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Queuing Theory 2

Queuing Theory 2. HSPM J716. Simple queue model assumes …. Constant average arrival rate λ and service rate μ Independence One arrival doesn’t make another arrival more or less likely The average length of service doesn’t change regardless of How many are waiting

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Queuing Theory 2

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  1. Queuing Theory 2 HSPM J716

  2. Simple queue model assumes … • Constant average arrival rate λ and service rate μ • Independence • One arrival doesn’t make another arrival more or less likely • The average length of service doesn’t change regardless of • How many are waiting • How busy the server has been

  3. Greek letters • ρ (“rho”) server utilization factor = average arrival rate λ ⁄ μ average service rate Probability of n in system = (1-ρ)ρn Probability of 0 to n in system = 1-ρn+1

  4. Customers in System and in Queue • L – mean number in system = • Lq – mean number 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.

  5. Mean and median in system • Median number in system = ln(.5)/ln(ρ) - 1 • Median number waiting = ln(.5)/ln(ρ) – 2 • If ρ = 2/3 • L = 2 Median is 0.7 • Lq = 1.33 Median queue is 0.

  6. System time and wait time Customers’ average time through system • W = L/λ = 1/(μ-λ) Customers’ average wait to be served • Wq = ρW = ρ/(μ-λ) • Most customers spend less than these times.

  7. Expand from basic model • More than one server • in parallel (one queue to many servers) • in series (queues in series or stages) Modify independent arrival assumption • Limited number in system • Limited customer population Modify service assumptions • Constant service time • Stages of service • Priority classes, rather than simple FIFO

  8. Multiple parallel servers

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

  10. 2 servers • ρ = λ/(2μ) ρ is how busy each server is • Probability of 0 in system:

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

  12. 2 servers • Probability of n in the system • If n = 1 (P(0))2ρ • If n ≥ 2 (P(0))4ρn/2

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

  14. 2 servers • Lq = • L = Lq + λ/μ • Wq = Lq/λ • W = Wq + 1/μ

  15. 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?

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

  17. 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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