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Understanding Queuing Models for Optimal Service Level Analysis

Learn about queuing theory, models, and analysis techniques to determine the best service level. Explore characteristics, types, and disciplines of queuing systems, as well as measuring performance and Kendall’s notation. Dive into a case study at Mary’s Shoes for practical application.

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Understanding Queuing Models for Optimal Service Level Analysis

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  1. Queuing Models

  2. Queuing or Waiting Line Analysis • Queues (waiting lines) affect people everyday • A primary goal is finding the best level of service • Analytical modeling (using formulas) can be used for many queues • For more complex situations, computer simulation is needed

  3. Classification of Queues • Queuing system can be classified by: • Arrival process. • Service process. • Number of servers. • System size (infinite/finite waiting line). • Population size. • Notation • M (Markovian) = Poisson arrivals or exponential service time. • D (Deterministic) = Constant arrival rate or service time. • G (General) = General probability for arrivals or service time. Example: M / M / 6 / 10 / 20

  4. Characteristics of aQueuing System The queuing system is determined by: • Arrival characteristics • Queue characteristics • Service facility characteristics

  5. Arrival Characteristics • Size of the arrival population – either infinite or limited • Arrival distribution: • Either fixed or random • Either measured by time between consecutive arrivals, or arrival rate • The Poisson distribution is often used for random arrivals

  6. Poisson Distribution • Average arrival rate is known • Average arrival rate is constant for some number of time periods • Number of arrivals in each time period is independent

  7. Poisson Distribution λ = the average arrival rate per time unit P(x) = the probability of exactly x arrivals occurring during one time period P(x) = e-λ λx x!

  8. Behavior of Arrivals • Balking refers to customers who do not join the queue • Reneging refers to customers who join the queue but give up and leave before completing service • Jockeying (switching) occurs when customers switch lines once they perceived that another line is moving faster

  9. Types of Queues Single Stage

  10. Multiple Stage - Manufacturing Plant

  11. Multi-channel Single Stage - Bank

  12. Parallel Single Stage - Supermarket

  13. Customer Discrimination – Bus station

  14. Converging Arrivals - Walk-in and Drive-thru

  15. Queue Discipline • First Come First Served - FCFS • Most customer queues. • Last Come First Served - LCFS • Packages, Elevator. • Served in Random Order - SIRO • Entering Buses • Priority Service • Multi-processing on a computer. • Emergency room.

  16. Service Facility Characteristics • Configuration of service facility • Number of servers • Number of phases 2. Service distribution • The time it takes to serve 1 arrival • Can be fixed or random • Exponential distribution is often used

  17. Measuring Queue Performance • ρ = utilization factor (probability of all servers being busy) • Lq = average number in the queue • Ls = average number in the system • Wq = average waiting time • Ws = average time in the system • P0 = probability of 0 customers in system • Pn = probability of exactly n customers in system

  18. Kendall’s Notation A / B / s A = Arrival distribution (M for Poisson, D for deterministic, and G for general) B = Service time distribution (M for exponential, D for deterministic, and G for general) S = number of servers

  19. Models Covered

  20. Single Server Queuing System (M/M/1) • Poisson arrivals • Arrival population is unlimited • Exponential service times • All arrivals wait to be served • λ is constant • μ > λ (average service rate > average arrival rate)

  21. Operating Characteristics for M/M/1 Queue • Average server utilization ρ = λ / μ • Average number of customers waiting Lq = λ2 μ(μ – λ) • Average number in system Ls = Lq + λ / μ

  22. Average waiting time in the queue Wq = Lq = λ λμ(μ – λ) • Average time in the system Ws = Wq + 1/ μ • Probability of 0 customers in system P0 = 1 – λ/μ • Probability of exactly n customers in system Pn = (λ/μ )n P0

  23. MARY’s SHOES • Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process. • Service time is exponentially distributed with an average of 8 minutes per customer. • Management is interested in determining the performance measures for this service system.

  24. Pw = l/m =0.6667 r = l/m =0.6667 MARY’s SHOES - Solution • Input l = 1/12 customers per minute = 60/12 = 5 per hour. m = 1/ 8 customers per minute = 60/ 8 = 7.5 per hour. • Performance Calculations • P0 = 1 - (l/m) = 1 - (5/7.5) = 0.3333 • Pn = [1 - (l/m)](l/m)n = (0.3333)(0.6667)n • L = l/(m - l) = 2 • Lq = l2/[m(m - l)] = 1.3333 • W = 1/(m - l) = 0.4 hours = 24 minutes • Wq = l/[m(m - l)] = 0.26667 hours = 16 minutes

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