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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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customers in system and in queue
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
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
M servers
  • ρ = λ/(Mμ) ρ is how busy each server is
  • Probability of 0 in system:
2 servers
2 servers
  • ρ = λ/(2μ) ρ is how busy each server is
  • Probability of 0 in system:
m servers1
M servers
  • Probability of n in the system
  • If n ≤ M (P(0))(Mρ)n/n!
  • If n ≥ M (P(0))MMρn/M!
2 servers1
2 servers
  • Probability of n in the system
  • If n = 1 (P(0))2ρ
  • If n ≥ 2 (P(0))4ρn/2
m servers2
M servers
  • Lq =
  • L = Lq + λ/μ
  • Wq = Lq/λ
  • W = Wq + 1/μ
2 servers2
2 servers
  • Lq =
  • L = Lq + λ/μ
  • Wq = Lq/λ
  • W = Wq + 1/μ
examples
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
Cookbook
  • Pdf version – cell references
  • Named cells version
cookbook contents
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