Queuing Theory 2

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# Queuing Theory 2 - PowerPoint PPT Presentation

Queuing Theory 2. HSPM J716. 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. Expand from basic model.

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### Queuing Theory 2

HSPM J716

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.
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)
M servers
• ρ = λ/(Mμ) ρ is how busy each server is
• Probability of 0 in system:
2 servers
• ρ = λ/(2μ) ρ is how busy each server is
• Probability of 0 in system:
M servers
• Probability of n in the system
• If n ≤ M (P(0))(Mρ)n/n!
• If n ≥ M (P(0))MMρn/M!
2 servers
• Probability of n in the system
• If n = 1 (P(0))2ρ
• If n ≥ 2 (P(0))4ρn/2
M servers
• Lq =
• L = Lq + λ/μ
• Wq = Lq/λ
• W = Wq + 1/μ
2 servers
• Lq =
• L = Lq + λ/μ
• Wq = Lq/λ
• W = Wq + 1/μ
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?
Cookbook