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14. Queuing (OR Models) - Topics Performancemeasures...

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Lecture 13 – Queuing Systems Topics • Basic structure and components • Performance measures • Steady-state analysis and Little’s law • Birth-death processes • Single-server and multi-server examples • Flow balance equations
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Input source Queue Service mechanism arriving customers exiting customers Structure of Single Queuing Systems Note: 1. Customers need not be people   parts, vehicles, machines,  jobs. 2. Queue might not be a physical line   customers on hold, jobs waiting to be printed, planes circling airport.
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Input Source   The size of the “calling population”           may be modeled as infinite or finite.             Calculations are easier in the infinite case and  in many cases this is a reasonable approximation  (bank, pizza parlor, blood bank). Queuing Discipline   First-come first-served (FIFO) is most frequent assumption, but priority  ordering is important in some settings. Components of Model Service Mechanism   One or more servers may be  placed in parallel.
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System Arrival Process Service Process Bank Customers Arrive Tellers serve customers Pizza  Orders are phoned  Deliveries driven to    parlor    in  customers  Blood  Pints of blood arrive Patients use up    bank    via donation   pints of blood Shipyard Damaged ships sent to   Ships are repaired   shipyard for repair   & return to sea Printers Jobs arrive from  Documents are   computers    printed Queuing Applications
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What is the ...     1. average number of customers in the system?     2. average time a customer spends in the system?     3. probability a customer is rejected?     4. fraction of time a server is idle? These questions are aimed at characterizing complex systems .   Analyses used to support  decision-making . Typical Performance Questions In queuing (and most analyses of complex stochastic  systems), OR takes the form of asking “ what if  ”  questions rather than trying to  optimize  the design.
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Multiple Servers, Single Queue What is average wait in the queue? What is average time in the system?
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Multiple Servers, Multiple Queues What is average wait in the queue? What is average time in the system?
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N ( t   )    = # of customers in the system at time  t    0 P k ( t   = probability exactly  k  customers in system       at time  t , given # in system at time 0 s   = # of parallel servers λ k = mean arrival rate (expected # of arrivals per unit time) μ k = mean service rate  (expected # of departures per unit time)   (Both  λ k  and  μ k  assume  k  customers are in system) Notation and Terminology
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If there are s servers, each with the same service rate, then μ n    =   s μ  for  n     s
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