Queuing systems are the backbone of service operations, from bank tellers to emergency rooms. They consist of customers, servers, and waiting lines, with arrival and service processes dictating flow. Understanding these components is crucial for optimizing efficiency and customer satisfaction.

This topic dives into various queuing models, from basic single-server setups to complex priority systems. We'll explore key performance measures like Little's Law and system utilization, essential for analyzing and improving real-world service operations. These concepts form the foundation for tackling more advanced queuing scenarios.

Queuing System Components

Fundamental Structure and Processes

• Queuing systems consist of customers, servers, and a waiting line or queue, forming the fundamental structure for analyzing service operations
• Arrival process describes how customers enter the system
  • Characterized by arrival rate and inter-arrival time distribution
  • Typically modeled using probability distributions (Poisson process for random arrivals)
• Service process represents how customers are served
  • Defined by service time distribution and number of servers
  • Often assumed to be exponential for mathematical tractability
• Queue discipline determines the order in which customers are served
  • Common types include First-Come-First-Served (FCFS), Last-Come-First-Served (LCFS), and priority-based systems
  • Impacts system performance and fairness

System Capacity and Population

• System capacity refers to the maximum number of customers that can be in the system, including those in service and waiting
  • Can be finite or infinite, affecting analysis methods
• Customer population influences arrival process and system analysis
  • Finite population models assume a limited customer base (M/M/c/N/N)
  • Infinite population models assume an unlimited customer base (M/M/1, M/M/c)

Queuing Notation and Customer Behavior

• Kendall's notation (A/B/C) concisely describes queuing systems' characteristics
  • A: arrival process distribution
  • B: service time distribution
  • C: number of servers
  • Additional parameters may include system capacity (K) and population size (N)
• Customer behaviors can complicate queue discipline analysis
  • Balking: customers decide not to join the queue upon arrival (long lines at a restaurant)
  • Reneging: customers leave the queue after joining (impatient customers at a bank)
  • Jockeying: customers switch between queues (moving to a shorter line at a grocery store)

Queuing Model Types

Basic Queuing Models

• Single-server queuing models (M/M/1) assume:
  • Poisson arrivals
  • Exponential service times
  • One server with infinite capacity and population
  • Example: Single teller at a small bank branch
• Multi-server queuing models (M/M/c) extend single-server models to c identical servers
  • Maintain Poisson arrivals and exponential service times
  • Example: Multiple checkout counters at a supermarket

Advanced Queuing Models

• Finite capacity models (M/M/c/K) limit the total number of customers in the system to K
  • Includes customers in service and waiting
  • Example: Waiting room at a doctor's office with limited seating
• Non-Markovian models relax assumptions on service time or inter-arrival time distributions
  • M/G/1 or G/G/c models require more complex analysis techniques
  • Example: Manufacturing process with variable production times
• Priority queuing models incorporate different customer classes with varying service priorities
  • Affect queue discipline and system performance
  • Example: Emergency room triage system

Specialized Queuing Models

• Finite population models (M/M/c/N/N) assume a finite customer population of size N
  • Affects arrival process as population in the system increases
  • Example: Machine repair shop serving a fixed number of machines
• Bulk arrival or bulk service models consider customers arriving or being served in groups
  • Alters arrival and service processes
  • Examples:
    • Bulk arrival: Groups of tourists arriving at a theme park
    • Bulk service: Elevator serving multiple passengers simultaneously

Analyzing Queuing System Elements

Arrival Process Analysis

• Inter-arrival times in a Poisson process follow an exponential distribution
  • Characterized by the memoryless property
  • $P(X > t + s | X > s) = P(X > t)$
• Arrival rate (λ) represents the average number of customers arriving per unit time
  • For Poisson process: $P(n \text{ arrivals in time } t) = \frac{(λt)^n e^{-λt}}{n!}$
• Non-Poisson arrival processes may use other distributions
  • Erlang distribution for more regular arrivals
  • Hyperexponential distribution for more variable arrivals

Service Process Evaluation

• Service time distribution often assumed exponential for mathematical tractability
  • $f(t) = μe^{-μt}, \text{ where } μ \text{ is the service rate}$
• Non-exponential service time distributions model more realistic scenarios
  • Deterministic: Fixed service time (automated car wash)
  • Erlang: Sum of exponential phases (multi-step service process)
  • General: Any other distribution (complex manufacturing operations)
• Service rate (μ) represents the average number of customers served per unit time
  • For multiple servers: $μ_c = c μ, \text{ where } c \text{ is the number of servers}$

Queue Discipline Impact

• FCFS (First-Come-First-Served) most common and easiest to analyze
  • Fair for customers, but may not optimize system performance
• Priority disciplines can be preemptive or non-preemptive
  • Preemptive: High-priority customers interrupt service of low-priority customers (emergency surgeries)
  • Non-preemptive: High-priority customers move to front of queue but don't interrupt ongoing service (VIP ticket holders at an event)
• Impact of queue discipline on system performance
  • Affects waiting times for different customer classes
  • Influences overall system efficiency and customer satisfaction

Performance Measures for Queuing Systems

Fundamental Relationships

• Little's Law relates average number of customers in system (L) to arrival rate (λ) and average time spent in system (W)
  • $L = λW$
  • Applies to both queue and entire system
  • Example: If 10 customers arrive per hour and spend 30 minutes in system, average number in system is 5
• System utilization (ρ) calculated as ratio of arrival rate to service rate
  • $ρ = \frac{λ}{cμ}, \text{ where } c \text{ is number of servers}$
  • Indicates proportion of time servers are busy
  • Example: If λ = 5 customers/hour and μ = 6 customers/hour, ρ = 5/6 ≈ 0.83 or 83% utilization

Queue and System Metrics

• Average queue length (Lq) represents expected number of customers waiting in queue
  • For M/M/1: $L_q = \frac{ρ^2}{1-ρ}$
• Average system length (L) includes customers in service
  • For M/M/1: $L = \frac{ρ}{1-ρ}$
• Average waiting time in queue (Wq) and average time in system (W) measure customer experience
  • For M/M/1: $W_q = \frac{ρ}{μ(1-ρ)}, W = W_q + \frac{1}{μ}$
• Probability of n customers in system (Pn) and probability of empty system (P0) provide insights into system state distributions
  • For M/M/1: $P_n = (1-ρ)ρ^n, P_0 = 1-ρ$

Advanced Performance Analysis

• Server utilization and idle time percentages aid in capacity planning
  • Idle time percentage = 1 - ρ
  • Example: If ρ = 0.8, servers are idle 20% of the time
• Performance measures for priority queuing systems include:
  • Average waiting times for each priority class
  • Queue lengths for different priority levels
• Sensitivity analysis examines impact of parameter changes on system performance
  • Example: How does doubling arrival rate affect average waiting time?
• Cost-benefit analysis balances service improvements against operational costs
  • Example: Determining optimal number of servers to minimize total cost (waiting cost + server cost)