14. Queuing (OR Models) - Lecture 13 Queuing Systems Topics...

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Unformatted text preview: Lecture 13 Queuing Systems Topics Basic structure and components Performance measures Steady-state analysis and Littles law Birth-death processes Single-server and multi-server examples Flow balance equations 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. 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. 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 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. Multiple Servers, Single Queue What is average wait in the queue? What is average time in the system? Multiple Servers, Multiple Queues What is average wait in the queue? What is average time in the system? 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 k = mean arrival rate (expected # of arrivals per unit time) k = mean service rate (expected # of departures per unit time) (Both k and k assume k customers are in system) Notation and Terminology If there are s servers, each with the same service rate, then...
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14. Queuing (OR Models) - Lecture 13 Queuing Systems Topics...

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