1 / 17

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

twila
Download Presentation

Queuing Theory 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related