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Chapter 18: Queuing Models. C*. Capacity decisions: a trade-off . Total cost. Cost of Service (provider). Cost . Cost of Waiting (customer). Capacity to serve. Model Nomenclature & Terms . A/B/C = arrival dist / service dist / # of servers Poisson arrivals (M)

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capacity decisions a trade off

C*

Capacity decisions: a trade-off

Total cost

Cost of

Service

(provider)

Cost

Cost of

Waiting

(customer)

Capacity to serve

model nomenclature terms
Model Nomenclature & Terms
  • A/B/C = arrival dist/ service dist/ # of servers
    • Poisson arrivals (M)
      • Distribution of independent arrivals over time
    • Exponential service (M) (time between poisson arrivals)
    • General service (G) (other distributions)
m m 1
M/M/1

Customers waiting

Customer being served

  • λ = average customer arrivals per hour
  • μ = average service rate per hour

λ

μ

Lq = length of queue

Ls =length of (# of customers in) system

m m 15
M/M/1

λ

μ-λ

Length in System (Ls) =

1

μ-λ

Time in System (Ws) =

In an effort to maximize efficiency, the manager of the airport tried to get the average time per take off to equal the time between arrivals. What will happen?

definitions
Definitions
  • n = number of customers in the system
  • ρ = mean number of customers in system (λ/μ)
  • N = maximum number of customers in system
  • c = number of servers
  • Pn = probability of exactly n customers in system
  • L = number of customers in (s=system,q=queue, b=busy system)
  • W= mean waiting time of a customer in (s=system,q=queue, b=busy system)