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Topic – 13

Topic – 13. Waiting Line: Queuing Models. Real World: Waiting in Queues. People in Queues: Customers waiting in Bank/Supermarket/Gas Station/.... Children waiting in Disney World/Six Flag/...... Patients waiting in Hospitals/Clinics/...... .................... Jobs in Queue:

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Topic – 13

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  1. Topic – 13 Waiting Line: Queuing Models

  2. Real World: Waiting in Queues • People in Queues: • Customers waiting in Bank/Supermarket/Gas Station/.... • Children waiting in Disney World/Six Flag/...... • Patients waiting in Hospitals/Clinics/...... • .................... • Jobs in Queue: • Printing orders waiting in printing shops. • Airplane waiting for runway at airports. • Applications waiting in a processing center. • Cases waiting in a federal court. • Tax Returns waiting in the IRS processing center. • Telephone callers waiting on a phone line. • ........................ • Waiting in a queue can be seen everywhere and anytime. For managers, "waiting in queue" may imply significant managerial concerns (e.g., costs of inventory, customer service levels). • Basic Question: Why is there a queue lining up? • How can we cut the queue short?

  3. Structure of Queuing Models • Basic Structure of A Queuing System: • Input "Customers" who need service and are generated from a "Population" (finite vs. infinite). • Arrival Process: Deterministic vs. Stochastic • Deterministic: Constant Inter-arrival Time • Stochastic: Random Arrival following A Distribution Pattern • Service Facility Mechanism: • Single "Server" vs. Multiple "Servers" • Series "Servers" vs. Parallel "Servers" • Service Discipline: dispatching rule for selecting next customer/job from the queue. • FCFS/Preemptive Priority/SPT/LPT/..... • Queuing Behavior: "Normal" customers - enter the system, waiting in queue, serviced and leave. (No Balking, Reneging, Jockeying, Combining/Dividing, Cycling)

  4. Basic Queuing System Configuration

  5. Basic Terms in Queuing Models λ: mean customer arrival rate (No. of customers/unit time) (1/ λ): mean interarrival time (mean time between customer arrivals) u: mean "service" time for one customer (time/per-customer) μ=(1/u): mean service rate (No. of customer/unit time) ρ=( λ / μ): the utilization rate of the facility (λ < μ - so ρ < 1 ) ρø=(1-ρ): the "idle" rate of the facility. N(t): No. of customers (both in service and in queue) in system P(N): the probability of exactly N customers in system Queuing Statistics: (performance measures of queuing systems) L: Average No. of customers (both in service and queue) in system Lq: Average No. of customers in queue (average queue length) W: Average time (both in service and in queue) in system Wq: Average time in queue (mean waiting time) Steady State - Little's Equation: Relationships among L/Lq/W/Wq L = /PW = Lq + ρ Lq = /PWq W = Wq + (1/ μ)

  6. Basic Characteristics of Queuing Models Queuing Models are primarily "analytical" and "descriptive": • Predicating system queuing behavior: [P(N), L, Lq, W, Wq] • Assisting managerial decision making with other necessary information: • the "optimal" service level (queue length) • the "optimal" number of "servers", or • the "optimal" policy that minimizes total queuing cost while improving service level (e.g., better training). "U-Shape" Cost Curve Consideration: There is an "optimal" policy that can minimize total cost of: • Cost of Waiting (Cw): work-in-process inventory costs/loss of potential customers/... (may be very hard to estimate) • Cost of Increasing Service Level (Cs): cost of facility/labor, cost of idle resources/...... Primary Objectives of Queuing Models: • Minimizing total queuing cost, while • Meeting desired service level.

  7. Queuing Cost and Service Level

  8. Classification of Queuing Models Queuing Models are expressed and classified based on the following 6 factors: • [ M/D/k/FCFS/N/∞ ], where: • Distribution of Arrival Process [M, D, G, GI] • Distribution of service time [M, D, G, GI] • Number of "Servers" in the system [K = 1, 2, ....S] • Service Discipline: [FCFS, or others] • Size of Queue Length: [infinite ( ) or limited (N)] • Size of the Population: [infinite ( ) or limited (N)] Since FCFS and infinite queue length and population are usually assumed, the last three expressions are generally omitted. Classification of Single-Facility Queuing Models: Based on four major distribution patterns: • M - Exponential (Possion) Distribution • D - Deterministic Distribution • G/GI - General (any) Distribution of Service/Arrival Time • Ek - Erlang Distribution

