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Advanced Queueing Theory Applications in Service Systems

Explore queuing theory beyond basic models with multiple servers, priority classes, limited system size, and more. Learn about scenarios such as parallel servers, limited customer populations, and constant service times. Find examples like fast-food chains, cafeterias, and office setups to understand real-world applications of queuing theory.

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Advanced Queueing Theory Applications in Service Systems

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  1. Queuing Theory 2 HSPM J716

  2. Customers in System and in Queue • L – mean customers in system = • Lq – mean customers in queue = L-ρ (not L-1) • There are usually fewer in the system than L, and fewer in line than Lq ,because the probability of n in system is skewed.

  3. Expand from basic model • More than one server • in parallel (one queue to many servers) • in series (queues in series or stages) • Limited number in system • Limited customer population • Constant service time • Priority classes, rather than simple FIFO • (Arrivals in clumps -- not this year)

  4. Multiple parallel servers

  5. M servers • ρ = λ/(Mμ) ρ is how busy each server is • Probability of 0 in system:

  6. 2 servers • ρ = λ/(2μ) ρ is how busy each server is • Probability of 0 in system:

  7. M servers • Probability of n in the system • If n ≤ M (P(0))(Mρ)n/n! • If n ≥ M (P(0))MMρn/M!

  8. 2 servers • Probability of n in the system • If n = 1 (P(0))2ρ • If n ≥ 2 (P(0))4ρn/2

  9. M servers • Lq = • L = Lq + λ/μ • Wq = Lq/λ • W = Wq + 1/μ

  10. 2 servers • Lq = • L = Lq + λ/μ • Wq = Lq/λ • W = Wq + 1/μ

  11. Examples • a 2nd pharmacist • Burger King vs. McDonald’s: • 1 line to 2 servers vs. 2 lines to 2 servers. • 2 slow servers vs. 1 server who is twice as fast • How many seats in the cafeteria? • E.g. 1 customer per minute, 15 min. to eat, 15 seats? • How they save when you eat faster • Comfortable chairs?

  12. Cookbook • Pdf version – cell references • Named cells version

  13. Cookbook contents • One server (like assignment 7A) • One server, arrivals from limited group • One server, limited queue length (“balking”) • One server, constant service time • Stages of service, queue only at start • Parallel servers, one queue (Post Office) • Parallel servers, no queue (hotel) • Priority classes for arrivals

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