Queuing Theory

Queuing Theory

Description also available in video format (attached below), for better experience use your desktop.

Introduction

• Queuing Theory is the mathematical study of waiting lines or queues.
  It helps in analyzing:

  • Waiting time
  • Queue length
  • System efficiency
  • Resource allocation

  Used in: Hospitals, banks, airports, call centers, supermarkets, manufacturing units, etc.

  Key Elements of a Queuing System

  Element	Description
  Arrivals (λ)	The rate at which customers arrive
  Service rate (μ)	The rate at which service is completed
  Queue discipline	Rule for serving (e.g., FIFO, LIFO, priority)
  Number of servers	One or more service channels
  System capacity	Max number of customers allowed in the system
  Population size	Source from which customers come (finite/infinite)

  Types of Queuing Models (Kendall’s Notation)

  Format: A/S/c

  • A = Arrival distribution
  • S = Service time distribution
  • c = Number of servers

  Common Models:

  • M/M/1 → Poisson arrivals, Exponential service, 1 server
  • M/M/c → Poisson arrivals, Exponential service, c servers
  • M/G/1 → Poisson arrivals, General service time, 1 server

  Queue Disciplines

  • FIFO (First In First Out) – Most common (used in hospitals)
  • LIFO (Last In First Out) – Stack-like
  • Priority Queue – Emergency patients get treated first
  • Random selection

  Key Formulas (for M/M/1 Queue)

  Let:

  • λ = Arrival rate
  • μ = Service rate
  • ρ = Traffic intensity = λ / μ (must be < 1 for stability)

  Measure	Formula
  Average number in system (L)	L = λ / (μ - λ)
  Average number in queue (Lq)	Lq = λ² / μ(μ - λ)
  Average time in system (W)	W = 1 / (μ - λ)
  Average time in queue (Wq)	Wq = λ / μ(μ - λ)

  Applications in Hospitals

  • OPD registration counters
  • Pharmacy dispensing
  • Emergency room triage
  • Lab test counters
  • Billing desks
  • Operation theatre scheduling

  Advantages of Queuing Theory

  • Reduces waiting time
  • Optimizes staff allocation
  • Improves patient satisfaction
  • Enhances decision-making
  • Helps design efficient service layouts

  Limitations

  • Assumes mathematical distributions (real-world may vary)
  • Doesn’t account for human emotions (e.g., patient frustration)
  • Complexity increases with more variables

  Real-life Example (Hospital Scenario)

  Imagine a single doctor (1 server) in the OPD.
  Patients (customers) arrive at a rate of λ = 10/hour, and the doctor sees patients at a rate of μ = 12/hour.

  Using formulas:

  • ρ = 10/12 = 0.83
  • W = 1 / (12 - 10) = 0.5 hours (30 minutes average waiting time)

  Tools Used for Queuing Analysis

  • Simulation software (e.g., Arena, Simul8)
  • Excel-based models
  • Python or R for advanced modeling

Video Description

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