Queuing theory analyzes waiting lines using probability and statistics. It's applied in telecommunications, manufacturing, healthcare, and transportation to optimize processes and improve efficiency. Understanding queuing theory helps engineers tackle real-world problems involving resource allocation and customer flow.

Queuing systems consist of arrival processes, service processes, and queue disciplines. These components are modeled using probability distributions like Poisson and exponential. Various queuing models, described by Kendall's notation, help analyze different scenarios from single-server to multi-server systems with varying assumptions.

Introduction to Queuing Theory

Queuing theory and applications

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• Studies waiting lines or queues using probability theory and statistics to analyze queuing system behavior
• Telecommunications applications model call centers (customer support), network traffic (internet congestion), and resource allocation (bandwidth distribution)
• Manufacturing applications optimize production lines (assembly processes) and inventory management (stock levels)
• Healthcare applications manage patient flow (emergency room triage), resource allocation (medical equipment), and appointment scheduling (doctor's office)
• Transportation applications analyze traffic congestion (highway bottlenecks), toll booths (payment processing), and airport security checkpoints (passenger screening)
• Customer service applications improve waiting times (retail checkout) and resource utilization (bank tellers) in various service industries (hospitality, finance)

Components of queuing systems

• Arrival process describes the pattern or distribution of customers or entities entering the queuing system
  • Inter-arrival time measures the duration between consecutive arrivals
  • Arrival rate $\lambda$ represents the average number of arrivals per unit time
• Service process describes the pattern or distribution of service times for customers or entities in the system
  • Service time measures the duration required to serve a single customer or entity
  • Service rate $\mu$ represents the average number of customers served per unit time
• Queue discipline determines the order in which customers or entities are selected for service
  • First-Come, First-Served (FCFS) serves customers in the order of their arrival (supermarket checkout)
  • Last-Come, First-Served (LCFS) serves the most recent arrival first (stack data structure)
  • Priority serves customers with higher priority before those with lower priority (emergency room triage)
  • Processor Sharing equally divides service capacity among all customers in the system (computer CPU scheduling)

Probability distributions in queuing

• Model the randomness in arrival and service processes to analyze queuing system behavior
• Arrival process distributions:
  • Poisson distribution models the number of arrivals in a fixed time interval, assuming a constant arrival rate (call center volume per hour)
  • Exponential distribution models the inter-arrival times, assuming a memoryless property (time between bus arrivals at a stop)
• Service process distributions:
  • Exponential distribution models the service times, assuming a memoryless property (duration of customer service calls)
  • Deterministic distribution used when service times are constant and known (automated car wash)
  • General distribution allows for any arbitrary probability distribution of service times (complex repair tasks)
• The choice of probability distribution depends on the characteristics and assumptions of the queuing system being modeled (short vs. long service times, consistent vs. variable arrival rates)

Types of queuing models

• Kendall's notation describes queuing models using the format A/S/c/K/N/D
  1. A: Arrival process distribution (M for Markov or Poisson, G for General)
  2. S: Service process distribution (M for Markov or Exponential, D for Deterministic)
  3. c: Number of servers or service channels
  4. K: Maximum number of customers allowed in the system (default: $\infty$)
  5. N: Population size from which customers arrive (default: $\infty$)
  6. D: Queue discipline (default: FCFS)
• Common queuing models:
  • M/M/1: Single server (bank teller), Poisson arrivals, exponential service times, infinite queue capacity
  • M/M/c: Multiple servers (call center), Poisson arrivals, exponential service times, infinite queue capacity
  • M/G/1: Single server, Poisson arrivals, general service time distribution (complex repairs), infinite queue capacity
  • M/D/1: Single server, Poisson arrivals, deterministic service times (automated car wash), infinite queue capacity
• Assumptions and characteristics of queuing models:
  • Steady-state behavior assumes the system reaches an equilibrium over time (long-run average performance)
  • Infinite population assumes customers arrive from an unlimited source (large customer base relative to system capacity)
  • Infinite queue capacity assumes no limit on the number of customers waiting in the queue (theoretical simplification)
  • Independent arrivals and service times assume no correlation between consecutive arrivals or service times (simplifies analysis)