ch-MD-PPTaccessible

Operations Management: Sustainability and Supply Chain Management

Module D: Waiting-Line Models

  • Focus: The study of waiting-line models and their practical applications in operations management.

  • Theoretical Background: Understanding queuing theory is essential in both manufacturing and service environments.

Outline

  • Queuing Theory

  • Characteristics of a Waiting-Line System

  • Queuing Costs

  • Variety of Queuing Models

  • Other Queuing Issues

Learning Objectives

Key Skills After Completing This Chapter

  • D.1 Describe characteristics of arrivals, waiting lines, and service systems.

  • D.2 Apply single-server queuing model equations.

  • D.3 Conduct cost analysis for a waiting line.

  • D.4 Apply multiple-server queuing model formulas.

  • D.5 Apply constant-service-time model equations.

  • D.6 Perform finite-population model analysis.

Queuing Theory

  • Definition: The study of waiting lines, important in various domains.

  • Real-Life Applications: Common scenarios include service environments like supermarkets, doctor's offices, and banks.

Common Queuing Situations

Situation

Arrivals in Queue

Service Process

Supermarket

Grocery shoppers

Checkout clerks

Highway toll booth

Automobiles

Collection of tolls

Doctor's office

Patients

Treatment by doctors/nurses

Computer system

Programs to run

Computers processing jobs

Telephone company

Callers

Call forwarding equipment

Bank

Customers

Transactions by teller

Machine maintenance

Broken machines

Repair technicians

Harbor

Ships/barges

Dock workers loading/unloading

Characteristics of Waiting-Line Systems

  • Arrivals or Inputs: Factors include population size, behavior, and statistical distribution.

  • Queue Discipline: Characteristics of the waiting line including length and order of service.

  • Service Facility: Includes design and statistical distribution of service times.

Arrival Characteristics

  • Population Size: Can be unlimited or limited.

  • Behavior of Arrivals: Understanding customer behavior in queues (no switching lines, balking, or reneging).

  • Arrival Pattern: Could be scheduled or random, often following a Poisson distribution.

Poisson Distribution

  • Formula: P(x) = e^(-λ) * (λ^x) / x!

  • Represents the probability of a certain number of arrivals in a specified time frame.

  • Key Variables:

    • λ = average arrival rate.

    • e = base of the natural logarithm (≈ 2.7183).

Waiting-Line Characteristics

  • Queue Length: Can be limited or unlimited.

  • Queue Discipline: Often follows FIFO (First In First Out) rule, with other priority rules possible under specific conditions.

Service Characteristics

  • System Designs:

    • Single-server vs. multiple-server.

    • Single-phase vs. multiphase systems.

  • Service Time Distribution: Consists of constant and random service times.

Measuring Queue Performance

  • Key performance indicators include:

    • Average time in queue.

    • Average queue length.

    • Average time in system.

    • Average number in system.

    • Probability of idle service facilities.

    • Utilization factor.

Queuing Costs

  • Impact of Queuing Costs: Balancing the costs of providing service and waiting time on total cost.

  • Optimal Service Level: Finding a balance between service cost and waiting time costs to minimize total expected costs.

Queuing Models Overview

Common Assumptions Across Models

  • Poisson distribution of arrivals.

  • FIFO discipline.

  • All models assume a single-service phase.

Model Examples

  • Model A: Single-server system (M/M/1) - e.g., Checkout at a convenience store.

  • Model B: Multiple-server system (M/M/S) - e.g., Airline ticket counter.

  • Model C: Constant-service system (M/D/1) - e.g., Automated car wash.

  • Model D: Finite population model (M/M/1 with finite source) - e.g., Shops with limited machines.

Little's Law

  • Relates average number of customers in the system, arrival rate, and average time in the system.

  • Formula: L = λW

    • L = average number of customers in the system.

    • λ = arrival rate.

    • W = average time a customer spends in the system.

  • Utilized in steady-state conditions.

The Psychology of Waiting

  • Effective management of customer expectations during waiting periods can enhance satisfaction.

  • Tactics include:

    • Making waiting comfortable (seating, refreshments).

    • Establishing virtual queues.

    • Distracting customers (videos or mirrors).

    • Providing transparent estimates of wait times.

    • Fair queue management to ensure equal wait times.

Conclusion

  • Understanding queue dynamics enhances operational efficiency and improves customer satisfaction.

  • Application of queuing theory aids in both service and manufacturing settings.