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Queuing models

The Fundamentals. Queuing models. Collection of entities kept in order Addition of entity at the rear of the terminal and removal at the front terminal. What is a queue?. Used to approximate a real queuing situation to be analyzed mathematically

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Queuing models

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  1. The Fundamentals Queuing models

  2. Collection of entities kept in order Addition of entity at the rear of the terminal and removal at the front terminal What is a queue?

  3. Used to approximate a real queuing situation to be analyzed mathematically • Allow a number of useful performance measures to be determined: • Average number in the queue • Average time spent in queue Queuing model

  4. Elements of queuing systems

  5. Population Arrival Service/Servers Queue Output Elements

  6. FIFO (First in First out) LIFO (Last in First out) SIRO (Serve in Random Order) Priority Queue Queuing discipline

  7. Parallel Series Queuing system characteristics

  8. Parallel servers Customers in line

  9. series servers Customers waiting in line

  10. Time Averages: L = expected no. of customers in the system LQ = expected no. of customers in the queue LS = expected no. of customers in service P (all idle) = probability that all servers are idle P (all busy) = probability that all servers are budy Queuing relationships

  11. Customer Averages: W = expected time spent in the system WQ = expected time spent in the queue WS = expected time spent in service Queuing relationships

  12. λ = average rate at w/c customers enter the system L = expected number of customers in the system W = expected time a customer spends in the system Therefore: L = λW LQ =λWQ LS =λWS Little’s formula

  13. Single server model • = arrival rate • 1/= mean time between arrivals • = service rate • 1/= mean service time per customer • = traffic intensity = / •  x 100 = % service utilization

  14. Typical Front Desk Queuing: 30 customers per hour Each representative spends 1.5 minutes/customer Manager’s objective is to decide whether to improve the system or not Example

  15. 30 customers per hour  = 0.5 cx/min 1.5 mins/cx  = 1 cx / 1.5 mins/cx = 0.67 cx/min Excel File solution

  16. M/M/s model M – means that interarrival times are exponentially distributed M- service times for each server are exponentially distributed s- denotes the number of servers Multiple Server Mode

  17. Customers arrive at a rate of 150 customers per hour Branch employs 6 tellers Average service time is 2 minutes to serve each customer All customers performs all tasks Customers arrived and finds 6 tellers busy serving First Come First Serve fashion Manager’s objective = to find the “best” numbers of tellers given that tellers are paid $8 per hour Queueing at a Country Bank

  18. Thank you The End

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