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Understanding Simple Queue Models in Queuing Theory

This article provides an overview of the fundamental principles of simple queue models in queuing theory. We discuss assumptions such as constant average arrival and service rates, independence of arrivals, and the effects of server utilization. Key metrics like mean and median numbers of customers in the system and queue, as well as average times spent waiting and served, are highlighted. The document also explores variations of the basic model, including multiple servers and priority classes. Practical examples illustrate these concepts in real-world contexts.

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Understanding Simple Queue Models in Queuing Theory

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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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