1. Queuing Theory 2 HSPM J716

2. Customers in System and in Queue L – mean customers in system = L q – mean customers in queue = L-ρ (not L-1) There are usually fewer in the system than L, and fewer in line than L q,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))M M ρ n /M!

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

9. M servers L q = L = L q + λ/μ W q = L q /λ W = W q + 1/μ

10. 2 servers L q = L = L q + λ/μ W q = L q /λ W = W q + 1/μ

11. Examples a 2 nd 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 1.One server (like assignment 7A) 2.One server, arrivals from limited group 3.One server, limited queue length (“balking”) 4.One server, constant service time 5.Stages of service, queue only at start 6.Parallel servers, one queue (Post Office) 7.Parallel servers, no queue (hotel) 8.Priority classes for arrivals