  9. Classification of Single Facility Queuing Model

  10. Notation System Used to Label Queuing Models

  11. The Most Common Queuing System

  12. Queuing Models: Solution Methods Analytical vs. Simulation: • Analytical Solutions: Given the complexity of queuing process, only a few queuing models have analytical solutions of steady state results. • Simulation Solutions: Solutions to most queuing models are only attainable through computer simulation studies. Analytical Queuing Model Solutions: • Model Input Parameters: (λ, μ, σ2, N, ....) • Desired Output Queuing Statistics: W, Wq, L, Lq, P{T>t}, P(N) • Deterministic Model: (D/D/K) Three situations • when λ = μ, there is no queue. • when λ < μ, no queue, fixed server idle time. • when λ > μ, explosive queue length, server(s) is overloaded. (Example)

  13. Basic Model: M/M/1 Queuing Model Formula: W = 1/ (μ-λ) = 1/ [μ (1- ρ)] Wq= λ/ [μ (μ- λ)] = ρ / [μ (1- ρ)] L = λ/ (μ- λ) = ρ / (1- ρ) Lq= λ2/ [μ (μ- λ)] = ρ2 / (1- ρ) P{T > t} = e(λ - μ)*t (e = 2.7182) [e(λ- μ)*t] (The Probability of being in the system longer than time t.) P{T > t'} = ρ *e(λ - μ)*t' (e = 2.7182) [ρ*e(λ- μ)*t’] (The Probability of waiting in queue longer than time t'.) P(N) = ρ N(1 - ρ) [ρN*(1- ρ)] (The probability of funding exactly N customers in the system.) P{N > n} = ρn+1 [ρN+1] (The probability that the number of customers in the system N, will be larger than desired number, n) Note: P{N < n} = 1 - P{N > (n-1)}

  14. Waiting Line ModelsSingle Channel

  15. The Multifacility Waiting Line System

  16. A Serial Queue Situation (Combined with Parallel Servers)

  17. Other Queuing Models • M/M/K Model: Parallel/Identical Queues: Examples: Bank Service/Supermarket Checkout/.... Formula: Note: There is a nonlinear (anti-commonsense) relationship between the number of "servers" and queuing performance. • M/M/1 with Limited Customer Source: Examples: Machine Repairing/Tool room Service/... Key Difference: Limited Source - Non-Constant Arrival Rate. Formula: • M/G/1 Model (General Service Time Distribution): Examples: Job Shop Processing/Self-Service Facility/..... Application: In many cases, service time distribution can not be known or identified, only the mean and variance can be calculated from past observations. Formula: • M/D/1 Model (Deterministic Service Time Distribution): Examples: Repair Shop/Order Processing/....... Application: Customer arrivals are uncontrollable, but the service time variations can be minimized through certain better education and training. Formula:

  18. Complex Queuing Systems Serial Queues (or Queuing Network) Models: Application: Most real world complex queuing systems can be modeled as Serial or Network queuing models based on: • Decomposition Principle: a system is decomposed as several single-server models (M/M/1, or M/G/1, or M/D/1). • The whole system is solved by integrating all single-server system results. • Identify interarrival distributions among facilities in the system • (e.g., Possion-In/Possion-Out). Important Properties of Exponential Distribution: • Lack of Memory: earlier customers have no impact on later customer arrival patterns. • Small Service Times (one-sided distribution): more than half (50%) of customer service times are less than the mean time. • Relation to Possion Distribution: When customer interarrival times follow Exponential distribution, then customer arrivals follow Possion distribution. Summary: Why queues are built up when 6 < 100%? - Variations. • Eliminating variations in both arrival and service times is the key to reduce queues in all types of systems. • When customer arrival/service times do not exactly follow known distributions, computer simulation method must be used to obtain solutions.

  19. Waiting Line ModelMultiple Channels

  20. Waiting Line ModelsA Single Channel with Arbitrary Service Times

  21. Assumptions of the Basic Queuing Model • Arrivals are served on a first come, first served basis • Arrivals are independent of preceeding arrivals • Arrivals rates are described by the Poisson probability distribution, and customers come from a very large population • Service times vary from one customer to another, and are independent of one and other, the average service time is known • Service times are described by the negative exponential probability distribution • The service rate is greater than the arrival rate

  22. Measures of Performance of Queuing System • Average time each customer spends in the queue • Average length of the queue • Average time each customer spends in the system • Average number of customers in the system • Probability that the service facility will be idle • Utilization factor for the system • Probability of a specific number of customers in the system

  23. Two Examples of the Poisson Distribution for Arrival Times

  24. Two Examples of the Exponential Distribution for Service Time

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