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Queuing Models

Queuing Models. 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.

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Queuing Models

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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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