Queuing Theory 2

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Queuing Theory 2

• HSPM J716

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Customers in System and in Queue

• L mean customers in system
• Lq mean customers in queue L-? (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.

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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)

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Multiple parallel servers
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M servers

• ? ?/(Mµ) ? is how busy each server is
• Probability of 0 in system

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2 servers

• ? ?/(2µ) ? is how busy each server is
• Probability of 0 in system

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M servers

• Probability of n in the system
• If n M (P(0))(M?)n/n!
• If n M (P(0))MM?n/M!

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2 servers

• Probability of n in the system
• If n 1 (P(0))2?
• If n 2 (P(0))4?n/2

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M servers

• Lq
• L Lq ?/µ
• Wq Lq/?
• W Wq 1/µ

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2 servers

• Lq
• L Lq ?/µ
• Wq Lq/?
• W Wq 1/µ

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Examples

• a 2nd pharmacist
• Burger King vs. McDonalds
• 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?

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Cookbook

• Pdf version cell references
• Named cells version

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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